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Title: Model-Independent Test for Gravity using Intensity Mapping and Galaxy Clustering

Abstract: We propose a novel method to measure the $E_G$ statistic from clustering alone. The $E_G$ statistic provides an elegant way of testing the consistency of General Relativity by comparing the geometry of the Universe, probed through gravitational lensing, with the motion of galaxies in that geometry. Current $E_G$ estimators combine galaxy clustering with gravitational lensing, measured either from cosmic shear or from CMB lensing. In this paper, we construct a novel estimator for $E_G$, using only clustering information obtained from two tracers of the large-scale structure: intensity mapping and galaxy clustering. In this estimator, both the velocity of galaxies and gravitational lensing are measured through their impact on clustering. We show that with this estimator, we can suppress the contaminations that affect other $E_G$ estimators and consequently test the validity of General Relativity robustly. We forecast that with the coming generation of surveys like HIRAX and $\textit{Euclid}$, we will measure $E_G$ with a precision of up to 7% (3.9% for the more futuristic SKA2).

https://export.arxiv.org/pdf/2208.10419

\title{Model-Independent Test for Gravity using Intensity Mapping and Galaxy Clustering} \author{Muntazir M. Abidi} \affiliation{% Universit\'e de Gen\`eve, D\'epartement de Physique Th\'eorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Gen\`eve 4, Switzerland }% \author{Camille Bonvin} \affiliation{% Universit\'e de Gen\`eve, D\'epartement de Physique Th\'eorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Gen\`eve 4, Switzerland }% \author{Mona Jalilvand} \affiliation{% Universit\'e de Gen\`eve, D\'epartement de Physique Th\'eorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Gen\`eve 4, Switzerland }% \affiliation{% Department of Physics, McGill University, 3600 rue University, Montreal, QC H3A 2T8, Canada }% \affiliation{% McGill Space Institute, McGill University, 3550 rue University, Montreal, QC H3A 2A7, Canada }% \author{Martin Kunz} \affiliation{% Universit\'e de Gen\`eve, D\'epartement de Physique Th\'eorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Gen\`eve 4, Switzerland }% \date{\today}% \pacs{Valid PACS appear here}% \section{Introduction} One of the main goals of the coming generation of large-scale structure surveys is to test the consistency of General Relativity (GR) at cosmological scales. Since the observation of the accelerated expansion of the Universe in 1998 \citep{Riess:1998fmf, Perlmutter:1998vns}, a large number of theories of gravity beyond General Relativity have been constructed. Testing these theories one by one by confronting them with observations is not anymore a feasible option. This complexity in the landscape of models beyond GR has led the community to build model-independent tests of gravity, i.e.\ tests that can be applied to data without relying on particular modelling and whose outcome will either confirm or rule out the validity of GR (e.g.\ \citep{Kunz:2012aw,Amendola:2012ky,Pinho:2018unz,Franco:2019wbj,Bonvin:2020cxp,Boschetti:2020fxr,Sobral-Blanco:2021cks,Raveri:2021dbu}). One particularly useful test is the so-called $E_G$ statistic, first proposed by Zhang et al.~\citep{Zhang:2007nk}. The idea of this test is to combine galaxy-lensing correlations with galaxy-velocity correlations to test the relation between the sum of the metric potentials, $\Phi+\Psi$, and the galaxy peculiar velocity $V$: \be E_G=\frac{\rm{galaxy-lensing}}{\rm{galaxy-velocity}}\propto \frac{\langle \delta_\g (\Phi+\Psi)\rangle}{\langle \delta_\g V\rangle}\, . \label{eq:EGdef} \ee In GR, these two quantities are related via Einstein's equations so that $E_G$ takes a specific scale-independent value. In modified theories of gravity, however, the relation between $\Phi$ and $\Psi$ is generically modified, as is the growth rate of structures $f$, which governs the evolution of peculiar velocities \citep{Pogosian2008PhRvD}: $E_G$ is consequently modified and becomes potentially scale-dependent. Measurements of $E_G$ provide, therefore, a direct test of the validity of GR. This test has the advantage of being independent of galaxy bias $b$, since the galaxy density appears both in the numerator and denominator of Eq.~\eqref{eq:EGdef}, and also of the initial conditions, which cancel out in the ratio \cite{Amendola:2012ky}. Different methods have been used to measure $E_G$ in practice. First, since peculiar velocities are not straightforward to measure, the galaxy-velocity correlation in the denominator has been replaced by the product of the galaxy-galaxy correlations and the parameter $\beta=f/b$, which can be measured directly from redshift-space distortions (RSD)~\citep{Kaiser1987}. Concerning the numerator, two observables have been used: 1) the correlations between galaxy clustering and cosmic shear, and 2) the correlations between galaxy clustering and CMB lensing. The first measurement of $E_{G}$ was carried out by Reyes et al.~\citep{Reyes:2010tr} using the first method applied to luminous red galaxies from the Sloan Digital Sky Survey (SDSS)~\citep{York2000}. They measured $E_G = 0.39 \pm 0.06$, confirming the $\Lambda$CDM predictions, on the scales of tens of Mpc. Later, Amon et al.~\citep{Amon2018MNRAS.479.3422A} measured $E_G$, again with the first method, by combining deep imaging data from the Kilo-Degree Survey \citep{Kuijken:2015vca} with the overlapping spectroscopic 2-degree Field Lensing Survey~\citep{Blake2016MNRAS.462.4240B}, the Baryon Oscillation Spectroscopic Survey~\citep{Dawson2013AJ} and the Galaxy Mass Assembly Survey~\citep{Driver2011MNRAS} and found some tension of their results with the GR predictions. Pullen et al.~\citep{Pullen:2015vtb} used the second method, cross-correlating the Planck CMB lensing map with the SDSS III CMASS galaxy samples and found an almost 2.6$\sigma$ deviation from the GR prediction. Other works on the $E_G$ measurements found similar tensions~\citep{Alam:2016qcl, Blake2015, delaTorre:2013rpa}. Measuring $E_G$ more precisely with the coming generation of surveys will reveal whether these tensions persist, possibly indicating a breakdown of GR at cosmological scales. In this context, minimising the impact of systematic effects on measurements of $E_G$ is crucial. Recently, it has been shown that one important contamination, which is negligible for current surveys, will contaminate $E_G$ at high redshift, invalidating its use to test the consistency of GR with the next generation of surveys~\cite{MoradinezhadDizgah:2016pqy}. This contamination is the contribution of lensing magnification to galaxy clustering. Since lensing magnification correlates strongly with cosmic shear (method 1) and with CMB lensing (method 2), it inevitably leads to an extra contribution in the numerator of Eq.~\eqref{eq:EGdef}. At $z=1.5$ this contamination reaches 25-40\%, leading to an $E_G$ which is neither scale-independent nor bias-independent. In~\cite{Ghosh2019a} a method has been proposed to remove this contamination by measuring additional correlations, namely shear-shear correlations and shear-CMB lensing correlations, and subtracting them from $E_G$. In this paper, we propose an alternative way of measuring $E_G$ \emph{without contamination}, using only clustering information. We use two different tracers of the large-scale structure (LSS): galaxy clustering and 21\,cm intensity mapping (IM). Intensity mapping is a novel technique to map the LSS by measuring the intensity fluctuations of some emission line (typically the 21\,cm line emitted by neutral hydrogen) with radio surveys. These fluctuations, which follow the dark matter distribution, can be used as a new tracer of the LSS. Although the presence of large foregrounds make IM auto-correlation measurements very challenging, this problem is mitigated in cross-correlation measurements of IM with galaxy clustering, which have already been successfully performed ~\cite{Chang:2010jp,Masui:2012zc,Switzer:2013ewa,Anderson:2017ert,Tramonte:2020csa,Li:2020pre,Wolz:2021ofa,CHIME:2022kvg,Cunnington:2022uzo}. In~\cite{Jalilvand2019}, we built a new observable, called GIMCO, which combines IM with galaxy clustering to obtain a direct measurement of the galaxy-lensing correlation. Here we propose to use GIMCO in the numerator of $E_G$. Instead of measuring the galaxy-lensing correlation from cosmic shear or CMB lensing, we measure it directly from clustering. As we will see, this method has the strong advantage of being unaffected by the lensing contamination described above, which affects both galaxy-shear correlations and galaxy-CMB lensing correlations. Moreover, since it relies only on clustering information, it is unaffected by potential inconsistencies between low redshift and high redshift data sets (as is the case for galaxy-CMB lensing correlations) or systematics affecting the measurement of cosmic shear. It provides, therefore, a robust way of testing the consistency of GR. The rest of the paper is organised as follows: In Sec.~\ref{EG_theory} we introduce our new method to measure $E_G$ using intensity mapping. In Sec.~\ref{sec:SNR} we study the contaminations to $E_G$, and we determine the optimal redshift binning to reduce these contaminations to a negligible level. In Sec.~\ref{sec:forecasts} we forecast the precision with which $E_G$ will be measured with the coming generation of galaxy and IM surveys and the constraints expected on modifications of gravity. We conclude in Sec.~\ref{sec:conclusion}. \section{ ${\emph E}_G$ from intensity mapping} \label{EG_theory} We build $E_G$ using two different biased tracers of the matter clustering: the fluctuations in galaxy number counts, $\Delta_{\rm g}$, and the fluctuations in the 21\,cm brightness temperature, $\deltaim$. The fully relativistic expressions at linear order for these two quantities have been derived in~\cite{Yoo:2009au,Bonvin:2011bg,Challinor:2011bk,Hall:2012wd}. Here we consider only the terms that dominate in the angular power spectrum of thick redshift bins. For the galaxy number counts, we have \begin{equation} \Delta_{\rm g}(\mathbf{n}, z) = b_{\rm g}(z)\delta(\mathbf{n},z) + \big(5s(z)-2\big)\kappa(\mathbf{n},z)\, , \label{eq:numbercount} \end{equation} where $\bn$ denotes the direction of observation, $z$ is the redshift, $b_{\rm g}$ is the linear galaxy bias, $\delta$ is the matter density contrast, and $s$ is the magnification bias. The first term in Eq.~\eqref{eq:numbercount} is the density contribution, and the second term is the lensing magnification, proportional to the lensing convergence $\kappa$, \be \kappa(\mathbf{n}, z)= \frac{1}{2}\int_{0}^{\chi(z)}\! d\chi' \,\frac{\chi(z) -\chi' }{\chi(z) \chi'} \, \nabla^2_{\Omega} (\Phi+\Psi)(\mathbf{n},\chi')\, , \label{eq:kappa} \ee where $\nabla^2_{\Omega}$ denotes the angular Laplacian~\footnote{Note that we use the following metric convention $\text{d}s^2 = a(\eta)^2\big[-(1+2\Psi)\text{d}\eta^2 + (1-2\Phi)\delta_{ij}\text{d}x^i\text{d}x^j\big]$, where $\eta$ denotes conformal time, so that $\chi=\eta_0-\eta$ for the conformal time today $\eta_0$.}. Contrary to galaxy number counts, the fluctuations in the brightness temperature are not affected by lensing magnification at linear order in perturbation theory due to conservation of surface brightness~\cite{Hall:2012wd}, and we have \be \deltaim(\mathbf{n}, z) = b_{\rm HI}(z)\delta(\mathbf{n},z)\, , \label{eq:IMcount} \ee where $b_\im$ is the bias of neutral hydrogen. In Eqs.~\eqref{eq:numbercount} and~\eqref{eq:IMcount} we have neglected redshift-space distortions (RSD) since they are subdominant for thick redshift bins. In our forecasts, we will choose the binning such that this contribution is always negligible and does not contaminate $E_G$, see Section~\ref{sec:SNR}. We expand the galaxy number counts and brightness temperature in spherical harmonics \begin{align} & \Delta_{\rm g}(\bn,z) = \sum_{\ell m}a_{\ell m}^{\rm g}(z)Y_{\ell m}(\boldsymbol{n})\, ,\\ & \deltaim(\bn,z) = \sum_{\ell m}a_{\ell m}^{\rm HI}(z)Y_{\ell m}(\boldsymbol{n})\, , \end{align} and consider their angular power spectra \begin{equation} C^{\text{XY}}_{\ell}(z,z') = \big\langle a_{\ell m}^{\text{X}}(z)a_{\ell m}^{*\text{Y}}(z')\big\rangle\, , \end{equation} for $\rm{X,Y}=g, {\im}$. We can then build $E_G$ from these spectra in the following way: the numerator of $E_G$ is given by the galaxy-lensing correlation. Since galaxy number counts are affected by lensing magnification whereas intensity mapping is not, the following estimator, proposed for the first time in~\cite{Jalilvand2019}, is directly proportional to the galaxy-lensing correlation \begin{align} E^\times_{\ell}(z_f, z_b) &\equiv C_\ell^{\im {\rm g}}(z_f, z_b)-C_\ell^{{\rm g}\im}(z_f, z_b)\label{eq:El}\\ &= \big(5 s(z_b)-2\big)b_{\im}(z_f)C_{\ell}^{\delta\kappa}(z_f, z_b) \nonumber\\ &-\big(5s(z_f)-2\big)b_{\im}(z_b)C_{\ell}^{\kappa\delta}(z_f, z_b)\nonumber\\ &+\big[b_{\text{g}}(z_b)b_{\im}(z_f)-b_{\text{g}}(z_f)b_{\im}(z_b)\big] C_{\ell}^{\delta \delta}(z_f, z_b)\, . \nonumber \end{align} The term we are interested in is the one in the second line, $C_{\ell}^{\delta\kappa}(z_f,z_b)$, which represents the lensing magnification of background galaxies generated by a foreground density at $z_f$. The other lensing term in the third line, $C_{\ell}^{\kappa\delta}(z_f, z_b)$, is always negligible since it is due to correlations of foreground lensing magnification with background density. Finally, the density term in the last line is suppressed by two effects: first by the bias difference, which would exactly vanish if the biases were redshift independent, and second by the fact that density correlations, $C_{\ell}^{\delta \delta}(z_f, z_b)$, quickly decrease with redshift separation. In Sec.~\ref{sec:contaminations} we will choose the redshift bins such that this density contamination is negligible. This estimator, called GIMCO~\cite{Jalilvand2019}, provides therefore a robust way of isolating lensing magnification, and we will use it in the numerator of $E_G$ \be E^\times_{\ell}(z_f, z_b) \simeq \big(5 s(z_b)-2\textbf{})b_{\im}(z_f)C_{\ell}^{\delta\kappa}(z_f, z_b)\, .\label{eq:GIMCO_app} \ee In the denominator of $E_G$ we need the galaxy-velocity correlation. In~\cite{Pullen2016}, this correlation was replaced by the product of $\beta_{\rm g}(z_f)=f(z_f)/b_{\rm g}(z_f)$ and the galaxy-galaxy correlations. In our case, instead of the galaxy-galaxy correlations, we use the galaxy-intensity mapping correlations, such that the $\im$ bias in Eq.~\eqref{eq:GIMCO_app} cancels. We therefore have \begin{align} E_{G}&=\frac{E_\ell^\times(z_f,z_b)}{\beta_{\g}(z_{f}) \Cgh \left(z_{f}, z_{f}\right)}= \frac{ C_\ell^{{\rm HI} \rm g} (z_{f}, z_{b}) - \Cgh (z_f,z_b)}{\beta_{\rm g}\left(z_{f}\right) \Cgh\left(z_{f}, z_{f}\right)}\nonumber\\ &=\frac{\big(5 s(z_b)-2\big)C_{\ell}^{\delta\kappa}(z_f, z_b)}{f(z_f)C_{\ell}^{\delta\delta}(z_f, z_f)}\, . \label{eq:Eg} \end{align} We see from Eq.~\eqref{eq:Eg} that we only need two types of correlators to measure $E_G$: the cross-correlation between galaxy clustering and intensity mapping and the auto-correlation of galaxy clustering, from which $\beta_\g$ is measured. This is particularly interesting to test deviations from GR: Eq.~\eqref{eq:Eg} tests the relation between density, velocity and lensing potential measured from the same correlators. Therefore, the outcome of this test is not subject to potential inconsistencies between different data sets, as can be the case for standard versions of $E_G$ that rely on galaxy clustering and shear or on galaxy clustering and CMB lensing. As discussed above, the numerator in Eq.~\eqref{eq:Eg} is not affected by the lensing contamination computed in~\cite{MoradinezhadDizgah:2016pqy}: the density-magnification contribution is the \emph{signal} in our estimator, whereas the magnification-magnification contamination is absent since intensity mapping is insensitive to lensing magnification. The denominator in Eq.~\eqref{eq:Eg} contains density-magnification contamination and an RSD contamination. In Section \ref{sec:SNR} we will show that we can choose the redshift binning such that these two contaminations remain negligible. Note that an $E_G$ estimator using intensity mapping has already been proposed in~\cite{Pourtsidou_2016}. However, this estimator differs from ours: in~\cite{Pourtsidou_2016} the lensing signal is not measured from clustering as we do here, but from CMB lensing cross-correlated with intensity mapping. This estimator, like ours, is unaffected by magnification contamination. It relies, however on the auto-correlation of intensity mapping to measure $\beta_{\im}$ as well as $C_\ell^{\im\, \im}$. The auto-correlations of intensity mapping will be very challenging to measure due to the difficulty of subtracting the foregrounds accurately. Our estimator relies instead only on cross-correlations of intensity mapping with galaxy clustering, which are unaffected by intensity mapping foregrounds and have already been measured~\cite{Chang:2010jp,Masui:2012zc,Switzer:2013ewa,Anderson:2017ert,Tramonte:2020csa,Li:2020pre,Wolz:2021ofa,CHIME:2022kvg,Cunnington:2022uzo}. \section{Signal-to-Noise and Contamination} \label{sec:SNR} We now study the detectability of $E_G$ with the coming generation of intensity mapping and galaxy surveys. For intensity mapping, we consider the 21\,cm intensity mapping survey HIRAX, which will measure the neutral hydrogen distribution in the redshift range of $z=0.775$ to $2.55$ covering 15'000 square degrees of the southern sky~\citep{Newburgh:2016mwi,Crichton:2021hlc}. For the galaxy survey, we study two examples, one modelled on SKA phase 2, covering 30'000 square degrees, based on the specifications in ~\cite{Villa:2017yfg, Bull:2015lja}, and a 15'000 square-degree \emph{Euclid}-like survey~\cite{Euclid:2021icp}, based on~\cite{Euclid:2019clj,Euclid:2021rez}. We perform separate forecasts for the spectroscopic and photometric samples for the \emph{Euclid}-like scenario. The spectroscopic sample has the advantage of providing an accurate measurement of $\beta_\g$, through RSD, whereas the photometric sample will measure $\beta_\g$ with relatively large error bars. On the other hand, the photometric sample has a significantly lower shot noise due to the much larger number of galaxies detected and extends to higher redshifts. It is, therefore, interesting to study how the measurement of $E_G$ differs in these two samples. The overlapping redshift range between HIRAX and the galaxy surveys are $z\in [0.9, 1.8]$ for \emph{Euclid}-like spectroscopic, $z\in [0.78, 2.16]$ for \emph{Euclid}-like photometric and $z\in [0.78, 2.0]$ for SKA2-like. We assume that \emph{Euclid}-like has a $2/3$ aerial overlap with HIRAX, leading to $f_{\text{sky}}=0.242$, whereas the SKA2-like survey fully overlaps with HIRAX leading to $f_{\text{sky}}=0.363$. For HIRAX correlated with the \emph{Euclid}-like spectroscopic and the SKA2-like surveys, we consider top-hat redshift bins since the accuracy in the redshift determination is excellent. For the \emph{Euclid}-like photometric surveys, on the other hand, we use the bins given in~\cite{Euclid:2021rez} (see Fig.\ 3), which we approximate with Gaussian window functions according to Table 4 in~\cite{Nistane:2022xuz}. Since we are using the code {\sc class}~\citep{class2,DiDio:2013bqa} to compute the angular power spectra, and it does not allow us to choose two different windows at two different redshifts, we also use Gaussian windows for HIRAX in our forecasts in this case. The signal depends on the galaxy magnification bias, $s$. For \emph{Euclid}-like photometric, we use the magnification bias measured from the flagship simulation, see Table 1 of~\citep{Euclid:2021rez}. For SKA2, we use the expression from \citep{Jelic-Cizmek:2020pkh}, and for \emph{Euclid}-like spectroscopic, we use the model developed in~\cite{Montanari:2015rga}. The signal is by construction independent of galaxy bias, however, the variance of $E_G$ depends on the galaxy bias. Explicit expressions for the biases used to model the different surveys can be found in Appendix~\ref{app:surveys}. \subsection{Limber approximation} $E_G$ depends on the foreground and background redshifts, $z_f$ and $z_b$. However, in Limber approximation, the background dependence goes away, and $E_G$ directly probes deviations from GR at redshift $z_f$. We first relate the metric potentials (that enter in the convergence) to the matter density contrast, $\delta$, using the Poisson equation. Since we want to test the validity of GR, we allow for deviations, encoded in the parameter $\Sigma$, see e.g.~\cite{Kunz:2006ca,Amendola:2007rr,Pogosian:2010tj,Kunz:2012aw} \begin{equation} k^2(\Phi + \Psi ) =- 3 H^2_0\Omega_{\text{m}}(1+z)\Sigma(z)\delta(k,z)\, . \label{eq:mod_Poisson} \end{equation} Here $\Omega_{\text{m}}$ denotes the matter density parameter today, and $H_0$ represents the present-time value of the Hubble parameter. In $\Lambda$CDM, $\Sigma=1$ on the scales and redshifts of interest for us. Deviations from $\Lambda$CDM can be encoded into a modification to the Poisson equation, and a difference between the two metric potentials, which combine into a $\Sigma$ generically different from 1~\cite{Kunz:2006ca,Amendola:2007rr,Pogosian:2010tj,Kunz:2012aw}. Plugging this into the convergence~\eqref{eq:kappa}, and using Limber approximation~\cite{Limber:1954zz} we obtain the density-magnification correlation, see e.g.~\cite{Euclid:2021rez} \begin{align} &C_{\ell}^{\delta \kappa}(z_f,z_b) = \frac{\ell(\ell+1)}{(\ell+1/2)^2}\frac{3H_0^2 \Omega_{\text{m}}}{2}\int_{z^{\rm min}_f}^{z^{\rm max}_f}\!\!\!\text{d}z\,n_{\im}(z) \Sigma(z)\nonumber\\ &\times(1+z)P_{\delta\delta}\left[\frac{\ell+1/2}{\chi(z)},z\right]\int_{z^{\rm min}_b}^{z^{\rm max}_b}\!\!\!\text{d}z'\,n_{\g}(z') \frac{\chi(z')-\chi(z)}{\chi(z)\chi(z')} \label{eq:clkg} \end{align} where $n_\g(z)$ is the normalised galaxy distribution, $n_\im(z)$ is the normalised intensity distribution and the integrals run over the size of the redshift bins. Note that the $\ell$-dependent coefficient in front of the integral can be set to one at large $\ell$, where Limber approximation is valid. Moreover, we assume that $\Sigma$ is independent of $k$, which is a common assumption in large-scale structure analyses (e.g.~\cite{Planck:2015bue}), and is motivated by the fact that various theories of modified gravity obey this assumption in the quasi-static approximation, e.g.~\cite{Brans:1961sx,Dvali:2000hr,Babichev:2009ee}. The density-density correlation can also be simplified using Limber approximation, leading to \begin{equation} C^{\delta\delta}_{\ell}(z_f,z_f) = \int_{z^{\rm min}_f}^{z^{\rm max}_f}\!\!\!\text{d}z\, \frac{H(z)}{\chi^2(z)}n_{\g}(z)n_\im(z)P_{\delta\delta}\left[\frac{\ell+1/2}{\chi(z)},z\right] \label{eq:cldd} \end{equation} Inserting Eqs.~\eqref{eq:clkg} and~\eqref{eq:cldd} in $E_G$~\eqref{eq:Eg} and assuming that the functions $\Sigma(z), \chi(z), H(z)$ and $n_\g(z)$ vary slowly with redshift inside each redshift bin, we obtain \begin{align} E_G(\ell, z_f, z_b) = \Gamma(z_f, z_b)\frac{\Omega_m \Sigma(z_f)}{f(z_f)}\, , \end{align} where \begin{align} \Gamma(z_f,z_b)\equiv& \big(5s(z_b)-2\big)\frac{3 H_0^2}{2H(z_f) }(1+z_f) \\ & \times \int_{z^{\rm min}_f}^{z^{\rm max}_f}\!\!\!\text{d}z\,n_{\im}(z) \int_{z^{\rm min}_b}^{z^{\rm max}_b}\!\!\!\text{d}z'\,n_{\g}(z') \frac{\chi(z')-\chi(z)}{\chi(z)\chi(z')} \nonumber \\ & \times \left( \int_{z^{\rm min}_f}^{z^{\rm max}_f}\!\!\!\text{d}z\,n_{\g}(z)n_\im(z) \right)^{-1}\nonumber \\ \simeq&\big(5s(z_b)-2\big)\frac{3 H_0^2}{2H(z_f) }(1+z_f)\label{eq:gamma}\\ &\times\frac{n_\g(z_b)}{n_\g(z_f)}\frac{1}{\chi(z_f)}\int_{z^{\rm min}_b}^{z^{\rm max}_b}\!\!\!\text{d}z\frac{\chi(z)-\chi(z_f)}{\chi(z)}\, .\nonumber \end{align} The second expression corresponds to top-hat bins, while the first one allows for more general redshift bins. The coefficient $\Gamma$ depends on the background cosmology, the galaxy distribution and the magnification bias. A common assumption in large-scale structure analyses is to fix the background to $\Lambda$CDM (since the CMB has extremely well constrained it) and to test for deviations from General Relativity at the level of the perturbations. This is, for example, the strategy adopted in RSD analyses to measure $\beta_\g$ in a model-independent way, see e.g.~\cite{BOSS:2016psr}. Here we follow the same procedure. The galaxy distribution and the magnification bias are quantities that can directly be measured for a given population of galaxies and a given survey. Consequently, $\Gamma$ is a parameter we can predict. Note that we do not include $\Omega_{\text{m}}$ in $\Gamma$ as it cannot be measured in a model-independent way when allowing for general dark energy or modified-gravity scenarios \cite{Kunz:2007rk,Amendola:2012ky}. We then define a re-scaled $E_G$ variable, by dividing it with $\Gamma (z_f, z_b)$ \begin{align} \hat{E}_G(\ell, z_f, z_b)\equiv&\frac{E_\ell^\times(z_f,z_b)}{\Gamma(z_f,z_b)\beta_\g(z_f)C_\ell^{\g\im}(z_f,z_f)}\nonumber \\ =&\frac{\Omega_{{\text{m}}}\Sigma(z_f)}{f(z_f)}\, . \label{eq:EGhat} \end{align} We have tested the validity of the approximations used to obtain Eq.~\eqref{eq:EGhat}. In Fig.~\ref{fig:EGratioPlots} we compare $\hEG$ using the full-sky angular power spectra, with the result using Limber approximation, and with the second line of Eq.~\eqref{eq:EGhat}, which further neglects redshift evolution within each bin. Limber approximation induces a sub percent error on $\hEG$ for $\ell$ above 100 (and less than $0.1\%$ for $\ell \geq 400$). Neglecting redshift evolution induces a $\sim1\%$ percent error at all $\ell$'s. This is well below the statistical uncertainties on $\hEG$, as we will see in Section~\ref{sec:forecasts}. The model-independent and $\ell$-independent function $\Gamma(z_f,z_b)$ defined in Eq.~\eqref{eq:gamma} is therefore perfectly adapted and allows us to define $\hEG$ in a model-independent way, in contrast to the function $C_\Gamma(\ell)$ used in~\cite{Pullen:2015vtb}, which depends on the density power spectrum and is thus model-dependent.~\footnote{Note that to properly test the validity of Eq.~\eqref{eq:EGhat}, it is necessary to use the same distribution functions, $n_\g$ and $n_\im$, in Eqs.~\eqref{eq:clkg}, \eqref{eq:cldd} and~\eqref{eq:gamma}. In our case, we use top-hat window functions approximated by a smooth function in CLASS, with a sharpness that can be adjusted.} \subsection{Signal-to-noise ratio} We now compute the signal-to-noise ratio (SNR) of $\hEG$ for the three combinations of surveys. Since the signal is independent of the background redshift, as discussed above, we sum over all pairs at fixed $z_f$: \begin{align} \label{eq:sn_signal} &\left(\frac{\rm S}{\rm N}\right)^2(z_f)=\sum_{\ell,\ell'=\ell_{\rm min}}^{\ell_{\rm max}}\sum_{z_b, z_b'=z_{\rm min}}^{z_{\rm max}}\hat{E}_G(\ell, z_f, z_b) \\ &\times {\rm cov}^{-1} \big[\hat{E}_G(\ell, z_f, z_b),\hat{E}_G(\ell', z_f, z_b')\big]\hat{E}_G(\ell', z_f, z_b')\, .\nonumber \end{align} The sum runs over all multipoles used in the analysis and all background redshift bins. In Section~\ref{sec:contaminations} we will select those such that the contamination from RSD, lensing and density are negligible. The full expression for the covariance of $\hEG$ is given in Appendix~\ref{app:covariance}. It depends on the covariance of $C_\ell^{\g\im}$, on the variance of $\beta_\g$ and the covariance between the two. The covariance of $C_\ell^{\g\im}$ is given by \begin{align} &{\rm cov}\left[C_\ell^{\g\im}(z_1,z_2), C_\ell^{\g\im}(z_3,z_4)\right]=\frac{1}{f_{\rm sky}(2\ell+1)} \label{eq:covClgim}\\ &\times\left[C_\ell^{\g\g}(z_1,z_3)C_\ell^{\im\im}(z_2,z_4) +C_\ell^{\g\im}(z_1,z_4)C_\ell^{\im \g}(z_2,z_3)\right]. \nonumber \end{align} All terms are affected by cosmic variance. In addition, the galaxy-galaxy correlation is affected by shot noise when $z_1=z_3$ \begin{equation} C_\ell^{\g\g\,{\rm sn} }(z_1,z_3)=\frac{\delta_{z_1,z_3}}{\bar N(z_1)}, \end{equation} where $\bar N(z_1)$ denotes the mean number of galaxies per redshift bin and steradian. The $\im$-$\im$ correlation is affected by shot noise and interferometer noise; however, it has been shown in~\cite{Shaw:2013wza, Shaw:2014khi} that the former is always subdominant with respect to the latter. In~\cite{Jalilvand2019}, we derived an expression for interferometer noise for \mbox{HIRAX}, based on an analytical expression derived in~\citep{Bull:2014rha}, adapted to the outcome of numerical simulations for HIRAX~\citep{Shaw:2013wza,Shaw:2014khi}. Finally, the cross-correlation between the galaxy and intensity mapping is not affected by shot noise or interferometer noise. As shown in Appendix~\ref{app:covariance}, the covariance of $\hEG$ is also affected by the variance of $\beta_\g$. This quantity is measured from the multipoles of the correlation function (or the power spectrum), which are directly sensitive to RSD. Spectroscopic surveys are designed to measure $\beta_\g$ very precisely and are expected to reach a precision of 1\%~\cite{Amendola:2016saw}. Photometric surveys can also measure $\beta_\g$ but with much less precision, of the order of 10\%~\cite{Asorey:2013una}. We use these two values for the spectroscopic and photometric forecasts. Note that, as shown in Appendix~\ref{app:covariance}, the variance of $\beta_\g$ generates non-diagonal contributions, with $\ell\neq\ell'$, to the covariance of $\hEG$. This is because $\beta_\g$ is a parameter measured from the full correlation function in each redshift bin. Its error is, therefore, fully correlated for different values of the angular multipoles $\ell$ and $\ell'$. Neglecting these correlations, as has been done in~\cite{Pourtsidou:2015ksn}, would underestimate the $\beta_\g$ contribution to the variance by a factor of roughly $2\ell+1$. Finally, the covariance of $\hEG$ depends on the covariance between $C_\ell^{\g\im}$ and $\beta_\g$. Even though this covariance is not precisely zero (since $\beta_\g$ and $C_\ell^{\g\im}$ can be measured from the same volume), it is highly suppressed and can be neglected: as has been shown in~\cite{Taylor:2022rgy}, $C_\ell^{\g\im}$ is mainly insensitive to small radial modes, due to the size of the redshift bins, which washes out small-scale information, whereas $\beta_\g$ is measured from small radial modes. This leads to a negligible covariance between these two types of measurements. \subsection{Contaminations} \label{sec:contaminations} From Eq.~\eqref{eq:sn_signal} we see that the SNR depends on the range of multipoles used in the analysis and the redshift bins. We determine those to reduce the contaminations to $\hEG$. As discussed previously, the numerator in $\hEG$ is not affected by the lensing contamination that affects standard $\hEG$ estimators~\cite{MoradinezhadDizgah:2016pqy}. However, when $z_f$ is close to $z_b$, the numerator is affected by density-density contamination, given by the last line in Eq.~\eqref{eq:El}. In order to reduce this contamination to acceptable levels, once the redshift bins have been fixed, we will compute the contamination for each pair and remove those for which it is too high. The denominator in $\hEG$ is affected by contamination from lensing magnification and by contamination from RSD. The first one has been discussed in~\cite{MoradinezhadDizgah:2016pqy}, whereas the second one is usually ignored. We aim to choose the widths of the redshift bins to minimize these two contaminations without compromising the SNR. The contribution from RSD and lensing magnification to the galaxy-HI spectrum at equal redshift can be expressed as \begin{align} &\Delta C^{\g\im}_{\ell}(z_f,z_f) = b_\im(z_f) \big(5 s(z_f)-2\big)C_{\ell}^{\delta\kappa}(z_f,z_f) \label{eq:cont}\\ &\!\!\!+ \big[b_\g(z_f)+b_\im(z_f)\big]C_{\ell}^{\delta\,{\rm RSD}}(z_f,z_f) + C_{\ell}^{{\rm RSD\,RSD}}(z_f,z_f)\, .\nonumber \end{align} The term in the first line is the lensing magnification contamination, and the two terms in the second line are the contaminations from RSD. In Fig.~\ref{fig:Cl_cont0.8} we show the relative contamination from lensing magnification and RSD for $z_f=1.1$ and $z_f = 1.6$ and for two values of the width: $\Delta z_f=0.1$ and $\Delta z_f=0.3$. We choose here the \emph{Euclid}-like spectroscopic survey specifications. We see that for thin bins (dashed curves), the contamination due to lensing is less than $0.1\%$ for both redshifts, $z_f =1.1$ and $z_f = 1.6$, while RSD contributes more than 1\% on a wide range of scales. For thick bins (solid curves), the situation is different. Here RSD is still the dominant contamination on large scales, but on small scales ($\ell \gtrsim 300$ for $z_f=1.1$ and $\ell \gtrsim 550$ for $z_f = 1.6$) the lensing magnification contamination dominates. We see that this contamination does not increase with redshift. This is because, in Limber approximation, only the intra-bin lensing contributes. This contribution decreases with redshift roughly at the same rate as the density-density contribution. However, the bias increases faster with redshift than the magnification bias factor $5s-2$ (see Fig.~\ref{fig:gal-mag-bias} in Appendix~\ref{app:surveys}), leading, overall, to a slight decrease with redshift of the relative lensing contamination. Overall the wider redshift bin choice is clearly better in terms of total contamination, which remains below 3\% for $\ell \gtrsim 100$ for $z_f=1.1$ and $\ell \gtrsim 200$ for $z_f = 1.6$. Based on Fig.~\ref{fig:Cl_cont0.8} we see that for each foreground redshift $z_f$, we can choose a width $\Delta z_f$ that would minimise the total contamination. Here we have compared the contamination with the density contribution $b_\g b_\im C^{\delta\delta}_{\ell}$. In practice, however, what will determine if our estimator is biased or not is the size of the contamination with respect to the variance: as a rule of thumb, if the contamination leads to a contribution in $\hEG$ which is of the same order as its variance, we expect a bias of the order of 1$\sigma$ on the best-fit parameters extracted from $\hEG$. We define, therefore, the following SNR associated with the contaminations \begin{align} \label{eq:sn_contmn} &\left(\frac{\Delta \rm S}{\rm N}\right)^2(z_f)\equiv\sum_{\ell,\ell'=\ell_{\rm min}}^{\ell_{\rm max}}\sum_{z_b, z_b'=z_{\rm min}}^{z_{\rm max}}\Delta \hat{E}_G(\ell, z_f, z_b)\\ &\times {\rm cov}^{-1} \big[\hat{E}_G(\ell, z_f, z_b),\hat{E}_G(\ell', z_f, z_b')\big]\Delta \hat{E}_G(\ell', z_f, z_b')\, .\nonumber \end{align} Here $\Delta \hat{E}_G$ is the difference between the contaminated signal, and the uncontaminated one: \begin{align} \Delta &\hat{E}_G (\ell, z_f, z_b)= \\ &\Bigg\{\frac{E_\ell^\times(z_f,z_b)}{\beta_{\g}\left(z_{f}\right) \left(b_\g \left(z_{f}\right)b_\im\left(z_{f}\right) C_\ell^{\delta \delta}\left(z_{f}, z_{f}\right)+\Delta C^{\g\im}_\ell \left(z_{f}, z_{f}\right)\right)}\nonumber\\ &-\frac{E_\ell^\times(z_f,z_b)}{\beta_{\g}\left(z_{f}\right) b_\g(z_f) b_\im(z_f) C_\ell^{\delta \delta} \left(z_{f}, z_{f}\right)}\Bigg\}\frac{1}{\Gamma(z_f,z_b)}\, , \end{align} where $\Delta C^{\g\im}_\ell$ contains the contamination from RSD and lensing, defined in Eq.~\eqref{eq:cont}. Our strategy to choose optimal redshift bins is based on two factors: minimising the contamination~\eqref{eq:sn_contmn} and maximising the total signal to noise~\eqref{eq:sn_signal}. There are four variables that we need to optimise: $\ell_{\text{min}}$, $\ell_{\text{max}}$, $\Delta z_f$, and $\Delta z_b$. We choose $\ell_{\rm max}=1000$, since above this value, the non-linear modelling of lensing magnification and density fluctuations is uncertain. In Fig.~\ref{fig:SNR_contlmax} (left panel) we plot the SNR contamination as a function of $\ell_{\rm min}$ and $\Delta z_f$, for the redshift pair $(z_f=1.1, z_b=1.6)$, and for fixed $\Delta z_b=0.22$. We show the results for the case of the \emph{Euclid}-like spectroscopic survey. We see that at fixed $\ell_{\rm min}$, the contamination decreases as we increase the bin size. This is because the RSD contamination decreases for thick bins. The lensing contamination, on the other hand, increases with the size of the bin, but it never overcomes the contamination from RSD once we sum over multipoles. Moreover, the contamination decreases quickly when increasing $\ell_{\rm min}$. This is again due to RSD, which peaks at low $\ell$ as can be seen from Fig.~\ref{fig:Cl_cont0.8}. To optimally choose $\Delta z_f$ and $\ell_{\rm min}$, we need to put Fig.~\ref{fig:SNR_contlmax} in balance with the SNR for $\hEG$. Clearly, decreasing $\ell_{\rm min}$ will increase the SNR. A good compromise is to choose $\ell_{\rm min}=100$. This is also motivated by the fact that IM surveys like HIRAX may not have good enough calibration to precisely measure lower multipoles~\cite{Bull:2014rha}. Concerning the size of the bins: the SNR at each $z_f$ increases with the size of the bins, but if we want to test the evolution of $\hEG$ with redshift, we want to have a reasonable number of foreground bins. In Fig.~\ref{fig:SNR_contlmax} (right panel), we plot the SNR summed over all background bins as a function of foreground redshift $z_f$ and for different sizes of the bins. We see that $\Delta z_f =0.24$ is optimal: it allows us to measure $\hEG$ for three foreground redshifts (versus two for $\Delta z_f =0.3$) and has a total SNR of 18.4, larger than for all other cases. The thinner choice, $\Delta z_f=0.14$, is interesting since it allows us to measure $\hEG$ in a higher number of redshift bins. However, the overall SNR (summed over redshift) is 8 percent lower than for $\Delta z_f =0.24$. We therefore choose $\ell_{\rm min}=100$ and $\Delta z_f = 0.24$, which lead to a small SNR contamination of 0.17. This procedure needs to be performed for each value of the foreground redshift $z_f$. We find similar results at all redshifts, with a maximum SNR contamination of 0.21 for the pair $z_f=1.02$, and $z_b=1.71$, which is still perfectly acceptable. We then repeat the same procedure for the \emph{Euclid}-like photometric and the SKA2-like surveys. In Table \ref{tab:bins}, we list the optimal binning for each case. \begin{table}[t] \centering \caption{Optimal redshift bin configurations for \emph{Euclid}-like spectroscopic, photometric and SKA2-like surveys.} \label{tab:bins} \begin{tabular}{|C{1cm} C{1cm}| C{1cm} C{1cm}| C{1cm} C{1cm}|} \hline \multicolumn{2}{|c|}{\emph{Euclid} spectro} & \multicolumn{2}{c|}{\emph{Euclid} photo} & \multicolumn{2}{c|}{SKA2} \\ $z$ & $\Delta z$ & $z$ & $\Delta z$& $z$&$\Delta z$ \\ \hline 1.02 & 0.24 & 0.84 & 0.2& 1.02& 0.24 \\ 1.26 & 0.24 & 1.00 & 0.2& 1.26& 0.24 \\ 1.50 & 0.24 & 1.14 & 0.2& 1.50& 0.24 \\ 1.71 & 0.18 & 1.30 & 0.2& 1.74& 0.24 \\ & & 1.44 & 0.2& 1.93& 0.14 \\ & & 1.62 & 0.2& & \\ & & 1.78 & 0.5& & \\ & & 1.91 & 0.5& & \\ \hline \end{tabular} \end{table} \begin{table}[t] \centering \caption{SNR contamination from density for the \emph{Euclid}-like spectroscopic survey.} \label{table:densSNRspec} \begin{tabular}{c |C{1.6cm} C{1.6cm} C{1.6cm}} \hline $z_f$ & $z_b = 1.26$ & $z_b = 1.50$ & $z_b = 1.71$ \\ [0.5ex] % \hline $1.02$ & 0.09& $3 \times 10^{-16}$ & $7 \times 10^{-16}$ \\ $1.26$ & & $0.006$ & $4 \times 10^{-16}$ \\ $1.50$ & & & $0.04$ \\ \hline \end{tabular} \end{table} \begin{table}[t] \centering \caption{SNR contamination from density for the SKA2-like survey.} \label{table:densSNR-SKA} \begin{tabular}{c |C{1.6cm} C{1.6cm} C{1.6cm} C{1.6cm}} \hline $z_f$ & $z_b = 1.26$ & $z_b = 1.50$ & $z_b = 1.74$ & $z_b = 1.91$\\ [0.5ex] % \hline $1.02$ & 1.40& $10^{-16}$ & $10^{-16}$ & $ 10^{-16}$\\ $1.26$ & & $0.94$ & $10^{-16}$ &$ 10^{-16}$ \\ $1.50$ & & & $0.60$ & $10^{-17}$\\ $1.74$ & & & & $1.54$\\ \hline \end{tabular} \end{table} \begin{table}[t] \centering \caption{SNR contamination from density for the \emph{Euclid}-like photometric survey.} \label{table:densSNR-photo} \begin{tabular}{c |C{1.0cm} C{1.0cm} C{1.0cm} C{1.0cm} C{1.2cm} C{1.0cm} C{1.0cm}} \hline $z_f$ & $z_b = 1.0$ & $z_b = 1.14$ & $z_b = 1.30$ & $z_b = 1.44$ & $z_b = 1.62$ & $z_b = 1.78$ & $z_b = 1.91$\\ [0.5ex] % \hline $0.84$ & $40.8$ & $3.7$ & $0.25$& $ 0.004$ & $ 10^{-6}$ & $ 10^{-3}$ & $10^{-3}$ \\ $1.00$ & & $22.6$ & $1.8$& $0.15$& $ 10^{-3}$ & $ 0.1$ & $0.03$\\ $1.14$ & & & $27.1$& $4.3$ & $0.08$ & $ 0.24$ & $0.14$\\ $1.30$ & & & & $12.8$ & $1.4$ & $2.7$ & $1.3$\\ $1.44$ & & & & & $5.7$ & $10.3$ & $5.8$\\ $1.62$ & & & & & & $66.8$& $51.8$\\ $1.78$ & & & & & & &$28.7$ \\ \hline \end{tabular} \end{table} Once the binning has been fixed, we compute the SNR contamination from density in the numerator of $\hEG$ (due to the last line in Eq.~\eqref{eq:El}), for each pair $(z_f, z_b)$. The results for the \emph{Euclid}-like spectroscopic survey are shown in Table \ref{table:densSNRspec}. This SNR contamination is at most 9\% (for neighbouring pairs). We can therefore keep all pairs in our forecast. For SKA2, the density contamination for adjacent pairs is more significant, as seen in Table~\ref{table:densSNR-SKA}. This is because the variance of $\hEG$ is smaller for SKA2 due to the broader sky coverage and lower shot noise. Consequently, the contamination from density relative to the variance is larger for SKA2 than for \emph{Euclid} spectroscopic. To avoid bias in our estimator, we remove adjacent pairs for SKA2. For \emph{Euclid}-like photometric, the situation is different since we are using Gaussian window functions. In this case, there is still a significant overlap of non-consecutive bins, especially at high redshift, leading to non-negligible density contamination, as seen from Table~\ref{table:densSNR-photo}. Based on these results, we remove all pairs with contamination larger than 0.25. With this, we ensure that the bias from such contamination is below 0.25$\sigma$, which is acceptable. We see that with this criteria, we have only three foreground bins for \emph{Euclid}-like photometric. Note that in reality, we expect the contamination to be smaller: here, we have used a Gaussian window for HIRAX as well, due to the limitation of {\sc class} to have two different window functions. In practice, however, HIRAX can select bins with sharp edges for which the bins overlap, and consequently, the density contamination will be reduced. Overall, our choice of binning is such that the total contamination from RSD, lensing and density is at most 0.3$\sigma$ for \emph{Euclid}-like spectroscopic (for $z_f=1.02$), 0.6$\sigma$ for SKA2-like (for $z_f=1.02$), and 0.4$\sigma$ for Euclid-like photometric (for $z_f=1.14$). Our estimator is therefore robust and unbiased. \section{Modified gravity forecasts} \label{sec:forecasts} From Eq.~\eqref{eq:EGhat}, we see that constraints on $\hEG(z_f)$ directly translate into constraints on the combination of parameters $\Omega_{\rm m}\Sigma(z_f)/f(z_f)$. For each $z_f$, the Fisher element for this combination reads \begin{align} \mathcal{F}&=\sum_{\ell,\ell'=\ell_{\rm min}}^{\ell_{\rm max}}\sum_{z_b, z_b'=z_{\rm min}}^{z_{\rm max}}\frac{\text{d} \hat{E}_G(\ell, z_f, z_b)}{\text{d} (\Omega_{\text{m}}\Sigma/f)}\\ &\times {\rm cov}^{-1} \big[\hat{E}_G(\ell, z_f, z_b),\hat{E}_G(\ell', z_f, z_b')\big]\frac{\text{d} \hat{E}_G(\ell', z_f, z_b')}{\text{d} (\Omega_{\text{m}}\Sigma/f)}\nonumber\\ &=\sum_{\ell,\ell'=\ell_{\rm min}}^{\ell_{\rm max}}\sum_{z_b, z_b'=z_{\rm min}}^{z_{\rm max}}\!\!{\rm cov}^{-1} \big[\hat{E}_G(\ell, z_f, z_b),\hat{E}_G(\ell', z_f, z_b')\big]\, .\nonumber \label{eq:fisher} \end{align} In Fig.~\ref{fig:Fisher} we show the relative error on $\Omega_{\rm m}\Sigma/f$ for the three combinations of surveys, using the optimal redshift bins defined in Table.~\ref{tab:bins}. The constraints expected from a \emph{Euclid}-like spectroscopic survey are very good at low redshift, around 7\%. The precision decreases as the foreground redshift increases, reaching 32\% at $z_f=1.50$. The fact that the constraints are significantly better at low redshift is due to two reasons: firstly, when $z_f$ is small, we can correlate it with a larger number of background redshift bins, which increases the precision of the measurement. Secondly, at low redshift, the number density for a \emph{Euclid}-like spectroscopic survey is significantly larger than at high redshift, leading to a strong suppression of shot noise. On the other hand, a \emph{Euclid}-like photometric survey has a precision of $\sim 11\%$ for all foreground redshifts. The primary source of error for photometric surveys is the error on $\beta_\g$ (of 10\%) since this parameter cannot be well constrained from a photometric survey. This consequently limits the precision with which $\hEG$ can be measured. To model this uncertainty correctly, it is crucial to include the $\beta_\g$ covariance fully correlated in $\ell$ as in \eqref{eq:covarianceEG}. Otherwise, one can obtain an uncertainty on $\hEG$ that is smaller than the uncertainty on $\beta_g$ alone, like for example in \cite{Pourtsidou:2015ksn}, where $\hEG$ is predicted to be measured with a precision of 1\% with LSST, even though LSST can measure $\beta_\g$ with a precision of 10\% only, which is not possible. The constraints predicted for SKA2 are tighter by a factor of 2 with respect to the constraints from \emph{Euclid}-like spectroscopic. This is mainly due to the higher number density of SKA2, the broader sky coverage, and the larger redshift range. However, the constraints are quite poor in the last redshift bin, of 54\% only. This is due to the large shot noise at high redshift, which significantly degrades the constraints, and to the fact that only one pair can be used for this case. In Fig.~\ref{fig:Eg-mu-sigma}, we compare our forecasted constraints with current measurements of $\hEG$ from~\citep{Reyes:2010tr, Pullen:2015vtb, Amon2018MNRAS.479.3422A, Blake2015}. The orange dashed line represents the GR prediction, $\hat{E}_G(z) = \Omega_{{\text{m}}}/f(z)$, computed with our fiducial choice of cosmology\footnote{Note that the measurements depend on the choice of background cosmology through $\Gamma$ and can therefore not directly be compared with our prediction to assess if there is a tension or not. In Table~\ref{table:eg} we provide for this reason a summary of the tension (or agreement) quoted from the respective studies.}. The best measurement for $\hEG$ is the one from~\citep{Alam:2016qcl} at $z_f=0.57$ which has a precision of 13\%. We see from Fig.~\ref{fig:Eg-mu-sigma} that our method to measure $\hEG$ from clustering of galaxies and intensity mapping has the potential to significantly tighten these constraints over a wide range of redshifts, allowing us to test the validity of GR robustly on cosmological scales. \begin{table}[t] \centering \caption{Previous measurements of $\hat{E}_{G}$.} \label{table:eg} \begin{tabular}{c |c |c |c} \hline Authors & $z$ & $\hat{E}_G(z)$& Agreement \\ [0.5ex] \hline \multirow{1}{9em}{Reyes et al \citep{Reyes:2010tr} } & $0.37$& $0.392 \pm 0.065$& agrees with GR\\[1ex] \hline \multirow{2}{9em}{Blake et al \citep{Blake2016MNRAS.462.4240B} } & $0.32$ & $0.48\pm 0.10$ & agrees with GR \\[1ex] & $0.57$ & $0.30 \pm 0.07$ & 1.4$\sigma$ deviation \\ [1ex] \hline \multirow{3}{9em}{Amon et al \citep{Amon2018MNRAS.479.3422A} } & $0.27$ & $0.43\pm 0.13$ & agrees with GR \\[1ex] & $0.31$ & $0.27 \pm 0.08$ & $2.3\sigma$ deviation \\ [1ex] & $0.55$ & $0.26 \pm 0.07$ & $2 \sigma$ deviation \\ [1ex] \hline \multirow{1}{9em}{Alam et al \citep{Alam:2016qcl} } & $0.57$& $0.42 \pm 0.056$ & agrees with GR \\[1ex] \hline \multirow{1}{9em}{Pullen et al \citep{Pullen2016} } & $0.57$& $0.24 \pm 0.060$ & $2.6\sigma$ deviation\\[1ex] \hline \end{tabular} \end{table} \section{Conclusion} \label{sec:conclusion} Testing the consistency of GR at cosmological scales is one of the main goals of modern cosmology. Since a large number of theories have been constructed where gravity is modified or where a dark energy component has been added, confronting each theory individually with observations is no longer feasible. It is, therefore, necessary to build model-independent tests. In this paper, we focused on one of these tests: the $\hEG$ statistic, which compares the evolution of the sum of the potentials with that of the velocity. The $\hEG$ statistic has been successfully measured with various data sets, showing mild tensions with the GR predictions in some cases. It is therefore of great interest to see if future surveys will confirm or not these tensions. However, the currently used estimators of $\hEG$ suffer from two problems: first, they are affected by lensing magnification, a contamination that will become important for future surveys and may invalidate their use. Second, they mix different observables (clustering and shear, or clustering and CMB lensing), making them sensitive to tensions between these different data sets, due for example to different systematics. In this paper, we have proposed an alternative estimator for $\hEG$, which relies {\it only on clustering measurements} from two different tracers: galaxies and intensity mapping. We have shown that this estimator is very robust: it only uses cross-correlations that are less affected by systematics, and by choosing the binning in which $\hEG$ is measured appropriately, the contaminations are below 0.3$\sigma$ for a \emph{Euclid}-like spectroscopic survey, below 0.6$\sigma$ for a SKA2-like survey, and below 0.4$\sigma$ for a \emph{Euclid}-like photometric survey. Moreover, our estimator depends on a choice of fiducial cosmology only through background parameters, namely through $H(z)$ and distances. Contrary to some estimators used in the literature, it does not depend on the evolution of perturbations, like in~\cite{Pullen:2015vtb}. Using Fisher forecasts, we have found that $\hEG$, and consequently deviations from GR, can be measured over a wide range of redshifts and with a precision of up to 7\% with a \emph{Euclid}-like spectroscopic survey and 3.9\% with a SKA2-like survey. Such measurements will, therefore, dramatically improve current constraints and extend them to higher redshifts, allowing us to test deviations from GR very accurately. \section{Acknowledgments} It is a pleasure to thank Patrick Simon for useful comments on the manuscript. MA and CB acknowledge financial support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant agreement No.~863929; project title ``Testing the law of gravity with novel large-scale structure observables"). MJ and MK acknowledge financial support from the Swiss National Science Foundation. \appendix \section{Covariance of $\hEG$} \label{app:covariance} We compute the covariance of $\hEG(\ell,z_f,z_b)$ for a fixed value of $z_f$ since we measure it independently in each foreground redshift bin. However, since we combine different background redshift bins in the measurements, we need to account for the covariance of $\hEG$ between different background redshifts $z_b$ and $z_b'$. Moreover, since $\beta_\g$, which enters into $\hEG$, is measured by combining all scales (and not independently for each value of $\ell$), it will induce correlations between $\hEG$ at different values of $\ell$. In general, for $X=f(A,B)$ and $X'=f(A',B')$, where $f$ is a generic function, the covariance of $X$ and $X'$ can be expressed as: \begin{align} \operatorname{Cov}(X,X') = \sum_{\substack{\alpha= A,B \\ \beta=A',B'}} \frac{\partial X}{\partial \alpha} \operatorname{Cov}(\alpha,\beta) \frac{\partial X'}{\partial \beta}\, . \end{align} Using this formula, the covariance of $\hEG$ becomes \begin{align} \operatorname{Cov}&[\hEG(\ell,z_f,z_b),\hEG(\ell',z_f,z_b')] =\label{eq:covarianceEG}\\ &\Bigg\{ \frac{\text{Cov}\left[\ec(z_f,z_b),\ec(z_f,z_b')\right]}{\ec(z_f,z_b)\hec(z_f,z_b')} \nonumber\\ &\quad+ \frac{\operatorname{Var}\left[\Cgh(z_f,z_f)\right]}{\left(\Cgh(z_f,z_f)\right)^2} \nonumber \\ &\quad- \frac{\operatorname{Cov}\left[\ec(z_f,z_b),\Cgh(z_f,z_f)\right]}{\ec(z_f,z_b)\Cgh(z_f,z_f)} \nonumber \\ & \quad- \frac{\operatorname{Cov}\left[\ec(z_f,z_b'),\Cgh(z_f,z_f)\right]}{\ec(z_f,z_b')\Cgh(z_f,z_f)} \nonumber \Bigg\}\\ &\times E_{G}(\ell,z_f,z_b)E_{G}(\ell,z_f,z_b')\,\delta^{\text{D}}_{\ell,\ell'}\nonumber \\ & + \hEG(\ell,z_f,z_b) \hEG(\ell',z_f,z_b') \frac{\operatorname{Var}\left[\beta_{\text{g}}\right]}{\beta_{\text{g}}^2}\, ,\nonumber \label{eq:covEq} \end{align} where $\delta^{\text{D}}_{\ell,\ell'}$ represents the Dirac delta function. Here, we have neglected the covariance between $\beta_\g$ and both $C_\ell^{\g\im}$ and $\ec$, as discussed in the main text. The covariance of GIMCO, which enters in the second line of Eq.~\eqref{eq:covarianceEG} is given by \begin{align} \text{Cov}\Big[&E^{\times}_{\ell}(z_f,z_b),E^{\times}_{\ell}(z_f,z_b')\Big]= \\ &\text{Cov}\left[ C_{\ell}^{\g\text{HI}}(z_b,z_f), C_{\ell}^{\g\text{HI}}(z_b',z_f) \right]\nonumber\\ &- \text{Cov}\left[ C_{\ell}^{\g\text{HI}}(z_b,z_f), C_{\ell}^{\g\text{HI}}(z_f,z_b')\right]\nonumber\\ &- \text{Cov}\left[ C_{\ell}^{\g\text{HI}}(z_f,z_b), C_{\ell}^{\g\text{HI}}(z_b',z_f)\right]\nonumber\\ & +\text{Cov}\left[ C_{\ell}^{\g\text{HI}}(z_f,z_b), C_{\ell}^{\g\text{HI}}(z_f,z_b') \right]\, ,\nonumber \end{align} where the covariance of $C_\ell^{\g\im}$ is given in Eq.~\eqref{eq:covClgim}. Similarly, the covariance of $\ec$ with $C_\ell^{\g\im}$, which enters in the 4th and 5th lines of Eq.~\eqref{eq:covarianceEG} can be written as \begin{align} \operatorname{Cov}\Big[&\hec(z_f,z_b), \Cgh(z_f,z_f)\Big]=\\ &\text{Cov}\left[ C_{\ell}^{\g\text{HI}}(z_b,z_f), C_{\ell}^{\g\text{HI}}(z_f,z_f) \right]\nonumber\\ &- \text{Cov}\left[ C_{\ell}^{\g\text{HI}}(z_f,z_b), C_{\ell}^{\g\text{HI}}(z_f,z_f)\right]\nonumber\, . \end{align} \section{Galaxy and Intensity Mapping Surveys} \label{app:surveys} This paper uses three example surveys for galaxy clustering: \emph{Euclid}-like spectroscopic, \emph{Euclid}-like photometric, and SKA2-like; and one intensity mapping survey: HIRAX. The bias of the intensity mapping from HIRAX is given by \citep{Jalilvand2019} \begin{equation} b_{\text{HI}}(z) = 0.677\Big(1 + 3.8 \times 10^{-1} z + 6.7 \times 10^{-2} z^2\Big)\, . \label{eq:bIM} \end{equation} For the \emph{Euclid}-like photometric survey, we use the bias and magnification bias that have been measured from the flagship simulation in each of the redshift bins we are using, see Table 1 of~\citep{Euclid:2021rez}. For the SKA2-like survey, we use the bias, and magnification bias modelled in~\cite{Villa:2017yfg}: \begin{align} b_{\text{g}}(z) &= 0.5887 \exp(0.813 z)\, , \\ s(z) &= s_0 + s_1 z + s_2 z^2 + s_3 z^3\, , \label{eq:bias_magSKA2} \end{align} with $s_0 = -0.1068$, $s_1=1.359$, $s_2=-0.620008$, and $s_3 = 0.1885$. Finally, for the \emph{Euclid}-like spectroscopic survey we use the bias given in~\citep{Euclid:2019clj} (Table 3). Since we are not using the same bins as there, we interpolate this bias with the following function \begin{equation} b_\g(z) = -1.094 + 1.813 \sqrt{1+z}\, .\label{eq:Euclidspectro} \end{equation} The magnification bias for \emph{Euclid}-like spectroscopic has never been modelled nor measured in simulations. We use, therefore, for simplicity, the expression given in~\citep{Montanari:2015rga} (which is another model for SKA2) \begin{equation} s(z) = 0.9329 - 1.562\exp(-2.437 z)\, . \label{eq:mag_Euclidspec} \end{equation} Galaxy and magnification biases are plotted in Fig.~\ref{fig:gal-mag-bias}. \section{Window Functions} We use the top-hat window function for spectroscopic surveys and a Gaussian window for the photometric survey. The reason is that in spectroscopic surveys, we can measure precisely the redshift, while in photometric surveys, we have larger errors on the redshift. For the top-hat, we choose a smooth version of the top-hat filter: \begin{equation} W_{\text{TH}}(z) = \begin{cases} \frac{1}{2}\Big[1+\tanh\Big(\frac{z - \mu+\Delta z /2}{\alpha}\Big)\Big] & \text{if } z \leq \mu\\ \frac{1}{2}\Big[1+\tanh\Big(-\frac{(z - \mu-\Delta z /2)}{\alpha}\Big)\Big] & \text{otherwise} \end{cases} \end{equation} where $\mu$ is the mean of the redshift bin, $\Delta z$ is the width of the bin, and $\alpha$ is the speed of the transition which we choose to be $\alpha=\Delta z/10$. The expression for the Gaussian window function is given by \begin{equation} W_{\text{G}}(z) = \frac{1}{\Delta z\sqrt{2\pi}}\exp\left[-\frac{(z-\mu)^2}{2\Delta z^2}\right]\, . \end{equation} \bibliographystyle{unsrt} \bibliography{gimco_Egbib}

Title: Missing Metals in DQ Stars; a Compelling Clue to their Origin

Abstract: White dwarf stars frequently exhibit external pollution by heavy elements, and yet the intrinsically carbon-enriched DQ spectral class members fail to experience this phenomenon, representing a decades-old conundrum. This study reports a high-resolution spectroscopic search for Ca II in classical DQ white dwarfs, finding that these stars are stunted both in pollution frequency and heavy element mass fractions, relative to the wider population. Compared to other white dwarf spectral classes, the average external accretion rate is found to be at least three orders of magnitude lower in the DQ stars. Several hypotheses are considered which need to simultaneously account for i) an apparent lack of accreted metals, ii) a dearth of circumstellar planetary material, iii) an observed deficit of unevolved companions in post-common envelope binaries, iv) relatively low helium mass fractions, and remnant masses that appear smaller than for other spectral classes, v) a high incidence of strong magnetism, and vi) modestly older disk kinematics. Only one hypothesis is consistent with all these constraints, suggesting DQ white dwarfs are the progeny of binary evolution that altered both their stellar structures and their circumstellar environments. A binary origin is already suspected for the warmer and more massive DQ stars, and is proposed here as an inclusive mechanism to expose core carbon material, in a potential evolutionary unification for the entire DQ spectral class. In this picture, DQ stars are not descended from DA or DB white dwarfs that commonly host dynamically-active planetary systems.

https://export.arxiv.org/pdf/2208.05990

\begin{keywords} circumstellar matter--- planetary systems--- stars: abundances--- stars: evolution--- white dwarfs \end{keywords} \section{Introduction} White dwarf stars whose spectra exhibit molecular carbon features have been known for over 60 years, when the first pair were discovered by carefully constructed spectrophotometry \citep{greenstein1957,bell1962}. Owing to instrumentation limits on wavelength at that time, only the weaker C$_2$\,(1,0) feature near 4670\,\AA \ was observable \citep{eggen1965,greenstein1970}, and not the stronger C$_2$\,(0,0) feature with bandhead near 5165\,\AA \ that would later be found in many more stars (e.g.\ \citealt{oke1974,liebert1977,wesemael1993}). It has been understood for nearly half a century that white dwarfs exhibiting C$_2$ Swan bands (or other carbon features) in their spectra -- denoted as spectral class DQ \citep{mccook1987} -- have atmospheres that are dominated by helium, with only trace abundances of carbon, and little or no hydrogen \citep{bues1973,grenfell1974}. Early pioneering work to understand the carbon abundances in DQ stars pointed out that convective mixing in a gravitationally-stratified atmosphere was problematic, because full mixing would likely result, and thus lead to a carbon-dominated atmosphere by mass \citep{vauclair1979a,dantona1979}, in contrast with the most successful models for the observed spectra. There were already theoretical indications that the mass of the helium layer in DQ stars should be significantly thinner than in their non-carbon bearing, helium atmosphere counterparts \citep{muchmore1977}, and this has been corroborated by successful models that dredge up carbon, where it emerges through the diffusive (upward) tail in a stratified atmosphere \citep{koester1982,fontaine1984}. The first time-dependent modeling of this process showed that the inferred trace carbon abundances are best reproduced with helium layer mass fractions in the range $-4.0 \lesssim \log q({\rm He}) \lesssim -3.5$, and demonstrated that higher mass fractions, such as those predicted by stellar evolution, fail to dredge sufficient carbon to reproduce the observations \citep{pelletier1986}. On a similar timescale, it was already known that white dwarfs sometimes exhibited metals such as Ca, Mg, and Fe in their spectra, and that these were also trace abundances in helium-dominated atmospheres (e.g.\ vMa\,2; \citealt{weidemann1960,strittmatter1971,wegner1972}. In this way, the metal-rich DZ spectral class was shown to share basic characteristics, such as temperature and atmospheric composition, with the DQ stars, based on the earliest discoveries and successful modeling. In contrast with the DQ stars, however, there was already a scientific consensus that the metals observed in DZ stars could not have been brought to the surface from their interiors \citep{fontaine1979,vauclair1979b}. Over a wide range of temperatures, white dwarfs should be gravitationally sedimented (via diffusion), on timescales much faster than any potentially competing process \citep{dupuis1992}, including convective mixing -- a mechanism that actually enhances downward diffusion \citep{alcock1980}. Thus it has been clear for many decades that an external source is necessary for DZ stars, but that DQ stars can be understood as polluted via their own interiors. \citet{fontaine1984} first mentioned the striking problem that is the subject of this work, ``{\it... it is interesting to speculate on the reasons why metals and carbon still appear to be mutually exclusive in the atmospheres of cool helium-rich white dwarfs.}'' The DQ stars are around 9--10 per cent of the local population as defined by the 20\,pc sample, which is well constrained if not complete \citep{hollands2018}, and thus are not outliers or unusual by any means. Indeed, more than 60 years after the first discovery, this fundamental question about their nature remains unanswered, despite recent and ongoing work with far superior data and modeling, and based on hundreds of DQ and DZ stars \citep{koester2019,coutu2019,koester2020}. This paper reports a deep spectroscopic search for external, heavy element pollution in DQ stars using one of the largest ground-based telescopes available. A dramatic lack of metal pollution is revealed for DQ white dwarfs, as compared to the similarly helium-rich DZ stars. Section~2 reports on the high-resolution spectroscopic observations, and Section~3 quantifies the differences between the DQ stars and DZ stars in terms of pollution frequencies and accretion rates, using both the high-resolution spectral data as well as available SDSS spectroscopy. Section~4 examines other threads of available evidence on DQ stars, and in particular where there are distinct properties as compared to DZ-type and other white dwarfs (mass and stellar structure, circumstellar matter, duplicity, magnetism, and kinematics). Section~5 examines several hypotheses that might account for all the available evidence, and concludes that the DQ white dwarfs are the result of binary evolution, where mass loss has resulted in a thinner helium layer than otherwise expected, and which also depleted any nearby planetary material that might pollute the star. Section~6 discusses this possibility in detail, and provides suggestions for future work that can test this novel proposed origin for DQ stars. \section{Spectroscopy and metal abundances} \begin{table} \begin{center} \caption{Target parameter summary, Ca upper limits and detections.\label{targs}} \begin{tabular}{@{}crlcrl@{}} \hline WD\# &$T_{\rm eff}$ &Mass &[C/He] &[Ca/He] &Notes\\ &(K) &($M_{\odot}$) & & &\\ \hline 0038$-$226 &5210 &0.51 &$-$8.4 &$<-$12.3 &1\\ % 0042$-$238 &10500 &0.5: &$-$2.7 &$<-$11.8 &2\\ % 0312$-$084 &9080 &0.5: &$-$4.2 &$<-$12.0 &3\\ % 0435$-$088 &6400 &0.55 &$-$6.3 &$<-$12.8 &4\\ % 0548$-$001 &6080 &0.66 &$-$6.6 &$<-$12.5 &1\\ % 0806$-$661 &10210 &0.58 &$-$5.5 &$<-$12.5 &5,9\\ % 0811$+$250 &7920 &0.54 &$-$5.4 &$<-$12.2 &4\\ % 0856$+$331 &9490 &0.87 &$-$3.5 &$<-$11.8 &4\\ % 0913$+$103 &8410 &0.53 &$-$5.2 &$ -$11.6 &4\\ % 0935$-$371 &9380 &0.78 &$-$4.2 &$<-$12.3 &6\\ % 1015$+$088 &7580 &0.53 &$-$6.0 &$<-$12.5 &4\\ % 1142$-$645 &7950 &0.58 &$-$5.5 &$<-$12.9 &4\\ % 1149$-$272 &6440 &0.56 &$-$6.7 &$<-$12.8 &4\\ % 1708$-$147 &9280 &0.54 &$-$3.9 &$ -$11.7 &7,9\\ % 1831$+$197 &7120 &0.56 &$-$6.2 &$<-$12.8 &4\\ % 1837$-$619 &8500 &0.5: &$-$5.0 &$<-$12.5 &2,9\\ % 1917$-$077 &10400 &0.62 &$-$5.8 &$<-$12.8 &5,10\\ % 2059$+$316 &9100 &0.66 &$-$5.0 &$<-$12.5 &4\\ % 2140$+$207 &7520 &0.50 &$-$6.3 &$<-$12.8 &4\\ % 2147$+$280 &11000 &0.5: &$-$7.0 &$<-$12.0 &2,10\\ % 2154$-$512 &7190 &0.60 &$-$4.4 &$<-$12.8 &5\\ % 2311$-$068 &7350 &0.55 &$-$6.1 &$<-$12.5 &4\\ % 2317$-$173 &10800 &0.5: &$-$6.5 &$<-$12.3 &2,10\\ % \hline \multicolumn{2}{l}{Other DQ(Z) stars:}\\ 0208$-$510 &8180 &0.59 &$-$4.8 &$<-$11.8 &8\\ % 0736$+$053 &7590 &0.55 &$-$5.9 &$- $11.8 &4\\ % \hline \end{tabular} \end{center} {\em Notes and references}: (1) \citet{blouin2019}; (2) \citet{weidemann1995}; (3) \citet{sayres2012}; (4) \citet{coutu2019}; (5) \citet{giammichele2012}; (6) \citet{dufour2005}; (7) \citet{subasavage2017}; (8) \citet{farihi2013a}; (9) C$_2$ band detected in the optical for the first time; (10) carbon detected only in ultraviolet. \end{table} \subsection{Target selection and observations} The observed DQ stars are listed in Table~\ref{targs}, and taken from the literature based on favorable brightness and position in the sky. Science targets were selected to have $T_{\rm eff}\la11\,000$\,K, and thus are primarily classical DQ stars with C$_2$ Swan bands, where trace atmospheric carbon is attributed to the dredge up of core material via successful modeling in helium-rich white dwarfs \citep{pelletier1986,koester2020}. These cooler DQ white dwarfs were selected because they represent the bulk of all objects in this spectral class, and form a modest but significant fraction of the local white dwarf population \citep{hollands2018}. One known DQZ star (0913+103; \citealt{wegner1985,kleinman2013}) was included in the study in order to have at least one confident detection of Ca\,{\sc ii} H and K, and to search for other metal species and thereby better assess the rare, externally polluted DQ stars. The overall science program was executed in service mode as priority B during ESO periods 95, 96, and 97, where data were ultimately received for 23 out of 30 possible targets. Each white dwarf was observed with UVES \citep{dekker2000} on the VLT, using the 1\,arcsec slit and $2\times2$ binning on-chip, thereby achieving a nominal resolving power $R\approx40\,000$. Spectroscopy was performed using a standard dichroic configuration, with central wavelengths of 3900\,\AA \ on the blue side, and 5640\,\AA \ on the red side. Stars were observed using two identical integrations taken in sequence, where the individual exposure times varied between roughly 300\,s and 1800\,s (typically 900\,s) in order to achieve signal-to-noise (S/N) $>20$ at 4000\,\AA. Raw science and calibration frames for all targets were retrieved from the ESO archive and reduced using the standard {\sc reflex} UVES pipeline. Spectra were flat-fielded, bias- and dark-subtracted, extracted, and wavelength-calibrated following parameter optimizations recommended in the {\sc reflex} UVES documentation. For the DA+DQ visual binary 0935$-$371, frames were reduced and calibrated using the standard processes, however for spectral extraction the slit length and offset parameters were adjusted to isolate each star. As observations were taken in pairs, individual spectra were combined via a weighted average to increase the S/N, and then normalized in each arm. For the entire sample, S/N values of $10-70$ in the blue, and $20-120$ in the red were found over the wavelength ranges 3500--3600\,\AA \ and 5200--5300\,\AA \ respectively. To provide confidence in the {\sc reflex} reductions, the ESO archive phase 3 spectra for all targets were retrieved and combined in an identical manner, where it was found that both the S/N and wavelength RMS values agree within 1 per cent. \subsection{Abundances and inferred accretion rates} Abundances\footnote{Abundance notation in this paper corresponds to [Z/He] = $\log(n_{\rm Z}/n_{\rm He})$.} and upper limits for Ca\,{\sc ii} were derived using state-of-the-art models of helium atmosphere white dwarfs, and specifically for stars that are enhanced in either carbon or metals typically found as external pollutants \citep{coutu2019}. This was achieved by the method of line profile fitting, where, after a good fit was approximated, models were recalculated with $\pm0.3$\,dex abundances, and visual inspection of the lines then determined the best abundance (or upper limit). Basic stellar parameters such as mass, effective temperature, and carbon abundance were taken from the literature and are listed in Table~\ref{targs}, together with the results for calcium. For each DQ star in the UVES sample, the [Ca/He] abundance determination or upper limit was converted into a time-averaged accretion rate following standard methodology. For helium atmosphere white dwarfs with temperatures in the range considered here, metal sinking timescales are sufficiently long that a steady state cannot be inferred with any confidence \citep{koester2009a}. However, the steady-state accretion rate is equivalent to a time-averaged value over a single sinking timescale, and is sufficiently informative in several contexts \citep{farihi2009a,farihi2016a,hollands2018}. This is simply the mass of a particular metal in the outer, fully-mixed layers of the star, divided by the diffusion timescale for this element \citep{dupuis1993}. To estimate the total mass accreted based on a single element, a correction factor is applied; in this case Ca is assumed to be 0.016 of the total mass as in the bulk Earth \citep{allegre1995,farihi2012b}. Up-to-date diffusion calculations for pure helium atmospheres with no convective overshoot have been used \citep{koester2020}. \section{Pollution inferences and implications} The following section makes use of the metal abundances and upper limits from the UVES sample, as well as from a larger samples of DQ stars available from the Sloan Digital Sky Survey (SDSS). While the UVES observations represent the most sensitive search for Ca\,{\sc ii} in DQ stars performed to date, the sample size is only 23 objects. In contrast, there are an order of magnitude more DQ white dwarfs with SDSS spectroscopy, and although these data are somewhat less sensitive to Ca\,{\sc ii} (Section~3.3), the resulting detection frequency is more robust than for the smaller UVES sample. First, accretion rates and upper limits are inferred for the white dwarfs with UVES spectroscopy (and for two DQ stars observed with {\em HST}). Second, pollution frequencies are discussed in the context of robust samples with available SDSS spectroscopy. Third, the combined evidence is evaluated and compared with what is known for the wider population of polluted white dwarfs. \subsection{Accretion rates: UVES} Figure~\ref{mdots} explores the inferred total (time-averaged) accretion rates for the observed DQ stars, alongside a sample of DZ stars with similar effective temperatures \citep{dufour2007,farihi2010b}. Both sets of stars are understood to have helium-dominated atmospheres, and can be seen to occupy the corresponding evolutionary track on the Hertzsprung-Russell diagram shown in the Figure. Although there appear to be a few DQ stars that are outliers in $G_{\rm BP}-G_{\rm RP}$ color, this is simply the result of objects with strong Swan band absorption in the blue, and not from any actual difference in the range of temperatures or cooling ages. The two DQZ stars in the UVES sample lie directly on the helium atmosphere cooling sequence and do not otherwise distinguish themselves. In stark contrast to their shared effective temperatures and luminosities, in terms of metal accretion rates, {\em the DQ and DZ stars show no overlap whatsoever}. It may appear that there are no DZ stars with abundances and inferred accretion rates as low as the DQ(Z) objects, but there has not yet been any search for weak metal lines in DC white dwarfs using large telescopes and high-resolution spectroscopy. The conventional detection of DZ stars is straightforward using low-resolution spectroscopy of modest quality, based on the fact that cool helium atmospheres are largely transparent owing to a lack of major opacity sources \citep{koester2009a}. Notably, the Ca\,{\sc ii} lines in vMa\,2 were detected with photographic plate technology over 100 years ago \citep{vanmaanen1919}, and, as can be seen in Figure~\ref{uves}, using an 8\,m class telescope and a high-resolution spectrograph is clearly overkill, even for [Ca/He] $=-10.0$ ($\dot M_{\rm z} = 1\times10^7$\,g\,s$^{-1}$). For context, DZ stars with metal abundances 30 times smaller than vMa\,2 are known, and the corresponding Ca\,{\sc ii} line strengths are readily detected using the $R\approx2000$ resolving power of SDSS spectroscopy \citep{dufour2007,coutu2019}. The use of the VLT and UVES, together with a sample size of 23 targets, demonstrates conclusively that the pollution in DQZ stars never approaches the levels typically attained in DAZ, DBZ, and DZ stars. For systems that are likely in accretion-diffusion equilibrium (i.e.\ a steady state), instantaneous accretion rates can exceed $10^9$\,g\,s$^{-1}$ \citep{farihi2016b}, and roughly speaking, that is 5600 times higher than the rates inferred for polluted DQ white dwarfs. And while steady-state accretion cannot be assumed for helium-rich atmospheres, if the time-averaged mass accretion rates are compared between the DQZ and the DZ stars, the top accretors in each class are separated by a factor of 18\,000(!). Furthermore, most DQ stars have only upper limit metal abundances and accretion rates, and therefore the gulf between these two populations is vast. \subsection{Pollution frequency: SDSS} Although Ca\,{\sc ii} in the stars 0913$+$103 and 1707$-$147 was detected with UVES, these lines should be detectable in DQ stars via SDSS spectroscopy, for the same reasons outlined above for DZ stars; transparent atmospheres and broad lines. Because DZ and DQ stars both have atmospheres dominated by helium, and similar effective temperatures, the much larger SDSS datasets of DQ white dwarfs will result in lower yet comparable sensitivity to Ca\,{\sc ii} absorption. For this purpose, two SDSS spectroscopic samples were analyzed, where DQZ candidates were visually inspected, all of which are shown in Figure~\ref{sdss}. First, there are 164 stars designated as DQ (160) or DQZ (4) in the SDSS DR7 white dwarf catalog, ignoring those spectral types deemed uncertain \citep{kleinman2013}. On further scrutiny, three of these four are genuine DQZ (J091602.83$+$101109.7, J133205.62 +274003.9, J153447.54$+$414559.4), and one is likely a false positive with noisy data (J140256.39 +111332.3; Figure~\ref{sdss}). This represents 3:164 or $1.8${\raisebox{0.5ex}{\tiny$^{+1.7}_{-0.6}$} per cent of DQ stars that are DQZ. Second, a sample of 221 DQ white dwarfs with reliable SDSS spectroscopic identifications based on S/N, and robust {\em Gaia} parallaxes were taken from the literature \citep{koester2019}, where each spectrum was examined for the possible presence of Ca\,{\sc ii} lines. Of these, there are four DQZ stars; two are in common with the DR7 catalog, plus the objects J075230.82$+$444749.9 and J131953.49$+$084422.0, yielding 4:221 or $1.8${\raisebox{0.5ex}{\tiny$^{+1.4}_{-0.5}$} per cent. During these visual searches, weaker candidates with noisy spectra were also considered, and the strongest of these are shown in Figure~\ref{sdss}; they are not considered further, and even if real the preceding frequency statistics would not be significantly altered. There is a modestly larger sample of DQ stars that has been analyzed by \citet{coutu2019}. The catalog is comprised of all stars designated as DQ and DZ in the Montr\'eal white dwarf database and thus prone to selection effects, but is largely comprised (80 per cent) of sources with SDSS spectroscopy. Despite any possible biases within this heterogeneous sample, the numbers are revealing. Of the 319 DQ white dwarfs studied, only four DQZ stars are noted; Procyon\,B and three SDSS objects re-evaluated in Figure~\ref{sdss}, where J090051.91$+$033149.3 is rejected here due to insufficient signal. The remaining three sources with metal lines represent 3:319 or $0.9${\raisebox{0.5ex}{\tiny$^{+0.9}_{-0.3}$} per cent of DQ stars. It is also noteworthy that in the same database, there are 1023 DZ white dwarfs with no evidence for carbon enrichment \citep{coutu2019}. Again keeping in mind that this catalog has selection effects, it seems that external pollution favors helium atmospheres without carbon at a ratio of 1026:3 or in 99.7 per cent of cases. \subsection{0913+103: a bridge connecting UVES and SDSS} The previous section takes advantage of large samples of DQ white dwarfs for statistical purposes, but a priori, $R\approx2000$ spectroscopy with the 2.5\,m SDSS telescope may not be readily comparable to $R\approx40\,000$ data from the VLT. However, and perhaps surprisingly, the UVES target 0913+103 is weakly but clearly detected as a DQZ in the SDSS with designation J091602.83$+$101109.7, and was selected for the VLT survey based on this foreknowledge (Section~2). Figure~\ref{0913} compares the Ca\,{\sc ii} lines detected in both SDSS and UVES, where the broad width of the weak lines -- K in particular -- enables their detection at moderately low spectral resolution. The SDSS spectrum has an average S/N $=53$ (per resolution element) over its entire wavelength range from an exposure that is close to 3600\,s (4 integrations). Although the UVES exposure time is comparable at 3960\,s (2 integrations), the raw spectrum has only S/N $=23$ in the blue arm where Ca\,{\sc ii} is found. In the Figure, the UVES data are rebinned to the same number of points contained in the SDSS wavelength array, yielding a spectrum that is remarkably comparable in terms of S/N. However, in these rebinned data, the H line appears more robustly detected, and both the H \& K features have modestly greater line depths, a likely result of the original higher resolution. This example demonstrates that, owing to the intrinsic width of the metal absorption features in cool, helium atmosphere white dwarfs -- exemplified by vMa\,2 in Figure~\ref{uves} -- the detection of DQZ stars using SDSS spectroscopy is broadly comparable to their detection with UVES, at least to the same degree as found in 0913+103. Using this DQZ white dwarf as a benchmark, it can be inferred that abundances and time-averaged accretion rates as low as [Ca/He] $=-11.6$ and $\dot M_{\rm z} = 3\times10^5$\,g\,s$^{-1}$ are likely detectable, and thus absent in the $N>200$ DQ stars in the SDSS. This conclusion adds considerable weight to the stark difference in pollution between the DQ and DZ white dwarfs. \subsection{DQZ stars vs. other polluted white dwarfs} In Table~\ref{rates}, the different types of polluted white dwarfs are compared by 1) $f_{\rm z} \equiv$ the frequency at which pollution is detected and 2) $\langle \dot M_{\rm z} \rangle \equiv$ a typical mass accretion rate inferred for a given spectral class. \begin{enumerate} \item{For the spectroscopic class of DA stars (= hydrogen Balmer lines strongest, typically but not always hydrogen-dominated atmospheres), pollution frequency and accretion rate statistics are taken from \citet{zuckerman2003} and \citet{koester2014}. Polluted spectral types include DAZ and DZA, where the latter implies the metal lines are the stronger than the Balmer lines.} \item{For the spectroscopic class of DB stars (= He\,{\sc i} lines strongest, helium-dominated atmospheres), the occurrence rate of pollution and accretion statistics are taken from \citet{zuckerman2010}. Polluted spectral types include DBZ and DBAZ, where the latter indicates weak hydrogen lines, and even weaker metal lines.} \item{For the spectroscopic class of DC stars (= continuum only, no lines, typically but not always helium-dominated atmospheres), pollution statistics and accretion rate inferences are taken from \citet{farihi2010b} and \citet{hollands2018}. Polluted spectral type is DZ.} \end{enumerate} \begin{table} \begin{center} \caption{White dwarf pollution as a function of spectral class.\label{rates}} \begin{tabular}{@{}r@{\hspace{2pt}}lrlc@{}} \hline \multicolumn{2}{c}{Spectral Class} &$f_{\rm z}$ &$\langle \dot M_{\rm z} \rangle$ &Ref\\ & & &(g\,s$^{-1}$) &\\ \hline DAZ / &[ DA + DAZ ] &0.25 &$\sim10^8$ &1,2\\ DBZ / &[ DB + DBZ ] &0.30 &$\sim10^8$ &3\\ DZ / &[ DC + DZ ] &0.28 &$\sim10^8$ &4,5\\ DQZ / &[ DQ + DQZ ] &$<0.09$ &$\lesssim10^{5}$ &6\\ DQZ / &[ DQ + DQZ ] &$0.02$ &$\lesssim10^{5.5}$ &7\\ \hline \end{tabular} \end{center} {\em References}: (1) \citealt{zuckerman2003}; (2) \citealt{koester2014}; (3) \citealt{zuckerman2010}; (4) \citealt{farihi2010b}; (5) \citealt{hollands2018}; (6) This study, based on UVES; (7) This study, based on SDSS. \end{table} The UVES sample studied here consists of 23 targets, including a known DQZ, and one new discovery. The nominal fraction of externally polluted DQ stars is thus 1:22 or $4.5${\raisebox{0.5ex}{\tiny$^{+9.1}_{-1.4}$} per cent. It can be argued, however, that the previously known DQZ is actually a part of the UVES sample, because it is possible it could have been a selected target without prior knowledge. In that case, the pollution frequency for DQ stars could be as high as 9.1\, per cent. However, on the basis of the much larger, unbiased SDSS samples of DQ white dwarfs discussed above, it is far more likely the true fraction is substantially smaller, and closer to 2 per cent. As can be seen from the Table, there is a general coherence in the picture of polluted white dwarfs among the spectral classes DA(Z), DB(Z), and DC/Z, where this is now a well-understood story of dynamically active planetary systems in the post-main sequence \citep{veras2016}. Most of the aforementioned pollution frequencies and accretion rate inferences belong to studies more than a decade old, and in the intervening years those interpretations have been strengthened. After cooling for 20\,Myr or so (during which the sensitivity to external pollution is poor, owing to hotter atmospheres; \citealt{koester2014,barstow2014}), a given white dwarf may exhibit metals at the observed frequency as a fraction of its remaining cooling lifetime (i.e.\ a duty cycle), or a subset of the entire population may stay constantly polluted, while others are never enriched by remnant planetary systems. With a few notable exceptions, polluted white dwarfs are a population dominated by isolated stars \citep{wilson2019}, together with a small fraction of wide binaries where each star has evolved in effective isolation \citep{zuckerman2014}. Although stars with helium-rich atmospheres often have time-averaged accretion rates that exceed those inferred from DAZ stars in accretion-diffusion equilibrium \citep{girven2012,farihi2012b}, it is still the case that typical rates are comparable with those of their hydrogen-rich counterparts. This story breaks down completely for DQ stars; not only is pollution rare, but when it is present it is stunted by several orders of magnitude. The frequency of external pollution in DQ stars is drastically lower than for known polluted white dwarfs in Table~\ref{rates}. On top of this stark difference in the frequency of detected metals, the discrepancy between the mass accretion rates is also severe; in fact, there is no overlap between these populations. {Therefore, it is highly improbable that DQ stars are directly related to, or descended from DA or DB white dwarfs that frequently retain active planetary systems.} \section{Additional anomalies} In the previous sections, it was shown that DQ stars do not share any characteristics in common with either hydrogen or helium atmosphere stars that commonly evidence their remnant planetary systems via external pollution. However, the striking lack of metal pollution in DQ stars is just one empirical perspective that suggests a distinct origin for these white dwarfs. In this section, several other of their likely or potentially disparate characteristics are explored in detail. \subsection{Lack of core dredge-up in DZ white dwarfs} While the statistics have been clear for decades, the present study reiterates the fact that DZ and DQ stars are essentially two mutually exclusive populations, and determines the size of the intervening gulf in terms of atmospheric metals. But, importantly, there are two independent aspects that underly this lack of common spectral features; the fact that DQ stars are rarely DQZ (Section~3), and separately that DZ stars essentially never manifest as DZQ (\citealt{bergeron1997}; see below). Assuming both are the result of single star evolution, then why don't any helium-rich white dwarfs with planetary system pollution, also dredge up their interior carbon? In a scenario of isolated stellar evolution, the unavoidable interpretation is that these two spectral classes have distinct interior structures. That is, there is no plausible mechanism for a planetary system orbiting a DZ white dwarf to prevent the host star from dredging up carbon from its core. In fact, it is already well-established that core material dredge-up is most efficient when the outer envelope of helium is somewhat reduced compared to the predictions of standard stellar evolution. The earliest calculations of carbon dredge-up indicated that efficiency peaked with helium mass fractions $-4.0 \lesssim \log q({\rm He}) \lesssim -3.5$ \citep{pelletier1986}, while a decade later, carbon enhancements were produced with $-3 \lesssim \log q({\rm He}) \lesssim -2$ \citep{macdonald1998}. More recently, a similar range of helium layer masses has been corroborated, and with predictions that mostly bracket the classical, observed [C/He] sequence as a function of effective temperature \citep{dufour2005}, where state-of-the-art models indicate that $-3.5 \lesssim \log q({\rm He}) \lesssim -2.5$ best reproduces this abundance pattern \citep{koester2020}. However, there are two known [C/He] sequences at the cool end \citep{dufour2005,koester2006b}, one of which requires even smaller helium layers $\log q({\rm He}) \lesssim -3$ \citep{coutu2019}, and the warmer DQ stars require drastically lower $\log q({\rm He}) \lesssim -5$ \citep{koester2020}. It is well known that some DQ white dwarfs exhibit carbon features in the optical, while others do so only in the ultraviolet, and that this can be ascribed to differences in helium mass fractions rather than effective temperature \citep{weidemann1995}. In fact, as noted in Table~\ref{targs}, several of the DQ targets were found to have an optical C$_2$ feature for the first time in their UVES spectra (0806--661, 1708--147, 1837--619), while others only exhibit carbon features in the ultraviolet (1917--077, 2147+280, 2317--173) despite the new, highly-sensitive observations. Therefore, it is possible or even likely that there is a continuous distribution of helium layer masses \citep{koester2020}, where thinner layers will lead to optical carbon features, modestly thicker layers result in ultraviolet carbon features only, and if sufficiently thick, no carbon will be effectively dredged. It should be mentioned that there is one DZ star with ultraviolet carbon features (2216--657; \citealt{weidemann1989,wolff2002}), but the abundance has been shown to be consistent with carbon-rich planetary material, externally accreted, and thus not from core dredge-up \citep{swan2019b}. And while it has been shown that the most extreme cases of metal pollution can mask carbon features in both the optical and ultraviolet, in general this is not the case \citep{hollands2022}. Therefore, the number of known cases where carbon features can be confidently ascribed to core dredge-up in DZ stars is likely (currently) zero. That is, DZ stars have helium mass fractions that are significantly and consistently larger than the range found for DQ white dwarfs. There are two related points to be made from these successful and independent modeling efforts of [C/He] in DQ stars. The helium atmosphere, DZ and DC stars appear to have larger $q({\rm He})$ so that core dredge-up is inefficient, and this stellar evolutionary outcome represents the majority of all cool, helium-rich white dwarfs (40:54 = 0.74 in the 20\,pc sample; \citealt{hollands2018}). The modest to drastic reduction in $q({\rm He})$, necessary so that core carbon is effectively dredged in DQ stars, is sufficiently common that an explanation is needed, where possibilities are discussed in Section~5. \subsection{Lack of circumstellar material orbiting DQ stars} To date, all white dwarfs suspected or confirmed to have orbiting circumstellar disks associated with planetary debris are also observed to have atmospheres polluted by heavy elements \citep{farihi2016a}. There are currently eight systems exhibiting irregular transits from their debris disks, all of which are polluted with the exception of a few, where available data are insufficiently sensitive \citep{vanderbosch2020,guidry2021}. Confirmation of photospheric metals may require a high spectral dispersion sufficient to resolve or rule out interstellar absorption components \citep{koester2005a,zuckerman2013}, or ultraviolet spectroscopy for modest to low metal abundances in hotter stars \citep{farihi2013b,xu2015}. While {\em Spitzer} observations were ground-breaking for debris disk detections towards white dwarfs, over its 16 year lifetime, targets were strongly biased towards stars already known to be polluted. Owing to this bias, DQ stars were not searched for infrared excess with comparable sensitivity, with one possible exception \citep{farihi2010a}. However, unbiased studies using the {\em WISE} satellite have been able to probe much larger, unbiased samples \citep{debes2011,hoard2013,dennihy2017}, now including nearly every sufficiently bright white dwarf identified by {\em Gaia} and utterly blind to spectral type \citep{xu2020,lai2021}. Optical spectroscopic follow-up of debris disk candidates is still ongoing, but so far there is not a single DQ star candidate (E.~Dennihy, private communication) among $N\gtrsim100$ white dwarfs where orbiting debris is detected either by infrared excess, optical emission or absorption lines, or transits. The dearth of detected circumstellar material orbiting DQ white dwarfs may appear -- superficially -- redundant with the lack of atmospheric pollution, as there is a strong connection established between these two hallmarks of evolved planetary systems. As a rule for white dwarfs, photospheric heavy elements require ongoing or recent accretion; that is, pollution implies closely-orbiting circumstellar material, where it is understood that the bulk of debris disks are simply undetectable owing to sensitivity limits \citep{rocchetto2015,bonsor2017}. But the converse is not necessarily true, and planetary matter can exist in a wide range of orbits and sizes that would not necessarily lead to the chemical enrichment of the host star. Little is known about the architecture of planetary systems orbiting polluted white dwarfs outside of a few tens of R$_{\odot}$ \citep{xu2013,farihi2014}, but there are at least two examples of the aforementioned scenario: there is a giant planet candidate, transiting the non-polluted white dwarf WD\,1856+534 (spectral type DC; \citealt{vanderburg2020}), and a 2500\,AU, planetary-mass companion orbiting WD\,0806-661 (spectral type DQ, Table~\ref{targs}; \citealt{luhman2012}). As mentioned above, existing observations may be insufficiently sensitive to photospheric metals, yet able to detect circumstellar material. For example, PG\,0010+280 is a $T_{\rm eff}\approx27\,000$\,K DA white dwarf with an apparent infrared excess from a debris disk, but lacks photospheric metal lines at optical wavelengths, presumably due to its relatively high effective temperature and opacity \citep{xu2015,walters2022}. However, current models do not predict significant opacity in DQ stars, which are understood to have helium-dominated atmospheres, possibly with trace abundances of hydrogen that modestly impact models at the lowest effective temperatures \citep{koester2020}. Atmospheric modeling of DQ white dwarfs already includes opacities for carbon molecules and atoms \citep{dufour2005,coutu2019}, and thus there is currently no other suspected opacity source that might otherwise mask metal lines if such heavy elements were indeed present in their atmospheres. Another means of masking the detectable signature of external pollution, where the detection of circumstellar matter might be more straightforward, would be to dilute the accreted material into a much larger stellar reservoir. While speculative, if the mass of the outer, fully mixed layers of DQ white dwarfs were orders of magnitude larger than expected from current theory, then the strength of any metal absorption features would be correspondingly weaker than predicted, and the derived abundances commensurately smaller. All else being equal, the DZ and DQ accretion rates (e.g.\ Figure \ref{rates}, treating upper limits as detections for heuristic purposes) could be made to agree, if the mass of the convection zones in DQ white dwarfs are roughly $3000\times$ more massive. While this possibility is almost certainly pathological, in this case DQ stars could accrete from circumstellar disks that might be detected, yet the resulting pollution would be drastically diminished. However, no circumstellar matter has been detected towards DQ white dwarfs, and this partly mitigates the possibly of masking any pollution as typically observed in a DZ star. Statistics on infrared excesses indicate that robust detections are rare below 9000\,K or cooling ages significantly longer than 1\,Gyr \citep{bergfors2014,farihi2016a}, and thus the lack of infrared excess is not the strongest constraint on the lack of orbiting material, but it is suggestive. \subsection{Observed lack of close DQ+dM binaries} Low-mass main-sequence stars dominate the initial mass function, and the mass functions for companions to white dwarfs and to their A-type star progenitors \citep{farihi2005b,derosa2014}. Thus, from a formation and evolutionary point of view, it is expected that white dwarfs with M dwarf companions are abundant, which is observed to be the case \citep{rebassa2007,hollands2018}. Of these prevalent stellar pairs, based on both theory and observations, it is expected that roughly two-thirds will be in wide binaries where the original semimajor axis increased in response to post-main sequence mass loss, and around one-third will be in short-period orbits that resulted from common envelope evolution \citep{farihi2010c,nebot2011,ashley2019}. Another property of DQ stars, that implies deviation from typical pathways of isolated stellar evolution, is the fact that there are no known examples in post-common envelope binaries with an unevolved companion (e.g.\ an M dwarf). To characterize this deficit in more detail, the largest available spectroscopic catalog of white dwarf -- main sequence binaries was searched for any DQ white dwarfs. This catalog is a compendium of all sources with composite (white dwarf + main-sequence star) spectra within the SDSS \citep{rebassa2012,rebassa2016}, and contains over 3200 entries, 91 per cent of which are spatially-unresolved, plus a small fraction of partly-resolved pairs that contribute their combined light into a single spectroscopic fiber. The catalog was searched for all systems containing an M dwarf companion, which represents 98 per cent of all entries, and the primary spectral type was noted in each case. Only confident spectral type identifications were used for both the white dwarf and the companion, and only those systems that are spatially unresolved. Of the unresolved composites, 2263 have a white dwarf of spectral type DA (strongest lines from H), 93 contain a DB star (strongest lines from He\,{\sc i}), and 70 host a DC white dwarf (no spectral lines). Not a single DQ white dwarf is present in the catalog, including entries with uncertain spectral type identifications. Because DC and DQ white dwarfs share similar stellar properties -- with the likely exception of the atmospheric helium mass fraction -- one would expect a comparable number of each in the SDSS catalog, adjusted for their relative occurrence rates as field stars. Within the 20\,pc sample, there are 29 DC and 15 DQ white dwarfs \citep{hollands2018}, and if their frequency in unresolved (including short-period) binaries were similar, one would expect close to 35 DQ spectral types within the composite binary catalog. Therefore, the lack of DQ stars in this catalog is striking. While identifying a weak C$_2$ 5165\,\AA \ feature may be difficult in low S/N spectra, and in the presence of flux from an M dwarf companion, many DQ white dwarfs exhibit deep and strong Swan bands that are unmistakable \citep{dufour2005}. It is important to keep in mind that the stellar wind from a low-mass, main-sequence star can donate hydrogen to a helium-atmosphere white dwarf via wind capture, if the orbit is sufficiently close, as can be the case in post-common envelope binaries. In this way, a helium atmosphere white dwarf with a sufficiently thin convection zone (or fully mixed layer) can be transformed into one where hydrogen is more abundant. This scenario is likely responsible for the dominance of DA spectral types in close, white dwarf -- main sequence pairs, including the extensive SDSS catalog, and is consistent with the lack of any known, post-common envelope systems containing a helium atmosphere white dwarf with a thin convection zone (e.g.\ DB white dwarfs; \citealt{vandenesselaar2005,rebassa2010}). To demonstrate the plausibility of this atmospheric transformation, consider a $T_{\rm eff}\approx20\,000$\,K, pure helium atmosphere white dwarf, which will manifest as a DB spectral type, and where the fully mixed outer layer has mass $q\lesssim10^{-10}$\,M$_{\odot}$\citep{koester2009a,koester2020}. If this white dwarf accretes solar composition wind from a companion at a modest rate of $\dot M = 10^{-14}-10^{-16}$\,M$_{\odot}$\,yr$^{-1}$ \citep{pyrzas2012,parsons2012,ribiero2013}, hydrogen will come to dominate by number within a fraction of a Myr. With sufficient sensitivity, a hydrogen(-enriched) atmosphere white dwarf accreting stellar wind should appear as a DAZ spectral type, and there are several well-documented cases among post-common envelope binaries \citep{zuckerman2003,debes2006,tappert2011}. However, for the aforementioned, pure helium atmosphere case, once the convection zone deepens due to further cooling, any superficial hydrogen layer may be easily overwhelmed. By the time such a white dwarf has cooled to $T_{\rm eff}\approx10\,000$\,K, the size of the fully mixed layer should be $q\sim10^{-5}$\,M$_{\odot}$, where this applies both to DQ white dwarfs and their helium atmosphere DC star counterparts \citep{koester2009a,koester2020}. With convection zones of this depth, it would take several Gyr or longer for the atmosphere to become dominated by hydrogen that is captured from a companion stellar wind, with a strong dependence on the binary separation (i.e.\ wind capture rate). Therefore, it is expected that some DC, DZ, and DQ white dwarfs should be hosts to post-common envelope, main-sequence companions. Indeed, there are a handful of such systems known to host DC stars \citep{nebot2011,parsons2013}, and at least one candidate DZ accreting metals in a stellar wind from an M dwarf \citep{fajardo2016}. But there are none with a DQ white dwarf. In contrast, there are many examples of DQ stars in wider binaries. Within the 20\,pc sample alone, there are several: Procyon\,B (F5IV+DQZ), GJ\,86B (K0V+DQ), LHS\,290 (DQp+dM), 2154-512 (DQ+dM), 1633+572 (DQ+dM), and 1917-077 (DBQ+dM). Thus, the lack of DQ white dwarfs in close or post-common envelope systems is likely a clue to their origin, and consistent with the destruction of close stellar companions during prior evolutionary phases. \subsection{Apparently low DQ white dwarf masses} The left-hand panel of Figure~\ref{mhist} plots the mass distribution for the sample of 221 classical DQ white dwarfs discussed in Section~3.2 \citep{koester2019}. The masses of DQ stars have been studied recently by two independent teams, based on sources with reliable parallax data from {\em Gaia}, finding that their average mass is roughly 0.05\,M$_{\odot}$ lower than those found for DA and DB stars \citep{coutu2019,bedard2022}, and also lower than DZ and DC white dwarfs \citep{koester2019}. There have been suggestions that there may be model imperfections that prevent better agreement between DQ stars and other white dwarfs, but to date, no other studies have considered enhanced mass loss that may result from binary evolution. It should be noted that two iconic DQ white dwarfs in the 20\,pc sample have masses derived independently of models, and these values are also lower than the typical $\langle M \rangle=0.62$\,M$_{\odot}$ found for spectral type DA and DB white dwarfs \citep{genest2019}. In fact, the dynamical mass of Procyon\,B (0.59\,M$_{\odot}$) is in modest tension with the overall age estimated for this well-studied binary system \citep{liebert2013,bond2015}. Adopting a comprehensive, recent, and empirical initial-to-final mass relation, the predicted mass of Procyon\,B should be closer to 0.66\,M$_{\odot}$ \citep{cummings2018}. Similarly, for the planet-hosting binary GJ\,86, the DQ white dwarf has a mass of 0.54\,M$_{\odot}$ \citep{zeng2022}. And while it is likely this system is relatively old (either thick disk or somewhat younger; \citealt{fuhrmann2014}), any recent initial-to-final mass relation will yield a progenitor that should not have evolved off the main sequence in a Hubble time. Therefore, it is possible that the masses of DQ stars are actually low compared to other white dwarfs, and not a result of model shortcomings, but as a consequence of their evolutionary history. Among the 221 DQ white dwarfs plotted in Figure~\ref{mhist}, there are 23 objects with derived masses $M<0.45$\,M$_{\odot}$, where 16 of these have $\upvarpi/\upsigma_\upvarpi > 20$, and thus excellent astrometric data. It is often the case that (apparently) over-luminous white dwarfs are two unresolved stars with comparable brightness, and therefore {\em Gaia} DR3 was searched in a 5\,arcsec radius around the positions of these 23 candidate low-mass DQ stars; none were found to be resolved into two sources. However, two were found to have renormalized unit weight error (RUWE) suggestive of a possible or likely non-single star: SDSS\,J094058.34+384350.2 (RUWE $=1.37$) and SDSS\,J125245.44+194311.2 (RUWE $=2.33$). Thus, the inferred low masses of these 23 DQ white dwarfs may be partly accounted for as binary light sources, but the bulk currently lack indications of duplicity. Similarly, the argument is often made that a DQA-type spectrum must be two unresolved white dwarfs, one a DQ and one a DA star, but currently there is only verification of one such binary (GD\,73; \citealt{vennes2012}), where it is modestly overluminous relative to single star expectations \citep{gentile2019}, and has an RUWE consistent with a single astrometric source. Of the 23 DQ white dwarfs with implied low masses, only one source shows unambiguous (yet weak) absorption from hydrogen in its spectrum (SDSS\,J150126.78+210056.9), but has an RUWE in the range considered normal for single stars, and no other {\em Gaia} sources nearby. It can be argued that these low-mass DQ candidates are unresolved DQ+DC white dwarfs, but again one might expect a preponderance of high {\em Gaia} RUWE values, which is not yet observed. Given that all DQ stars must have atmospheric carbon by definition, and thus carbon cores, it may seem oxymoronic to speculate on the possibility of low-mass members of this class. Owing to the finite age of the Galaxy, white dwarfs with $M\lesssim0.45$\,M$_{\odot}$ cannot yet have formed by single star evolution. Indeed the majority of low-mass white dwarfs are found to be single- or double-lined binaries \citep{marsh1995,brown2011,rebassa2011}, whose evolution is thought to be terminated prior to helium ignition, and thus such stars would have helium cores as opposed to carbon cores. However, it is possible to form a low-mass carbon core in stars with $M_{\rm MS}\gtrsim2.3$\,M$_{\odot}$, where helium ignites non-degenerately \citep{bauer2021}. Binary interactions would then be required to truncate stellar evolution and further core growth, in order to later be observed as a low-mass white dwarf with a carbon core. While speculative, this is a distinct possibility for some DQ white dwarfs, as their binary properties are only weakly constrained at present, and their molecular bands are relatively insensitive to radial velocity changes induced by companions, especially compared to white dwarfs with atomic spectral lines. The confirmation of any DQ white dwarf with $M<0.45$\,M$_{\odot}$ would be a strong indication of a binary origin. The expectations for single star evolution are illustrated in the right-hand panel of Figure~\ref{mhist}, where main-sequence progenitor masses have been calculated using the previously mentioned initial-to-final mass relation. Given the low masses of the DQ white dwarfs, it is thus not surprising that the predicted main-sequence predecessors are also of relatively low mass, but this important detail has so far escaped attention in previous studies \citep{coutu2019,koester2019,bedard2022}. The age of thin disk stars should be less than around 8\,Gyr \citep{bensby2014,kilic2017,sharma2019}, but Figure~\ref{mhist} indicates that most single star parents of DQ white dwarfs would have main-sequence lifetimes longer than this. For the DQ stars plotted in the Figure, $\langle T_{\rm eff} \rangle = 7800$\,K and $\langle M \rangle = 0.55$\,M$_{\odot}$, where white dwarf cooling would add an additional 1.2\,Gyr to their inferred total ages. This picture would imply that DQ stars are dominated by thick disk and halo members, which, as shown below, is inconsistent with their kinematical and companion metallicity properties. \subsection{Contrasts with DZ and DC stars} It is useful to compare evolutionary indicators of DQ white dwarfs with their helium atmosphere counterparts, the DZ and DC stars. While the mass distribution of DQ stars has been discussed above, and in particular the potential of a low-mass tail, this could simply be the result of unresolved binaries. If that is indeed the case, and DQ stars are the product of isolated stellar evolution, then it is expected that other white dwarf classes would also exhibit the same basic properties in their mass distributions. In Figure~\ref{chist}, the cumulative mass distributions are shown for the 221 DQ white dwarfs discussed above, together with SDSS spectroscopic DZ and DC samples from the same study \citep{koester2019}. Only those DZ and DC stars with $T_{\rm eff} < 10\,000$\,K, and which thus overlap with the DQ white dwarfs, have been used in order to compare these three spectral classes of helium atmosphere stars using identical criteria. The most obvious features of the cumulative mass histogram is that DQ white dwarfs have a larger fraction of lower masses, the DC stars have a notable fraction of higher masses, while the planetary system hosting DZ stars are somewhat in the middle of these two. Another feature is the concentration of DQ white dwarf masses in the range $0.5-0.6$\,M$_{\odot}$, shown by a steepness in the distribution that is not observed for DZ or DC stars. One possible interpretation of this figure panel is as follows. Planetary systems form and evolve commonly around the progenitors of DZ stars, which leave remnants with a given mass distribution as observed. The DQ white dwarfs originate in a distinct population, with lower remnant masses (reasons explored below), and rarely formed planetary systems that lead to white dwarf pollution. While speculative, the DC white dwarfs may descend from stars that are, on average, somewhat more massive than DZ star progenitors, and which may indicate that planet formation is inhibited to some degree in higher-mass main-sequence stars. The tangential velocity distributions tell a similar story but with less distinction between the DZ and DC white dwarfs. The DZ stars appear to include few stars with relatively high velocities, consistent with planetary hosts coming from a primarily young disk population, where star formation likely benefitted from a comparatively metal-rich environment. The DQ stars are modest outliers compared to their helium atmosphere, DZ and DC white dwarf counterparts, and seem to be missing stars at the lowest velocities, below around 20\,km\,s$^{-1}$. This may imply the DQ white dwarfs are somewhat older than the DZ and DC stars. Another possibility is that DQ white dwarfs have received significant velocity kicks from asymmetric mass loss, and which was not experienced by the DZ and DC stars. Only modest white dwarf recoil, no larger than 1\,km\,s$^{-1}$, has been inferred using a large and pristine catalog of wide {\em Gaia} binaries, via comparison of those pairs with two main-sequence stars, versus those containing at least one white dwarf \citep{elbadry2018}. However, based on the large separation between binary components in that study, this modest recoil may apply only to stars that evolved in effective isolation. In contrast, close binary evolution may induce larger kicks that may account for the lack of low space velocities in the DQ stars. The space motions of DQ white dwarfs are far from that expected for Population II stars, and neither are there any indications of a Galactic thick disk or halo origin. Under the assumption of zero radial velocity, the spectral classes plotted in Figure~\ref{chist} have calculated mean space motions in the direction of Galactic rotation ($V$) given in Table~\ref{galvs}. The three samples are without any kinematical biases, and were selected based on spectral classification via SDSS observations \citep{koester2019}. As can be seen, the DQ stars may exhibit a modest fraction of older -- or kicked -- thin disk stars within their population, but for the most part, the DQ, DZ, and DC white dwarfs are members of the relatively young Galactic disk \citep{bensby2014}. Thus, the low masses of the DQ white dwarfs, and the older ages implied by single star evolution, are not supported by their space velocities. \subsection{Magnetism and rotation among DQ stars} Magnetism has long been associated with DQ white dwarfs, including detections of strong fields \citep{schmidt1995,schmidt1999,vornanen2010}, as well as spectral indications of magnetism via modeling \citep{hall2008,blouin2019a}. In a pioneering effort to characterize magnetism among all white dwarfs within 20\,pc \citep{bagnulo2021}, it has been recently found that DQ stars are likely outliers. First, the nominal frequency of magnetism in DQ white dwarfs is higher than in other spectral classes. Second, the actual frequency is likely to be higher, because the sensitivity is restricted to magnetic fields on the order of MG or larger, while spectropolarimetry can readily detect fields of 100\,kG or somewhat smaller in white dwarfs with atomic lines (i.e.\ DA, DB, or DZ stars). Third, it is noteworthy that, in fact, the DQ stars have the highest magnetic fields of white dwarfs within 20\,pc \citep{bagnulo2021}. Rapid rotation has been associated with some magnetic white dwarfs, especially in the scenario where close stellar binaries or mergers may generate the observed fields \citep{liebert2005,tout2008}. Thus, relatively rapid rotation might be expected in some DQ white dwarfs, and appears to be well documented for the warmer examples of this spectral class \citep{dufour2008b,lawrie2013,williams2016}. Based on these modest correlations, a more thorough investigation of the entire DQ spectroscopic class would likely identify additional magnetic examples, as well as stars rotating more rapidly than the tens to hundreds of hour spin periods typical for isolated white dwarfs \citep{hermes2017}. \begin{table} \begin{center} \caption{Space velocities in the direction of Galactic rotation.\label{galvs}} \begin{tabular}{@{}ccc@{}} \hline Spectral &$\langle V \rangle$ &$\upsigma_V$\\ Class &(km\,s$^{-1}$) &(km\,s$^{-1}$)\\ \hline DQ &$-13$ &30\\ DZ &$-5$ &22\\ DC &$-7$ &25\\ \hline Thin disk &$-15$ &20\\ Thick disk &$-46$ &38\\ \hline \end{tabular} \end{center} {\em Notes}: Values for the thin and thick disk are taken from \citet{bensby2014}, and $v_{\rm rad}=0$ is assumed for all white dwarf calculations. \end{table} \section{Possible origins of DQ stars} The previous section forms a list of independent yet circumstantial evidence that may help to constrain the origin of DQ white dwarfs. And with the exception of the dramatic lack of metal pollution, the empirical data are suggestive but far from conclusive. In this section, the various threads are combined and addressed with concrete hypotheses, which can be tested against the existing evidence, now and in the near future, and will hopefully set the stage to reveal the origin of all DQ white dwarfs; classical examples, warmer stars with atomic lines, spectral oddballs, as well as peculiar and magnetic examples. \subsection{Summary of observations} Below is a concise list of observational constraints, which can directly be compared to any hypothetical evolutionary possibilities. Any successful hypothesis for the origin of DQ stars should be consistent with these data and indications, where Table~\ref{hypoths} tracks the various scenarios and their potential success in addressing each. \begin{description} \item{(1) DQ stars and DZ planetary hosts are essentially mutually exclusive populations, with hundreds of known examples of each. }\smallskip \item{(2) There are no known DQ stars with infrared, optical, or transit indications of circumstellar, planetary material.}\smallskip \item{(3) DQ stars are not found in post-common envelope binaries with unevolved companions.}\smallskip \item{(4) The DQ spectral class is characterized by low $q({\rm He})$, where the bulk of members may have relatively low masses.}\smallskip \item{(5) DQ white dwarfs are commonly magnetic, and likely more frequently or with stronger fields than other spectral classes.}\smallskip \item{(6) The kinematics of DQ white dwarfs reveal they are only slightly older, or kicked, relative to DZ and DC stars.} \end{description} \subsection{Metals are masked by unknown stellar opacity} This hypothesis states that the heavy element pollution in DQ white dwarfs is present, and similar to that observed e.g.\ in DZ stars, but is masked by an unknown stellar opacity source. This possibility can in principle account for the fact that DQ stars rarely exhibit {\em detectable} metal lines, but does not address the bulk of relevant issues discussed in the previous section. The derived masses of metals in the fully mixed outer layers of the star, as well as the inferred accretion rates, are tied to line strengths. Then, if metal lines were reduced in strength within DQ stars and not in DZ stars, owing to a previously unaccounted for opacity source, there would be an expected trend with effective temperature or carbon abundance (or both). Neither the deep UVES observations, nor the hundreds of stars observed with the SDSS, show any trends in detected Ca\,{\sc ii} line strengths with the DQ stellar parameters. This hypothesis fails to address most of the other outstanding characteristics of the DQ stars, and would not, for example, prevent the detection of circumstellar material in the infrared, nor account for their distinctive stellar properties. \subsection{Metals are diluted in deeper convective layers} This idea is similar to that discussed immediately above, but instead dilutes atmospheric metals within significantly deeper convection zones. A scenario where the relative sizes of the fully mixed outer layers of DQ and DZ stars are orders of magnitude different can, in principle, account for the offset seen in the right hand panel of Figure~\ref{mdots}. This could be achieved by increasing the size of convection zones in DQ stars, or decreasing those in DZ stars. However, any order of magnitude changes in the depth of these layers will also directly effect the heavy element diffusion timescale, as these two are strongly correlated \citep{koester2009a}. For example, if DQ convection zones were $1000\times$ larger than currently predicted, and crudely extrapolating linearly from current models, one might expect diluted metals in DQ stars to be seen for timescales of Gyr, and thus more frequently than in DZ stars, all else being equal. Unfortunately, this hypothesis faces all the same problems as the previous, and would have to be reconciled with the current underlying framework of diffusion in helium atmosphere white dwarfs. \subsection{Progenitors of DQ stars have high-intermediate masses} A plausible theory for the lack of planetary material in and around DQ white dwarfs is inhibited planet formation, and one possibility is via higher-mass main-sequence stars. If DQ white dwarfs are the progeny of stars with masses greater than $3-4$\,M$_{\odot}$, then it could be argued that this is consistent with their lack of circumstellar matter, and pollution from planetary debris. Around such luminous stars, protoplanetary disk lifetimes may be insufficient to form planetary precursors and full-fledged planets \citep{kennedy2009,kunimoto2021}, and there are currently few confirmed cases of planets detected towards stars with masses above 3\,M$_{\odot}$ (e.g.\ \citealt{reffert2015}). However, the expectations of this hypothesis would be that the remnant masses of DQ white dwarfs would be higher than the field, not lower as observed and modeled. And while such progeny may be prone to detectable magnetism \citep{caiazzo2020}, they should be a relatively young kinematical population, having spent little time on the main sequence. These predictions fail to match the properties of DQ stars as shown in Figure~\ref{chist}. Furthermore, higher-mass parent stars would not address the lack of post-common envelope binaries, unless it were shown that this leads more commonly to companion destruction during the post-main sequence. \begin{table} \begin{center} \caption{Consistency of DQ hypotheses vs. observational constraints.\label{hypoths}} \begin{tabular}{@{}cccccc@{}} \hline Constraint &Metals &Metals &High &Metal &Binary\\ &Masked &Diluted &Mass &Poor &Evol\\ \hline Pollution &+ &+ &+ &+ &+\\ Circumstellar &- &- &+ &+ &+\\ Companions &- &- &- &- &+\\ $q({\rm He})$ / Mass &- &- &- &+ &+\\ Magnetism &- &- &+ &- &+\\ Kinematics &+ &+ &- &- &+\\ \hline \end{tabular} \end{center} \end{table} \subsection{Progenitors of DQ stars are (very) metal poor} Similar to the hypothesis above, a lack of planetary matter orbiting and polluting DQ stars could be attributed to an origin as Population II stars that are truly metal poor. There are numerous planetary occurrence rate studies that focus on stellar metallicity for hosts of transiting or Doppler planets, but in general these are restricted to metallicities higher than 0.1\,$Z_{\odot}$. The most ancient planetary systems confirmed to date orbit thick disk stars that are only modestly metal poor \citep{mortier2012,campante2015}, and similarly debris disk masses are thought to be strongly linked to metallicities comparable to solar \citep{gaspar2016}. Truly metal poor stars have not yet revealed planetary systems, and theoretical models suggest there is a critical metallicity, below which planet formation becomes challenging, with values in the range $0.01 < Z/Z_{\odot} < 0.1$ \citep{johnson2012,hasegawa2014}. If DQ white dwarfs descend from such metal-poor stars, that could account for the lack of planetary material as pollution and circumstellar debris, and might also be consistent with their low masses, as the remains of stars that spent the bulk of their lifetime on the main sequence. However, as shown above, their kinematics utterly fail to support this interpretation. Furthermore, the known widely-bound companions to DQ stars are not significantly metal poor, and one would expect some fraction to host e.g.\ M subdwarf companions \citep{monteiro2006,zhang2013}. \subsection{Progenitors of DQ stars are binaries} A binary origin for DQ white dwarfs has several advantages over other hypotheses. This scenario can immediately be invoked to account for the apparently low masses, and more acutely, their relatively thin helium layer mass fractions, which are arguably their defining characteristics according to current models \citep{coutu2019,koester2019,bedard2022}. Furthermore, it is already generally accepted that binary evolution is responsible for the warmer examples of DQ stars \citep{dunlap2015,cheng2020}, and while precise evolutionary modeling has yet to reproduce this inference based on observations, at face value there is a clear pathway to reduce $q({\rm He})$ in DQ star progenitors via binary interactions. During close binary evolution, a post-main sequence precursor star will often lose mass via unstable Roche-lobe overflow, initiating a common envelope that shrinks the semimajor axis, and which is known to lead to lower remnant masses \citep{marsh1995,rebassa2011}. This scenario relieves the tension between the relatively young space motions of the DQ stars, and the long single-star lifetimes implied by their low masses. A high incidence of (strong) magnetism might also be expected for a binary population, as is observed for DQ white dwarfs. Of all the theories discussed above, only a binary population might account for the apparent lack of unevolved, post-common envelope companions, and it requires that such companions are either consumed, or remain hidden. The major challenge for this hypothesis is to show that binary evolution does not favor the formation or retention of planetary architectures that commonly pollute the surfaces of white dwarfs, and theoretical work is needed to investigate this aspect in particular. It can be speculated that asymmetric mass loss, especially that which might occur during the cannibalization of low-mass stellar or substellar companions, might assist in the removal or destabilization of planetary material within the system. \section{A DQ binary population outlook} This study has shown that DQ white dwarfs originate in a stellar and circumstellar population that is distinct from the DAZ, DBZ, and DZ hosts of remnant planetary systems, and highlighted several other, potentially disparate characteristics of these carbon-enriched stars that point to a binary origin. It is noteworthy that, for the most part, this work has focussed on the classical and cool DQ stars that are defined spectroscopically by the presence of Swan bands. However, there is a miniature zoo of DQ spectral subtypes, and in particular a binary origin has already been suggested for those DQ stars that have warmer temperatures and even thinner, or absent, helium layers \citep{dunlap2015,cheng2020}, as well as unusual spectral types such as DAQ \citep{hollands2020}, all of which appear to have higher remnant masses, and thus superficially amenable to merger hypotheses. These deductions have yet to be supported by dedicated evolutionary modeling efforts, but if correct, then one clear outcome is that binary interactions can result in reduced $q({\rm He})$ in white dwarf remnants. Therefore, if binary evolution can reduce the helium mass fraction in white dwarfs, the outlook is immediately positive for the entire class of DQ stars. The only major difference for the cool and classical DQ stars with Swan bands, those that form the bulk of the known population, is their lower or more typical remnant masses. For this reason, the merger of two white dwarfs is unlikely to be the dominant, binary formation channel for DQ stars. Moreover, the classical DQ white dwarfs do not have kinematics of a blue straggler population, with an added delay in cooling for the time it takes to merge. Binary evolution does not necessarily imply older ages, as two stars can begin to interact as soon as one has evolved sufficiently for mass transfer to begin, and thus a binary DQ population can have ages similar to their helium atmosphere, DZ and DC white dwarf counterparts, as observed. The common envelope is thought to be a rapid process, and should not add significantly to the inferred total ages of DQ stars in the case that unevolved companions are immediately consumed; the age bottleneck is still the evolutionary timescale of the white dwarf progenitor. Planetary companions are also a possibility in a scenario of fatal engulfment and binary evolution for DQ white dwarfs. Two basic outcomes are required, one is that the white dwarf experiences enhanced mass loss compared to single star evolution (possibly in terms of total mass and the outer helium envelope), and that any substellar companions prohibit or diminish planetary material that might later precipitate atmospheric pollution. There is some evidence that giant planets in close orbits (i.e.\ hot Jupiters) are rarely found with additional planetary companions, and may have cleared away planet-forming disk material during migration, or ejected other planets dynamically \citep{latham2011,mustill2015}. While it is unclear if a giant planet (or two) is sufficient to amplify mass loss during stellar evolution, the absence of hot Jupiters around subgiant hosts argues that giant planet ingestion does indeed occur \citep{schlaufman2013}. A key question is exactly how binary evolution might prevent the eventual pollution of a white dwarf, where theoretical modeling is clearly needed. From observations, it is noteworthy that for known white dwarfs in short-period binaries with low-mass stellar and substellar companions, there is only a single case of circumbinary material and pollution (SDSS\,J155720.77+091624.6; \citealt{farihi2017a}). Despite this apparent rarity, there is a general lack of sensitivity to such circumbinary dust systems, because the cool companion can itself cause a large infrared excess, and thus additional emission from dust would be challenging to identify. Binary evolution modeling is needed to test the basic premise raised here for DQ white dwarfs. The lack of unevolved companions in close orbits is a key observable; on the one hand, if they are destroyed then the remnant might be expected to be somewhat more massive, and on the other hand if they survive, it is unclear why they are not detected. It is theoretically possible for a companion to merge and eject more material than it provides to the merged stellar core, but such outcomes are best addressed via dedicated modeling. It is important to note that both Procyon\,B and GJ\,86B may have transferred mass during their evolution, as the originally more massive stars within their binary systems. Current data and modeling do not require such mass transfer in these iconic systems, but it may account for remaining inconsistencies, especially from an evolutionary perspective \citep{bond2015,zeng2022}. Binary or white dwarf -- main-sequence merger candidates may be challenging to detect given the lack of atomic lines in many DQ stars, but relatively rapid stellar rotation, photometric or astrometric variation owing to unseen companions are likely one way forward, observationally. Lastly, what is the nature of the rare and weak pollution in DQ white dwarfs? The default hypothesis should be remnant planetary systems\footnote{Appealing to the interstellar medium for the weak DQZ pollution may be tempting, but there are myriad issues such as their utter lack of hydrogen.}, and there is no a priori reason to expect other mechanisms, even if the stars are currently or formerly binary \citep{rafikov2013,martin2013,bromley2015}. There are dynamical issues that are unique to binary systems, and circumbinary planet hosts in particular, with additional instabilities as compared to single stars, during formation and on the main sequence, where planetary ejection is one of the likely outcomes \citep{smullen2016,sutherland2016,sutherland2019}. It might be speculated that such instabilities might be catastrophically amplified during a common envelope or merger event where mass loss may be asymmetric or impulsive, thus quickly reducing or removing planets and associated planetesimal belts. For context, the solar zodiacal cloud is replenished at a rate in the range $10^7-10^8$\,g\,s$^{-1}$ \citep{nesvorny2011,rigley2022}, and the rare DQZ exhibit accretion rates $100\times$ lower, not to mention the typical DQ upper limits at least $1000\times$ lower, which approaches the accretion rate of extraterrestrial material onto the Earth \citep{love1993,yada2004}. It may be possible to detect multiple heavy elements in the rare DQZ stars, and if successful, their ratios can be compared to the planetary abundances associated with DAZ, DBZ, and DZ white dwarfs. While challenging owing to the low abundances and line strengths, it may provide an important clue to the nature of these stars, which are now, hopefully, less of an astrophysical enigma. \section*{Acknowledgements} J.~Farihi thanks D.~Koester and G.~Fontaine for key discussions over the years that helped to form the foundation of this work, E.~Dennihy for confirming the fact that no DQ star has a suspected infrared excess from circumstellar dust, A.~Rebassa-Mansergas for sharing his full white dwarf-main sequence binary catalog, and valuable exchanges with E.~B.~Bauer, R.~R.~Rafikov, and J.~J.~Eldridge. Several colleagues provided feedback on an earlier version of the manuscript, including those mentioned above, as well as T.~von Hippel, A.~J.~Mustill, A.~Swan, and B.~Zuckerman. The authors acknowledge the European Southern Observatory for the award of telescope time via programs 095.D-0706, 096.D-0076, and 097.D-0063. This work has made use of the ESA {\em Gaia} mission, processed by the {\em Gaia} DPAC. J.~Farihi acknowledges support from STFC grant ST/R000476/1. T.~G.~Wilson acknowledges support from STFC consolidated grants ST/R000824/1, ST/V000861/1, and UKSA grant ST/R003203/1. \section*{Data Availability} Data acquired at ESO facilities for the PI programs listed above are available through their archive. \bibliographystyle{mnras} \bibliography{/Users/jfarihi/papers/references} \bsp % \label{lastpage}

Title: Polarimetry and Photometry of Gamma-Ray Bursts Afterglows with RINGO3

Abstract: We present photometric and polarimetric measurements of gamma-ray burst (GRB) optical afterglows observed by the RINGO3 imaging polarimeter over its $\sim$7 year lifetime mounted on the Liverpool Telescope. During this time, RINGO3 responded to 67 GRB alerts. Of these, 28 had optical afterglows and a further ten were sufficiently bright for photometric and polarimetric analysis ($R\lessapprox{17}$). We present high quality multicolour light curves of ten sources: GRB 130606A, GRB 130610A, GRB 130612A, GRB 140430A, GRB 141220A, GRB 151215A, GRB 180325A, GRB 180618A, GRB 190114C, and GRB 191016A and polarimetry for seven of these (excluding GRB 130606A, GRB 130610A, and GRB 130612A, which were observed before the polarimetry mode was fully commissioned). Eight of these ten GRBs are classical long GRBs, one sits at the short-long duration interface with a $T_{90}$ $\sim$ 4 seconds and one is a classical short, hard burst with extended emission. We detect polarization for GRB 190114C and GRB 191016A. While detailed analyses of several of these GRBs have been published previously, here we present a uniform re-reduction and analysis of the whole sample and investigation of the population in a broad context relative to the current literature. We use survival analysis to fully include the polarization upper limits in the comparison with other GRB properties, such as temporal decay rate, isotropic energy and redshift. We find no clear correlation between polarization properties and wider sample properties and conclude that larger samples of early time polarimetry of GRB afterglows are required to fully understand GRB magnetic fields.

https://export.arxiv.org/pdf/2208.01729

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} (transients:) gamma-ray bursts -- techniques: polarimetric -- techniques: photometric -- magnetic fields \end{keywords} \section{Introduction} Gamma-ray bursts (GRBs) are extremely energetic transients occurring at cosmological distances. They have been observed at various wavelengths ranging from gamma-rays to radio and these observations help us to piece together the puzzle of GRB physics. One of the proposed scenarios is that accretion onto a compact object powers the relativistic outflow and other prediction is the magnetar spindown as the powering source for the GRB \citep{Metzger_2011}. Internal dissipation in the outflow causes the prompt gamma-rays, and external shocks (interaction of the jet with local ambient medium) produce afterglow emission at various frequencies ranging from x-ray to radio \citep{Piran_1999, Zhang_2004}. One of the most puzzling aspects of GRBs is the magnetic field properties of their jets which can shed light on the driving mechanism of the explosion \citep{Lazzati_2006, Toma_2013,Covino_2016,Kobayashi_2019}. Since GRBs are cosmological in nature, we cannot obtain the spatial resolution to understand the magnetic field strength and structure of the jets with most observational methods. Photometric observations of the forward and reverse shock afterglows can constrain the relative strength of the magnetic fields in the two shock regions, whereas polarimetry can provide information about the structure of the magnetic field in the original ejecta from the GRB central engine. Thus, polarization studies of early afterglows (few minutes after the burst) are an important technique to better understand the magnetic field properties of GRB jets. Most polarimetric studies have focused on the observation of polarization near the jet break to understand the jet opening angle \citep{Ghisellini_1999, Sari_1999, Rossi_2004}. These jet breaks occur $\sim$ 1 day after the burst, at times when we observe mostly the forward shock emission which is not highly polarized \citep{steele-ringo2-2017, kopac_2015, Jordana_2020}. Increased availability of robotic telescopes and instruments designed to observe polarization of transients have facilitated the observation of earlier time signals of these GRBs \citep{Steele_2004} (minutes to hours after the burst). The early afterglow includes the signatures of reverse shocks and the magnetic field structure of the jet itself; these reverse shock afterglows have higher levels of polarization compared to forward shocks \citep{Steele_2009,Mundell_2013,steele-ringo2-2017,Shrestha_2021}. These data can be utilized with current theoretical models to narrow down the physics of GRB jets. The Liverpool telescope (LT) which is a 2.0 meter fully autonomous robotic telescope at Observatorio del Roque de los Muchachos, La Palma \citep{Steele_2004}. LT is equipped with polarimeters designed for rapid observations. RINGO (2006 - 2009), RINGO2 (2010 - 2012), and RINGO3 (2013 - 2020) are a series of polarimeters that used a rapidly rotating polaroid analyser \citep{Jermak_2016,Arnold_2017} to observe rapidly fading sources with high accuracy. RINGO and RINGO2 observed very high polarization signals of early-time optical polarization in some GRBs \citep{Steele_2009, Mundell_2013}. All the RINGO2 GRB observations are presented in \citet{steele-ringo2-2017}. In this paper we present a unified analysis of all of the GRBs observed by final generation of the RINGO polarimeters, RINGO3. This increases the sample size of polarization data which will help us better understand the magnetic properties of GRB jets. In this paper, we present the results of the complete set of GRBs observed by RINGO3. We present photometric analysis of ten GRBs and polarimetric analysis of seven GRBs. The paper is arranged as follows; in Section \ref{sec:obs}, we present the design of RINGO3 and different observations performed during its run time. We describe the data reduction process in Section~\ref{sec:reduction} and present the polarimetric and photometric results in Section~\ref{sec:results}. We discuss the implications of these observations in Section~\ref{sec:discussions}. Finally, we provide concluding remarks in Section~\ref{sec:conclusions}. \section{Instrument and Observations}\label{sec:obs} In this paper we present observational data from three different instruments: RINGO3 (polarimeter), IO:O (imager), and RATCam (imager) on board the LT. Since it is a fully robotic telescope, it is optimal for time-domain astrophysics including GRB studies \citep{Guidorzi_2006}. For all the instruments, basic CCD reductions such as bias subtraction, dark subtraction, flat fielding and World Coordinate system fitting is done via an internal common pipeline \footnote{https://telescope.livjm.ac.uk/TelInst/Pipelines/}. \subsection{RATCam and IO:O Imaging Cameras} RATCam\footnote{https://telescope.livjm.ac.uk/TelInst/Inst/RATCam/} \citep{steele_2001} and IO:O \footnote{http://telescope.livjm.ac.uk/TelInst/Inst/IOO/} were used for photometric observations. RATCam (field of view $4.6\times4.6$ arcmin) and IO:O ($10\times10$ arcmin) are optical CCD cameras equipped with $u'g'r'i'z'$ filters. In this paper, we present results from camera using the $r'$ filter because the wavelength range of this filter is closest to the $R$ band data of RINGO3 thus we can make a better comparison to the rest of the data set. \subsection{RINGO3 polarimeter}\label{subsec:inst-ringo3} RINGO3 \citep{Arnold_2012} was the third generation of fast-readout optical imaging polarimeters on board the LT and was observing from early 2013 to January 2020. It had a field of view of $4 \times 4$ arcmin, and used a polaroid that rotated at $\sim 0.4$-Hz. The instrument was designed using three separate electron multiplying CCDs to simultaneously observe polarized images in three different wavebands. The three wavebands have wavelength ranges of $7700-10000 $ \AA, $6500-7600$ \AA, and $3500-6400$ \AA. We convert these filters roughly corresponding to the standard astronomical $I$, (with $\lambda_{\rm eff} \sim 8500 $ \AA), $R$ ($\lambda_{\rm eff} \sim 7050 $ \AA ), and $V$ ($\lambda_{\rm eff} \sim 5300 $ \AA ) bands. Each camera obtained eight exposures per rotation which were synchronised with the phase of the polaroid's rotation. RINGO3 produced 24 CCD frames (8 per camera) every 2.3 seconds which were stacked per camera into 1 minute and 10 minutes blocks for each eight rotor position image. Data from these eight exposures were utilized to deduce linear Stokes vectors; explained in detail in Section~\ref{sec:reduction}. For 10 minute stacked data, we obtain a polarization accuracy up to $2.5\%$, $1.5\%$, and $0.5 \%$ for a 17 mag source in $I$, $R$, and $V$ filters respectively. Thus, we create a 17 mag cut off for robust polarimetric analysis. \subsection{Observations}\label{subsec:inst-obs} Between 2013 and 2020, a total of 67 GRB alerts as shown in Table~\ref{tab:alert} were observed by RINGO3 and 28 of them had optical counterpart. Out of 28 GRBs with optical counterparts, two had only one data point so they were excluded from this analysis. Three observations experienced instrumental issues and had incorrect pointing. Thirteen were too faint, with an $R$ magnitude greater than 17, to attempt RINGO3 photometric and polarimetric analysis. Thus, ten alerts had optical afterglows which were bright enough to perform RINGO3 photometry and polarimetry. Figure~\ref{fig:time_coverage} shows the observational time coverage of these GRBs in the observer's and time-dilation corrected time range, along with $T_{90}$ which is the duration between $5\%$ to $95\%$ of counts is measured and the Burst Alert Telescope (BAT) peak time in the observer's reference frame. Three of these afterglows were observed before December 2013; during that time period there were problems constraining the instrumental polarization induced by the two dichroic mirrors. Thus, we can only perform photometric analysis of these GRBs. In this paper, we present polarimetric results for seven out of ten bright afterglows observed by RINGO3. The properties of GRBs analysed in this paper are presented in Table~\ref{tab:general} which contains names of the GRBs, RA, DEC, RINGO3 observation duration, $T_{90}$, Galactic extinction, redshift, and related references. Results for GRB 140430A, GRB 141220A, GRB 190114C, and GRB 191016A have already been published separately in \cite{kopac_2015, Jordana_2021,Jordana_2020, Shrestha_2021} respectively and a detailed analysis of GRB~180618A is submitted for publication (Jordana-Mitjans et al, 2022, submitted). In this paper we re-analyse these bursts as well as the data on the other unpublished events in order to allow a more homogeneous analysis of the entire sample. \begin{table*} \center \caption{Properties of all the triggers observed by RINGO3 in $\sim$ 7 years. OT stands for optical transient. GCN stands for gamma-ray burst coordinates network which is a system that distributes the information about GRBs. } \resizebox{!}{.95\height}{\begin{tabular}{c|c|c|c|c|c|c|c} \hline \hline GRB & RA ($\degr$) & DEC ($\degr$) & OT & $T - T_0$ (s) & IO/RATCAM & LT GCN & Note \\ \hline 130216A& 67.90 & 14.67 & NO & 678 & YES &- &- \\ 130328A & - & - & - & 246 & - & - &No GCN \\ 130408A & 134.40 & -32.36 & YES & 517 & YES & 14362 &Only one observation \\ 130427A & 173.13 & 27.69 & YES & 47158 & YES & - &Limited number of observation\\ 130504A &272.45 & -16.31 & NO & 193 & YES &- &Only upper limit in GCN\\ 130606A & 249.3964 & 29.7963 & YES & 2097 & YES & 14785 & Visible only in I band\\ 130610A & 224.4203 & 28.2072 & YES & 207 & YES &14843 &Analysis in this paper\\ 130612A & 259.7941& 16.7200 & YES & 178 & YES & 14875&Analysis in this paper\\ GRB:130702:1 & 217.308 & 15.774& - & - & - & -&No other information \\ 130824 & 288.805 & 10.956 & NO & 2253 & YES& - & Not a GRB\\ 131004A & 296.11 & -2.95 & YES & 95 & YES & 15306&Too faint for RINGO3(R>17) \\ GRB:576238:0 & - & - & - & - & - & -\\ 131030A & 345.065 & -5.36 & YES & 3748 & YES &15406& WCS error in RINGO3 observations \\ 140206A & 145.33 & 66.76 & YES & 147 & YES & 15806& Issue with RINGO3 observations \\ GRB:592204:0 & - & - & - & - & - & -&Not a GRB\\ INTEGRAL:GRB:6599:0 & - & - & - & - & - &-& Not a GRB \\ GRB:596958:0 & 202.928 &29.258 & - & 206 & YES & -&WCS error \\ 140430A &102.9359& 23.0237 &YES & 123 & YES & 16192 &Analysis in this paper \\ 140516A &252.98 & 39.96 & NO & 3158 & YES & -&- \\ 140709A & 304.66 & 51.22 & YES & 101 & YES &- &Faint for RINGO3 (R>17)\ \\ 141026A & 44.084 & 26.928 & Maybe & 196 & YES &- & Faint for RINGO3 (R>17)\\\ 141220A & 195.0657 & 32.1464 & YES & 128 & YES & 17199& Analysis in this paper \\ 141225A &138.77 & 33.79 & YES & 278 & YES & 17231 &Faint for RINGO3 (R>17)\\ 150302A & 175.53 & 36.811 & NO & 169 & YES & - &-\\ 150309A & 277.10 & 86.42 & NO & 210 & YES & 17556 &-\\ 150317A & 138.98 & 55.46 & Maybe &147& NO & -&No source in the image \\ 150428B & 292.63 & 4.125 & NO & 172 & YES & - &-\\ GRB:650221:0 & 7.256& 59.596 & NO & 310& YES & - &Not a GRB \\ 150831B &271.03 & -27.25 & NO & 183& YES & - & -\\ 150908 & 288.80 & 10.94 & NO & 1273 & YES & - &Not a GRB\\ 151118A & 57.17 & 65.90 & NO & 182 & YES & - &- \\ 151215A & 93.5844 & 35.5159 & YES & 181 & YES & - &Analysis in this paper\\ 160119A & 211.92 & 20.46 & Maybe & 216& YES & - &Faint for RINGO3 (R>17)\\ 160313A & 183.79 & 57.28 & NO & 208& YES & 19177 &- \\ 160316A & 118.92 & -29.56 & NO & 169 & YES & - & Not a GRB \\ 160401A & 89.73 & 26.68 &NO & 168 & YES & 19254 &-\\ 160401B & - & - & NO & 798 & YES & - & Same field as 160401A\\ 160401C &- & - & NO & 3592 & YES &- & Same field as 160401A\\ GRB:702630:0 & 299.64 & 35.22 & NO & 247& YES &- & Not a GRB\\ 160705B & 168.10 & 46.69 & Maybe & 192 & YES & 19658 & Faint for RINGO3 (R>17) \\ 160714A & 234.49 & 63.80 & NO & 156 & NO & - &-\\ GRB:704327:0 & 272.61 & 72.05 & NO & 158 & NO & -\\ 160821B & 279.97 & 62.39 & YES & 181 & YES & - & Faint for RINGO3 (R> 17) \\ 161022A & 129.00 & 54.34 & Maybe & 203 & YES & 20090 & Faint for RINGO3 (R> 17) \\ 161214A & 190.72 & 6.83 & YES & 114& YES & 20252 & Faint for RINGO3 (R> 17)\\ INTEGRAL:GRB:7644:2 & 190.72 & 6.82 & - & 3534 & YES &- & Same as 161214A \\ 170208B & 127.14 & -9.02 & YES &126 & YES &- &Faint for RINGO3 (R> 17)\\ 170604B & 200.80 & 64.19 & NO & 177 & YES & - & -\\ 170728B & 237.98 & 70.12 & YES & 213 & YES & 21375 &Faint for RINGO3 (R> 17)\\ 171003A & 40.91 & 61.43 & NO & 871 & YES & 21961 & Galactic Transient\\ GRB:778435:0 & 84.08 & 34.44 & NO & 159 & YES & -&Not a new GRB \\ 171020A & 39.24 & 15.20 & YES & 184 & YES & 22033 &Faint for RINGO3 (R> 17)\\ GRB:782859:0 &40.93 & 61.43 & NO & 934 & YES & - &Not a GRB\\ 171115A & 278.38 & 9.12 & NO & 211 & YES & - & -\\ 180325A & 157.4275& 24.4635 & YES & 146 & YES & 22534 & Analysis in this paper\\ 180512A & 201.93 & 21.40 & NO & 332 & YES &22716 & - \\ GRB:841583:0 & 245.06 & -15.70 & NO & 1318 & YES & - &Not a GRB\\ 180618A & 169.9410& 73.8371 & YES & 200 & YES &22792 & Analysis in this paper\\ 180704A & 32.66 & 69.96 & NO & 176 & YES & - & - \\ 180720C & 265.63 & -26.62 & NO & 265 & YES & 22991 &- \\ 180904A & 274.24 & 46.62 & NO & 224& YES & 23199 &-\\ 190114C & 54.5048& -26.9464 & YES &201 & YES & - & Analysis in this paper\\ 190427A & 280.21 & 40.30 & NO & 229& YES & - & -\\ 190624A & 144.52 & 46.47 & - & 272 & YES & - &Not a new source\\ 191011A & 44.72 & -27.84 & YES & 140 & NO & - & Faint for RINGO3 (R> 17)\\ 191016A & 30.2695& 24.5099 & YES & 533 & YES & - & Analysis in this paper\\ \hline \end{tabular}} \label{tab:alert} \end{table*} \begin{table*} \centering \caption{Properties of the GRBs observed by RINGO3 for which we could perform photometry. Bursts from 2014 onwards could also be analysed polarimetrically. We list co-ordinates (J2000) in the second and third column, the fourth column gives MJD start time of our observations, the fifth column gives the time range observed by RINGO3, sixth is the $T_{90}$ for the GRB and the seventh column provides the reference for these numbers, eighth column is the galactic extinction, ninth column is for jet break time ($T_{JB}$), and their references is given in tenth column, redshift is given in eleventh column and their reference is provided in twelfth, and the last column gives the GCN reference of GRB detection.} \resizebox{\textwidth}{!}{\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c} \hline \hline GRB & RA ($\degr$) & DEC ($\degr$) & MJD start &$T$-$T_0$ (s) & $T_{90}$ (s) &Ref. &$E(B-V)^{GAL}$ &$T_{JB}$(days) & Ref.& z & Ref. & GCN reference \\ \hline 130606A & 249.3964 & 29.7963& 56449.90 &2097-2697 & 276.6 $\pm$ 19.6 &L16 & 0.021& >1.3 & Y16 &5.91&CT13 &\citet{Ukwatta_2013}\\ 130610A & 224.4203 & 28.2072& 56453.13 &207 -807 & 47.7 $\pm$ 10.7 &L16 &0.0181& >2.9 & E09 &2.0920&S13 &\citet{Cummings_2013}\\ 130612A & 259.7941& 16.7200 & 56455.14& 176-776 & 4.0 $\pm$ 1.4 & L16 &0.065&>1.0 & E09 &2.0060& T13&\citet{GCN14874}\\ 140430A & 102.9359& 23.0237&56777.85 &124-800&173.6 $\pm$ 3.7&L16 &0.12&>1.15 & K15 &1.6 & K14 &\citet{GCN16190}\\ 141220A & 195.0657 & 32.1464&57011.25& 129-1929 &7.2 $\pm$ 0.47& L16&0.011 &>0.35 & J21 &1.3195 & U14&\citet{GCN17196}\\ 151215A & 93.5844 & 35.5159&57371.12 &182-1982& 17.8 $\pm$ 1.0&G15 & 0.34& >2.3 & E09 &2.59 & X15&\citet{Gibson_2015}\\ 180325A & 157.4275& 24.4635& 58202.07 &147-1947& 94.14 $\pm$ 1.47& T18 & 0.0147& >0.4 & E09 &2.25 & He18 & \citet{troja_2018}\\ 180618A & 169.9410& 73.8371& 58287.02 &200-1400& 3.71$^1$ $\pm$ 0.58 & H18 &0.058&>0.02 & TW & $<$1.2 & S18, J22 &\citet{GCN22790}\\ 190114C & 54.5048& -26.9464& 58497.87 &201-2000 &116.4 $\pm$ 2.56 & H19&0.01& 0.21 & J20 &0.4245 & S19& \citet{GCN23688}\\ 191016A & 30.2695& 24.5099& 58772.17 &3987-7587& 219.70 $\pm$ 183.35&E09 & 0.09& 0.52 & S22,P22 &3.29$\pm$0.4$^2$ & S21 &\citet{GCN26008}\\ \hline \end{tabular}} {References: L16 - \citet{Lien_2016}; G15 - \citet{Gibson_2015}; T18 - \citet{troja_2018}; H18 - \citet{Hamburg_2018}; H19 - \citet{Hamburg_2019}; E09 - \citet{Evans_2009}; CT13 - \citet{Castro-Tirado_2013}; S13 - \citet{Smette_2013}; T13 - \citet{Tanvir_2013}; K14 - \citet{Kruehler_2014}; U14 - \citet{postigo_2014};X15 - \citet{Xu_2015}; He18 - \citet{Heintz_2018};S18 - \citet{GCN22810}; J22 - Jordana-Mitjans et al. 2022, submitted ;S19 - \citet{Selsing_2019}; S21 - \citet{Smith_2021}; Y16 - \citet{Yasuda_2017}; K15 - \citet{kopac_2015}; J21 - \citet{Jordana_2021}; TW - This Work; J20 - \citet{Jordana_2020}; S22-\citet{Shrestha_2021}; P22-\citet{Pereyra_2022}.\\ 1-This is a short GRB with and extended emission \\ 2-This is photometric redshift all other are spectroscopic redshift.} \label{tab:general} \end{table*} \section{Data reduction}\label{sec:reduction} In this section, we present the data reduction technique used to extract counts and uncertainties in counts from eight different images; these values are used to calculate both the photometric and polarization signals. \subsection{Photometry and calibration}\label{subsec:reduction-Photometry} First we perform photometric reduction on the images and extract counts and uncertainties for the eight different images. We perform aperture photometry using the Python package Astropy Photutils \citep{Bradley_2019}. We first detect sources in the field of view (FOV) with a minimum of 15 times the standard deviation of the image signal-to-noise ratio (SNR) using DAOStarFinder. Once we identify these sources, we estimate background noise using Background2D function of Photutils and subtract this from the data. After the background is subtracted, we perform aperture photometry, which requires the selection of the appropriate aperture size. We use two different methods to calculate the best aperture size; 1) calculate full width at half-maximum (FWHM) of the source, and 2) calculate counts and error in counts with respect to different aperture size for one stacked image per observation. We use 2 to 3 times FWHM of the target and the aperture that produces the best counts to counts error ratio as our aperture size to perform photometry per target. We obtain eight different counts and error in counts. Error in counts are calculated via root mean square sum of background noise and Poisson noise of the source \citep{Bradley_2019}. We perform relative photometry with USNO-B1.0 catalogue stars to calculate the magnitude of the GRBs. The sum of eight polarized images in a RINGO3 observations provides the total intensity of the source. For each GRB, we get simultaneous observations for three different wavebands. In the same FOV, we select one or two stars whose magnitude is already known and use those sources to calibrate the magnitude of the GRB being observed. Colour transforms from \citet{kopac_2015,Jordana_2020} were used to convert the RINGO3 magnitudes to the standard Johnson-Cousins system. In order to correct for Galactic extinction we used \citet{Schlafly_2011} to correct the magnitude of the GRBs. To convert magnitudes to fluxes we used zeropoint values from \citet{Bessell_1998}. Properties such as redshift and high energy burst duration from literature for all ten GRBs are presented in Table~\ref{tab:general}. In Fig.~\ref{fig:time_coverage} we present the time coverage of GRB observations by RINGO3 in observer frame in red and in co-moving frame in grey. The BAT peak time and $T_{90}$ in the observer frame are presented as triangles and circles in the same plot for all ten GRBs. The light-curves of all ten GRBs observed are presented in Fig.~\ref{fig:photometry_1} and Fig.~\ref{fig:photometry_2}. We fit the optical light curves of RINGO3 data with either a simple (PL) or broken power (BPL) and provide reduced $\chi^2$ value for each fit in the Tables~\ref{tab:photometry_pl} and ~\ref{tab:photometry_bpl} respectively. We perform a PL or BPL fit for each wave band observation separately, thus giving us different decay indices ($\alpha$) for different wavelength observations. For GRB 191016A, we perform a PL fit for IO:O data as well to see the earlier time decay index compared to later time observations made by RINGO3. \begin{table*} \centering \begin{tabular}{c|c|c|c|c|c|c|c|c} \hline \hline GRB & Model & $\alpha I $ & $\alpha R $ & $\alpha V$ & $\chi_{r}^2 (I)$ &$\chi_{r}^2 (R) $ &$\chi_{r}^2 (V)$ & d.o.f\\ \hline 130606A & PL & 1.55 $\pm$0.5 & - & - & 6.1 & - & - & 8\\ 130610A & PL & 0.90$\pm$0.13 & 1.02 $\pm$ 0.1 & 0.85 $\pm$ 0.05& 2.2 & 0.85 & 2.7 & 8\\ 130612A & PL & 0.77 $\pm$0.07 & 0.80 $\pm$ 0.07 & 0.79 $\pm$ 0.06& 1.2 & 1.4 & 1.3 & 8 \\ 140430A & PL & 0.71 $\pm$ 0.06& 0.57 $\pm$ 0.02 & 0.55 $\pm$ 0.02 & 0.4 & 0.2 & 0.2 &28\\ 141220A & PL & 1.09 $\pm$ 0.02& 1.10 $\pm$ 0.02& 1.03 $\pm$ 0.02& 1.05 & 2.4 & 4.3 &28 \\ 151215A & PL & 0.68 $\pm$ 0.03 & 0.98 $\pm$ 0.03 & 0.92 $\pm$ 0.03 & 1.7 & 2.9 & 6.0 & 16 \\ 180325A & PL & 0.58 $\pm$ 0.04 & -& -& 2.04 &- & - &48 \\ \hline \end{tabular} \caption{Light curve fitting results for GRBs that can be fitted with single power-law (PL) model. PL is given by $F \propto t^{-\alpha}$ For each fit we provide reduced $\chi_{r}^2$ values and degree of freedom (d.o.f). } \label{tab:photometry_pl} \end{table*} \begin{table*} \centering \resizebox{\textwidth}{!}{\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c} \hline \hline GRB & Model & $T_{b}$ (m) &$\alpha_1 I $ & $\alpha_2 I $ & $\alpha_1 R $ & $\alpha_2 R $ & $\alpha_1 V$ & $\alpha_2 V$ & $\chi_{r}^2 (I)$ &$\chi_{r}^2 (R) $ &$\chi_{r}^2 (V)$ & d.o.f\\ \hline 180618A & BPL & 22.8 &0.48$\pm$ 0.08 & 2.32 $\pm$ 0.8 & 0.53$\pm$ 0.04 & 2.45$\pm$ 0.4 & 0.57$\pm$ 0.05 & 2.26$\pm$ 0.5& 1.2 & 1.4 &0.6 &18 \\ 190114C & BPL & 6.7 &1.43$\pm$ 0.03 & 0.87 $\pm$ 0.02 & 1.50$\pm$ 0.02 & 0.94 $\pm$ 0.01 & 1.47$\pm$ 0.02 & 0.99 $\pm$ 0.02 & 0.46 & 0.3 & 0.08 & 45\\ 191016A & BPL & 102.4 ($I$), 101.4 ($R$), 87.5 ($V$) &0.97$\pm$ 0.07 & 0.04$\pm$ 0.17 & 0.98$\pm$ 0.07 & -0.44$\pm$ 0.17 & 1.25$\pm$ 0.1 & 0.01$\pm$ 0.09 & 0.9 & 1.01 & 1.04 & 56 \\ \hline \end{tabular}} \caption{Light curve fitting results for GRBs whose light-curve is fitted by broken power-law (BPL) model. $\alpha_1$ and $\alpha_2$ denotes the decay index before and after the break time respectively. For each fit we provide reduced $\chi_{r}^2$ values and degree of freedom (d.o.f). } \label{tab:photometry_bpl} \end{table*} \subsection{Polarization signal}\label{subsec:reduction-polarimetry} The sky-subtracted counts of the eight different images are used to extract polarization of the source using recipe shown by \citet{clarke_2002}. The same process is followed for RINGO, RINGO2, and previous RINGO3 analysis (e.g. \citet{Jermak_2016, Shrestha_2021}). Using this technique, we can get linear Stokes parameters $q$ and $u$. In every case we need to correct for polarization introduced by the instrument itself, therefore we observe different unpolarized and polarized standards. We take the average of Stokes q and u for the unpolarized standard star, with the assumption that the unpolarized standard star has Stokes $q \sim 0$ and $u \sim 0$, and calculate the average values introduced by the instrument. With the instrumental Stokes parameters being $q_{inst}$ and $u_{inst}$, the instrument corrected Stokes parameters of the target are given by: \begin{align} & q_c = q - q_{inst} \\ & u_c = u - u_{inst}. \end{align} We perform error propagation in $q$ and $u$ to calculate the error value $q_e$ and $u_e$. Finally, these $q_c$ and $u_c$ are used to calculate a raw percentage polarization and position angle by using: \begin{align} & \%p = \sqrt{q_c^2+u_c^2} \times 100. \label{eq:pol} \\ & \psi =\frac{1}{2} \arctan\left ( \frac{u}{q} \right). \label{eq:pa} \end{align} Analysis of RINGO3 data of polarized standard shows no significant instrumental depolarization \citep{Jermak_2017}. We take measures to get the correct quadrant for the position angle. The position angle needs to be rotated based on the telescope Cassegrain axis sky position angle (SKYPA), measured east of north which gives electron vector position angle (EVPA). \begin{equation} EVPA = \psi + SKYPA + K. \label{eq:evpa} \end{equation} Here $K$ is a calibration factor which gives the position angle offset combined of the angles between the orientation of the polarizer, the telescope focal plane, and the trigger position of the angle measuring sensor. This angle offset was calculated using polarized standards observed during various time periods and the values of $K$ are provided in Table~\ref{tab:inst}. Some parts of the analysis are taken from \citet{Jermak_2017}. The last step in the polarization calculation is bias correction and error calculation for the polarization degree and EVPA. Noise in \textit{q} and \textit{u} introduces a polarization signal which is not intrinsic. In order to correct for this and to calculate the error in polarization degree, we use the prescription developed by \cite{Plaszczynski_2014}. The error in EVPA is calculated using standard error propagation applied to Eq.~\ref{eq:pa}. \begin{table*} \centering \begin{tabular}{c|c|c|c|c|c|c} \hline \hline MJD Range & $q_{in}$ (I)& $\sigma$ & $u_{in}$(I) & $\sigma$ & K (I $\degr$) & K (I $\degr$ $\sigma$) \\ \hline 56658-56816& -0.0119$\pm$0.0005 &0.003 & -0.0410 $\pm$ 0.0006 & 0.0036& 57.39 & 4.26 \\ 56816-57202 & -0.0154 $\pm$ 0.0004 & 0.004 & 0.0295$\pm$ 0.0015& 0.014 & 115.15 & 3.75\\ >57202 & -0.0131 $\pm$ 0.004& 0.025 & -0.0336 $\pm$ 0.001 &0.006 & 125.61 & 4.63 \\ \hline MJD Range & $q_{in}$ (R)& $\sigma$ & $u_{in}$(R) & $\sigma$ & K (R $\degr$) & K (R $\degr$ $\sigma$) \\ \hline 56658-56816& -0.01163 $\pm$ 0.0004& 0.0024 & -0.0371 $\pm$ 0.0048& 0.029 & 55.58 & 3.6 \\ 56816-57202 & -0.0157 $\pm$ 0.0003 & 0.002 & 0.0333 $\pm$ 0.0014 & 0.013 & 115.95 & 3.25\\ >57202 & -0.0105 $\pm$ 0.0024&0.014 & -0.0356 $\pm$ 0.009& 0.059 & 124.8 & 5.05 \\ \hline MJD Range & $q_{in}$ (V)& $\sigma$ & $u_{in}$(V) & $\sigma$ & K (V $\degr$) & K (V$\degr$ $\sigma$) \\ \hline 56658-56816& -0.0096 $\pm$ 0.0003& 0.002 &-0.0177 $\pm$ 0.0003& 0.002& 54.93 & 3.94 \\ 56816-57202 & -0.0077 $\pm$ 0.0002& 0.0018 & 0.0215 $\pm$ 0.0009 & 0.008 & 115.42 & 2.92\\ >57202 & -0.0047 $\pm$ 0.004& 0.026 & -0.022 $\pm$ 0.0058& 0.035 & 124.90 & 4.89 \\ \hline \end{tabular} \caption{Table of instrument $q$, $u$, and K factor values for different time periods of observations using RINGO3. We quote standard error ($\frac{\sigma}{\sqrt{N}}$) where $N$ is the total number of observations as the error in instrument $q$ and $u$. Standard deviation ($\sigma$) is also presented. These values are much smaller than error in Stokes $q$ and $u$ of GRB. Instrumental $q$, $u$, and $K$ values are from \protect\cite{Jermak_2017}. } \label{tab:inst} \end{table*} \begin{table*} \centering \begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c} \hline \hline GRB & $T$-$T_0$ (s) & P (\%) ($I$) & P (\%) ($R$) & P (\%) ($V$) & Rank ($I$) & Rank ($R$) & Rank ($V$)\\ % \hline 140430A &124-724& <35.9&<18.9 & <18.50.97 & 0.67 & 0.20 \\%& <124 &173.6&0.46&1.6 \\ 140430A &724-1324& <42.2&<58.6 & <32.3& 0.78 & 0.68 & 0.66 &\\%& \\%<124 &173.6&0.46&1.6 \\ 140430A &1324-1924 & <96.91&<24.1 &<23.0& 0.78 & 0.54 & 0.30 \\%&\\% <124 &173.6&0.46&1.6 \\ \hline 140430A &124-1924 & <12.6 &<9.6 &<16.5& 0.97 & 0.11 & 0.099 \\%& \\%<124 &173.6&0.46&1.6 \\ \hline \hline 141220A & 129-729 & <17.9& <10.8& <7.5& 0.88 & 0.67 & 0.79 \\%&\\%<129 &7.616 &0.033 &1.3195\\ 141220A & 729-1329 & <15.3& <6.7& <11.1& 0.45 & 0.69 & 0.21 \\%&\\%<129 &7.616 &0.033 &1.3195\\ 141220A & 1329-1929 & <69.5& <31.7& <11.2 & 0.84& 0.76 & 0.57\\% &<129 &7.616 &0.033 &1.3195\\ \hline 141220A & 129-1929 & <3.52& <3.23& <1.74 & 0.33 & 0.13 & 0.73 \\% &<129 &7.616 &0.033 &1.3195\\ \hline \hline 151215A & 182-782& <7.8& <5.9 & <3.3 & 0.31 & 0.199 & 0.065 \\%&<182 &17.8& 1.32& 2.59\\ 151215A & 782-1382& <14.0& <20.27 & <4.2& 0.16 & 0.73 & 0.051 \\%& -&<182 &17.8& 1.32& 2.59\\ 151215A & 1382-1982& <22.34&<28.9 & <5.6& 0.67 & 0.37 & 0.95 \\%&<182 &17.8& 1.32& 2.59\\ 151215A & 1982-2582& <15.2& <22.5 & <8.2 & 0.047 & 0.044 & 0.32 \\%&<182 &17.8& 1.32& 2.59\\ \hline 151215A & 182-2582& <6.45& <6.64 & <6.05 & 0.059 & 0.91&0.91 \\%&<182 &17.8& 1.32& 2.59\\ \hline \hline 180325A & 147-747&<18.2 &-& -& 0.70 &-&-\\%& <147& 94.1& 0.055*& 2.25\\ 180325A & 747-1347&<6.8 &-&-& 0.20 &-&-\\%& <147& 94.1& 0.055*& 2.25\\ 180325A & 1347-1947&<30.1 &-&-& 0.24 &-&-\\%& <147& 94.1& 0.055*& 2.25\\ \hline 180325A & 147-1947&<12.29 &-&-& 0.342 &-&- \\%& <147& 94.1& 0.055*& 2.25\\ \hline \hline 180618A &800-1400& <66.0& <25.7 &<10.7 & 0.90 & 0.95 & 0.54 &\\%& <200&3.712 & 0.237&<1.2\\ 180618A &1400-2000& <71.4&<19.5 &<25.5 & 0.95 & 0.55 & 0.107 &\\%& <200&3.712 & 0.237&<1.2\\ 180618A &2000-2600&<100&<100 &<26.0 & 0.88 & 0.78 & 0.53 \\%& <200&3.712 & 0.237&<1.2\\ \hline 180618A &800-2600& <5.26&<12.24 &<6.78 & 0.38 &0.61 & 0.078 \\%& <200&3.712 & 0.237&<1.2\\ \hline \hline \hline \hline \end{tabular} \caption{Table of polarization degree measurements from the observation done by RINGO3. The upper limit of polarization for each time interval is presented along with the upper limit for stacked data of the whole observation is presented for each GRB. The presented data are not Milky Way ISP corrected. Columns are GRB identifier, time range, polarization degree for $I$, $R$, and $V$ bands, permutation rank for detected polarization.} \label{tab:polarization_nondetection} \end{table*} \begin{table*} \centering \begin{tabular}{c|c|c|c|c|c|c|c|c|c|c} \hline \hline GRB & $T$-$T_0$ (s) & P (\%) ($I$) & P (\%) ($R$) & P (\%) ($V$) & EVPA (\degr) ($I$) & EVPA (\degr) ($R$) &EVPA (\degr) ($V$) & Rank ($I$) & Rank ($R$) & Rank ($V$)\\ % \hline \hline 190114C &203-803& {$\bf 2.9 \pm 0.8$}&{$\bf 3.2 \pm 0.8$} &{$\bf 2.0 \pm 1.2 $} & 26 $\pm$ 9& 48 $\pm$ 9 & 25 $\pm$ 25 & 0.98 &0.997& 0.99 \\%& <200&3.712 & 0.237&<1.2\\ 190114C &803-1403& 2.0 $\pm$ 1.5 & 2.5 $\pm$ 2 &<3.7 & - & -&-& 0.147 &0.32 & 0.87 \\%& <200&3.712 & 0.237&<1.2\\ 190114C &1403-2003& 3.7 $\pm$ 2.6&<4.8 &<5.0 & - &-&- & 0.32 & 0.07 & 0.801 \\%& <200&3.712 & 0.237&<1.2\\ 190114C &2354-2954& <4.1&<3.1 &<6.5 &- &-&-& 0.156 & 0.82 & 0.99 \\%& <200&3.712 & 0.237&<1.2\\ 190114C &2954-3554& <8.2&<10.5 &<6.4 &- &-&-& 0.87 & 0.81 & 0.98 \\%& <200&3.712 & 0.237&<1.2\\ \hline 190114C &203-2003& <2.7&<2.8 &<2.22 &- &-&-& 0.82 & 0.79 & 0.53 \\%& <200&3.712 & 0.237&<1.2\\ \hline \hline 191016A & 3987-4587& $\bf 4.7 \pm 4.1$ & $<9.1$ & $\bf 7.8 \pm 5.6$ & 93 $\pm$ 22 &-& 90 $\pm$ 18 & 0.99 & 0.63 & 0.99\\%&<3987 &219.7& 0.29& 3.29 \\ 191016A & 4587-5187&$<5.2$ &$\bf 11.2 \pm 6.6$ & $\bf 5.7 \pm 5.6$&-& 90 $\pm$ 15 & 82$\pm$ 26 & 0.59 & 0.99 &0.98 \\%&<3987 &219.7& 0.29& 3.29 \\ 191016A & 5187-5787&<14.0 & $<5.5$& <10.8&-&-&- & 0.72 & 0.11 & 0.73 \\%&<3987 &219.7& 0.29& 3.29 \\ 191016A & 5787-6387& $\bf 14.6 \pm 7.2$ & $\bf 6.1 \pm 6.1$ & <9.2 & 100 $\pm$ 12 & 90 $\pm$ 30 &- &0.99 & 0.95 & 0.74 \\%&<3987 &219.7& 0.29& 3.29 \\ 191016A & 6387-6987& <10.7 & <12.0& <13.5&-&-&-& 0.67 & 0.83 & 0.86 \\%&<3987 &219.7& 0.29& 3.29 \\ 191016A & 6987-7587&<17.6 & <11.0& <9.2 & - &-&- & 0.76 & 0.86 & 0.94 \\% &<3987 &219.7& 0.29& 3.29 \\ \hline 191016A & 3987-7587&<3.82 & <5.23&<3.7&-&-&- & 0.34 & 0.141 & 0.097\\% &<3987 &219.7& 0.29& 3.29 \\ \hline \hline \end{tabular} \caption{ Table for two GRBs with polarization detection. The polarization degree for each time interval is presented and the upper limit for stacked data for the whole observation time range is also presented. The values presented are not corrected for Milky Way ISP contribution. Columns are GRB identifier, time range, polarization degree for $I$, $R$, and $V$ bands, Position angle, permutation rank for detected polarization. Detections that pass both error bar and permutation analysis are highlighted in bold. } \label{tab:polarization_detection} \end{table*} \begin{table*} \centering \begin{tabular}{c|c|c|c|c|} \hline \hline GRB & $E(B-V)^{GAL}$ & $I-ISP^{GAL} (\%)$ & $R-ISP^{GAL} (\%)$ & $V-ISP^{GAL} (\%)$ \\ \hline 140430A & 0.14 & 1.03& 1.17&1.26\\ 141220A & 0.01&0.07 &0.08 &0.09\\ 151215A & 0.40&2.95&3.35 & 3.6\\ 180325A & 0.02&0.14 &0.16 &0.18\\ 180618A & 0.07&0.51 &0.58 &0.63\\ 190114C & 0.01& 0.07& 0.08 & 0.09\\ 191016A & 0.09&0.66 & 0.75 & 0.8\\ \hline \end{tabular} \caption{Table of upper limit of Galactic interstellar polarization estimates based on extinction values from \citet{Schlafly_2011}} and using\ \protect\cite{serkowski_1975}. \label{tab:isp} \end{table*} In Figures~\ref{fig:pol_results_1} and ~\ref{fig:pol_results_2} , we present the evolution of polarization with time for all seven GRBs for the three different wavebands $I$, $R$ and $V$ (top to bottom panels). We present $1\sigma$ error bars in these plots for the polarization results. When the error bars in polarization degree crosses $0 \%$, then we consider the polarization to be an upper limit and upper limit value is presented by the upper end of the error bar. When we have error bars not crossing $0\%$, we consider it to be a possible detection. \subsection{Polarization Detections} Table \ref{tab:polarization_detection} identifies possible polarization detections in GRB 190114C at early times (before 2003-s) and in GRB 191016A at various epochs as from consideration of their error bars as outlined above. In order to confirm the possible polarization detections, we implemented ``permutation analysis" (described in detail in \cite{steele-ringo2-2017}) to rigorously investigate the probabilities of the detection by looking at the individual counts at the 8 rotor positions for a source. Before doing this we correct for instrumental polarization using a bright star in the field of view as an unpolarized source and dividing the GRB source counts by the unpolarized source counts at the corresponding rotor position. These corrected counts from the eight rotor positions are shuffled into all possible ordered permutations. This procedure will destroy any coherent polarization signal encoded in the data and generates $(8-1)!$ (5040) permutations of the corrected counts for the GRB source. Each of these permutations will have identical noise characteristics to the original data (being generated directly from it) and are then used to calculate a polarization degree. By sorting the resulting polarization values we can then generate a rank which tells us the probability of the detected polarization degree being true (as opposed to being artificially created by the transformation of noise into polarization signal due to polarization bias). We can then check the null hypothesis; if the source is unpolarized then what is the chance of getting some polarization signal due to noise in the data? For example if the rank is greater than 0.9 it means the probability of being an unpolarized source $p = 1-rank$ will be $<0.1$. Since we have carried out a total of 79 tests over the sample, using the threshold $p<0.05$, we could of course expect $\sim4$ false positives to have arisen from this testing procedure under the null hypothesis that all GRBs do not show polarization. Overall we find a total of 13 such positives in the sample. The binomial cumulative probability of such an outcome is highly significant ($p<1.e\times10^{-4}$) indicating that at least some of our detections ($\sim 9$) should be true polarization signals. Further evidence of the validity of the detections can be inferred from the correspondence between individual detections made by the two techniques (error bar analysis and permutation analysis). From the error bar analysis we find 13 measurements are identified as possible detections, 9 of which have permutation analysis with $p<0.05$. Testing against the null hypothesis of no correspondence, we find the cumulative binomial probability of this outcome is $p<1\times10^{-6}$, indicating a strongly significant association. In comparison only 4 out of the 65 measurements which have only error bar upper limits show a permutation $p<0.05$ - an outcome with a cumulative binomial $p<0.41$ i.e. entirely consistent with the high permutation rank values in this case being spurious due to the multi-trial nature of the test. Overall we are therefore confident that the 9 measurements that pass both techniques (error bars and permutation analysis) are true detections of polarization. We highlight these measurements in bold text in Table \ref{tab:polarization_detection}. \subsection{Galactic Interstellar Polarization estimate} We use known GRB Milky Way Galactic extinction values to estimate the Milky way Galactic interstellar polarization (GISP) following the formulation by \citet{serkowski_1975}. First we used $p^V(GISP) \leq 9 E_{B-V}$ \footnote{https://irsa.ipac.caltech.edu/cgi-bin/bgTools/nph-bgExec} to calculate the upper limit in polarization induced by GISP in the $V$ band. For each GRB in our observed sample we used $5\degr \times 5\degr$ statistics from \citet{Schlafly_2011}. After this calculation, we used $p/p_{max} =\exp[-\kappa {\rm ln}^2(\lambda_{max}/\lambda)]$ to calculate GISP in $R$ and $I$ bands using $V$ band as the $\lambda_{max}$ and $p^V$ as the $p_{max}$; where $p$ is the polarization induced by GISP at the wavelength $\lambda$, $p_{max}$ is the maximum polarization induced by GISP at the wavelength $\lambda_{max}$, and $\kappa$ (normally $K$ is used but here to avoid confusion with $K$ of EVPA constant we use $\kappa$)is a constant given to be 1.15 in \citet{serkowski_1975} and \citet{Wilking_1982} later modified it to be $\kappa = -0.10+1.86 \lambda_{max}$, this is used for our analysis in this paper. The GISP estimates for the seven GRBs are presented in Table~\ref{tab:isp}. We do not correct for GISP in our polarization results as the GISP values are very low and can only be used for reference. We also do not correct for the ISP contribution from the host galaxy for all the GRBS. However, we present the host galactic ISP contribution for GRB 191016A and GRB 190114C, for which we have a probable detection. \section{Results}\label{sec:results} In this section we present the photometric results of ten GRBs and polarimetric results of seven GRBs. In Table~\ref{tab:flux}, we provide a subset of flux values for GRB 130606A and full data for all the GRBs are available in machine readable format online. \begin{table} \begin{tabular}{llrrrr} \hline GRB & Filter & $T_{start}$(s) & Exp(s) & $F_\nu$ (mJy) & $F_\nu{err}$(mJy) \\ \hline 130606A & I & 2097.0 & 60 & 0.615 & 0.061 \\ 130606A & I & 2157.0 & 60 & 0.681 & 0.070 \\ 130606A & I & 2217.0 & 60 & 0.603 & 0.063 \\ 130606A & I & 2277.0 & 60 & 0.605 & 0.061 \\ 130606A & I & 2337.0 & 60 & 0.543 & 0.060 \\ 130606A & I & 2397.0 & 60 & 0.563 & 0.062 \\ 130606A & I & 2457.0 & 60 & 0.453 & 0.050 \\ 130606A & I & 2517.0 & 60 & 0.467 & 0.056 \\ 130606A & I & 2577.0 & 60 & 0.558 & 0.060 \\ 130606A & I & 2637.0 & 60 & 0.441 & 0.054 \\ \hline \end{tabular} \caption{Sample photometry of GRB 130606A. Here columns are GRB name, RINGO3 Filter, $T_{start}$ is the start time of exposure in seconds since the trigger time, $T_{exp}$ is exposure time in seconds, $F_\nu$ is flux in mJy, and $F_\nu err$ is error in flux in mJy. This table is available in its entirety in machine-readable form.} \label{tab:flux} \end{table} \subsection{GRB 130606A} RINGO3 observations of this burst were obtained $\sim$35 minutes after the trigger time of 21:04:39 UTC \citep{Ukwatta_2013} and only the $I$ band has SNR high enough to perform photometry. The gamma-ray duration is $T_{90} = 276 s \pm 20$ \citep{Lien_2016} and spectroscopic redshift of $z = 5.91$ as observed by GTC \citep{Castro-Tirado_2013}. We corrected for Galactic extinctions for the GRB corresponding to $A_V =0.064 $, $A_R =0.051$, and $A_I = 0.036 $. We fit a power-law to the light curve and get a decay index of 1.55 $\pm$ 0.5 as shown in the Fig.~\ref{fig:photometry_1} and Table~\ref{tab:photometry_pl}. Swift XRT data are best fitted by 2 breaks at $482^{+243}_{-393}$ s and $2.1 ^{+0.6}_{-0.3} \times 10^4 $ s with a decay index of $0.63^{+0.09}_{-1.77}$, $1.09 \pm 0.05$, and $1.79^{+0.22}_{-0.19}$ \citep{Evans_2009}. \citet{Yasuda_2017} constraint the jet break time to be greater than 1.3 days. Polarimetric analysis are not presented here because we could not constrain the instrument polarization during this period. \subsection{GRB 130610A} RINGO3 photometric observations were obtained in $I$, $R$, and $V$ bands $\sim$3 minutes after the trigger time of 3:12:13 UTC and RATCam observations were obtained in SDSS $g'$,$r'$, and $i'$ band $\sim$18 minutes after the trigger. We used stars in the field to do photometric calibrations and corrected for Galactic extinction corresponding to $A_V =0.058 $, $A_R =0.046$, $A_I = 0.033 $, $A_{g'} = 0.071 $, $A_{r'} =0.049 $, and $A_{i'} =0.037 $. A simple power-law is fitted to all the data in three different wave bands. We get $\alpha = 0.90 \pm 0.13, 1.02 \pm 0.1,$ and $0.85 \pm 0.05$ for $I$, $R$ and $V$ filters. We note that for R band images there is a dark line running through the image where the GRB is located which has been seen previously in RINGO3 R images. We found that the light-curve behaviour is dependent on how we subtract the background noise. When the median of the image is considered the background, we get the decay index to be $0.56 \pm 0.05$ which is much lower than the value we get when we use a 2D background estimate.We present results in this paper using 2D background estimate. Swift XRT and RATCam data are also presented in the Fig.~\ref{fig:photometry_1} along with RINGO3 data and the power-law fit. This GRB has $T_{90} =47 \pm 11 s$ \citep{Lien_2016} and a spectroscopic redshift of $z = 2.092$ \citep{Smette_2013}. The Swift XRT light-curve was fitted by one break at $242^{+27}_{-33}$ seconds with a decay index of $2.47^{+0.44}_{-0.29}$ and $1.09 \pm 0.03$ before and after the break \citep{Evans_2009}. From the Swift XRT light-curve, we assume the minimum jet break time to be 2.9 days \citep{Evans_2009}. \subsection{GRB 130612A} This is the one of the GRBs in the sample that might be argued to be a short burst ($T_{90} = 4.0 \pm 1$ s \citep{Lien_2016}) and was observed around 3 minutes after the trigger time of 3:22:23.361 UTC as reported by Swift. RINGO3 photometric observations were made in $I$, $R$, and $V$ bands and RATCam observations were obtained in SDSS $g'$, $r'$, and $i'$ band. Calibration was done using stars in the field and the Galactic extinction correction corresponded to $A_V =0.204 $, $A_R =0.161$, $A_I = 0.115$, $A_{g'} = 0.251 $, $A_{r'} =0.174 $, and $A_{i'} =0.129 $. Fig.~\ref{fig:photometry_1} shows the light curve of GRB 130612A for all three filters of RINGO3 including RATCam and Swift XRT data. Power-law fits are applied to RINGO3 data with $\alpha$ = $0.77 \pm 0.09$, $0.85 \pm 0.09$, and $0.80 \pm 0.06$ for $I$, $R$ and $V$ filters respectively. Swift XRT data was best fitted by single power-law with a decay index of $1.03 \pm 0.06$ \citep{Evans_2009} and this gives the minimum jet break time to be 1 day. The redshift of the GRB was established spectroscopically to be $z = 2.006$ \citep{Tanvir_2013}. \subsection{GRB 140430A} \citet{kopac_2015} presented polarization and photometric results for GRB 140430A. Here we performed a simple power-law fit as we did for other GRBs for consistency. For RINGO3 data, we find a decay index of $\alpha = 0.71 \pm 0.06, 0.57 \pm 0.02,$ and $0.55 \pm 0.02$ for $I$, $R$ and $V$ filters respectively (Fig~\ref{fig:photometry_1}). We note that here we are using 60 second stacked data whereas \cite{kopac_2015} used 10 second exposure data, hence there is difference in the light curve and decay index values. Swift XRT data was best fitted with three breaks at $320^{+17}_{-16}$, $412^{+26}_{-15}$, and $3.4^{+3.6}_{-1.6} \times 10^4$,seconds with a decay indices of $3.64^{+0.23}_{-0.19}, 8^{+0.0}_{-1.46},$, $0.64^{+0.06}_{-0.08}$, and $1.14 ^{+0.29}_{-0.23}$ \citep{Evans_2009}. The minimum jet break time is assumed to be 1.15 days from \citet{kopac_2015}. This burst is a relatively long burst with $T_{90} = 174 \pm 4$ and the X-ray light curve shows early flares which have been suggested to originate due to internal dissipation processes \citep{ Zhang_2006,Troja_2015, kopac_2015}. We performed a permutation analysis on the polarimetric data and found no probable detection. For the stacked data we find polarization upper limits of $<12.6\%$, $<9.6\%$, and $<16.5\%$ for $I$, $R$, and $V$ filters respectively. We find a similar upper limit of polarization as \citet{kopac_2015} for all the different wavelengths. We note that our upper limit values, as stated in Table~\ref{tab:polarization_nondetection}, are slightly different because we implement a different technique for error calculations and the time intervals of these measurements are different. The GISP estimates for this GRB are $1.03\%$, $1.17\%$, and $1.03\%$ for $I$, $R$ and $V$ band respectively. All the results presented in Fig.~\ref{fig:pol_results_1} are for 10 minute stacked data. \subsection{GRB 141220A} RINGO3 made observations of the GRB about 3 minutes after the trigger time of 6:02:52 UTC in all the three USNO $I$, $R$, and $V$ bands. IO:O SDSS $r'$ band observations were made 33 minutes after the trigger. We used field stars to calibrate magnitude and flux. Galactic extinctions of $A_V =0.035 $, $A_R =0.027$, $A_I = 0.02 $, $A_{r'} =0.029$ were corrected. The redshift of 1.3195 was inferred from spectroscopic observations done using OSIRIS at the 10.4 m GTC \citep{postigo_2014} and gamma-ray burst duration is $T_{90} = 7 \pm 0.5 s$ \citep{Lien_2016}. We fit a power-law function to RINGO3 observations of GRB 141220A and get a decay index of $\alpha$ = $1.09 \pm 0.02, 1.10 \pm 0.02$, and $1.03 \pm 0.02$ for $I$, $R$ and $V$ filters respectively as shown in Fig.~\ref{fig:photometry_1} and these values match well with the decay indices reported in \citet{Jordana_2021} of $1.105\pm 0.013$, $ 1.067 \pm 0.009$, and $1.095\pm 0.005$ for $I$, $R$, and $V$ filters. The Swift XRT light-curve could be fitted by broken power-law with a time break at $207^{+101}_{-45}$\,s and decay index of $-0.3 \pm 0.6$ and $1.375^{+0.104}_{-0.099}$ before and after the break \citep{Evans_2009}. \citet{Jordana_2021} reports a jet break time of 0.35 days or longer. We present upper limit on bias-corrected polarization degree in Fig.~\ref{fig:pol_results_1} for all three wavelengths for 10 minute stacked data. \citet{Jordana_2021} found polarization detection for the first epoch in $V$ band and upper limits for the rest. However, in our analysis we do not find any detection and only upper limits for all the cases. This could be due to the difference in time bin of the presented results. Though the upper limit values from this analysis doesn't match exactly with results from \citet{Jordana_2021}, the trend of polarization for different filters i.e. $V$, $R$, and $I$ bands have an upper limit in incremental order is the same. In addition, the behaviour with time is consistent for all the filters with results from \citet{Jordana_2021}. The observed polarization degree data for the 30 minutes stacked data are $< 3.52\%, <3.23\%$, and $1.74\%$ for $I$, $R$ and $V$ bands respectively. For this GRB, the GISP is estimated to be $0.07\%, 0.08\%$, and $0.09\%$ for $I$, $R$, and $V$ bands thus contribution from GISP is negligible. Since we are observing forward shock dominated emission, the low level of polarization detection is in line with theoretical predictions \citep{Rossi_2004,Kobayashi_2019}. \subsection{GRB 151215A} LT observations started within 3 minutes of the trigger time 3:01:28 UTC. Spectroscopic analysis of NOT observations gave the redshift of $z = 2.59$ \citep{Xu_2015}. The gamma-ray burst duration is $T_{90} = 18 \pm 1$\,s \citep{Gibson_2015}. We fit a single power-law and get $\alpha = 0.68 \pm 0.03, 0.98 \pm 0.03,$ and $0.92 \pm 0.03$ for $I$, $R$ and $V$ filters respectively. In Fig.~\ref{fig:photometry_1}, the decay index is slightly different for the $I$ band compared to the $R$ and $V$ bands. We note that there is another light source close to the target which could contaminate the GRB's aperture photometry in some cases. Thus, we cannot confirm colour evolution of the GRB. Swift XRT data is best fitted by a power-law of decay index $0.95^{+0.06}_{-0.05}$ \citep{Evans_2009}and this data put a lower limit on the jet break time to be 2.3 days. We present bias-corrected polarization degree values at different times for GRB 151215A in Fig.~\ref{fig:pol_results_1}. We do not detect polarization and upper limit of observed polarization is presented. The polarization values for 40 minutes of stacked data are $< 6.45\%, <6.64\%$, and $6.05\%$ for $I$, $R$ and $V$ bands respectively. The estimated GISP values are $2.95\%, 3.35\%$, and $3.6\%$ for $I$, $R$, and $V$ bands respectively, which is a significant factor compared to the upper limit values. \subsection{GRB 180325A} RINGO3 observations started $\sim$13 minutes from the trigger time 01:53:02 and reliably detected the transient in the $I$ filter only. In Fig.~\ref{fig:photometry_2} we present the $I$ band light curve of the GRB along with limited, late-time IO:O $r'$ data. We find a decay index of $\alpha = 0.58 \pm 0.04$ which is less than 1; we attribute it to the GRB forward shock for RINGO3 I data with possibility of energy injection. Swift constrained the gamma-ray burst duration to $T_{90}=94 \pm 2 s$ \citep{troja_2018}. The redshift was obtained spectroscopically by NOT as $z = 2.25$ \citep{Heintz_2018}. The Swift XRT best-fitting light-curve has 3 breaks at $238^{+31}_{-116}, 2128^{+812}_{-461}, 3.5^{+0.6}_{-0.4}\times 10^4$ seconds with decay indices of $-0.75^{+0.22}_{-0.68},0.24^{+0.27}_{-0.25},1.99\pm 0.08,$ and $5^{+3}_{-2}$ \citep{Evans_2009}. Using this XRT light-curve, we assume a lower limit on jet break time to be 0.4 days. The polarization degree values after bias correction are presented in Fig.~\ref{fig:pol_results_1} for $I$ band. Permutation analysis on the polarization values did not show any significance for all the observed data points. For the first 30 minutes of stacked data we obtain $< 12.29\%$ for $I$ band. The GISP for this case is $0.15\%$ in $I$ band. \subsection{GRB 180618A} RINGO3 made observations of the GRB about 3 minutes after the trigger time of 0:43:13 UTC in $I$, $R$, and $V$ bands. After 30 minutes of observations by RINGO3, IO:O was online and made follow-up observations of the GRB in SDSS $r$ band filter. Stars in the field of view were used to calibrate the magnitude and flux of the GRB and Galactic extinction of $A_V =0.182 $, $A_R =0.144$, $A_I = 0.103 $, $A_{r'} =0.155$ correction was implemented in the results. The gamma-ray duration is $T_{90} = 3.7 \pm 0.6 $ s from Fermi GBM observations and Swift UVOT filter detection put the upper limit on the redshift to be $z<1.2$ \citep{GCN22810}. The light curve from Swift-BAT data shows a short multi-peak at $T_0$ to $\sim T_0+0.3 s$ and extended emissions lasting until $\sim T_0+50 s$ \citep{Sakamoto_2018}. They also did further analysis to get power-law index and fluence which are consistent with a short GRB with extended emission \citep{Sakamoto_2018}. The Swift XRT light curve is best fitted by a power-law with 3 breaks at $147^{+21}_{-22}, 296^{+185}_{-55},$ and $5483^{+1.63\times 10^3}_{-2015}$ s with decay indices of $0.80^{+0.17}_{-0.20}, 1.48\pm 0.19, 1.876^{+0.132}_{-0.063},$ and $1.04^{+0.18}_{-0.15}$ \citep{Evans_2009}. The best fit for the RINGO3 data at all three different wavelengths is a broken power law with the same break time of $1370$ s. We assume this break time to be the lower limit of jet break time as well. The best fit has $\alpha = 0.48 \pm 0.08, 0.53 \pm 0.04,$ and $0.57 \pm 0.05$ for $I$, $R$ and $V$ filters respectively before the break and $\alpha = 2.32 \pm 0.8, 2.45 \pm 0.4,$ and $2.26 \pm 0.5$ for $I$, $R$ and $V$ filters respectively after the break. GRB 180618A is a short GRB with extended emission as discussed by \cite{Sakamoto_2018}. Our upper limits on polarization are large for $I$ and $R$ filters at a later time because the source is fainter and the noise is high as shown in Fig.~\ref{fig:pol_results_2}. For the $V$ filter the upper limit values are better constrained because of the higher signal-to-noise ratio in this filter for the source. For 30 minutes of stacked data we get polarization values of $< 5.26\%, <12.24\%$, and $<6.78\%$ and GISP values of $0.51\%, 0.58\%$, and $0.63\%$ for $I$, $R$, and $V$ bands respectively. Further detailed analysis of GRB 180618A RINGO3 data will be presented by Jordana-Mitjans et al., 2022 (submitted). \subsection{GRB 190114C} The LT observed this GRB $\sim$ 3 minutes after the burst time of 20:57:02.341 UTC and made observations using RINGO3 in $I$, $R$, and $V$ bands. After the first thirty minutes of RINGO3 observations, IO:O was triggered and made observations in SDSS $r$ band. Since the GRB was bright, more RINGO3 observations were taken after the IO:O observations. Detailed analysis of the LT follow-up observations and data from other telescopes has been presented in \citep{Jordana_2020}. Here we present a similar analysis to other GRBs. The light curve and power-law fit agree well with the results from \citet{Jordana_2020} as seen in Fig.~\ref{fig:photometry_2}. The best-fit for the RINGO3 light curve is a broken power-law with a break at $401$ seconds (we find the best fit to have the same break time for all the filters unlike in \citet{Jordana_2020}). We get $\alpha = 1.43 \pm 0.03, 1.50 \pm 0.02,$ and $1.47 \pm 0.02$ for $I$, $R$ and $V$ filters respectively before the break and $\alpha = 0.87 \pm 0.02, 0.94 \pm 0.01,$ and $0.99 \pm 0.02$ for $I$, $R$ and $V$ filters respectively after the break. The jet break time is 0.21 days \citep{Jordana_2020}. \citet{Jordana_2020} have presented detailed polarimetric analysis of GRB 190114C using RINGO3 data. Here we perform our polarimetric RINGO3 GRB analysis for 10 minute stacked data. We obtain a polarization detection for the earlier time period and the polarization values are low; mostly coming from the ISP of the host galaxy as seen by \citet{Jordana_2020}. We note a slight difference in polarization measurements compared to \citet{Jordana_2020} due to a difference in time intervals of our measurements and a different error calculation technique. For these data points we performed permutation analysis and found detections for a few points as presented in Table~\ref{tab:polarization_detection}. There are two points in the $I$ band for the time interval 803-1403 and 1403-2003 seconds and one data point in the $R$ band for the time interval 803-1403 seconds whose error values do not cross the zero point, however, their permutations ranks are lower than 0.95. Hence, we do not consider these values as detection and present them as the upper limit in Fig.~\ref{fig:pol_results_2}. For 30 minutes of stacked data we get polarization values of $< 2.65\%, <2.78\%$, and $<2.21\%$ and GISP values of $0.07\%, 0.08\%$, and $0.09\%$ for $I$, $R$, and $V$ bands respectively. \cite{Jordana_2020} estimated the polarization contribution of the host galaxy to be $< 3.9\%$, $< 4.5\%$, and $< 4.5\%$ (larger than the detected polarization) therefore the detected polarization could easily be interpreted as simply coming from the dust in the host galaxy confirming our earlier work \citep{Jordana_2020}. % \subsection{GRB 191016A} There was a delay in LT observations of this GRB and initial IO:O observations were made in SDSS $r$ band 40 minutes after the trigger time of 04:09:00 UTC. We made RINGO3 follow-up observations 66 minutes after the trigger time in $I$, $R$, and $V$ filter. Even though it was observed much later, the afterglow was bright enough to be detected in all the filters. Magnitude and flux calibrations were done with stars in the field and Galactic extinction of $A_V =0.281 $, $A_R =0.222$, $A_I = 0.159 $, $A_{r'} =0.239$ was also corrected. The gamma-ray burst duration was inferred to be $T_{90} = 220 \pm 183$ seconds from Swift and the photometric redshift of the burst is 3.29 $\pm$ 0.40 \citep{Smith_2021}. Detailed analysis for the GRB is presented in \citet{Shrestha_2021}. Briefly, the light curve is best fitted by broken power-law with different break point for different filters. For initial IO:O data the best fit model shows a simple power-law decay with decay index of 1.24. For RINGO3 $I$, $R$, and $V$ bands the best fit models have decay index of $0.97 \pm 0.07, 0.98 \pm 0.07,$ and $1.25 \pm 0.1$ before the break time of 6146, 6087, and 5247 seconds. After the break the decay indices are $0.04 \pm 0.17, -0.44 \pm 0.17,$ and $0.01 \pm 0.09$ respectively. This plateau phase is also seen by \citet{Pereyra_2022}. The difference in decay indices during the plateau phase is hard to explain. One possibility presented by \citet{Shrestha_2021} is the difference in electron energy distribution indexes in the blast wave and the reverse shock because the reverse shock is sub-relativistic. Further analysis could be found in \citep{Shrestha_2021}. \citet{Shrestha_2021} calculated the jet break time for this GRB to be 0.53 days with a limited number of data points. Later, \citet{Pereyra_2022} calculated the jet break time using a larger number of data and found it to be between 0.24 to 0.52 days after the trigger. Using these two values along with Swift XRT data, we present 0.52 days as jet break time for this GRB. We present results of bias corrected polarization degree in Fig.~\ref{fig:pol_results_2}. We get polarization detection at 1 sigma level in all three filters at different time period. One hour stacked data shows polarization values of $<3.83\%$, $<5.23\%$, and $<3.7\%$ respectively. The GISP estimates for this GRB are $0.66\%$, $0.75\%$, and $0.8\%$ in $I$, $R$, and $V$ bands, thus a very negligible contribution to our polarization measurements. We also calculated the ISP contribution of the host galaxy for this GRB. The best fit model from \citet{Smith_2021} shows $A_V =0.354$ for the host galaxy, and calculating the ISP using this extinction value for Milky Way-like dust gives $1.0\%$, $1.1\%$, and $1.2\%$ for $I$, $R$, and $V$ bands. However, in \citet{Smith_2021} the best fit model shows small magellanic cloud (SMC)-like dust for the host galaxy and using SMC-like dust \citep{Rodrigues_1992} to estimate host galaxy ISP gives $0.92\%$, $1.18\%$, and $1.45\%$ for $I$, $R$, and $V$ bands respectively. Hence, the detected polarization is intrinsic polarization. Our polarization result matches well with our previous analysis presented in \citet{Shrestha_2021} which showed that the combination of polarimetry and photometry favours scenarios with energy injection from the central engine. In this case, slower magnetized ejecta from the central engine catches up with the decelerating blast wave and causes forward and reverse shocks. This short-lived reverse shock can explain the polarization detection we see near the plateau phase of the light-curve. \section{Discussions} \label{sec:discussions} We have presented polarimetric and photometric analysis of ten GRBs observed by RINGO3. Four of them, GRB 140430A, GRB 141214A, GRB 190114C, and GRB 191016A have been published separately in \citet{kopac_2015,Jordana_2020, Jordana_2021, Shrestha_2021}, in this paper we have carried out a uniform re-reduction and analysis of the whole sample. Our analysis produces similar values of polarization degree and EVPA for these GRBs as previously noted in the published papers. We have presented light curves of GRB 130606A, 130610A, and 130612A which have been fit with a power-law; we have presented the decay indices ($\alpha$) for these GRBs for all the three RINGO3 wave bands except for GRB 130606A for which we only have good SNR for the $I$ band as shown in Fig~\ref{fig:photometry_1}. The XRT light-curve for GRBs in the same time period is also presented. For all sources, other than GRB 140430A, we see the XRT light curve is similar to their optical light curves. Hence the X-ray photons in these events should originate from the forward shocks. For most of the cases, the decay indices for the photometric light curve are less than 1.5 which shows we are observing the forward shock dominated light-curve \citep{Sari_1999, Kobayashi_2000, Zhang_2004, Gomboc_2009, Japelj_2014} in some cases with energy injection (decay indices can be closer to 0.5). However, for the case of GRB 130606A and GRB 180618A (after the break) we observed decay index values greater than 1.5. For GRB 130606A it could be the reverse shock dominated emission we are seeing but we do not have polarization degree calculations due to the instrument not being well calibrated. And for GRB 180618A, the steeper decay could be due to the jet break instead of a reverse shock emission. However, we note that the flattening in the X-ray at later times (alpha = 1.04 at $t>5483s$) can not be explained in a simple jet model. GRB 151215A and GRB 180325A have light curves which could be modelled by a single power-law. Their decay indices are smaller than 1.5; suggesting that the forward shock emission is dominating or suggestive of some energy injection. For these cases, we get polarization upper limits in three wavelengths, which is expected in the case of forward shock dominated emission \citep{Rossi_2004}. Thus, it is possible that most of the observed polarization is contributed by dust in the host galaxy. GRB 180618A is a short GRB with extended emission. The light-curve of this GRB is best fit by a broken power-law with a break at $1370$ seconds. Initially the light-curve showed a shallow decay of $0.48$, $0.53$, and $0.57$ for $I$, $R$, and $V$ band respectively. After the break the decay is sharper with $2.32$, $2.45$, and $2.26$ for $I$, $R$, and $V$ band respectively. There are few points after the break, thus the sharp decay is not well modelled. We do not detect any polarization and get upper limits in polarization. Further discussion on GRB 180618A will be presented in Jordana-Mitjans et al. 2022 (submitted). There are few polarization observations of GRB early afterglows in the literature. \citet{Uehara_2012} detected polarization in the early afterglow of GRB 091208B and \citet{King_2014} reported early-time polarization of GRB 131030A. We investigated the relationship between the polarization signal and various properties of GRBs such as decay index in Fig.~\ref{fig:alpha}, isotropic energy $E_{iso}$, peak energy $E_p$, BAT peak, $T_{90}$, redshift, and extinction of the Milky Way as shown in Fig.~\ref{fig:pol-properties}. In addition, we also checked the relation between polarization and temporal distance of jet break from our observation. As most of the data points are upper limits, we performed survival analysis \citep{Feigelson_1985} using the Python package lifelines \citep{lifeline_2020} to check for any co-relation between polarization signal and different GRB parameters as noted. For all of the cases we get a concordance index close to 0.5, which is the expected results from random predictions, hence, we cannot conclude any relation from our data set. In order to get a better relation between polarization and various properties we need to increase the number of observations of GRB early afterglows. The increased sensitivity of the new polarimeter MOPTOP \citep{Shrestha_2020} on the LT will improve the number of polarization observations of early afterglows in the future. \subsection{Polarization and decay index} In the literature we find most of the high degree of polarization values to be observed for the case of reverse-shock emission \citep{Steele_2009, Mundell_2013}. When the GRB jet interacts with the local ambient medium, there are forward and reverse shock components: the reverse shock is short lived emission and decays faster than the forward shock emission. The decay index for the light-curves where reverse shock is dominating is expected to be greater than 1.5 \citep{Sari_1999, Kobayashi_2000, Zhang_2004, Gomboc_2009, Japelj_2014}. Previously, \citet{steele-ringo2-2017} presented how polarization varies with decay index for 9 different GRBs observed when RINGO2 was online (see Figure. 13 in their paper). Here we add to this data set and study how polarization changes with decay index for $I$, $R$, and $V$ bands in Fig.~\ref{fig:alpha} top, middle, and bottom panels respectively. In the middle panel, we also include results from RINGO2 observations as presented in \citet{steele-ringo2-2017}. All the upper limits presented here are for stacked data shown in Tables~\ref{tab:polarization_nondetection} and ~\ref{tab:polarization_detection} and B 190114C and GRB 191016A detected polarization is also presented. For GRB 191016A, polarization has been detected before the break in light-curve and after the break which is also plotted in Fig.~\ref{fig:alpha}. In the RINGO3 data set we do not find any cases with a decay-index close to 2, thus we suggest that our observed cases are for forward shock dominated emission, which is not highly polarized as shown in the Figure~\ref{fig:alpha}. To get a better relation between polarization and decay-index we need to increase the number of observations of GRB early afterglows, where the reverse shock is dominated which will be possible thanks to new polarimeter MOPTOP \citep{Shrestha_2020} on the LT. \section{Conclusions} \label{sec:conclusions} We have presented photometric and polarimetric results and analysis of ten GRBs out of 67 GRBs triggered by RINGO3 during the time period of 2013 to 2020. For the first three GRBs, instrument polarization was not well constrained so we only present photometry results. For the subsequent seven GRBs we present both photometric and polarimetric results with polarization degree (or upper limits) and EVPA values in the case of detection. Out of these GRBs, polarization was detected for GRB 190114C and GRB 191016A. Further analysis of GRB 190114C showed that detected polarization was contributed by the host galaxy dust. For GRB 191016A the contribution from the host galaxy, assuming SMC-like dust, is negligible and thus the detected polarization is considered to be intrinsic. We created light-curves of all ten GRBs using RINGO3 for $I$, $R$, and $V$ band (where available), IO:O $r'$ band and RATCam $r'$ band data. We performed best fits for these RINGO3 light-curve using either single power-law or broken power-law and report the decay indices for these light curves. We analyzed the relation between decay index and polarization degree, since we mostly observed slowly decaying events we cannot provide clear correlation between decay index and polarization degree. We performed survival analysis to investigate if there is any co-relation between decay index and polarization. From our survival analysis we get concordance index of 0.47 which shows that with our limited data, we do not see any co-relation. Hence, we need more early time observations of GRB events to study the relation between polarization and decay index. We make an intrinsic detection of polarization for GRB 191016A which has a late peak of at least 1000 seconds after the BAT trigger \citep{Smith_2021}. The source is bright enough to perform polarimetery and photometry even 66 minutes after the BAT trigger. The light curve is best fitted by a broken power-law at 5500 seconds after the BAT trigger and we get a shallow decay index close to 1 for all three wavelengths before the break time and it plateaus after the break time. With a high level of detected polarization ($>9\%$) and no jet-break like feature, we deduce that the light-curve has reverse shock emission and the shallow decay is due to the energy injection to the forward shock/ blast wave. The GRB 190114C case shows that even a detection of low polarization degree can help us understand the afterglow emission mechanism. For the GRB 191016A case, polarization and EVPA calculations along with the light-curve allowed us to carry out detailed analysis of the afterglow emission. In the absence of polarization analysis, the GRB 191016A afterglow would have been considered as forward shock emission. However, polarization and EVPA measurements point towards the possibility of reverse shock emission in the afterglow. Thus, polarization observations of GRBs can provide crucial clues to getting detailed information about the event along with photometric and spectroscopic observations. RINGO3 polarimeter have successfully observed various early optical afterglows of GRBs within few hundred seconds of trigger. The results presented in this paper shows the importance of simultaneous multicolour photometry and polarimetry (colours>2) which helps to determine the underlying emission mechanism. New polarimeters with increased sensitivity to probe a larger statistical sample over a significant time period of their evolving emission would open new windows on GRB physics. \section*{Acknowledgements} Operation of LT on the island of La Palma by Liverpool John Moores University at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias is financially supported by the UK Science and Technologies Facilities Council (STFC). Financial support for the development of MOPTOP was provided by the STFC PRD scheme. MS is supported by an STFC consolidated grant number (ST/R000484/1) to LJMU. We acknowledge with thanks the variable star observations from the AAVSO International Database contributed by observers worldwide and used in this research. This research made use of Photutils, an Astropy package for detection and photometry of astronomical sources (\cite{Bradley_2019}). AG acknowledges the financial support from the Slovenian Research Agency (research core funding P1-0031, infrastructure program I0-0033, project grants J1-8136, J1-2460) and networking support by the COST Actions CA16104 GWverse and CA16214 PHAROS. CGM and NJM thank Hiroko and Jim Sherwin for financial support. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. We would like to thank our anonymous referee for thorough and thoughtful comments that have greatly improved the paper. \section*{Data Availability} All the observational data are freely available online in the LT archive at https://telescope.livjm.ac.uk/. \bibliographystyle{mnras} \bibliography{grb_catalog} % \bsp % \label{lastpage}

Title: Metallicity Ceiling in Quasars from Recycled Stellar Winds

Abstract: Optically luminous quasars are metal rich across all redshifts. Surprisingly, there is no significant trend in the broad-line region (BLR) metallicity with different star formation rates (SFR). The average N V/ C IV metallicity does not exceed $9.5~Z_\odot$ and the average Si IV/ C IV metallicity is similarly $\sim10~Z_\odot$. Combined, these observations are indicative of a metallicity ceiling. Here, we study whether a metallicity ceiling can exist in quasar disks due to the evolution of embedded stars. We develop a simple model that starts with gas in a closed box, which is enriched by cycles of stellar evolution until eventually newly formed stars undergo significant mass loss before they reach the supernovae stage and no further enrichment is possible. Using the MESA code, we create a grid over a parameter space of masses ($>8~M_\odot$) and metallicities ($1-10~Z_\odot$), and locate portions of the parameter space where mass loss via winds occurs on a timescale shorter than the lifetime of the stars. We find that for reasonable assumptions about stellar winds, sufficiently massive ($8-22~M_\odot$) and metal-rich ($\sim9~Z_\odot$) stars lose significant mass via winds and fail to evolve to the supernovae stage, thereby failing to enrich and increase the metallicity of their surroundings. This suggests that a metallicity ceiling is the final state of a closed-box system of gas and stars.

https://export.arxiv.org/pdf/2208.04337

\title{Metallicity Ceiling in Quasars from Recycled Stellar Winds} \correspondingauthor{Shelley J. Cheng} \email{shelley.cheng@cfa.harvard.edu} \author[0000-0002-5462-0294]{Shelley J. Cheng} \affiliation{Harvard-Smithsonian Center for Astrophysics, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA} \author{Abraham Loeb} \affiliation{Harvard-Smithsonian Center for Astrophysics, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA} \section{Introduction} \label{sec:intro} Embedded quasars are enshrouded by dust and gas, and may emit enough energy from accretion to clear the surrounding gas \citep[e.g.][]{Rieke:88}. If this occurs, an optically luminous quasar is left behind. These optically luminous quasars usually have a low star formation rate (SFR) since the surrounding gas has already been consumed by star formation with the leftovers blown away by the quasar \citep[e.g.][]{Izumi:18}. Metallicity is often used as an important probe of past star formation, since cycles of star formation progressively enrich the interstellar medium (ISM) with metal-rich material. Therefore, it is interesting that despite the low SFR of optically luminous quasars, quasars are metal rich across all redshifts \citep{simon:10,shemmer:04,warner:04,nagao:06,jiang:07,juarez:09}. Previous studies regarding this puzzle have concluded that broad-line region (BLR) metallicity is independent of the ongoing star formation of the host galaxy, which is responsible for the far-infrared (FIR)-bright quasars. Instead, the BLR metallicity is correlated to enrichment from an earlier stage of star formation \citep{simon:10}. Also, interestingly, there is no significant trend in the BLR metallicity with SFR, showing an average N V/ C IV metallicity not exceeding 9.5 solar metallicities \citep{simon:10}. This metallicity ceiling of $\sim 10$ solar metallicities is also seen in other metallicity probes such as average Si IV/ C IV metallicity (see Section~\ref{sec:ceil}). Incidentally, this corresponds to $\sim 20~\%$ of the mass gas budget composed by heavy elements. Line-driven winds are driven by photon momentum transfer through metal line absorption and provides a source of mass loss for hot stars \citep{kudritzki:00}. Since these winds involve metal line absorption, it is not surprising that wind momenta and thus mass loss rate directly correlates with metallicity \citep{lucy:70,castor:75,vink:21}, with stars of higher metallicities experiencing greater mass loss. In some cases, this mass loss can be so great that stars lose their envelope and never burn heavier metal elements, ending their lives as Wolf-Rayet stars \citep{sander:20}. As cycles of stellar formation take place and the ISM is continually enriched, stars will eventually experience significant mass loss due to winds. In this paper, we create a grid of models using the \texttt{MESA}\footnote{\href{https://docs.mesastar.org/en/release-r21.12.1/index.html}{https://docs.mesastar.org/en/release-r21.12.1/index.html}} code that span a range of masses and metallicities (Section~\ref{sec:methods}) and determine which systems fail to reach the supernova minimum mass due to winds, where the mass loss timescale is shorter than the stellar lifetime (Section~\ref{sec:results}). Additional tests are presented in Section~\ref{sec:extra} that show more details about the dependence of the results on wind model parameters and metallicity. Finally, we offer a discussion of our results before drawing our conclusions (Section~\ref{sec:discussion}). \section{Metallicity Ceiling} \label{sec:ceil} For our purposes of determining systems where enrichment may prevent supernovae, it is necessary to assess whether a metallicity ceiling exists with emission line ratios involving elements primarily produced in the final stages of stellar evolution. A good emission line ratio for this purpose is the Si IV/C IV ratio, since Silicon is generally created in advanced stellar evolutionary stages. For a metallicity ceiling, it is necessary to establish a relatively constant metallicity for systems with different SFR \citep[e.g.][]{simon:10}. In \cite{simon:10}, the metallicity of systems was determined by a conversion from N V/C IV emission line ratios using theoretical relationships of secondary enrichment (CNO nucleosynthesis) processes \citep{Hamann:2002}. In their sample, \cite{simon:10} categorized systems based on FIR luminosity ($L_{\text{FIR}}$), which is a proxy for SFR. FIR emission in quasars is generally thought to be due to dust heated by star formation \citep[see][]{simon:10, Haas:2003, Lutz:2007, Lutz:2008} and remains uncontaminated from any hotter dust heated by quasar emission \citep[e.g.][]{Beelen:2006}. Though the N V/C IV emission line ratio is commonly used in the literature to determine metallicity, the constituent elements of N and C are produced in large amounts throughout the evolution of stars of varying masses (via the CNO cycle, winds etc.). Therefore, following the general methods of \cite{simon:10}, we sought to determine the existence of a Si IV/C IV metallicity ceiling. Using data from \cite{Wang:2022, DeRosa:2014, juarez:09}, we first converted each paper's observed luminosities to FIR luminosity by using relations such as Equation $17$ in \cite{DeRosa:2014}, and Figure $6$ in \cite{Li:2020}. After comparing the resulting FIR luminosities to those in \cite{simon:10}, we categorized \citep[following][]{simon:10} the systems into the FIR categories of ``FIR Faint'',``FIR Intermediate'', and ``FIR Bright''. \begin{table} \begin{centering} \hspace{0.5em} \begin{tabular}{llc} \\ \hline FIR Category & $\log(L_{\text{FIR}}/L_\odot)$ & Z ($Z_\odot$) \\ \hline FIR Faint & $12.4(\pm0.2)$ & $10.5(\pm2.5)$ \\ FIR Intermediate & $12.7(\pm0.2)$ & $9.6(\pm3.0)$\\ FIR Bright & $13.02(\pm0.01)$& $10.3(\pm2.8)$\\ \hline \end{tabular} \end{centering} \caption{\textbf{Si IV/C IV data.} $L_{\text{FIR}}$ is based on Si IV/C IV data from \cite{Wang:2022, DeRosa:2014, juarez:09} and categorized by FIR following \cite{simon:10}. Values presented are averages along with the range around it. } \label{table:ceiling} \end{table} To determine the metallicity of each system, we applied the theoretical emission line ratio relations in Figure $29$ of \cite{nagao:06} to the Si IV/C IV metallicities presented by the authors. We only considered systems where the Si IV/C IV line ratios had measurement uncertainties of less than $0.1$, since the theoretical emission line ratio relations in \cite{nagao:06} are sensitive to small uncertainties in line ratios. For each FIR category, the average Si IV/C IV metallicity was approximately $\sim10~Z_\odot$ (see Table~\ref{table:ceiling}). This shows that a super-solar metallicity of $\sim10~Z_\odot$ (derived from Si IV/C IV line ratio) is independent of FIR luminosity and thus SFR. Figure ~\ref{fig:ceiling} demonstrates that the Si IV/C IV metallicity clusters between $7-13~Z_\odot$ across a wide range of FIR luminosities. The dispersion in metallicity of a factor of $2$ is similar to that in \cite{simon:10}, and exposes the theoretical uncertainties of metallicity determination from line ratios \citep[e.g.][]{Dietrich:2003}. The range in FIR luminosities here covers around $75-80\%$ of the range in \cite{simon:10}. Therefore, using Si IV/C IV line ratio observation data \citep{Wang:2022, DeRosa:2014, juarez:09}, we demonstrated that the metallicity ceiling seen in \cite{simon:10} indeed exists and is not dependent on the specific choice of emission line ratio. A high metallicity of $\sim10~Z_\odot$ that is independent of SFR (i.e. a metallicity ceiling) exists with the Si IV/C IV emission line ratio probe. \section{Methods} \label{sec:methods} Since quasars feature deep gravitational potential wells and have short dynamical times, a closed box model that allows for many generations of star formation is a good approximation. Within this closed box framework, we assume many cycles of star formation without any loss of enriched material. Therefore, to explore the possibility of a metallicity ceiling, it is sufficient to study the evolution of the last cycle of stars. If all the stars that could potentially end their lives with a Type II supernova ($M>8~M_\odot$) lose enough mass due to winds in their lifetime (reaching $M\lesssim8~M_\odot$), then they will no longer be massive enough to end their lives as a Type II supernova and enrich the ISM. We created a grid over a parameter space of varying masses ($8-22~M_\odot$) and metallicities ($3~Z_\odot$, $5~Z_\odot$ and $9~Z_\odot$), and located portions of the parameter space where mass loss via winds occurs on a timescale shorter than the lifetime of stars. These regions represent configurations where stars will fail to enrich the surrounding gas with heavy elements, leading to a metallicity ceiling in the surroundings. To evolve the stars in our parameter space, we used version \texttt{r21.12.1} of the Modules for Experiments in Stellar Astrophysics code (\texttt{MESA}) \citep{paxton:11,paxton:13,paxton:15,paxton:18,paxton:19}. The \texttt{MESA} equation of state (EOS) combines OPAL \citep{Rogers2002}, SCVH \citep{Saumon1995}, FreeEOS \citep{Irwin2004}, HELM \citep{Timmes2000}, PC \citep{Potekhin2010}, and Skye \citep{Jermyn2021} EOSes. Radiative opacities combines OPAL \citep{Iglesias1993, Iglesias1996} and data from \citet{Ferguson2005} and \citet{Poutanen2017}. Electron conduction opacities are from \citet{Cassisi2007}. Nuclear reaction rates are from JINA REACLIB \citep{Cyburt2010}, NACRE \citep{Angulo1999} and \citet{Fuller1985, Oda1994, Langanke2000}. Screening is included via the prescription of \citet{Chugunov2007}. Thermal neutrino loss rates are from \citet{Itoh1996}. We focused on stars with masses $>8~M_\odot$, since only these massive stars can end their evolution in a Type II supernova and increase the metallicity of the surrounding gas (following canonical stellar evolution). The different metallicity choices of $3~Z_\odot$, $5~Z_\odot$ and $9~Z_\odot$ were implemented in \texttt{MESA} by altering the \texttt{initial\_z} and \texttt{Zbase} parameters, with chosen \texttt{Zbase} values of $0.04$, $0.06$, and $0.12$ for the respective \texttt{initial\_z} metallicity choices. We approximate the lifetime of stars as the time until the start of core Helium burning using the analytic expressions for $t_{\text{He}}$ presented in \cite{hurley:2000}. This was chosen since \texttt{MESA} was unable to reliably evolve these extremely high-$Z$ models to advanced burning stages (likely due to the limitation of opacity grids). For winds, we adopted the \texttt{Dutch} wind model, which follows \cite{Glebbeek:2009} in combining the models of \cite{Vink:2001} and \cite{Nugis:2000}. The \texttt{Dutch} wind model is sophisticated and depends on a variety of parameters including metallicity, stellar mass, luminosity, temperature, terminal velocity, and Helium mass fraction. We set the \texttt{Dutch\_scaling\_factor} to $2.0$. This choice was motivated by previous work from \cite{Belczynski:22} and \cite{Agrawal:22}, which indicated that \texttt{MESA}'s \texttt{Dutch} wind model may underestimate winds for massive stars, and which discussed a \texttt{Dutch\_scaling\_factor} of $2-3$ for \texttt{MESA} models \citep[e.g. in][]{Belczynski:22}. The effect of a reduced \texttt{Dutch\_scaling\_factor} is described in Section~\ref{sec:extra_wind}. Additionally, mass loss due to super-Eddington luminosity was included for completeness, with the standard \texttt{super\_eddington\_wind\_Ledd\_factor} of $1.0$. We note that this \texttt{super\_eddington\_wind\_Ledd\_factor} is a division scaling factor (refer to \href{https://docs.mesastar.org/en/release-r21.12.1/reference/controls.html}{MESA documentation} for more details) and the choice of $1.0$ indicates standard calculation of super-Eddington luminosity. The inlists used for the models were adapted from the \texttt{high\_z} models available in the \texttt{MESA} test suites. \newpage \section{Results} \label{sec:results} We determine whether mass loss due to winds is sufficient to significantly impact stellar evolution by plotting the ratio between the wind mass loss timescale, $t_{\text{wind}}$, and the lifetime left until the start of Helium core burning, $t_{\text{to\_He\_core}}$. $t_{\text{to\_He\_core}}$ decreases in value as the star evolves. $t_{\text{wind}}$ is defined as \begin{equation} t_{\text{wind}} = \frac{M}{\dot{M}_{\text{wind}}} \end{equation} with the mass of the star as $M$ and the mass loss rate due to winds as $\dot{M}_{\text{wind}}$. The full result is shown in Figure~\ref{fig:wind}. For clarity, an example evolutionary track is shown in Figure~\ref{fig:evol} with the stages labeled. When this wind-lifetime timescale ratio is less than $1$ ($t_{\text{wind}} < t_{\text{to\_He\_core}}$), the mass loss due to winds is significant enough to potentially prevent supernova (if the star's mass falls below $\sim8~M_\odot$). As seen in Figure~\ref{fig:wind} (a), models with $5~Z_\odot$ require initial stellar masses of $M>13~M_\odot$ for the wind-lifetime timescale ratio to dip below $1$. Conversely, as shown in Figure~\ref{fig:wind} (b), the wind-lifetime timescale ratio of models with $9~Z_\odot$ across all modeled masses ($8-22~M_\odot$) falls below $1$ as the stars transition from Hydrogen core burning to Helium burning. This indicates that mass loss due to winds is significant for systems at very high metallicities. The mass at which the timescale ratio equals $1$ is defined as the crossover mass $M_{\text{cross}}$. If $M_{\text{cross}}\lesssim8~M_\odot$, then the system loses sufficient mass and will no longer end its life in a Type II supernova and would therefore fail to further enrich the ISM. We adopted this notion of a crossover mass due to the limitations of \texttt{MESA} in modelling massive stars at high metallicities at more advanced evolutionary stages. \newpage \newpage We emphasize that the existence of a crossover mass alone is not sufficient to make conclusions about whether the system undergoes a Type II supernova. The mass \textit{value} at which crossover occurs must be below $8~M_\odot$ for the system to no longer undergo a supernova. Thus, the crossover mass value acts as a simple test of whether a star undergoing wind-driven mass loss will lose enough mass to prevent a supernova. Figure~\ref{fig:crossover} shows the crossover mass of all models where the timescale ratio reached below $1$, for all modeled metallicities. We clearly see that $3~Z_\odot$ and $5~Z_\odot$ systems have $M_{\text{cross}}>8~M_\odot$, and Type II supernovae and enrichment can continue to occur. Therefore metallicities below $5~Z_\odot$ are insufficient to produce a metallicity ceiling. However, for $9~Z_\odot$ systems, we see that $M_{\text{cross}}\lesssim8~M_\odot$ for all modeled masses with a decreasing trend towards the higher masses. This indicates that, at very high metallicities of around $9~Z_\odot$, massive stars with $M>8~M_\odot$ fail to retain enough mass to end their lives as Type II supernovae. Therefore, stars will no longer undergo supernova and cannot further enrich the ISM once the ISM metallicity reaches around $9~Z_\odot$. A metallicity ceiling is thus possible once stars begin to form in a $9~Z_\odot$ ISM. Given the intrinsic uncertainties of modelling massive stars at high metallicities with wind-driven mass-loss, this result is consistent with the observed average BLR metallicity of no more than $9.5~Z_\odot$ \citep{simon:10} as well as the Si IV/C IV metallicity ceiling of $\sim10~Z_\odot$ established in Section~\ref{sec:ceil}. \newpage \section{Additional Tests} \label{sec:extra} \subsubsection{Metallicity} \label{sec:extra_metallicity} Figure~\ref{fig:metals_explicit} explicitly shows the effect of different metallicities on the evolution and crossover mass of a $17~M_\odot$ star (with $\texttt{Dutch\_scaling\_factor}=2$). Other than metallicity, all other model parameters were identical between the $3$ models shown in Figure~\ref{fig:metals_explicit}. It is clear that higher $Z$ leads to more efficient winds and greater mass loss before the onset of helium burning in the stars (refer to Figure~\ref{fig:evol} for evolutionary stages). This can be explained by the direct dependence on $Z$ of the \texttt{Dutch} wind model as well as the effect of $Z$ on other stellar parameters relevant in the wind model (refer to Section~\ref{sec:methods} for wind model references and further details). The crossover masses likewise decrease with increasing $Z$. \subsection{Wind Model} \label{sec:extra_wind} As mentioned in Section~\ref{sec:methods}, the scaling factor of the \texttt{Dutch} wind model for massive stars is typically understood to be $\sim2$. However, since the modeling of massive star evolution at high metallicities using \texttt{MESA} is overall uncertain, we conducted a test with a $25\%$ reduction in the \texttt{Dutch} wind scaling factor on a $12~M_\odot$ star at $9~Z_\odot$. When \texttt{Dutch\_scaling\_factor} was set to $2$, the crossover mass of the system was $7.3~M_\odot$ (see the blue line in Figure~\ref{fig:crossover}). Since this crossover mass is below the minimum supernovae mass of $8~M_\odot$, the system would not undergo supernova in its lifetime. However, when this \texttt{Dutch\_scaling\_factor} was reduced by $25\%$, the crossover mass for this $12~M_\odot$ star at $9~Z_\odot$ increased to $8.1~M_\odot$ and therefore the star would undergo supernova. At first glance, this would seem to suggest that our results are sensitive to the choice of \texttt{Dutch\_scaling\_factor}. However, this is not the case, since a small increase in metallicity would decrease the crossover mass and compensate for the effect of a lower wind scaling factor. As shown in Figure~\ref{fig:lesswind}, an increase of metallicity by $0.5~Z_\odot$ to a value of $9.5~Z_\odot$ is sufficient to decrease the crossover mass down to $7.2~M_\odot$ (see green line in Figure~\ref{fig:lesswind}). Since the metallicity ceilings from both \cite{simon:10} and Section~\ref{sec:ceil} have sizable uncertainties, this $0.5~Z_\odot$ change in metallicity is acceptable. Therefore, the results of this work remain qualitatively valid with changes in the \texttt{Dutch} wind scaling factor. \section{Discussion and Conclusions} \label{sec:discussion} We investigated whether stars embedded near quasars (quasar stars) can lead to a metallicity ceiling based on simple closed box model. We assumed that the stars evolve over many cycles and progressively increase the metallicity within the closed system, and explicitly studied the last cycle of stars that may remain massive enough to evolve and enrich the ISM via Type II supernovae. Observationally, evidence of a metallicity ceiling in quasars was presented in \cite{simon:10} who found that the broad-line region metallicity averages at no more than $9.5~Z_\odot$ with no significant dependence on star formation rate (SFR). Additionally, using observation data of Si IV/C IV line ratios \citep{Wang:2022, DeRosa:2014, juarez:09}, we repeated the analysis done in \cite{simon:10} and found metallicity averages of $\sim10~Z_\odot$ across a wide range of SFR (via the far-infrared luminosity proxy probe). We evolved stars of masses $8-22~M_\odot$ and $3~Z_\odot$, $5~Z_\odot$ and $9~Z_\odot$ using \texttt{MESA} to explore the mass-metallicity parameter space required for a metallicity ceiling. We found that, in an existing ISM metallicity of $9~Z_\odot$, stars with masses $8-22~M_\odot$ lose enough mass before core Helium burning to fall below the $\sim8~M_\odot$ Type II supernova threshold, and will therefore lead to a metallicity ceiling. We determined that lower ISM metallicities of $5~Z_\odot$ and $3~Z_\odot$ are insufficient for a metallicity ceiling, and therefore conclude that a minimum ISM metallicity of $9~Z_\odot$ is needed for saturation of the enrichment process. Consequently, a metallicity ceiling may be a general phenomenon in environments that can be approximated as stellar formation and evolution within a closed-box. The idea of a metallicity ceiling in quasars has implications to our understanding of black hole accretion and nuclear star clusters (NSCs). For instance, stellar formation at the outer regions of quasar disk \citep{goodman:04} may be a possible explanation for periods of lower-than-expected black hole accretion rates, since the gas that otherwise would be accreted instead fragments and form stars (and accretion is choked). However, as the metallicity of stars become too high to maintain stability after many cycles, gas may no longer be trapped in stars, and thus black hole accretion rates increase to expected levels (i.e. choked accretion is lifted). If we adopt the metallicity ceiling idea for star formation in the outer regions of the disk, changes in the black hole accretion rates may reveal information about the fraction of inflowing gas that reaches the inner accretion disk versus the fraction that remains in the outer star-forming region. Additionally, there is an almost-constant ratio between the mass of NSCs and the mass of the central black hole \citep{neumayer:20} in observations. If NSCs form with the presence of a metallicity ceiling that lifts choked accretion, this almost-constant ratio of black hole mass and NSC mass may be explained. The metallicity ceiling discussed in this paper is still valid in binary and multiple stellar systems. Though most massive stars are in binary or multiple-star systems \citep[e.g.][]{Sana:2012}, binary interactions would only serve to increase metallicity through mixing, leading to the system reaching a metallicity ceiling quicker. As the accretor star gain mass, it experiences stronger winds, loses mass more rapidly than it gains mass (in \texttt{MESA}'s \texttt{Dutch} wind model), and therefore will more likely fall below $\sim8~M_\odot$. Likewise, the donor star will lose mass and more likely fall below $\sim8~M_\odot$. Therefore, the presence of multiple-star systems would only lead to the system suppressing supernova more effectively. Additionally, we can interpret our result through the lens of stellar masses. When supernovae are suppressed, a mass cut-off past the minimum supernova mass of $\sim8~M_\odot$ is expected. Therefore, the presence of a highly supersolar metallicity ceiling of $\sim 10~Z_\odot$ suggests a mass cut-off above $\sim8~M_\odot$. We emphasize that the metallicity ceiling and suppression of supernova described in this work only applies to environments that can be approximated as a closed-box. For more realistic environments, in-flowing fresh gas will be able to maintain $Z<8~Z_\odot$ and allow supernova, and a metallicity ceiling will not be present. \section*{Acknowledgments} We thank Evan Bauer for advice and comments on \texttt{MESA} wind modeling, and Matteo Cantiello, Charlie Conroy, Lars Hernquist, and Mark Reid for helpful comments on the manuscript. \bibliography{sample631}{} \bibliographystyle{aasjournal}

Title: Hints of a universal width-energy relation for classified fast radio bursts

Abstract: The total available sample of fast radio bursts (FRBs) has been growing steadily in recent years, facilitating the study of FRBs from a statistical point of view. At the same time, the classification of FRBs is currently an imperative issue. We propose that the brightness temperature of bursts can serve as an ideal criterion for classification. In this work, we gather the available data for all localized FRBs and we find a positive relation between the intrinsic pulse width and burst energy, $T_{\rm i}\propto E_\nu^{0.25}$, for three repeating FRBs that is similar to that of our previous work using FRB 20121102A data alone. The critical line $T_{\rm B,cri}$ is found to vary for different FRBs, which may reflect the differences in source properties. This relation can put strong constraints on mainstream radiation mechanisms. It is evident that neither the coherent curvature radiation or synchrotron maser radiation have the capability to reach the high brightness temperature required to reproduce this relation.

https://export.arxiv.org/pdf/2208.13972

\title{Hints of a universal width-energy relation for classified fast radio bursts} \author{Di Xiao\inst{\ref{inst1},\ref{inst3}} \and Zi-Gao Dai \inst{\ref{inst2},\ref{inst3}}} \institute{Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, People's Republic of China \label{inst1} \\ \email{dxiao@pmo.ac.cn} \and Department of Astronomy, University of Science and Technology of China, Hefei 230026, China \label{inst2} \and School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China \label{inst3}} \date{Received XXX / Accepted XXX} \abstract{The total available sample of fast radio bursts (FRBs) has been growing steadily in recent years, facilitating the study of FRBs from a statistical point of view. At the same time, the classification of FRBs is currently an imperative issue. We propose that the brightness temperature of bursts can serve as an ideal criterion for classification. In this work, we gather the available data for all localized FRBs and we find a positive relation between the intrinsic pulse width and burst energy, $T_{\rm i}\propto E_\nu^{0.25}$, for three repeating FRBs that is similar to that of our previous work using FRB 20121102A data alone. The critical line $T_{\rm B,cri}$ is found to vary for different FRBs, which may reflect the differences in source properties. This relation can put strong constraints on mainstream radiation mechanisms. It is evident that neither the coherent curvature radiation or synchrotron maser radiation have the capability to reach the high brightness temperature required to reproduce this relation.} \keywords{Methods: statistical -- Radiation mechanisms: non-thermal} \titlerunning{Width-Energy Relation for Classified FRBs} \authorrunning{D. Xiao and Z. G. Dai} \section{Introduction} \label{sec1} Fast radio bursts (FRBs) are flashing radio signals discovered fifteen years ago \citep{Lorimer2007}. Following a decade of research, the number of known FRBs is now growing rapidly thanks to the development of radio facilities and detection technology. There are hundreds of events recorded at present, as the Canadian Hydrogen Intensity Mapping Experiment (CHIME) Collaboration reported in their first FRB catalog last year \citep{CHIME2021cat}. With such a large sample, it is now possible to study FRB in a statistical context. Since this field is still at an early stage, there are many unknowns around the nature and properties of FRBs, and population study is expected play an important role in the future. Many aspects could be explored statistically, for instance, the (cumulative) distribution of observed quantities can give us some information on FRB luminosity function, event rate, or local environment \citep{Macquart2018a, Macquart2018b}. Currently, individual FRBs appear distinct in many aspects such as their pulse morphology, polarization property, and spectro-temporal behavior, however, they should have something in common if they belong to a single population. These notions point to a crucial open question of population study regarding the classification of FRBs and whether there are two or more FRB populations. The existing classification based on repeating behavior is phenomenological. At the moment, the majority of FRBs in the catalog are one-off events. However, selection bias could play a role here, as it may make a repeater seem non-repeating. If subsequent bursts are narrower or weaker than the original one, they could possibly being missed from an observational point of view \citep{Connor2018, Palaniswamy2018}. For instance, the repetition of FRB 20171019A could not be identified until two very faint bursts were detected, which were found to be $\sim590$ times weaker than the discovery burst \citep{KumarPra2019}. This strongly implies that some repeaters may have been misidentified as non-repeaters. It is now under debate whether all FRBs do indeed exhibit repetitive behaviors \citep{Caleb2019, Lu2020a}. Several works have proposed using a repetition fraction to study this issue, however, it will take some time to verify whether this fraction approaches unity or not with the accumulation of observing time \citep{Ai2021,Gardenier2021}. In this sense, it is not very useful to use repetition in classifying FRBs \citep[e.g.,][]{ZhangKJ2022,ZhongSQ2022} and it is necessary to find a more physical criterion as a basis \citep[e.g.,][]{GuoHY2022}. As we pointed out in a recent paper \citep[][hereafter Paper I]{XiaoD2022a}, brightness temperature could serve this function since it is related to the radiation mechanism directly. Delving further into the spectral luminosity-duration phase space, different radio transients cluster in different regions characterized by their brightness temperature \citep{Nimmo2022}. We have drawn a dividing line for ``classical'' FRBs and applied this classification method to the large sample of FRB 20121102A observed by Five-hundred-meter Aperture Spherical radio Telescope \citep[FAST,][]{LiD2021}. A positive power-law relation has been found for burst width versus fluence (or energy). However, there are doubts as whether this method can be extended to a larger number of FRBs and the universality of this relation has been challenged. This work is aimed at exploring this issue and the paper is organized as follows. We introduce the statistical method and present the results in Sect. \ref{sec2}, whereby hints of a universal relation between pulse width and energy for classified FRBs are found. The implication of this relation on FRB radiation mechanism is explored in Sect. \ref{sec3}. We conclude with our discussion and conclusions in Sect. \ref{sec4}. \section{Method of FRB classification} \label{sec2} The brightness temperature of an FRB is: \bea T_{\rm B}&=&F_\nu d_{\rm A}^2/2\pi k (\nu T)^2\nonumber\\ &=&1.1\times10^{35}{\,\rm K}\,\left(\frac{F_\nu}{\rm Jy}\right)\left(\frac{\nu}{\rm GHz}\right)^{-2}\left(\frac{T}{\rm ms}\right)^{-2}\left(\frac{d_{\rm A}}{\rm Gpc}\right)^2,\nonumber\\ \label{eq:T_B} \ena where $F_\nu$ is the flux density, $\nu$ is the emission frequency, $T$ is the pulse width, and $d_{\rm A}$ is the angular diameter distance \citep{Zhang2020c,XiaoD2021}. Distance information is a prerequisite for obtaining $T_{\rm B}$, therefore we adopted all localized FRBs with measured redshifts in the FRB host database\footnote{\url{http://frbhosts.org}}. Our sample thus consists of 7 repeating and 12 non-repeating FRBs. The data were primarily taken from the FRBSTATs catalog \citep{FRBSTAT}\footnote{\url{https://www.herta-experiment.org/frbstats/ }} and cross checked with original research papers. Details of the data sources are as follows: FRB 20180301A \citep{Price2019}; FRB 20180916B \citep{CHIME2021cat,Chawla2020,Marcote2020,Pilia2020}; FRB 20180924B \citep{Bannister2019}; FRB 20181030A \citep{CHIME2019b}; FRB 20181112A \citep{Prochaska2019}; FRB 20190102C, 20190608B, 20190611B, 20190711A \citep{Macquart2020}; FRB 20190614D \citep{Law2020}; FRB 20191228A, FRB 20200906A \citep{Bhandari2022}; FRB 20200120E \citep{Bhardwaj2021,Majid2021,Kirsten2022}; FRB 20201124A \citep{Farah2021, Herrmann2021, Hilmarsson2021, Marthi2021, KumarPra2022, Lanman2022}. We have included all bursts with reliable $(T, F_\nu, \mathcal{F}_\nu)$ measurements for these 14 localized events \footnote{Four localized one-off FRBs 20190523A \citep{Ravi2019}, 20190714A \citep{Bhandari2019}, 20191001A \citep{Shannon2019} and 20200430A \citep{KumarPra2020b} are not adopted in the sample due to lacking in $T$ or $F_\nu$ value.}, where $\mathcal{F}_\nu$ is the burst fluence. In accordance with Paper I, we used the same data for FRB 20121102A \citep{LiD2021,Hessels2019,Cruces2021}. The FRB distance was obtained from the redshift information in the FRB host database, using the cosmological parameters $H_0=67.7\,\rm km\,s^{-1}\,Mpc^{-1}$, $\Omega_m=0.31$, $\Omega_\Lambda=0.69$ \citep{Planck2016}. Inspired by Paper I, we first investigate whether there is a direct correlation between observed pulse width and fluence for this large sample of bursts. However, since the redshifts of these FRB events are distinct, the cosmological expansion effect should be corrected and we obtain the rest-frame pulse width, $T_{\rm rest}$, after dividing $T$ by the $(1+z)$ factor. Furthermore, the fluence is an observed quantity, hence, here the radiated enengy should be relevant instead. We calculated the specific burst energy as $E_\nu=4\pi d_{\rm L}^2\mathcal{F}_\nu/(1+z)$. Typically, the total integrated energy can be estimated by multiplying $E_\nu$ with the observing bandwidth or the central frequency \citep{ZhangB2018,Aggarwal2021}, however, in our large sample, some bursts are in the extreme narrow band, while some others show continued emission beyond the observing bandwidth. We did not multiply either frequency in order to avoid extra bias. Figure \ref{fig1} shows the overall distribution of these FRBs on the $T_{\rm rest}-E_\nu$ plane. Obviously, this distribution is quite scattered, with no evidence of any correlation between the two quantities. Then we calculated the brightness temperature for each burst in the big sample. For three ``bursty'' FRBs, 20121102A, 20180916B, and 20201124A, we plotted their number distributions for $T_{\rm B}$ in Fig. \ref{fig2}. As in Paper I, we drew a dividing line by $T_{\rm B}$ for ``classical'' high-$T_{\rm B}$ FRBs. There is a sign that the brightness temperature distributions of FRBs 20180916B and 20201124A reach minimums around $10^{33}\,\rm K$, being close to the critical value for FRB 20121102A in Paper I. Therefore, as a trial, we adopted $T_{\rm B, cri}=10^{33}\,\rm K$ for all the FRB events in the sample. We note that the critical value is not necessarily the same for all FRBs, as can be seen in Table \ref{table1}. We go on to investigate whether the pulse width-energy relation discovered in Paper I can be generalized to this large sample. After dropping the bursts with $T_{\rm B}<T_{\rm B,cri}$, we can plot the remaining bursts on the $T_{\rm rest}-E_\nu$ plane in Fig. \ref{fig3}. The blue, green, and magenta symbols represent ``bursty'' FRBs 20121102A, 20180916B, and 20201124A, respectively, while the grey dots are for other localized FRBs. We can clearly see that $T_{\rm rest}$ is in positive relation with $E_\nu$, however, the slope varies for different events. We fit the correlation with simple power-laws and obtain the best-fitting lines for these three FRBs: \bea \log\left(\frac{T}{\rm ms}\right)&=& 0.30\log\left(\frac{E_\nu}{\rm erg\,Hz^{-1}}\right)-8.54, \quad\text{(20121102A),}\nonumber\\ \log\left(\frac{T}{\rm ms}\right)&=& 0.56\log\left(\frac{E_\nu}{\rm erg\,Hz^{-1}}\right)-15.88, \quad\text{(20180916B),}\nonumber\\ \log\left(\frac{T}{\rm ms}\right)&=& 0.18\log\left(\frac{E_\nu}{\rm erg\,Hz^{-1}}\right)-4.98, \quad\text{(20201124A),} \label{eq:plfit} \ena respectively. These power-law relations are similar to that discovered with FAST sample alone in Paper I. Furthermore, we sought to establish a universal relation for all FRBs. If this relation exists, it should be totally intrinsic and we need to remove all the propagation effects. As we know, the observed pulse width is broadened and should be a combined result \citep{Petroff2019}, \bea T=\sqrt{T_{\rm i}^2(1+z)^2+t_{\rm samp}^2+\Delta t_{\rm DM}^2+\tau^2}, \label{eq:Tobs} \ena where the four terms on the right represent intrinsic width, data sampling time, dispersion smearing, and scattering time, respectively. Also, $t_{\rm samp}$ can be found for each survey. Dispersion smearing can be important if bursts are incoherently dispersed. For coherently dispersed bursts, the smearing timescale is usually tens of microseconds and can be neglected. We carefully picked out the incoherently-dispersed bursts for different surveys and calculated the relevant smearing. The scattering timescale $\tau$ is the most complicated step. For a large sample of different FRB events, the intervening medium between source and observer could make a huge difference, therefore, their scattering should be quite varied. Moreover, the motion of scattering medium is highly uncertain, leading to the observational fact that some bursts show clear scattering tails in their pulse profiles, while many others do not. In principle, the scattering of a burst is determined by Monte-Carlo fitting its pulse profile with an assumption of Gaussian intrinsic shape \citep{Ravi2019b, Qiu2020}. However, this is too time-consuming and almost unachievable for our large sample of more than two thousand bursts. Instead, we deal with scattering in a simple way below. The scattering is scaled with the frequency as a power law of $\tau\propto \nu^{\alpha}$, where $\alpha$ depends on the property of the scattering medium \citep{Lohmer2001,Cordes2003,Xu2016}. Furthermore, we assumed that the scattering time for bursts in a same survey does not vary greatly. For instance, we assumed a single scattering time, $\tau_{\rm FAST}$, for the FAST sample of FRB 20121102A at the observing central frequency of 1.25 GHz. For other bursts of this event observed by Arecibo, GBT, and Effelsberg, the observing frequencies are different and the corresponding scattering can be determined by the frequency dependence. For FRBs 20180916B and 20201124A, we took the scattering of CHIME and GMRT observations as reference values, respectively. To avoid introducing too many free paremeters, we only adopted these three events here. Therefore, we introduced four parameters ($\tau_{\rm FAST},\,\tau_{\rm CHIME},\,\tau_{\rm GMRT}$, $\alpha$) to deal with scattering for our sample of bursts. If a universal power-law relation between intrinsic pulse width and energy exists, we can parameterize it as: \bea \log T_{\rm i}=A\log E_\nu+B, \label{eq:Tin} \ena where $A$ and $B$ are two free parameters that are yet to be determined. Furthermore, as we point out above, the critical brightness temperature for classifying FRBs can vary for different FRB events, hence, we have three critical $T_{\rm B}$ values for these three FRBs. Therefore, we have a total of nine free parameters, with which the observed pulse width can be expressed as: \bea T(E_\nu)=T(E_\nu;A,B,\tau_{\rm FAST},\tau_{\rm CHIME},\tau_{\rm GMRT}, T_{\rm B,20121102A}, \nonumber\\T_{\rm B,20180916B},T_{\rm B,20201124A},\alpha) \label{eq:paras} .\ena Next we carried out a Markov-Chain Monte-Carlo (MCMC) fitting of the observed pulse width using the emcee package \citep{Foreman-Mackey2013}. The prior (i.e., the allowed ranges in Table \ref{table1}) is set to a log uniform. We did not consider the intrinsic dispersion mainly because most of the bursts in the big sample are lacking in trustable error bars. Therefore, we defined a likelihood function that returns the sum of the squared difference. In this sense, the MCMC fitting result is nearly equivalent to that of the linear least-squares method. The results are shown in Table \ref{table1} and Fig. \ref{fig4} gives the relevant corner plot. We note that the best-fit values of three critical brightness temperatures always approach their higher ends, since this will eliminate more data points and the correlation is expected to be tighter with fewer bursts. To ensure there are enough data points left (at least five) for each repeater, we set the upper bounds for these three $T_{\rm B}$ artificially; this leads to low uncertainties. Using these best-fitting parameters, we can obtain the intrinsic width of bursts for the three FRB events. We plotted the classified bursts on $T_{\rm i}-E_\nu$ plane in Fig. \ref{fig5} and different symbols represent different events, with the discovery telescope also being marked. The red line corresponds to Eq.(\ref{eq:Tin}) with $A$ and $B$ given in Table \ref{table1}. Obviously there is evidence to assume a universal relation of $T_{\rm i}\propto E_\nu^{0.25}$ for these repeating FRBs. \begin{table} \centering \caption{Best-fit values for the nine parameters.} \label{table1} \begin{tabular}{ccc} \toprule Parameter &Allowed range & Best-fitting value \\ \midrule$A$ &[$0.0,\,1.0$]& $0.25_{-0.021}^{+0.022}$ \\ $B$ &[-10.0, 0.0] & $-7.22_{-0.69}^{+0.66}$ \\ $\tau_{\rm FAST}$ &[0.01, 0.1] & $0.059_{-0.027}^{+0.027}$ \\ $\tau_{\rm CHIME}$ &[2.0, 3.0]& $2.81_{-0.13}^{+0.067}$ \\ $\tau_{\rm GMRT}$ &[$5.0,\,6.0$]& $5.47_{-0.18}^{+0.21}$ \\ $\log T_{\rm B,20121102A}$ &[$32.0,\,36.82$]& $36.80_{-0.045}^{+0.011}$\\ $\log T_{\rm B,20180916B}$ &[$32.0,\,32.52$]& $32.41_{-0.076}^{+0.076}$\\ $\log T_{\rm B,20201124A}$ &[$32.0,\,33.40$]& $33.37_{-0.024}^{+0.019}$\\ $\alpha$ &[$-5.0,\,0.0$]& $-4.02_{-0.66}^{+0.68}$\\ \bottomrule & & \end{tabular}% \end{table} The best-fit values in Table \ref{table1} are physically meaningful. The power-law index $A$ is close to the value we found using only FAST sample of FRB 20121102A \citep{XiaoD2022a}. By adopting the intrinsic width in this work, the positive correlation between $T$ and $E_\nu$ can be verified. The critical lines $T_{\rm B,cri}$ for FRB 20180916B and 20201124A are close to the nominal value of $\sim10^{33}\,\rm K$, implying that the sources for these two FRBs could be normal magnetars with a typical surface magnetic field $B_{\rm s}\sim10^{15}\,\rm G$ and rotational period of $P\sim 1\,\rm s$ \citep[see discussions in][]{XiaoD2022a}. However, the critical line for FRB 20121102A is nearly four orders of magnitude higher, corresponding to that the number of electrons in a single bunch is two orders of magnitude more abundant. Therefore the source of this FRB might be a more energetic magnetar for which the combination of $B_{\rm s}P^{-1}$ is greater by a factor of $\sim100$ \citep[see Eqs.8,9 of][]{XiaoD2022a}. The scattering power-law index is quite close to the expected values of $-4$ (for a thin extended scattering screen) and $-4.4$ (for a Kolmogorov spectrum of scattering medium) \citep{Xu2016}. This implies that the majority of bursts may be scattered with a scattering timescale that is only dependent on frequency. \section{Consequences of the established correlation} \label{sec3} As we pointed out in Paper I, we can classify FRBs based on brightness temperature because there is an upper limit on $T_{\rm B}$ for each radiation mechanism. Generally, the critical lines for coherent curvature radiation and synchrotron maser radiation lie around $T_{\rm B,cri}\sim10^{33}\,\rm K$ \citep{XiaoD2022a}, which is consistent with the best-fit critical $T_{\rm B}$ values of FRB 20180916B and 20201124A (shown in Table \ref{table1}). Below $T_{\rm B,cri}$, the correlation of $T_{\rm i}\propto E_\nu^{0.25}$ is buried since multiple mechanisms can produce those bursts. Beyond $T_{\rm B,cri}$, we can expect one single mechanism to be at work, therefore, the above correlation emerges. This correlation certainly gives us some hints on the FRB radiation mechanism and we can explore this issue in a preliminary fashion:\ basically, for the coherent curvature radiation, the duration of emission is in proportion to the single bunch length, $l_\parallel$, and the number of bunches $N_{\rm b}$ \citep{Kumar2017}: \bea T_{\rm i}\sim N_{\rm b}\frac{\lambda}{c}\propto N_{\rm b}\nu^{-1}, \label{eq:duration} \ena where the bunch length, $l_\parallel$, is on the same order as the emission wavelength $\lambda$. Meanwhile, the specific energy can be approximated as \bea E_\nu\sim L_{\rm curv}T_{\rm i}/\nu, \label{eq:fluence} \ena where the luminosity of coherent curvature radiation is \bea L_{\rm curv}\sim N_{\rm b}N_{\rm e}^2\gamma^2 P_{\rm curv}, \label{eq:Lcurv} \ena assuming $N_{\rm e}$ electrons moving with Lorentz factor $\gamma$ in each bunch. Substituting the single electron emission power $P_{\rm curv}=\frac{2}{3}\frac{\gamma^4e^2c}{\rho^2}$ and replacing the curvature radius $\rho$ with the characteristic frequency, $\nu=\frac{3c\gamma^3}{4\pi\rho}$, we obtain: \bea E_\nu\propto N_{\rm b}N_{\rm e}^2\gamma^2\frac{\gamma^4}{(\gamma^3/\nu)^2}N_{\rm b}\nu^{-1}/\nu\propto N_{\rm b}^2N_{\rm e}^2. \ena The number of electrons in the coherently emitting region is in proportion to its volume: $V_{\rm coh}=\pi l_\parallel \min[l_\perp^2,(\gamma\lambda)^2]$, where $l_\perp$ is the transverse size of the bunch \citep{Kumar2017}. Therefore, we have \bea E_\nu\propto \begin{cases} N_{\rm b}^2l_\perp^4\nu^{-2}, &l_\perp\leq\gamma\lambda,\\ N_{\rm b}^2\gamma^4\nu^{-6}, &l_\perp>\gamma\lambda. \end{cases}\label{eq:curv_corr} \ena Assuming that $l_\perp$ does not vary substantially, then we can expect $T_{\rm i}\propto E_\nu^{0.5}$ for the case $l_\perp\leq\gamma\lambda$. Similarly, $T_{\rm i}\propto E_\nu^{1/6}$ is obtained for the other case if we assume both $N_{\rm b}$ and $\gamma$ have narrow value ranges (although somewhat rigorous). The above two scalings deviate from the correlation we found, implying once again that coherent curvature radiation is unlikely to be the mechanism for high-$T_{\rm B}$ ``classical'' bursts. Currently, the FRB radiation mechanism is largely unknown and it was only recently that \citet{Zhang2022} proposed that coherent inverse Compton scattering (ICS) could produce very high-$T_{\rm B}$ bursts attributed to enhanced single electron emission power. Here, we examine the expected scaling for this mechanism in a similar way. The duration of emission is the same with Eq. \ref{eq:duration}. The luminosity of coherent ICS has the similar form to Eq. \ref{eq:Lcurv}, with $P_{\rm curv}$ substituted by the ICS power, $P_{\rm ICS}=\frac{4}{3}\gamma^2\sigma_{\rm ICS}cU_{\rm ph}$. The scattering cross-section is proportional to $\gamma^{-2}$ and the photon energy density does not vary greatly if we assume the scale of crust oscillation basically remain unchanged \citep{Zhang2022}. Therefore, for this mechanism, we have: \bea E_\nu\propto N_{\rm b}^2N_{\rm e}^2\nu^{-1}\propto\begin{cases} N_{\rm b}^2l_\perp^4\nu^{-3}, &l_\perp\leq\gamma\lambda,\\ N_{\rm b}^2\nu^{-5}, &l_\perp>\gamma\lambda, \end{cases}\label{eq:ICS_corr} \ena with the ICS emission frequency, $\nu\propto\gamma^2$, is substituted. If we assume $N_{\rm b}$ and $l_\perp$ remain unchanged, then we can expect $T_{\rm i}\propto E_\nu^{1/3}$ and $T_{\rm i}\propto E_\nu^{1/5}$ for the above two cases respectively. In reality, the number of bunches and bunch size can not be all the same from burst to burst. However, as long as they have relative small value ranges, the positive correlation between $T_{\rm i}$ and $E_\nu$ is still expected for both the coherent curvature radiation and ICS radiation mechanisms. The power-law index of $\sim$0.25 that we found lies well between $1/5$ and $1/3$, which might be ascribed to the variation of $N_{\rm b}$. Therefore, we consider coherent ICS radiation as a probable mechanism for inducing high-$T_{\rm B}$ ``classical'' FRBs. The scaling of $T_{\rm i}$ with $E_\nu$ is not straight-forward for synchrotron maser radiation. In the scenario described by \citet{Metzger2019}, the maser emission has been reprocessed by the external medium, then the observed peak frequency is higher than the intrinsic maser frequency. Therefore, the observed FRB energy strongly depends on the density of the ambient medium \citep{XiaoD2020}. The calculated burst energy does not vary monotonously with the medium density \citep[see Fig. 6 in][]{Metzger2019}. Also, the intrinsic maser spectrum strongly depends on the upstream magnetization and can be only given by detailed particle-in-cell simulations \citep{Plotnikov2019}. It is for these reasons that we do not expect a simple relation between width and energy for this mechanism. A more careful treatment is needed to find out whether there exists any correlation between other physical quantities for synchrotron maser mechanism, as well as for other proposed radiation mechanisms \citep{Waxman2017,Wadiasingh2019,Lyubarsky2020,Lyutikov2021a} \section{Discussion and conclusions} \label{sec4} In this paper, we present a detail analysis of the burst width-energy correlation using a large sample of FRBs. We found a positive relation of $T_{\rm i}\propto E_\nu^{0.25}$ for three bursty FRBs, confirming the results given in Paper I, based on a single repeating FRB. We note that we used the observed width from the previous paper. In a physical sense, this is not very accurate since our classification criterion $T_{\rm B}$ is an intrinsic property that relates directly to the radiation mechanism. In order to look for any physical correlation, we need to remove all the propagation effects and get the intrinsic width. However, the relation between the observed width and fluence in Paper I is still tenable if the scattering time between individual bursts does not vary greatly at a given observation frequency for FRB 20121102A. Based on the claim that brightness temperature is a more feasible classification criterion for FRBs than\ repetition since it is determined by the radiation mechanism, we investigated two mainstream mechanisms. We found that both coherent curvature radiation and synchrotron maser radiation cannot easily reach high brightness temperature. In addition, we explain the $T_{\rm i}-E_\nu$ relation, finding that coherent ICS by bunches can meet these requirements, but still remain to be verified by further observational evidence. There may be several factors that can influence the power-law index of this correlation. First, the method of obtaining intrinsic pulse width has been handled in a simple way. We assumed that all bursts are scattered, with their scattering time dependent only on the frequency in a power-law form. In fact, many bursts do not show scattering tails in their burst profiles, therefore, the scattering time should be fit from burst to burst in order to be accurate. Second, selection effects and observational bias can play a role. The measured width and fluence are only lower limits if FRB radiation spectrum partially lies in the observing band. Observationally, many bursts show a trend of continued emission beyond the observing band in their waterfall plots, while statistical analyses of narrow-banded bursts fully within the observing band may approach the realistic scenario more closely \citep{Aggarwal2021}. In addition, a ``tip-of-iceberg'' effect might also be at work for weak bursts. The measured pulse width could be shorter than the intrinsic value for a weak burst as its flux falls below the background noise. In this sense, bursts with high signal-to-noise ratios are preferred. Except for the $T_{i}-E_\nu$ relation, other empirical two-parameter correlations of classified FRBs can be searched for in future studies, in a similar to those of supernovae and gamma-ray bursts \citep[e.g.,][]{Amati2002}. We note that the luminosity-duration relation has already been discussed \citep{Hashimoto2019,Hashimoto2020}. These correlations make FRBs potential ``standard candles'' that can be very useful in cosmological studies, as long as the scatter of the correlation can be effectively reduced when using a greater store of data in the future. \begin{acknowledgements} This work is supported by the National Key Research and Development Program of China (Grant No. 2017YFA0402600), the National SKA Program of China (grant No. 2020SKA0120300), and the National Natural Science Foundation of China (Grant No. 11833003, 11903018). DX is also supported by the Natural Science Foundation for the Youth of Jiangsu Province (Grant NO. BK20180324) and Shanghai Sailing Program (Grant No. 19YF1420300). \end{acknowledgements} \bibliographystyle{aa} % \bibliography{FRBlatest} %

Title: Neutron-Mirror-Neutron Oscillation and Neutron Star Cooling

Abstract: It was pointed out in a recent paper that the observed cooling rate of old, cold neutron stars (NS) can provide an upper limit on the transition rate of neutron to mirror neutron ($n-n'$). This limit is so stringent that it would preclude any discovery of $n \to n'$ oscillation in the current round of terrestrial searches for the process. Motivated by this crucially important conclusion, we critically analyze this suggestion and note an interesting new effect present in nearly exact mirror models for $n \to n'$ oscillation, which significantly affect this bound. The new element is the $\beta$ decay $n' \to p'+ e' +\bar{\nu}'_{e}$, which creates a cloud of mirror particles $n'$, $p'$, $e'$ and $D'$ inside the NS core. The $e'$ can "rob" the energy generated by the $n \to n'$ transition via $e-e'$ scattering enabled by the presence of a (minute) milli-charge in mirror particles. This energy is emitted as unobserved mirror photons via fast mirror bremsstrahlung leading to a relaxation of this upper limit.

https://export.arxiv.org/pdf/2208.03771

\title{\large Neutron-Mirror-Neutron Oscillation and Neutron Star Cooling } \author{Itzhak Goldman} \affiliation{Afeka College and Tel Aviv University, 6195001 Tel Aviv, Israel} \author{Rabindra N. Mohapatra} \affiliation{Maryland Center for Fundamental Physics and Department of Physics, University of Maryland, College Park, Maryland 20742, USA} \author{Shmuel Nussinov} \affiliation{Tel Aviv University, 6195001 Tel Aviv, Israel} \author{Yongchao Zhang} \affiliation{School of Physics, Southeast University, Nanjing 211189, China} \date{\today} {\bf Introduction:--} Neutron stars (NSs) and their origin from Supernovae have played an important role in constraining physics beyond the standard model (BSM)~\cite{raffelt}. One class of BSM scenarios which can lead to new effects in NSs are the mirror models, which consist of a mirror sector coexisting with standard model (SM) and which contains a parity symmetric duplicate of the particles and forces of the SM~\cite{mirror}. When the mirror parity is nearly exact, all particles in the two sectors including the neutron and mirror neutron are nearly degenerate. This raises the possibility of neutrons oscillating to mirror neutrons ($n\to n'$)~\cite{bere1} if the sum of ordinary ($B$) and mirror ($B'$) baryon numbers is conserved. This phenomenon has been proposed as a solution to the neutron lifetime anomaly~\cite{bere3}. There are a number of experiments already carried out or planned to search for this {$n \to n'$ oscillation~\cite{expts}}. It is therefore important to know if there are any constraints on the $n-n'$ mixing parameter $\epsilon_{nn'}$ from astrophysical settings. Since NSs are extremely rich in neutrons, they are a perfect laboratory for testing implications of {$n\to n'$ oscillation}. {The transition of an ordinary neutron $n$ to a mirror neutron $n'$ is followed by a migration of the latter towards the NS center under gravity. The hole left will then be filled by another neutron at the Fermi level, and in the process energy is liberated~\cite{bere2}}. If the process is fast enough, it would lead to a fully mixed star. The resulting mass loss of an NS will not only lead to changes in the orbital period of a binary pulsar~\cite{GN}, but also affect the luminosity of {a single NS}~\cite{bere2, posp}. The observational constraints on the rate of the binary periods for several binary pulsars were shown to lead to upper bounds on $\epsilon_{nn'}$ of $10^{-13}$ eV~\cite{GMN}. On the other hand, taking the coldest NS, i.e. PSR J2144 -- 3933~\cite{Guillot:2019ugf}, it was argued in Refs.~\cite{bere2, posp} that one gets $\epsilon_{nn'}\leq 10^{-17}$ eV. Both the bounds are valid for $n-n'$ mass difference up to 15 MeV~\cite{GMN}. This luminosity limit is particularly important, since currently planned terrestrial experiments are sensitive to $\epsilon_{nn'}$ at the level of $10^{-17} $ eV~\cite{expts}. Note that in terrestrial searches for $n-n'$ oscillation, to maintain coherent build-up of the mirror neutron wave function along the neutron beam, and allow for such sensitive measurements, one must require a remarkably precise degeneracy between the neutron and its mirror partner of ${\delta_{nn'}}/m_n \leq 10^{-26}$ with $\delta_{nn'}\equiv{|m_{n'}-m_n|}$. In this letter, we critically analyze the luminosity bound, by following the evolution of the $n'$ generated in {$n \to n'$ transition} a bit longer. We observe that in almost exact mirror models, the mirror neutrons generated inside the NS $\beta$ decay producing mirror fermions $e'$, $p'$ and $\bar{\nu}'_e$ leading eventually to a cloud of $e'$ and deuterons $D'$. These mirror particles then provide a competing cooling channel via the emission of mirror photons $\gamma'$, and reduce the photonic signal claimed in Ref.~\cite{posp} considerably, relaxing the upper bounds on $\epsilon_{nn'}$. For a relatively wide acceptable range of interactions between the ordinary and mirror sectors, mediated by the millicharge of mirror particles~\cite{holdom}, the nucleons and electrons of the visible sector in this core region of the NS can transfer their energy to the mirror particles. The latter then emit this energy via mirror photons $\gamma'$, which do not interact with the ordinary nucleons and electrons and can freely escape. The philosophy of this paper is similar to that in Ref.~\cite{zurabb}. The millicharge on mirror particles arises if $\gamma$ and $\gamma'$ have kinetic mixing. {\bf $n \to n'$ transition:--} Initially, shortly after its birth, a NS is relatively hot and cools down via volume emission of neutrino pairs. At the time of observation, the star may be still cooling off or, if some other sources of energy exist, it may have settled into a thermal steady state, with the thermal energy emitted as electromagnetic radiation often as a black body radiation~\cite{Yakovlev1,Yakovlev2}. Let us apply this scenario to the pulsar PSR J2144 -- 3933. In a steady state, the NS black body luminosity is given by the Stefan-Boltzmann formula ${\cal L}_{\rm NS} = 4\pi \sigma_{\rm SB} R^2 T_s^4$, where $\sigma_{\rm SB}$ is the Stefan-Boltzmann constant, $R$ is the radius of the NS, and its external surface temperature $T_s$ is maintained by the constant internal energy source. If we have observational limits on the luminosity, this implies upper bounds on the rate of internal heat production. It is important to note that there is a $\sim$100 meter thick nuclear ``thermal blanket'' just under the surface~\cite{Yakovlev3}. It causes the internal temperature, which is almost uniform over the NS, to drop dramatically by a factor of $\sim$100 as we move out from the inside across the blanket towards the surface. The estimated upper bound on surface temperature $T_s\sim 42000$ K of the coldest pulsar PSR J2144 -- 3933 would then correspond to the internal temperature $ T_{\rm int} \simeq 0.35$ keV, which would play an important role in obtaining upper bounds on any heat generating mechanism. If the $n \to n'$ processes were the only source of heat supply, then in a steady state the overall $n-n'$ transition rate would be given by ${d{\cal N}_{n'}}/{dt} = {{\cal L}_{\rm NS}}/{\Delta E}$, where $\Delta E\sim 30 $ MeV is the energy initially gained by ordinary nucleons in each $n\to n'$ transition. For PSR J2144 -- 3933, taking $R=11$ km, the rate of generating new mirror neutrons turns out to be: \begin{eqnarray} \label{eqn:dNpdt} \frac{d {\cal N}_{n'}}{dt} \sim 0.45\times 10^{32} \left( \frac{T_s}{42000\ {\rm K}} \right)^4 \; {\rm sec}^{-1} \,. \end{eqnarray} During its long lifetime of 330 million years, about $ {\cal N}_{n'} \sim 10^{48}$ neutrons would have converted into mirror neutrons. This comprises a tiny ${\cal N}_{n'}/{\cal N}_{n} \sim 10^{-9}$ fraction of the total neutron number ${\cal N}_n \sim 2 \times 10^{57}$ in the star, with no change of the gravity fields and of the local density profile of the ordinary NS. Some pulsars have temperatures up to 100 times higher yielding $d{\cal N}_{n'}/d{t} \sim 10^{40}$ sec$^{-1}$, and were also used to bound high $\epsilon_{nn'}$ values~\cite{Yakovlev1, Yakovlev2}. Neighboring neutrons rush into the ``hole'' formed by {$n \to n'$ transition}, and the work done in the process is $\sim 30$ MeV on average and becomes the kinetic energy of these nucleons. The nucleons collide with neighboring neutrons with density $ n_{N} \sim 10^{39}$ cm$^{-3}$, and very quickly settle into the spatially and temporally fixed internal temperature $T_{\rm int}$ ($ \sim 0.35 $ keV). It should be noted that only the $f=kT/E_{F}$ fraction of nucleons and electrons in the high energy tail of the degenerate Fermi-Dirac energy distribution are not Pauli blocked and can be excited (or de-excited) to higher (or lower) empty energy states, reducing the specific heat and the heat content $Q^*$ of the NS by a factor of $f$. It is then given by \begin{eqnarray} \label{eqn:Qstar} Q^*= {\cal N}_n f^2 E_F \,. \end{eqnarray} Upon using $kT \simeq 0.35$ keV for PSR J2144 -- 3933 and $E_F = 30$ MeV, we find $Q^* \sim 10^{52}$ keV, with only the $f \sim 10^{-5}$ fraction of these end point ``active'' electrons partaking in electron scattering or any other dynamic processes, which will play an important role in the following calculations. {\bf $n'$ decay and the $e'-D'$ fluid:--} In connection with the extreme degeneracy of $n$ and $n'$, there are three extra light neutrinos and the mirror photon in exact mirror models. To bring about consistency between three extra neutrinos and an extra photon contributing to the energy density in the Big Bang Nucleosynthesis (BBN) epoch of the universe with the Planck data~\cite{Planck:2018nkj}, we require that there be asymmetric inflation implemented~\cite{BDM}. This will remove the BBN problem by lowering the reheat temperature in the mirror sector by a factor of three, thus diluting the impact of the extra mirror neutrinos and the mirror photon on BBN. % The $\beta$ decay $n'\to p' +e' +\bar{\nu}'_{e}$ of $n'$ proceeds in the same manner as $n$, and will have the the same rate of $\sim (800 \; \rm sec)^{-1}$ as $n$ decay in vacuum, so long as the Fermi energy of the electron is much smaller than the $Q$ value of 0.7 MeV of the $\beta$ decay.\footnote{As in the normal sector, there might be Zeeman effect in the mirror sector which lead to corrections to the $n'$ mass thus affecting the mirror $\beta$ decay. However, such effect is much leas important than the $n-n'$ mass splitting $\Delta M \sim {\cal O} ({\rm MeV})$ under gravity in the star, and can be safely neglected.} % The $p'$s, like the $n'$s, are gravitationally bound to the NS, and local mirror charge neutrality forces {the number densities $n_{e',\,p'} (r)$ of $e'$ and $p'$ to be the same at all $r < R$, i.e. $n_{e'}(r) = n_{p'}(r)$}. The mirror neutrons and mirror protons slow down and form mirror deuterons $D'$, since the process $p'+n'\to D'+\gamma'$ is faster than the inverse beta decay $ e'+p' \to n' +\nu'$. All the $p'$s are ``eaten up'' to form $D'$, and the number of $e'$s that will remain is only half the number of $n'$ produced. The resulting $\gamma'$s escape taking away part of the energy released in $n \to n'$ transition but it does not drain the energy generated by neutrons falling from the Fermi surface, which is drained away via $e-e'$ scattering between the two sectors. Charge neutrality requires that $n_{e'}(r) = n_{D'}(r)$, {with $n_{D'}(r)$ the number density of $D'$}. The new processes we consider are depicted schematically in Fig.~\ref{fig:1}. The $e'$ and $D'$ constitute a fluid that is supported against the gravity of the ordinary NS by degenerate pressure, which is dominated by that of the $e'$. The mass density of the fluid is dominated by the $D'$. The corresponding hydrostatic equation is \begin{eqnarray} \label{eqn:hydrostatic} \frac{\partial}{\partial r}P_{e'}(r) = - \rho(r) g (r) \,, \end{eqnarray} where $\rho(r)= n_{e'} (r)m_{D'}$ is the mass density of the $D'$, and the $e'$ pressure for a given Fermi momentum $p_F$ is \begin{eqnarray} P_{e'} = \frac{8\pi}{3 m_e \hbar^3}\int_0^{p_F}dp \frac{p^4}{\sqrt{1 +({p/m_{e'} c)^2} }} \,. \end{eqnarray} For the small radii considered, the gravitational acceleration can be approximated by \begin{eqnarray} \label{eqn:g(r)} g(r) = \frac{G_N M(r)}{r^2} = \frac{4\pi}{3}G_N \rho_0 r \,, \end{eqnarray} where $G_N$ is the Newtonian constant of gravitation. For $r< 2$ km the density $\rho_0 \simeq 10^{15} \, {\rm gr}\, {\rm cm}^{-3}$ in the center of the NS is almost a constant. The general relativistic modifications of the hydrostatic equation are very small, at the level of $10^{-3}$. We can solve the hydrostatic equation~(\ref{eqn:hydrostatic}) analytically and get \begin{eqnarray} \label{eqn:ne(r)} n_{e'}(r)= \frac{8\pi}{3m_{e'}^3 c^3 \hbar^3} \left[ \left( \sqrt{X_F^2(0)+1} - \frac{r^2}{2r_0^2} \right)^2 - 1 \right]^{3/2} \,, \end{eqnarray} where $r_0 = (3m_{e'} c^2/4\pi G_N \rho_0 m_{D'})^{1/2} \simeq 0.296$ km, and $X_F (0) ={p_F} (0)/{m_{e'} c}$ . Then the number of the $e'$ up to the radius $r$ is \begin{eqnarray} {\cal N}_{e'}(r)= \int_0^{r}4\pi n_{e'}(x )x^2 dx \,. \end{eqnarray} The fluid is confined inside a sphere with radius $R_c$ so that $n_{e'}( R_c) =0$. Once $X_F(0)$ is given, $R_c$, $n_{e'}(r)$ and the total number ${\cal N}_{e'} = {\cal N}_{D'}$ are determined by pure numbers and fundamental constants. This resembles the case of the Chandrasekhar mass. The dimensionless constant $X_F(0)$ is determined by ${\cal N}_{e'}(R_c)= 5\times 10^{47}$ so that ${\cal N}_{e'} = {\cal N}_{D'}$ is half of the total $n'$ generated. We obtain $X_F(0) \simeq 8.9$, implying $E_F(0) \simeq 4$ MeV and $R_c \simeq 1.18$ km. More details can be found in the supplemental material~\cite{supplemental}. {\bf Energy drain from the visible sector to the mirror fluid:--} In deriving the strict bound by using the electromagnetic luminosity ${\cal L} = dW/{dt}$ of the NS, a key point is that the rate of $n\to n'$ transition is constant and independent of any thermal or other variations (except for stopping when the mixed star forms, which happens after many Hubble times for the small values of $\epsilon_{nn'}$ considered). The $\sim 50\%$ of the heat generated which resides in the SM component is then radiated via a fixed black body luminosity~\cite{posp}. Having all the mirror particles segregated in a ``core region'' {(the orange region in Fig.~\ref{fig:1})} comprising $\sim 0.1\%$ of the star volume would have seemed to minimize their ability to intercept and impede ordinary heat emission and photon radiation from the mirror free, large outer region. % This, in turn, would have suggested only minor luminosity reduction and no relaxing of the bounds on $\epsilon_{nn'}$. However, a more careful scrutiny shows that this simplistic argument is misleading. The energy emission from the core will be dominated by the radiation of mirror photons, while the heat is continuously transferred from the normal sector to the mirror sector by scatterings of the normal and mirror electrons in the core region. For sufficiently large millicharge $\epsilon$, the heat emission rate from the mirror particles may overtake the normal emission rate from the external surface by an appreciable factor. The ordinary photonic energy may then account only for a small part of the energy generated inside the star. Furthermore, the cumulative effect of this over most of the star's history will reduce its heat content and push the internal and external surface temperatures to zero, quenching the photonic emission and destroying the steady state model envisioned. Thanks to the mutual mirror electromagnetic scattering of the mirror particles inside the core region and attendant emission of the fast escaping mirror photons, the time required for their cooling off and equilibrating at a temperature $T'$ is very short on typical thermal timescale of $t_{\rm thermal} = W^*/(dW/{dt})$, where $W^*=Q^*$ is the total heat content of the star. Using Eq.~(\ref{eqn:Qstar}) we find $t_{\rm thermal} \sim 3\times10^{15}$ sec, which happens to be close to the age of the star. Since the emission of heat from the mirror sector is much faster than heat transfer between the sectors, any amount of heat in the mirror sector will be emitted rather than go back to the normal sector, which also implies that $T' \leq T$. To avoid detailed discussion at the particle scattering level, we first view the core region as a black body for the {\it mirror} photons with temperature $T'$, as indeed it absorbs any such photon falling on it . The surface of area $4\pi R_c^2$ of the inner ``core region'' serves effectively as an additional boundary, through which the heat in the normal component of the surrounding star can be emitted. The mirror electrons in the core will then radiate their heat content to the outside with the rate of black body luminosity: ${\cal L}' = 4\pi \sigma_{\rm SB} R_c^2 T'^4$. Relative to the internal core region surface $4\pi R_c^2$, the stellar surface is larger -- by roughly a factor of 100. However, the thermal blanket makes the internal temperature about hundred-fold bigger than the surface temperature. Thanks to the possibility that $T^{'4} \geq 10^8 T_s^4$, even if we keep $T' < T$ to make $e\to e'$ energy transfers more than the reverse transfer, we can still, in principle, have the rate of mirror photon emission almost six orders of magnitude bigger than that of the ordinary photons, so long as $R_c\geq 1$ km. However, to verify that this indeed happens, we need to check how many $e-e'$ collisions occur per second (which we denote by $\dot{\cal N}_{\rm col}$) between the ${\cal N}_{e'}( r<R_c)\sim 10^{38}R^3_c \ {\rm cm}^{-3}$ electrons in the core region and the ${\cal N}_{e'} \sim 5 \times 10^{47}$ mirror electrons. If the total energy transferred per second via these collisions from the ordinary to mirror electrons much exceeds the stellar luminosity, namely the inequality \begin{eqnarray} \label{eqn:inequality} \dot {\cal N}_{\rm col} \Delta E \sim \dot{\cal N}_{\rm col} \Delta T \gg {\cal L}_{\rm NS} \sim 2\times 10^{36} ~ {\rm keV}~{\rm sec}^{-1} \end{eqnarray} holds, then the mirror luminosity dominates and the scenario envisioned in deriving the strict upper bounds on $\epsilon_{nn'}$ becomes inoperative. On the other hand, if the inequality in Eq~(\ref{eqn:inequality}) is (strongly) reversed, then the scenario above involving the $\beta$ decay of the mirror neutron will be irrelevant. For the average energy transfer of $\Delta T \sim 0.35$ keV, Eq.~(\ref{eqn:inequality}) becomes $\dot {\cal N}_{\rm col} \geq10^{37}$ sec$^{-1}$. Each electron and also each mirror electron move with the speed of light $c$. Then we can express $\dot{\cal N}_{\rm cal}$ with energy transfer of $\sim$0.35 keV in a manner, which is symmetric between the ordinary and mirror sectors: \begin{eqnarray} \dot{\cal N}_{\rm cal} = \frac{c f f'{\cal N}_e (r< R_c) {\cal N}_{e'} \sigma_{ee'} } {(4\pi/3) R_c^3} \,, \end{eqnarray} where $f' = kT'/{E'_F} \sim 10^{-4}$ is the fractions of the ``active'' mirror electrons. For $R_c=1.2$ km, the condition $\dot{\cal N}_c \gg 10^{37}$ sec$^{-1}$ translates into the following requirement on the $e-e'$ scattering cross section: \begin{eqnarray} \sigma _{ee'} \simeq \epsilon^2 \sigma_{ee} \gg 10^{-50} ~{\rm cm}^{2} \,, \end{eqnarray} where $\sigma _{ee}$ is the standard Rutherford scattering cross section of electrons in the same kinematic configuration. For the formula for $\sigma_{ee'}$, see the supplemental material~\cite{supplemental}. Including only the Feynman diagram for the $t$-channel photon exchange, the cross section $\sigma_{ee'}$ is calculated by having the relativistic $e$ and $e'$ with energies $E_F \simeq 10 E'_F\simeq 35$ MeV collide at random relative direction in the laboratory frame and transferring an energy of $T \sim 0.35$ keV between them. Using a plasmon mass as the cutoff, we estimate this cross section to be $\sigma_{ee'} \simeq {4\pi \alpha^2\epsilon^2}/{E_FT} \simeq 10^{-23} \epsilon^2$ cm$^2$, which leaves us with the rather weak, easy to satisfy requirement \begin{eqnarray} \label{eqn:epsilon} \epsilon^2 \gg 10^{-27} \,. \end{eqnarray} The strongest upper bounds $\epsilon\leq 10^{-12}$~\cite{Redondo} do not apply here, as in mirror models the dark matter (DM) is made of neutral objects such as the $p'-e'$ composite mirror Hydrogen, deuteron or Helium. On the other hand, $\epsilon \leq 10^{-9}$ required for cosmological constraint consistent with BBN limits is more directly applicable here~\cite{angela} whereas a weaker limit of $\epsilon \leq 10^{-7}$ comes from the consistency of asymmetric inflation~\cite{BM}. This still leaves nine orders of magnitude margin for satisfying Eq.~(\ref{eqn:epsilon}). We also note that even though the photonic cooling of ultra-cold NS (UCNS) is not a reliable way to set bound on the $n \to n'$ transition rate for near exact mirror models and slow $n \to n'$ transition, there are situations when it works: e.g. (i) we could have a near exact mirror symmetry but the millicharge of the mirror fermions $\epsilon \leq 10^{-13}$ or, (ii) an asymmetric mirror model with $m_{p'} \geq m_{n'}$ where $n'$ is the DM of the universe, so that $\beta$ decay of mirror neutron is forbidden. It can also work in other dark baryon contexts, such as those suggested in connection with the neutron lifetime anomaly. An advantage of the heating up argument as compared with the orbital period stability method~\cite{GMN} is that: in the former one can use all pulsars, whereas the latter case requires binary pulsars~\cite{GN, GMN}. Unfortunately, unlike the misquote in Ref.~\cite{posp}, the spinning period changes of single pulsars -- which, as part of the ambitious nano-gravity project, determined in many cases with stunning accuracy -- {\it cannot} be used, as it is affected by relatively large and incalculable changes due to magnetic braking etc. This is the reason why binary pulsars were used in Refs.~\cite{GN,GMN}. {\bf Conclusion.--} To summarize our main result is that the photonic luminosities of UCNSs do not necessarily imply robust bounds on $\epsilon_{nn'}$. In particular, they do NOT exclude discovery via terrestrial measurements of the tiny $ \epsilon_{nn'}\sim {\cal O}(10^{-17}$ eV). This happens due to the beta decay of $n'$ following {$n\to n'$ transition} {and the subsequent} deuteron formation. Our main assumption, the existence of a millicharge $\epsilon$, is definitely allowed and possibly even favored within mirror models. In this scenario, under the joint effect of the weight of the mirror deuterons and the Fermi energy of the mirror electrons, the mirror deuterons and electrons form a configuration resembling that of a ``mini white dwarf'' inside the NS. A remarkable feature of this configuration is its universality stemming from, and in analogy with, the features of NSs and actual white dwarfs. Within this structure, heat is transferred relatively fast (on characteristic thermal time scales of the NS) from the heat reservoir in the normal matter of the NS to the mirror sector, and is radiated via mirror photons. \acknowledgments {\bf Acknowledgements.--} The work of R.N.M. is supported by the US National Science Foundation grant no. PHY- 1914631. Y.Z. is supported by the National Natural Science Foundation of China under grant No.\ 12175039, the 2021 Jiangsu Shuangchuang (Mass Innovation and Entrepreneurship) Talent Program No.\ JSSCBS20210144, and the ``Fundamental Research Funds for the Central Universities''. \setcounter{equation}{0} \setcounter{figure}{0} \setcounter{table}{0} \makeatletter \renewcommand{\theequation}{S\arabic{equation}} \renewcommand{\thefigure}{S\arabic{figure}} \renewcommand{\bibnumfmt}[1]{[S#1]} \renewcommand{\citenumfont}[1]{S#1} \begin{widetext} \begin{center} \textbf{\large Supplemental Material} \vspace{10pt} \\ \textbf{ Itzhak Goldman, Rabindra N. Mohapatra, Shmuel Nussinov and Yongchao Zhang} \end{center} \end{widetext} \setcounter{equation}{0} \setcounter{figure}{0} \setcounter{table}{0} \makeatletter \renewcommand{\theequation}{S\arabic{equation}} \renewcommand{\thefigure}{S\arabic{figure}} \renewcommand{\bibnumfmt}[1]{[S#1]} \renewcommand{\citenumfont}[1]{S#1} \section{Solving the hydrostatic equation analytically} From Eqs.~(\ref{eqn:hydrostatic}) to (\ref{eqn:g(r)}), we obtain \begin{eqnarray} && \frac{8\pi}{3 m_{e'} \hbar^3}\frac{p_F^4}{\sqrt{1 +({{p_F}/{m_{e'} c})^2}}} \frac{\partial}{\partial r}p_F(r) \nonumber \\ &=& - \frac{4\pi}{3}G_N \rho_0 m_{D'} n_{e'} (r) r \,. \end{eqnarray} Substituting in % \begin{eqnarray} \label{eqn:ne_r} n_{e'}(r)= \frac{8\pi}{ 3}\left(\frac{p_F(r)}{\hbar}\right)^3 \,, \end{eqnarray} and introducing the dimensionless quantity $X_F (r) \equiv {p_F} (r)/{m_{e'} c}$, we arrive at a very simple equation for $X_F(r)$: \begin{eqnarray} \frac{X_F}{ \sqrt{1+ X_F^2} } \frac{d}{dr}X_F(r) = - \frac{r}{r_0^2} \,. \end{eqnarray} The solution is \begin{eqnarray} \sqrt{ 1+X^2_F(0) } - \sqrt{ 1+X^2_F(r) } = \frac{r^2}{2 r_0^2} \,. \end{eqnarray} Then from Eq.~(\ref{eqn:ne_r}) we can obtain the solution for $n_{e'}(r)$ in Eq.~(\ref{eqn:ne(r)}), which is shown in Fig.~\ref{fig:2} as function of $r$. \section{$e - e'$ scattering cross section} In this section we give the formulae for $e-e'$ scattering cross section. The scattering of $e$ and $e'$ is very similar to the $e^- - e^-$ M\o ller scattering, with the $e-e'$ scattering having only the $t$ channel diagram, since $e$ and $e'$ are not identical particles. The amplitude square for $e-e'$ scattering is given by \begin{eqnarray} \frac14 |{\cal M}|^2 = \frac{2e^2 e^{\prime2} \epsilon^2}{t^2} \Big[ s^2 + u^2 -8m^2(s+u) + 24m^4 \Big] \,, \end{eqnarray} with $m$ the mass of $e$ and $e'$. In the relativistic limit, \begin{eqnarray} \frac14 |{\cal M}|^2 = 32 \pi^2 \alpha^2 \epsilon^2 \left[ 1 - \frac{2}{\sin^2\left(\theta/2\right)} + \frac{2}{\sin^4\left(\theta/2\right)} \right] \,, \end{eqnarray} with $\theta$ the scattering angle in the center-of-mass frame. Then the differential cross section reads \begin{eqnarray} \frac{d\sigma}{d\Omega} &=& \frac{1}{4E_1 E_2} \frac{1}{32\pi^2} \frac14 |{\cal M}|^2 \nonumber \\ &=& \frac{\alpha^2 \epsilon^2}{4E_1 E_2} \left[ 1 - \frac{2}{\sin^2\left(\theta/2\right)} + \frac{2}{\sin^4\left(\theta/2\right)} \right] \,, \end{eqnarray} where $E_{1,\,2}$ are the energies of $e$ and $e'$ in the initial state in the star frame. The presence of $\sin(\theta/2)$ in the denominator implies a mostly forward scattering. The expression is divergent for $\theta=0$ and we put the cutoff at the plasmon mass in the fluid in our estimate. In the dense $e-p$ fluid the (ordinary) photon behave as a plasmon with a mass $m_\gamma$ equal to the plasma frequency $\omega$, i.e. $m_\gamma=\omega$. The $e-e'$ scattering cross section is therefore proportional to $m_\gamma^{-2} = \omega^{-2}$ instead of $(TT')^{-1} \sim T^{-2}$. In normal metals with $n\sim10^{24}/{\rm cm^3}$, the standard expression \begin{eqnarray} \label{eqn:omega} \omega^2 = 4\pi e^2 n_e/m_e \end{eqnarray} yields a plasma frequency corresponding to an energy of $\sim$15 eV. Having here $n_e \sim 10^{38} \; {\rm cm}^{-3}$, i.e. $10^{13}$ times higher, leads to a modification of $e-e'$ cross section, which is $2\times10^{-11}$ times smaller. This dramatically reduces the range of $\epsilon$, for which the basic constraint of ${\cal N}_{\rm col} > 10^{37}$ is satisfied. However, the plasma frequency in Eq.~(\ref{eqn:omega}) is invalid here. As mentioned in the main text, in the highly degenerate electron fluid only a small fraction $f=T/E_F$ of ``active'' electrons can respond to an external oscillating electric field, much like the fact that only the electrons at the top of the conduction band in metals can freely respond. In the NS, the electron density $n_e \sim 4k_F^3/{9\pi}$. With the high Fermi energy $E_F \sim 20\ {\rm MeV} \gg m_e$ of the electrons in the NS, the electrons are relativistic, and we can write the density of the relevant ``active'' electrons as: \begin{equation} n_{e,\,{\rm active}} = f \times \frac{4E_F^3}{9\pi} = \frac{4 E_F^2 T}{9\pi} \,. \end{equation} Also, the electron mass $m_e$ representing the inertial resistance of the system to oscillating is no longer relevant, and should be replaced by its Fermi enegy $E_F$ in Eq.~(\ref{eqn:omega}). Making these two changes in Eq.~(\ref{eqn:omega}), we have \begin{equation} m_{\gamma}^2 = \omega^2 = \frac{16}{9} E_F T \,. \end{equation} The $e-e'$ Rutherford cross section will then be reduced by \begin{equation} \frac{T}{E_F} \sim \frac{0.35 \rm \; keV}{20 \rm \; MeV} \sim 10^{-5} \,, \end{equation} which still allows ${\cal N}_{\rm col} > 10^{37}$ so long as $\epsilon > 10^{-13}$.

Title: New strong lensing modelling of SDSS J2222+2745 enhanced with VLT/MUSE spectroscopy

Abstract: SDSS J2222+2745, at z = 0.489, is one of the few currently known lens clusters with multiple images of a background (z = 2.801) quasar with measured time delays. We combine imaging from the Hubble Space Telescope (HST) with recent Multi Unit Spectroscopic Explorer (MUSE) spectroscopic data to securely identify 34 cluster members and 12 multiple images from 3 background sources. We measure the stellar velocity dispersions of 13 cluster galaxies, enabling an independent estimate of the contribution of the sub-halo mass component to the lens total mass. The projected total mass distribution of the lens cluster is best modelled with a single large-scale mass component, a galaxy-scale component, anchored by the MUSE kinematic information, and an external shear. The best-fit strong lensing model yields a root mean square separation between the model-predicted and observed positions of the multiple images of 0".29. When analysing the impact of systematic uncertainties, stemming from modelling assumptions and used observables, we find that the projected total mass profile, relative weight of the sub-halo mass component, and critical lines are consistent, within the statistical uncertainties. The predicted magnification and time delay values are, instead, more sensitive to the local details of the lens total mass distribution, and vary significantly among lens models that are similarly good at reproducing the observed multiple image positions. Due to its complex morphology, the low number of point-like multiple images, and current model degeneracies, it becomes clear that additional information (from the observed surface brightness distribution of lensed sources and the measured time delays) needs to be included in the modelling of SDSS J2222+2745 for accurate and precise cosmological measurements. The full MUSE secure spectroscopic catalogue presented in this work is made publicly available.

https://export.arxiv.org/pdf/2208.13788

\title{New strong lensing modelling of SDSS J2222+2745 enhanced with VLT/MUSE spectroscopy} \author{ A.~Acebron \inst{\ref{unimi},\ref{inafmilano}} \fnmsep\thanks{E-mail: \href{mailto:ana.acebron@unimi.it}{ana.acebron@unimi.it}} \and C.~Grillo \inst{\ref{unimi},\ref{inafmilano}} \and P.~Bergamini \inst{\ref{unimi}, \ref{inafbo}} \and G.~B.~Caminha \inst{\ref{max_plank}} \and P.~Tozzi \inst{\ref{inaffi}} \and A.~Mercurio \inst{\ref{inafna}} \and P.~Rosati \inst{\ref{unife},\ref{inafbo}} \and G.~Brammer \inst{\ref{dawn},\ref{niels}} \and M.~Meneghetti \inst{\ref{inafbo}} \and M.~Nonino \inst{\ref{inafts}} \and E.~Vanzella \inst{\ref{inafbo}} } \institute{ Dipartimento di Fisica, Universit\`a degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy \label{unimi} \and INAF - IASF Milano, via A. Corti 12, I-20133 Milano, Italy \label{inafmilano} \and INAF -- OAS, Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, I-40129 Bologna, Italy \label{inafbo} \and Max-Planck-Institut f\"ur Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany \label{max_plank} \and INAF – Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125, Firenze, Italy \label{inaffi} \and INAF -- Osservatorio Astronomico di Capodimonte, Via Moiariello 16, I-80131 Napoli, Italy \label{inafna} \and Dipartimento di Fisica e Scienze della Terra, Universit\`a degli Studi di Ferrara, via Saragat 1, I-44122 Ferrara, Italy \label{unife} \and Cosmic Dawn Center (DAWN), Copenhagen, Denmark \label{dawn} \and Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark \label{niels} \and INAF -- Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143, Trieste, Italy \label{inafts} } \date{\today} \abstract {SDSS J2222+2745, at $z = 0.489$, is one of the few currently known lens clusters with multiple images (six) of a background ($z = 2.801$) quasar with measured time delays between two image pairs (with a sub-percent relative error for the longer time delay). Systems of this kind can be exploited as alternative cosmological probes through high-precision and accurate strong lensing models.} {We present recent observations from the Multi Unit Spectroscopic Explorer (MUSE) on the Very Large Telescope (VLT) and new total mass models of the core of the galaxy cluster SDSS J2222+2745.} {We combine archival multi-band, high-resolution imaging from the \textit{Hubble} Space Telescope (HST) with our VLT/MUSE spectroscopic data to securely identify 34 cluster members and 12 multiple images from 3 background sources. We also measure the stellar velocity dispersions of 13 cluster galaxies, down to HST $\rm F160W = 21~mag$, enabling an independent estimate of the contribution of the sub-halo mass component to the lens total mass. By leveraging the new spectroscopic dataset, we build improved strong lensing models. } {The projected total mass distribution of the lens cluster is best modelled with a single large-scale mass component, a galaxy-scale component, anchored by the VLT/MUSE kinematic information, and an external shear component. The best-fit strong lensing model yields a root mean square separation between the model-predicted and observed positions of the multiple images of $0\arcsec.29$. When analysing the impact of systematic uncertainties, stemming from modelling assumptions and used observables, we find that the resulting projected total mass profile, the relative weight of the sub-halo mass component, and the critical lines are consistent, within the statistical uncertainties. The predicted magnification and time-delay values are, instead, more sensitive to the local details of the lens total mass distribution, and vary significantly among lens models that are similarly good at reproducing the observed multiple image positions. In particular, the model-predicted time delays can differ by a factor of up to $\sim 1.5$.} {SDSS J2222+2745 is a promising lens cluster for cosmological applications. However, due to its complex morphology, the relatively low number of secure {`point-like'} multiple images, and current model degeneracies, it becomes clear that additional information (from the observed surface brightness distribution of lensed sources and the measured time delays) needs to be included in the modelling for accurate and precise cosmological measurements. The full VLT/MUSE secure spectroscopic catalogue presented in this work is made publicly available.} \keywords{Gravitational lensing: strong -- Galaxies: clusters: individual: SDSS J2222+2745 -- Cosmology: observations} \section{Introduction} Exquisite multi-wavelength observations of galaxy clusters have recently allowed for a new generation of high-precision and accurate strong lensing (SL) models. High-resolution \textit{Hubble} Space Telescope (HST) imaging, combined with Multi Unit Spectroscopic Explorer \citep[MUSE;][]{Bacon2010, Bacon2014} spectroscopic follow-up, has enabled the identification of a large number of secure multiple images that critically constrain the SL models \citep[see e.g.][]{Richard2015, Kawamata2016, Caminha2019, Jauzac2019, Lagattuta2019, Bergamini2021}. In particular, the inclusion in the models of multiply lensed emission knots within extended sources provides crucial information on the position of the critical lines and therefore on the high-magnification regions \citep{Grillo2016, Bergamini2021, Bergamini2022}. In parallel, MUSE observations allow for the secure identification of large sets of cluster galaxies \citep{Mercurio2021, Lagattuta2022}. Their stellar kinematics can be used effectively in SL models to independently weight the contribution of the sub-halo mass component, thus reducing inherent model degeneracies \citep[][]{Bergamini2019, Bergamini2021, Pignataro2021}. \citet{Granata2022}, exploiting the MUSE kinematic measurements of cluster members, together with their structural parameters determined from HST photometry, adopted the Fundamental Plane relation \citep{Dressler1987} to obtain a more realistic description of the total mass distribution of cluster galaxies in the galaxy cluster Abell S1063. Cluster lenses, such as SDSS J1004+4112 \citep{Inada2003}, SDSS J1029+2623 \citep[][hereafter \citetalias{Acebron2022}]{Oguri2013, Acebron2022}, MACS J1149.5+2223 \citep{Grillo2016}, and SDSS J2222+2745, where a variable background source is multiply imaged, are in addition of particular interest for their cosmological applications \citep[see e.g.][]{Grillo2018, Grillo2020}. They represent emergent, independent probes for measuring the expansion rate \citep{Refsdal1964} and the geometry of the Universe. In the era of precision cosmology, these cluster systems can offer important insights into unknown systematic effects and help clarify current tensions in cosmology \citep[see][for a recent review]{Moresco2022}. SDSS J2222+2745 (SDSS 2222, hereafter), at a redshift of $z=0.489$, was discovered within the Sloan Giant Arcs Survey \citep{Hennawi2008, Bayliss2011, Sharon2020} from the Sloan Digital Sky Survey (SDSS) Data Release 8 \citep[DR8;][]{Aihara2011}. Subsequent photometric and spectroscopic follow-up, with the MOsaic CAmera (MOSCA) and the Andalusia Faint Object Spectrograph and Camera (ALFOSC) at the Nordic Optical Telescope (NOT), revealed that SDSS 2222 is a lens cluster that produces six multiple images of a background quasi-stellar object \citep[quasar or QSO;][]{Dahle2013}. These observations provided a spectroscopic confirmation of the six images of the QSO (labelled A, B, C, D, E, and F) and of the southern arc, which were used to build a first SL model of the galaxy cluster. Following its discovery, SDSS 2222 was photometrically monitored with the NOT to measure the time delays between the QSO multiple images. Observations taken between September 2012 and January 2019 yielded the time-delay values of $\Delta t_{AB}=-42.44^{+1.44}_{-1.36}$ days and $\Delta t_{AC}=696.65^{+2.10}_{-2.00}$ days \citep{Dahle2015, Dyrland2019}. High-resolution imaging obtained with the HST and spectroscopic follow-up data from the Gemini Multi-Object Spectrograph (GMOS) on the Gemini-North Telescope were then used to create an updated SL model in \citet[][hereafter \citetalias{Sharon2017}]{Sharon2017}. Additional multiple image candidates were identified in the HST images, while GMOS spectroscopy refined the redshift estimates for the six quasar images and the southern arc and yielded new redshift measurements for one multiple image and 11 cluster galaxies. In this work we present recent spectroscopic observations of SDSS 2222 obtained with the MUSE integral field spectrograph, mounted on the Very Large Telescope (VLT). We exploit the newly obtained data to build an improved SL model of the cluster. The paper is organised as follows. Section \ref{sec:data} describes the HST imaging and the VLT/MUSE data used to develop the new lens model of SDSS 2222. Section \ref{sec:SLM} presents the selection of the multiple images and the cluster members, as well as the adopted total mass parametrisation of the cluster. We discuss our results in Sect. \ref{sec:results} and draw the main conclusions of this work in Sect. \ref{sec:conclusions}. Throughout this work we adopt a flat $\Lambda$ cold dark matter cosmology with $\Omega_{\rm m} = 0.3$ and $H_0= 70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$. Within this cosmology, a projected distance of $1\arcsec$ corresponds to a physical scale of 6.03 kpc at the cluster redshift of $z=0.489$. All magnitudes are given in the AB system \citep{Oke1974}. The quoted uncertainties correspond to the $68\%$ confidence interval, unless otherwise stated. \section{Data} \label{sec:data} We present here the data exploited to develop a new SL model of SDSS 2222: the high-resolution imaging in Sect. \ref{sec:hst} and the newly obtained VLT/MUSE spectroscopic follow-up in Sect. \ref{sec:vlt}. In Sect. \ref{sec:chandra} we briefly present a new analysis of the {\sl Chandra} X-ray observations, although they are not used in the subsequent analysis. \subsection{HST imaging} \label{sec:hst} We used archival HST multi-colour imaging (GO-13337; P.I.: Sharon), from the {Advanced Camera for Surveys} (ACS) and the {Wide Field Camera 3} (WFC3), taken in August and October 2014. SDSS 2222 was imaged over two orbits in each of the ACS filters (F475W, F606W, and F814W), while the WFC3 imaging was allocated one single orbit (in F110W and F160W). A detailed description of the observations and data reduction process is provided in \citetalias{Sharon2017}. We extracted our HST/F160W photometric catalogue with the public software \texttt{SourceExtractor} \citep{Bertin1996}. \subsection{VLT/MUSE spectroscopy} \label{sec:vlt} SDSS 2222 was targeted with the integral field spectrograph MUSE at the VLT, under programme 0103.A-0554(A) (P.I.: Grillo) between 2019 June 3 and July 10. The lens galaxy cluster was observed with a single pointing ($\sim$~1~arcmin$^2$ field of view; see the footprint in Fig. \ref{Fig:MUSE}), for a total of 11 exposures of 1440 seconds each, resulting in a cumulative exposure time of $4.4$ hours on target. Following the procedure described in \citet{Caminha2017, Caminha2017b, Caminha2019}, we used the standard reduction pipeline \citep[version 2.6;][]{Weilbacher2020} to process the raw MUSE exposures and create the final stacked datacube. In addition, the `{auto-calibration'} method and the Zurich Atmosphere Purge \citep[ZAP;][]{Soto2016} were applied to mitigate slice-to-slice flux variations and improve the sky-subtraction. The data have a spatial pixel size of $0\arcsec.2$ and the value of the full width half maximum (FWHM) measured from the white image is $0\arcsec.8$. Redshifts were then measured by extracting one-dimensional spectra of all sources with HST detections within circular apertures of $0.\arcsec8$ radius. To improve the signal to noise ratio ($S/N$) of the extracted spectra for faint galaxies, we adopted customised apertures based on their estimated morphology from the HST imaging. Following \citet{Balestra2016} and \citet{Caminha2016}, we assigned a quality flag (QF) to each redshift measurement in order to quantify its reliability: `{insecure'} (QF = 1), `{likely'} (QF = 2), `{secure'} (QF = 3), or `{based on a single emission line'} (QF = 9). The full MUSE spectroscopic catalogue contains 118 reliable (i.e. QF $\ge$ 2) redshift measurements, of which 11 are stars, 7 are foreground galaxies ($z < 0.474$), 34 are cluster members ($0.474\le z \le 0.504$; see Sect. \ref{sec:cm}), 59 are background galaxies ($z>0.504$), and 7 are multiple images (see Fig. \ref{Fig:histoMUSE}). The foreground and background objects are identified in Fig. \ref{Fig:MUSE}, and their coordinates and redshifts are listed in Table \ref{tab:fgbg}. The multiple image and cluster member catalogues are presented in Sects. \ref{sec:multim} and \ref{sec:cm}, respectively. \subsection{X-ray data} \label{sec:chandra} SDSS 2222 was observed by the {\sl Chandra} telescope between 2016 April 24 and 29 with the Advanced CCD Imaging Spectrometer (ACIS-S; Observation IDs 17048, 18831, and 18832; P.I.: Pooley), for a total exposure time of 66.06 ks after data reduction. Thanks to the exquisite arcsecond resolution of {\sl Chandra}, the six multiple images of the quasar are clearly detected (unblended) and can therefore be removed. A visual inspection of the image allowed us to identify a diffuse emission with a very low surface brightness, and no significant concentration towards the cluster centre, that is associated with the central position of the brightest cluster galaxy (BCG) identified in the HST images. This is shown in the left panel of Fig. \ref{Fig:Xray}, where the soft-band diffuse emission is clearly seen inside a radius of $33\arcsec$ (or 200 kpc at the cluster redshift; red circle). However, some emission may also be present out to a radius of $70\arcsec$ (or 420 kpc at the cluster redshift; blue circle). Aperture photometry provides $217\pm 22$ and $73 \pm 22$ net counts in the soft and hard band, respectively, within a radius of $33\arcsec$, which maximises the $S/N$ of the diffuse emission in the total band (0.5-7.0 keV). This corresponds to a flux of $1.9\times 10^{-14}$ and $1.7\times 10^{-14}$ ergs~cm$^{-2}$~s$^{-1}$ in the 0.5-2.0 and 2.0-10.0 keV bands, respectively. The presence of an emission beyond $33\arcsec$ is confirmed by the values of $359\pm 38$ and $135 \pm 42$ net counts in the soft and hard band, respectively, measured within a radius of $70\arcsec$. These results critically depend on the background subtraction, which was performed by sampling the source-free regions beyond that radius. We verified that the positive photometry obtained in both bands, within $70\arcsec$, is robust against the uncertainty on the soft and hard background. We also note that masking out the quasar images implies a loss of less than 2\% of the solid angle with respect to a full circle with a radius of $33\arcsec$ and therefore does not affect our measurements. We find that about 65\% of the signal within $33\arcsec$ in the soft band is contributed by the six quasar images. A spectral analysis of the diffuse emission shows that a single-temperature {\tt mekal} model provides a best-fit temperature of $kT=3.5_{-0.8}^{+1.2}$ keV. The luminosity of this diffuse emission, corrected for Galactic absorption, is $\sim\! 2 \times 10^{43}$ ergs~s$^{-1}$ within $30\arcsec$ in the 0.5-2.0 band ($\sim\! 3 \times 10^{43}$ ergs~s$^{-1}$ within $70\arcsec$). We note, however, that, due to the low S/N, the thermal model is statistically equivalent to a power law with slope $\Gamma = 2.2$. We conclude that, with the current X-ray data, it is not possible to test the thermal nature of the diffuse emission and apply the hydrostatic equilibrium to derive a robust X-ray mass profile. Therefore, even though a diffuse intracluster medium component is clearly detected, we do not use the X-ray data to further constrain the virial mass of SDSS 2222 in the following analysis. \begin{table} \caption{Coordinates and spectroscopic redshifts, with the corresponding quality flag, of the multiple image systems.} \label{tab:multim} \centering \begin{tabular}{cccccc} \hline\hline ID & R.A. & Decl & $\rm z_{spec}$\tablefootmark{a} & QF & MUSE ID \\ & deg & deg & & & \\ \hline A & 335.537698 & 27.760538 & 2.801 & 3 & 1679\\ B & 335.536677 & 27.761115 & 2.801 & 3 & 1844\\ C & 335.532955 & 27.760503 & 2.801 & 3 & 1789\\ D & 335.536191 & 27.758897 & 2.801 & 2 & 1885\\ E & 335.535998 & 27.758252 & 2.801 & - & - \\ F & 335.535866 & 27.759723 & 2.801 & - & - \\ 1a & 335.537495 & 27.760796 & 2.801 & - & - \\ 1b & 335.536859 & 27.761150 & 2.801 & - & - \\ 1c & 335.532917 & 27.760339 & 2.801 & - & - \\ 2a & 335.538397 & 27.758230 & 4.560 & 3 & 392 \\ 2b & 335.534815 & 27.757629 & 4.560 & 3 & 435 \\ 2c & 335.533866 & 27.757973 & 4.560 & 3 & 398 \\ 3a & 335.538412 & 27.760379 & - & - & - \\ 3b & 335.535499 & 27.761519 & - & - & - \\ 3c & 335.533613 & 27.760871 & - & - & - \\ 4a & 335.534535 & 27.754882 & 4.505 & 1 & 438 \\ 4b & 335.534083 & 27.754935 & 4.505 & 1 & 2153 \\ 4c & 335.533520 & 27.755171 & 4.505 & 1 & 555555\\ \hline \end{tabular} \tablefoot{ \tablefoottext{a}{Redshift values within a system are averages of the measurements with the highest QFs.} } \end{table} \section{Strong lensing modelling} \label{sec:SLM} We modelled the total mass distribution of SDSS 2222 with the public software \texttt{lenstool}\footnote{\url{https://projets.lam.fr/projects/lenstool}} \citep[see][for a detailed description]{Kneib1996, Jullo2007}. The SL modelling methodology closely follows that presented in \citetalias{Acebron2022} for another galaxy cluster with multiple images of a background QSO, SDSS J1029+2623. We thus refer the reader to that publication for an in-depth description of the adopted methodology. In Sect. \ref{sec:multim} we present the multiple images that are used as constraints in the SL model. Section \ref{sec:cm} describes the spectroscopic and photometric selection and the measurement of the stellar velocity dispersion of the cluster members. The general modelling formalism is discussed in Sect. \ref{sec:method} and our adopted parametrisation of SDSS 2222 in Sect. \ref{sec:MassModel}. \subsection{Multiple image systems} \label{sec:multim} The multiple images identified in SDSS 2222, whose positions and redshifts constitute the constraints for our lensing model, were identified thanks to MOSCA observations at the NOT \citep{Dahle2013} and high-resolution HST imaging (\citetalias{Sharon2017}). In this work we adopt the same notation as that introduced in the literature for the QSO system (\cite{Dahle2013}, \citetalias{Sharon2017}) but choose to use the \texttt{lenstool} convention for the other systems (i.e. a number and a letter identifying, respectively, the family and the image within the family). All systems were analysed using the new MUSE datacube and are discussed below. The background QSO is multiply lensed into six images, labelled A, B, C, D, E, and F (see Fig. \ref{Fig:multimg}). The multiple images were first identified with NOT/MOSCA observations in \citet{Dahle2013}. NOT/ALFOSC follow-up provided a spectroscopic confirmation for images A, B, C, D, and E, while a tentative Ly-$\alpha$ emission line was detected for image F \citep{Dahle2013}. GMOS data spectroscopically confirmed the lensed nature of images A, B, C, and D (\citetalias{Sharon2017}). Based on our MUSE spectra, we refined the spectroscopic redshift value of this system to $z=2.801$ (see Table \ref{tab:multim}). Due to the strong light contamination from bright cluster members, a redshift measurement for images E and F could not be obtained. System 1 is associated with a knot identified in the galaxy that hosts the QSO and is detected three times in the HST images (close to the most magnified images of the QSO, images A, B, and C). Its redshift value was chosen to be equal to that of the QSO. The southern arc, labelled A1 in \citetalias{Sharon2017}, is securely confirmed to be at a redshift of $z=2.295$ (see Table \ref{tab:fgbg}). While the extended image clearly shows several emission regions, it is not possible to robustly identify counter-images (misidentifications would in turn introduce potential biases into the lens model). For that reason, this system is not used here. The extended surface brightness modelling of the arc could provide useful constraints on the total mass distribution of the cluster \citep[see][for instance]{Wang2022}. We defer that analysis to a future work. System 2 is composed of three multiple images, only one of which had a spectroscopic confirmation until now (see system B in \citetalias{Sharon2017}). Thanks to the MUSE observations, all three images are now securely identified at a redshift of $z=4.560$ (see Table \ref{tab:multim}). System 3 (labelled C in \citetalias{Sharon2017}) is formed by three faint multiple images in the northern region of the cluster. No secure redshift estimate was possible based on the GMOS data (\citetalias{Sharon2017}), and no emission line was detected in the MUSE data. For the three images of system 4, the GMOS data provided an uncertain measurement of $z=0.86$, possibly contaminated by a foreground object (see system D in \citetalias{Sharon2017}). The analysis of the MUSE datacube results in a tentative (QF = 1) redshift estimate of $z \sim 4.51$, based on a possible Ly-$\alpha$ emission (at the noise level). The sample of secure multiple image systems spans the redshift range $z=2.801$ to $z=4.560$, with a total of 12 multiple images from three background sources. Systems 3 and 4, which add six additional multiple images, are instead considered photometric, and their redshift values were optimised in the modelling, when included. We measured the coordinates of the luminosity peaks of the multiple images in the HST F606W band and used them as observables in the lens models. We show the measured positions of the 18 multiple images in Fig. \ref{Fig:multimg}, and their properties are summarised in Table \ref{tab:multim}. Finally, Fig. \ref{fig:spec} presents the extracted spectra for the images with a QF $\geq$ 2, together with small cutouts of the HST colour-composite image. \subsection{Selection and internal kinematics of cluster members} \label{sec:cm} The cluster member catalogue was built by considering only secure cluster galaxies. Spectroscopic cluster members were selected mainly based on the analysis of the MUSE datacube. From the MUSE data, we identified 34 member galaxies in the cluster inner region with a reliable redshift estimate (i.e. with a QF $\geq$ 2), 26 of which are newly spectroscopically confirmed. We also included three cluster galaxies outside the MUSE field of view that had previous archival redshift measurements from the SDSS DR9 catalogue (see Fig. \ref{Fig:MUSE} and Table \ref{tab:cm}). The redshift distribution of these galaxies, shown in Fig. \ref{Fig:histoMUSE}, can be fit with a Gaussian distribution with mean and standard deviation values of $\overline{z}=0.489 \pm 0.004$. These spectroscopic cluster members are selected as the galaxies with rest-frame relative velocities within $\Delta V = 3000 ~ \rm km\,s^{-1}$ of the cluster mean velocity, corresponding to the redshift range $z=0.474$-$0.504$. Their properties are summarised in Table \ref{tab:cm}. The redshift and F160W magnitude distributions of the member galaxies are shown in Fig. \ref{Fig:histoMUSE}. In a second step, we took further advantage of the MUSE data to measure the line-of-sight stellar velocity dispersion for the brightest cluster members. Including additional, independent information, such as stellar kinematics, in SL models can help reduce degeneracies between the different cluster mass components. We followed the methodology presented in \citet{Bergamini2019, Bergamini2021}, which has already been applied to several other lens clusters \citep[][]{Pignataro2021, Granata2022, Acebron2022, Bergamini2022}. Spectra of the confirmed cluster members were extracted from the MUSE datacube within $0\arcsec.8$ radius apertures and are consistent with the median FWHM value of the MUSE observations (see Sect. \ref{sec:vlt}). We performed a visual inspection and verified, using a small sub-set of galaxies with angularly close neighbours, that reducing the extraction apertures to radii with values of $0\arcsec.6$ yields consistent measurements, given the uncertainties. We eventually only decreased the value of the aperture radius to $0\arcsec.4$ for the BCG (ID 2010 in Table \ref{tab:cm}) due to the contamination from the light of multiple image F of the QSO and of an angularly close satellite galaxy. We then measured the line-of-sight stellar velocity dispersion values with the public penalised pixel-fitting method software pPXF \citep{Cappellari2004, Cappellari2017}. The cross-correlation between the observed spectra and an extended set of stellar templates was performed in the rest-frame wavelength range [3600, 5000] \r{A}. In order to only exploit reliable measurements in the subsequent lensing analysis, we limited the sample to galaxies with $\langle S/N \rangle >10$ and $\sigma_0 > 80~ \mathrm{km~s^{-1}}$ \citep[as discussed in][]{Bergamini2019, Bergamini2021}. Of the 34 MUSE cluster galaxies, 13 of them, down to $ \mathrm{F160W} \sim 21$, satisfy these criteria. The measured stellar velocity dispersion values of the 13 cluster members were then fitted following the Bayesian approach presented in \citet{Bergamini2019, Bergamini2021}. In this way, we derived the best-fit values of the logarithmic slope, $\alpha$, and of the reference velocity dispersion, $\sigma_{0}^{\star}$, of the $\sigma_0$--F160W relation, which were then adopted as prior information for the scaling relations in our SL model of SDSS 2222 (see Sect. \ref{sec:MassModel}). We show in Fig. \ref{Fig:sigma} the measured stellar velocity dispersion as a function of the F160W magnitude values for the 13 selected cluster galaxies and the resulting best-fit relation. \begin{table} \caption{Figure-of-merit estimators for the spectroscopic and photometric (top) and the spectroscopic-only (bottom) best-fit models.} \label{tab:FoM} \centering \begin{tabular}{|c|ccccc|} \hline Model & rms [\arcsec]& $\nu$ & $\chi^2$ & BIC & AIC \\ \hline \texttt{NoES-Model} & 0.34 & 16 & 32.70 & 71.3 & 39.7 \\ \texttt{ES-Model} & 0.29 & 14 & 24.60 & 69.8 & 35.0 \\ \hline \texttt{NoES-Model-zspec} & 0.21 & 10 & 8.27 & 30.3 & 13.8 \\ \texttt{ES-Model-zspec} & 0.20 & 8 & 7.98 & 36.4 & 17.5 \\ \hline \end{tabular} \tablefoot{ We note that these values correspond to the models before the re-scaling of the multiple image positional uncertainty (see Sect. \ref{sec:MassModel}). } \end{table} \subsection{Modelling methodology} \label{sec:method} The pipeline \texttt{lenstool} allows for parametric mass reconstructions of a lens, where the total mass of the lens can be separated into several components. In this work we consider cluster-scale and galaxy-scale mass components. We chose to model all halos with dual pseudo-isothermal elliptical mass density (dPIE) profiles \citep{Eliasdottir2007}. There are seven free parameters associated with the dPIE profile in \texttt{lenstool}: the coordinates of the centre, $x,~y$; the ellipticity, $e=(a^2-b^2)/(a^2+b^2)$, where $a$ and $b$ are the values of the major and minor semi-axes, respectively; the orientation, $\theta$ (counted anti-clockwise from the $x$-axis); the core and truncation radii, $r_{\rm core}$ and $r_{\rm cut}$; and a velocity dispersion, $\sigma_{\rm LT}$, which is linked to the central velocity dispersion of the dPIE profile according to the relation $\sigma_0 = \sqrt{3/2}~ \sigma_{\rm LT}$. Cluster-scale halos are described by non-truncated, elliptical dPIE profiles. Galaxy-scale halos, which are associated with the cluster galaxies, are instead modelled with singular, circular dPIE profiles. To significantly reduce the number of free parameters, the following two scaling relations \citep{Jullo2007} between the galaxy total mass and its corresponding luminosity (as measured in the HST F160W band; see Sect. \ref{sec:cm}) are typically adopted: \begin{equation} \label{eqsigma} \sigma_0=\sigma_0^{\star} \left(\frac{L}{L^{\star}}\right)^{\alpha} \text{and}~ r_{\rm cut}=r_{\rm cut}^{\star}\left(\frac{L}{L^{\star}}\right)^{\beta}, \end{equation} where $L^{\star}$ represents the reference luminosity value of a galaxy at the cluster redshift, which we associated with the BCG (with a magnitude value in the HST F160W band of $17.56$). The two free parameters in the lens model are then $\sigma_0^{\star}$ and $r_{\rm cut}^{\star}$. The (fixed) parameters $\alpha$ and $\beta$ correspond to the slopes of the $\sigma_0$ and $r_{\rm cut}$ scaling relations, respectively. We adopted values of $\alpha$ and $\beta$ so that the galaxy total mass-to-light ratio varies with the luminosity as $M^{\rm tot}_i L_i^{-1} \propto L ^{\gamma}_i$, with $\gamma=0.2$ (i.e. a relation that is compatible with the so-called tilt of the Fundamental Plane; \citealt{Faber1987, Bender1992, Ciotti1996, Bernardi2003, Grillo2010}). To introduce further flexibility into the lens modelling of galaxy clusters, an external shear component can be considered. The external shear is described by two additional free parameters: its magnitude, $\gamma_{\rm ext}$, and orientation, $\phi_{\rm ext}$. The best-fitting values of the model parameters that describe the total mass distribution of the lens, $\mathbf{p}$, were obtained by minimising on the image plane the distance between the observed , $\boldsymbol{\theta}^{\rm obs}$, and model-predicted, $\boldsymbol{\theta}^{\rm pred}$, positions of the multiple images through a $\chi^2$ function (see e.g. Eq. 4 in \citetalias[]{Acebron2022}). More generally, we quote the root mean square (rms) value of the difference between the observed and model-predicted positions of the multiple images to quantify the goodness of our models: \begin{equation} {\rm rms}=\sqrt{\frac{1}{N_{\rm tot}}\sum_{i=1}^{N_{\rm tot}}\left|\boldsymbol{\theta}^{\rm obs}_{i}-{\boldsymbol{\theta}}^{\rm pred}_{i}(\mathbf{p})\right|^2}, \end{equation} where $N_{\rm tot}$ is the total number of images. In addition, other statistical estimators can be used to compare lens models with different numbers of observables or free parameters, especially for models with a low number of observable quantities, as in the case of SDSS 2222. Thus, we also consider in the following the Bayesian information criterion \citep[BIC;][]{Schwarz1978} and the Akaike information criterion \citep[AIC;][]{Akaike1974}. The BIC is defined as ${\rm BIC}=-2\ln({\mathcal{L}}) + k \times \ln(n) $, and the value of the AIC is obtained as ${\rm AIC}=2k -2\ln({\mathcal{L}})$, where $\mathcal{L}$ is the maximum value of the likelihood, $k$ is the number of free parameters, and $n$ is the number of observational constraints. \begin{table*} \caption{Median values and 68\% (statistical) confidence level intervals of the lens mass parameters for the SL models discussed in this work.} \label{tab:massparams} \centering \renewcommand{\arraystretch}{1.2} \begin{tabular}{c|ccccccccccc} \hline Model & Component & $x$ & $y$ & $e$ & $\theta$ & $\sigma_0$ & r$_{\rm cut}$ & r$_{\rm core}$ & $\gamma_{\rm ext}$ & $\phi$\\ & & [\arcsec] & [\arcsec] & & [$\deg$] & [km\ s$^{-1}$] & [\arcsec] & [\arcsec] & & [$\deg$]\\ \hline \texttt{NoES-Model} & DM & $0.0^{+0.2}_{-0.2}$ & $1.8^{+0.1}_{-0.4}$ & $0.58^{+0.06}_{-0.06}$ & $94.9^{+1.0}_{-1.0}$ & $705^{+36}_{-25}$& [2000] & $5.6^{+1.0}_{-0.8}$& -- & --\\ & $L^{\star}$ Galaxy & -- & -- & -- & -- & $345^{+15}_{-13}$ & $3.4^{+0.7}_{-0.5}$ & [0] & -- & --\\ \hline \texttt{ES-Model} & DM & $0.0^{+0.2}_{-0.2}$ & $1.7^{+0.1}_{-0.4}$ & $0.46^{+0.11}_{-0.13}$ & $103.2^{+8.9}_{-3.6}$ & $620^{+56}_{-79}$& [2000] & $4.2^{+1.5}_{-1.2}$& -- & --\\ & Ext. Shear & -- & -- & -- & -- & -- & -- & -- & $0.12^{+0.05}_{-0.08}$ & $169.2^{+2.7}_{-26.4}$\\ & $L^{\star}$ Galaxy & -- & -- & -- & -- & $345^{+16}_{-11}$ & $4.1^{+1.5}_{-0.7}$ & [0] & -- & --\\ \hline \texttt{NoES-Model-zspec} & DM & $0.0^{+0.1}_{-0.1}$ & $1.1^{+0.6}_{-0.5}$ & $0.35^{+0.09}_{-0.08}$ & $92.9^{+0.9}_{-0.8}$ & $805^{+68}_{-39}$& [2000] & $6.3^{+1.6}_{-1.4}$& -- & --\\ & $L^{\star}$ Galaxy & -- & -- & -- & -- & $288^{+24}_{-23}$ & $2.1^{+0.9}_{-0.8}$ & [0] & -- & --\\ \hline \texttt{ES-Model-zspec} & DM & $-0.2^{+0.2}_{-0.3}$ & $1.3^{+0.1}_{-0.9}$ & $0.47^{+0.15}_{-0.15}$ & $98.6^{+3.9}_{-5.2}$ & $756^{+67}_{-56}$& [2000] & $5.4^{+1.7}_{-1.4}$& -- & --\\ & Ext. Shear & -- & -- & -- & -- & -- & -- & -- & $0.07^{+0.06}_{-0.04}$ & $134.1^{+17.1}_{-38.9}$\\ & $L^{\star}$ Galaxy & -- & -- & -- & -- & $326^{+36}_{-31}$ & $2.1^{+0.8}_{-0.7}$ & [0] & -- & --\\ \hline \end{tabular} \tablefoot{ Parameter values in square brackets are kept fixed in the optimisation. We note that the $L^{\star}$ corresponds to the reference luminosity adopted to be that of the BCG with a magnitude value in the HST F160W band of $17.56$. } \end{table*} \subsection{Mass parametrisation of SDSS 2222} \label{sec:MassModel} In this work we consider two sets of cluster total mass parametrisations and two sets of observables in order to find which model best fits the data and to investigate the impact of systematic uncertainties on our results. \noindent \textit{Mass models:} The cluster-scale mass component of SDSS 2222 was modelled with a single, non-truncated dPIE profile. The values of the ellipticity, position angle, core radius, and velocity dispersion associated with this halo were optimised within large flat priors, while the truncation radius was fixed to a very large value. The coordinate values of the halo were free to vary within $2\arcsec$ of those of the BCG. For the galaxy-scale mass component, we considered the 37 spectroscopically confirmed cluster members (see Sect. \ref{sec:cm} and Table \ref{tab:cm}), which were all modelled within the adopted scaling relations (see Eq. \ref{eqsigma}). As presented in Sect. \ref{sec:cm}, we made use of the stellar velocity dispersion measurements obtained for a sub-set of cluster galaxies. In particular, in the lens model optimisations we used the normalisation and slope values of the best-fit $\sigma_0$--F160W relation. We thus chose a Gaussian distribution centred on the measured value of $321~\rm km~s^{-1}$ and with a standard deviation value of $41 \mathrm{\, km \, s^{-1}}$ as a prior for the value of the normalisation $\sigma_{0}^{\star}$. Instead, since no independent information is available, a large flat prior for the $r_{\rm cut}^{\star}$ value was considered (between 0 and 250 kpc). The value of the slope, $\alpha$, was fixed to the fitted one of 0.295 (based on the stellar kinematic measurements; see Sect. \ref{sec:cm}), and the value of $\beta$ was inferred such that $M^{\rm tot}_i/L_i \propto L ^{0.2}_i$ following \begin{equation} \label{eqbeta} \beta=\gamma-2\alpha+1. \end{equation} This cluster total mass parametrisation is referred to as \texttt{NoES-Model}. In the second mass model, labelled \texttt{ES-Model}, we included an additional external shear component, as described in Sect. \ref{sec:method}. \texttt{NoES-Model} has a total of 8 free parameters related to the mass parametrisation, while \texttt{ES-Model} has 10.\\ \noindent \textit{Observables:} The full catalogue of multiple images is presented in Sect. \ref{sec:multim} and Table \ref{tab:multim}. In total, we have five multiply imaged sources, three of which are spectroscopically confirmed. To investigate potential biases coming from the inclusion of less secure information \citep[see e.g.][]{Grillo2015, Johnson2016}, we considered the two lens total mass models presented above, including either the full spectroscopic and photometric multiple image sample or the spectroscopic-only one (labelled \texttt{NoES-Model-zspec} and \texttt{ES-Model-zspec}). In the former case, the redshift values of systems 3 and 4 were optimised in the models, assuming large flat priors. These catalogues thus provide 36 or 24 observational constraints, and the lens models depend on 12 or 6 free parameters for the positions and redshifts of the corresponding sources, respectively.\\ Initially, a positional uncertainty of $0\arcsec.25$ was adopted for all images and models. For each final model run, the multiple image positional uncertainty was then re-scaled in order to obtain a minimum $\chi^2$ value comparable with the number of degrees of freedom ($\nu$) such that $\chi^2/\nu\sim1$. In particular, we re-scaled the positional uncertainty for both \texttt{NoES-Model} and \texttt{ES-Model} from $0\arcsec.25$ to $0\arcsec.35$, while for \texttt{NoES-Model-zspec} and \texttt{ES-Model-zspec} we used the value of $\chi^2/\nu\sim1$ without any re-scaling of the positional uncertainty (see Table \ref{tab:FoM}). \section{Results and discussion} \label{sec:results} In this section we present the results from the model optimisations and statistical analyses described in Sect. \ref{sec:MassModel} and discuss the impact on our results of systematic uncertainties related to the modelling assumptions and choices of observables. We refer to two cluster total mass parametrisations (with or without an external shear component) and two sets of observational constraints (a full or spectroscopic-only multiple image sample). The values of the statistical estimators introduced in Sect.~\ref{sec:method} for these four SL models are summarised in Table \ref{tab:FoM}, and the resulting median values of the model free parameters, and the associated $1\sigma$ statistical uncertainties, are given in Table \ref{tab:massparams}. Considering the values of all figures of merit, \texttt{ES-Model} was favoured when considering the full sample of multiple images, despite having two additional free parameters related to the external shear field. However, when considering only the spectroscopic sample of lensed sources, the inclusion of an external shear component is not supported, given the larger BIC and AIC values with respect to the model without. The resulting rms value for the best-fit model with the full sample of images (i.e. \texttt{ES-Model}) is $0\arcsec.29$. Figure \ref{Fig:multimg} shows the resulting critical lines at the redshift of the QSO system ($z=2.801$) for the four models. The main difference arises between models that consider different sets of observables (see the solid and dashed coloured lines), in the southern region of the cluster core, and they can be attributed to the degeneracy between the cluster-scale and galaxy-scale mass components. As previously mentioned, including as additional information the surface brightness distribution of the southern arc, A1, should help robustly reconstruct the total mass distribution of the lens and distinguish between different total mass parameterisations. The obtained values of the model mass parameters are generally consistent, within the statistical uncertainties. In addition, Fig. \ref{Fig:sigma} highlights the importance of an independent determination and implementation in our lens models of the $\sigma_0$--F160W sub-halo scaling relation in reducing inherent model degeneracies between the cluster- and galaxy-scale mass components. When contrasted with the previous SL analysis of SDSS 2222 by \citetalias{Sharon2017}, the values of the mass parameters related to the large-scale dark matter (DM) halo are consistent, within the errors. However, a comparison of the sub-halo contribution to the total mass is not straightforward: in \citetalias{Sharon2017}, the value of the normalisation of the velocity dispersion--luminosity scaling relation was fixed to a given arbitrary value and the value of the reference luminosity, $L^{\star}$, was not provided. We find that when an external shear component is included, the \texttt{ES-Model}(\texttt{-zspec}) models require a non-negligible amplitude (i.e. $\gamma_{\rm ext}=0.12(0.07))$, albeit with a large statistical uncertainty. Similar values have been quoted in the literature \citep[see e.g.][]{Caminha2016, Caminha2019, Lagattuta2019}. The inclusion of this additional (non-localised) term introduces further flexibility into the lens modelling and can account for several non-modelled lensing effects, such as the cluster environment, line-of-sight mass structures, or asymmetries in the total mass distribution \citep[see e.g.][and \citetalias{Acebron2022}, for a discussion]{Lagattuta2019}. We also find consistent model predictions for the redshift value of system 3: $z_{\rm S3}=3.3_{-0.2}^{+0.4}$ for \texttt{NoES-Model} and $z_{\rm S3}=3.2_{-0.2}^{+0.3}$ for \texttt{ES-Model}. These values are also in agreement with the tentative GMOS spectroscopic measurement and the lens model predictions from \citetalias{Sharon2017}. On the other hand, the model-predicted redshift value for system 4 is completely unconstrained in both lens models. System 4 being the farthest identified system from the cluster centre, but angularly close to the third brightest cluster member ($\sim3\arcsec$ from galaxy ID 2064 in Table \ref{tab:cm}), this can be explained by model degeneracies between the system redshift and the sub-halo mass component. We show in Fig. \ref{fig:massprof} the cumulative projected total mass profile of the cluster as a function of the distance from the BCG centre for the four lens models (top panel) and the ratio of the profiles from the different models with respect to \texttt{NoES-Model} (bottom panel). Within the approximate average distance of the multiple images from the BCG centre, we measure a precise projected total mass value of $M(<40~\rm kpc)= 1.02_{-0.02}^{+0.02} \times 10^{13}~ M_{\odot}$ for \texttt{ES-Model}. Given the errors, this mass estimate is perfectly in agreement for the four models, with a statistical plus systematic relative uncertainty (determined from the different mass parametrisations) of only approximately $3\%$. The projected total mass of the cluster enclosed within a circle with a radius equal to the average multiple image distance from the lens centre (indicated as vertical lines in Fig. \ref{fig:massprof}) is thus robustly measured. Well beyond the region where the observables are available, model extrapolations and systematic uncertainties become more relevant, but all model projected total mass profiles remain consistent, within $2\sigma$. SDSS 2222 is a relatively low-mass lens cluster, with an X-ray luminosity in the group regime, a complex morphology, and a fairly small number of securely identified {`point-like'} multiple image constraints. We note that current SL models suffer from important degeneracies between several model parameters. In particular, we verified that the $y$ coordinate value of the main DM halo is correlated to those of its velocity dispersion and ellipticity, and the mass contribution of the sub-halo component to the lens total mass. We thus further investigated the impact of relaxing or tightening the prior on the $y$ coordinate value of the main DM halo (from $1\arcsec$ to $10\arcsec$ from the BCG). We find that models for which a larger $y$ value is favoured (i.e. with a cluster DM halo more distant from the luminous counterpart) require a rounder (i.e. less elliptical) DM halo, with a more massive sub-halo component. Despite these degeneracies (as illustrated in Fig. \ref{fig:posterior} for \texttt{ES-Model} and \texttt{NoES-Model-zspec}), the different lens models yield projected total mass profiles, normalisations of the $\sigma_0$--F160W sub-halo scaling relation, and critical lines (e.g. at the redshift of the QSO) that are consistent within the statistical uncertainties, showing that these global quantities and features are robustly reconstructed. However, we remark that the model-predicted magnification and time-delay values are much more sensitive to the modelling assumptions and considered constraints, and thus to systematic uncertainties. Magnifications and time delays of the multiple images of a source do indeed depend on the local details, close to the multiple-image-observed positions, of a lens total mass density distribution. This is illustrated in Fig. \ref{fig:magzqso}, where we compare the magnification and Fermat potential (or the arrival time-delay surface) contours for a source at the position and redshift of the QSO, obtained from the two best-fit \texttt{ES-Model} and \texttt{NoES-Model-zspec} models. Clearly, they provide significantly different predictions. In particular, we find that the model-predicted time delays between the QSO multiple image pairs A and B and A and C, which have measured time delays \citep{Dahle2015, Dyrland2019}, can vary by factors of approximately 1.4 and 1.5, respectively. When different priors on the $y$ coordinate value of the DM cluster-scale halo are adopted, we obtain similar variations for the model-predicted time delays, showing how sensitive these quantities are to the modelling details. Therefore, we stress that if one is interested in magnifications and time delays, it is not possible to distinguish among the different predictions of disparate models that can reproduce the observed positions of a set of {point-like} multiple images similarly well. We thus report similar findings to those presented in \citetalias{Acebron2022}, in the sense that cluster SL models with comparable rms and statistical estimator values, referring to only the positions of multiple images, can provide contrasting values of predicted time delays, which would then result in considerably different estimates of the Hubble constant. These results further highlight the need to include the measured values and associated uncertainties of the time delays between the multiple images of a variable source as observational constraints for cluster lensing cosmological applications \citep[see e.g.][]{Grillo2018, Grillo2020}. \section{Conclusions} \label{sec:conclusions} SDSS J2222+2745, at $z=0.489$, is one of the few currently known lens clusters that host multiple images (six) of a background quasar ($z=2.801$) with measured time delays between two image pairs \citep{Dahle2015, Dyrland2019}. In order to exploit this particular lens cluster as a robust cosmological probe, a high-precision and accurate lens model is crucial. As a first step towards future cosmological analyses, in this work we have presented and used recent VLT/MUSE spectroscopic observations of the SDSS J2222+2745 core. In combination with archival multi-band HST imaging, we have been able to securely identify more than 30 cluster members and confirm the redshift values of three multiple image families. Based on these new data, we have built a refined SL model of the galaxy cluster SDSS J2222+2745 with the parametric software \texttt{lenstool} \citep{Jullo2007}. Our findings can be summarised as follows: \begin{enumerate} \item Thanks to the MUSE spectra, we have provided an updated redshift value for the lensed quasar. In addition, we have measured secure redshifts for all images in system 2, for which only one of the three images had previously been spectroscopically confirmed (\citetalias{Sharon2017}). Our SL models have included five families (three of them spectroscopic), with a total of 18 multiple images (see Table \ref{tab:multim} and Figs. \ref{Fig:multimg} and \ref{fig:spec}). \\ \item We have spectroscopically identified 34 cluster galaxies with a QF $\geq$ 2 (see Table \ref{tab:cm} and Fig. \ref{Fig:MUSE}), a pure sample that was then included in the SL modelling. By further exploiting the new MUSE data, we have reliably measured the stellar velocity dispersion values of a sub-sample of 13 cluster members, down to HST F160W $\sim$ 21. These values have been used to independently calibrate the scaling relations of the sub-halo population in the lens modelling (see Fig. \ref{Fig:sigma}). \\ \item We have performed a parametric SL modelling of SDSS J2222+2745. The observed positions of the multiple images are reproduced with an rms value of $0\arcsec.29$ for the best-fit \texttt{ES-Model} model (see Table~\ref{tab:FoM}). Within the average projected distance of the multiple images from the BCG centre, we have measured a precise cluster total mass value of $M(<40~\rm kpc)= 1.02 \times 10^{13}~ M_{\odot}$, with a statistical and systematic relative uncertainty of approximately $3\%$. \\ \item We have investigated the impact of systematic uncertainties arising from different mass parametrisations, sets of observational constraints, and model degeneracies. We find that the lens projected total mass profile, the normalisation of the $\sigma_0$--F160W sub-halo scaling relation, and the critical lines (at the redshift of the quasar) are all robustly measured. \\ \item We have verified that the model-predicted magnification and time-delay values are very sensitive to the reconstructed local distribution of the lens total mass density, and thus to systematic uncertainties. In particular, we have shown, from similar lens total mass models and sets of observational constraints, that the predicted time delays between the quasar multiple image pairs A and B and A and C (both with measured time delays) can vary by more than 30\%. This finding further stresses the fact that time-delay predictions obtained from different {point-like} lens models that do not include the measured time delays as constraints should not be used to estimate the value of the Hubble constant. \end{enumerate} SDSS J2222+2745 is a complex cluster with a relatively small number of secure multiply imaged sources. Going beyond {point-like} SL models, by including the surface brightness distribution of the multiple images of the quasar host galaxy and of the southern tangential arc, will alleviate current model degeneracies and result in a significant improvement of the modelling robustness. This will in turn provide more accurate and precise lens total mass reconstructions and predictions for the multiple image systems. As already remarked in \citetalias[]{Acebron2022}, to fully exploit cluster-scale lensing systems such as SDSS J2222+2745 as cosmological probes, it is key that all available lensing observables, including the measured values and errors of the time delays \citep{Grillo2018, Grillo2020}, be incorporated in the analyses. The full MUSE spectroscopic catalogue of SDSS J2222+2745 presented in this work is made publicly available\footnote{The catalogue is available at \url{www.fe.infn.it/astro/lensing}.}. \begin{acknowledgements} We kindly thank the anonymous referee for the useful comments that have helped improving the manuscript. This work is based in large part on data collected at ESO VLT (prog. ID 0103.A-0554(A)) and NASA \textit{HST}. AA has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101024195 — ROSEAU. We acknowledge financial support through grants PRIN-MIUR 2017WSCC32 and 2020SKSTHZ. MN aknowledges INAF Mainstream 1.05.01.86.20. GBC thanks the Max Planck Society for support through the Max Planck Research Group for S. H. Suyu and the academic support from the German Centre for Cosmological Lensing. \end{acknowledgements} \bibliographystyle{aa} \bibliography{bibliography} \begin{appendix} \section{Multiple images in SDSS 2222} We present in Fig. \ref{fig:spec} the MUSE spectra of the multiply imaged background sources securely identified in SDSS 2222, together with the colour-composite HST counterparts. \clearpage \section{Cluster members included in the SL modelling of SDSS 2222} We present in Table \ref{tab:cm} the coordinates, spectroscopic redshifts, and the associated QF for the cluster galaxies that are considered in the SL model. \begin{table}[h] \caption{Catalogue of the spectroscopic cluster members included in the SL modelling of SDSS 2222. } \label{tab:cm} \centering \begin{tabular}{ccccc} \hline\hline ID & R.A. & Decl & $\rm z_{spec}$ & QF\\ & deg & deg & & \\ \hline 2010 & 335.535746 & 27.759866 & 0.4900 & 3\\ 1974\tablefootmark{a} & 335.546683 & 27.758010 & 0.4833 & -\\ 2064 & 335.534337 & 27.755753 & 0.4917 & 3\\ 1682 & 335.536303 & 27.759224 & 0.4926 & 3\\ 1351\tablefootmark{a} & 335.535539 & 27.772641 & 0.4833 & -\\ 1879 & 335.535913 & 27.758409 & 0.4910 & 3\\ 2514 & 335.527403 & 27.751236 & 0.4899 & 3\\ 1614 & 335.532449 & 27.762223 & 0.4873 & 3\\ 254\tablefootmark{a} & 335.536865 & 27.744699 & 0.4945 & -\\ 3675 & 335.533015 & 27.765765 & 0.4899 & 3\\ 2392 & 335.533646 & 27.753336 & 0.4861 & 3\\ 1859 & 335.534695 & 27.759614 & 0.4977 & 3\\ 2277 & 335.535656 & 27.755222 & 0.4895 & 3\\ 1782 & 335.533871 & 27.760461 & 0.4890 & 3\\ 2175 & 335.538561 & 27.755814 & 0.4857 & 3\\ 2285 & 335.535389 & 27.754893 & 0.4871 & 3\\ 1676 & 335.535248 & 27.759868 & 0.4904 & 3\\ 2459 & 335.534338 & 27.752224 & 0.4839 & 3\\ 2129 & 335.531878 & 27.756635 & 0.4834 & 3\\ 2051 & 335.537469 & 27.757450 & 0.4927 & 3\\ 1959 & 335.536778 & 27.761426 & 0.4872 & 3\\ 3603 & 335.533086 & 27.767028 & 0.4882 & 3\\ 2523 & 335.532022 & 27.751610 & 0.4914 & 3\\ 3201 & 335.540378 & 27.765003 & 0.4906 & 3\\ 2021 & 335.529668 & 27.757910 & 0.4928 & 3\\ 1791 & 335.528265 & 27.760577 & 0.4935 & 3\\ 1835 & 335.530911 & 27.759113 & 0.4894 & 3\\ 232 & 335.540238 & 27.766579 & 0.4829 & 3\\ 328 & 335.530760 & 27.761539 & 0.4889 & 3\\ 1631 & 335.537592 & 27.762424 & 0.4800 & 3\\ 1993 & 335.537994 & 27.758516 & 0.4862 & 3\\ 423 & 335.528033 & 27.756864 & 0.4967 & 3\\ 294 & 335.533328 & 27.762795 & 0.4906 & 3\\ 2022 & 335.539222 & 27.757964 & 0.4984 & 3\\ 2407 & 335.529515 & 27.753380 & 0.4898 & 2\\ 265 & 335.543839 & 27.764291 & 0.4896 & 2\\ 1801 & 335.538663 & 27.760608 & 0.4803 & 3\\ \hline \end{tabular} \tablefoot{ \tablefoottext{a}{Redshift values from the SDSS DR9 catalogue.}\\ } \end{table} \clearpage \section{Foreground and background galaxies identified in SDSS 2222} We present in Table \ref{tab:fgbg} the coordinates and spectroscopic redshifts, with the associated QF, of foreground ($z<0.474$) and background ($z>0.504$) galaxies with respect to the cluster redshift. \begin{table}[h] \caption{Catalogue of the foreground (top) and background (bottom) galaxies with a secure spectroscopic measurement based on the MUSE data. } \label{tab:fgbg} \centering \begin{tabular}{cccccc} \hline\hline ID & R.A. & Decl & $\rm z_{spec}$ & QF\\ & deg & deg & & \\ \hline \vspace{-0.2cm}\\ 2494 & 335.527883 & 27.752258 & 0.1666 & 3\\ 2060 & 335.535654 & 27.756604 & 0.1731 & 3\\ 1836 & 335.531605 & 27.759843 & 0.2844 & 3\\ 1566 & 335.533400 & 27.763878 & 0.2846 & 3\\ 2161 & 335.531964 & 27.759812 & 0.2852 & 3\\ 2219 & 335.545558 & 27.755106 & 0.4657 & 3\\ 2218 & 335.545291 & 27.754859 & 0.4659 & 3\\ \hline 2520 & 335.546073 & 27.751610 & 0.603 & 3\\ 1787 & 335.539611 & 27.760557 & 0.624 & 3\\ 341 & 335.539542 & 27.760744 & 0.624 & 3\\ 2455 & 335.529934 & 27.752699 & 0.666 & 3\\ 2167 & 335.532660 & 27.756038 & 0.686 & 3\\ 555556 & 335.534018 & 27.755143 & 0.686 & 3\\ 1579 & 335.533869 & 27.762893 & 0.687 & 3\\ 1565 & 335.540698 & 27.763180 & 0.718 & 3\\ 482 & 335.542613 & 27.754734 & 0.754 & 3\\ 2196 & 335.541723 & 27.755398 & 0.832 & 3\\ 1673 & 335.535118 & 27.757902 & 0.832 & 3\\ 1733 & 335.545648 & 27.761332 & 0.834 & 2\\ 3021 & 335.526249 & 27.751953 & 0.853 & 3\\ 2217 & 335.544759 & 27.754999 & 0.908 & 3\\ 462 & 335.544976 & 27.755142 & 0.909 & 3\\ 3464 & 335.527385 & 27.767330 & 0.910 & 3\\ 297 & 335.528105 & 27.762531 & 0.981 & 3\\ 2598 & 335.535789 & 27.750941 & 1.024 & 2\\ 2382 & 335.531780 & 27.753580 & 1.070 & 3\\ 480 & 335.528616 & 27.754721 & 1.071 & 3\\ 2401 & 335.545781 & 27.752973 & 1.173 & 2\\ 1696 & 335.534418 & 27.761781 & 1.200 & 3\\ 1753 & 335.536373 & 27.761947 & 1.201 & 3\\ 55555151 & 335.530780 & 27.765559 & 1.269 & 9\\ 2300 & 335.539642 & 27.754210 & 1.272 & 3\\ 513 & 335.532463 & 27.753210 & 1.273 & 3\\ \hline \end{tabular} \end{table} \begin{table}[h] \vspace{4.05cm} \centering \begin{tabular}{cccccc} \hline\hline ID & R.A. & Decl & $\rm z_{spec}$ & QF \\ & deg & deg & & \\ \hline \vspace{-0.2cm}\\ 1461 & 335.538489 & 27.764606 & 1.295 & 3\\ 387 & 335.530384 & 27.758833 & 1.314 & 2\\ 1927 & 335.544663 & 27.759414 & 1.510 & 2\\ 1731 & 335.528905 & 27.761430 & 1.536 & 2\\ 3557 & 335.533823 & 27.767280 & 2.039 & 2\\ 2261 & 335.534169 & 27.757256 & 2.176 & 2\\ 2150 & 335.536000 & 27.756825 & 2.295 & 3\\ 2061 & 335.536858 & 27.756943 & 2.295 & 3\\ 2134 & 335.535668 & 27.756852 & 2.296 & 3\\ 555559 & 335.529544 & 27.757045 & 3.060 & 3\\ 533 & 335.542052 & 27.752321 & 3.131 & 3\\ 999991 & 335.532275 & 27.757333 & 3.277 & 9\\ 55555141 & 335.536801 & 27.765589 & 3.280 & 9\\ 528 & 335.529304 & 27.752584 & 3.454 & 9\\ 5555514 & 335.527340 & 27.761884 & 3.495 & 3\\ 5555513 & 335.527428 & 27.762041 & 3.495 & 9\\ 999993 & 335.545637 & 27.765787 & 3.680 & 3\\ 2091 & 335.539297 & 27.757099 & 3.869 & 9\\ 3642 & 335.538114 & 27.766197 & 3.870 & 9\\ 55555154 & 335.529808 & 27.764538 & 3.909 & 3\\ 555557 & 335.537407 & 27.754958 & 4.531 & 3\\ 555552 & 335.538607 & 27.754902 & 4.538 & 3\\ 55555115 & 335.533177 & 27.755314 & 4.538 & 2\\ 479 & 335.538535 & 27.754866 & 4.540 & 3\\ 478 & 335.538719 & 27.754932 & 4.542 & 2\\ 488 & 335.536164 & 27.754437 & 4.548 & 2\\ 308 & 335.527973 & 27.762278 & 4.559 & 2\\ 472 & 335.542556 & 27.755287 & 4.562 & 3\\ 55555113 & 335.529161 & 27.754992 & 4.710 & 3\\ 5555514 & 335.539716 & 27.766822 & 4.711 & 3\\ 999994 & 335.543832 & 27.751557 & 4.725 & 9\\ 55555155 & 335.534001 & 27.763796 & 5.185 & 9\\ 999995 & 335.527068 & 27.753547 & 6.274 & 9\\ \hline \end{tabular} \end{table} \clearpage \section{Posterior probability distributions} We present in Fig. \ref{fig:posterior} the posterior probability distributions of the parameter values of the cluster-scale and sub-halo mass components for \texttt{ES-Model} and \texttt{NoES-Model-zspec}. \end{appendix}

Title: Properties of dense molecular gas along the major axis of M 82

Abstract: Dense gas is important for galaxy evolution and star formation. Optically-thin dense-gas tracers, such as isotopologues of HCN, HCO+, etc., are very helpful to diagnose excitation conditions of dense molecular gas. However, previous studies of optically-thin dense-gas tracers were mostly focusing on average properties of galaxies as a whole, due to limited sensitivity and angular resolution. M82, a nearby prototype starburst galaxy, offers a unique case for spatially-resolved studies with single-dish telescopes. With the IRAM 30-m telescope, we observed the J = 1 - 0 transition of H13CN, HC15N, H13CO+, HN13C, H15NC, and SiO J = 2 - 1, HC3N J= 10 - 9, H2CO J = 2 - 1 toward five positions along the major axis of M82. The intensity ratios of I(HCN)/I(H13CN) and I(HCO+)/I(H13CO+) show a significant spatial variation along the major axis, with lower values in the central region than those on the disk, indicating higher optical depths in the central region. The optical depths of HCO+ lines are found to be systematically higher than those of HCN lines at all positions. Futhermore, we find that the 14N/15N ratios have an increasing gradient from the center to the outer disk.

https://export.arxiv.org/pdf/2208.01259

command. \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \def \um {$\rm \mu m$} \def \HCNto {HCN\,$J$=2$\rightarrow$1 } \def \HCOto {HCO$^+$\,$J$=2$\rightarrow$1 } \def \COoz {CO\,$J$=1$\rightarrow$0 } \def \kms {$\rm km\,s^{-1}$} \def \Lsun {$\rm L_\odot$} \def\purple#1 {{\textcolor{purple}{#1}}\ } \def\red#1 {\textcolor{red}{#1}} \def\new#1 {{\bf #1 }} \def\blue#1 {{\textcolor{blue}{#1}}\ } \def \Ctw {$^{12}$C} \def \Cth {$^{13}$C} \graphicspath{{./}{figures/}} \accepted{31 May, 2022} \shorttitle{Dense gas in M$\,$82} \shortauthors{Li et al.} \begin{document} \title{ Properties of dense molecular gas along the major axis of M$\,$82}% \email{lifei.astro@gmail.com;zzhang@nju.edu.cn} \author[0000-0003-3353-7106]{Fei Li} \affiliation{School of Astronomy and Space Science, Nanjing University, Nanjing 210093, People’s Republic of China\\} \affiliation{Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, People’s Republic of China\\} \affiliation{Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People’s Republic of China\\} \author[0000-0002-7299-2876]{Zhi-Yu Zhang} \affiliation{School of Astronomy and Space Science, Nanjing University, Nanjing 210093, People’s Republic of China\\} \affiliation{Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, People’s Republic of China\\} \author[0000-0001-6106-1171]{Junzhi Wang} \affiliation{Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People’s Republic of China\\} \affiliation{School of Physical Science and Technology, Guangxi University, Nanning 530004, People’s Republic of China \\} \author[0000-0002-2581-9114]{Feng Gao} \affiliation{Hamburger Sternwarte, Universitaet Hamburg, Gojenbergsweg 112, 21029, Hamburg, Germany\\} \author[0000-0003-1275-5251]{Shanghuo Li} \affiliation{Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea\\} \author[0000-0002-0818-1745]{Jing Zhou} \affiliation{School of Astronomy and Space Science, Nanjing University, Nanjing 210093, People’s Republic of China\\} \affiliation{Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, People’s Republic of China\\} \author[0000-0003-0477-1606]{Yichen Sun} \affiliation{School of Astronomy and Space Science, Nanjing University, Nanjing 210093, People’s Republic of China\\} \affiliation{Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, People’s Republic of China\\} \author[0000-0002-7532-1496]{Ziyi Guo} \affiliation{School of Astronomy and Space Science, Nanjing University, Nanjing 210093, People’s Republic of China\\} \affiliation{Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, People’s Republic of China\\} \author[0000-0001-6016-5550]{Shu Liu} \affiliation{CAS Key Laboratory of FAST, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China} \keywords{Starburst galaxies; Interstellar medium; Star formation}% \section{Introduction} % Observations show that star-formation activities are closely connected with dense molecular gas both in the Milky Way and in other galaxies. Heiderman2010,Lada2010,Lada2012. The dense molecular gas are directly involved into star formation activities, and it can be probed with molecular lines with critical densities ($n_{\rm crit}$) greater than $10^4$ $\rm{cm}^{-3}$, such as multi-$J$ transitions of HCN and HCO$^+$ \citep{Lada1992,Kohno1999,Kennicutt2012}. With observations of HCN $J = 1\rightarrow0$ toward 65 galaxies, \cite{Gao2004a} found a tight linear correlation between the luminosities of HCN $J$ = 1$\rightarrow0$ and total infrared emission. However, most dense gas tracers are normally optically thick both in Galactic giant molecular clouds (GMCs) and in external galaxies \citep{Wang2014,Meier2015,Jimenez2017,Li2017,Li2020}. Therefore, there is a large uncertainty in estimating the dense gas mass from a single transition line of a high dipole moment molecule, which is similar to the issue of the CO-to-H2 conversion factors \citep{Narayanan2012,Bolatto2013}. Optically thin dense-gas tracers, such as the isotopologues of HCN and HCO$^+$, could help better revealing dense gas properties in galaxies, such as volume density, temperature, and excitation conditions. One can determine optical depths of dense gas tracers, using their isotopologue line ratios \citep{Henkel1998,Martin2010,Wang2014,Li2020}. Because of their low abundances, isotopologue lines are mostly optically thin, and thus they can be used to accurately determine dense molecular gas properties and help study dense gas--star formation relations in different galaxies \citep{Li2020}. In most galaxies, however, isotopologue lines of dense gas tracers are too faint to be detected. Only a few detections of such lines have been reported in nearby galaxies \citep{Henkel1998,Wang2014,Wang2016,Tunnard2015,Li2020}, which either are limited within only the central regions of galaxies, or take global properties of galaxies as a whole \citep{Wang2014,Li2020}. Toward one of the nearest starburst galaxy, M$\,$82, \cite{Li2020} observed multi-$J$ HCN lines in the central region. Intriguingly, they found that the line profiles of H$^{13}$CN $J$=1$\rightarrow$0 and $J$=3$\rightarrow$2 are not consistent with each other. The $J$=1$\rightarrow$0 and $J$=3$\rightarrow$2 transitions of H$^{13}$CN are dominated by the red-shifted and the blue-shifted velocity components, respectively. Because H$^{13}$CN $J$=3$\rightarrow$2 has a higher upper energy level and a higher critical density than those of H$^{13}$CN $J$=1$\rightarrow$0, such a difference may indicate velocity-dependent excitation conditions, isotope abundance variations, or exotic radiative transfer processes. It is natural to rise the following questions: How much would dense gas properties vary at different locations and velocity components of a galaxy? How would dense gas isotopologue abundances change along the galactic disk? What physical mechanisms would regulate dense gas properties on galactic scales? To address these questions above, the best way is to obtain spatially resolved dense gas isotopologue measurements, with deep integration. In this paper, we present new IRAM 30-m observations of the $^{13}$C and $^{15}$N isotopologues of HCN, HCO$^+$, and HNC, along the major axis of M$\,$82. We describe observations and data reduction in Section \ref{sec:obs}. The new spectra, their intensities, optical depths, and abundance ratios are presented in Section \ref{sec:results}. Section \ref{sec:discussion} discussed possible mechanisms that may dominant or bias these line ratios. Final conclusions are summarized in Section \ref{sec:summary}. \section{Observations and data reduction} \label{sec:obs} Observations were performed with the IRAM 30-m telescope, at Pico Veleta, Spain during February 2019 (Project number: 186-18, PI: Feng Gao). A total number of four different pointings were used to sample the off-center region along the major axis of M82, with a typical pointing offset of 15$^{\prime\prime}$. We list these pointing positions in Table \ref{table:M82_position} below. Figure \ref{fig:M82_cor} shows the observed four off-center positions overlaid on the velocity-integrated flux (moment 0) map of HCN \citep{Salas2014}, where positions-1 and -2 are located in the South-Western side of M$\,$82 while positions-4 and -5 are located in the North-Eastern side \citep{Aladro2011b}. The Eight Mixer Receiver (EMIR) with dual-polarization and the Fourier Transform Spectrometers (FTS) backend with a 8-GHz frequency coverage for each band and a 195-kHz spectral resolution were used. EMIR was configured to the mutual observing mode for both E-90 (at 3 mm) and E-150 (at 2 mm) receivers. To verify that the signal is from the sky frequency rather than radio-frequency interference (RFI) from the Earth or from the backend, two different local oscillator (LO) tuning setups were used during the observations (see Table \ref{tab:parameter_obs}). All observations were performed with the wobbler switching mode, which has a beam throw distance of $\pm$60$^{\prime\prime}$ and a switching frequency of 0.5 Hz. Telescope pointing was checked every two hours with nearby strong quasi-stellar objects, while focus was checked at the beginning of each observation. The beam sizes of the IRAM 30-m telescope are approximately 29$^{\prime\prime}$ and 16$^{\prime\prime}$ at 87 GHz and 145 GHz, respectively. Typical system temperature are $\le$ 100 K and $\sim$ 120 K, at 3-mm and 2-mm band, respectively. The antenna temperature ($T_{\rm A}^{*}$) is converted to the main beam temperature ($T_{\rm mb}$), using $T_{\rm mb}$=$T_{\rm A}^{*}\times F_{\rm eff}$/$B_{\rm eff}$, where the forward efficiencies ($F_{\rm eff}$) are 0.95 and 0.93, beam efficiencies ($B_{\rm eff}$) are 0.81 and 0.74, at 3-mm and 2-mm band, respectively. Each spectrum was read out every 2 minutes. The on-source time ranges from 3.5 to 4 hours towards each position. We adopt GILDAS/CLASS\footnote{http://www.iram.fr/IRAMFR/GILDAS} \citep[GILDAS team 2013]{Pety2005} to reduce all spectral data as the following order: First, we check all spectra by eye and removed questionable ones. Second, for each position we average all reliable spectra into one spectrum. We then fit a first-order polynomial baseline and subtract it from the averaged spectrum. Last, the baseline-subtracted spectra are smoothed to a velocity resolution of $\sim$ 35 km s$^{-1}$. \begin{table} \scriptsize \begin{center} \caption{Observed positions along the major axis of M$\,$82}\label{tab:source} \label{table:M82_position} \begin{tabular}{lllllllll} \\ \hline \hline Position & R.A. & Dec. & $R_{\rm GC}$ \\ & (J2000) & (J2000) & kpc \\ \hline 1 & 09:55:48 & 69:40:40 & 0.45 \\ 2 & 09:55:50 & 69:40:43 & 0.28\\ 3(center) & 09:55:53.1 & 69:40:41 & 0\\ 4 & 09:55:55 & 69:40:50 & 0.23\\ 5 & 09:55:57 & 69:40:55 & 0.42\\ \hline \end{tabular} \end{center} Positions-1 and -2 locate in the SW lobe. Positions-4 and -5 locate in the NE lobe. $R_{\rm GC}$ is the distance from the Galactic Centre. \end{table} \begin{table*} \centering \caption{The basic properties of these five positions} \label{tab:parameter_obs} \begin{minipage}{160mm} \footnotesize \begin{tabular}{p{1.2cm}p{2cm}p{1.6cm}p{1.6cm}p{1.6cm}p{1.8cm}p{1.8cm}} \hline \hline {Position} & {Observing dates} & {$t_{\rm on\,source}$} & {$T^{\rm 3 mm}_{\rm sys}$} & {$T^{\rm 2 mm}_{\rm sys}$} & {LO(3 mm) } & {LO(2 mm)$^{~\rm a}$ } \\ & YYYY-MM-DD & (min) & (K) & (K) & (GHz) & (GHz) \\ \hline Position-1 & 2019-Feb-06 & 47 & 83 & 120 & 86.34 & 146.97 \\ & 2019-Feb-07 & 20 & 93 & 130 & 86.29 & 146.92 \\ & 2019-Feb-10 & 48 & 110 & $-$ & 86.15 & $-$ \\ Position-2 & 2019-Feb-06 & 26 & 93 & 120 & 86.34 & 146.97 \\ & 2019-Feb-07 & 25 & 94 & 127 & 86.29 & 146.92 \\ & 2019-Feb-08 & 39 & 105 & $-$ & 86.24 & $-$ \\ & 2019-Feb-09 & 26 & 106 & $-$ & 86.19 & $-$ \\ Position-4 & 2019-Feb-06 & 22 & 96 & 134 & 86.34 & 146.97 \\ & 2019-Feb-07 & 25 & 90 & 121 & 86.29 & 146.92 \\ & 2019-Feb-08 & 45 & 109 & $-$ & 86.24 & $-$ \\ & 2019-Feb-09 & 26 & 104 & $-$ & 86.19 & $-$ \\ Position-5 & 2019-Feb-06 & 16 & 101 & 142 & 86.34 & 146.97 \\ & 2019-Feb-07 & 35 & 92 & 125 & 86.29 & 146.92 \\ & 2019-Feb-10 & 38 & 116 & $-$ & 86.15 & $-$ \\ \hline \end{tabular}\\ \begin{list}{}{} \item[${\mathrm{a}}$]Tuning frequencies of the Local oscillator (LO) during the observations. \end{list} \end{minipage} \end{table*} \begin{table*} \begin{center} \caption{Integrated Intensities at All Positions} \label{tab:all_position} \scriptsize \begin{tabular}{p{16.3mm}p{11mm}lllllllllllllllllllll} \hline \hline & & \multicolumn{10}{c}{$I$(K km s$^{-1}$)} \\ \cline{3-13} {Line} & {$\nu _{\rm rest}$} & {Position-1} & \multicolumn{2}{c}{Position-2} & & \multicolumn{2}{c}{Position-3} & & \multicolumn{2}{c}{Position-4 } & & \multicolumn{2}{c}{Position-5 }\\ \cline{4-5} \cline{7-8} \cline{10-11} & {(GHz)} & Blue-shifted & Blue-shifted & Red-shifted & & Blue-shifted & Red-shifted & & Blue-shifted & Red-shifted & & Red-shifted \\ \hline HC$^{15}$N 1-0 & 86.055 & $0.08\pm0.025$ & $0.29\pm0.04$ & $<0.11$ & & $0.29\pm0.02$ & $0.24\pm0.02$ & & $- $ & $<0.08 $ & & $<0.08$ \\ H$^{13}$CN 1-0 & 86.340 & $0.23\pm0.05$ & $0.38\pm0.04$ & $0.27\pm0.04$ & & $0.21\pm0.04$ & $0.33\pm0.04$ & & $<0.15$ & $0.30\pm0.05$ & & $0.14\pm0.02$ \\ SiO 2-1 & 86.847 & $<0.20$ & $<0.12$ & $-$ & & $0.17\pm0.04$ & $0.25\pm0.04$ & & $- $ & $0.25\pm0.07$ & & $0.29\pm0.07$ \\ H$^{13}$CO$^+$ 1-0 & 86.754 & $0.53\pm0.07$ & $0.80\pm0.04$ & $0.12$ & & $0.50\pm0.04$ & $0.71\pm0.04$ & & $0.26\pm0.07$ & $0.68\pm0.07$ & & $0.36\pm0.07$ \\ HCO 1-0 & 86.670 & $0.45\pm0.07$ & $0.46\pm0.04$ & $-$ & & $0.27\pm0.04$ & $0.34\pm0.04$ & & $- $ & $-$ & & $<0.20 $ \\ HN$^{13}$C 1-0 & 87.091 & $<0.10$ & $<0.12$ & $-$ & & $<0.08 $ & $0.13\pm0.03$ & & $- $ & $0.15\pm0.05$ & & $0.16\pm0.05$ \\ HCN 1-0 & 88.632 & $14.46\pm0.26$ & $19.63\pm0.04$ & $6.04\pm0.04$ & & $11.57\pm0.04$ & $15.47\pm0.04$ & & $3.71\pm0.03$ & $21.09\pm0.03$ & & $14.26\pm0.06$ \\ H$^{15}$NC 1-0 & 88.866 & $<0.09$ & $<0.12$ & $-$ & & $- $ & $<0.15$ & & $- $ & $<0.10 $ & & $<0.18 $ \\ HCO$^+$ 1-0 & 89.189 & $22.29\pm0.04$ & $30.23\pm0.04$ & $8.14\pm0.04$ & & $18.16\pm0.05$ & $22.64\pm0.05$ & & $6.18\pm0.04$ & $29.89\pm0.04$ & & $19.69\pm0.05$ \\ HNC 1-0 & 90.663 & $6.60\pm0.03$ & $9.67\pm0.05$ & $2.45\pm0.05$ & & $6.00\pm0.05$ & $7.13\pm0.05$ & & $1.74\pm0.03$ & $9.68\pm0.03 $ & & $5.82\pm0.05 $ \\ HC$_3$N 10-9 & 90.979 & $0.64\pm0.03$ & $1.19\pm0.04$ & $0.40\pm0.04$ & & $0.65\pm0.03$ & $1.06\pm0.03$ & & $0.11\pm0.03$ & $1.14\pm0.03 $ & & $0.61\pm0.04 $\\ H$_2$CO 2-1 & 145.603 & $1.73\pm0.06$ & $2.86\pm0.08$ & $-$ & & $0.67\pm0.07$ & $2.35\pm0.07$ & & $0.17\pm0.05$ & $3.22\pm0.05 $ & & $1.40\pm0.06 $ \\ \hline \end{tabular} \end{center} Notes: Velocity-integrated intensities are calculated with fixed velocity ranges of 0--200 km s$^{-1}$ and 200--400 \kms\ for the blue-shifted and the red-shifted component, respectively. Errors of the velocity-integrated intensities are calculated with $\Sigma (I)$ = $\sigma_{\rm line-free}^{\rm chan} \times \sqrt{\Delta V \delta \varv}$, where $\sigma_{\rm line-free}^{\rm chan}$ is the r.m.s. noise level obtained from the line-free channels at the corresponding velocity resolution; $\delta v$ is the velocity resolution (36 \kms); $\Delta V$ is the line width (200 km s$^{-1}$). \end{table*} \section{results} \label{sec:results} From the calibrated spectra data, we identify different transitions at each pointing position according to their rest frequency, which is listed in Table \ref{tab:all_position}. Figures \ref{fig:H13CN_H13CO+}-\ref{fig:HC3N_H2CO_Dense} present spectra of H$^{13}$CN $J$=1-0, HC$^{15}$N $J$=1-0, H$^{13}$CO$^+$ $J$=1-0, HN$^{13}$C $J$=1-0, H$^{15}$NC $J$=1-0, and HC$_3$N $J$=10-9 covered by the 3-mm band, while H$_2$CO $J$=2-1 is covered by the 2-mm band. The intensities of all spectra are plotted on the $T\rm_{MB}$ scale. For comparison, spectra of the major isotopologues of HCN $J$=1-0, HCO$^+$ $J$=1-0 or HNC $J$=1-0 are overlaid. Most spectra, especially those from main isotopologues, at position-3 (hereafter P3; the central position) show two velocity components, which has been also shown in \citet{Salas2014} and \citet{Li2020}. Only a single velocity component can be identified from the line profiles at all four off-centre positions. Among them, positions-1 and -2 (hereafter P1 \& P2; at the south-west side) are dominated by the blue-shifted velocity component, while positions-4 and -5 (hereafter P4 \& P5; at the north-east side) are dominated by the red-shifted velocity component, respectively. \subsection{Detected lines} As shown in \cite{Li2020}, the spectra at P3 can be fitted with two-component Gaussian profiles. However, the velocity difference between the two components may not be constant among all five positions, and the outcome of the two-component Gaussian fitting strongly depends on the data quality. Therefore, we calculate the velocity-integrated intensities of all molecular lines in two fixed velocity ranges, i.e., 0--200 \kms\ and 200--400 \kms, for the blue-shifted and the red-shifted component, respectively. Table \ref{tab:all_position} presents the velocity-integrated intensities at all positions. For P1 and P5, only the major velocity component is considered, because of the weakness of the minor component. Although some lines show emission line features on $\sim$ 2-$ \sigma$ levels, we only adopt 3-$\sigma$ at their upper limits. In the following, we present properties of each isotopologue line. \begin{itemize} \item {\bf H$^{13}$CN $J$=1$\rightarrow$0, HN$^{13}$C $J$=1$\rightarrow$0, and H$^{15}$NC $J$=1$\rightarrow$0:} We present the line profile of H$^{13}$CN $J$=1-0 in the left panel of Figure \ref{fig:H13CN_H13CO+}. H$^{13}$CN $J$=1$\rightarrow$0 is detected at all five positions. Two velocity components are detected at three central positions, P2, P3, and P4. At P4, the blue-shifted component only shows a weak feature at $\sim$ 2-$\sigma$ level. Therefore, we present 3-$\sigma$ upper limits for it in Table \ref{tab:all_position}. HN$^{13}$C $J$=1-0 is detected in P3, P4, and P5, which is shown in the middle panel of Figure \ref{fig:HC15N_HN13C_H15NC}. The north-east (NE) side shows stronger HN$^{13}$C emission than those at the south-west (SW) side. H$^{15}$NC $J$=1-0 is not detected at all positions, which is shown in the right panel of Figure \ref{fig:HC15N_HN13C_H15NC}. So, we show 3-$\sigma$ upper limits of the velocity-integrated intensities in Table \ref{tab:all_position}. \item {\bf HC$^{15}$N $J$=1$\rightarrow$0:} We present the line profile of HC$^{15}$N $J$=1-0 in the left panel of Figure \ref{fig:HC15N_HN13C_H15NC}. HC$^{15}$N $J$=1$\rightarrow$0 is blended with SO $J$=2-1, with only a 200 \kms\ offset in velocity. This line blending effect is mostly severe at P3, where SO $J$=2-1 peak is at the same level of HC$^{15}$N $J$=1$\rightarrow$0 (see Figure \ref{fig:HC15N_HN13C_H15NC}). At all other four positions, however, HC$^{15}$N $J$=1-0 emission seems not heavily polluted by the SO $J$=2-1 emission, possibly because the SO emission is only enhanced in the central shocked region \citep{Pineau1993,Aladro2011b}. Therefore, we use the HCN line profile at P3 as a template and fit the SO profile at the same position, then remove the fitted SO contribution. We then use the residual to derive the integrated intensity of HC$^{15}$N at P3. For P4 \&\, P5, we estimate 3-$\sigma$ upper limits for the velocity-integrated intensities. Comparing with the spatial variation of H$^{13}$NC $J$=1-0, HC$^{15}$N $J$=1$\rightarrow$0 show a contrary trend along the major axis. \item {\bf H$^{13}$CO$^+$ $J$=1$\rightarrow$0, HCO $J$=1$\rightarrow$0 and SiO $J$=2$\rightarrow$1:} As shown in the right panel of Figure \ref{fig:H13CN_H13CO+}, H$^{13}$CO$^+$ $J$=1-0 is detected at all five positions. HCO $J$=1-0 and H$^{13}$CO$^+$ $J$=1-0 have a velocity offset of 294 \kms, therefore they show a weak blending at P2 and P3. The HCO $J$=1-0 line is well detected in the SW side, while in the NE side it was only marginally detected at P5. SiO $J$=2-1 seems not blended with H$^{13}$CO$^+$ $J$=1-0. SiO $J$=2-1 is stronger at the NE side than that at the SW side, where we only obtain 3-$\sigma$ upper limits at P1\&\,P2. This asymmetric distribution of SiO indicates a fast shock at the NE side, possibly driven by outflow \citep{Gusdorf2008,Gibb2007,Lopez2011}. \item {\bf HC$_3$N $J$=10$\rightarrow$9:} We present the line profile of HC$_3$N $J$=10-9 in the left panel of Figure \ref{fig:HC3N_H2CO_Dense}. Being optically-thin in most galactic environments \citep{Morris1976}, HC$_3$N 10-9 is detected at all positions. This line is stronger than those isotopologue lines of other dense gas tracers at the same position. The line profile of HC$_3$N is similar to that of HCN $J$=1$\rightarrow$0 at each position. The velocity-integrated intensity of HC$_3$N $J$=10-9 at P-2 is consistent with that reported by \cite{Aladro2015}. \item {\bf H$_2$CO $J$=2(0,2)-1(0,1):} We present the line profiles of H$_2$CO $J$=2(0,2)-1(0,1) in the middle panel of Figure \ref{fig:HC3N_H2CO_Dense}. Most of the emission feature around 145.603 GHz seems from H$_2$CO $J$=2(0,2)-1(0,1), whose frequency fits the line profile very well. However, HC$_3$N $J$=16-15 might also contribute part of the flux to the red-shift emission feature. \item {\bf HCN, HCO$^+$, and HNC $J$=1-0:} HCN, HCO$^+$ and HNC $J$=1-0 are all detected with high signal to noise level, at all five positions. Their line profiles agree well with each other at the same positions (see Figure \ref{fig:HC3N_H2CO_Dense}). All these lines show double-peak profiles at the central position, which is not always seen in those weak isotopologue lines. \end{itemize} \subsection{ Line Intensity Ratios} With all measured velocity-integrated intensities, we obtained line intensity ratios between the main and rare isotopologue lines of HCN, HCO$^{+}$, and HNC at each position, and we list results in Table \ref{tab:Isotopic_ratio}. At the central position (P-3), we calculated ratios obtained from three velocity-integrated intensities, i.e., blue-shifted, red-shifted , and total velocity components of these lines. As shown in Table \ref{tab:Isotopic_ratio}, the ratios of $I$(H$^{13}$CO$^{+}$)/$I$(H$^{13}$CN) and $I$(H$^{13}$CN/$I$(HC$_{3}$N) seem relatively constant among all positions, with a variation $\sim$20-30\%. However, most other intensity ratios show a significant spatial variation, which can reach up to a factor of 4--5 (see Figure \ref{fig:Ratio_position}). The spatial variation of $I$(HCN)/$I$(H$^{13}$CN), $I$(HCO$^{+}$)/$I$(H$^{13}$CO$^{+}$), and $I$(HCN)/$I$(HC$^{15}$N) ratios follow a similar trend, with higher values at the NE side than those at the SW side. These ratios in the central region are lower than those on the disk. These trends are consistent with \textbf{what has been} found in $I(^{12}$CO)/$I(^{13}$CO) $J$=1-0 \citep{Kikumoto1998}. However, the $I$(HNC)/$I$(HN$^{13}$C) intensity ratios show an opposite trend along the major axis, compared with others. \subsection{Optical depths of the main isotopologue lines} Under the assumptions of local thermal equilibrium (LTE), uniform gas excitation, constant isotopic abundance ratios, and neglecting radiative transfer by the cosmic microwave background, we can derive optical depths of the main isotopic lines. We adopt the same method as reported in \cite{Li2020}, in which the line ratio $R$ and the optical depth for a specific isotopologue line $\tau$, is related as$R$ = (1-$e^{-\tau_{12}}$)/(1-$e^{-\tau_{13}}$) Due to the large uncertainty of absolute abundance ratio of isotopes in external galaxies, we adopt various $^{12}$C/$^{13}$C and $^{14}$N/$^{15}$N abundance ratios reported in the literature: $^{12}$C/$^{13}$C = 89 \citep[the Solar system,][]{Clayton2004} and 200 \citep[Ultra-Infrared-luminous ULIRGs,][] {Romano2017}), and $^{14}$N/$^{15}$N = 100 \citep[nearby starburst galaxies,][]{Chin1999}, 200 \citep[Galactic massive star-forming regions,][]{Li2017}, and 290 \citep[the Solar neighbourhood at $R_{\rm gc}\sim 7.9$ kpc,][]{Adande2012}. Table \ref{tab:depth_HCN} shows the derived optical depths of HCN, HCO$^+$, and HNC $J$ = 1$\rightarrow$0. The optical depths of HCN $J$=1-0 and HCO$^{+}$ $J$=1-0 only have a slight variation along the major axis of M$\,$82, with higher optical depths in the central region than those on the two sides of the disk. The observed I(HCN)/I(H$^{13}$CN) $J$=1-0 line ratio could exceed the assumed $^{12}$C/$^{13}$C abundance ratio occasionally, meaning that the actual $^{12}$C/$^{13}$C ratio should be even higher. Therefore, the optical depths calculated with $^{12}$C/$^{13}$C = 89 are the lower limits, making $\tau_{\rm HCN\, J=1-0}>1$ at most positions. This also applies to the $\tau_{\rm HCN\, J=1-0}$ calculated with the assumptions of various $^{14}$N/$^{15}$N abundance ratios. Due to limited sensitivity, we can only obtain optical depths of HNC $J$=1-0 at three positions. \subsection{Constraints \textbf{on} $^{12}$C/$^{13}$C and $^{14}$N/$^{15}$N Abundance Ratios} To obtain optical depths accurately, one should also consider the possible radial gradient of $^{12}$C/$^{13}$C abundance ratio, which increases radially in our Milky Way. This is currently not possible due to the lack of abundance measurements. However, our newly measured line ratios between the isotopologues can still be used to set tight constraints on the abundance ratios. The observed HCN/H$^{13}$CN line ratio reaches 101$\pm$15, which is higher than the $^{12}$C/$^{13}$C abundance ratio in the Solar neighbourhood ($\sim$ 89). But it is consistent with the values measured in the outer Galactic disk regions \citep{Milam2005}. With the double isotope method \citep{Adande2012}, the $^{14}$N/$^{15}$N ratio can be calculated using the line ratio of $I$(H$^{13}$CN)/$I$(HC$^{15}$N) and the $^{12}$C/$^{13}$C abundance ratio, by $^{14}$N/$^{15}$N = $I$(H$^{13}$CN)/$I$(HC$^{15}$N) $\times$ $^{12}$C/$^{13}$C. Assuming $^{12}$C/$^{13}$C=89 in at SW side and at the center, average values of $^{14}$N/$^{15}$N ratios are 186 $\pm$ 113 and 91$\pm$11, respectively. On the other hand, at the NE side, adopting $^{12}$C/$^{13}$C=89, the lower limits of $^{14}$N/$^{15}$N ratios range from 156 to 334. \begin{table*} \centering \caption{Intensity ratios of the isotopologues of HCN, HNC and HCO$^+$} \label{tab:Isotopic_ratio} \begin{minipage}{160mm} \begin{tabular}{llllllllllllllll} \hline \hline {Position} &{$\frac{I( \rm HCN)}{I( \rm H^{13}CN)}$} & {$ \frac{I( \rm HCN)}{ I( \rm HC^{15}N)}$} & {$ \frac{ I( \rm HCO^{+})}{ I( \rm H^{13}CO^{+})}$} & {$\frac{ I( \rm HNC)}{ I( \rm HN^{13}C)}$} & {$ \frac{ I( \rm H^{13}CN)}{ I( \rm HC^{15}N)}$} & {$ \frac{ I( \rm H^{13}CO^+)}{ I( \rm H^{13}CN)}$} & {$ \frac{ I( \rm H^{13}CN)}{ I( \rm HC_{3}N)}$} \\ \hline &$J$=1-0 & 1-0 & 1-0 & 1-0 & 1-0 & 1-0 & 1-0/10-9\\ \hline 1 &63 $\pm$ 14 & 180$\pm$ 68 & 42 $\pm$ 6 & $>66 $ & 2.89$\pm$1.25 & 2.30 $\pm$ 1.25 & 0.36 $\pm$ 0.08 \\ 2 &52 $\pm$ 5 & 68 $\pm$ 9 & 38 $\pm$ 2 & $>81 $ & 1.31$\pm$0.23 & 2.10 $\pm$ 0.23 & 0.32 $\pm$ 0.04 \\ 3$_{~\rm a} $ &55 $\pm$ 11 & 40 $\pm$ 3 & 36 $\pm$ 3 & $>75 $ & 0.72$\pm$0.15 & 2.38 $\pm$ 0.15 & 0.32 $\pm$ 0.06 \\ 3$_{~\rm b} $ &46 $\pm$ 6 & 64 $\pm$ 5 & 32 $\pm$ 2 & 55$\pm$13 & 1.37$\pm$0.20 & 2.15 $\pm$ 0.20 & 0.31 $\pm$ 0.04 \\ 3$_{~\rm total} $ &51 $\pm$ 6 & 52 $\pm$ 3 & 34 $\pm$ 2 & $>65$ & 1.02$\pm$0.13 & 2.24 $\pm$ 0.13 & 0.32 $\pm$ 0.02 \\ 4 &70 $\pm$ 12 & $>264 $ & 44 $\pm$ 5 & 65$\pm$22 & $>3.75 $ & 2.26 $\pm$ 0.44 & 0.26 $\pm$ 0.04 \\ 5 &101$\pm$ 15 & $>178 $ & 55 $\pm$ 11 & 36$\pm$11 & $>1.75 $ & 2.57 $\pm$ 0.62 & 0.23 $\pm$ 0.04 \\ \hline \end{tabular}\\ Note. To keep uniformity, we only calculate the intensity ratio of one velocity component. \\ $^{~\rm a} $ The ratios of the blue-shifted emission are estimated for positions-1 and -2.\\ $^{~\rm b} $For positions-3, -4, and -5, the red-shifted emission are used to calculate the ratios.\\ \end{minipage} \end{table*} \begin{table*} \footnotesize \centering \tabcolsep 1mm \caption{Optical depths for HCN, HNC, and HCO$^+$} \label{tab:depth_HCN} \begin{minipage}[c]{160mm} \begin{tabular}{llllllllllllll} \hline \hline {Position} & \multicolumn{2}{c}{${\tau}$(HCN)$^{~\rm a}$ (1-0) } & & \multicolumn{2}{c}{ ${\tau}$(HNC)$^{~\rm b}$ (1-0)} & & \multicolumn{2}{c}{$\tau$(HCO$^+$)$^{~\rm c}$ (1-0) } & & \multicolumn{3}{c}{${\tau}$(HCN)$^{~\rm d}$ (1-0)} \\ \cline{2-3} \cline{5-6} \cline{8-9} \cline{11-13} & 89$^{~\rm e}$ & 200$^{~\rm e}$ & & 89 & 200 & & 89 & 200 & & 100$^{~\rm f}$ & 200$^{~\rm f}$ & 290$^{~\rm f}$ \\ \hline 1 & 0.75$\pm$0.52 & 3.05$\pm$0.82 & & $<$0.64 & $<$2.88 & & 1.78$\pm$0.38 & 4.78$\pm$0.68 & & $-$ & 0.22$\pm$0.01 & 1.05$\pm$0.13 \\ 2 & 1.21$\pm$0.27 & 3.79$\pm$0.44 & & $<$0.19 & $<$2.21 & & 2.07$\pm$0.15 & 5.31$\pm$0.28 & & 0.84$\pm$0.39 & 2.77$\pm$2.18 & 4.23$\pm$1.40 \\ 3$_{~\rm a} $ & 1.07$\pm$0.48 & 3.57$\pm$0.79 & & $<$0.36 & $<$2.45 & & 2.24$\pm$0.26 & 5.61$\pm$0.75 & & 2.27$\pm$1.40 & 5.03$\pm$3.10 & 7.34$\pm$2.28 \\ 3$_{~\rm b} $ & 1.52$\pm$0.34 & 4.34$\pm$0.58 & & 1.07$\pm$0.02 & 3.57$\pm$0.03 & & 2.62$\pm$0.18 & 6.34$\pm$0.56 & & 0.98$\pm$0.45 & 2.99$\pm$2.27 & 4.51$\pm$1.50 \\ 3$_{~\rm total} $ & 1.26$\pm$0.19 & 3.88$\pm$0.47 & & $<$0.75 & $<$3.05 & & 2.41$\pm$0.15 & 5.95$\pm$0.31 & & 1.51$\pm$1.21 & 3.80$\pm$2.61 & 5.61$\pm$1.82 \\ 4 & 0.51$\pm$0.38 & 2.68$\pm$0.58 & & 0.67$\pm$0.81 & 2.93$\pm$1.3 & & 1.65$\pm$0.28 & 4.55$\pm$0.5 & & $-$ & $-$ & $<$0.19 \\ 5 & $-$ & 1.58$\pm$0.40 & & 2.24$\pm$0.03 & 5.61$\pm$0.05 & & 1.07$\pm$0.49 & 3.57$\pm$0.8 & & $-$ & $<$0.24 & $<$1.07 \\ \hline mean & 1.11 $\pm$ 1.51 & 3.64 $\pm$ 2.64 & & $<0.75$ & $<3.05$ & & 1.78 $\pm$ 0.98 & 4.78 $\pm$1.79 & & $<1.11$ & $<0.9$ & $<2.12$\\ whole & 1.16 $\pm$ 0.15 & 3.72 $\pm$ 0.24 & & $<0.64$ & $<2.88 $ & & 1.92 $\pm$ 0.11 & 4.03 $\pm$0.21 & & $<1.87$ & $<1.9$ & $<3.05$\\ \hline \end{tabular}\\ \begin{list}{}{} \item[${\mathrm{a}}$] The optical depth of HCN is calculated by the ratio of HCN/H$^{13}$CN. \item[${\mathrm{a}}$] The optical depth of HNC $J$=1-0 is calculated by the ratio of HNC/HN$^{13}$C $J$=1-0. \item[${\mathrm{b}}$] The optical depth of HCO$^+$ 1-0 is calculated by the ratio of HCO$^+$/H$^{13}$CO$^+$ 1-0. \item[${\mathrm{d}}$] The optical depth of HCN is calculated by the ratio of HCN/HC$^{15}$N. \item[${\mathrm{e}}$] Abundance ratios of $^{12}$C/$^{13}$C: 89 (Solar system \citep{Clayton2004} ) and 200 (ULIRGs \citep{Romano2017}). \item[${\mathrm{f}}$] Abundance ratios of $^{14}$N/$^{15}$N: 100 (nearby starburst galaxies \citep{Chin1999}), 200 ( Galactic massive star-forming regions \citep{Li2017}) and 290 (local interstellar medium \citep{Adande2012}). \end{list} \end{minipage} \end{table*} \section{Discussion} \label{sec:discussion} \subsection{Comparison with data in the literature} Table \ref{tab:Isotopic_ratio_other_galaxies} summarizes literature measurements of $I$(HCN)/ $I$(H$^{13}$CN) $J$=1-0, $I$(HCO$^+$)/ $I$(H$^{13}$CO$^+$) $J$=1-0, and $I$(HNC)/ $I$(HN$^{13}$C) $J$=1-0 from nearby galaxies, including starburst galaxies, ULIRGs and active galactic nucleus (AGN)-dominated galaxies. All these ratios in M$\,$82 are higher than those found in the literature, except for M~83. Previous observations of dense gas tracers have shown that they are mostly optically thick in both Galactic giant molecular clouds (GMCs) and external galaxies \citep{Wang2014,Meier2015,Jimenez2017,Li2017,Li2020}. If we assume that the abundance ratio of $^{12}$C/$^{13}$C is 40, which is obtained as the average condition of nearby galaxies \citep{Henkel1994,Henkel2010}, the HCN, HCO$^+$ and HNC lines of M~82 would be optically thin. At all five positions \textbf{in} M$\,$82, ratios of $I$(H$^{13}$CN)/$I$(H$^{13}$CO$^+$) $J$=1-0 are lower than those of $I$(HCN)/$I$(HCO$^+$) $J$=1-0. Similar results for the $J$=2-1 transition were also reported by \cite{Aladro2011b} at P2, where the $I$(H$^{13}$CN)/$I$(H$^{13}$CO$^+$) $J$=2-1 ratio is lower than that of $I$(HCN)/$I$(HCO$^+$) $J$=2-1. This indicates that HCN lines should have lower optical depths than those of HCO$^+$ lines. \subsection{Optical Depths } Given the high intensity ratios of $I$(HCN)/$I$(H$^{13}$CN) $J$=1-0 in a range from 51$\pm$6 to 101$\pm$15 at all five positions, a $^{12}$C/$^{13}$C abundance ratio of 40 \citep{Henkel1998} can not be valid. \cite{Kikumoto1998} and \cite{Tan2011} also found similar results, and they adopted a $^{12}$C/$^{13}$C abundance ratio of 60. No matter which $^{12}$C/$^{13}$C abundance ratio is adopted, $^{12}$C-bearing dense gas tracer lines need to have at least moderate optical depths. The optical depths of dense gas tracers may be decreased by the feedback of supernova explosions in M~82, which has been found to in previous studies \citep{Allen1998, Mattila2001}. Such supernova feedback strongly affect the molecular gas environments, which can be seen from strong SiO emission (see Figure \ref{fig:H13CN_H13CO+}), high H$_2$ 1-0 $S$(1)/Br$\gamma$ ratio, and high ratio of $I$($^{12}$CO)/$I$($^{13}$CO) \citep{Lester1990, Mouri1989}. Therefore, the $^{12}$C-bearing molecular lines would be expected to be more optically thin, because the shock conditions could broaden the lines, which have an increased escape probability in radiative transfer. Such effects have been seen in Galactic supernova remanent, which are interacting with molecular clouds \citep[e.g.,][]{Zhang2010}, To mimic spatially unresolved galaxies, we further derive the weighted mean optical depths of dense gas tracers in M~82, by averaging optical depths among all positions, weighted with their line fluxes (see Table \ref{tab:depth_HCN} marked as ``mean''). The unweighted mean optical depth, which was obtained directly from the total flux ratios, is also listed in the same tables (marked as ``whole''). Both ``mean'' and ``whole'' optical depths agree well with those obtained on the disk, within the error bars. Thus, for galaxies without spatially information, optical depths of dense gas tracers obtained from a whole galaxy could generally represent average conditions on the disk. \subsection{ The \Ctw/\Cth\ abundance ratio } \label{sec:12C_13C_Interprete} The two isotopes of Carbon, \Ctw\ and \Cth, have different mechanisms of nucleosynthesis. The main isotope, \Ctw, could be partly produced by the triple-$\alpha$ reaction in massive stars \citep[$>$ 8 M$_\odot$,][]{Wilson1992, Nomoto2013}, and partly by lower mass stars. Therefore, \Ctw\ can increase quickly after the starburst starts, due to the short lifetime of massive stars. Most of \Cth, on the other hand, is formed in low- and intermediate-mass stars ($<$ 8 M$_\odot$), which means that most \Cth\ should be released to the ISM on much longer timescales than that of \Ctw\ \citep{Wilson1992,Hughes2008}. As a result, the \Ctw/ \Cth\ abundance ratio could represent the star-formation history \citep{Wilson1994, Henkel2010} --- high ratios for long-term starbursts (a few hundreds of Myrs) and low ratios for young starbursts (tens of Myrs), before the low-mass stars start to release \Cth. Given the relatively short starburst timescales of M$\,$82 \citep[$\sim 5 \times 10^7$yr,][]{Konstantopoulos2009}, the majority of the newly born low-mass stars are still alive. So, the \Ctw/\Cth\ ratio can be enhanced in the central regions, where starburst is more intensive compared to that in the outer disk. Therefore, the current starbursts generate the \Ctw\ / \Cth\ decreasing trend from center to outskirt along the major axis of M$\,$82. However, the observed \Ctw/ \Cth\ ratios are not only contributed from the current starbursts contribution, but also produced from all the past starburst activities in history. Besides, Galactic chemical evolution models predict that the \Ctw/ \Cth\ ratio would vary by a factor of two, even if a strong starburst produce half of the stellar mass in a secular evolving galaxy \citep{Romano2017}. Even if the undergoing starburst could slightly enhance \Ctw/ \Cth\ abundance ratio in a short timescale in the central region during the starburst, it would not change the increasing \Ctw/ \Cth\ gradient in M~82, similar to the Milky Way. \begin{table*} \centering \caption{Integrated intensity ratios from the literature} \label{tab:Isotopic_ratio_other_galaxies} \begin{minipage}{160mm} \begin{tabular}{llllllllll} \hline \hline {Galaxy} & {$ \frac{\rm HCN}{\rm H^{13}CN}$} & {$ \frac{\rm HCO^{+}}{\rm H^{13}CO^{+}}$} & {$\frac{\rm HNC}{\rm HN^{13}C}$} & Type & Reference \\ & 1-0 & 1-0 & 1-0 \\ \hline NGC~3079 & 10$\pm$ 5 & 7$\pm$2 & $>$28 & SB/AGN & 1\\ Mrk~231 & 16$\pm$5 & 12$\pm$5 & $>$7 & SB & 1\\ NGC~4418 & 8$\pm$3 & $-$ & $-$ & SB/AGN & 2\\ NGC~1068 & 16$\pm$1 & 20$\pm$1 & 38$\pm$6 & AGN & 3\\ NGC~3351 & 21$\pm$3 & $>$17 & $-$ & SB(r)b & 4 \\ NGC~3627 & 7$\pm$1 & $>$12 & $-$ & SB/AGN & 4 \\ NGC~253 & 17$\pm$1 & 24$\pm$2 & $-$ & SB & 4\\ M~83 & 41$\pm$7 & 44 $\pm$13 & $-$ & SAB(s)c & 5 \\ NGC~5194 & 27 $\pm$18 & 34$\pm$29 & $>$16 & SB/AGN & 6\\ M$\,$82 &46$\pm$6$-$101$\pm$15 &32$\pm$2$-$55$\pm$11 &36$\pm$11$-$65$\pm$22 &SB &this work\\ \hline \end{tabular}\\ {References: (1)\cite{Li2020} (2) \cite{Costagliola2011} (3) \cite{Wang2014} (4)\cite{Jimenez2017} (5)\cite{Aladro2015} (6)\cite{Watanabe2014} (7)\cite{Muller2011}.} \end{minipage} \end{table*} \subsection{ The $^{14}$N/$^{15}$N abundance ratio} In external galaxies, the $^{14}$N/$^{15}$N abundance ratios have been measured in a range from $\sim$ 100 to $\sim$ 450 \citep{Henkel1998,Henkel2018,Wang2014,Wang2016,Adande2012}. This ratio measured in two starbursts, NGC~4945 \citep[$\sim$200-- 400,][]{Henkel2018} and ULIRG Arp~220 \citep[$440^{+140}_{-82}$,][]{Wang2014}, are higher than those found in M~82 ($\sim 80 --250$, see Figure \ref{fig:14N_15N}). In our Milky Way, the $^{14}$N/$^{15}$N ratios distribute in a large range \citep[100--600,][]{Li2017}, with a possible positive gradient from the center to out disk \citep{Chen2021}. Figure \ref{fig:14N_15N} shows estimated $^{14}$N/$^{15}$N ratios as a function of galactocentric distance. If we adopt an assumption of constant $^{12}$C/$^{13}$C abundance ratio of 89, $^{14}$N/$^{15}$N abundance ratios tend to increase with galactocentric distance, which is consistent with the $^{14}$N/$^{15}$N gradient found in the Milky Way \citep{Dahmen1995,Adande2012,Romano2003,Romano2017,Chen2021}. On the other hand, the $\rm ^{12}C/^{13}C$ abundance ratio shows an increasing gradient from the center to the outer disk in the Milky Way \citep{Wilson1994,Savage2002,Milam2005,YtYan2019}. If we adopt a similar positive $^{12}$C/$^{13}$C abundance gradient to M~82, which is consistent with the prediction in Section \ref{sec:12C_13C_Interprete}, the $^{14}$N/$^{15}$N abundance ratio would show an even stronger gradient, with lower values in the center and higher ratios on the disk. \subsection{Astrochemical effects} Selective photon dissociation prefers to destroy $^{13}C$- and $^{15}C$-bearing molecules, which would increase $\rm ^{12}C/^{13}C$ and $\rm ^{14}N/^{15}N$ abundance ratios in high UV fields \citep{Wilson1992,Savage2002}. However, the inner region of M$\,$82 has lower $I$(HCO$^+$)/$I$(H$^{13}$CO$^+$) and $I$(HCN)/$I$(HC$^{15}$N) line values, indicating that photo-dissociation does not play a key role. Isotope fractionation, which could enhance H$^{13}$CO$^+$/HCO$^+$ and HC$^{15}$N/HCN ratios, is only effective at a very low temperature \citep{Smith1980,Woods2009,Rollig2013,Loison2019}. In M~82, the temperature is relatively high, especially in the center. However, both $I$(HCO$^+$)/$I$(H$^{13}$CO$^+$) and $I$(HCN)/$I$(HC$^{15}$N) ratios increases in the off-center regions, meaning that fractionation effect does not play a key role as well. \section{Summary} \label{sec:summary} In this paper we present results from IRAM 30-m telescope observations along the major axis of M$\,$82 in the 2-mm band and 3-mm band. Four positions are selected to measure the isotopic lines of HCN, HCO$^+$, and HNC. The spatial distribution of these optically thin dense gas tracers are obtained, including H$^{13}$CN, H$^{13}$CO$^+$, HC$^{15}$N, HN$^{13}$C, H$^{15}$NC 1-0. A few other species of SiO $J$=2-1, HCO $J$=1-0, H$_2$CO $J$=2-1, and HC$_3$N $J$=10-9 are also detected. We obtain the following results: (1) H$^{13}$CN $J$=1-0, H$^{13}$CO$^+$ $J$=1-0, HC$_3$N $J$=10-9 and H$_2$CO $J$=2-1 are detected in all four positions along the major axis of M$\,$82. Spectral lines of the transition HC$^{15}$N $J$=1-0 are not detected at the NE side and HN$^{13}$C $J$=1-0 emissions are not detected at the SW side. We did not obtain any detection of H$^{15}$NC $J$=1-0 at all positions For the tentative detection of the isotopic lines, the 3-$\sigma$ upper limits are presented. (2)For spectral line intensity ratios, $I$(HCN)/$I$(H$^{13}$CN) $J$=1-0, $I$(HCO$^+$)/$I$(H$^{13}$CO$^+$) $J$=1-0 and $I$(HCN)/$I$(HC$^{15}$N) $J$=1-0 show a large spatial variation along the major axis of M$\,$82, which are higher at the NE side than those at the SW side and the value in the central region is lower than that on the disk. However, the $I$(HNC)/$I$(HN$^{13}$C) ratio seems to show an opposite trend along the major axis. (3) % The optical depths of HCN $J$=1-0 and HCO$^{+}$ $J$=1-0 only have a slight variation along the major axis of M$\,$82, with higher optical depth in the central region than those on the two sides of the disk. Due to limited sensitivity, we can only obtain optical depths of HNC $J$=1-0 at three positions, thus could not summarise any trend for its optical depth variation. (4) Our measured line ratios between the isotopologues set a lower limit for the abundance ratios of $^{12}$C/$^{13}$C. Using the double method and $I$(H$^{13}$CN/$I$(HC$^{15}$N) ratio, the derived $^{14}$N/$^{15}$N abundance ratios have an increasing gradient from the center to the outer disk. \section*{Acknowledgements} This work is supported by the National Natural Science Foundation of China grant (12041305, 12173067 and 121030243), and the fellowship of China Postdoctoral Science Foundation 2021M691531. We would like to thank P. Salas to provide their data for Figure \ref{fig:M82_cor}. We are grateful to the staff of IRAM 30-m telescope for their kind help and support during our observations. This study is based on observations carried out under project number 186-18 (PI: Feng Gao) with the IRAM 30-m telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain). We acknowledge the Program for Innovative Talents, Entrepreneur in Jiangsu. We acknowledge the science research grants from the China Manned Space Project with NO.CMS-CSST-2021-A08. \appendix \bibliographystyle{mnras} \bibliography{M82} \appendix \section{Other lines:SiO $J$=2-1, HCO $J$=1-0, H$_2$CO $J$=2-1, and HC$_3$N $J$=10-9 } \label {sec:Appendix A} Apart from the isotopic lines, we also detected a few unique molecular lines, such as SiO $J$ = 2$\rightarrow$1, HCO $J$ = 1$\rightarrow$0, H$_2$CO $J$ = $\rightarrow$1, and HC$_3$N $J$ = 10$\rightarrow$9. These lines offer an excellent opportunity to reveal physical conditions of the ISM of M$\,$82. \subsection{SiO $J$=2-1} SiO, as the shock tracer \citep{Usero2006}, is detected in both the central region and at the NE side, while only 3-$\sigma$ upper limits are obtained at the SW side. Such result indicates that the shock at the NE side is stronger than that at the SW side. This is consistent with \cite{Garcia-Burillo2001}, who found extended off-nuclear SiO $J$=2-1 emission on the NE side, indicating strong molecular shock on large scales. On the other hand, \cite{Lester1990, Mouri1989} found higher ratio of H$_2$ 1-0 $S$(1)/Br$\gamma$ at the NE side than that at the SW side, consistent with the scenario that the NE shock is stronger. However, the linewidths of HCN, HCO$^+$ and HNC do not show obvious differences at both sides, indicating that the shocks may not heavily broaden the linewidth and influence the global properties of dense gas. \subsection{HCO $J$=1-0} As shown on the right panel of Figure \ref{fig:H13CN_H13CO+}, HCO $J$=1-0, as a good tracer of PDR \citep{Garcia-Burillo2002, Gerin2009, Martin2009b}, is stronger at the SW side than that at the NE side. In addition, \cite{Garcia-Burillo2002} also found a giant PDR of $\sim$650 pc size in M$\,$82 with HCO 1-0 mapping observation, using the IRAM Plateau de Bure Interferometer. The result suggests that the chemistry of the SW molecular side is dominated by the PDR, which shows typical features of an evolved starburst \citep{Aladro2011b, Fuente2008}. \subsection{H$_2$CO $J$=2-1} With high critical density of 1.6$\times$10$^6$ cm$^{-3}$ \citep{Kennicutt2012}, the H$_2$CO $J$=2-1 transition can also trace dense gas \citep{Bayet2008}, which traces the high-excitation component of the molecular gas in M$\,$82 very well \citep{Muhle2007}. The H$_2$CO 2-1 emission at the SW side is stronger than that at the NE side, indicating higher excitation conditions at the SW side. Such difference may be caused by the asymmetric outflow \citep{Shopbell1998}, which may also impact the NE disk \citep{Seaquist2001, Veilleux2009}. \subsection{HC$_3$N $J$=10-9} As a warm and dense gas tracer\citep{Tanaka2018}, HC$_3$N can be easily destroyed by UV radiation and Cosmic Rays \citep{Rico2020,Costagliola2010}. Therefore, HC$_3$N is mainly excited by collision. HC$_3$N lines are bright enough to be detected in the local galaxies NGC 4418, NGC 253, IC 342 NGC 6240 and so on \citep{Costagliola2011,Aladro2011a,Li2019}. These detections all indicate that the optical depth of HC$_3$N is very small ($\tau\ll1$) in most cases, because it has many energy populations. Assuming similar excitation temperature, the abundance ratio of $I$(H$^{13}$CN)/$I$(HC$_3$N) can be estimated by their intensity ratio. This ratio does not show large variation among different positions, indicating that the distribution of H$^{13}$CN and HC$_3$N might be uniform along the major axis. The results also implied that the HC$_3$N emission is either spatially separated from PDRs, or the PDRs might be weak, because UV photons from PDRs can dissociate HC$_3$N.

Title: Radio monitoring of transient Be/X-ray binaries and the inflow-outflow coupling of strongly-magnetized accreting neutron stars

Abstract: Strongly-magnetized ($B\geq10^{12}$ G) accreting neutron stars (NSs) are prime targets for studying the launching of jets by objects with a solid surface; while classical jet-launching models predict that such NSs cannot launch jets, recent observations and models argue otherwise. Transient Be/X-ray binaries (BeXRBs) are critical laboratories for probing this poorly-explored parameter space for jet formation. Here, we present the coordinated monitoring campaigns of three BeXRBs across four outbursts: giant outbursts of SAX 2103.5+4545, 1A 0535+262, and GRO J1008-57, as well as a Type-I outburst of the latter. We obtain radio detections of 1A 0535+262 during ten out of twenty observations, while the other targets remained undetected at typical limits of $20$-$50$ $\mu$Jy. The radio luminosity of 1A 0535+262 positively correlates with its evolving X-ray luminosity, and inhabits a region of the $L_X$-$L_R$ plane continuing the correlation observed previously for the BeXRB Swift J0243.6+6124. We measure a BeXRB $L_X$-$L_R$ coupling index of $\beta = 0.86 \pm 0.06$ ($L_R \propto L_X^\beta$), similar to the indices measured in NS and black hole low-mass X-ray binaries. Strikingly, the coupling's $L_R$ normalisation is $\sim 275$ and $\sim 6.2\times10^3$ times lower than in those two comparison samples, respectively. We conclude that jet emission likely dominates during the main peak of giant outbursts, but is only detectable for close-by or super-Eddington systems at current radio sensitivities. We discuss these results in the broader context of X-ray binary radio studies, concluding that our results suggest how supergiant X-ray binaries may host a currently unidentified additional radio emission mechanism.

https://export.arxiv.org/pdf/2208.14903

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} accretion, accretion discs -- stars: neutron -- X-rays: binaries -- radio continuum: transients \end{keywords} \section{Introduction} What mechanism underlies the formation of jets remains an important but poorly-understood question across a wide range of astrophysical objects. Jets are observed in combination with accretion processes onto sub-Solar mass objects up to supermassive black holes, as well as during the end-phases of the lives of massive stars, mergers of compact objects, and formation of stars. Accretion-driven, stellar-mass jet sources, such as neutron stars (NSs) and black holes accreting in binary systems, can be studied to observe the jet launching process and its evolution over time-scales from weeks to months. In such systems, X-ray and radio observations probe distinct, but coupled components of the system; the inflow of matter in the accretion flow by the former, and the collimated jet outflow by the latter. The coupling between inflows and outflows in accreting NS and black hole systems has predominantly been established for systems with low-mass donor stars. Among such systems, called low-mass X-ray binaries (LMXBs), black holes were first discovered to show a correlation between their X-ray and radio luminosity \citep{hannikainen98,corbel00,corbel03,merloni03,gallo03}, often spanning multiple orders of magnitude during the quiescent and hard accretion states of outbursts \citep{fender04}. Monitoring of a large sample of black holes has revealed a radio-bright (shallower) and radio-quiet (steeper) track of this correlation, merging as sources decay into quiescence. However, debate remains regarding the origin of these different correlations for black hole systems \citep{gallo2014,soleri11,dincer14,meyer14,drappeau15,espinasse18}. The behavior of NS LMXB in the X-ray--radio (hereafter $L_X$--$L_R$) diagram is more complicated and varied. In individual NS systems, a correlation between the X-ray and radio luminosity has been observed \citep{migliari03,migliari06,tudor17,gusinskaia17,russell18}; however, such couplings across a range of X-ray luminosities have been seen in only a small number of sources, and, surprisingly, different couplings have been observed between different outbursts of the same target \citep[e.g.][]{gusinskaia20}. As a sample, as a sample, the NS LMXBs are significantly radio-fainter than the sample of black hole systems, complicating the study of their $L_X$--$L_R$ behaviour \citep{fender01,vandeneijnden2021}. The source class, as a sample, shows an overall correlation in the $L_X$--$L_R$ diagram whose coupling index is similar to the black hole sample \citep{gallo18}; however, individual sources have been observed to follow steeper indices \citep{migliari06,gusinskaia17,gusinskaia20}. Therefore, while the in- and outflow in individual NS LMXBs are thought to be coupled, it remains unclear whether a single correlation can describe the entire source class. Similar monitoring in the X-ray and radio band has, to date, rarely been performed for high-mass X-ray binaries (HMXBs): binary systems wherein the compact object accretes from a massive, early-type O/B donor star, with a mass typically exceeding $10$ $M_{\odot}$. Based on the donor star type and the mode of accretion, HMXBs are often divided into three broad categories. Be/X-ray binaries (BeXRBs), Supergiant X-ray Binaries (SgXBs), and Superfast X-ray Transients (SFXTs). BeXRBs are systems with a Be-type donor star \citep{porter03}, characteristically showing optical emission lines and an IR excess due to the presence of a decretion disk around the star \citep{reig11}. The great majority of BeXRBs are transient systems, with typically long and eccentric orbits, . Accretion outbursts can occur close to periastron passages, as the compact object moves through the stellar decretion disk, in what is called a Type-I outburst \citep{okazaki01}. Alternatively, giant outbursts (also known as Type-II outbursts) can occur at any orbital phase, may last longer than the orbital period, and reach higher X-ray luminosities than Type-I outbursts. The trigger of giant outbursts currently remains debated, but may be related to the properties of the Be-star disk, increased Be-star activity, and instabilities driven by the interaction between the NS and decretion disk \citep{okazaki01,moritani13,martin14,monageng17,laplace17}. The second and third general types of HMXBs both host supergiant O/B stars as the secondary, but differ in their X-ray properties. The SgXBs persistently emit at X-ray energies, typically between $10^{35}$ erg/s and several times $10^{37}$ erg/s, although they can vary in luminosity and accretion state \citep{sidoli2018}. SFXTs, on the other hand, are typically X-ray faint, with luminosities between $L_X \sim 10^{32}$--$10^{34}$ erg/s \citep{sidoli17}. However, as their name suggests, SFXTs show occasional flares lasting a number of kilo-seconds, reaching up to $\sim 10^{37}$ erg/s. The origin of the difference in X-ray properties between SgXBs and SFXTs remains debated, with possible explanations involving the quasi-spherical settling accretion regime \citep{shakura2012} or the influence of the NS magnetosphere \citep{bozzo2008}. Alternatively, HMXBs can be divided based on their compact object. A small minority of known HMXBs host a black hole, with only few confirmed Galactic systems such as Cyg X-1 and MWC 656 (and, e.g. LMC X-1 and LMC X-3 in the Large Magellanic Cloud). The former persistently accretes from the stellar wind of their donor star, showing strong variability in X-rays. MWC 656, on the other hand, is the only confirmed black hole BeXRB \citep{casares2014}, and has just one recorded X-ray outburst \citep{williams2010}. Both black hole systems have been detected at radio frequencies and monitored across limited ranges of X-ray luminosities \citep{ribo17}. The remainder of the HMXB class, instead, hosts a NS as its primary with a strong, $B>10^{12}$ G magnetic field and a slow, typically $P>1$ second, rotation period. However, despite their larger numbers, monitoring campaigns including multiple detections have only been obtained for two systems\footnote{Here, we ignore the famous systems Cir X-1 and SS 433. While both have been studied extensively in radio and X-rays, neither is confirmed to be a NS HMXB. For the former, the donor star nature remains debated \citep{johnston16}, while for the latter, the primary is likely to be a black hole \citep[see e.g.][]{fabrika04}. In addition, even if Cir X-1 is a HMXB, its inner accretion flow displays many characteristics of LMXBs instead.}. Firstly, the persistent but X-ray variable NS HMXB GX 301-2 was monitored at radio frequencies at different orbital phases by \citet{pestalozzi09}. The system is (marginally) detected in most observations, regardless of orbital phase, with variable radio flux density and spectrum. The authors postulate that the emission is dominated by stellar emission, with a possible intermittent contribution from a short-lived and weak jet. Secondly, the BeXRB Swift J0243.6+6124 was monitored extensively in radio during its super-Eddington discovery outburst in 2017/2018. Comparing the coupled X-ray and radio properties during the main peak of the giant outburst, \citet{vandeneijnden2018_swj0243} attributed the radio emission to a relativistic jet. Surprisingly, during a later X-ray re-brightening at significantly lower X-ray luminosity, the radio emission re-brightened to similar levels as the main outburst peak \citep{vandeneijnden2019_reb}. Additionally, \citet{vandeneijnden2021} presented a sample of radio observations of active SgXBs and BeXRBs in quiescence, including several detections of the former class\footnote{As well as symbiotic X-ray binaries: strongly-magnetized NSs accreting from the stellar wind of a low-mass, evolved donor in a wide orbit.}. However, those samples did not include monitoring campaigns, but instead consisted of single observations. In this work, we present coordinated radio and X-ray monitoring campaigns of three BeXRBs across four outbursts. Our main goal is to better understand the possibility and properties of jet launching in these systems, in order to constrain jet launching in general. This is critical as currently, no jet model can fully account for the launch and properties of jets by NSs with magnetic fields exceeding $\sim 10^{9}$--$10^{10}$ G \citep{massi08,parfrey16,vandeneijnden2021}. With just a single transient strongly-magnetized accreting NS detected and monitored at radio frequencies, few constraints on such models currently exist. Observing more BeXRBs, across their different outburst types, we aim to obtain more stringent constraints on jet properties across a wider range of source properties and X-ray luminosity. \subsection{Targets: GRO J1008-57, SAX J2103.5+4545, 1A 0535+262} \label{sec:targets} For this study, we obtained radio and X-ray monitoring of the 2019 Type-I outburst and the 2020 giant outburst of GRO J1008-57, the 2020 giant outburst of SAX J2103.5+4545, and the 2020--2021 giant outburst of 1A 0535+262. GRO J1008-57 is a regularly-outbursting BeXRB, hosting a NS with a spin period of $\sim 93.7$ seconds in an orbit of $\sim 249.5$ days with an eccentricity of $e\sim0.68$ around the donor Be star \citep{kuhnel2013}. GRO J1008-57 shows a cyclotron resonance scattering feature (CRSF) at 78 keV \citep{shrader1999,yamamoto2014}, confirming directly that the NS is strongly magnetized --- the line energy implies a field strength around $B\sim(6$--$7)\times10^{12}$ G \citep[see e.g.][for a review]{staubert19}. GRO J1008-57 shows a Type-I outburst at almost every periastron passage, as well as less frequent giant outbursts \citep{kuhnel2013}. The remarkable consistency of its outbursts at each periastron passage allowed us to target its Type-I outburst in June 2019 with a pre-planned observing campaign. Just less than one year later, \textit{NICER} \citep{reynolds2020} and \textit{MAXI} \citep{nakajima2020}, on 2020 May 21 and 22 respectively, reported the onset of a giant outburst, which we subsequently monitored at radio frequencies as well. As the giant outbursts of GRO J1008-57 are typically preceded by an enhanced flux state, this outburst was expected and we were able to trigger early during the outburst rise. SAX J2103.5+4545 is a BeXRB with a NS, spinning at a period of approximately $346$ seconds \citep{hulleman1998}\footnote{See also the \textit{Fermi}/GBM pulse frequency monitoring at \url{https://gammaray.msfc.nasa.gov/gbm/science/pulsars.html}.}). \citet{brumback2018} presented the possible detection of a cyclotron line at $12$ keV, consistent with a $\sim 10^{12}$ G magnetic field. While this detection has not been repeated independently \citep[see][]{staubert19}, we assume such a magnetic field in this work given the HMXB type and long spin period of the NS. SAX J2103.5+4545 is not a typical BeXRB. Compared to other systems of similarly long spin, it has a relatively short orbit of 12.7 days \citep{baykal2007,camero2007,corbet86}. Secondly, its super-orbital X-ray behaviour is atypical. As summarized by \citet{reig2014}, it displays low and high X-ray flux states lasting months \citep{reig2010}, typically around $\sim 1.5\times10^{35}$ erg/s in the former and one order of magnitude higher in the latter state. During the high flux state, which starts with a bright and short flare, SAX J2103.5+4545 shows outbursts, similar to other BeXRBs. In August 2020, \citet{grishina2020} reported the optical brightening of this BeXRB. We subsequently triggered radio and X-ray monitoring, which we halted after a single radio non-detection, as the outburst peak had already passed, making later radio detections unlikely. While not strictly a monitoring campaign, we include this radio observation in our study. Finally, 1A 0535+262 is a BeXRB showing both Type-I and giant outbursts. Due to its proximity, the Type-I outbursts can typically reach fluxes close to one Crab, while the giant outbursts easily reach several Crab, even when the X-ray binary reaches only $\sim 10\%$ of the Eddington luminosity. Time scales of years separate the giant outbursts, with previous ones occurring in 1980, 1983, 1989, 1994, 2005, 2009, and 2011. The NS has a spin period of $\sim 103$ seconds, while the orbit has an eccentricity of $\sim 0.47$ \citep{finger1994} and a period of $\sim 111$ days \citep{motch1991}. Its CRSF energy suggests a magnetic field strength of $\sim 5\times10^{12}$ keV. In November 2020, \citet{mandal2020} reported the onset of a new giant outburst, which reached the highest peak X-ray flux observed from this source ($\sim 11$ Crab). During previous outbursts, \citet{tudose10} and \citet{migliari11} obtained radio observations of 1A 0535+262 but did not detect a radio counterpart, with 4.9-GHz flux density upper limits of $210$ $\mu$Jy and $160$ $\mu$Jy, respectively. At the start of the 2020-2021 outburst, we reported the detection of the radio counterpart at $39\pm4$ $\mu$Jy \citep{vandeneijnden2020_a0535atel}. In this paper, we report on the remainder of that observing campaign. \section{Observations and data analysis} In this section, we describe the radio and X-ray observations, data reduction, and analysis. Given the number of observations, targets, and observatories, we discuss these topics in relatively general terms here. The actual fitted parameters and fluxes/flux densities, as well as observation details such as ObsID, dates, and observing times, are tabulated in the Online Supplementary Materials. \label{sec:obs} \subsection{Radio observational campaign setup and analysis} The radio observations of the three targets were taken with the Karl G. Jansky Very Large Array (hereafter VLA; for SAX J2103.5+4545 and 1A 0535+262) and the Australia Telescope Compact Array (ATCA; for GRO J1008-57). SAX J2103.5+4545 was observed once, 1A 0535+262 was observed over twenty epochs, and the giant and Type-I outbursts of GRO J1008-57 were targeted with five and six radio observations, respectively. All VLA observations were taken in 3-bit mode at C band, with a central frequency of $6.0$ GHz and a bandwidth of $4.0$ GHz. These observations span a range of different configurations, including several non-standard configurations in transitions (i.e. BnA$\rightarrow$A and A$\rightarrow$D)\footnote{\url{https://science.nrao.edu/facilities/vla/proposing/configpropdeadlines}}. The ATCA data were obtained simultaneously at two central frequencies of 5.5 GHz and 9.0 GHz, both with $2.0$ GHz of bandwidth. Again, multiple array configurations were used\footnote{\url{https://www.narrabri.atnf.csiro.au/operations/array\_configurations/configurations.html}}: 6A for the Type-I outburst monitoring of GRO J1008-57; and both 1.5C and H214 for its giant outburst. For the observation of SAX J2103.5+4545, the primary and nearby secondary calibrator were 3C 286 and J2102+4702. For both campaigns of GRO J1008-57, 0939-608 was the secondary calibrator. In the Type-I outburst campaign, only 0823-500 was used as the primary calibrator; in the giant outburst observations, either 0823-500 or 1934-638 was used as primary calibrator, depending on their visibility. Similary, for the 1A 0535+262 observations, we used either 3C 286 or 3C 48 as the primary calibrator, depending on the time of the observation. For this final campaign, we used J0547+2721 as the secondary calibrator. For all four observing campaigns, we used standard practices in \textsc{common astronomy software application} \citep[\textsc{casa};][]{mcmullin07} v5.4.1 to flag, calibrate, and image the radio data. We used a combination of manual inspection and automated routines to flag RFI and other data quality issues, before performing standard calibration steps. We then used the multi-scale, multi-frequency \textsc{tclean} task to image the field. For this final step, we use robust parameters of $1.0$ for 1A 0535+262, $1.0$ for SAX J2102.5+4545, and $-0.5$/$0.0$ (5.5/9 GHz) for GRO J1008-57, chosen per target and frequency to optimize the balance between sensitivity and imaging artefacts. If the target was detected, we used the \textsc{imfit} task to measure its flux density by fitting a 2D elliptical Gaussian profile with FWHMs and position angle equal to the synthesized beam's minor and major axis and angle. Using \textsc{imfit}, we also measured the source position. We measured the RMS sensitivity of the observation over a nearby region devoid of point sources. We then also re-imaged individual sub-bands of $1$-GHz width to determine the radio spectral index. If no radio emission was detected from the source, we determined the RMS sensitivity over the source position and tripled this value to obtain the $3-\sigma$ flux density upper limit. \subsection{X-ray data reduction} In order to assess the X-ray properties and measure the X-ray flux at or close to the time of the radio observations, we used publicly available observations from four instruments: the X-ray Telescope \citep[XRT;][]{burrows04} and Burst Alert Telescope \citep[BAT;][]{barthelmy05} aboard the \textit{Neil Gehrels Swift Observatory} \citep[hereafter \textit{Swift};][]{gehrels04}, as well as the \textit{Monitor of All-sky X-ray Image}/Gas-Slit Camera \citep[\textit{MAXI}/GSC;][]{matsuoka09} and \textit{Neutron Star Interior Composition ExploreR}/X-ray Timing Instrument \citep[\textit{NICER}/XTI;][]{gendreau2016} mounted on the International Space Station. The \textit{Swift}/BAT data were obtained via the Hard X-ray Transient Monitor webpage\footnote{\url{https://swift.gsfc.nasa.gov/results/transients/}}, which hosts light curves of the average daily and orbital X-ray flux in the $15$--$50$ keV band for known X-ray sources. We did not perform additional analysis of the \textit{Swift}/BAT data but instead employed it as long-term reference light curves of the three sources and as evenly-spaced monitoring observations for the targeted outbursts. For the other three observatories, we extract and model the X-ray spectra to measure the X-ray flux. For the \textit{NICER} observations, only available for 1A 0535+262, we downloaded the observations from the \textsc{heasarc}\footnote{\url{https://heasarc.gsfc.nasa.gov}} and re-ran the level 2 data reduction tool \textsc{nicerl2} v1.6 to apply the latest version of the calibration, accessed via the online \textsc{caldb}. We then extracted the source spectrum using \textsc{xselect}, selecting all counts in the energy range $0.5$-$10$ keV across the entire field-of-view, since \textit{NICER} is a non-imaging instrument. Due to the very high count rates of the target \cite[i.e. often greatly exceeding $10^3$ ct/s, compared to a typical background rate $< 1$ ct/s;][]{remillard2021}, we did not generate a background spectrum. Finally, when fitting the spectra, we used the pre-calculated instrument response files \textsc{nixtiref20170601v002.rmf} and \textsc{nixtiaveonaxis20170601v004.arf} for each observation\footnote{As discussed in the Online Supplementary Materials, we tested whether using the recently developed \textsc{nicerarf}, \textsc{nicerrmf}, and \textsc{nicer\_bkg\_estimator} tools to determine observation-specific response and background files, resulted in significant differences in the fits. As it did not -- and, importantly, since they also left soft instrumental features remaining -- we instead used the pre-calculated response files in our analysis.}. For the \textit{Swift}/XRT observations, we used the online data reduction pipeline \citep{evans09}\footnote{\url{https://www.swift.ac.uk/user\_objects/}} to extract X-ray source and background spectra, as well as instrument response files. This pipeline automatically corrects for pile-up at high count rates, which is particularly relevant for our analysis of 1A 0535+262. Finally, for \textit{MAXI}, we used the On-demand Process tool (\url{http://maxi.riken.jp/mxondem/}) to extract spectrum and response files. Given the lower sensitivity and spatial resolution of \textit{MAXI} compared to \textit{Swift} and \textit{NICER}, we only used \textit{MAXI} spectra for the giant outburst of GRO J1008-57, where no pointed observations by the latter two observatories are available. \subsection{X-ray spectral fitting} After extracting the X-ray spectra, we used \textsc{xspec} v12.10.1 to model the emission, using the \textsc{tbabs} model with abundances from \citet{wilms00} and cross-sections from \citet{verner96} to account for interstellar absorption. All errors and confidence intervals quoted in this paper, appendices, and supplementary material, are calculated at the $1-\sigma$ level. \subsubsection{SAX J2103.5+4545 and GRO J1008-57} For SAX J2103.5+4545, only two pointed X-ray observations are available close in time to the single VLA radio observation: \textit{Swift}/XRT observations were taken $5.2$ and $6.8$ days after the radio observation. Due to the low number of total counts in both Photon Counting-mode (PC) observations, we used C-statistics \citep{cash1979} in an energy range of $1$-$10$ keV to attempted fits with two simple, phenomenological models: an absorbed black body (\textsc{tbabs*bbody}) and an absorbed power law (\textsc{tbabs*powerlaw}). For both spectra, the latter model fitted significantly better and returned a non-zero absorption column, contrary to the black body model and consistent with earlier X-ray analyses of this target \citep[$N_H = (2.9-4.4)\times10^{22}$ cm$^{-2}$;][]{brumback2018}. During the 2019 Type-I outburst of GRO J1008-57, \textit{Swift}/XRT observed the target in a cadence coordinated with the ATCA radio observations. Of these six observations, the first two were taken only in PC mode due to the low source flux at the start of the outburst. For the third and sixth observations, both PC and Window Timing (WT) mode data are available with a sufficient exposure time, while for observation four and five, only WT-mode data were taken. Similar to SAX J2103.5+4545, we use C-statistics in the $1$-$10$ keV range and attempt fits with the same two phenomenological models (an absorbed black body or power law). We find that, in all six spectra, the absorbed power law model provides a statistically better fit. Additionally, making use of the larger number of total counts compared to SAX J2103.5+4545, we also attempt to fit a combined model, \textsc{tbabs*(bbody+power law)}. However, in none of the six spectra does the addition of a second spectral component significantly improve the fit: in all cases, $\Delta C < 3$ with two additional parameters, which does not correspond to a significant improvement\footnote{For a single extra free parameter, a $3-\sigma$ improvement would require $\Delta C \geq 9$ \citep{cash1979}; for two extra free parameters, this change should be even larger.}. During the 2020 giant outburst of GRO J1008-57, no pointed X-ray observations were taken. Therefore, we instead analysed the \textit{MAXI}/GSC spectra to measure the flux during or close to the five radio observations. For each MJD with available \textit{MAXI} data during the radio-monitored part of the outburst (i.e. MJD 58983--59025), we extracted a spectrum by combining all source counts from that MJD. During the final six days of the above period, the X-ray flux had dropped significantly; therefore, at those times, we instead generated two spectra by combining three days of observations (i.e. combining MJD 59020--59022 and MJD 59023--59025). We then fitted these spectra using $\chi^2$ statistics, as the \textit{MAXI} pipeline automatically bins the spectra to a sufficient number of counts for this approach. To constrain the absorption column despite the low number of counts and energy band of $2$--$10$ keV, we fit all seventeen spectra jointly, tying $N_H$ between the spectra. We find that the spectra are better described by an absorbed power law than a black body model ($\chi^2_\nu = 1.04$ vs $1.16$ for $466$ free parameters in both cases). \subsubsection{1A 0535+262} Finally, the 2020/2021 giant outburst of 1A 0535+262 was monitored in extensive detail by both \textit{NICER} and \textit{Swift}. We analyse observations up to MJD 59300, since the final radio observation was taken on MJD 59279. Given the extremely bright nature of the outburst (brighter than any previous outburst monitored by \textit{Swift}/BAT; cf. Figure \ref{fig:LC_A0535}), simple phenomenological models do not yield statistically acceptable fits. Instead, we follow the approach applied by \citet{jaisawal2019} to fit joint \textit{NICER} and \textit{NuSTAR} spectra of the bright giant outburst of Swift J0243.6+6124. As we solely intend to accurately measure the flux, and do not study the spectral evolution in this work, we employ their Model I, a phenomenological model defined as \textsc{tbabs*(bbodyrad + cutoffpl + gauss + gauss + gauss)}. The three Gaussian components in this model correspond to three narrow iron lines in the range $6.4$--$7$ keV\footnote{Note that we leave out the iron edge included in the original model by \citet{jaisawal2019} as we do not find significant evidence for its necessary inclusion in our spectral model.}. However, fitting this, or any other model, to the \textit{NICER} spectra, runs into issues for the observations at the highest flux. At those fluxes, instrumental response residuals appear below $3$ keV, particularly between $1$ and $2$ keV. One can see this effect in the five \textit{NICER} spectra, taken around the peak of the 1A 0535+262 giant outburst, shown in the top panel of Figure \ref{fig:nicerfits}. The bottom panel shows the $\Delta \chi^2$ residuals compared to a model fit between $1$ and $10$ keV, vertically offset for clarity, highlighting how at higher fluxes, the instrumental residuals are present at high significance. While the model fit, as shown in the top panel, clearly describes the broadband spectral shape and can provide a reliable flux estimate, it is not formally statistically acceptable. Moreover, for poor fits with $\chi^2_\nu \geq 2$, \textsc{xspec} does not allow for the calculation of errors on parameters and fluxes. Therefore, we employ the following approach to fitting the \textit{NICER} spectra: using the above model as a starting point, we run an automated fitting script written in \textsc{tcl}\footnote{Available with the data reproduction notebook for this paper; see the Data Availability Section.} to first fit the model in the $1$--$10$ keV range. If no decent broad-band fit is obtained due to the presence of instrumental residuals (i.e. $\chi_\nu^2 > 2$) the energy range is limited instead to $2$--$10$ keV. We note again that using \textit{NICER}-observation-specific response files and backgrounds does not alleviate this issue. For the \textit{Swift} spectra, we employ the same fitting script, but find that the fitting range does not require restriction. With this automated approach, further inspired by the large number of both \textit{Swift} and \textit{NICER} observations, two issues require careful attention. Firstly, we confirmed that all poor fits are indeed driven by instrumental residuals, instead of an incorrect or incomplete model, by searching for asymmetric residual structures and unphysical parameters. This conclusion is further confirmed by comparing the \textit{NICER} flux measurement with \textit{Swift} results, revealing that both instruments show a consistent flux evolution and therefore that instrument-specific effects do not significantly change the measured fluxes. Secondly, at the highest fluxes, all three narrow Gaussian lines are clearly present and fitted in the \textit{NICER} spectra. However, the inclusion of three such components may cause \textsc{xspec} to diverge at lower fluxes, as such narrow features can then also be fitted to any local noise deviation. Therefore, when we notice that a fit does not converge, we instead apply a similar scripted fit, first trying a single Gaussian instead, and if the issue persists, no Gaussian at all. For this reason, we ended up not including a Gaussian component in any of the \textit{Swift} spectra. The number of Gaussian components fitted to each \textit{NICER} spectrum is listed in Table 1 in the Online Supplementary Materials. \subsubsection{Flux measurements and cross-checks} After finishing each of the aforementioned spectral fits, we use the \textsc{cflux} convolution model to measure the unabsorbed flux and its error in the standard energy band of $0.5$--$10$ keV. As stated earlier, all relevant remaining details regarding the X-ray analysis, such as the exact ObsID, date and length of the observations, instrument setup, fitted model parameters with errors and quality of the final fit, as well as all the measured fluxes and errors, are presented in the Online Supplementary Materials. In addition, we calculate the $1$-$10$ keV and $2$-$10$ keV fluxes, which we include as machine-readable Online Supplementary Data tables to allow comparison with other works using these different energy ranges. Since \textit{MAXI} is not a pointed instrument, we explicitly checked whether fitting the $2$--$10$ keV \textit{MAXI} spectrum introduces biases in the flux determination for the giant outburst of GRO J1008-57. Therefore, we also extracted a \textit{MAXI}/GSC spectrum on the same day as the brightest Swift observation of the Type-I outburst of this source (ObsID 31030152; MJD 58652). We then jointly fit the \textit{Swift}/XRT PC and WT mode $1$-$10$ keV spectra and the \textit{MAX}/GSC $2$-$10$ keV spectrum with an absorbed power law model. The inferred power law indices are consistent at $1-\sigma$ between the two instruments. However, the \textit{MAXI} fit systematically overestimates the flux by $70$\% compared to \textit{Swift}, in each of the three considered flux energy bands. Given the pointed nature of the \textit{Swift} observations, and the significantly lower spatial resolution of \textit{MAXI}, we expect the \textit{Swift} flux measurement to be more accurate. Since the giant and Type-I outburst of GRO J1008-57 traverse similar X-ray regimes (with peak BAT rates within a factor $\sim 2$), we correct the giant outburst \textit{MAXI} fluxes by dividing by a factor 1.7. \subsection{Matching X-ray and radio observations} To place the observations on the X-ray binary $L_X$--$L_R$ diagram, the radio observations need to be matched up with the best estimate for the quasi-simultaneous X-ray flux. For the Type-I outburst of GRO J1008-57, this is straighforward, since each radio observation has an associated \textit{Swift}/XRT pointing, coordinated to be taken within a day. For the giant outburst of this same source, we use the \textit{MAXI} flux from the spectrum extracted from the day of (or the range of days covering) the radio observation. The only exception is the first radio observation, which was not covered by \textit{MAXI} data. Therefore, we instead performed a linear interpolation between the measured logarithmic fluxes measured as close as possible before and after the radio observation. For 1A 0535+262, the majority of radio observations were obtained on a day where either \textit{Swift} or \textit{NICER} (or both) observed the target. Then, we associated the radio observation with the closest X-ray observation. Otherwise, we again performed a linear interpolation of the logarithmic flux measured one day earlier and later. The final case, of SAX J2103.5+4545, is somewhat more complicated, as X-ray measurements are only available after the radio observation, preventing interpolation. Therefore, we instead take the X-ray flux measured by \textit{Swift} closest to the radio observation, and scale it by the ratio of \textit{Swift}/BAT daily count rates between those dates. This implies an increase in flux of $\sim 66$\%, under the assumption that the X-ray spectrum did not significantly change shape. \subsection{A note on distances} For all three targets considered in this work, several distances estimates exist in the literature. For consistency, we calculate distances based on the Gaia eDR3 parallax measurements \citep{bailerjones2020}. Following \citet{atri2019}, we apply the Galactic distribution of LMXBs as a prior when converting parallaxes to distances\footnote{We find consistent results when using the more standard exponentially decreasing space density prior instead.} and apply a zero-point correction. At a $68$\% confidence level, we find distances of $1.79^{+0.08}_{-0.07}$ kpc for 1A 0535+262, $3.55^{+0.17}_{-0.15}$ kpc for GRO 1008-57, and $6.23^{+0.55}_{-0.47}$ kpc for SAX J2103.5+4545. Finally, we find a distance of $5.21^{+0.32}_{-0.28}$ kpc for the BeXRB Swift J0243.6+6124, which will be relevant in Section \ref{sec:results}. These derived distances can be included in our modelling while taking their uncertainties fully into account. We note that they are consistent with the aforementioned literature estimates based on other techniques: $\sim 2$ kpc \citep{bailerjones18} for 1A 0535+262; $3.6^{+0.4}_{-0.5}$ kpc \citep{arnason2021} and $\sim 5$ kpc \citep{coe1994} for GRO J1008-57; and $\sim 6.5$ kpc \citep{reig2004} and $\sim 4.5$ kpc \citep{baykal2007} for SAX J2103.5+4545. \section{Results} \label{sec:results} \subsection{Light curves} In Figures \ref{fig:LC_A0535}, \ref{fig:LC_J2103}, and \ref{fig:LC_J1008}, we show the X-ray and radio monitoring light curves of our three targets, 1A 0535+262, SAX J2103.5+4545, and GRO J1008-57, respectively. All plotted pointed observations, i.e. radio flux densities and X-ray fluxes, are listed in the Online Supplementary Materials. The top panel in Figure \ref{fig:LC_A0535} shows the long-term \textit{Swift}/BAT daily monitoring light curve of 1A 0535+262, re-scaled to Crab units, over the past sixteen-and-a-half years. The dashed and dotted black lines indicate the times of the earlier radio observations, by \citet{tudose10} and \citet{migliari11}, respectively (non-detections with $210$ and $160$ $\mu$Jy upper limits). The red lines indicate the time range plotted in the zoomed-in bottom light curve. The 2020-2021 giant outburst was clearly brighter than any observed previously with \textit{Swift}/BAT, reaching $\sim 11$ Crab, compared to $\sim 6$ Crab in 2009, during the second-brightest outburst. Type-I outbursts can be seen as the short-duration spikes, reaching up to $\sim 1$ Crab fluxes. In the bottom panel, we plot the \textit{Swift}/XRT, \textit{NICER}, and VLA light curves of the 2020-2021 outburstof 1A 0535+262, in red, black, and blue, respectively. The \textit{Swift}/BAT light curve is shown as well, scaled by a single arbitrary factor to allow for the comparison of its shape to the light curve of pointed X-ray observations. While the outburst becomes visible in the \textit{Swift}/BAT monitoring between MJD 59140 and 59150, pointed X-ray observations start around MJD 59160, when the X-ray flux had already reached $\sim 7\times10^{-9}$ erg/s/cm$^2$. As shown by both the daily \textit{NICER} and the less-frequent \textit{Swift}/XRT observations, the outburst rise continued for two weeks, peaking at a flux of $\sim 5.8\times10^{-8}$ erg/s/cm$^2$ on MJD 59174. Subsequently, the initial outburst decayed gradually until MJD 59194, where the \textit{Swift}/BAT monitoring reveals an acceleration of the flux decrease during a gap in the \textit{NICER} monitoring. 1A 0535+262 later reaches a relatively stable, low-flux plateau, decaying the flux from $\sim 2.5\times10^{-10}$ to $8\times10^{-11}$ erg/s/cm$^2$ between MJD 59214 and 59246. Finally, as shown predominantly by the \textit{Swift} monitoring, the X-ray flux reaches a higher-flux plateau, stabilizing in the range of $(5$--$10)\times10^{-10}$ erg/s/cm$^2$. While the \textit{Swift}/XRT monitoring has continued after MJD 59300, we end the light curve due to the lack of radio data. In the VLA radio monitoring, 1A 0535+262 is detected in the first $9$ observations at $\geq 3-\sigma$ significance, with flux densities between $12.5\pm3.9$ $\mu$Jy and $39.2\pm4.0$ $\mu$Jy. In Figure \ref{fig:image_A0535}, we show the 6-GHz field around the target in the second observation, highlighting the faint but clear counterpart of the BeXRB. The best-fit source position in this image is: \begin{align*} &\text{RA (J2000)} = 05\text{h } 38\text{m } 54.571\text{s } \pm 0.008\text{s} \\ &\text{Dec (J2000)} = 26^{\rm o} 18\text{' } 56.79\text{" } \pm 0.09\text{"} \end{align*} \noindent The aforementioned flux densities, observed in the first and second radio observations, show how the radio flux density increased during the outburst rise, peaking close to the time of the X-ray peak. During radio observations 3--9, the flux density globally decayed, as the X-rays peaked and subsequently decayed as well. This radio flux density decrease is, however, very gradual, and the difference in peak times in X-rays and radio may be affected by a changing radio spectral shape. During the tenth observation, radio emission is still observed at the position of 1A 0535+262, albeit at less than $3\-\sigma$ significance: $9.8\pm4.1$ $\mu$Jy. We detect no radio emission from the source position in any of the remaining ten observations (i.e., starting on MJD 59191). The resulting upper limits on the radio flux density, below the typical radio levels during the outburst peak, show that the radio emission is enhanced during the accretion outburst and is not present (at detectable levels) for X-ray fluxes below $2.4\times10^{-8}$ erg/s/cm$^2$. Finally, based on our results, the earlier radio-non-detections by \citet{tudose10} and \citet{migliari11} can be attributed to the lower sensitivity in those observations. To study the spectral shape, we divided the full 4-8 GHz observing band into four sub-bands of 1 GHz width, thereby roughly halving the sensitivity per band. During the brightest radio epoch (observation two), we obtain the best single-observation constraint on the spectral index $\alpha$ (where the flux density scales with frequency as $S_\nu \propto \nu^{\alpha}$). Even in this observation, however, we measure a relatively poor measurement of $\alpha = -0.9 \pm 1.1$, encompassing the range of expected indices for both optically-thin discrete ejecta (i.e. $\alpha = -0.7$) and unresolved, compact jets ($\alpha \geq 0$) \citep{fender04,russell13}. To increase signal-to-noise, we then repeated this procedure combining the three brightest observations (2, 3, and 4), all taken in the same array configuration. There, we measure a spectral index $\alpha = -0.1 \pm 0.1$, consistent with a flat-spectrum radio jet. The radio spectrum for both cases is shown in the Online Supplementary Materials. We now turn to SAX J2103.5+4545, shown in Figure \ref{fig:LC_J2103}. Here, we plot the \textit{Swift}/BAT light curve for the one hundred days around the VLA observation, scaled to arbitrary units similar to the bottom panel of Figure \ref{fig:LC_A0535}. The X-ray fluxes, measured from the two Swift/XRT pointed observations, are shown in red. These two observations clearly capture the behaviour during the decay of the outburst, just before the outburst cannot be clearly identified anymore in the \textit{Swift}/BAT light curve. The VLA observation occurs slightly earlier, albeit also during the outburst decay, and returns a non-detection. The 6-GHz image RMS is $6$ $\mu$Jy, resulting in a $3-\sigma$ upper limit on the flux density of $18$ $\mu$Jy. No further pointed observations, in the radio or X-ray band, were performed afterwards. Finally, in Figure \ref{fig:LC_J1008}, we show the long-term \textit{Swift}/BAT monitoring and our pointed X-ray and radio monitoring during two outbursts, of GRO J1008-57, in similar fashion to Figure \ref{fig:LC_A0535}. The upper light curve confirms that GRO J1008-57 is a prolific outbursting source, showing Type-I outbursts every orbital period (separated by $249.5$ days), as well as multiple giant outbursts. On two occasions since 2012, we can identify the occurrence of two giant outbursts in between successive Type-I outbursts: around the start of 2015 and in 2017. Moreover, in 2012 and 2020, the regular Type-I outburst is followed by a variable state of enhanced X-ray flux and then followed by a giant outburst. Evidence for this effect can also be seen before the giant burst in early 2015 and the brightest of the two giant bursts in 2017. Our radio and X-ray monitoring campaigns targeted the 2019 Type-I outburst and the 2020 giant outburst, where we were able to catch the outburst rise early due to the the enhanced flux state before the latter outburst. In the two bottom panels for Figure \ref{fig:LC_J1008}, we show the zoomed in light curves of these two outbursts, with X-ray fluxes measured from the \textit{Swift}/XRT observations in the left panel, and those measured from the \textit{MAXI}/GSC spectra in the right. In both cases, the radio monitoring cadence samples the outburst evolution well. However, we note the difference in the scaling on the horizontal axis; the separation between the radio observations in the Type-II outburst is longer and less regular, due to the triggered nature of the campaign. In none of the eleven ATCA observations, we detect any significant radio emission at either $5.5$ or $9$ GHz at RMS sensitivities usually ranging between $\sim 7$ and $10$ $\mu$Jy (at $9$ GHz), leading to typical $3-\sigma$ upper limits of $\sim 20$--$30$ $\mu$Jy. The higher upper limit in the final observation during the giant outburst ($66$ $\mu$Jy) results from a change in ATCA configuration to a compact H214 configuration. When we stack the four first giant outburst observations, or all Type-I observations (i.e. those taken in the same setup and configurations), we also do not detect a counterpart, down to slightly deeper levels than the individual observations (See the Online Supplementary Materials). \subsection{The X-ray -- radio luminosity plane} Combining the X-ray and radio flux (density) measurements taken close in time (see Section \ref{sec:obs}), we can place our three targets on the X-ray binary $L_X$--$L_R$ plane. For this purpose, we calculated the radio luminosity (upper limit) at $6$ GHz, assuming a flat spectrum, and we assume the distances listed in Section \ref{sec:targets}. In Figure \ref{fig:lxlr_full}, we show the resulting luminosities alongside three comparison samples, all taken from \citet{vandeneijnden2021}: black hole LMXBs shown as the grey crosses, NS LMXBs shown as grey circles, and persistent NS HMXBs shown as black squares. The transient BeXRBs are shown as the colored and filled-in data points: SAX J2103.5+4545, GRO J1008-57, and 1A 0535+262 as octagons of different colors per source and outburst type, and archival Swift J0243.6+6124 data as purple squares. Below X-ray luminosities of $10^{37}$ erg/s, the BeXRB sample is dominated by radio non-detections. The only exceptions are a single detection of 1A 0535+262, as well as the four radio detections of Swift J0243.6+6124 obtained after its main giant outburst \citep{vandeneijnden2019_reb}. To highlight the observations of GRO J1008-57 within the cluster of data points surrounding it, we show a zoomed version of the radio -- X-ray luminosity plane in Figure \ref{fig:lxlr_J1008} with the other sources faded out. Between the Type-I and giant outbursts, our observations spanned a factor $\sim 25$ in X-ray luminosity. The two outbursts also overlap in X-ray luminosity. However, no radio emission is detected in either outburst. When we consider this crowded region of the radio -- X-ray luminosity plane for all targets, the archival radio detections of Swift J0243.6+6124 stand out. These data points lie above the majority of radio upper limits for the other three targets at similar X-ray luminosity. This discrepancy hints towards a difference in radio behaviour between main outbursts and X-ray re-flares, which we will discuss in more detail in Section \ref{sec:discussion}. The sensitivity limits of current radio observatories in common monitoring observation lengths are clearly visible in Figure \ref{fig:lxlr_full}: approximately $2$--$3\times10^{27}$ erg/s for sources located at distances of the order of $\sim5$ kpc and approximately $3$--$5\times10^{26}$ erg/s for sources located at $\sim2$ kpc. The close distance to 1A 0535+262 has been essential in detecting its radio emission. From this sample of four BeXRBs, it appears that current radio telescopes may detect radio emission if: i) the source is located at relative close distances (i.e. $\leq 2$ kpc); ii) the source accretes close to, or at, super-Eddington luminosities; \textit{or} iii) the source is observed during re-flaring activity after a main giant outburst. \subsection{A possible X-ray/radio correlation for transient BeXRBs} Based on the radio monitoring of the main outburst of Swift J0243.6+6124, \citet{vandeneijnden2018_swj0243} argued that its X-ray and radio luminosity display a global coupling during the outburst decay, measuring the relation $L_R \propto L_X^{0.54 \pm 0.16}$. This measurement, however, did not take into account the initial radio non-detection. In addition, the later observations of Swift J0243.6+6124, presented in \citet{vandeneijnden2019_reb}, were not taken into account. In our extended sample, that now includes four sources, we can expand on this analysis. From Figure \ref{fig:lxlr_full}, it appears that during the peak of its outburst, 1A 0535+262 also followed a coupling between its X-ray and radio luminosity. Moreover, this apparent correlation visually seems to continue the relation seen during the giant outburst decay in Swift J0243.6+6124. Under the simplest assumption that giant BeXRB outbursts follow a similar relation between their X-ray and radio luminosity, we can attempt to determine a global $L_X$--$L_R$ coupling index for this source class and outburst type. To properly measure a giant outburst $L_X$--$L_R$ coupling, we combine the data from the main outburst of Swift J0243.6+6124, all data from 1A 0535+262 and SAX J2103.5+4545, and the giant outburst data from GRO J1008-57. While most radio upper limits for the latter two sources, as well as those for 1A 0535+262 at low X-ray luminosities, are likely unconstraining when assuming a single correlation, other radio limits will have a more significant effect. The early non-detection of Swift J0243.6+6124 and the three X-ray-brightest radio upper limits of 1A 0535+262 appear, by eye, to lie close to any global correlation. Therefore, improving upon \citet{vandeneijnden2018_swj0243}, it is important to properly account for all giant outburst radio non-detections. All radio detections and upper limits used in the fit, are shown in Figure \ref{fig:lxlr_fitted}. We follow the approach originally developed by \citet{kelly2007}, introduced to the study of the $L_X$--$L_R$ plane by \citet{gallo2014}, in the \textsc{LinMix} method: a Bayesian Markov-Chain Monte-Carlo fit of a linear model, fully accounting for upper limits in the dependent variable. We specifically applied the \textsc{python}-version of this method\footnote{Publicly available via \href{https://github.com/jmeyers314/linmix}{https://github.com/jmeyers314/linmix}.}, to fit a model of the form \begin{equation} \frac{L_R}{L_{R,0}} = \xi \left(\frac{L_X}{L_{X,0}}\right)^\beta\text{ ,} \end{equation} where $\xi$ is an arbitrary scaling factor and $\beta$ is the coupling index between the luminosities. Luminosities with a subscript $0$ denote the mean luminosity of the radio-detected observations: $L_{X,0} = 2\times10^{38}$ erg/s and $L_{R,0}=4.2\times10^{27}$ erg/s. To apply the \textsc{LinMix} method, we linearize the model as \begin{equation} \log L_R - \log L_{R,0} = \log \xi + \beta\left(\log L_X - \log L_{X,0} \right) + \epsilon\text{ ,} \label{eq:linmodel} \end{equation} where $\epsilon$ is an additional parameter describing the intrinsic Gaussian scatter around the best fit correlation. To measure the fitted parameters $\log \xi$, $\beta$, and $\epsilon$, we followed \citet{gallo18} and calculate the mean parameter from $10^4$ draws from the posterior distribution (instead using the median returns equivalent results). We repeat this fit $500$ times, following \citet{gusinskaia20}, and report the mean of the 500 parameter estimates as the fitted values. To take into account distance uncertainties, we draw a random distance for each source from the Gaia distance distribution for each of these 500 iterations. However, we find the uncertainties in the fit are dominated by the low signal-to-noise of the radio observations. The $1-\sigma$ errors are calculated by taking, for each of the 500 runs, the $16^{\rm th}$ and $84^{\rm th}$ percentile from the $10^4$ parameter draws, and subsequently averaging those 500 values. We show an example distribution from a single run and the distributions after 500 runs, for $\log \xi$ and $\beta$ in the Online Supplementary Materials. Following the above approach, we measure $\log \xi = 0.057 \pm 0.052$, $\beta = 0.86 \pm 0.06$, and $\epsilon = 0.17^{+0.05}_{-0.03}$. Figure \ref{fig:lxlr_fitted} shows the best fit version of Equation \ref{eq:linmodel} and its uncertainty range. The index $\beta$ is steeper than that measured for Swift J0243.6+6124 alone, although the exclusion of the radio upper limit may have pushed that earlier fit to shallower slopes. Comparing with \citet{gallo18}, we find that the measured index is steeper that that seen in black holes ($\beta = 0.59 \pm 0.02$) and the full sample of NS LMXBs ($\beta = 0.44^{+0.05}_{-0.04}$; however, we note that individual NS systems have been observed to show significantly different coupling indices). The scatter seen for the BeXRBs is smaller, compared to $\epsilon = 0.46 \pm 0.02$ and $\epsilon = 0.43^{+0.05}_{-0.04}$, in the other two samples, respectively. However, this is hardly surprising, as our sample includes only four BeXRBs and the fit was motivated specifically by the similarity in correlation between Swift J0243.6+6124 and 1A 0535+262. The scatter seen for these BeXRBs may, to some degree, be driven by short-time-scale ($< 1$ day) variability in the radio flux density, especially in combination with the association of non-simultaneous X-ray and radio observations separated by up to a day. We cannot directly compare the intercept, $\log \xi$, to \citet{gallo18}, due to the different values of $L_{X,0}$ and $L_{R,0}$. Re-scaling the measured $\log \xi$ value to their values, e.g. $L_{X,0,\rm G+18} = 2.00\times10^{36}$ erg/s and $L_{R,0,\rm G+18}=3.72\times10^{28}$ erg/s, we find that $\log \xi_{\rm G+18} = -2.61$. This value is significantly lower than measured for the black hole and NS LMXB populations, i.e $\log \xi = 1.18 \pm 0.03$ and $\log \xi = -0.17 \pm 0.05$, respectively. This makes the transient BeXRB population, while similar in its inferred coupling index, $\sim275$ times radio fainter than the NS LMXB population, at an X-ray luminosity of $2\times10^{36}$ erg/s. Due to the steeper index for the transient BeXRBs, this difference decreases towards higher X-ray luminosities, as is visible in Figure \ref{fig:lxlr_full}. The two BeXRBs with radio detections, Swift J0243.6+6124 and 1A 0535+262, do not overlap in the X-ray luminosity during the radio monitoring of the main outburst. Therefore, physical differences or observational uncertainties can systematically affect the measured correlation for the entire BeXRB sample. For instance, an incorrect distance measurement for one of the sources may affect the measured slope of the correlation. Similarly, if the magnetic field strength or spin affects the radio luminosity (as discussed in Section \ref{sec:disc_BS}), this would affect the slope and normalization of the coupling: the spin period of Swift J0243.6+6124 is more than 10 times smaller than 1A 0535+262. While this issue plays a role in NS LMXBs as well, their overlap in X-ray luminosity and small differences in spin \citep{patruno17} implies that any distance, spin, and magnetic field effects, affect the scatter more than the slope. With the above considerations in mind, it is an interesting exercise to assess Swift J0243.6+6124 and 1A 0535+262 separately (using $L_{X,0,\rm G+18}$ and $L_{R,0,\rm G+18}$). When we repeat our \textsc{LinMix} fits for 1A 0535+262 individually, we find $\beta = 0.80 \pm 0.26$, $\log \xi = -2.59 \pm 0.24$, and $\epsilon = 0.18\pm0.08$. For Swift J0243.6+6124, we find $\beta = 0.65^{+0.17}_{-0.14}$, $\log \xi = -2.13^{+0.32}_{-0.38}$, and $\epsilon = 0.26^{+0.16}_{-0.02}$. The slopes $\beta$ for the sources individually are consistent with each other, and with the NS LMXBs. However, we note that the reduced number of data points and the smaller range in X-ray luminosity contribute to significantly enhanced uncertainties on $\beta$ \citep[see][for a discussion on the effects of small ranges in X-ray luminosity]{corbel13}. \section{Discussion} \label{sec:discussion} In this paper, we have presented coordinated radio and X-ray monitoring of three transient BeXRBs during four outbursts. No radio counterpart was detected in the eleven ATCA observations of GRO J1008-57 across two outbursts of different types, nor in the single VLA observation of SAX J2103.5+4545. 1A 0535+262, on the other hand, was detected in the first ten of twenty VLA giant outburst monitoring observations. Here, we will discuss the origin and properties of this detected radio emission and its apparent relation to the X-ray luminosity of the BeXRBs, before investigating the origin of their radio faintness. Finally, we will compare our results with the behaviour of Swift J0243.6+6124 during X-ray re-flares and with the radio properties of persistently accreting NS HMXBs. \subsection{The origin of transient radio emission in BeXRBs} \label{sec:whatdowesee} The launch of a relativistic jet from the inner accretion flow can explain the observed radio properties of the three targets. As the only detected source, 1A 0535+262 naturally dominates this interpretation. Firstly, the set of ten non-detections in the tail of the outburst cannot be explained by a decrease in radio sensitivity. Instead, these non-detections appear to be linked to the decrease in X-ray luminosity, connecting the prior radio detections to the presence of accretion. In other words, we do not expect that the radio emission originates from either the NS or the donor star itself, or their interaction, via processes that also operate in quiescence. Secondly, the global correlation between the X-ray and radio luminosity of 1A 0535+262 is consistent with a coupling between an in- and outflow, as commonly observed in LMXBs. Finally, this coupling appears to extend the correlation observed during the Swift J0243.6+6124 giant outburst. A major difference, however, with that Swift J0243.6+6124 data set, is the lack of accurate spectral shape measurements for 1A 0535+262 across the outburst, due to its relative faintness; the only constraining measurement of $\alpha = -0.1 \pm 0.1$ could be obtained by combining the three brightest observations. This value is consistent with a flat-spectrum radio jet and the spectral shape that Swift J0243.6+6124 tended towards as it decreased in X-ray luminosity. In a radio jet scenario, the non-detections of GRO J1008-57, SAX J2103.5+4545, and 1A 0535+262 below $L_X = 5\times10^{36}$ erg/s, can be explained by the limits of observational sensitivity in combination with a coupling between X-ray and radio luminosity. As can be seen in Figure \ref{fig:lxlr_fitted}, the distance to GRO J1008-57 and SAX J2103.5+4545 makes both BeXRBs undetectable in radio below $L_X \approx 0.5$--$1\times10^{38}$ erg/s, given their radio upper limits. These limits are representative of current radio sensitivities in standard monitoring observations (i.e. 4 hours with ATCA). Future observatories are therefore likely needed to detect giant outbursts below $\sim10$\% $L_{\rm Edd}$ for BeXRBs located beyond $\sim 4$ kpc. This statement assumes that all giant outbursts follow a single X-ray -- radio luminosity coupling (but see Section \ref{sec:disc_BS}). In addition to a jet origin, it is essential to consider alternative emission origins. We first consider the Be star in a BeXRB, itself. Isolated Be stars at distances of, typically, tens to hundreds of parsecs, have been observed extensively at radio wavelengths \citep[e.g.][]{taylor1987,taylor1990,drake1990,dougherty1991,clark1998}. The majority of these observations did not yield radio detections, while several sources were detected only in a subset of observations \citep{dougherty1991}. The detections of thermal radio emission of Be stars typically reveal specific radio luminosities between $\sim 1.3\times10^{15}$ to $2.7\times10^{16}$ erg/s/Hz, where the maximum was observed in $\beta$ Mon A by \citet{taylor1990}. Assuming a flat spectral shape, these specific luminosities correspond to $\sim 8\times10^{24}$ to $\sim 1.6\times10^{26}$ erg/s at $6$ GHz, firmly below the deepest upper limits and the radio detections for our four BeXRB targets. An interesting exception is EW Lac, which was observed by \citet{taylor1990} at a specific luminosity of $\sim 10^{17}$ erg/s/Hz, or a luminosity of $\sim 6\times10^{26}$ erg/s at $6$ GHz. While fainter than the majority of radio detections of 1A 0535+262, it is higher than the upper limits on the radio flux of this target during its late outburst decay. However, \citet{dougherty1991} did not detect EW Lac, with an upper limit $5$ times lower than the previous radio detection. This highlights the intrinsic radio variability of Be stars, and the need for coordinated X-ray and radio observations during BeXRB outbursts, to connect radio emission to the presence of accretion in BeXRBs. It similarly shows how, for BeXRBs at small ($<1$ kpc) distances at low X-ray luminosities, the radio flux and variability from the Be star may create a limit to our ability to track the relation between X-ray and radio luminosity into quiescence. We next consider shock interactions between the Be-star's outflow or circumstellar disk and a pulsar wind, thought to be responsible for the radio emission in $\gamma$-ray binaries. These interactions can be ruled out on three grounds. Firstly, it is typically assumed that no pulsar wind is launched by actively accreting pulsars, which would imply that shock radio emission should become visible towards very low accretion rates; in 1A 0535+262, we observe the opposite \citep[although the recent detection of possibly-spin-powered optical/UV pulsations in the accreting millisecond X-ray pulsar SAX J1808.4-3658 could suggest that the pulsar mechanism may still operate during accretion episodes in some accreting NSs;][]{ambrosino2021}. Secondly, since the spin evolution of the pulsar in BeXRBs is (during outburst) regulated by the transfer of angular momentum between accretion flow and NS, we can estimate what their spin down energy $\dot{E}_{\rm spin}$ would be as isolated pulsars. The spin down energy is $\dot{E}_{\rm spin} = 4\pi^2 I \dot{P}/P^3$, for a NS with period $P$, period derivative $\dot{P}$, and moment of inertia $I$. $\dot{P}$ scales with the magnetic field as $B \propto \sqrt{P \dot{P}}$. Combined, this yields \begin{equation} \dot{E}_{\rm spin} \leq 4\times10^{31} \left( \frac{B}{10^{12} {\rm G}} \right)^2 \left( \frac{P}{1 {\rm s}} \right)^{-4} {\rm erg/s}\text{ .} \end{equation} \noindent Using magnetic field estimates from their cyclotron line measurements, we find upper limits for 1A 0535+262 and GRO J1008-57 of $\dot{E}_{\rm spin} \leq 5\times10^{24}$ erg/s and $\dot{E}_{\rm spin} \leq 3\times10^{25}$ erg/s, respectively. The radiative luminosity of a shock between a pulsar wind and the Be-disk will be a fraction of this spin down energy. Therefore, the energetics of a pulsar wind, even if launched, are not sufficient to account for the observed radio luminosities. For SAX J2103.5+454, no cyclotron lines have been detected. However, with its slow spin ($P \sim 346$ s), its upper limit will be even lower. Thirdly, given their (measured or assumed) magnetic field strengths and spin periods, all three sources fall beyond the pulsar death line, implying they are not expected to launch a pulsar wind even if isolated \citep[e.g.][]{ruderman1975,zhang2000}. Recently, \citet{chatzis2021} formulated a new model for the radio emission from BeXRBs hosting a strongly-magnetized NS. In this shock-based model, the radio emission consists of a superposition of a thermal stellar-wind component and a non-thermal synchrotron component. In an X-ray binary analogy to colliding wind binaries, this shock takes place between the stellar wind from the Be star and a non-relativistic outflow launched from the accretion flow. Assuming spherical morphologies for both outflow and stellar wind, and a constant presence of disk outflows at all accretion rates (both sub- and super-Eddington, without any requirements on the exact mechanism), this shock-model derives the resulting radio luminosity, spectrum, and coupling to the X-ray luminosity. The latter is found to be steep in the sub-Eddington regime ($\beta = 12/7$), and dependent on the electron number density distribution in the super-Eddington regime --- $\beta = 2(p-1)/7$, where p is the the power-law index of the electron distribution. Interestingly, the analytical nature of this model allows for fits to the observed X-ray -- radio behaviour of BeXRBs. When fitting the full outburst behaviour of Swift J0243.6+6124, \citet{chatzis2021} find that the sub-Eddington behaviour can be consistently explained via this model for reasonable binary and outflow parameters, as we will return to in Section \ref{sec:biggerpicture}. The super-Eddington properties, on the other hand, cannot be explained in this model, as the inferred shock location is too close to the Be star (i.e. within $\sim 20$ Solar radii). We can consider whether this new approach could account for the radio emission observed in 1A 0535+262. The most reliable method to assess this question, is to perform a full fit to the new data, similar to \citet{chatzis2021}. As such a fit is beyond the scope of this work, we will instead consider some qualitative lines of thought. There are a number of arguments suggesting that, similarly to the super-Eddington phase of Swift J0243.6+6124, these observations may be challenging to explain via such shocks. The similarity in $L_X$--$L_R$ coupling between these two sources presents several issues. Firstly, this coupling does not fit with the predicted $\beta = 12/7$ for the sub-Eddington regime, and secondly, no large change in coupling is observed between the super- and sub-Eddington regime. Observing a similar coupling index in those two regimes requires $p \approx 7$, which is inconsistent with typical values for diffusive shock acceleration \citep[i.e. $p\approx2$--$2.2$;][]{bell1978,matthews2020}. Finally, such a single slope is significantly steeper than the observed $\beta = 0.84 \pm 0.06$. However, we reiterate that this inference is based on only two sources. Another challenge is that the apparently similar correlation between X-ray and radio luminosity is suggestive of a single underlying mechanism. If the super-Eddington Swift J0243.6+6124 data cannot be explained through such shocks, this line of reasoning would also argue against that origin in 1A 0535+262. However, we reiterate that a full fit will shed more light on this question. \subsection{A BeXRB X-ray -- radio luminosity correlation: effects of magnetic field and spin?} \label{sec:disc_BS} Considering the X-ray -- radio luminosity plane, our observations confirm the inference in \citet{vandeneijnden2018_swj0243} that giant BeXRB outbursts are significantly radio underluminous compared to LMXB outbursts. Such a striking difference makes one naturally wonder about its origin. Remarkably, \citet{ribo17} showed how the only-known black hole BeXRB, MWC 656, falls on the black hole X-ray -- radio luminosity correlation at quiescent X-ray luminosities (where the two black hole tracks have converged). While it is rather speculative to extrapolate from a single source, this may suggest that the radio faintness of BeXRBs does not result from the binary or accretion flow properties. Instead, the compact object properties would then appear to play a more significant role in the low radio luminosity of NS BeXRBs. A similar conclusion follows from a comparison with NS LMXBs. During giant outbursts, the NS may accrete from an accretion disk in a similar fashion to accreting NSs in LMXBs. Therefore, the fundamental difference between BeXRB giant outbursts and LMXB hard states, appears to lie in the NS properties: the strong magnetic field in BeXRB truncates the accretion disk at hundreds to thousands of gravitational radii \citep[e.g.][]{tsygankov17}, compared to maximally a few to tens of $R_g$ in LMXBs \citep{degenaar17,vandeneijnden2018_igr17379,ludlam16,ludlam17a,ludlam17b}. Moreover, the NSs in BeXRBs typically have orders of magnitude slower spins than those of NSs in LMXBs \citep{reig11,patruno17}. To quantitatively compare NS jet launching between LMXBs and BeXRBs, we fundamentally assume a single jet launch mechanism underlies this process in both cases. This argues against Blandford-Payne-type jet launch models, i.e. classical magneto-centrifugal jet launch models, as those suggest a maximum NS magnetic field for jet formation that excludes BeXRB NSs \citep[i.e. $\sim 10^9$ G;][]{massi08,kylafis2012}. As an alternative, spin-powered jet launching models do not carry this restriction. From that category of models, we will consider the model proposed by \citet{parfrey16}, although we stress that currently, the main argument for assuming a single model for both source classes comes from Occam's razor instead of direct observational evidence. In the jet-launching model by \citet{parfrey16}, the jet power is provided by magnetic field lines from the spinning NS, opened up by the accretion flow. Importantly, this model proposes that the jet power $L_J$ scales with three physical parameters: the NS spin period $P$, the NS magnetic field $B$, and the mass accretion rate $\dot{M}$: \begin{equation} L_J \propto P^{-2} B^{6/7} \dot{M}^{4/7}\text{ .} \label{eq:L_J} \end{equation} \noindent This equation immediately reveals the fundamental difference with magneto-centrifugal models, as the jet power \textit{increases} with magnetic field strength. This may, however, still be reconciled with the radio faintness of strongly-magnetic BeXRBs, as their NSs spin slowly. More recently, \citet{das2022} presented GRMHD simulations of accreting NSs with a complex magnetic field morphology \citep[inspired by e.g.,][]{riley19}, finding the same relation between jet power, magnetic field, and spin period. In the remainder of this discussion, we will consider this class of models as \textit{magneto-rotational} models. To review these models' scaling with NS and accretion parameters more quantitatively, we can introduce an additional piece of information and an assumption. Firstly, out of the three parameters setting the jet power in Equation \ref{eq:L_J}, only the accretion rate is related to the X-ray luminosity; $L_X \propto \dot{M}$ if we consider a range in mass accretion rate where the accretion flow does not transition between, e.g., radiatively efficient and inefficient. Secondly, we can pose the assumption that the jet power is correlated to jet radio luminosity, for all jet powers relevant to BeXRBs, in similar fashion to black hole jets: $L_R \propto L_J^{1.4}$ \citep{blandford79,markoff01,falcke1996}. With those scalings, Equation \ref{eq:L_J} can be written as \begin{equation} L_R \propto P^{-14/5} B^{6/5} L_X^{4/5} \label{eq:L_R} \end{equation} \noindent This equation takes the same functional form as Equation \ref{eq:linmodel}, fitted to the $L_X$--$L_R$ relation, with $\beta = 0.8$ and $\xi~\propto~P^{-14/5} B^{6/5}$. With our measurements of both $\beta$ and $\xi$ for BeXRBs and the results from \citet{gallo18} for hard-state NS LMXBs, we can assess whether the magneto-rotational models are able to approximately explain the differences between both classes of accreting NSs. Firstly, we can consider the full sample of giant BeXRB outbursts studied in this work. For these combined data sets, we measure a coupling index of $\beta = 0.84 \pm 0.06$, consistent with a slope of $\beta = 0.8$. However, as discussed, we should be careful when we apply Equation \ref{eq:L_R} to any sample of accreting NSs. When doing so, we implicitly assume a single spin and magnetic field for all sources within this sample. In other words, we neglect the dependence of the magneto-rotational models on spin and magnetic field. For 1A 0535+262 and J0243.6+6124 individually, we instead measured $\beta = 0.81 \pm 0.27$ and $\beta = 0.65^{+0.17}_{-0.15}$, respectively. Both slopes remain consistent with each other and with the $\beta=0.8$ value from Equation \ref{eq:L_R}. We can also assess the normalization, and possible effects of the magnetic field and spin, which can be parameterized in this model as $\xi = \xi_0 (P/\text{1 sec})^{-14/5} (B/10^{12}\text{ G})^{6/5}$. For 1A 0535+262, using its measured $\xi$, $P=103$ sec, and $B=5\times10^{12}$ G, we can infer that $\xi_0 = 168.7^{+124.5}_{-62.2}$. For Swift J0243.6+6124, no cyclotron line has been detected \citep{jaisawal2017,jaisawal2019,zhang2019,tao2019}. Instead, we can apply the scaling with spin and magnetic field strength to infer the magnetic field, required to explain the difference between its measured $\log \xi$ and that of 1A 0535+262, given their known difference in spin. Considering the $1-\sigma$ ranges in $\log \xi$ for both sources, one then finds that the magnetic field of Swift J0243.6+6124 should lie between $2\times10^{10}$ G and $1.4\times10^{11}$ G. Indirect estimates of this field have been obtained via pulse frequency evolution modelling and through searches for the transitional propeller X-ray luminosity or the critical X-ray luminosity, yielding contrasting results: while several authors report evidence for a field strength of $B \geq 10^{13}$ G \citep{doroshenko17,vandeneijnden2019_reb,kong2020}, others find $B < 10^{13}$ G \citep{doroshenko2020,sugizaki2020}, and finally some conclude both ranges are possible \citep{wilson18,tsygankov18}. Therefore, with current evidence, we cannot rule out a magnetic field in the range required to explain the difference between 1A 0535+262 and Swift J0243.6+6124 in the magneto-rotational models. However, this range is significantly lower than a subset of estimates for Swift J0243.6+6124 and the magnetic field typically observed in BeXRBs. For such, more typical, BeXRB fields i.e. ($B\geq10^{12}$ G), the magneto-rotational models would have predicted a larger difference between the two targets. It is worth briefly pointing out that the radio non-detections of two other BeXRBs, GRO J1008-57 and SAX J2103.5+4545, are not surprising in this model interpretation. The former target has similar spin and magnetic field to 1A 0535+262, but a significantly larger distance, while spin period of the latter is more than three times larger than that of 1A 0535+262. As both sources where observed at similar X-ray luminosities as the 1A 0535+262 outburst decay, we expect lower radio flux densities under the assumption of the magneto-rotational models. Combined with their higher radio luminosity limits, the non-detections are therefore consistent with this model. Finally, we can conduct a similar comparison between NS BeXRB and the full sample of NS LMXBs (ignoring for simplicity the intrinsic variations in the latter sample), for which \citet{gallo18} measure $\log \xi = -0.17 \pm 0.05$. That value, in combination with the measurement of $\xi_0$ from 1A 0535+262, does not fit well with typical spin and magnetic field values assumed or measured for NS LMXBs. For instance, it implies a maximum spin frequency of $\sim 7$ Hz for a $10^{8}$ G magnetic field, an order of magnitude below the typical spin frequencies of accreting millisecond X-ray pulsars (AMXPs). At a field strength of $10^7$ G, at the low end of what is typically invoked for AMXPs, this maximum spin frequency only increases to $\sim 20$ Hz. We therefore find that the magneto-rotational models cannot reproduce both the slope and normalisation of the observed $L_X$--$L_R$ relationship in a consistent fashion for the NS LMXBs and BeXRBs. Instead, the measured difference in radio luminosity normalisation is smaller than predicted, leading to the low inferred spin frequency for NS LMXBs mentioned above\footnote{We note that even for an individual radio-bright AMXP \citep[for instance IGR J17591-2342; see e.g.][]{russell18,gusinskaia20_igrj17591}, the larger difference in $\log \xi$ is not sufficient to be consistent with magneto-rotational models. Even if it were, it would go against our initial assumption underlying the comparison: a single magneto-rotational model holds for \textit{all} NS X-ray binaries}. We conclude that the magneto-rotational models can qualitatively account for the differences seen between the samples of NS LMXBs and BeXRBs, and between the two considered BeXRBs, but currently fails to quantitatively explain these for reasonable spin and magnetic field values. \subsubsection{Assessing the implicit assumptions} In the analysis above, we make a number of assumptions to compare BeXRBs and LMXBs. Therefore, it is important to assess whether the difficulty to explain the quantitative differences between these sources in the magneto-rotational models, arises due to these assumptions. We can start by discussing the role of the X-ray luminosity in these calculations. For instance, we assumed that the X-ray luminosity scales in a linear fashion with mass accretion rate across all considered X-ray luminosities, including the super-Eddington ones reached by Swift J0243.6+6124. When considering the radio-detected BeXRBs individually, we find their $L_X$--$L_R$ slopes to be consistent despite the different (but overlapping) ranges in $L_X$ they span, as expected in this scenario. However, this does not imply that the same inflow-outflow coupling necessarily operates in the sub- and super-Eddington regime. Especially given the small range in X-ray luminosity and uncertainties on $\beta$, that conclusion cannot be made. Another possible issue with the X-ray luminosity, as mentioned by \citet{chatzis2021}, may be the relatively low contribution of the accretion flow to the total X-ray emission. If the emission is dominated by the accretion column, we may need to consider instead only a fraction of the X-ray luminosity as input for Equation \ref{eq:L_R}. If this fraction is independent of the total luminosity, such a change only affects the normalisation, increasing the inferred value of $\xi_0$. As the accretion column emission is only expected to play such a significant role in BeXRBs, and not LMXBs\footnote{Fractional variabilities in accreting millisecond pulsars are typically of the order of a few per cent or less \citep{patruno18}.}, an increased value of $\xi_0$ exacerbates the issue that we measure a smaller normalisation difference between these source classes than expected in the magneto-rotational models. We note, on the other hand, that a substantial fraction of X-ray lumionosity of LMXBs can originate from a boundary layer, which is not present in BeXRBs. Moreover, both the accretion column and boundary layer luminosity, fundamentally, scale with accretion rate. A systematic exploration of these X-ray spectral decompositions on the tracks of NSs in the $L_X$--$L_R$ plane would help disentangle these effects. Another assumption, especially relevant for 1A 0535+262 and NS LMXBs without a spectral index measurements, is that the observed radio fluxes can be extrapolated to $5$ GHz radio luminosities without loss of information. Changes in spectral index are likely occurring throughout outbursts of most sources, based on the monitoring of sources where spectral index measurements were made \citep{vandeneijnden2019_reb,gusinskaia20,russell2021}. By ignoring or not measuring such changes, the radio flux density to luminosity conversion will introduce scatter into the relationship between X-ray and radio luminosity, and possible affect its slope. In addition, using the $5$-GHz radio luminosity ingores changes in spectral break frequency and the optically thin slope, which strongly affect the total jet power \citep{russell14}. \subsection{The radio properties of BeXRBs in the context of all X-ray binaries} \label{sec:biggerpicture} Having focused on the radio behaviour of BeXRBs in the previous two sections, we will now turn to a comparison with the broader class of X-ray binaries. Based on the assumption that giant BeXRB outbursts show a single $L_X$--$L_R$ correlation, we have drawn a schematic to summarize the $L_X$--$L_R$ plane for various types of X-ray binaries in Figure \ref{fig:schematic}. In this Figure, the solid regions indicate measured correlations between the two luminosities, while the dashed regions indicate extrapolated behaviour. The hard state black hole systems, regardless of donor mass, are radio-brightest \citep{fender01,migliari06}; in this schematic, we follow \citet{gallo18} and treat the entire black hole population as one and do not distinguish a radio-loud and radio-quiet track \citep[see e.g.][for more discussion]{soleri11,gallo2014,dincer14,meyer14,drappeau15}. Different types of NS X-ray binaries populate different regions in this diagram: combined into one class, the low-mass systems approximately trace the black hole correlation with similar coupling index \citep{gallo18}, although the sample's radio luminosity normalisation at $2\times10^{36}$ erg /s (i.e. $0.01L_{\rm Edd}$, where $L_{\rm Edd}$ is defined assuming a NS) is a factor $\sim 22$ lower and individual sources can show deviating behavior \citep[e.g.][]{migliari06,gusinskaia20}. As argued in this paper, the NS BeXRBs are even radio fainter, by a further factor of $\sim 275$ at $0.01L_{\rm Edd}$ compared to the NS LMXBs, while showing a slightly steeper index. These two NS classes show a large range of extrapolation, particularly at low X-ray luminosity; systematic radio detections have only been obtained down to $\sim 0.01L_{\rm Edd}$ for NS LMXBs, while this limit is $\sim 0.03L_{\rm Edd}$ for the NS BeXRBs. Therefore, we make the simplest assumption of a single powerlaw coupling down to low luminosities, although this remains to be confirmed observationally. The latter class does extend to significantly super-Eddington luminosities, due to the inclusion of Swift J0243.6+6124. Finally, we note that the hard boundaries drawn between classes in Figure \ref{fig:schematic} are, in reality, not clear-cut. Variations between individual sources mean that, e.g., compact object type cannot be determined beyond doubt from the position in this diagram. This simplistic picture is complicated by the inclusion of two additional pieces of information. Firstly, the outburst decay and especially re-flares of Swift J0243.6+6124 are significantly radio-brighter than the extrapolated NS BeXRB correlation, as indicated by the dashed arrow in the schematic. We stress that the exact path of this source during its outburst has not been fully constrained, due to several radio non-detections -- the dashed arrow shows the simplest route consistent with the observations. Secondly, the NS supergiant X-ray binaries (SgXRBs), where a NS in a tight orbit persistently accretes from the strong stellar wind of a supergiant donor star, are wedged in between the two aforementioned NS correlations \citep{vandeneijnden2021}. An important caveat to that statement is, however, that this only holds for the radio-detected NS SgXBs -- several of such sources are not radio detected and fall below the yellow area \citep{vandeneijnden2021}. This complicating behaviour of Swift J0243.6+6124 and the NS SgXBs, both radio-bright compared to the NS BeXRB correlation, is strongly suggestive of an additional radio-emission mechanism. This raises the question `What could such a mechanism be?' For the radio behaviour of Swift J0243.6+6124 during its outburst decay and X-ray re-flares, \citet{vandeneijnden2019_reb} suggested a two-fold explanation. The initial radio flaring might have originated in large-scale shocks, as jet material interacts with the ISM, while the radio properties during the X-ray re-flare could represent a rapidly re-establishing jet. While the latter scenario proposes a jet that would be, in terms of radio luminosity, remarkably similar to NS LMXBs, its inferred similarity to the jet observed in the super-Eddington outburst phase remained puzzling. The possible presence of an ultra-fast disk outflow during the super-Eddington phases has been proposed to play a role in regulating the maximum radio luminosity of the super-Eddington jet; however, such an explanation is quite speculative \citep{vandeneijnden2019_chandra}. An alternative answer may therefore lie in the model by \citet{chatzis2021} discussed previously, which is able to describe these sub-Eddington radio observations satisfactorily. A challenge for this model would be, then, to explain the launch of a (roughly) spherical outflow from the accretion disk at X-ray luminosities between the super-Eddington and propeller regimes. In addition, if this shock model indeed explains the outlying behaviour of Swift J0243.6+6124, it should similarly predict no or fainter radio emission in the two considered outbursts of GRO J1008-57, the giant outburst of SAX J2103.5+4545, and in the late giant outburst decay of 1A 0535+262. As the particle acceleration and shock emission properties in this model depend heavily on the system's geometry (i.e. orbital separation and viewing angle) and wind properties (mass loss rate and velocity), such differences may be expected. For instance, the wind properties of Be stars are very poorly constrained in BeXRBs, and could differ strongly, causing differences in the location and energetics of a shock. Another factor to consider is the difference between these transient states. An X-ray re-flare of Swift J0243.6+6124, Type-I outburst of GRO J1008-57, and giant outburst of 1A 0535+262 or SAX J2103.5+4545 do not necessarily respresent the exact same accretion flow state despite similar $L_X$. Turning briefly to the NS SgXBs, a logical next question is then whether a stellar wind or the \citet{chatzis2021} shock model could be the inferred additional radio emission process. As discussed in detail in \citet{vandeneijnden2021}, thermal stellar wind emission may play a role in a subset of targets. However, it is not expected to be the driving factor of this enhanced radio luminosity, as not all radio-detected NS SgXBs launch a stellar wind capable of explaining the radio emission, while some non-detected targets should have been detected in this scenario. Several lines of reasoning also argue against the model by \citet{chatzis2021}. Firstly, it is unclear whether an accretion disk, capable of launching one of the two shocking outflows, is present in all NS SgXBs \citep[see e.g.][for a recent discussion]{elmellah2019}. Secondly, the stellar wind of the massive star is significantly denser than those in BeXRBs, which makes it unlikely that emission from a shock deep in the massive stellar wind can be observed. However, the fundamental idea of this model -- shocks occur between the stellar wind and some other structure, causing the acceleration of relativistic electrons -- may still contribute. For instance, the presence of large scale accretion and photo-ionisation wakes in some NS SgXBs \citep{blondin1991,kaper1994} may provide sites for shocks with the stellar wind to develop on larger physical scales that are less affected by effects suppressing the radio emission \citep[i.e. free-free absorption and the Razin effect;][]{hornby1966}. A more detailed model, as well as further observations of radio NS SgXBs and a better understanding of the circumstances (e.g. binary and stellar wind properties) where accretion and photo-ionisation wakes are formed, are necessary to further consider such a scenario. With regards to the SgXBs, we will make two final comments. Firstly, in the above discussion, we have assumed that strongly-magnetized NSs in SgXBs are equally capable of launching jets as NS BeXRBs, and would do so via the same mechanism. While that may be a reasonable assumption in terms of the NS properties, the accretion flow itself differs significantly between these two source classes. For instance, if a smaller disk, or no disk at all, is present in a NS SgXB, magneto-rotational models may or may not operate. However, whether that predicts a lower radio luminosity, or instead allows for another (possibly radio-brighter) jet launch mechanism to take over, cannot be determined without adjusting strong-B jet launch models for spherical accretion flows or focused winds. Secondly, the above discussion regarding additional radio emission mechanisms, especially shocks, does not require a NS primary. However, for systems with BH primaries, such as Cyg X-1, any resulting radio emission is significantly fainter than the jet and would be virtually undetectable; for the radio-detected BH system MWC 656, on the other hand, this scenario does not apply, as it hosts a Be-star instead of supergiant donor. \subsection{Future Galactic and extragalactic prospects} In our own Galaxy, the advent of the next-generation VLA (ngVLA), as well as the SKA and SKA precursors in the Southern hemisphere, with their enhanced sensitivity, would greatly extend the range of X-ray luminosity and distances where BeXRB radio emission and jets may be probed with observed of reasonable length (see the blue lines in Figure \ref{fig:lxlr_fitted}). Given the typical range of radio luminosities observed in isolated Be stars, such future observations may probe down to the regime where this emission cannot simply be ignored. For instance, for sources within $\sim 2$ kpc, a one-hour ngVLA observation is sensitive down to $\sim 2\times10^{25}$ erg/s, reaching far into the range of isolated Be star radio emission (see Section \ref{sec:whatdowesee}). Not all Be stars are, however, detected at radio frequencies. Therefore, with coordinated X-ray observations and sufficiently dense radio monitoring, transient radio emission could still be tracked down to low accretion rates for those BeXRBs hosting radio-faint Be stars. In this work, we present evidence for the existence of an X-ray -- radio luminosity coupling for BeXRBs. If we assume that this holds more generally for strongly-magnetized accreting NSs, we can use this correlation to briefly move focus to ultra-luminous X-ray sources (ULXs). ULXs are extragalactic X-ray sources with X-ray luminosities exceeding the Eddington luminosity of a $\sim 10$ $M_{\odot}$ black hole \citep[i.e. $\sim 10^{39}$ erg/s;][]{kaaret17}. While the exact nature of ULX compact objects long remained unclear, with both stellar-mass compact objects and intermediate mass black holes considered as options, the detection of pulsations from multiple ULXs \citep{bachetti14,furst16b,israel17b} has unambiguously shown that at least a fraction of them host accreting NSs. The exact fraction remains unknown, although both observational \citep{walton18c} and theoretical considerations \citep{king16_ulx} are consistent with a significant proportion. ULXs also show evidence for outflows, both through the detection of resolved (feedback) structures \citep{pakull2003,kaaret03} and X-ray absorption lines from ultra-fast outflows \citep{pinto16}. However, unresolved radio counterparts from compact jets have not been detected unambiguously from ULX pulsars \citep{mezcua15,cseh15a,kaaret17}. The identified NSs in ULXs rotate slowly, similar to their strongly-magnetized Galactic counterparts. Extending our suggested BeXRB X-ray--radio luminosity relation to a typical ULX luminosity, one might ask what the prospects are for detecting radio point source emission? If we assume an X-ray luminosity of $\sim 10^{41}$ erg/s \citep[on the high end of their luminosity distribution;][]{kaaret17}, the predicted radio luminosity would be of the order $\sim 10^{30}$ erg/s. At a Mpc distance and $5$ GHz, this is equivalent to a $0.2$ $\mu$Jy flux density. Such depths are out of reach for any current facilities in reasonable observing times, but are approached by the planned ngVLA sensitivity \citep[$\sim 0.23$ $\mu$Jy at 8 GHz in 1 hour of observing time;][]{selina18}. At such depths, confusion limits, host galaxy emission, diffuse feedback structures and other extended, close-by sources may complicate any searches for radio point source emission (especially at low frequencies). However, given the $>3$ orders of magnitude difference in radio luminosity normalisation compared to the black hole systems, the detection and flux density of radio emission may help to understand the nature of the compact object accretor. \section{Acknowledgments} The authors thank the referee for a constructive report. JvdE is supported by a Lee Hysan Junior Research Fellowship awarded by St. Hilda's College. TDR acknowledges financial contribution from the agreement ASI-INAF n.2017-14-H.0. GRS is supported by NSERC Discovery Grants RGPIN-2016-06569 and RGPIN-2021-04001. The authors acknowledge the use of public data from the Swift data archive. This research has made use of MAXI data provided by RIKEN, JAXA and the MAXI team. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The Australia Telescope Compact Array is part of the Australia Telescope National Facility which is funded by the Australian Government for operation as a National Facility managed by CSIRO. We acknowledge the Gomeroi people as the traditional owners of the ATCA observatory site. This research has made use of data and software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC) and NASA's Astrophysics Data System Bibliographic Services. This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement. \section*{Data Availability} All radio observations can be accessed via the VLA Data Archive (\url{https://archive.nrao.edu/archive/archiveproject.jsp}) or the Australia Telescope Online Archive (\url{https://atoa.atnf.csiro.au}). Relevant project codes are listed in the Online Supplementary Materials. All X-ray data is publicly available via the HEASARC (pointed observations) or \textit{Swift}/BAT Hard X-ray Transient Monitor (\url{https://swift.gsfc.nasa.gov/results/transients/}). A Jupyter notebook reproducing the Figures and \textsc{LinMix} fits in this paper, will be publically accessible upon acceptance and publication via \url{https://github.com/jvandeneijnden/LxLrCouplingInBeXRBs}. \input{main.bbl} \appendix \section{Online Supplementary Materials} \section{Observational details and analysis} \label{app:observations_fluxes} In Tables \ref{tab:radio_obs} and \ref{tab:xrays_j1008_saxJ2103}, we list further details on the analysed radio and X-ray observations. In Figure \ref{fig:spectrum}, we show the radio spectra of 1A 0535+262 in the second and combined second to fourth observations. In Tables \ref{tab:NICER_0535_1} and \ref{tab:NICER_0535_2}, we list details regarding the analysed \textit{Swift} and \textit{NICER} observations of 1A 0535+262, as well as regarding the spectral fit. Note that we only analyse \textit{Swift} data up to MJD 59300, as the final radio observation was taken three weeks prior to that date. We refer to the main paper for full details, especially regarding the model choice, fitted energy band, and the number of included narrow Gaussian lines in the iron line complex. In two observations, the fitted \textit{NICER} band is highlighted by an asterisk. For those observations, even restricting the band to $2$--$10$ keV did not alleviate the issues with instrumental residuals sufficiently to yield $\chi^2_\nu < 2$. Therefore, in those two observations, we additionally added a $1$\% systematic uncertainty to the spectrum. As a result, the uncertainty on the flux and parameters may be enhanced. In Figures \ref{fig:swift_params_a0535} and \ref{fig:NICER_params_a0535}, we plot the evolution of the parameters fitted to the \textit{Swift} and \textit{NICER} spectra, respectively. In both data sets, the parameter uncertainties increase systematically towards lower X-ray flux, as expected. However, other effects cause large uncertainties in several observations as well: short exposure times, for instance, or the complexity of the \textit{NICER} model, especially regarding the Gaussian lines and cutoff energy. In the \textit{NICER} light curves, one can see how the Gaussian line energies are fitted at reasonably stable values of $\sim 6.4$, $\sim 6.67$, and $\sim 6.96$ keV at high X-ray flux, as expected for the iron K$\alpha$ complex. At lower fluxes, however, the energies, as well as their widths, are more poorly constrained. Finally, the $\chi^2_\nu$ of the \textit{NICER} fits clearly peaks at high flux, due to the appearance of significant instrumental residuals below $3$ keV -- we stress that the $\chi^2_\nu$ values in both Table and Figure are calculated \textit{after} applying the energy band restriction. All parameters values, with uncertainties, as well as the information from Tables \ref{tab:NICER_0535_1} and \ref{tab:NICER_0535_2}, are available in machine-readable format in the other files in these Online Supplementary Materials. Additionally, we include machine-readable files with the fluxes in three energy bands (0.5-10 keV, 1-10 keV, 2-10 keV) for all sources. Alternatively, all machine-readable table files can be accessed via the \textsc{github} repository of this paper at \url{https://github.com/jvandeneijnden/RadioMonitoringOfTransientBeXRBs}. As detailed in the main paper, the \textit{NICER} spectra were modelled using the standard, pre-calculated reponse files for the \textsc{rmf} and \textsc{arf}. However, we explicitly tested the use of \textsc{nicerarf} and \textsc{nicerrmf} to generate observation-specific response files, following the \textit{NICER} analysis threads\footnote{\url{https://heasarc.gsfc.nasa.gov/docs/nicer/analysis\_threads/arf-rmf/}}. In addition, we also generated background files using the pre-release version v0p6 of \textsc{nicer\_bkg\_estimator}\footnote{\url{https://heasarc.gsfc.nasa.gov/docs/nicer/tools/nicer\_bkg\_est\_tools.html}}. However, we found that this does not lead to significant differences in the analysed spectra. In Figure \ref{fig:resp_comp}, we show the ratio between the \textit{NICER} spectra, measured in counts/s/keV from ObsID 3200360135, plotted with the pre-calculated and observation-specific instrument response. While the energy-averaged ratio is slightly offset from unity, due to the subtraction of the background in the latter spectrum, no energy-dependent structures can be identified. In fact, at all energies, the ratio is consistent with unity. This observation, shown as the blue spectrum in Figure 1 of the main paper, shows strong instrumental effects below $2$ keV; evidently, these effects are not reduced by the alternative response and background approach. Finally, we also tested whether the binsize used in the analysis affected the presence of these residual features and the measured parameters. Rebinning spectra by a factor 3 or 10 did not alleviate the issues with residuals, but did naturally lead to slightly enhanced uncertainties on the parameters and higher $\chi^2_\nu$ values. Importantly, the fitted parameters and derived fluxes did not change significantly. However, we note that the derived errors and $\chi^2_\nu$ listed in these supplementary materials are only valid for the default \textit{NICER} binning. \section{MCMC run figures} \label{sec:appendix_MCMC} In Figure \ref{fig:MCMC_4panels}, we show the posterior distributions of the offset $\log \xi$ and slope $\beta$ of a single MCMC run to fit the behaviour of giant BeXRB outbursts in the $L_X$--$L_R$ plane (top left and right panels, respectively). In the bottom, we show the distribution of the 16$^{\rm th}$ and 84$^{\rm th}$ percentile (blue), as well as the mean (black), of the same two parameters, after 500 MCMC runs (bottom left and right panels, respectively). These percentiles and the mean are indicated in the top panels as well, for the single example run. \begin{table*} \caption{Details of the radio observations in this work. For each studied outburst, we list the observation number, date in MJD, observation length in hours, observing frequency $\nu$ and bandwidth $\Delta \nu$ in GHz, measured flux density or $3-\sigma$ upper limit in $\mu$Jy, the observatory and its configuration, and finally the program ID. These program IDs provide the most straightforward access to publicly available data in the observatory archives.} \label{tab:radio_obs} \begin{tabular}{lllllllll} \hline \multicolumn{9}{c}{\textit{1A 0535+262}} \\ \hline Number & MJD & $\Delta T$ [hr] & $\nu$ [GHz] & $\Delta\nu$ [GHz] & Flux density [$\mu$Jy] & Observatory & Configuration & Program ID \\ \hline 1 & 59163.42 & 1 & 6 & 4 & $12.5 \pm 3.9$ & VLA & BnA & 20A-171 \\ 2 & 59168.47 & 1 & 6 & 4 & $39.2 \pm 4.0$ & VLA & BnA$\rightarrow$A & 20A-171 \\ 3 & 59170.25 & 1 & 6 & 4& $27.9 \pm 4.0$ & VLA & BnA$\rightarrow$A & 20A-171 \\ 4 & 59172.23 & 1 & 6 & 4& $27.9 \pm 4.3$& VLA & BnA$\rightarrow$A & 20A-171 \\ 5 & 59177.12 & 1 & 6 & 4& $20.3 \pm 3.3$& VLA & BnA$\rightarrow$A & 20A-171 \\ 6 & 59178.4 & 1 & 6 & 4& $27.5 \pm 4.0$& VLA & BnA$\rightarrow$A & 20A-171 \\ 7 & 59178.45 & 1 & 6 & 4& $20.6 \pm 4.0$& VLA & BnA$\rightarrow$A & 20A-171 \\ 8 & 59185.09 & 1 & 6 & 4& $21.6 \pm 4.3$& VLA & BnA$\rightarrow$A & 20A-171 \\ 9 & 59186.47 & 1 & 6 & 4& $16.2 \pm 4.2$& VLA & BnA$\rightarrow$A & SM0612 \\ 10 & 59189.09 & 1 & 6 & 4& $9.8 \pm 4.1$& VLA & BnA$\rightarrow$A & 20A-171 \\ 11 & 59191.40 & 1 & 6 & 4& $ < 11.7$ & VLA & BnA$\rightarrow$A & 20A-171 \\ 12 & 59201.31 & 1 & 6 & 4& $ < 12.9 $ & VLA & A & SG9053\\ 13 & 59202.36 & 1 & 6 & 4& $ < 12.6$ & VLA & A & SG9053\\ 14 & 59220.01 & 1 & 6 & 4& $ < 18.0$ & VLA & A & SG9053\\ 15 & 59235.07 & 1 & 6 & 4& $ < 16.5$ & VLA & A & SG9053\\ 16 & 59247.93 & 1 & 6 & 4& $ < 14.0$ & VLA & A & SG9053\\ 17 & 59262.05 & 1 & 6 & 4& $ < 13.0$ & VLA & A & SG9053\\ 18 & 59266.89 & 1 & 6 & 4& $ < 13.0$ & VLA & A & SG9053\\ 19 & 59278.95 & 1 & 6 & 4& $ < 15.0$ & VLA & A$\rightarrow$D & SG9053\\ 20 & 59279.10 & 1 & 6 & 4& $ < 15.0$ & VLA & A$\rightarrow$D & SG9053\\ \hline \multicolumn{9}{c}{\textit{GRO J1008-57: giant outburst}} \\ \hline Number & MJD & $\Delta T$ [hr] & $\nu$ [GHz] & $\Delta\nu$ [GHz] & Flux density [$\mu$Jy] & Observatory & Configuration & Program ID \\ \hline \multirow{2}{*}{1} & \multirow{2}{*}{58984.43} & \multirow{2}{*}{4} & 5.5 & 1 & $< 33.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{1.5C} & \multirow{2}{*}{C3299} \\ & & & 9.0 & 1 & $< 22.5$ & & & \\ \cline{3-6} \multirow{2}{*}{2} & \multirow{2}{*}{58990.42} & \multirow{2}{*}{4}& 5.5 & 1 & $< 36.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{1.5C} & \multirow{2}{*}{C3299}\\ & & & 9.0 & 1 & $< 24.0$ & & & \\ \cline{3-6} \multirow{2}{*}{3} & \multirow{2}{*}{59001.19} & \multirow{2}{*}{4}& 5.5 & 1 & $< 33.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{1.5C} & \multirow{2}{*}{C3299} \\ & & & 9.0 & 1 & $< 22.5$ & & & \\ \cline{3-6} \multirow{2}{*}{4} & \multirow{2}{*}{59013.15} & \multirow{2}{*}{4}& 5.5 & 1 & $< 16.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{1.5C} & \multirow{2}{*}{C3299} \\ & & & 9.0 & 1 & $< 23.0$ & & & \\ \cline{3-6} \multirow{2}{*}{5} & \multirow{2}{*}{59025.00} & \multirow{2}{*}{4}& 5.5 & 1 & $< 180.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{H214} & \multirow{2}{*}{C3299} \\ & & & 9.0 & 1 & $< 66.0$ & & & \\ \cline{3-6} \multirow{2}{*}{\textit{1--4}} & \multirow{2}{*}{N/A} & \multirow{2}{*}{16}& 5.5 & 1 & $< 20.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{1.5C} & \multirow{2}{*}{C3299} \\ & & & 9.0 & 1 & $< 24.0$ & & & \\ \hline \multicolumn{9}{c}{\textit{GRO J1008-57: periastron outburst}} \\ \hline Number & MJD & $\Delta T$ [hr] & $\nu$ [GHz] & $\Delta\nu$ [GHz] & Flux density [$\mu$Jy] & Observatory & Configuration & Program ID \\ \hline \multirow{2}{*}{1} & \multirow{2}{*}{58639.17} & \multirow{2}{*}{4}& 5.5 & 1 & $< 33.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{6A} & \multirow{2}{*}{C3298} \\ & & & 9.0 & 1 & $< 27.0$ & & & \\ \cline{3-6} \multirow{2}{*}{2} & \multirow{2}{*}{58643.27} & \multirow{2}{*}{4}& 5.5 & 1 & $< 30.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{6A} & \multirow{2}{*}{C3298} \\ & & & 9.0 & 1 & $< 27.0$ & & & \\ \cline{3-6} \multirow{2}{*}{3} & \multirow{2}{*}{58647.15} & \multirow{2}{*}{4}& 5.5 & 1 & $< 30.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{6A} & \multirow{2}{*}{C3298} \\ & & & 9.0 & 1 & $< 30.0$ & & & \\ \cline{3-6} \multirow{2}{*}{4} & \multirow{2}{*}{58651.15} & \multirow{2}{*}{4}& 5.5 & 1 & $< 27.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{6A} & \multirow{2}{*}{C3298} \\ & & & 9.0 & 1 & $< 24.0$ & & & \\ \cline{3-6} \multirow{2}{*}{5} & \multirow{2}{*}{58655.13} & \multirow{2}{*}{4}& 5.5 & 1 & $< 33.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{6A} & \multirow{2}{*}{C3298} \\ & & & 9.0 & 1 & $< 30.0$ & & & \\ \cline{3-6} \multirow{2}{*}{6} & \multirow{2}{*}{58658.08} & \multirow{2}{*}{4}& 5.5 & 1 & $< 28.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{6A} & \multirow{2}{*}{C3298} \\ & & & 9.0 & 1 & $< 45.0$ & & & \\ \cline{3-6} \multirow{2}{*}{\textit{1--6}} & \multirow{2}{*}{N/A} & \multirow{2}{*}{4}& 5.5 & 1 & $< 27.0$ & \multirow{2}{*}{ATCA} & \multirow{2}{*}{6A} & \multirow{2}{*}{C3298} \\ & & & 9.0 & 1 & $< 19.5$ & & & \\ \hline \multicolumn{9}{c}{\textit{SAX J2103.5+4545: giant outburst}} \\ \hline Number & MJD & $\Delta T$ [hr] & $\nu$ [GHz] & $\Delta\nu$ [GHz] & Flux density [$\mu$Jy] & Observatory & Configuration & Program ID \\ \hline 1 & 59101.02 & 2 & 6 & 4 & $ < 18.0$ & VLA & B & 20A-171 \\ \hline \end{tabular}\\ \end{table*} \begin{table*} \caption{Full details of the X-ray observations and spectral fits for the two outbursts of GRO J1008-57 and the giant outburst of SAX J2103.5+4545. We refer to the main text for full details on data selection and model choice. When a joint fit is performed to the \textit{Swift}/XRT PC and WT mode data, the exposure times of both instruments are listed. * As detailed in the text, we apply a correction based on the limited soft energy coverage of \textit{MAXI} and have divided the measured X-ray flux by a factor 1.7. Here, we list these corrected fluxes. ** Joint fit between all seventeen spectra, returning a single $N_H$ measurement and fit statistic.} \label{tab:xrays_j1008_saxJ2103} \begin{tabular}{llllllllll} \hline \multicolumn{10}{c}{\textit{GRO J1008-57: periastron outburst}} \\ \hline Observatory/ & \multirow{2}{*}{\#} & \multirow{2}{*}{ObsID/mode} & \multirow{2}{*}{MJD} & Exposure & $N_H$ & \multirow{2}{*}{$\Gamma$} & 0.5-10 keV flux & \multirow{2}{*}{C-stat} & \multirow{2}{*}{DOF} \\ Instrument & & & & [seconds] & [$10^{22}$ cm$^{-2}$] & & [erg/s/cm$^2$] & & \\ \hline \textit{Swift}/XRT & 1 & 0003103149/pc & 58639.91 & 951 & $2.2\pm0.4$ & $1.0\pm0.2$ & $(1.9\pm0.1)\times10^{-10}$ & 288 & 269 \\ \textit{Swift}/XRT & 2 & 0003103150/pc & 58643.24 & 1079 & $2.6\pm0.5$ & $1.1\pm0.2$ & $(6.3\pm0.4)\times10^{-10}$ & 241 & 315 \\ \textit{Swift}/XRT & 3 & 0003103151/pc+wt & 58647.02 & 659+136 & $2.3\pm0.3$ & $0.76\pm0.11$ & $(1.12\pm0.05)\times10^{-9}$ & 718 & 865 \\ \textit{Swift}/XRT & 4 & 0003103152/wt & 58652.21 & 829 & $3.0\pm0.1$ & $0.92\pm0.04$ & $(1.92\pm0.02)\times10^{-9}$ & 718 & 865 \\ \textit{Swift}/XRT & 5 & 0003103153/wt & 58655.80 & 362 & $3.5\pm0.3$ & $1.15\pm0.09$ & $(9.3\pm0.3)\times10^{-10}$ & 757 & 711 \\ \textit{Swift}/XRT & 6 & 0003103154/pc+wt & 58658.45 & 687+258 & $2.7\pm0.3$ & $0.9\pm0.1$ & $(1.07\pm0.04)\times10^{-9}$ & 888 & 1063 \\ \hline \multicolumn{10}{c}{\textit{GRO J1008-57: giant outburst}} \\ \hline Observatory/ & \multirow{2}{*}{\#} & \multirow{2}{*}{ObsID/mode} & \multirow{2}{*}{MJD} & Exposure & $N_H$ & \multirow{2}{*}{$\Gamma$} & 0.5-10 keV flux* & \multirow{2}{*}{$\chi^2_\nu$} & \multirow{2}{*}{DOF} \\ Instrument & & & & [seconds] & [$10^{22}$ cm$^{-2}$] & & [erg/s/cm$^2$] & & \\ \hline \textit{MAXI}/GSC & 1 & -- & 58980 \& 58983 & 929 & \multirow{17}{*}{$4.5\pm0.8$**} & $1.6 \pm 0.3$ & $(1.2 \pm 0.5)\times10^{-9}$ & \multirow{17}{*}{1.036**} & \multirow{17}{*}{466**} \\ \textit{MAXI}/GSC & 2 & -- & 58986 & 310 & & $0.8 \pm 0.3$ & $(1.7 \pm 0.6)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 3 & -- & 58993 & 580 & & $1.1 \pm 0.1$ & $(4.0 \pm 0.7)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 4 & -- & 58994 & 581 & & $1.4 \pm 0.2$ & $(4.5 \pm 0.9)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 5 & -- & 58995 & 620 & & $1.6 \pm 0.2$ & $(5 \pm 1)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 6 & -- & 58996 & 1072 & & $1.3 \pm 0.1$ & $(5.0 \pm 0.7)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 7 & -- & 58997 & 1107 & & $1.2 \pm 0.1$ & $(4.3 \pm 0.6)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 8 & -- & 58998 & 2533 & & $1.3 \pm 0.1$ & $(4.1 \pm 0.5)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 9 & -- & 58999 & 2603 & & $1.5 \pm 0.1$ & $(4.4 \pm 0.6)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 10 & -- & 59000 & 1466 & & $1.4 \pm 0.1$ & $(3.8 \pm 0.6)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 11 & -- & 59012 & 2188 & & $1.5 \pm 0.1$ & $(1.6 \pm 0.3)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 12 & -- & 59013 & 2495 & & $1.5 \pm 0.1$ & $(1.5 \pm 0.3)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 13 & -- & 59014 & 2773 & & $1.4 \pm 0.1$ & $(1.2 \pm 0.3)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 14 & -- & 59015 & 2170 & & $1.5 \pm 0.2$ & $(1.0 \pm 0.3)\times10^{-9}$ & & \\ \textit{MAXI}/GSC & 15 & -- & 59016 & 1937 & & $1.6 \pm 0.2$ & $(8 \pm 3)\times10^{-10}$ & & \\ \textit{MAXI}/GSC & 16 & -- & 59020$\rightarrow$59022 & 7149 & & $2.1 \pm 0.4$ & $(5 \pm 3)\times10^{-10}$ & & \\ \textit{MAXI}/GSC & 17 & -- & 59023$\rightarrow$59025 & 10090 & & $2.1 \pm 0.4$ & $(4 \pm 2)\times10^{-10}$ & & \\ \hline \multicolumn{10}{c}{\textit{SAX J2103.5+4545: giant outburst}} \\ \hline \multirow{2}{*}{Observatory} & \multirow{2}{*}{\#} & \multirow{2}{*}{ObsID/mode} & \multirow{2}{*}{MJD} & Exposure & $N_H$ & \multirow{2}{*}{$\Gamma$} & 0.5-10 keV flux & \multirow{2}{*}{C-stat} & \multirow{2}{*}{DOF} \\ & & & & [seconds] & [$10^{22}$ cm$^{-2}$] & & [erg/s/cm$^2$] & & \\ \hline \textit{Swift}/XRT & 1 & 00030922079/wt & 59106.25 & 1026 & $1.5\pm0.2$ & $0.8\pm0.1$ & $(2.5\pm0.1)\times10^{-10}$ & 599 & 654 \\ \textit{Swift}/XRT & 2 & 00030922080/wt & 59107.84 & 953 & $1.5\pm0.3$ & $0.7\pm0.1$ & $(2.2\pm0.1)\times10^{-10}$ & 497 & 598 \\ \hline \end{tabular}\\ \end{table*} \begin{table*} \caption{Details of the X-ray spectral analysis of \textit{NICER} and \textit{Swift} monitoring of 1A 0535+262. In this Table, we list information about these observations, such as observatory, ObsID and observing mode, and time and exposure of the observation. We also list details about the spectral analysis, such as the fitted band, the measured flux, the number of narrow Gaussian model components fitted, and the fit statistic. Continues in Table \ref{tab:NICER_0535_2}. The flux error is listed at the $1-\sigma$ level.} \label{tab:NICER_0535_1} \begin{tabular}{lllllllllll} \hline Observatory/ & \multirow{2}{*}{Obs. \#} & \multirow{2}{*}{ObsID/mode} & \multirow{2}{*}{MJD} & Exposure & Fitted band & Flux & Error & \multirow{2}{*}{Gaussians} & \multirow{2}{*}{$\chi^2_{\nu}$} & \multirow{2}{*}{DOF} \\ Instrument & & & & [second] & [keV] & \multicolumn{2}{c}{[0.5-10 keV; erg/s/cm$^2$]} & & & \\ \hline \textit{NICER} & 1 & 3200360123 & 59160.23 & 8288 & 2-10 & 7.25e-09 & 6.60e-12 & 3 & 1.17 & 884 \\ \textit{NICER} & 2 & 3200360124 & 59161.26 & 3609 & 2-10 & 9.15e-09 & 1.05e-11 & 3 & 1.23 & 884 \\ \textit{NICER} & 3 & 3200360125 & 59162.29 & 2169 & 2-10 & 9.22e-09 & 1.54e-11 & 3 & 1.04 & 884 \\ \textit{NICER} & 4 & 3200360126 & 59163.13 & 5231 & 2-10 & 1.13e-08 & 1.08e-11 & 3 & 1.40 & 884 \\ \textit{NICER} & 5 & 3200360127 & 59164.36 & 12166 & 2-10 & 1.34e-08 & 7.51e-12 & 3 & 1.56 & 884 \\ \textit{NICER} & 6 & 3200360128 & 59165.0 & 1510 & 2-10 & 1.47e-08 & 2.19e-11 & 3 & 1.09 & 884 \\ \textit{NICER} & 7 & 3200360129 & 59166.1 & 4591 & 2-10 & 2.27e-08 & 1.60e-11 & 3 & 1.65 & 884 \\ \textit{NICER} & 8 & 3200360130 & 59167.34 & 2569 & 2-10 & 3.22e-08 & 2.49e-11 & 3 & 1.65 & 884 \\ \textit{NICER} & 9 & 3200360131 & 59168.05 & 3608 & 2-10 & 4.07e-08 & 3.03e-11 & 3 & 1.56 & 784 \\ \textit{NICER} & 10 & 3200360132 & 59169.15 & 3352 & 2-10 & 4.71e-08 & 3.36e-11 & 3 & 1.71 & 784 \\ \textit{NICER} & 11 & 3200360133 & 59170.04 & 1333 & 2-10 & 5.21e-08 & 4.79e-11 & 3 & 1.82 & 884 \\ \textit{NICER} & 12 & 3200360134 & 59171.08 & 2918 & 2-10 & 5.77e-08 & 3.71e-11 & 3 & 1.84 & 784 \\ \textit{NICER} & 13 & 3200360135 & 59172.17 & 1814 & 2-10 & 5.49e-08 & 4.19e-11 & 3 & 1.87 & 884 \\ \textit{NICER} & 14 & 3200360136 & 59173.01 & 3451 & 2-10 & 5.57e-08 & 3.17e-11 & 3 & 1.90 & 784 \\ \textit{NICER} & 15 & 3200360137 & 59174.11 & 3397 & 2-10 & 5.80e-08 & 2.93e-11 & 3 & 1.78 & 784 \\ \textit{NICER} & 16 & 3200360139 & 59176.11 & 8572 & 2-10* & 5.63e-08 & 4.99e-11 & 3 & 0.67 & 784 \\ \textit{NICER} & 17 & 3200360140 & 59177.14 & 6892 & 2-10* & 5.31e-08 & 4.93e-11 & 3 & 0.72 & 784 \\ \textit{NICER} & 18 & 3200360141 & 59179.66 & 2447 & 1-10 & 4.88e-08 & 3.36e-11 & 3 & 1.92 & 884 \\ \textit{NICER} & 19 & 3200360142 & 59180.04 & 3683 & 2-10 & 4.87e-08 & 3.13e-11 & 3 & 1.61 & 784 \\ \textit{NICER} & 20 & 3200360143 & 59181.46 & 5783 & 2-10 & 4.32e-08 & 2.26e-11 & 3 & 1.82 & 784 \\ \textit{NICER} & 21 & 3200360144 & 59182.24 & 5438 & 2-10 & 4.17e-08 & 2.42e-11 & 3 & 1.83 & 784 \\ \textit{NICER} & 22 & 3200360145 & 59183.27 & 1718 & 1-10 & 3.82e-08 & 3.52e-11 & 3 & 1.52 & 884 \\ \textit{NICER} & 23 & 3200360147 & 59186.05 & 2559 & 1-10 & 3.30e-08 & 2.65e-11 & 3 & 1.44 & 884 \\ \textit{NICER} & 24 & 3200360148 & 59187.08 & 2518 & 1-10 & 3.07e-08 & 1.66e-11 & 3 & 1.42 & 884 \\ \textit{NICER} & 25 & 3200360149 & 59188.11 & 3086 & 1-10 & 2.86e-08 & 2.21e-11 & 3 & 1.48 & 884 \\ \textit{NICER} & 26 & 3200360150 & 59189.08 & 1786 & 1-10 & 2.76e-08 & 2.86e-11 & 3 & 1.24 & 884 \\ \textit{NICER} & 27 & 3200360151 & 59190.05 & 3234 & 1-10 & 2.48e-08 & 1.98e-11 & 3 & 1.44 & 884 \\ \textit{NICER} & 28 & 3200360152 & 59191.47 & 3560 & 1-10 & 2.39e-08 & 1.85e-11 & 3 & 1.37 & 884 \\ \textit{NICER} & 29 & 3200360153 & 59192.37 & 1662 & 1-10 & 1.96e-08 & 2.42e-11 & 3 & 1.12 & 884 \\ \textit{NICER} & 30 & 3200360154 & 59193.09 & 1445 & 1-10 & 2.05e-08 & 2.66e-11 & 3 & 1.10 & 884 \\ \textit{NICER} & 31 & 3200360155 & 59194.18 & 852 & 1-10 & 1.83e-08 & 3.15e-11 & 3 & 1.07 & 884 \\ \textit{NICER} & 32 & 3200360156 & 59206.93 & 949 & 1-10 & 2.22e-09 & 9.68e-12 & 3 & 1.04 & 884 \\ \textit{NICER} & 33 & 3200360157 & 59206.99 & 1635 & 1-10 & 1.94e-09 & 6.91e-12 & 3 & 1.05 & 884 \\ \textit{NICER} & 34 & 3200360158 & 59208.8 & 901 & 1-10 & 1.26e-09 & 7.12e-12 & 3 & 0.97 & 884 \\ \textit{NICER} & 35 & 3200360159 & 59209.17 & 1948 & 1-10 & 8.81e-10 & 4.44e-12 & 3 & 1.09 & 884 \\ \textit{NICER} & 36 & 3200360160 & 59210.85 & 1609 & 1-10 & 4.71e-10 & 3.08e-12 & 3 & 1.17 & 884 \\ \textit{NICER} & 37 & 3200360161 & 59213.3 & 378 & 1-10 & 1.71e-10 & 5.97e-12 & 3 & 0.97 & 884 \\ \textit{NICER} & 38 & 3200360162 & 59214.27 & 7263 & 1-10 & 2.57e-10 & 1.12e-12 & 3 & 1.03 & 884 \\ \textit{NICER} & 39 & 3200360163 & 59214.99 & 3617 & 1-10 & 1.86e-10 & 1.28e-12 & 3 & 1.08 & 884 \\ \textit{NICER} & 40 & 3200360164 & 59216.34 & 12302 & 1-10 & 2.16e-10 & 7.70e-13 & 3 & 1.05 & 884 \\ \textit{NICER} & 41 & 3200360165 & 59217.37 & 6975 & 1-10 & 2.60e-10 & 1.15e-12 & 3 & 1.06 & 884 \\ \textit{NICER} & 42 & 3200360166 & 59218.21 & 3513 & 1-10 & 1.96e-10 & 1.38e-12 & 3 & 1.08 & 884 \\ \textit{NICER} & 43 & 3200360167 & 59219.0 & 6149 & 1-10 & 3.67e-07 & 3.61e-05 & 3 & 1.09 & 884 \\ \textit{NICER} & 44 & 3200360168 & 59220.08 & 1881 & 1-10 & 1.27e-10 & 1.41e-12 & 1 & 1.23 & 890 \\ \textit{NICER} & 45 & 3200360169 & 59221.05 & 5712 & 1-10 & 1.23e-10 & 9.17e-13 & 1 & 1.09 & 890 \\ \textit{NICER} & 46 & 3200360170 & 59222.21 & 2535 & 1-10 & 1.63e-10 & 1.36e-12 & 1 & 1.05 & 890 \\ \textit{NICER} & 47 & 3200360171 & 59223.05 & 3151 & 1-10 & 1.34e-10 & 1.12e-12 & 1 & 1.03 & 890 \\ \textit{NICER} & 48 & 3200360172 & 59224.02 & 2134 & 1-10 & 1.12e-10 & 1.27e-12 & 1 & 0.94 & 890 \\ \textit{NICER} & 49 & 3200360173 & 59225.05 & 3180 & 1-10 & 1.56e-10 & 1.32e-12 & 1 & 1.09 & 890 \\ \textit{NICER} & 50 & 3200360174 & 59226.21 & 2187 & 1-10 & 2.05e-10 & 1.71e-12 & 0 & 1.17 & 893 \\ \textit{NICER} & 51 & 3200360175 & 59227.05 & 3519 & 1-10 & 1.07e-10 & 9.54e-13 & 1 & 1.00 & 890 \\ \textit{NICER} & 52 & 3200360176 & 59228.73 & 885 & 1-10 & 1.33e-10 & 2.29e-12 & 1 & 1.00 & 890 \\ \textit{NICER} & 53 & 3200360177 & 59229.31 & 4755 & 1-10 & 1.10e-10 & 9.29e-13 & 1 & 1.10 & 890 \\ \textit{NICER} & 54 & 3200360178 & 59230.73 & 1380 & 1-10 & 1.11e-10 & 1.58e-12 & 1 & 0.89 & 890 \\ \textit{NICER} & 55 & 3200360179 & 59231.31 & 5382 & 1-10 & 9.47e-11 & 7.24e-13 & 1 & 1.15 & 890 \\ \textit{NICER} & 56 & 3200360180 & 59232.54 & 2473 & 1-10 & 1.13e-10 & 1.16e-12 & 1 & 0.98 & 890 \\ \textit{NICER} & 57 & 3200360181 & 59233.06 & 4095 & 1-10 & 1.08e-10 & 8.86e-13 & 1 & 1.09 & 890 \\ \hline \end{tabular}\\ \end{table*} \begin{table*} \caption{Continuation of Table \ref{tab:NICER_0535_1}. * The exceptionally low $\chi^2_\nu$ value is caused by the short, $15$ second exposure and resulting over-fitting of the spectrum.} \label{tab:NICER_0535_2} \begin{tabular}{lllllllllll} \hline \multirow{2}{*}{Observatory} & \multirow{2}{*}{Obs. \#} & \multirow{2}{*}{ObsID/mode} & \multirow{2}{*}{MJD} & Exposure & Fitted band & Flux & Error & \multirow{2}{*}{Gaussians} & \multirow{2}{*}{$\chi^2_{\nu}$} & \multirow{2}{*}{DOF} \\ & & & & [second] & [keV] & \multicolumn{2}{c}{[0.5-10 keV; erg/s/cm$^2$]} & & & \\ \hline \textit{NICER} & 58 & 3200360182 & 59234.28 & 2712 & 1-10 & 1.01e-10 & 1.15e-12 & 1 & 1.05 & 890 \\ \textit{NICER} & 59 & 3200360183 & 59234.99 & 8227 & 1-10 & 9.52e-11 & 6.94e-13 & 1 & 1.04 & 890 \\ \textit{NICER} & 60 & 3200360184 & 59236.03 & 4121 & 1-10 & 8.12e-11 & 7.73e-13 & 1 & 1.00 & 890 \\ \textit{NICER} & 61 & 3200360185 & 59237.38 & 2583 & 1-10 & 7.09e-11 & 9.16e-13 & 1 & 1.02 & 890 \\ \textit{NICER} & 62 & 3200360186 & 59238.09 & 3829 & 1-10 & 9.06e-11 & 8.76e-13 & 1 & 1.14 & 890 \\ \textit{NICER} & 63 & 3200360187 & 59240.67 & 703 & 1-10 & 7.09e-11 & 2.12e-12 & 1 & 0.81 & 890 \\ \textit{NICER} & 64 & 3200360188 & 59241.13 & 4247 & 1-10 & 8.09e-11 & 7.93e-13 & 1 & 1.07 & 890 \\ \textit{NICER} & 65 & 3200360189 & 59242.29 & 2938 & 1-10 & 7.87e-11 & 9.01e-13 & 1 & 1.08 & 890 \\ \textit{NICER} & 66 & 3200360190 & 59243.0 & 4707 & 1-10 & 9.09e-11 & 7.60e-13 & 1 & 1.03 & 890 \\ \textit{NICER} & 67 & 3200360191 & 59246.23 & 1420 & 1-10 & 8.77e-11 & 1.45e-12 & 1 & 1.01 & 890 \\ \textit{NICER} & 68 & 3200360192 & 59248.03 & 847 & 1-10 & 2.00e-10 & 3.74e-12 & 1 & 1.07 & 890 \\ \textit{NICER} & 69 & 3200360193 & 59250.03 & 989 & 1-10 & 1.39e-10 & 2.17e-12 & 1 & 1.05 & 890 \\ \textit{NICER} & 70 & 3200360194 & 59251.97 & 1410 & 1-10 & 3.44e-10 & 4.18e-12 & 3 & 1.27 & 884 \\ \textit{NICER} & 71 & 3200360195 & 59252.16 & 1176 & 1-10 & 2.50e-10 & 2.85e-12 & 3 & 1.05 & 884 \\ \textit{NICER} & 72 & 3200360196 & 59253.59 & 6149 & 1-10 & 6.96e-10 & 1.88e-12 & 3 & 0.98 & 884 \\ \textit{NICER} & 73 & 3200360197 & 59254.04 & 1441 & 1-10 & 9.24e-10 & 4.85e-12 & 3 & 1.04 & 884 \\ \textit{NICER} & 74 & 3200360198 & 59256.04 & 1413 & 1-10 & 9.65e-10 & 5.18e-12 & 3 & 1.03 & 884 \\ \textit{NICER} & 75 & 3200360199 & 59257.07 & 1496 & 1-10 & 8.99e-10 & 4.61e-12 & 3 & 1.15 & 884 \\ \textit{NICER} & 76 & 3200360201 & 59259.72 & 1402 & 1-10 & 1.00e-09 & 5.52e-12 & 3 & 1.09 & 884 \\ \textit{NICER} & 77 & 3200360202 & 59262.17 & 695 & 1-10 & 6.67e-10 & 6.05e-12 & 3 & 1.07 & 884 \\ \textit{NICER} & 78 & 3200360203 & 59263.01 & 1144 & 1-10 & 4.67e-10 & 3.46e-12 & 3 & 1.19 & 884 \\ \textit{NICER} & 79 & 3200360204 & 59264.37 & 298 & 1-10 & 6.03e-10 & 8.10e-12 & 0 & 1.07 & 893 \\ \hline \textit{Swift} & 1 & 00035066077/pc & 59162.31 & 2964 & 1-10 & 1.25e-08 & 1.86e-10 & 0 & 1.03 & 811 \\ \textit{Swift} & 2 & 00035066078/wt & 59174.07 & 344 & 1-10 & 6.43e-08 & 6.42e-10 & 0 & 1.36 & 863 \\ \textit{Swift} & 3 & 00035066080/wt & 59180.84 & 908 & 1-10 & 5.11e-08 & 2.61e-10 & 0 & 1.16 & 891 \\ \textit{Swift} & 4 & 00089186001/wt & 59186.8 & 1380 & 1-10 & 3.06e-08 & 1.46e-10 & 0 & 1.29 & 894 \\ \textit{Swift} & 5 & 00013945001/wt & 59202.61 & 1053 & 1-10 & 5.12e-09 & 3.32e-11 & 0 & 1.13 & 884 \\ \textit{Swift} & 6 & 00013945002/wt & 59205.87 & 554 & 1-10 & 2.72e-09 & 2.78e-11 & 0 & 1.06 & 814 \\ \textit{Swift} & 7 & 00013945003/wt & 59208.53 & 898 & 1-10 & 1.15e-09 & 1.55e-11 & 0 & 1.08 & 774 \\ \textit{Swift} & 8 & 00013945004/wt & 59214.7 & 903 & 1-10 & 1.64e-10 & 7.34e-12 & 0 & 0.81 & 623 \\ \textit{Swift} & 9 & 00013945005/wt & 59217.15 & 1053 & 1-10 & 2.22e-10 & 7.15e-12 & 0 & 0.97 & 649 \\ \textit{Swift} & 10 & 00013945006/wt & 59220.33 & 1098 & 1-10 & 2.10e-10 & 1.14e-11 & 0 & 0.93 & 606 \\ \textit{Swift} & 11 & 00013945007/wt & 59223.33 & 953 & 1-10 & 1.11e-10 & 5.41e-12 & 0 & 0.84 & 549 \\ \textit{Swift} & 12 & 00013945008/wt & 59226.65 & 354 & 1-10 & 6.77e-10 & 6.86e-11 & 0 & 0.63 & 324 \\ \textit{Swift} & 13 & 00013945010/wt & 59232.76 & 1038 & 1-10 & 4.52e-11 & 1.88e-12 & 0 & 0.81 & 431 \\ \textit{Swift} & 14 & 00013945011/wt & 59235.01 & 1099 & 1-10 & 6.45e-11 & 3.27e-12 & 0 & 0.74 & 512 \\ \textit{Swift} & 15 & 00013945012/wt & 59237.06 & 1043 & 1-10 & 4.51e-11 & 3.72e-12 & 0 & 0.74 & 436 \\ \textit{Swift} & 16 & 00013945013/wt & 59241.58 & 1128 & 1-10 & 4.54e-11 & 2.21e-12 & 0 & 0.67 & 489 \\ \textit{Swift} & 17 & 00013945014/wt & 59244.31 & 963 & 1-10 & 2.75e-10 & 8.65e-11 & 0 & 0.63 & 559 \\ \textit{Swift} & 18 & 00013945015/wt & 59247.02 & 668 & 1-10 & 7.28e-11 & 4.50e-12 & 0 & 0.78 & 396 \\ \textit{Swift} & 19 & 00013945016/wt & 59250.61 & 1048 & 1-10 & 2.16e-10 & 2.28e-11 & 0 & 0.86 & 614 \\ \textit{Swift} & 20 & 00013945017/wt & 59253.4 & 858 & 1-10 & 2.65e-10 & 8.46e-12 & 0 & 0.97 & 647 \\ \textit{Swift} & 21 & 00013945018/wt & 59256.26 & 917 & 1-10 & 7.79e-10 & 1.21e-11 & 0 & 1.05 & 751 \\ \textit{Swift} & 22 & 00013945019/wt & 59259.24 & 933 & 1-10 & 1.01e-09 & 1.15e-11 & 0 & 0.84 & 772 \\ \textit{Swift} & 23 & 00013945020/wt & 59263.43 & 790 & 1-10 & 3.58e-10 & 8.73e-12 & 0 & 0.99 & 645 \\ \textit{Swift} & 24 & 00013945021/wt & 59269.2 & 950 & 1-10 & 5.18e-10 & 9.78e-12 & 0 & 1.08 & 699 \\ \textit{Swift} & 25 & 00013945022/wt & 59272.58 & 908 & 1-10 & 5.59e-10 & 9.99e-12 & 0 & 0.92 & 739 \\ \textit{Swift} & 26 & 00013945023/wt & 59275.84 & 883 & 1-10 & 8.46e-10 & 1.18e-11 & 0 & 0.97 & 749 \\ \textit{Swift} & 27 & 00013945024/wt & 59278.95 & 1043 & 1-10 & 5.36e-10 & 7.67e-12 & 0 & 1.00 & 714 \\ \textit{Swift} & 28 & 00013945025/wt & 59281.35 & 15 & 1-10 & 1.00e-08 & 2.93e-09 & 0 & 0.11* & 248 \\ \textit{Swift} & 29 & 00013945026/wt & 59284.93 & 928 & 1-10 & 3.56e-10 & 1.01e-11 & 0 & 0.99 & 699 \\ \textit{Swift} & 30 & 00013945027/pc & 59292.1 & 350 & 1-10 & 1.34e-09 & 6.90e-11 & 0 & 0.80 & 345 \\ \textit{Swift} & 31 & 00013945027/wt & 59292.1 & 608 & 1-10 & 1.09e-09 & 2.12e-11 & 0 & 1.04 & 730 \\ \textit{Swift} & 32 & 00013945028/pc & 59298.27 & 567 & 1-10 & 9.52e-10 & 4.72e-11 & 0 & 0.67 & 380 \\ \hline \end{tabular}\\ \end{table*} \bsp % \label{lastpage}

Title: Planck and BICEP/Keck Array 2018 constraints on primordial gravitational waves and perspectives for future B-mode polarization measurements

Abstract: Current and future B-mode polarization data are the most powerful observables to constrain gravitational waves from the early Universe. We set conservative constraints on tensor modes when relaxing the inflationary consistency condition $n_t=-r/8$ between the tensor tilt $n_t$ and the tensor-to-scalar ratio r. By adding a power-law spectrum of tensor perturbations to $\Lambda$CDM, and parameterizing this tensor contribution by two independent primordial tensor-to-scalar ratios $(r_1,r_2)$ at $k_1 = 0.005$ Mpc$^{-1}$ and $k_2 = 0.02$ Mpc$^{-1}$, Planck and BICEP/Keck Array 2018 data (BK18) lead to constraints $r_{0.005} < 0.030$ and $r_{0.02} < 0.098$ at 95% CL. The corresponding upper bound $r_{0.01} < 0.039$ is by a factor of 2 tighter than the one obtained with Planck 2018 and the older BK15 data. We then study the perspectives for future CMB experiments that will measure both the reionization bump and recombination peak of the B-mode polarization angular power spectrum, such as LiteBIRD. We test the robustness of the results to the choice of the scales for $(r_1,r_2)$ in these future perspectives. Whereas distinguishing $n_t=-r/8$ from exact scale invariance is impossible as expected, we show how radical, theoretically motivated departures from $n_t=-r/8$, which are consistent with the current data, could be distinguished with LiteBIRD. Moreover, LiteBIRD will be able to shrink the allowed parameter space area in the $(r_{0.005},r_{0.02})$ plane to less than one hundredth of the currently allowed area by Planck 2018 and BK18.

https://export.arxiv.org/pdf/2208.10482

\title{Planck and BICEP/Keck Array 2018 constraints on primordial gravitational waves and perspectives for future B-mode polarization measurements} \author{Daniela Paoletti} \email{daniela.paoletti@inaf.it} \affiliation{INAF/OAS Bologna, Osservatorio di Astrofisica e Scienza dello Spazio, Area della ricerca CNR-INAF, via Gobetti 101, I-40129 Bologna, Italy}% \affiliation{INFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, Italy} \author{Fabio Finelli}% \email{fabio.finelli@inaf.it} \affiliation{INAF/OAS Bologna, Osservatorio di Astrofisica e Scienza dello Spazio, Area della ricerca CNR-INAF, via Gobetti 101, I-40129 Bologna, Italy}% \affiliation{INFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, Italy} \author{Jussi Valiviita} \email{jussi.valiviita@helsinki.fi} \affiliation{Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland} \author{Masashi Hazumi} \email{masashi.hazumi@kek.jp} \affiliation{International Center for Quantum-field Measurement Systems for Studies of the Universe and Particles (QUP), High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan} \affiliation{Institute of Particle and Nuclear Studies (IPNS), High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan} \affiliation{Japan Aerospace Exploration Agency (JAXA), Institute of Space and Astronautical Science (ISAS), Sagamihara, Kanagawa 252-5210, Japan} \affiliation{Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan} \affiliation{The Graduate University for Advanced Studies (SOKENDAI), Miura District, Kanagawa 240-0115, Hayama, Japan} \date{October 31, 2022}% \section{Introduction} Primordial gravitational waves generated during inflation \cite{Starobinsky:1979ty} have a characteristic shape in the Cosmic Microwave Background (CMB) angular power spectra of temperature and polarization anisotropies which distinguishes them from scalar curvature perturbations. On top of temperature ($T$) and $E$-mode ($E$) polarization (also produced by curvature perturbations), the distinctive imprint of primordial gravitational waves is $B$-mode polarization \cite{Kamionkowski:1996zd,Seljak:1996gy}. The increasing sensitivity of CMB polarization measurements has led to comparable constraints on primordial gravitational waves from $T$,$E$, and from $B$ mode separately by the joint analysis of BICEP2/\Keck\ Array and \Planck\ data \cite{BICEP2:2015nss}. Since then, $B$-mode polarization alone has given a tighter constraint than $T$,$E$ for the tensor-to-scalar ratio \cite{BICEP2:2015xme,BICEP2:2018kqh,BICEP:2021xfz}, \begin{equation} r(k) = \frac{{\mathcal P}_{\rm t}(k)}{{\mathcal P}_{\mathcal R}(k)}\,, \label{ratio} \end{equation} where ${\mathcal P}_{\rm t}(k)$ and ${\mathcal P}_{\mathcal R}(k)$ are the tensor and scalar power spectra, here assumed as power laws, \begin{equation} {\mathcal P}_{\rm t}(k) = A_{{\rm t}} \left( \frac{k}{k_{{\rm *}}}\right)^{n_{\rm t}}\,, \label{eq:PTk} \end{equation} \begin{equation} {\mathcal P}_{\mathcal R}(k) = A_{\mathcal R} \left( \frac{k}{k_{{\rm *}}}\right)^{n_{\rm s}-1}\,. \label{eq:PSk} \end{equation} In the above, $n_{\rm t}$ ($n_{\rm s}$) is the tensor (scalar) spectral index and $A_{{\rm t}}$ ($A_{{\rm s}}$) the tensor (scalar) amplitude at the pivot scale, typically chosen such that $k_*=0.05\,$Mpc$^{-1}$ \footnote{We denote the tensor-to-scalar ratio at this scale simply by $r$ or occasionally by $r_{0.05}$ and at any other scale by adding a subscript indicating the corresponding wavenumber. Integer subscripts $1$ and $2$ refer to the scales of our two-scale parametrization, explained in Sec.~\ref{sec:two-scale}.}. The recent release of BICEP-Keck Array data (BK18) \cite{BICEP:2021xfz} in combination with \Planck\ 2018 data has set a 95\% CL upper limit on the tensor-to-scalar ratio $r < 0.035$ in the case of a scale-invariant primordial spectrum of gravitational waves. When the tensor tilt $n_{\mathrm t}$ satisfies the so-called consistency condition, i.e., $n_{\mathrm t} = -r/8$, motivated by Bunch-Davies initial conditions for tensor modes during slow-roll inflation driven by a single real scalar field with a standard kinetic term (denoted in the following by SSSRI), the limit is unchanged. This limit leads to the 95\% CL upper bound on the scale of inflation \begin{equation} V_* = \frac{3 \pi^2 A_{\mathrm{s}}}{2} \, r \, M_{\mathrm {Pl}}^4 < (1.4 \times 10^{16}~{\mathrm{GeV}} )^4 \quad (95\%\ \text{CL}), \end{equation} or on the Hubble parameter during inflation \begin{equation} \frac{H_*}{M_{\mathrm{Pl}}} < 2.0 \times 10^{-5} \quad (95\%\ \text{CL}). \end{equation} The improvements with BK18 compared to BK15, when combined with \Planck\ 2018 data \cite{Planck:2018jri}, in terms of tighter constraints to slow-roll inflationary models can be seen in the $(n_{\mathrm{s}},\,r_{0.002})$ plane in Fig.~\ref{fig:nsr}. The availability of an accurate and precise $B$-mode polarization likelihood has also made it possible to derive data driven constraints when the theoretical prior $n_{\mathrm t} = -r/8$ is relaxed \cite{Planck:2015sxf,Planck:2018jri}. This more conservative and phenomenological approach is justified since deviations from $n_{\rm t}=-r/8$ are predicted in well-motivated theoretical inflationary models. These deviations occur, for example, with a non-standard kinetic term for a single scalar field \citep{Garriga:1999vw} or with a more general Lagrangian \citep{Kobayashi:2010cm}, with an initial vacuum state which is not Bunch-Davies \cite{Ashoorioon:2013eia}, when more than one scalar field is present \citep{Bartolo:2001rt,Wands:2002bn,Byrnes:2006fr} or these are coupled also through the kinetic terms \cite{DiMarco:2005nq,Achucarro:2012yr}, and in Gauge-flation when a non-Abelian gauge field in a particular isotropic configuration drives the accelerated stage \cite{Maleknejad:2011jw}. The relation $n_{\mathrm t} = -r/8$ is also violated when gravitational waves are not only amplified by the expansion from quantum fluctuations, but also sourced by spectator fields \citep{Cook:2011hg,Dimastrogiovanni:2016fuu,Agrawal:2017awz} present during inflation, an effect which also leads to significant primordial non-Gaussianity. More radical departures from a nearly scale-invariant power spectrum are predicted in alternatives to inflation \citep{Gasperini:1992em,Boyle:2003km,Brandenberger:2006xi}. In this paper we use the two-scale analysis for tensor perturbations \cite{Planck:2015sxf,CORE:2016ymi,Planck:2018jri} in order to present the updated BK18 conservative constraints and the perspectives for future CMB polarization measurements when the theoretical prior $n_{\mathrm t} = -r/8$ is relaxed. After a review of the two-scale analysis for tensors in Sec.~\ref{sec:two-scale}, we present the \Planck\ 2018 + BK18 results in Sec.~\ref{sec:realdata} with a comparison to those derived with BK15 in \cite{Planck:2018jri}. In Sec.~\ref{sec:forecasts}, we forecast the capability of future $B$-mode polarization measurements to constrain a power-law spectrum of gravitational waves, by taking as a representative example the specifications of the LiteBIRD mission \citep{LiteBIRD:2022cnt}. We asses the dependence on the scales chosen and test how much the constraints would degrade if the low-multipole $B$-mode data were missing. In Sec.~\ref{sec:conclusions}, we draw conclusions. \section{Beyond consistency condition on the tensor tilt\label{sec:two-scale}} In this paper, we relax the condition $n_\mathrm{t} = -r/8$ and let the data (either the real or simulated one) determine both the amplitude of tensor perturbations and the tilt of their spectrum $n_\mathrm{t}$. These analyses are highly motivated by two reasons: 1) Without testing for this possibility, one would not know if there was a better fit than the standard inflationary prediction somewhere in the parameter space. 2) If it turns out that the data are consistent with the standard inflationary prediction, it is important to know how large a difference from $n_\mathrm{t} = -r/8$ one would need before the data were able to discern it from the standard inflationary prediction. We assume the cosmological concordance model, i.e., the adiabatic $\Lambda$CDM model without tensor perturbations. On this playground, while varying all the standard $\Lambda$CDM parameters, we test a more complicated model that also has tensor perturbations. If we assumed the inflationary consistency condition, $n_\mathrm{t} = -r/8$, we would have one extra parameter, $r$. A single value $r=0$ (which automatically also fixes $n_\mathrm{t}$ to zero) would mean that we recover the underlying simpler, tensorless, model. Any positive value means that the model is the more complicated one that has tensor perturbations. However, once we relax the consistency condition, we cannot choose $r$ and $n_\mathrm{t}$ as our sampling parameters when finding parameter constraints against the data, if the data are consistent with $r=0$ or if this model is a relatively good fit to the data. Here the problem is that setting $r=0$ already reduces our model to the simpler, tensorless, model, no matter what value $n_\mathrm{t}$ has. Once we marginalize over the $n_\mathrm{t}$ direction of parameter space, e.g., in our Markov Chain Monte Carlo (MCMC) analysis, we get larger and larger weight the closer to zero $r$ is, since $n_\mathrm{t}$ can have here very large negative or positive values. In essence, how sharp a peak near zero $r$ in the posterior probability density we find, depends mainly on two things: how wide a prior range we allow for $n_\mathrm{t}$ and what bin size we use for plotting 1d pdf of $r$. With a fine binning, doubling the allowed range of $n_\mathrm{t}$ would roughly double the weight of the first bins in $r$. \emph{All this would lead to artificially tight constraints on $r$} (in the case where the data are consistent with $r=0$). As a result, we also would report over-optimistic constraining power for future experiments. We illustrate this $n_\mathrm{t}$-prior-range effect in Fig.~\ref{fig:ntprioreffect} using the BK15 data and keeping other than tensor parameters fixed to the \Planck\ best-fit $\Lambda$CDM values. The wider prior range we allow for $n_\mathrm{t}$, the closer to zero the tensor-to-scalar ratio at $k=0.01\,$Mpc$^{-1}$ will be. In addition, the posterior probability for $n_\mathrm{t}$ obtained by considering it as a primary parameter would depend on the pivot scale chosen \footnote{ In a similar manner, there is a dependence on the pivot-scale chosen in fitting the isocurvature spectral index, as demonstrated in the $n_\mathrm{iso}$ panel of figure 21 of Ref.~\cite{Kurki-Suonio:2004bou}.}. Fortunately, these problems with the combination $(r,n_\mathrm{t})$ can be overcome by a well-defined combination $\left[{\mathcal P}_{\rm t}(k_1),\, {\mathcal P}_{\rm t}(k_2)\right]$ or $(r_1,\,r_2)$, where the former are the amplitudes of tensor perturbations at two different wavenumbers $k_1$ and $k_2$ and the latter are the tensor-to-scalar ratios at these two wavenumbers. With either of these parametrizations, the simpler, tensorless, model reduces to one single point $(0,0)$, instead of being an infinitely long line as in the $(r,n_\mathrm{t})$ case. The very same problem was identified 20 years ago when studying scalar isocurvature perturbations with a free spectral index in \cite{Kurki-Suonio:2004bou} and the suggested two-scale solution was implemented for the first time in \cite{Keskitalo:2006qv}, there named as an amplitude parametrization. Since then the method has also been applied to tensor perturbations by the \Planck\ Collaboration in \cite{Planck:2015sxf,Planck:2018jri}. We adopt $(r_1,r_2)$ to describe the power-law spectrum of tensor modes \footnote{% Equation~(\ref{matInter}) specifies a straight line in the $(\ln k,\, \ln {\cal P}_\mathrm{t})$ plane. This line goes through points $(\ln k_1,\, \ln {\cal P}_{\mathrm{t}1})$ and $(\ln k_2,\, \ln {\cal P}_{\mathrm{t}2})$, where ${\cal P}_{\mathrm{t}1} = {\cal P}_{\mathrm{t}}(k_1) = r_1 {\cal P}_{\mathrm{\cal R}} (k_1)$ and ${\cal P}_{\mathrm{t}2} = {\cal P}_{\mathrm{t}}(k_2) = r_2 {\cal P}_{\mathrm{\cal R}} (k_2)$. }, \begin{align} \begin{split} {\cal P}_{\mathrm{t}}(k)=&\exp \Biggl\{ \frac% {\ln k -\ln k_1} {\ln k_2-\ln k_1} \ln\bigl[ r_2 {\cal P}_{\mathrm{\cal R}} (k_2)\bigr]\\ &\qquad - \frac% {\ln k -\ln k_2} {\ln k_2-\ln k_1} \ln\bigl[ r_1 {\cal P}_{\mathrm{\cal R}} (k_1) \bigr] \Biggr\}. \label{matInter} \end{split} \end{align} In order to obtain reasonable constraints on these tensor-to-scalar ratios at two different scales, the wavenumbers $k_1$ and $k_2$ should be chosen in such a way that $k_1$ corresponds roughly to the largest observable scale and $k_2$ to the smallest observable scale. The exact best choice could depend on the fraction of the sky observed and its coverage in multipoles (and therefore on wavenumbers). For this reason, we study in this paper different choices of the pair $(k_1,k_2)$, in particular, for future experiments. In case of the tensor perturbations, there are two clearly observable features in the $B$-mode polarization at different $k$s: the reionization and recombination peaks. Thus we pick $k_1$ and $k_2$ near these features, respectively. Once we have used $(r_1,r_2)$ as the primary sampling parameters in our MCMC runs, we can calculate so-called derived parameters that have non-uniform priors (unlike the primary parameters). We often report $r$ at some middle scale between $k_1$ and $k_2$. If we want to show how the results would look like if the derived parameter had a uniform prior, we can simply weight our MCMC results by the inverse of the determinant of the Jacobian transform from the primary parameters to the derived one(s). In particular, we compare many of our results by reporting $r_{0.01}$, i.e., $r$ at $k=0.01\,$Mpc$^{-1}$, which is close to the decorrelation scale of ($r$, $n_\mathrm{t}$). However, any fundamental conclusions, such as a detection or determining whether $n_\mathrm{t} = -r/8$ is consistent with the data,should be drawn from the joint two-dimensional posterior distribution of $(r_1,r_2)$. It should be kept in mind that even if $r_1$ and $r_2$ are drawn independently from a uniform distribution, i.e., $r_1$ and $r_2$ have flat priors in the MCMC runs, the individual posterior probability densities for $r_1$ alone, or for $r_2$ alone, or, in particular, for the derived parameter $r_{0.01}$ alone do not encode the full result. Instead, one should resort to the (marginalized) two-dimensional posterior of $(r_1,r_2)$ --- either its numerical or graphical representation. For example, if the best-fit point and the 95\% CL contour in the $(r_1,r_2)$ plane were clearly away from point $(0,0)$, but the 99.7\% CL contour just reached $(0,0)$, then we might claim a weak $3\sigma$ detection. \section{\Planck\ and BICEP/\Keck\ Array 2018 constraints\label{sec:realdata}} We now derive conservative constraints on primordial gravitational waves with the current data by adopting the two-scale parametrization described in the previous section. We use the \Planck\ 2018 data \citep{Planck:2019nip} and the latest BICEP/\Keck\ Array data release BK18 \citep{BICEP:2021xfz}. We employ the \Planck\ 2018 baseline likelihood consisting of: a Gibbs sampling likelihood based on the component separated CMB map for temperature at $\ell \le 30$; $E$-mode simulation likelihood at $\ell \le 30$ based on the $100\times143\,$GHz cross angular power spectrum; Plik TTTEEE binned likelihood at high multipoles, i.e., $\ell > 30$. We also include the lensing likelihood based on the four-point correlation function of the lensing signal in the conservative multipole range $8$--$400$. As the BICEP/\Keck\ likelihood for $B$-mode polarization, we use the recently released likelihood which includes BICEP, \Keck\ Array and BICEP3 data up to the 2018 observing season \citep{BICEP:2021xfz}. We use \texttt{cosmomc} \cite{Lewis:2002ah,Lewis:2013hha} as the MCMC sampler and as a Bolzmann solver a modified version of \texttt{camb} \cite{Lewis:1999bs,Howlett:2012mh}, which includes the two-scale treatment for the tensor modes. In this work, with the real data, we use $k_1 = 0.005$ Mpc$^{-1}$ and $k_2 = 0.02$ Mpc$^{-1}$ and we also project our results on $r$ at the scale $k=0.01$ Mpc$^{-1}$. In \Planck\ X 2018 \cite{Planck:2018jri}, the use of $k_1 = 0.002$ Mpc$^{-1}$ was motivated by considering one of the two most-used scales for the tensor-to-scalar ratio as a primary parameter, but here we instead prefer to use a slightly smaller scale $k_1 = 0.005$ Mpc$^{-1}$, which has a broader overlap with the lowest multipoles probed by the BICEP/\Keck\ Array likelihood. The 68\% CL and 95\% CL posterior constraints on our primary tensor parameters are shown by the blue shaded regions in the first panel of Fig.~\ref{fig:realdata}. For a comparison, we also show by blue dotted lines the constraints we obtained with an older BICEP/\Keck\ Array likelihood from 2015 (BK15) together with the \Planck\ 2018 data. BK18 data are consistent with no primordial gravitational waves also when relaxing $n_\mathrm{t}=-r/8$ or $n_\mathrm{t}=0$ and improve the constraints significantly over BK15 in combination with \Planck. As can be seen from the first panel of Fig.~\ref{fig:realdata}, the line $n_\mathrm{t}=-r/8$ is within the 68\% CL. Using the same methodology as in \Planck\ X 2018 \cite{Planck:2018jri}, we also repeat the analysis by adding the LIGO\&Virgo 2016 95\% CL upper bound on the energy density parameter from gravitational waves, $\Omega_\mathrm{GW} < 1.7\times 10^{-7}$ at $k=(1.3$--$5.5)\times 10^{16}\,$Mpc$^{-1}$ \cite{LIGOScientific:2016jlg}, which is 18 orders larger $k$ than probed by the CMB $B$ mode. If the tensor power spectrum followed the strict power law that we assume, then a large region of positive $n_\mathrm{t}$ values would lead to a direct detection of stochastic primordial gravitational wave background that LIGO\&Virgo has not seen. The results, when making this huge extrapolation, are indicated by light gray in Fig.~\ref{fig:realdata}. Finally, we reweigh our \Planck+BK18 MCMC chains to demonstrate that using $(r_{0.01},\,n_\mathrm{t})$ as primary parameters would artificially exaggerate the constraining power of the data by giving a large weight to the models that have $r$ near to zero (where $|n_\mathrm{t}|$ can be almost arbitrarily large and hence give extra weight to $r\sim 0$ upon marginalization). This case is indicated by the blue \emph{dashed} lines in Fig.~\ref{fig:realdata}. The second panel of Fig.~\ref{fig:realdata} is based on the same analysis as the first panel, but now we show the derived parameters $r_{0.01}$ and $n_\mathrm{t}$ as in \cite{Planck:2015sxf,Planck:2018jri}, while the third panel shows the one-dimensional posterior probability densities (1d pdf) with peak values normalized to a same constant. Our main result with the real data is \begin{equation} \left.\begin{aligned} r_{0.005} & < 0.030\\ r_{0.02} & <0.098 \end{aligned}\ \right\}\ \ \mbox{\text{\parbox{4.9cm}{\begin{flushleft} (95\% CL, \Planck\ TT,TE,EE\\+lowE+lensing+BK18). \end{flushleft}}}} \label{eq:LCDMr1r2BK18} \end{equation} These constraints improve on the corresponding ones obtained with BK15, i.e. $r_{0.005} < 0.041$ and $0.009 < r_{0.02} < 0.23$. The constraints on the derived tensor parameters are \mbox{$r_{0.01}< 0.039$} and $-0.61< n_{\mathrm t} < 2.73$ at 95\% CL, when using flat priors on the primary parameters. From the last two panels of Fig.~\ref{fig:realdata}, we notice that \Planck+BK18 gives by a factor of two a tighter constraint on $r_{0.01}$ compared to \Planck+BK15. Thus, BK18 represents a significant improvement also beyond the case of a fixed $n_\mathrm{t}$ studied in \cite{BICEP:2021xfz}. Naturally, % the 95\% CL contours on $n_\mathrm{t}$ do not improve since BK18 brings the constraint on the actual tensor contribution closer to zero. The mean of the posterior at $n_\mathrm{t} \sim 1$ is due to the transfer function of primordial gravitational waves (that strongly damps their contribution to $C_\ell^{BB}$ at $\ell \gtrsim 200$ unless the primordial tilt $n_\mathrm{t}$ is very large), in combination with the CMB lensing, noise plus foregrounds, and cosmic variance. The primordial signal which minimizes the $\chi^2$ have (the amplitude and) a tensor tilt that mimics the effective noise. This phenomenon is analogous to the apparent preference of $n_\mathrm{iso} \sim 3$ for the CDM isocurvature perturbations in the lack of a detection of such a component, as explained, e.g., in Refs.~\cite{Kurki-Suonio:2004bou,Valiviita:2009bp,Valiviita:2012ub}. The flat priors on $r_1$ and $r_2$ induce a non-flat prior on $n_\mathrm{t}$ \cite{Galloni:2022mok} with a peak at $n_\mathrm{t} \approx 0$, as shown in the upper panel of Fig.~\ref{fig:ntprior}. This might introduce a mild push on $n_\mathrm{t}$ toward zero, but in Sec.~\ref{sec:nullcase} we show by using simulated $r=0$ data that this push does not outweigh the above-mentioned natural preference of $n_\mathrm{t} \sim 1$ in the null case when using the CMB data alone. The symmetric posterior of $n_\mathrm{t}$ around $1$ is an implication of \Planck+BK18 being consistent with no tensors within the sensitivity of these data. The induced prior on $r_{0.01}$ (see the red dotted curve in the lower panel of Fig.~\ref{fig:ntprior}) mildly pushes $r_{0.01}$ away from zero, making our quoted upper bound a conservative one \footnote{As the two-dimensional analysis of $(r_1,\,r_2)$ does not indicate any detection of a non-zero tensor contribution, i.e., the best fit is very near to $(0,\,0)$ and $(0,\,0)$ is in the 68\% CL region, we report the conservative 95\% CL upper bounds on the tensor-to-scalar ratio by forcing a one-tail analysis in \texttt{getdist}.}. In Table~\ref{tab:cosmoparamsrealdata}, we do not find any statistically significant shift in the remaining cosmological parameters when the consistency relation between the tensor-to-scalar ratio and the tensor tilt is relaxed. We also do not observe major degeneracy among $r_{0.005}$, $r_{0.02}$, and the foreground/nuisance parameters of the BK18 likelihood in combination with \Planck. When $n_\mathrm{t}$ is allowed to vary, the low-$k$ constraint, $r_{0.005} < 0.03$, does not degrade compared to the derived constraint of the $n_\mathrm{t} = -r/8$ case. Indeed, as there is more allowed parameter-space volume at the positive $n_\mathrm{t}$, the low-$k$ constraint is slightly tighter than in the $n_\mathrm{t} = -r/8$ model. Once we pass the recombination bump, the data become less and less sensitive to the primordial tensor modes as they are damped by the transfer function. This is reflected by the fact that the constraint on $r_{0.02}$ is by a factor of three weaker than the corresponding bound when keeping $n_\mathrm{t}$ fixed. Finally, once projected on the standard pivot scale, $k_* = 0.05$ Mpc$^{-1}$, we have $r_{0.05} < 0.71$ at 95\% CL when $n_\mathrm{t}$ is allowed to vary, which is by nearly a factor of 20 weaker than the upper bound $0.035$ obtained with a fixed $n_\mathrm{t}=-r/8$. \begin{table} \begin{ruledtabular} \begin{tabular}{lccdr} \textrm{Parameters}& \textrm{\Planck+BK18}&\textrm{\Planck+BK18} \\ &$n_{\rm t}=-r/8$ & free $n_{\rm t}$\\ \colrule \noalign{\vskip 2pt} $\Omega_\mathrm{b} h^2$ & $0.0224\pm 0.0001$ & $0.0224 \pm 0.0001$ \\ $\Omega_\mathrm{c} h^2$ & $0.120\pm 0.001 $ & $0.120\pm 0.001$ \\ 100 $\theta$ & $1. 0409 \pm 0.0003$ & $1.0409 \pm 0.0003$\\ $\tau$ & $0.0546_{-0.0072}^{+0.0073}$& $0.0544\pm 0.0073$ \\ $\ln(10^{10} A_\mathrm{s})$ & $3.045\pm 0.014$ & $3.044 \pm 0.014$ \\ $n_\mathrm{s}$ & $0.9653\pm 0.0041 $ & $0.9656\pm 0.0041$ \\ \noalign{\vskip 2pt} \colrule \noalign{\vskip 2pt} $r_{0.005}$ & \textcolor{gray}{$(<0.032)$} & $\mathbf{<0.030}$ \\ $r_{0.02}$ & \textcolor{gray}{$( <0.034)$} & $\mathbf{<0.098}$ \\ $r_{0.05}$ & $\mathbf{<0.035}$ & \textcolor{gray}{$(<0.71)$} \\ \end{tabular} \end{ruledtabular} \caption{\label{tab:cosmoparamsrealdata} 68\% CL constraints for the $\Lambda$CDM parameters with \Planck+BK18 data. In bold are the 95\% CL upper bounds for the primary tensor parameters, and in parenthesis are the derived tensor parameters. } \end{table} \section{Forecasts for future experiments\label{sec:forecasts}} In the next years, there will be several experiments devoted to CMB polarization measurements and, in particular, to the $B$ modes \citep{SimonsObservatory:2018koc,LSPE:2020uos,LiteBIRD:2022cnt,CMB-S4:2016ple}. In this section, we compute forecasts when $n_\mathrm{t}$ is allowed to vary by using the two-scale parametrization (see also \cite{CORE:2016ymi}) and simulated $B$-mode data representative of the future CMB measurements, taking as an example the Lite (Light) satellite for the study of $B$-mode polarization and Inflation from cosmic background Radiation Detection (LiteBIRD) \citep{LiteBIRD:2022cnt}, selected by the Japan Aerospace Exploration Agency (JAXA) as a strategic large class mission to which, in addition to Japan, also Europe, the United States, and Canada contribute. \begin{table} \begin{tabular}{ p{1.95cm} p{1.95cm} p{1.95cm} p{1.95cm} } \hline \hline \noalign{\vskip 3pt} \multicolumn{4}{c}{LiteBIRD} \\ \hline \noalign{\vskip 3pt} Frequency & T-sens & P-sens &FWHM \\ (GHz) & ($\mu K\,$arcmin) & ($\mu K\,$arcmin) & (arcmin)\\ \noalign{\vskip 2pt} \noalign{\hrule\vskip 3pt} 78 & 8.53 &12.07&36.9\\ 89 &7.99 &11.30& 33.0\\ 100 &4.64 &6.56& 30.2\\ 119 &3.24 &4.58& 26.3\\ 140 &3.39 &4.79& 23.7\\ 166 &3.94 &5.57& 28.9\\ 195 &4.14 &5.85& 28.0\\ \hline \hline \end{tabular} \caption{\label{tab:LB}A LiteBIRD-like configuration of the instrument central frequency channels, following the characterization given in \cite{LiteBIRD:2022cnt}. } \end{table} \subsection{Simulated data and methodology} We consider LiteBIRD-like instrumental specifications given in Table~\ref{tab:LB}. We produce simulated data for $T,E$ by considering the inverse noise weighting of the central frequency channels and by assuming that the lowest and highest frequencies are used to separate the foreground emission as done in \cite{LiteBIRD:2022cnt} (see also \cite{CORE:2016ymi}). For the $B$-mode polarization (in addition to the instrumental noise), we include the following two sources of confusion: the lensing signal and a contribution which mimics the foreground residuals. These inputs are inserted in a Wishart-like likelihood with an effective sky fraction of 70\% (60\%) for $T,E$ ($B$). With these settings, we obtain for the $n_\mathrm{t}=-r/8$ model a $\sigma(r) \sim 0.0013$, which is 30\% larger than the LiteBIRD value \cite{LiteBIRD:2022cnt} which includes systematic effects. Our constraints can therefore be seen as a conservative assessment of the LiteBIRD capabilities. We create simulated data from three fiducial models. Our first fiducial model satisfies the inflationary consistency condition $n_\mathrm{t} = -r/8$ (IC) and is motivated by SSSRI with Bunch-Davies quantum initial condition. % As a second case, we consider a positive value for the tensor tilt ($n_\mathrm{t}+$), as occurs when the null-energy condition during inflation is violated \cite{Baldi:2005gk} or in Galileon inflation \cite{Kobayashi:2010cm}. We fix $n_\mathrm{t}=0.3$, which is allowed by the \Planck\ constraints on primordial non-Gaussianity \cite{2011PhRvD..83j3524K,Planck:2015zfm,Planck:2015sxf,Planck:2019kim} in Galileon inflation \cite{Kobayashi:2010cm}. As a third case, we consider $n_{\rm t}=-r/(8 c_\mathrm{s})$, theoretically predicted by slow-roll inflation with a non-canonical kinetic term, which leads to a non-trivial speed of sound $(0 < )\ c_\mathrm{s} < 1$, with $c_\mathrm{s} = \partial p (\phi, X)/\partial X$. Here $p$ is the Lagrangian for the inflaton and $X = - \partial_\mu \phi \partial^\mu \phi/2$. As a value for the inflaton speed of sound we consider the 95\% CL lower limit $c_\mathrm{s} = 0.02$ obtained by the constraint on primordial non-Gaussianity \cite{Planck:2015zfm,Planck:2019kim}, which is enhanced by the non-trivial speed of sound. This value leads to a negative value for the tensor tilt, i.e. $n_\mathrm{t} = -0.3$, and we denote this case by $n_\mathrm{t}-$. We set $r=0.05$ (at $k=0.05\,$Mpc) in all three fiducial models and assume the underlying $\Lambda$CDM cosmology to be consistent with the \Planck\ 2018 baseline results: $\Omega_\mathrm{b} h^2$ = 0.02237, $\Omega_\mathrm{c} h^2$ = 0.120, $100\,\theta$ = 1.04092, $\tau$ = 0.0544, $n_\mathrm{s}$ = 0.9649, $\ln(10^{10}A_\mathrm{s})$ = 3.044. Figure~\ref{fig:APS} shows the $B$-mode angular power spectra for these fiducial models and, for comparison, the instrumental noise and lensing signal. With each of the three simulated data sets, we run three separate full MCMC runs (i.e., nine runs in total), but choosing three different pairs $k_1$---$k_2$ for the two-scale parametrization. As explained in Sec.~\ref{sec:two-scale}, the best choice depends on the $k$-coverage (multipole coverage) of the data. In addition, the best choice may depend on the actual underlying model to be recovered. In this subsection our fiducial models described above are chosen to have a largish $r$ and/or $n_\mathrm{t}$ (allowed by the previous BK15 data release), in order to test/exaggerate the latter effect. In the next subsection we repeat the analysis using fiducials that would be allowed by the current constraints. We test the sensitivity of parameter estimation to the choice of $k_1$ and $k_2$ by using the pairs $0.001$---$0.01$, $0.002$---$0.02$, and $0.005$---$0.02$ Mpc$^{-1}$. All these scales effectively correspond to multipoles where the expected tensor signal is non-negligible, differently from the conventional $k_*=0.05\,$Mpc$^{-1}$, where the signal is damped by the tensor transfer function. We check the posteriors of the primary tensor sampling parameters $r_1$ and $r_2$ in Fig.~\ref{fig:largefid_r1_r2}. We note that for all three fiducial models the green case ($k_1$---$k_2$=$0.002$---$0.02$) performs worst since it leads to a degeneracy between $r_1$ and $r_2$ for this LiteBIRD-like configuration that we use, which degrades the determination of both these parameters. The derived tensor parameters from the same MCMC runs as above are shown in Fig.~\ref{fig:2D05}. We notice that for all three fiducial models considered, any of our choices of $k_1$---$k_2$ leads to an excellent recovery of the derived parameters $r_{0.05}$, $r_{0.01}$, and $n_\mathrm{t}$ in terms of the median of the posterior. From Fig.~\ref{fig:2D05}, we see that distinguishing $n_{\rm t}=-r/8$ from the exact scale invariance is out of reach as expected \cite{Knox:2002pe}. For the modified sound speed case, $n_\mathrm{t}-$ in the darkest colors, we observe significantly better constraints on the tensor spectral index. This result reflects the fact that we used a fixed $r_{0.05}=0.05$ as an input fiducial, which, in the $n_\mathrm{t}-$ case, translates into a large $r$ at the LiteBIRD sensitivity region, as is obvious from the values of $r_{0.01}$ in the right panel of Fig.~\ref{fig:2D05}. Having a large $r$ naturally leads to tighter constraints on $n_\mathrm{t}$. This should be kept in mind when interpreting the results. For the $n_{\rm t}=-r/8$ case, it does not make a big qualitative difference on what scale one quotes $r$, as $n_{\rm t} \approx 0$ (and $n_\mathrm{s}-1 \approx -0.04$). However, once we relax the consistency condition, we are required to be careful and explicit with the scales. What matters is $r$ in the sensitivity region of the experiment. At the standard pivot scale, $k=0.05\,$Mpc$^{-1}$, the tensor-to-scalar ratio can then be very large (small) in the $n_{\rm t}+$ ($n_{\rm t}-$) case. \subsection{Forecasts for realistic cases} We now present the forecasts by choosing fiducial models in such a way that $r_{0.01}$ and $n_\mathrm{t}$ are inside the 95\% CL region of \Planck+BK18 posterior. All other aspects of the analysis stay the same as in the previous subsection. \subsubsection{Inflationary consistency (IC)} We start with the inflationary consistency case where we assume $r_{0.05}=0.036$ (giving $n_\mathrm{t} = -0.0045$), compatible with the 95\% CL region by the \Planck+BK18 data. The constraints for the tensor parameters are presented in Table \ref{tab:IC036} and the posteriors in Fig.~\ref{fig:r0p036_1}. Again the area covered by the two-dimensional contours for the primary parameters $r_1$ and $r_2$ apparently depends on the choice of scales, but this is due to plotting $r$s at different scales on the same figure. The dependence disappears when projecting on an amplitude ($r_{0.01}$ or $r_{0.05}$) and tilt in the middle and last panels. The first panel indicates that the green case ($k_1$---$k_2$=$0.002$---$0.02$) has a degeneracy between $r_1$ and $r_2$. Table~\ref{tab:IC036} shows that marginally the best recovery of the input parameters is achieved by $k_1$---$k_2$=$0.001$---$0.01$. The measurement precision in this case is $\sigma(r_{0.001})\approx0.009$ and $\sigma(r_{0.01})\approx0.003$, which implies approximately a $10\,\sigma$ detection of our non-zero input $r_{0.01}=0.0343$. \renewcommand{\dblfloatpagefraction}{0.3} \subsubsection{Positive tensor tilt $(n_\mathrm{t}+)$} For the case of a positive tensor tilt $n_\mathrm{t}=0.3$ we set $r_{0.05}=0.07$ in our fiducial model, again compatible with 95\% CL of \Planck+BK18. We present the posterior constraints for the tensor parameters in Table~\ref{tab:NT07} and in Fig.~\ref{fig:fig:r0p07_1}. Again the couple $k_1$---$k_2$=$0.002$---$0.02$ leads to a degeneracy between $r_1$ and $r_2$, but any of the choices would recover the input value of $r_{0.01}$ equally well. The measurement precision is now $\sigma(r_{0.001})\approx0.0058$ and $\sigma(r_{0.01})\approx0.0028$, i.e., slightly better compared to the IC case. As for the IC, also in this case we reach $\sigma(n_\mathrm{t})\approx 0.10$. The measured tensor tilt is clearly positive and differs from zero by more than $2\,\sigma$. \subsubsection{Modified sound speed $(n_\mathrm{t}-)$} We conclude with a negative tensor tilt due to a modified sound speed. We use a value compatible with the current constraints, $c_s=0.02$, and assume $r_{0.05}=0.03$, which gives $n_\mathrm{t}=-0.1875$. We present the constraints for the tensor parameters in Table~\ref{tab:CS03} and in Fig.~\ref{fig:fig:r0p03}. Also in this case, $k_1$---$k_2$=$0.001$---$0.01$ Mpc$^{-1}$ performs marginally better than $0.005$---$0.02$ Mpc$^{-1}$, whereas $0.002$---$0.02$ Mpc$^{-1}$ leads again to a degeneracy for the primary tensor parameters. The forecasted measurement precision is $\sigma(r_{0.001})\approx0.0014$ and $\sigma(r_{0.01})\approx0.003$. The measured tensor tilt is clearly negative and differs from zero at $2\,\sigma$. \renewcommand{\dblfloatpagefraction}{0.7} \subsection{The importance of space-based mission\label{sec:spacebased}} We now study the impact of the low multipoles which are accessible only by observing a sufficiently large fraction of the sky, one of the main advantages of space missions. In order to mimic what could happen with a ground-based instrument, we use the same setup as above but with $\ell=[2,19]$ for the $B$-mode polarization removed. The results for $k_1$---$k_2$=$0.001$---$0.01$ Mpc$^{-1}$ in Fig.~\ref{fig:LBcut} show how the uncertainties on $r_{0.001}$ more than double for any of the three models. We observe this degradation of constraints in $r_1$ also for the other sets of scales. Naturally, $r_{0.01}$ (as well as $r_{0.02}$) stays largely unaffected since $k=0.01\,$Mpc$^{-1}$ corresponds to multipoles larger than 20 (indeed $\ell \sim 70$). Importantly, without the space-based low-multipole data even the 68\% CL posterior regions of our representative three fiducial models overlap, whereas with the low multipoles included the positive tensor tilt is clearly distinguishable from the negative tilt, and the IC case only marginally overlaps with the $n_\mathrm{t}-$ case at 68\% CL. \begin{table} \begin{tabular}{ p{1.1cm} p{2.3cm} p{2.3cm} p{2.3cm} } \hline \hline \noalign{\vskip 3pt} \multicolumn{4}{c}{Inflation consistency (IC)} \\ \noalign{\hrule\vskip 3pt} Pars & $0.001-0.01$&$0.002-0.02$&$0.005-0.02$\\ \noalign{\hrule\vskip 3pt} $r_1$ & $0.035_{-0.010}^{+0.006}$ & $0.040_{-0.007}^{+0.005}$ &$0.034 \pm 0.003$\\ \noalign{\vskip 2pt} $r_2$ & $0.034 \pm 0.003$ & $0.035_{-0.005}^{+0.004}$ & $0.035_{-0.005}^{+0.004}$\\ \noalign{\vskip 1pt} \noalign{\hrule\vskip 3pt} $r_{0.01}$ & $0.034 \pm 0.003$& $0.035\pm 0.003$ &$0.035 \pm 0.003$ \\ $n_\mathrm{t}$ & $-0.03\pm 0.10$ & $-0.02\pm 0.10$ &$-0.01 \pm 0.10$ \\ \hline \hline\\ \noalign{\vskip -15pt} \end{tabular} \caption{\label{tab:IC036}68\% CL results for the tensor parameters when jointly fitting the $\Lambda$CDM parameters and $r_1$ and $r_2$ to the simulated LiteBIRD-like data. In this case, the input model has $r_{0.05}=0.036$ and $n_\mathrm{t}=-r/8$ (i.e., $r_{0.01}=0.0343$ and $n_\mathrm{t} = -0.0045$).} \end{table} \begin{table} \begin{tabular}{ p{1.1cm} p{2.3cm} p{2.3cm} p{2.3cm} } \hline \hline \noalign{\vskip 3pt} \multicolumn{4}{c}{Positive tensor tilt ($n_\mathrm{t}+$) } \\ \noalign{\hrule\vskip 3pt} Pars & $0.001-0.01$&$0.002-0.02$&$0.005-0.02$\\ \noalign{\hrule\vskip 3pt} $r_1$ & $0.021_{-0.006}^{+0.004}$ & $0.025_{-0.005}^{+0.004}$ &$0.033 \pm 0.003$\\ \noalign{\vskip 2pt} $r_2$ & $0.041\pm 0.003$ & $0.051 \pm 0.005$ & $0.052_{-0.006}^{+0.005}$ \\ \noalign{\vskip 1pt} \noalign{\hrule\vskip 3pt} $r_{0.01}$ & $0.041\pm 0.003$& $0.041 \pm 0.003$ &$0.041 \pm 0.003$ \\ $n_\mathrm{t}$ & $0.27\pm 0.10$& $0.28 \pm 0.10$ &$0.30 \pm 0.10$ \\ \hline \hline\\ \noalign{\vskip -15pt} \end{tabular} \caption{\label{tab:NT07} The same as Table~\ref{tab:IC036}, but now simulating the LiteBIRD-like data by using a fiducial model with $r_{0.05}=0.07$ and $n_\mathrm{t}=0.3$ (i.e., $r_{0.01}=0.0408$). } \end{table} \begin{table} \begin{tabular}{ p{1.1cm} p{2.3cm} p{2.3cm} p{2.3cm} } \hline \hline \noalign{\vskip 3pt} \multicolumn{4}{c}{Negative tensor tilt ($n_\mathrm{t}-$)} \\ \noalign{\hrule\vskip 3pt} Pars & $0.001-0.01$&$0.002-0.02$&$0.005-0.02$\\ \noalign{\hrule\vskip 3pt} $r_1$ & $0.059_{-0.016}^{+0.010}$ & $0.050_{-0.009}^{+0.006}$ &$0.043_{-0.004}^{+0.003}$\\ \noalign{\vskip 2pt} $r_2$ & $0.038\pm 0.003$ & $0.035 \pm 0.004$ & $0.035 \pm 0.004$ \\ \noalign{\vskip 1pt} \noalign{\hrule\vskip 3pt} $r_{0.01}$ & $0.038\pm 0.003$& $0.039\pm 0.003$ &$0.039\pm 0.003$ \\ $n_\mathrm{t}$ & $-0.22 \pm 0.10$& $-0.19 \pm 0.10$ &$-0.19 \pm 0.10$ \\ \hline \hline\\ \noalign{\vskip -15pt} \end{tabular} \caption{\label{tab:CS03} The same as Table~\ref{tab:IC036}, but now simulating the LiteBIRD-like data by using a fiducial model with the modified sound speed, adopting currently allowed values $r_{0.05}=0.03$ and $n_\mathrm{t} = -0.1875$ (giving $r_{0.01}=0.0384$). } \end{table} \subsection{Null case \label{sec:nullcase}} In addition to the three cases with $r \ne 0$, we also test $r=0$, representative of the case that inflation or its alternatives generate gravitational waves with an amplitude below the threshold of detection of future $B$-mode polarization experiments. The results are shown in Fig.~\ref{fig:LB0} (and in Fig.~\ref{fig:final} where we compare to the constraints given by the real data). We obtain the following 95\% CL upper bounds: $r_{0.001} < 0.0032$ and $r_{0.01} <0.0048$ (or $r_{0.005}<0.0027$ and $r_{0.02} < 0.01$), whereas our result with the current \Planck+BK18 data was $r_{0.005}<0.030$ and $r_{0.02} < 0.098$. Thus, in the possible null case, LiteBIRD would lead to 10 times tighter constraints on both primary tensor parameters than achieved by \Planck+BK18. As seen in the first panel of Fig.~\ref{fig:final}, this means that the area covered by the 95\% CL region in the ($r_{0.005},\,r_{0.02}$) plane shrinks more than by a factor of 100. Equally impressively, any point outside of the red region in the first panel of Fig.~\ref{fig:final} can be regarded at least as a two-sigma detection zone of a non-zero tensor contribution. As we move to the outer limits of the blue region (the currently allowed 95\% CL region) the detection of a non-zero $r$ by a LiteBIRD-like experiment would be at the $10\,\sigma$ level, as we have seen in the previous subsections. Let us finally note that in the case of a fiducial with zero tensor contribution, the LiteBIRD-like data lead to only a marginally narrower 95\% CL range for $n_\mathrm{t}$ than the current data, and the posterior peaks again at $n_\mathrm{t} \approx 1$, confirming the explanation of Sec.~\ref{sec:realdata} and showing that the effect of the prior peaking at $n_\mathrm{t} \approx 0$ (see the upper panel of Fig.~\ref{fig:ntprior}) is negligible. % \section{Conclusions\label{sec:conclusions}} We have obtained constraints on the amplitude and tilt of the primordial tensor mode by the most recent \Planck\ and BICEP/\Keck\ Array 2018 data, employing a two-scale analysis where the sampling parameters for the tensor power spectrum are (independent) tensor-to-scalar ratios at two different scales. This is a minimal extension of the analysis with $n_\mathrm{t}$ fixed with respect to more ambitious reconstructions of the primordial tensor power spectrum \cite{Hiramatsu:2018nfa,Campeti:2019ylm}. Our 95\% CL constraints $r_{0.005} < 0.030$ and $r_{0.02} < 0.098$ improve by nearly a factor of 2 those obtained from \Planck\ 2018 data in combination with the previous $B$-mode polarization BK15 likelihood in \cite{Planck:2018jri}. The \Planck+BK18 95\% CL constraints on the derived tensor parameters are $r_{0.01} < 0.039$ and $-0.6 < n_{\rm t} < 2.7$. As in \cite{Planck:2018jri}, we also report the results in combination with the upper bound on the stochastic gravitational wave background at much smaller scales, provided by the LIGO\&Virgo 2016 observing season, which excludes most positive values of the primordial tensor tilt: $r_{0.01} < 0.039$ and $-0.8 < n_{\rm t} < 0.5$ at 95\% CL. We have then forecasted how a two-scale analysis performs with future $B$-mode polarization data. As a representative experiment for future polarization data, we have considered conservative specifications for a LiteBIRD-like space-based mission. Given its capability to probe both the reionization and recombination peaks in the $B$-mode power spectrum, we had the possibility to study different choices of the two scales and to show how the results depend on this choice. We have also considered different fiducial values for the primordial tensor power spectrum, including the tensor-to-scalar consistency condition and two cases with $n_\mathrm{t}$ positive and negative, respectively. Whereas distinguishing $n_{\rm t}=-r/8$ from the exact scale invariance is out of reach as expected \cite{Knox:2002pe}, we have shown how with these LiteBIRD-like specifications we could detect at $\sigma(n_{\rm t}) \sim 0.1$, largely independent from any reasonable choice of scales, theoretically motivated departures from $n_\mathrm{t}=-r/8$ consistent with the current bounds. Accessing the low multipoles, virtually doable only by CMB space missions, is essential for reaching these results, as discussed in Sec.~\ref{sec:spacebased} and shown in Fig.~\ref{fig:LBcut}. We have also shown in Fig.~\ref{fig:final} the huge LiteBIRD-like discovery space compared to the current bounds when $n_\mathrm{t}$ is allowed to vary. We conclude reminding that the results presented here are conservative with respect to Ref.~\cite{LiteBIRD:2022cnt} and could be further improved by de-lensing, but show that a space mission, such as LiteBIRD, accessing the low multipoles is the most suitable for characterizing the primordial tensor spectrum with the minimal assumptions. \section*{Acknowledgments} DP and FF acknowledge financial support by ASI Grant 2016-24-H.0 and the agreement n. 2020-9-HH.0 ASI-UniRM2 ``Partecipazione italiana alla fase A della missione LiteBIRD". JV acknowledges funding from The Finnish Cultural Foundation (2020--21) and Ruth och Nils-Erik Stenb\"acks stiftelse (2022). We acknowledge the use of the INAF-OAS institute HPC cluster, and we acknowledge the use of the computing centre of Cineca under the agreement INFN-InDark. Part of the analysis was performed using computational resources provided by CSC --- IT Center for Science, Finland. MH was supported by the World Premier International Research Center Initiative (WPI) of MEXT, and by JSPS KAKENHI Grant Number 22H04945. \bibliography{apssamp}%

Title: Airfall on Comet 67P/Churyumov-Gerasimenko

Abstract: We here study the transfer process of material from one hemisphere to the other (deposition of airfall material) on an active comet nucleus, specifically 67P/Churyumov-Gerasimenko. Our goals are to: 1) quantify the thickness of the airfall debris layers and how it depends on the location of the target area, 2) determine the amount of $\mathrm{H_2O}$ and $\mathrm{CO_2}$ ice that are lost from icy dust assemblages of different sizes during transfer through the coma, and 3) estimate the relative amount of vapor loss in airfall material after deposition in order to understand what locations are expected to be more active than others on the following perihelion approach. We use various numerical simulations, that include orbit dynamics, thermophysics of the nucleus and of individual coma aggregates, coma gas kinetics and hydrodynamics, as well as dust dynamics due to gas drag, to address these questions. We find that the thickness of accumulated airfall material varies substantially with location, and typically is of the order $0.1$-$1\,\mathrm{m}$. The airfall material preserves substantial amounts of water ice even in relatively small (cm-sized) coma aggregates after a rather long ($12\,\mathrm{h}$) residence in the coma. However, $\mathrm{CO_2}$ is lost within a couple of hours even in relatively large (dm-sized) aggregates, and is not expected to be an important component in airfall deposits. We introduce reachability and survivability indices to measure the relative capacity of different regions to simultaneously collect airfall and to preserve its water ice until the next perihelion passage, thereby grading their potential of contributing to comet activity during the next perihelion passage.

https://export.arxiv.org/pdf/2208.01089

\noindent {\bf Title: Airfall on Comet 67P/Churyumov--Gerasimenko} \bigskip\bigskip \begin{trivlist} \item {\bf Author (1):} Bj\"{o}rn J. R. Davidsson \item {\bf Affiliation:} Jet Propulsion Laboratory, California Institute of Technology \item {\bf Address:} M/S 183--401, 4800 Oak Grove Drive, Pasadena, CA 91109, USA \item {\bf Telephone:} (818)--393--8032 \item {\bf E--mail address:} bjorn.davidsson@jpl.nasa.gov \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (2):} Samuel Birch \item {\bf Affiliation:} Cornell University \item {\bf Address:} 426 Space Sciences Building, Cornell University, Ithaca NY, 14850, USA \item {\bf Telephone:} (510)--712--0270 \item {\bf E--mail address:} sb2222@cornell.edu \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (3):} Geoffrey A. Blake \item {\bf Affiliation:} Division of Geological \& Planetary Sciences, California Institute of Technology \item {\bf Address:} MC150--21, Pasadena, CA 91125, USA \item {\bf Telephone:} (626)--395--6296 \item {\bf E--mail address:} gab@gps.caltech.edu \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (4):} Dennis Bodewits \item {\bf Affiliation:} Department of Physics, Auburn University \item {\bf Address:} 201 Allison Laboratory, Auburn, AL 36849, USA \item {\bf Telephone:} (334)--844--4274 \item {\bf E--mail address:} dennis@auburn.edu \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (5):} Jason P. Dworkin \item {\bf Affiliation:} NASA Goddard Space Flight Center \item {\bf Address:} Greenbelt, MD 20771, USA \item {\bf Telephone:} (301)--286--8631 \item {\bf E--mail address:} jason.p.dworkin@nasa.gov \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (6):} Daniel P. Glavin \item {\bf Affiliation:} NASA Goddard Space Flight Center \item {\bf Address:} Greenbelt, MD 20771, USA \item {\bf Telephone:} (301)--614--6361 \item {\bf E--mail address:} daniel.p.glavin@nasa.gov \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (7):} Yoshihiro Furukawa \item {\bf Affiliation:} Department of Earth Science, Tohoku University \item {\bf Address:} 6--3, Aza--aoba, Aramaki, Aoba--ku, Sendai 980--8578, Japan \item {\bf Telephone:} +81-22-795-3453 \item {\bf E--mail address:} furukawa@tohoku.ac.jp \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (8):} Jonathan I. Lunine \item {\bf Affiliation:} Department of Astronomy and Carl Sagan Institute, Cornell University \item {\bf Address:} Cornell University, Ithaca, NY 14850, USA \item {\bf Telephone:} (607)--255--5911 \item {\bf E--mail address:} jlunine@astro.cornell.edu \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (9):} Julie L. Mitchell \item {\bf Affiliation:} NASA Johnson Space Center \item {\bf Address:} 2101 NASA Pkwy, Houston, TX 77058, USA \item {\bf Telephone:} (281)--483--2925 \item {\bf E--mail address:} julie.l.mitchell@nasa.gov \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (10):} Ann N. Nguyen \item {\bf Affiliation:} Jacobs, Houston TX; Astromaterials Research and Exploration Science, NASA Johnson Space Center, Houston TX \item {\bf Address:} 2101 NASA Parkway, Mail Code XI3, Houston TX 77058 \item {\bf Telephone:} 281--483--9446 \item {\bf E--mail address:} lan-anh.n.nguyen@nasa.gov \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (11):} Steve Squyres \item {\bf Affiliation:} Blue Origin, LLC \item {\bf Address:} 21218 76th Ave S, Kent, WA 98032, USA \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (12):} Aki Takigawa \item {\bf Affiliation:} The Hakubi Center for Advanced Research / Division of Earth and Planetary Science \item {\bf Address:} Kyoto University, Kitashirakawa--Oiwakecho, Sakyo, Kyoto 606--8502, Japan \item {\bf Telephone:} +81--75--753--4169 \item {\bf E--mail address:} takigawa@kueps.kyoto-u.ac.jp \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (13):} Jean--Baptiste Vincent \item {\bf Affiliation:} DLR Insitute of Planetary Research \item {\bf Address:} Rutherfordstrasse 2, 12489 Berlin, Germany \item {\bf Telephone:} +49--30--67055--7912 \item {\bf E--mail address:} jean-baptiste.vincent@dlr.de \end{trivlist} \bigskip \begin{trivlist} \item {\bf Author (14):} Kris Zacny \item {\bf Affiliation:} Honeybee Robotics \item {\bf Address:} 398 W Washington Blvd., Pasadena, CA 91103 \item {\bf Telephone:} 646--508--9807 \item {\bf E--mail address:} kazacny@honeybeerobotics.com \end{trivlist} \bigskip \begin{trivlist} \item {\bf Number of manuscript pages: }65 \item {\bf Number of figures: }11 \item {\bf Number of tables: }4 \end{trivlist} \newpage \noindent {\bf Running head: }Airfall on Comet 67P/Churyumov--Gerasimenko \begin{trivlist} \item {\bf Address for editorial correspondence:} \item Bj\"{o}rn Davidsson \item Jet Propulsion Laboratory \item M/S 183--401 \item 4800 Oak Grove Drive \item Pasadena, CA 91109 \item USA \end{trivlist} \noindent Email: bjorn.davidsson@jpl.nasa.gov\\ \noindent (818)--393--8032 (phone) \newpage \newpage \section{Introduction} \label{sec_intro} The initial characterization of Comet 67P/Churyumov--Gerasimenko (hereafter 67P) by the OSIRIS cameras \shortcite{kelleretal07} on the Rosetta orbiting spacecraft \shortcite{glassmeieretal07} revealed widespread smooth terrains on the northern hemisphere of the nucleus, primarily in Ma'at on the small lobe, in Ash, Anubis, and Imhotep on the large lobe, and in Hapi that constitutes the northern neck region between the two lobes (\shortciteNP{sierksetal15}; \shortciteNP{thomasetal15a}). The smooth material in the Ma'at and Ash regions formed a relatively thin coverage over partially--revealed consolidated landforms. This led \shortciteN{thomasetal15a} to propose that the smooth material, at least in those regions, constituted a rather recent veneer of dust that had rained down from the coma. They named these deposits and the process forming them ``airfall''. The presence of smooth deposits in isolated topographic lows, e.g., in Khepry \shortcite{elmaarryetal15}, or in large gravitational lows found primarily in the Imhotep and Hapi regions, suggest that airfall may not be the only mechanism responsible for the formation of smooth terrain. \shortciteN{augeretal15a} propose that the vast smooth plain in Imhotep formed through mass wasting the surrounding steep cliffs and by transport downhill toward the lowland. Mass wasting from cliffs revealed by taluses, gravitational accumulation deposits, and diamictons that extends into Hapi \shortcite{pajolaetal19} indicate that such processes are partially responsible for the smooth material in that region as well. The presence of deposit--free regions that sharply contrast with surrounding smooth terrain, best illustrated by the Aten depression on the large lobe or the Anuket region in the neck area \shortcite{elmaarryetal15}, suggests that the removal rate of airfall material through self--cleaning is locally high and that the net accumulation rate may be slow (e.g., in case the Aten depression formed recently in a massive outburst event).\\ \noindent At the time of the Rosetta orbit insertion around 67P in August 2014 the northern hemisphere was illuminated while the southern hemisphere experienced polar night because of the orientation of the spin axis \shortcite{sierksetal15}. Most activity, as revealed by prominent dust jets, came from Hapi (\shortciteNP{sierksetal15}; \shortciteNP{laraetal15}), that also was the brightest and bluest unit in terms of spectral slope \shortcite{fornasieretal15}, showing that the neck region had accumulated particularly ice--rich material that was strongly active. \shortciteN{thomasetal15a} noted that the airfall deposits on north--facing regions are more extensive than on south--facing regions and proposed that the major transport route went from Hapi to other suitably oriented regions on the northern hemisphere. During approach to the inbound equinox in May 2015, increasingly large portions of the southern hemisphere were illuminated and eventually received peak solar illumination at the perihelion passage in August 2015 while the northern hemisphere experienced polar night. \shortciteN{kelleretal15} demonstrated that $\sim 80\%$ of the solar radiation is absorbed by the northern hemisphere at a low rate over an extended amount of time, while the southern hemisphere receives the remaining $\sim 20\%$ during a much shorter time interval. Their thermophysical modeling showed that the strongly non--linear response of sublimation to solar flux levels led to a $\sim 3$ times higher erosion in the south compared to the north. They therefore suggested that the main transport route of airfall material went from the southern hemisphere to the northern hemisphere. The views of \shortciteN{thomasetal15a} and \shortciteN{kelleretal15} are not in contradiction, but suggest that activity in the south around perihelion leads to airfall in the cold and inactive north. Some of this material is then redistributed from Hapi to other northern regions from the time of the outbound equinox to the shutdown of activity at larger heliocentric distance, with a pause during the aphelion passage followed by resumed redistribution when activity reappears during inbound motion.\\ \noindent OSIRIS observations of the southern hemisphere showed a strong hemispherical dichotomy regarding smooth terrains -- they were essentially missing in the south (\shortciteNP{elmaarryetal16}; \shortciteNP{leeetal16}; \shortciteNP{birchetal17}), showing that any deposited or otherwise produced smooth material is rapidly cleaned off. Observations by the mass spectrometer ROSINA (\shortciteNP{hassigetal15}; \shortciteNP{fougereetal16}) and the near--infrared spectrometer VIRTIS \shortcite{finketal16} on Rosetta show that the two hemispheres display a strong chemical dichotomy as well. The northern hemisphere is predominantly outgassing $\mathrm{H_2O}$ and comparably small amounts of $\mathrm{CO_2}$, while the southern hemisphere is a source of both water and carbon dioxide. \shortciteN{kelleretal17} interpreted the chemical dichotomy as a result of airfall. In their view, the $\mathrm{CO_2}$ is either missing in the solid material being ejected into the coma from the south near perihelion (i.e., the $\mathrm{CO_2}$ sublimation front is located at some depth) or is lost on the way during transport toward the north. Because of the substantially higher sublimation temperature of $\mathrm{H_2O}$ compared to $\mathrm{CO_2}$ the water loss is significantly smaller. The airfall that is responsible for northern activity in other parts of the orbit is therefore rich in water ice but poor in supervolatiles like $\mathrm{CO_2}$, according to \shortciteN{kelleretal17}.\\ \noindent High--resolution imaging of smooth terrain in Ash by OSIRIS during low ($\sim 10\,\mathrm{km}$) Rosetta orbits \shortcite{thomasetal15b}, by the Philae/ROLIS camera during descent toward Agilkia in Ma'at (\shortciteNP{mottolaetal15}; \shortciteNP{pajolaetal16}), and by OSIRIS during the landing of Rosetta at Sais in Ma'at \shortcite{pajolaetal17b} revealed that the material in those locations consisted primarily of pebbles, cobbles, and boulders in the cm--m size range. In the following, such units are collectively referred to as ``chunks''. The observations of similarly sized chunks in the coma (e.g., \shortciteNP{rotundietal15}; \shortciteNP{davidssonetal15b}), of which a substantial fraction display acceleration toward the nucleus \shortcite{agarwaletal16}, show that the airfall concept is viable.\\ \noindent Smooth terrains are not featureless. Structures resembling aeolean ripples were observed in Hapi, dunes that in some cases contained pits (potentially formed by sublimation) were seen in Serqet and Maftet, and some boulders in Hapi and Maftet appeared to have wind tails \shortcite{thomasetal15b}, that are also seen in Ma'at \shortcite{mottolaetal15}. Whether such features primarily form during airfall deposition or are the result of lateral transport mechanisms is unclear. They do indicate that deposition is not a trivial phenomenon and that significant local transport may take place after deposition. The features seen in smooth terrain are also not static. The first reported observations of large--scale morphological changes concerned roundish scarps that formed and expanded at $\sim 6\,\mathrm{m\,day^{-1}}$ in Imhotep \shortcite{groussinetal15b}. Later, similar scarp retreats were also seen in Hapi, Anubis, and in Seth (\shortciteNP{elmaarryetal17}; \shortciteNP{huetal17}). The scarps frequently displayed strong brightening and spectral slope changes suggestive of the presence of abundant sub--surface water ice that mixed up to the surface during the retreat process. A dramatic and puzzling example of these morphological changes where the removal of the aeolean ripples in Hapi in April--July 2015 by retreating scarps, that was followed by ripple reformation in December 2015 \shortcite{elmaarryetal17}.\\ \noindent OSIRIS observations allowed for a documentation of inbound removal and outbound redeposition of airfall material, albeit restricted to specific locations and time instances because of resolution, viewing geometry, and imaging cadence restrictions. A comprehensive summary is found in \shortciteN{huetal17}. The presence of ``honeycomb'' features in Ash, Babi, Serqet, Seth, and particularly in Ma'at in late March 2015 (also see \shortciteNP{shietal16}) caused by the removal of overlying granular material is indicative of self--cleaning. The time--scale of their formation is uncertain because of observational biases but the earliest known honeycomb sighting occurred two months earlier. Other signs of airfall deposit removal include exposure of previously hidden outcrops and boulders, formation of shallow depressions, and smoothing of previously pitted deposits \shortcite{huetal17}. The OSIRIS images were used to create three--dimensional maplets of selected terrains before and after the removal of deposits. Although it is not possible to estimate the thickness of the expelled layers by simply subtracting two maplets, insight about the erosion can still be obtained by measuring changes in the degree of roughness inferred from such maplets. \shortciteN{huetal17} find, for regions where changes are more easily detected, that at least $0.5\,\mathrm{m}$ has been removed (similar to the pixel resolution) and an estimated average removal of $1.3\,\mathrm{m}$ for one specific honeycomb. Post--perihelion observations showed that the honeycombs had disappeared and were replaced by smooth terrain. \shortciteN{huetal17} do not quantify the thickness of the redeposited layer but a reasonable assumption is of order $0.1$--$1\,\mathrm{m}$, so that the net orbital change is small. It is not self--evident that the northern airfall deposits currently experience net growth, or if there is currently net erosion of deposits built up at an earlier time when conditions were different than today.\\ \noindent From the description above a basic qualitative understanding has emerged regarding the origin and behavior of smooth terrain on 67P; this new insight emphasizes the global redistribution of material on the nucleus enabled by its spin axis obliquity. This process creates a non--primordial chemical surface heterogeneity. If these processes are active on other comet nuclei as well it makes the interpretation of groundbased production rate patterns more difficult than previously envisioned. However, it is also evident that a detailed understanding of these processes requires substantial modeling efforts that can fill the gaps that are not immediately provided by available observations. Therefore, the first goal of this paper is to present a quantitative study of the south--to--north transfer process proposed by \shortciteN{kelleretal15} and elaborated on by \shortciteN{kelleretal17}, i.e., to estimate the thickness of the deposited layer built during the perihelion passage and how it varies across the northern hemisphere. Specifically, we focus on 31 target areas distributed within the Ma'at region on the small lobe and within the Ash and Imhotep regions on the large lobe. These sites were selected and investigated in the context of a Phase~A study for the New~Frontiers~4 candidate mission CAESAR (Comet Astrobiology Exploration SAmple Return). The sites were selected to cover a range of regions of smooth terrain in the northern hemisphere of 67P that were deemed to be accessible to the CAESAR touch and go sampling. The location of the 31 target areas are shown in Fig.~\ref{fig0}.\\ \noindent Our second goal is to quantify the loss of volatiles expected to take place during the transfer process, thereby enabling a comparison between the ice abundance of fresh airfall deposits with that of the sources of this material. Our third goal is to quantify the relative capacity of different airfall deposition sites to retain their water ice content after deposition, in order to better understand their capability to uphold comet activity as the comet approaches the Sun after the aphelion passage. In the context of the CAESAR mission this work contributed to locating high science value sampling sites in the smooth terrain on 67P for future Earth return.\\ \noindent Our work relies on a pipeline of different models and methods that are described in Sec.~\ref{sec_method}. In brief, we first find Keplerian orbits that connect specific source regions and target areas that are unobstructed by the rotating nucleus. Those orbits are characterized by a specific ejection velocity $v_{\rm d}$ that needs to be provided by gas drag, and are realized for a specific dust diameter $a_{\rm crit}$. We calculate $a_{\rm crit}$ by first determining the local surface temperature and outgassing rate, use that information to calculate the coma number density, translational temperature, and gas expansion velocity versus height, which in turn are used to evaluate the gas drag force needed for dust dynamics calculations. We also present our model for calculating the loss of $\mathrm{H_2O}$ and $\mathrm{CO_2}$ from chunks in the coma, as well as the longterm nucleus thermophysics modeling. Our results are presented in Sec.~\ref{sec_results}, we discuss these results and compare with previously published airfall investigations in Sec.~\ref{sec_discussion}, and summarize our conclusions in Sec.~\ref{sec_conclusions}. \section{Methods} \label{sec_method} Section~\ref{sec_method_kepler} describes the method for determining whether an unobstructed Keplerian orbit exists that connects a certain southern source region with a specific northern target area, taking the irregular shape of the rotating nucleus into account. The purpose of that calculation is to identify the dust ejection velocity $v_{\rm d}$ that is needed in order to realize a particular orbit. The reliability of this analytical method based on the point--mass assumption is tested using numerical integration of orbits around irregular bodies with complex gravity fields described in Sec.~\ref{sec_method_numorb}.\\ \noindent The thermophysical model described in Sec.~\ref{sec_method_nucthermo} is needed in order to calculate the nucleus outgassing rate as function of surface location and time. The outgassing rate is used to calculate how the gas density, drift (expansion) speed, and temperature vary with height, using procedures described in Sec.~\ref{sec_method_coma}. Those quantities are, in turn, necessary to determine the specific chunk size that will be accelerated to the particular $v_{\rm d}$ needed to reach a given target area, as summarized in Sec.~\ref{sec_method_eject}. With all of this information in hand, the amount of airfall material can be calculated, as described in Sec.~\ref{sec_method_airfall}.\\ \noindent Section~\ref{sec_method_grainthermo} describes the model used for modeling the thermophysical evolution of chunks in the coma, and the longterm thermophysical evolution of target sites are described in Sec.~\ref{sec_method_volatileloss}. The mathematical symbols used throughout this work are summarized in Tables~\ref{tab0}--\ref{tab0b}. \begin{table} \begin{center} {\bf Model parameters \#1} \end{center} \begin{center} \footnotesize \begin{tabular}{||l|l|l||} \hline \hline Symbol & Description & Unit\\ \hline $A$ & Bond albedo & --\\ $\mathcal{A}$ & Chunk cross section & $\mathrm{m^2}$\\ $\mathcal{A}_{\rm crit}$ & Cross section of maximum liftable chunk & $\mathrm{m^2}$\\ $\mathcal{C}$ & Eq.~(\ref{eq:25}) constant & $\mathrm{K\,(1-\gamma)^{-1}}$\\ $C_{\rm D}$ & Drag coefficient & --\\ $D$ & Dust chunk diameter & $\mathrm{m}$\\ $D_{\rm min},\,D_{\rm max},\,D_{\rm c}$ & Minimum, maximum, and intermediate dust chunk diameter & $\mathrm{m}$\\ $\mathcal{D}$ & Molecular number flux & $\mathrm{m^{-2}\,s^{-1}}$\\ $E_{\rm t}$ & Molecular kinetic energy & $\mathrm{J}$ \\ $E_{\rm r}$ & Molecular rotational energy & $\mathrm{J}$\\ $F_{\rm drag}$ & Drag force & $\mathrm{N}$\\ $\mathbf{F}_{\rm g}$ & Gravitational force vector & $\mathrm{N}$\\ $F_{\imath\jmath}$ & Fraction of latent heat consumed during segregation & --\\ $\mathcal{F}_{jk}$ & View factor & --\\ $\mathcal{F}$ & Fraction of dust mass at $D>D_{\rm c}$ & --\\ $F,\,G$ & Eq.~(\ref{eq:18}) parameters & --\\ $G_{\rm N}$ & Newtonian gravitational constant & $\mathrm{N\,m^2\,kg^{-2}}$\\ $H_{\imath}$ & Energy release during crystallization & $\mathrm{J\,kg}$\\ $K_{\rm t}$ & Knudsen layer thickness & $\mathrm{m}$\\ $\mathcal{L}$ & Latent heat of sublimation & $\mathrm{J\,kg^{-1}}$\\ $L'$ & Normalized relative ice loss of target area & --\\ $M,\,M_1,\,M_2$ & Mach number & --\\ $M_{\rm n}$ & Comet nucleus mass & $\mathrm{kg}$\\ $M_{\rm rot}$ & Rotation matrix & --\\ $P$ & Nucleus or chunk rotation period & $\mathrm{h}$\\ $\mathcal{P}$ & Comet orbital period & $\mathrm{yr}$\\ $Q_{\rm d}$ & Dust production rate & $\mathrm{kg\,m^{-2}\,s^{-1}}$\\ $Q_j,\, Q_{\rm s}$ & Water sublimation rate & $\mathrm{kg\,m^{-2}\,s^{-1}}$\\ $R_{\rm n}$ & Comet nucleus curvature radius & $\mathrm{m}$\\ $R_*$ & Radiogenic energy production rate & $\mathrm{J\,m^{-3}\,s^{-1}}$\\ $\mathcal{R}$ & Reachability index & --\\ $S_1$ & Speed ratio & --\\ $S_{\odot}$ & Solar constant & $\mathrm{J\,m^{-2},s^{-1}}$\\ $S'$ & Normalized relative ice retainment capacity of target area & --\\ $\mathcal{S}$ & Survivability index & --\\ $T,\,T_j(x,t),\,T_k(x,t),\,T_{\rm s},\,T_0,\,T_1,\,T_{\infty}$ & Temperature & $\mathrm{K}$\\ $T_{\rm fr}$ & Freeze--out temperature & $\mathrm{K}$\\ $T_{\rm r}$ & Rotational temperature & $\mathrm{K}$\\ $T_{\rm surf}$ & Surface temperature & $\mathrm{K}$\\ $U$ & Gravitational potential & $\mathrm{J\,kg^{-1}}$\\ $W,\,W_1,\,W_2,\,W_{\infty}$ & Gas drift speed & $\mathrm{m\,s^{-1}}$\\ $\mathcal{W}$ & Speed ratio & --\\ $Z$ & Hertz--Knudsen formula & $\mathrm{kg\,m^{-2}\,s^{-1}}$\\ \hline \hline \end{tabular} \caption{Functions and parameters with descriptions and units. Upper case Latin.} \label{tab0} \end{center} \end{table} \begin{table} \begin{center} {\bf Model parameters \#2} \end{center} \begin{center} \footnotesize \begin{tabular}{||l|l|l||} \hline \hline Symbol & Description & Unit\\ \hline $a,\,b,\,c$ & Eq.~(\ref{eq:05}) coefficients & $\mathrm{m^3\,s^{-1}}$\\ $a_1,\,a_2,\,a_3$ & Eq.~(\ref{eq:11}) coefficients & --\\ $a_{\rm crit}$ & Radius of maximum liftable chunk & $\mathrm{m}$\\ $a_{\rm o}$ & Orbital semi--major axis & $\mathrm{m}$\\ $c_{\rm s}$ & Specific heat capacity & $\mathrm{J\,kg^{-1}\,K^{-1}}$\\ $d$ & Distance between chunk and target location & $\mathrm{m}$\\ $f$ & Eq.~(\ref{eq:05}) coefficient & --\\ $f_i$ & Ice volume fraction & --\\ $g_{\imath}$ & Heat capacity of gaseous species $\imath$ & $\mathrm{J\,kg^{-1}}$\\ $h$ & Height above the comet nucleus surface & $\mathrm{m}$\\ $\mathbf{h}=\{h_{\rm x},\,h_{\rm y},\,h_{\rm z}\}$ & Chunk orbit normal vector & $\mathrm{m^2\,s^{-1}}$\\ $\imath$ & Chemical species identifier (host) & --\\ $\jmath$ & Chemical species identifier (guest) & --\\ $j,\,k$ & Facet number & --\\ $k_{\rm B}$ & Boltzmann constant & $\mathrm{J\,K^{-1}}$\\ $l$ & Spherical chunk latitudinal coordinate & $\mathrm{rad}$\\ $m$ & Water molecule mass & $\mathrm{kg}$\\ $m_{\imath}$ & Molecule mass of species $\imath$ & $\mathrm{kg}$\\ $m_{\rm d}$ & Mass of a dust chunk & $\mathrm{kg}$\\ $n,\,n_{\rm s},\,n_0,\,n_1,\,n_2$ & Gas number density & $\mathrm{m^{-3}}$\\ $n_{\rm sp}$ & Number of chemical species & --\\ $\hat{n}_{\rm s}$ & Surface normal unit vector & --\\ $p,\, q$ & Eq.~(\ref{eq:05}) coefficients & --\\ $p_{\imath}$ & Partial pressure of species $\imath$ & $\mathrm{Pa}$\\ $q_{\imath}$ & Volume mass production rate of species $\imath$ & $\mathrm{kg\,m^{-3}\,s^{-1}}$\\ $q'_{\imath}$ & Volume mass segregation or crystallization rate & $\mathrm{kg\,m^{-3}\,s^{-1}}$\\ $p_{\rm sat}$ & Water saturation pressure & $\mathrm{Pa}$\\ $p_{\rm v}$ & Vapor pressure & $\mathrm{Pa}$\\ $q_{\rm s}$ & Dust chunk size distribution power--law index & --\\ $r$ & Spherical chunk radial coordinate & $\mathrm{m}$\\ $\mathbf{r}_{\rm d}$ & Chunk position vector & $\mathrm{m}$\\ $r_{\rm g}$ & Constituent grain radius & $\mathrm{m}$\\ $r_{\rm h}$ & Heliocentric distance & $\mathrm{AU}$\\ $r_{jk}$ & Distance between facets & $\mathrm{m}$\\ $r_{\rm o}$ & Orbital distance between chunk and nucleus center & $\mathrm{m}$\\ $\mathbf{r}_{\rm s}$ & Source region position vector & $\mathrm{m}$\\ $\mathbf{r}_{\rm t}=\{r_{\rm tx},\,r_{\rm ty},\,r_{\rm tz}\}$ & Target area inertial position vector & $\mathrm{m}$\\ & with $\mathbf{r}_{\rm t}'=\mathbf{r}_{\rm t}(t=0)$ & \\ $s_k$ & Facet area & $\mathrm{m^2}$\\ $t$ & Time & $\mathrm{s}$\\ $t_{\rm pl}$ & Target area crossing time with chunk orbital plane & $\mathrm{s}$\\ $u_{\rm d}$ & Chunk speed & $\mathrm{m\,s^{-1}}$\\ $v_{\rm d}$ & Post--acceleration speed along surface normal & $\mathrm{m\,s^{-1}}$\\ $v_{\rm esc}$ & Escape velocity & $\mathrm{m\,s^{-1}}$\\ $v_{\rm o}$ & Orbital speed & $\mathrm{m\,s^{-1}}$\\ $v_{j\odot},\,v_{k\odot}$ & Sunlit fraction of facet & --\\ $\mathbf{v}_{\rm s}$ & Dust chunk ejection velocity vector & $\mathrm{m\,s^{-1}}$\\ $x,\,x_j$ & Depth below the comet surface & $\mathrm{m}$\\ $\{\hat{x},\,\hat{y},\,\hat{z}\}$ & Inertial system coordinate system unit vectors & --\\ & ($\hat{z}$ parallel to the nucleus rotation axis) & \\ \hline \hline \end{tabular} \caption{Functions and parameters with descriptions and units. Lower case Latin.} \label{tab0a} \end{center} \end{table} \begin{table} \begin{center} {\bf Model parameters \#3} \end{center} \begin{center} \footnotesize \begin{tabular}{||l|l|l||} \hline \hline Symbol & Description & Unit\\ \hline $\alpha$ & Right ascension & $^{\circ}$\\ $\alpha_{\rm bf}$ & Molecular surface backflux fraction & --\\ $\alpha_{\rm s}$ & Sublimation coefficient & --\\ $\beta_j,\,\beta_k$ & Angle between facet normal and direction to other facet & $\mathrm{rad}$\\ $\beta^-$ & Molecular backflux ratio & --\\ $\gamma$ & Gas heat capacity ratio & --\\ $\delta$ & Declination & $^{\circ}$\\ $\varepsilon$ & Emissivity & --\\ $\zeta$ & Number of rotational degrees of freedom & --\\ $\kappa$ & Heat conductivity & $\mathrm{W\,m^{-1}\,K^{-1}}$\\ $\lambda,\,\lambda_0$ & Molecular mean free path & $\mathrm{m}$\\ $\mu_j,\,\mu_k$ & Cosine of incidence angle & --\\ $\mu_{\rm o},\,\mu_{\rm}^*$ & Reduced mass & $\mathrm{kg}$\\ $\rho$ & Density & $\mathrm{kg\,m^{-3}}$\\ $\sigma$ & Stefan--Boltzmann constant & $\mathrm{W\,m^{-2}\,K^{-4}}$\\ $\sigma_{\rm c}$ & Molecular collisional cross section & $\mathrm{m^2}$\\ $\tau_{\imath}$ & Phase change rate & $\mathrm{kg\,m^{-3}\,s^{-1}}$\\ $\Phi_{\imath}$ & Radial gas diffusion rate of species $\imath$ & $\mathrm{kg\,m^{-2}\,s^{-1}}$\\ $\Psi_{\imath}$ & Latitudinal gas diffusion rate of species $\imath$ & $\mathrm{kg\,m^{-2}\,s^{-1}}$\\ $\psi$ & Porosity & --\\ $\omega$ & Angular velocity of nucleus rotation & $\mathrm{rad\,s^{-1}}$\\ \hline \hline \end{tabular} \caption{Functions and parameters with descriptions and units. Greek.} \label{tab0b} \end{center} \end{table} \subsection{Keplerian orbits} \label{sec_method_kepler} Let $\mathbf{r}_{\rm s}$ be the position vector of a source region on the southern nucleus hemisphere, in an inertial frame with its $\hat{z}$ axis aligned with the nucleus positive spin pole, at the time of dust ejection $t=0$. Let $\mathbf{v}_{\rm s}$ be the dust ejection velocity vector in the same frame with one component due to gas drag acceleration along the local surface normal $\hat{n}_{\rm s}$ and another due to nucleus rotation, \begin{equation} \label{eq:01} \mathbf{v}_{\rm s}=v_{\rm d}\hat{n}_{\rm s}+\hat{z}\times\mathbf{r}_{\rm s}\omega \end{equation} where the angular velocity is $\omega=2\pi/P$ and $P$ is the nucleus rotational period. Then the normal to the dust chunk orbital plane is $\mathbf{h}=\mathbf{r}_{\rm s}\times\mathbf{v}_{\rm s}=\{h_{\rm x},\,h_{\rm y},\,h_{\rm z}\}$.\\ \noindent Let $\mathbf{r}_{\rm t}'$ be the position vector of a target area on the northern nucleus hemisphere in the inertial frame at the dust ejection time $t=0$ (equivalently, the target area position vector in the body--fixed frame). At a later time the target area has position $\mathbf{r}_{\rm t}(t)=M_{\rm rot}(t)\mathbf{r}_{\rm t}'=\{r_{\rm tx},\,r_{\rm ty},\,r_{\rm tz}\}$ in the inertial frame because of nucleus rotation, where \begin{equation} \label{eq:02} M_{\rm rot}(t)=\left(\begin{array}{l} \displaystyle \cos\omega t\hspace{0.2cm} -\sin\omega t\hspace{0.5cm} 0\\ \displaystyle \sin\omega t\hspace{0.65cm} \cos\omega t\hspace{0.5cm} 0\\ \displaystyle 0\hspace{1.5cm} 0\hspace{1.35cm} 1\\ \end{array}\right). \end{equation} The criterion for the target area being located within the dust orbital plane is then \begin{equation} \label{eq:03} \mathbf{h}\cdot\mathbf{r}_{\rm t}(t_{\rm pl})=0, \end{equation} remembering that the orbital plane necessarily must pass through the center of mass of the nucleus that hosts the origin of the coordinate system. The crossing of the target area with the dust orbital plane will repeat twice during a nucleus rotation at instances of time $t_{\rm pl}$ obtained from Eq.~(\ref{eq:03}) and associated equations, \begin{equation} \label{eq:04} t_{\rm pl}=\frac{1}{\omega}\sin^{-1}\left(\sqrt{1-f^2}\right) \end{equation} where \begin{equation} \label{eq:05} \left\{\begin{array}{l} \displaystyle f=-\frac{p}{2}\pm \sqrt{\frac{p^2}{4}-q}\\ \\ \displaystyle p=\frac{2ac}{a^2+b^2}\\ \\ \displaystyle q=\frac{c^2-b^2}{a^2+b^2}\\ \\ \displaystyle a=h_{\rm x}r_{\rm tx}+h_{\rm y}r_{\rm ty}\\ \displaystyle b=-h_{\rm x}r_{\rm ty}+h_{\rm y}r_{\rm tx}\\ \displaystyle c=r_{\rm tz}h_{\rm z} \end{array}\right.. \end{equation} Solutions exist if $f$ is real and $f\in [-1,\,1]$ (i.e., the target area does cross the orbital plane).\\ \noindent The $\{\mathbf{r}_{\rm s},\,\mathbf{v}_{\rm s}\}$ pair can be used to calculate orbital elements for the dust trajectory in ecliptic space, following standard methods (e.g., \shortciteNP{boulet91}). These orbital elements can be used to calculate the location $\mathbf{r}_{\rm d}(t_{\rm pl})$ of the dust chunk at $t_{\rm pl}$. The distance $d=|\mathbf{r}_{\rm d}(t_{\rm pl})-\mathbf{r}_{\rm t}(t_{\rm pl})|$ is a function of $v_{\rm d}$ only, for any given combination of source and target location. We calculate $d$ as function of $v_{\rm d}$, starting at $v_{\rm d}=0$ and incrementing in steps of $0.001\,\mathrm{m\,s^{-1}}$. Our condition for the existence of a Keplerian trajectory that connects the source and target regions is $\min(d)\leq 25\,\mathrm{m}$. It means that the dust trajectory crosses the circle swept up by the target area during nucleus rotation (co--location in space) and that the dust chunk and target area arrives to this interception point simultaneously (co--location in time).\\ \noindent Co--location in space and time is a necessary but not sufficient criterion for airfall onto the target area. It is also required that the dust chunk does not intercept another part of the nucleus on its way to the target area. Therefore, all facets on the shape model that cross the dust trajectory orbital plane are located, their crossing times are determined and the dust chunk position at that time is calculated. If a co--location in space and time takes place prior to the dust chunk arrival time to the target area, the trajectory is discarded.\\ \noindent The trajectory search was made by considering the SHAP5 version 1.5 shape model \shortcite{jordaetal16} of comet 67P degraded to $5\cdot 10^4$ facets. Out of the $\sim 2.5\cdot 10^4$ facets on the southern hemisphere treated as potential source regions, there were $151$ facets that had the correct location and orientation to enable successful trajectories to the 31 northern hemisphere sites. \subsection{Realistic gravity and numerical orbit integration} \label{sec_method_numorb} The method in Sec.~\ref{sec_method_kepler} treats the nucleus as a point mass. That allows us to consider all $\sim 400$ million combinations of source regions ($\sim 2.5\cdot 10^4$), $v_{\rm d}$ values ($\sim 500$), and target areas ($\sim 30$) at a relatively low computational cost because the approach is essentially analytical. However, the nucleus is irregular and the gravity field deviates from that of a point source. We evaluate the error introduced by our point--mass assumption by comparing the Keplerian orbits with trajectories obtained by numerical integration of the equation of motion in a realistic gravity field.\\ \noindent The nucleus gravitational potential $U$ at the center of each nucleus facet, the gravitational force vector $\mathbf{F}_{\rm g}(x,\,y,\,z)=\nabla U$ at any location exterior to the body surface, and the Laplacian $\nabla^2U$ (taking the value 0 outside the surface of the nucleus and $-4\pi$ within it) were calculated using the method of \shortciteN{wernerscheeres97}. This method applies to bodies with an arbitrarily complex shape described by a surface represented by polyhedrons under the assumption of constant density in the interior. Validation of the implementation was made by verifying that $\nabla U$ conformed with the Newtonian solution for polyhedrons forming a sphere, and that $U$ calculated for the highly irregular shape of comet 67P reproduces the potential map of \shortciteN{kelleretal17} in their Fig.~4. In order to represent the shape of 67P we rely on the SHAP5 version 1.5 shape model \shortcite{jordaetal16} degraded to $5\cdot 10^3$ facets.\\ \noindent In order to solve the equation of motion numerically an $8^{\rm th}$ order Gauss--Jackson integrator was implemented (\shortciteNP{fox84}; \shortciteNP{berryhealy04}) using a $4^{\rm th}$ order Runge--Kutta method \shortcite{burdenfairs93} as a startup formula. In order to validate the implementation the orbit of Comet C/1995~O1 (Hale--Bopp) was integrated from January 1993 through 2010 including the perturbations of the eight planets. This included a $0.77\,\mathrm{AU}$ encounter with Jupiter on April 5, 1996 that drastically modified the orbit, e.g., changed the orbital period from $\mathcal{P}=4221\,\mathrm{yr}$ to $2372\,\mathrm{yr}$. The solution was compared to that obtained with the RMVS3 version of the integrator \texttt{SWIFT} \shortcite{levisonduncan94} and the final positions of the comet according to the two codes differed by merely $70\,\mathrm{m}$. \subsection{Nucleus thermophysics} \label{sec_method_nucthermo} \noindent The search for uninterrupted dust trajectories described in Sec.~\ref{sec_method_kepler} identified 151 unique source facets on the southern hemisphere responsible for airfall at the 31 target regions. The purpose of the thermophysical nucleus modeling is to calculate the local water production rates for those source regions as a function of nucleus rotational phase during a 315 day period from the inbound equinox on May 11, 2015, throughout the perihelion passage up to the outbound equinox on March 21, 2016. At this time the southern hemisphere was illuminated and the airfall in the north would have peaked. We assume that all regions have equal potential for activity when subjected to identical illumination sequences. This is consistent with the surface $\mathrm{H_2O}$ activity distribution of \shortciteN{fougereetal16b} that shows little variability within the southern hemisphere.\\ \noindent All facets on the shape model that have an unobstructed view of at least one source facet were identified (here referred to as ``surrounding terrain''). The $\sim 3\cdot 10^4$ facets in surrounding terrain have the capacity of shadowing the source facets by temporarily switching off the direct solar flux, and radiation that is scattered in the visual or emitted in the infrared from these facets will illuminate the sources (such nucleus self heating elevates illumination levels at all times and are particularly important when the sources are in shadow or experiencing night--time). We used the model presented by \shortciteN{davidssonandrickman14} in order to calculate the illumination conditions, including shadowing and self heating. For a given source facet $j$ we evaluate whether the Sun is fully visible ($v_{j\odot}=1$) or if the facet is partially shadowed (rounded to $v_{j\odot}=1/3$ or $v_{j\odot}=2/3$) or fully shadowed by topography ($v_{j\odot}=0$) in $10^{\circ}$ increments of nucleus rotational phase throughout the time period, applying the spin axis in equatorial right ascension and declination $\{\alpha,\,\delta\}=\{69.54^{\circ},\,64.11^{\circ}\}$ of 67P \shortcite{preuskeretal15} and taking the time--dependent nucleus rotation period into account (e.g., \shortciteNP{kelleretal15b}). For simplicity we assume that the combined scattering and emission of a surrounding terrain facet $k$ equals the local incident direct illumination (i.e., there is no loss or gain because of heat conduction from or to the surface). The fraction of the radiation emanating from $k$ that reaches a source facet $j$ is given by the view factor (e.g., \shortciteNP{ozisik85}) \begin{equation} \label{eq:08} \mathcal{F}_{jk}=\frac{s_k\cos\beta_j\cos\beta_k}{\pi r_{jk}^2}. \end{equation} Here, $s_k$ is the surface area of facet $k$, the angles between the surface normals of $j$ and $k$ and their connection line are $\beta_j$ and $\beta_k$, respectively, and $r_{jk}$ is the distance between the two facets. For each source facet $j$ we solve the energy conservation equation \begin{equation} \label{eq:06} \rho c_{\rm s}\frac{\partial T_j(x,t)}{\partial t}=\frac{\partial}{\partial x_j}\left(\kappa\frac{\partial T_j(x,t)}{\partial x_j}\right) \end{equation} for the temperature $T_j$ by using the Finite Element Method. The surface boundary condition of Eq.~(\ref{eq:06}) is given by \begin{equation} \label{eq:07} \begin{array}{c} \displaystyle \frac{S_{\odot}v_{j\odot}(1-A)\mu_j(t)}{r_{\rm h}^2}=\varepsilon\sigma T_j(0,t)^4- (1-A)\sum_{k\not=j}\mathcal{F}_{jk}\left(\frac{S_{\odot}v_{k\odot}A\mu_k(t)}{r_{\rm h}^2}+\varepsilon\sigma T_k(0,t)^4 \right)\\ \displaystyle \\ \displaystyle +f_i\alpha_sZ(T_j)\mathcal{L}-\kappa\frac{\partial T}{\partial x_j}\Big|_{x_j=0}.\\ \end{array} \end{equation} From left to right, the terms in Eq.~(\ref{eq:07}) denote the absorbed flux from direct solar illumination; thermally emitted infrared radiation; self heating in the form of scattered optical and thermally emitted radiation from other facets; energy consumption by sublimation of near--surface ice; and heat conducted from the surface to the interior or vice versa depending on the sign of the temperature gradient. The boundary condition of Eq.~(\ref{eq:06}) at the bottom of the calculational domain (typically placed at ten times the diurnal thermal skin depth below the surface) is a vanishing temperature gradient; \begin{equation} \label{eq:09} \frac{\partial T_j}{\partial x_j}\Big|_{x=x_{\rm max}}=0. \end{equation} Solving Eqs.~(\ref{eq:06}) with its boundary conditions yields $T_j=T_j(x,t)$ and the vapor production rate then is calculated as \begin{equation} \label{eq:10} Q_j=f_i\alpha_sZ(T_j(0,t)) \end{equation} where $f_i$ is the volume fraction of water ice, $\alpha_{\rm s}$ is the sublimation coefficient \shortcite{kossackietal99b}, \begin{equation} \label{eq:11} \alpha_{\rm s}=1-\frac{1}{a_1}+\frac{1}{a_1}\tanh\left(-a_3\tan\left(\pi\frac{T-a_2}{273-a_2}-\frac{\pi}{2}\right)\right), \end{equation} where $a_1=2.342$, $a_2=150.5$, and $a_3=4.353$, and the Hertz--Knudsen formula is \begin{equation} \label{eq:12} Z=p_{\rm sat}(T)\sqrt{\frac{m}{2\pi k_{\rm B}T}} \end{equation} where the saturation pressure of water vapor (in $\mathrm{Pa}$) is given by \shortciteN{fanaleandsalvail84} \begin{equation} \label{eq:13} p_{\rm sat}(T)=3.56\times 10^{12}\exp\left(-\frac{6141.667}{T}\right). \end{equation} \subsection{Coma structure} \label{sec_method_coma} The purpose of this Section is to present the methods used for calculating the number density $n(h)$, translational temperature $T(h)$, and drift (expansion) speed $W(h)$ as functions of height $h$ above the nucleus surface (along the local surface normal). This calculation is non--trivial because outgassing from a solid surface into vacuum yields a vapor that initially is deviating drastically from thermodynamic equilibrium (e.g., \shortciteNP{cercignani00}), i.e., the velocity distribution function does not resemble the Maxwell--Boltzmann function, and hydrodynamic relations as such do not apply because they are based on a simplified Boltzmann equation where the collision integral is zero. Molecular collisions will gradually evolve the gas toward thermodynamic equilibrium, but only if molecular collisions are common. Therefore, there is a kinetic region known as the Knudsen layer above the nucleus surface that may transit into a hydrodynamic coma at some height, unless the Knudsen layer extends to infinity. We need to deal with both cases, treating the first possibility (strong sublimation) in Sec.~\ref{sec_method_coma_strong} and the second (weak sublimation) in Sec.~\ref{sec_method_coma_weak}. For various discussions about cometary Knudsen layers see, e.g., \shortciteNP{skorovandrickman98}, \citeyearNP{skorovandrickman99}; \shortciteNP{huebnerandmarkiewicz00}; \shortciteNP{crifoetal02}; \shortciteNP{davidssonandskorov04}; \shortciteNP{davidsson08}; \shortciteNP{davidssonetal10}.\\ \noindent In the discussion that follows, we use the following subscripts to distinguish various regions; a sub--surface region responsible for feeding the coma ``$\mathrm{s}$''; the bottom of the Knudsen layer ``0''; the top of the Knudsen layer ``1''; a location within the hydrodynamic part of the coma ``2''; and infinity ``$\infty$''. We start by defining the difference between strong and weak sublimation.\\ \noindent At a given moment the nucleus outgassing rate and surface temperature of a given facet are given by $\{Q_{\rm s},\,T_{\rm s}\}$. The sub--surface reservoir number density capable of sustaining the production rate $Q_{\rm s}$ is \begin{equation} \label{eq:14} n_{\rm s}=Q_{\rm s}\sqrt{\frac{2\pi}{mk_{\rm B}T_{\rm s}}} \end{equation} and the Knudsen layer bottom number density is \begin{equation} \label{eq:15} n_0=\frac{1}{2}(1-\alpha_{\rm bf})n_{\rm s} \end{equation} where $\alpha_{\rm bf}$ is the surface backflux fraction (i.e., the fraction of molecules that return to the surface). The molecular mean free path is \begin{equation} \label{eq:16} \lambda=\frac{1}{\sigma_{\rm c} n_0} \end{equation} where $\sigma_{\rm c}$ is the molecular collisional cross section. The Knudsen layer is taken to have a thickness of $K_{\rm t}=30\lambda$ (90--99\% of thermodynamic equilibrium is achieved at 20--$200\,\lambda$ according to \shortciteNP{cercignani00}). For a nucleus curvature radius $R_{\rm n}$ we define strong sublimation as $R_{\rm n}/(R_{\rm n}+30\lambda)>0.9$, i.e., the Knudsen layer is relatively thin compared to the nucleus radius and the degree of gas expansion within the Knudsen layer is small. Such conditions are typical on the dayside at small heliocentric distances. We define weak sublimation as $R_{\rm n}/(R_{\rm n}+30\lambda)\leq 0.9$, i.e., the gas expansion is substantial within the Knudsen layer and downstream hydrodynamic conditions may not be reached. Such conditions are typical at large heliocentric distances, and on the nightside at any point in the orbit. \subsubsection{Strong sublimation} \label{sec_method_coma_strong} We first describe our applied solution for the Knudsen layer, and then for the downstream hydrodynamic coma. Classical gas kinetic theory (\shortciteNP{anisimov68}; \shortciteNP{ytrehus77}) shows that the temperature and number density at the top of the Knudsen layer are related to the sub--surface parameters (the so--called ``jump conditions'') through \begin{equation} \label{eq:17} \left\{\begin{array}{c} \displaystyle \frac{T_1}{T_{\rm s}}=\left(-\frac{1}{8}\sqrt{\pi}S_1+\sqrt{1+\frac{\pi}{64}S_1^2}\right)^2\\ \\ \displaystyle \frac{n_1}{n_{\rm s}}=\frac{F+\sqrt{\frac{T_1}{T_{\rm s}}}G}{2\exp(-S_1^2)} \end{array}\right. \end{equation} where \begin{equation} \label{eq:18} \left\{\begin{array}{c} \displaystyle F=-\sqrt{\pi}S_1{\rm erfc}(S_1)+\exp(-S_1^2)\\ \\ \displaystyle G=(2S_1^2+1){\rm erfc}(S_1)-\frac{2}{\sqrt{\pi}}\exp(-S_1^2) \end{array}\right. \end{equation} where $\mathrm{erfc}()$ is the complimentary error function and the speed ratio is $S_1=\sqrt{5/6}M_1$ where the Mach number is \begin{equation} \label{eq:19} M=W\sqrt{\frac{m}{\gamma k_{\rm B}T}}. \end{equation} Here, the heat capacity ratio $\gamma$ and the number of rotational degrees of freedom $\zeta$ are related by \begin{equation} \label{eq:20} \gamma=\frac{\zeta+5}{\zeta+3}. \end{equation} The ratio of the number of molecules traveling back towards the surface at the bottom relative to the top of the Knudsen layer is \begin{equation} \label{eq:21} \beta^-=\frac{2(2S_1^2+1)\sqrt{\frac{T_1}{T_{\rm s}}}-2\sqrt{\pi}S_1}{F+\sqrt{\frac{T_1}{T_{\rm s}}}G}. \end{equation} Furthermore, the analysis showed that the Mach number approached unity at the top of the Knudsen layer ($M_1\approx 1$), which means that $T_1/T_{\rm s}\approx 0.67$, $n_1/n_{\rm s}\approx 0.21$ and that $\beta^-$ implies $\alpha_{\rm bf}\approx 0.18$ (see Eq.~\ref{eq:15}). However, these results apply for monatomic vapor and are not directly applicable for water molecules that have $\zeta=3$ and $\gamma=4/3$. Assuming that $T_1/T_{\rm s}\approx 0.67$ still holds, requiring that $M_1=1$ for $\gamma=4/3$, recognizing that mass conservation yields $W_1=Q_{\rm s}/mn_1$ on top of the Knudsen layer while applying Eqs.~(\ref{eq:14}) and (\ref{eq:19}) yields the following jump condition for water vapor, \begin{equation} \label{eq:22} \left\{\begin{array}{c} \displaystyle \frac{T_1}{T_{\rm s}}=0.67\\ \\ \displaystyle \frac{n_1}{n_{\rm s}}=\sqrt{\frac{T_{\rm s}}{2\pi\gamma T_1}}\approx 0.42. \end{array}\right. \end{equation} Using the ratios in Eq.~(\ref{eq:22}) we apply \begin{equation} \label{eq:23} \left\{\begin{array}{l} \displaystyle n(h)=\frac{n_1-n_0}{K_{\rm t}}h+n_0 \\ \\ \displaystyle W(h)=\frac{Q_{\rm s}}{mn(h)}\\ \\ \displaystyle T(h)=T_1\\ \end{array}\right. \end{equation} for heights $0\leq h\leq K_{\rm t}$ above the nucleus surface, i.e., we assume a linear reduction of the number density from $n_0$ near the surface (Eq.~\ref{eq:15}) to $n_1$ at $K_{\rm t}$, evaluate the drift speed according to mass conservation, and assume that the translational temperature remains quasi--constant throughout the Knudsen layer. The last property is motivated by numerical solutions to the Boltzmann equation obtained with the Direct Simulation Monte Carlo (DSMC) technique, see \shortciteN{davidsson08}.\\ \noindent For $h>K_{\rm t}$ we apply an analytical hydrodynamic solution valid for isentropic expansion summarized by \shortciteN{davidssonetal10}, see their Eq~(1)--(6), that merges smoothly with the Knudsen layer solution. Specifically, we first find the Mach number $M_2$ at height $h$ by numerically solving \begin{equation} \label{eq:24} \left(\frac{h+R_{\rm n}}{K_{\rm t}+R_{\rm n}}\right)^2=\frac{1}{M_2}\left(\frac{1+\frac{1}{2}(\gamma-1)}{1+\frac{1}{2}(\gamma-1)M_2^2}\right)^{-\frac{\gamma+1}{2(\gamma-1)}}. \end{equation} Next, the constant $\mathcal{C}$ given by \begin{equation} \label{eq:25} \mathcal{C}=Tn^{1-\gamma} \end{equation} is determined by using the known $\{n_1,\,T_1\}$ values and the parameter $\mathcal{D}(h)$ is evaluated which yields the product of number density and drift speed at the corresponding height from mass conservation, \begin{equation} \label{eq:26} \mathcal{D}=\frac{Q_{\rm s}}{m}\left(\frac{R_{\rm n}}{R_{\rm n}+h}\right)^2=nW. \end{equation} Equations~(\ref{eq:25})--(\ref{eq:26}) are inserted into Eq.~(\ref{eq:19}) which is solved for $n$, thereby allowing that parameter to be determined, \begin{equation} \label{eq:27} n(h)=\left(\frac{M_2}{\mathcal{D}}\sqrt{\frac{\gamma k_{\rm B}\mathcal{C}}{m}}\right)^{\left[\frac{1}{2}(1-\gamma)-1\right]^{-1}}. \end{equation} With $n_2=n(h)$ known, $W_2=W(h)$ follows from Eq.~(\ref{eq:26}) and $T_2=T(h)$ from Eq.~(\ref{eq:25}), thus the gas parameters for the strong sublimation case can be evaluated. \subsubsection{Weak sublimation} \label{sec_method_coma_weak} When collisions are rare, the gas is unable to achieve thermodynamic equilibrium. An accurate description of the coma in this case requires numerical solutions to the Boltzmann equation. Such sophisticated treatment of the problem is beyond the scope of this paper and we apply a simple parameterized description of the coma that reproduce general characteristics of DSMC solutions reasonable well (see Sec.~\ref{sec_results}).\\ \noindent The average kinetic energy of a water molecule upon release from the nucleus is \begin{equation} \label{eq:28} E_{\rm t}=\frac{3}{2}k_{\rm B}T_{\rm s} \end{equation} and its average internal energy because of molecular rotation is \begin{equation} \label{eq:28b} E_{\rm r}=\frac{\zeta}{2}k_{\rm B}T_{\rm s} \end{equation} (see, e.g., \shortciteNP{bird94}). If collisions are very common ($\lambda\ll R_{\rm n}$) the far downstream molecular velocity vectors are radially aligned (i.e., essentially parallel to each other) and all speed variability has been removed through energy equipartitioning. Because there are no random velocities about the local mean, $T_{\infty,\,\lambda\ll R_{\rm n}}=0$ and all kinetic energy is in bulk drift. Therefore, an ideal hydrodynamic gas that expands towards infinity ($n\rightarrow 0$) has $T\rightarrow 0$ (so that Eq.~\ref{eq:25} remains valid at arbitrarily low $n$). Because expansion leads to translational temperature cooling, and because collisions are common, energy is transferred from the internal to the kinetic mode and the rotational temperature cools to $T_{\rm r}=0$ at infinity as well. Thus, all available energy is in bulk drift kinetic energy $mW_{\infty,\,\lambda\ll R_{\rm n}}^2/2=E_{\rm t}+E_{\rm r}$ and therefore \begin{equation} \label{eq:29} W_{\infty,\,\lambda\ll R_{\rm n}}=\sqrt{\frac{(3+\zeta)k_{\rm B}T_{\rm s}}{m}}. \end{equation} If collisions are very few ($\lambda\gg R_{\rm n}$) the downstream velocity vectors are still radially aligned purely because of geometric reasons. However, a molecular speed dispersion remains that collisions have not been able to remove. Therefore, the downstream flow is characterized by a non--zero translational \emph{freeze--out} temperature $T_{\infty,\,\lambda\gg R_{\rm n}}=T_{\rm fr}\not=0$. Because most translational energy still is in bulk drift kinetic energy ($T_{\rm fr}$ is low) and because very little rotational energy has been transferred ($T_{\rm r}$ remains quasi--constant) the bulk drift kinetic energy is $mW_{\infty,\,\lambda\gg R_{\rm n}}^2/2=E_{\rm t}$ and therefore \begin{equation} \label{eq:30} W_{\infty,\,\lambda\gg R_{\rm n}}=\sqrt{\frac{3k_{\rm B}T_{\rm s}}{m}}. \end{equation} For the transition regime $0<\lambda<R_{\rm n}$ we assume that there is a linear transition from $\{T_{\infty},\,W_{\infty}\}_{\lambda\ll R_{\rm n}}$ to $\{T_{\infty},\,W_{\infty}\}_{\lambda\gg R_{\rm n}}$. The temperature and drift speed change with height, starting at their near--surface values $\{T_0,\,W_0\}$ and gradually approaching their downstream values $\{T_{\infty},\,W_{\infty}\}$. The rate of change depends on the number of molecular collisions that take place per time unit. That number depends on the local gas density, here assumed to follow $n\propto h^{-2}$. We realize these assumptions through the expressions \begin{equation} \label{eq:31} \left\{\begin{array}{l} \displaystyle W(h)=\left[1-\left(\frac{R_{\rm n}}{R_{\rm n}+h}\right)^2\right]\left(\sqrt{\frac{(3+\zeta\max\{1-\lambda_0/R_{\rm n},\,0\})k_{\rm B}T_{\rm s}}{m}}-W_0\right)+W_0\\ \\ \displaystyle T(h)=\left[1-\left(\frac{R_{\rm n}}{R_{\rm n}+h}\right)^2\right]\Big(T_{\rm fr}-T_{\rm fr}\max\{1-\lambda_0/R_{\rm n},\,0\}-T_0\Big)+T_0\\ \\ \displaystyle n(h)=\frac{\mathcal{D}(h)}{W(h)} \end{array}\right. \end{equation} where $\lambda_0=(n_0\sigma)^{-1}$ and $\mathcal{D}(h)$ is evaluated as in Eq.~(\ref{eq:26}). Inspired by the numerical DSMC solutions to the Boltzmann equation by \shortciteN{davidssonetal10} we set $T_{\rm fr}=20\,\mathrm{K}$. The expression for $W(h)$ in Eq.~(\ref{eq:31}) can be understood as follows. The first term within the curved bracket is the downstream terminal drift speed. It is close to Eq.~(\ref{eq:29}) when $\lambda_0\approx 0$ and falls linearly with $\lambda_0$ towards Eq.~(\ref{eq:30}) as $\lambda_0\rightarrow R_{\rm n}$, and remains at that value for even more diluted gas ($\lambda_0\geq R_{\rm n}$) because of the maximum function. The expression within the curved bracket therefore measures the largest considered increase above $W_0$, the near--surface value. The expression within the square bracket (confined to the $[0,1]$ interval) regulates how quickly $W(h)$ grows from $W_0$ at $h=0$ to $W_{\infty}$ as $h\rightarrow\infty$. It is an inverse--square law of height, because of the previously mentioned relations between collision frequency, rate of change of the kinetic properties, and number density. The expression for $T(h)$ follows a similar logic. \subsection{Dust ejection} \label{sec_method_eject} In order to calculate the velocity reached by a dust chunk of a particular size we need to solve the equation of motion along the local surface normal, \begin{equation} \label{eq:32} m_{\rm d}\frac{d^2\,h}{d\,t^2}=F_{\rm g}(h)+F_{\rm drag}(h,u_{\rm d}) \end{equation} where the chunk mass is $m_{\rm d}$ and the chunk speed is $u_{\rm d}=dh/dt$. We assume that the initial velocity is zero, i.e., that the mechanism that liberates dust particles from their attachment to the nucleus does not transfer momentum to the dust. For the current application we do not need to specify the mechanism that is responsible for the chunk release. However, we note that in laboratory experiments \shortciteN{ratkekochan89} observed oscillation of dust aggregates at a frequency of 1--$100\,\mathrm{Hz}$ that lasted for up to 10 minutes before their final release. This suggests that gas drag ultimately is responsible not only for the acceleration of dust but also for its liberation, through a fatigue process that gradually reduces the initially strong cohesion of the granular nucleus material to the point the chunk breaks free. We evaluate the local gravitational force $F_{\rm g}(h)$ (directed towards the nucleus) at height $h$ by using the \shortciteN{wernerscheeres97} formalism discussed in Sec.~\ref{sec_method_numorb}. Therefore, the point--mass assumption is used to determine the orbital speed $v_{\rm d}$ that is necessary to reach the target site, but realistic gravity is used when calculating the initial acceleration from rest to the point that $v_{\rm d}$ is reached. This approach is reasonable, given that realistic gravity rapidly approaches the point--mass solution with height. We evaluate the gas drag force $F_{\rm drag}(h,\,u_{\rm d})$ by utilizing the local coma solution $\{n(h),\,T(h),\,W(h)\}$ obtained in Sec.~\ref{sec_method_coma} for chunks of cross section $\mathcal{A}$ and applying \ (e.g., \shortciteNP{gombosi94}), \begin{equation} \label{eq:33} \left\{\begin{array}{c} \displaystyle F_{\rm drag}(h,\,u_{\rm d})=\frac{1}{2}\mathcal{A} C_{\rm D}(h,\,u_{\rm d}) n(h)\left(W(h)-u_{\rm d}(h)\right)^2\\ \\ \displaystyle C_{\rm D}(h,\,u_{\rm d})=\frac{\left(2\mathcal{W}^2+1\right)\exp\left(-\mathcal{W}^2\right)}{\sqrt{\pi}\mathcal{W}^3}+\frac{\left(4\mathcal{W}^4+4\mathcal{W}^2-1\right){\rm erf}\left(\mathcal{W}\right)}{2\mathcal{W}^4}. \end{array}\right. \end{equation} Here, $\mathrm{erf()}$ is the error function and $\mathcal{W}$ is given by \begin{equation} \label{eq:34} \mathcal{W}=\left(W(h)-u_{\rm d}(h)\right)\sqrt{\frac{m}{2k_{\rm B}T(h)}}. \end{equation} Equation~(\ref{eq:32}) is solved numerically for every source region, for every $10^{\circ}$ advancement of the nucleus rotational phase during the considered $\sim 11\,\mathrm{month}$ period around the perihelion passage, for 19 chunk size classes (with dimensions that are $0.05\mathcal{A}_{\rm crit}$--$0.95\mathcal{A}_{\rm crit}$ in increments of $0.05\mathcal{A}_{\rm crit}$, where $\mathcal{A}_{\rm crit}$ is the critical maximum liftable chunk cross section for which $F_{\rm g(0)}+F_{\rm drag}(0,\,0)=0$ is valid at the considered location and time). We here consider chunks shaped as oblate ellipsoids with axis ratio 4, exposing their largest cross section to the gas flow at $h=0$ ($\sim 2.5$ times that of an equal--mass sphere), that turn with constant radial velocity to expose their smallest cross section ($\sim 0.63$ times that of an equal--mass sphere) during the first $500\,\mathrm{m}$ of flight. The chunk shape is close to the axis ratio of 5 needed in order to match the particle speeds measured by Rosetta/GIADA according to \shortciteN{ivanovskietal17a}. The time needed for the chunks to reach the height of $500\,\mathrm{m}$ is similar to that needed by the aerodynamic torque to set non--spherical chunks in rotation according to simulations by \shortciteN{ivanovskietal17b}. The assumed decrease of the chunk cross section with time shortens the time needed for the particle to reach its terminal velocity, and makes $F_{\rm drag}$ diminish in strength faster than $F_{\rm g}$, thereby illustrating how the assumed dominance of nucleus gravity over drag (after the initial acceleration) can be motivated. We select the chunk size $a_{\rm crit}(t)$ that reaches a quasi--terminal velocity of $u_{\rm d,\,max}$ that is as close to $v_{\rm d}$ as possible. This is considered the best representative of the chunk type that feed a particular target area at a given time. Note that this size changes continuously during nucleus rotation and orbital motion. \subsection{Airfall} \label{sec_method_airfall} In order to estimate the amount of mass that is channeled toward a certain target region it is necessary to know the total amount of dust that is being ejected from a source region at a given time, and the fraction of that mass carried by chunks of size $a_{\rm crit}(t)$. The total dust mass flux is taken as $Q_{\rm d}=4Q_{\rm s}$ \shortcite{rotundietal15}. The estimate of the fraction of this mass carried by $a_{\rm crit}(t)$--chunks requires that we specify a dust size distribution function.\\ \noindent For $3.7\,\mathrm{mm}\leq a_{\rm crit}\leq 1.62\,\mathrm{cm}$ we use the size distribution for chunks observed in Philae/CIVA images at Abydos, consisting of three segments with different slopes \shortcite{pouletetal17}. We extrapolate the $\leq 5\,\mathrm{mm}$ part of this distribution to smaller sizes. For $4\,\mathrm{cm}\leq a_{\rm crit}\leq 1\,\mathrm{m}$ we use the size distribution for chunks observed in Philae/ROLIS images at Agilkia \shortcite{mottolaetal15}. We use this distribution to bridge the $1.62$--$4\,\mathrm{cm}$ gap in the measured data, and to extrapolate to larger sizes. With a differential size distribution ($dN/dD\propto D^{-q_{\rm s}}$) slope of $q_{\rm s}=3.8$ \shortcite{mottolaetal15} truncated at $D_{\rm min}=0.0162\,\mathrm{m}$ and $D_{\rm max}=1\,\mathrm{m}$ the amount of mass at diameters $D\geq D_{\rm c}=0.1\,\mathrm{m}$ is given by \begin{equation} \label{eq:35} \mathcal{F}=\frac{\int_{D_{\rm c}}^{D_{\rm max}}D^{3-q_{\rm s}}\,dD}{\int_{D_{\rm min}}^{D_{\rm max}}D^{3-q_{\rm s}}\,dD}=\frac{D_{\rm max}^{4-q_{\rm s}}-D_{\rm c}^{4-q_{\rm s}}}{D_{\rm max}^{4-q_{\rm s}}-D_{\rm min}^{4-q_{\rm s}}} \end{equation} and amounts to $\mathcal{F}=66\%$. If this slope is extrapolated to $D_{\rm min}=1\,\mathrm{mm}$ then $\mathcal{F}=80\%$ for $D_{\rm c}=1\,\mathrm{cm}$ and $\mathcal{F}=49\%$ for $D_{\rm c}=0.1\,\mathrm{m}$. It is clear that a majority of the mass is carried by chunks that are cm--sized or larger.\\ \noindent We truncate the size distribution function at the maximum ejectable size at a given source location and time, normalize the size distribution so that it carries the entire mass being ejected, and determine the fraction of mass carried by chunks with sizes in the range $a_{\rm crit}(t)/2$--$2a_{\rm crit}(t)$.\\ \noindent The total airfall on a given target area is calculated by integrating the mass over all contributing source regions and time. We consider the activity of most target areas to be so low during the mass accumulation period that we assume the airfall has free access to the surface. The exception is target areas \#30 and \#31 (see Table~\ref{tab1}) that themselves are located on the southern hemisphere and are strongly illuminated at times. We therefore calculate the maximum liftable size in those locations versus time and only accept airfall chunks that are larger (i.e., we assume that the smaller chunks will be re--ejected within a short period of time if they manage to reach the surface in the first place). Therefore, sites \#30 and \#31 in Imhotep are partially self--cleaning. The thickness of the accumulated layer at each site is calculated from the incident mass flux assuming that chunks build a deposit with the same bulk density ($\rho=535\,\mathrm{kg\,m^{-3}}$) as the nucleus \shortcite{preuskeretal15}. \subsection{Thermophysics of coma chunks} \label{sec_method_grainthermo} During their flight through the coma the airfall chunks will lose some of the ice they carry. In order to calculate the typical volatile loss from chunks and to estimate the ice abundance in fresh airfall deposits we apply the code \texttt{NIMBUS} (Numerical Icy Minor Body evolUtion Simulator) developed by \shortciteN{davidsson20}. \texttt{NIMBUS} considers a rotating spherical body consisting of a porous mixture of dust and ice, and tracks the internal ice sublimation, vapor condensation, and diffusion of gas and heat in the radial and latitudinal directions over time during body rotation. We refer to \shortciteN{davidsson20} for a detailed description of the model but here state and briefly discuss the governing equations. The energy conservation equation is given by \begin{equation} \label{eq:36} \begin{array}{c} \displaystyle \rho c(T)\frac{\partial T}{\partial t}=\frac{1}{r^2}\frac{\partial}{\partial r}\left(\kappa(\psi,\,T)r^2\frac{\partial T}{\partial r}\right)+\frac{1}{r\sin l}\frac{\partial }{\partial l}\left(\kappa (\psi,\,T)\frac{\sin l}{r}\frac{\partial T}{\partial l}\right)\\ \\ \displaystyle -\sum_{\imath=4}^{n_{\rm sp}}q_{\imath}(p_{\imath},\,T)\mathcal{L}_{\imath}(T)+\sum_{\imath=2}^{n_{\rm sp}}\sum_{\jmath=5}^{n_{\rm sp}}\left( q'_{\imath}(T)\left\{H_{\imath}-F_{\imath\jmath}\mathcal{L}_{\jmath}(T)\right\}\right)-\sum_{\imath=4}^{n_{\rm sp}}g_{\imath}\left(\Phi_{\imath}\frac{\partial T}{\partial r}-\frac{\Psi_{\imath}}{r}\frac{\partial T}{\partial l}\right)+R_*\\ \end{array} \end{equation} with the following terms going left to right; 1) change of the internal energy; 2) radial heat conduction; 3) latitudinal heat conduction; 4) sublimation of ice and recondensation of vapor; 5) energy release during crystallization of amorphous water ice and energy consumption during its release of occluded $\mathrm{CO}$ and $\mathrm{CO_2}$, as well as energy consumption during $\mathrm{CO}$ segregation from its partial entrapment in condensed $\mathrm{CO_2}$; 6) advection during radial and latitudinal gas diffusion; 7) heating by radioactive decay. The indices in Eq.~(\ref{eq:36}) refer to refractories ($\imath=1$), amorphous water ice ($\imath=2$), cubic water ice ($\imath=3$), hexagonal (crystalline) water ice ($\imath=4$), carbon monoxide ($\imath=5$), and carbon dioxide ($\imath=6$). Species denoted by $\jmath$ are hosted within a more abundant and less volatile species denoted by $\imath$. The upper boundary condition to Eq.~(\ref{eq:36}) is given by \begin{equation} \label{eq:37} \frac{S_{\odot}(1-A)\mu(l,t)}{r_{\rm h}^2}=\sigma\varepsilon T_{\rm surf}^4-\kappa\frac{\partial T}{\partial r}\Big|_{r=R_{\rm n}}, \end{equation} and balances absorbed solar radiation with net thermal radiation losses and heat conduction. The boundary condition at the body center is a vanishing temperature gradient. The mass conservation equation for vapor ($\imath\geq 4$) is given by \begin{equation} \label{eq:38} \begin{array}{c} \displaystyle \psi m_{\imath}\frac{\partial n_{\imath}}{\partial t}=-\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\Phi_{\imath} \right)-\frac{1}{r\sin l}\frac{\partial }{\partial l}\left(\Psi_{\imath}\sin l\right)+q_{\imath}(p_{\imath},\,T)+\sum_{\jmath=2}^{n_{\rm sp}}F_{\jmath,\imath}q'_{\jmath}(p_{\jmath},\,T), \end{array} \end{equation} with terms left to right; 1) changes to the gas density; 2) radial gas diffusion; 3) latitudinal gas diffusion; 4) release during sublimation and consumption during condensation; 5) release of $\mathrm{CO}$ and $\mathrm{CO_2}$ during crystallization of amorphous water ice, and of $\mathrm{CO}$ from sublimating $\mathrm{CO_2}$ ice. The boundary conditions to Eq.~(\ref{eq:38}) consist of a quantification of the gas venting to space at the surface and a vanishing gas flux at the center. Finally, the mass conservation equation for ice ($i\geq 2$) is given by \begin{equation} \label{eq:39} \begin{array}{c} \displaystyle \frac{\partial \rho_{\imath}}{\partial t}=-q_{\imath}(p_{\imath},\,T)+\tau_{\imath}(T), \end{array} \end{equation} with terms left to right; 1) changes to the amount of ice; 2) changes due to sublimation of ice and recondensation of vapor; 3) changes due to other phase transitions (crystallization of amorphous water ice, transition from cubic to hexagonal water ice).\\ \noindent In the current simulations we nominally only consider one volatile -- hexagonal (crystalline) water ice that initially constitutes $20\%$ of the body mass. However, one simulation also has condensed $\mathrm{CO_2}$ at a 5\% level relative to water by number. The specific heat capacity $c(T)$ and the heat conductivity $\kappa(\psi,\,T)$ are mass--weighted averages of temperature--dependent values measured in the laboratory for water ice (\shortciteNP{klinger80}, \citeyearNP{klinger81}), carbon dioxide (\shortciteNP{giauqueandegan37}; \shortciteNP{kuhrt84}) and refractories (forsterite from \shortciteNP{robieetal82} for $c$ and ordinary chondrites from \shortciteNP{yomogidamatsui83} for $\kappa$). The heat conductivity is corrected for porosity following \shortciteN{shoshanyetal02} and includes both solid state and radiative conduction. The volume mass production rates $q_{\imath}$ are standard expressions (e.g., \shortciteNP{mekleretal90}; \shortciteNP{prialnik92}; \shortciteNP{tancredietal94}) with $\mathrm{H_2O}$ and $\mathrm{CO_2}$ saturation pressures taken from \shortciteN{huebneretal06} and the gas diffusion fluxes ($\Phi_{\imath}$, $\Psi_{\imath}$) are evaluated as in \shortciteN{davidssonandskorov02b}. The radiogenic heating from a number of long--lived isotopes at chondritic abundances in the refractory component is included by default but has a completely negligible influence on the results in the current application. \subsection{Longterm loss of volatiles} \label{sec_method_volatileloss} In order to better understand the fate of water ice after it has been deposited as airfall we perform thermophysical simulations for a selection of target areas. These simulations rely on the same model as described in Sec.~\ref{sec_method_nucthermo}. The major difference is that the illumination conditions at the target areas (calculated for every $10^{\circ}$ of nucleus rotation) are not restricted to the orbital arc between the inbound and outbound equinoxes, but are made for the entire orbit. \section{Results} \label{sec_results} \subsection{The thickness of airfall debris layers} \label{sec_results_layerthickness} In order to exemplify the application of the models presented in Sec.~\ref{sec_method} we here describe sample results. The thermophysical modeling of the nucleus (Sec.~\ref{sec_method_nucthermo}) provides surface temperature $T_{\rm surf}(t)$ and water production rate $Q_{\rm s}$ during nucleus rotation. Figure~\ref{fig1} shows an example of diurnal temperature and production curves for a southern hemisphere source region\footnote{This randomly selected source is located in the Seth region at longitude $105.5^{\circ}\,\mathrm{W}$, latitude $21.1^{\circ}\,\mathrm{S}$, at a distance $1.33\,\mathrm{km}$ from the nucleus center (origin of the Cheops coordinate system, see \shortciteNP{preuskeretal15})}. that contribute with material to target area \#3, at a time near the inbound equinox. In this particular case the surface temperature peaks near $T_{\rm surf}=211\,\mathrm{K}$ at local noon and falls to $T=129\,\mathrm{K}$ just prior to sunrise. The waviness of the temperature curve is a consequence of changes in shadowing and self heating conditions during nucleus rotation, in addition to the continuous changes of the solar height above the local horizon. The gas production rate is a strongly non--linear function of surface temperature (see Eqs.~\ref{eq:12}--\ref{eq:13}). Therefore, it varies by seven orders of magnitude at this particular location and time. This has strong implications for the properties of the near--nucleus coma.\\ \noindent Figure~\ref{fig2} shows the number density, translational temperature, and expansion velocity of the gas emanating from the same region as in Fig.~\ref{fig1} near noon (see the right circle in that figure). Due to the high gas production rate, this case illustrates the strong sublimation case described in Sec.~\ref{sec_method_coma_strong}, i.e., a Knudsen layer separates the nucleus and the hydrodynamic coma. Here, the Knudsen layer is $\sim 2.15\,\mathrm{m}$ thick. The initial gas drift speed is $\sim 303\,\mathrm{m\,s^{-1}}$ and the gas has an initial translational temperature of $\sim 141\,\mathrm{K}$, much lower than the surface temperature because of the kinetic jump condition (Eq.~\ref{eq:22}). As the gas recedes from the nucleus surface, the expansion results in a number density reduction which is nearly an order of magnitude at $1\,\mathrm{km}$ from the surface. The gas acceleration to $\sim 605\,\mathrm{m\,s^{-1}}$ is a consequence of mass conservation (a thinner gas needs to flow faster to carry the same amount of mass over a boundary). The adiabatic cooling of the gas is evident and the translational temperature drops to $\sim 66\,\mathrm{K}$ within the lower kilometer of the coma.\\ \noindent The conditions in the nighttime coma are drastically different. Figure~\ref{fig3} shows the number density, translational temperature, and expansion velocity of the gas emanating from the same region as in Fig.~\ref{fig1} just before dawn (see the left circle in that figure). Because of the low production rate, Fig.~\ref{fig3} illustrates the weak sublimation case described in Sec.~\ref{sec_method_coma_weak}. The coma never achieves thermodynamic equilibrium and has to be modeled as an infinite Knudsen layer. Over the inner $1\,\mathrm{km}$ of the coma, the gas translational temperature drops from $\sim 87\,\mathrm{K}$ to $\sim 34\,\mathrm{K}$ while the expansion velocity increases from $\sim 238\,\mathrm{m\,s^{-1}}$ to $\sim 385\,\mathrm{m\,s^{-1}}$. The number density is $\sim 7$ orders of magnitude lower than on the dayside, as expected.\\ \noindent The source region in question has such a position, local rotational velocity in the inertial frame, and surface normal that it would be capable of launching chunks to a particular target area on the opposite nucleus hemisphere if these chunks are accelerated to $v_{\rm d}\approx 0.63\,\mathrm{m\,s^{-1}}$ in the vicinity of the nucleus. For comparison, the escape velocity of 67P is $v_{\rm esc}=\sqrt{2G_{\rm N}M_{\rm n}/R_{\rm n}}=0.83\,\mathrm{m\,s^{-1}}$, where the nucleus mass is $M_{\rm n}=9.982\cdot 10^{12}\,\mathrm{kg}$ \shortcite{patzoldetal16} and the area--equivalent nucleus radius is $R_{\rm n}=1932\,\mathrm{m}$ \shortcite{jordaetal16}. The size of the chunks that will reach $v_{\rm d}\approx 0.63\,\mathrm{m\,s^{-1}}$ (see Sec.~\ref{sec_method_eject}) varies drastically during the day. Figure~\ref{fig4} shows results from the numerical integration of Eq.~(\ref{eq:32}) for both the noon and near--dawn conditions discussed previously. At noon, the gas flow is sufficiently strong to lift chunks with diameters just under $1.9\,\mathrm{m}$. However, those chunks are moving too slowly to reach the particular target area in question. Chunks with $\sim 0.3\,\mathrm{m}$ diameter do reach the desired velocity (Fig.~\ref{fig4}, lower panel) and are expected to reach the target area, while chunks smaller than this will travel too fast. When the gas production rate wanes after noon, the critical chunk radius $a_{\rm crit}$ capable of entering the correct transport route to the target area diminishes rapidly. Figure~\ref{fig4} (upper panel) shows that only nano--sized grains would reach the desired velocity, if such particles exist.\\ \noindent Throughout the orbit the capability of this and other source regions to launch large chunks towards the northern hemisphere changes significantly. Seen from many source regions, the Sun is circumpolar at perihelion and will continuously send large chunks to the north. For the region discussed above the solar radiation intensity actually diminishes because of its location on the nucleus. The surface temperature fluctuates between $147$--$206\,\mathrm{K}$ and gas release can eject chunks with up to $1.3\,\mathrm{m}$ diameter range. It would launch chunks with $v_{\rm d}\approx 0.63\,\mathrm{m\,s^{-1}}$ for diameters in the $1.4\,\mathrm{\mu m}$--$0.2\,\mathrm{m}$ range. These numbers serve the purpose of illustrating the drastic variations in airfall chunk sizes a given target region may experience in just a few months during perihelion approach, because of changing illumination conditions at the source during nucleus rotation and orbital motion.\\ \noindent The substantial variability of the critical chunk size during the comet day is shown more clearly in Fig.~\ref{fig5}. Here, $a_{\rm crit}(t)$ is shown for a number of different source regions that all contribute to target area \#3 (see Table~\ref{tab1}), for a single nucleus rotation near perihelion. One source is continuously feeding cm--to dm--sized chunks to the target area because the Sun is circumpolar at that location. A couple of other sources experience day/night variations and their contributions fluctuate between micron and $\sim 0.1\,\mathrm{mm}$ contributions at night to cm--to dm--sized chunks in the day. Other sources are poorly illuminated and are only capable of providing sub--micron particles. The material feeding a given target area therefore originates from a variety of locations and arrives in chunks of drastically different size.\\ \noindent Before proceeding to the main results of this paper we first discuss the accuracy of the orbital solutions for the chunks. The orbit calculations in this paper are made assuming a nucleus point mass and corresponding Keplerian orbits (Sec.~\ref{sec_method_kepler}). We now compare such solutions to numerically integrated orbits in the realistic gravity field of a rotating irregular nucleus (Sec.~\ref{sec_method_numorb}). The black curves in Fig.~\ref{fig6} shows the $\{x(t),\,y(t),\,z(t)\}$ coordinates in the ecliptic system of a Keplerian orbit that connected a particular source region with target area \#24. In this particular case, airfall at that location could be realized for $v_{\rm d}=0.855\,\mathrm{m\,s^{-1}}$ according to the idealized point mass solution and a time of flight of $9.7\,\mathrm{h}$. The left panel of Fig.~\ref{fig6} shows that the numerically integrated trajectory in a realistic gravity field rapidly diverges from the idealized one and after $9.7\,\mathrm{h}$ the difference amounts to $7.7\,\mathrm{km}$.\\ \noindent The main reason for this drastic difference is that the nucleus potential at the launch site in question is merely $\sim 90\%$ of that obtained if all nucleus mass was concentrated at the origin. The chunk injects into an orbit that perhaps would have been similar to that around a point mass carrying $\sim 10\%$ less mass than 67P. If we consider the vis--viva equation (e.g., \shortciteNP{danby89}) that relates the orbital speed $v_{\rm o}$ and distance $r_{\rm o}$ from the origin for any elliptical orbit with semi--major axis $a_{\rm o}$, \begin{equation} \label{eq:visviva} v_{\rm o}^2=\mu_{\rm o}\left(\frac{2}{r_{\rm o}}-\frac{1}{a_{\rm o}}\right) \end{equation} we realize that any reduction of the reduced mass $\mu_{\rm o}$ to $\mu_{\rm o}^*<\mu_{\rm o}$ at a fixed $r_{\rm o}$ could be compensated for by lowering the velocity by a factor $\sqrt{\mu_{\rm o}^*/\mu_{\rm o}}$, thereby achieving exactly the same semi--major axis. Taking the velocity component due to nucleus rotation into account, this suggests that if the chunk velocity due to gas drag was reduced by $\sim 5\%$ compared to its original value (i.e., from $0.855\,\mathrm{m\,s^{-1}}$ to $0.800\,\mathrm{m\,s^{-1}}$) we would expect the numerically integrated chunk trajectory to follow the Keplerian one. The right panel of Fig.~\ref{fig6} shows a numerical orbit integration when $v_{\rm d}=0.8\,\mathrm{m\,s^{-1}}$ and the Keplerian orbit is indeed reproduced very well. It therefore seems like very modest modifications to the ejection speed could compensate for differences between the real and the idealized gravity field. In principle, an adjustment of the ejection velocity should be compensated for by a corresponding change of $a_{\rm crit}$. However, considering that the cross--section to mass ratio will vary among real chunks because they ought to have some density dispersion, we do not attempt such a correction but assume that our $a_{\rm crit}$--values are representative (particularly considering the factor 4 margin used in Sec.~\ref{sec_method_airfall} when evaluating the fraction of the ejected mass that contribute to airfall).\\ \noindent The final results of our simulations regarding airfall accumulation are tabulated in Table~\ref{tab1} and are shown graphically in Fig.~\ref{fig7}. The thicknesses of accumulated layers during a perihelion passage vary rather substantially across the nucleus surface. Some regions collect only small amounts of material, amounting to a few centimeters or less. Most regions accumulate a few decimeters. However, some regions (particularly \#5--6, 13--15, 18--20, and 29) collect substantial layers that are 1.5--$3.6\,\mathrm{m}$ thick. The average thickness for all 31 target areas is $0.87\,\mathrm{m}$. Target areas \#7, 8, and 10 are the ones located closest to Agilkia, where Philae bounced. The average airfall layer thickness for those target sites is $0.27\,\mathrm{m}$ according to our investigation. We therefore predict that the thickness of the airfall layer is comparably thin at this part of the nucleus (about a third of the average thickness) and we note that \shortciteN{bieleetal15} estimate a thickness of $\stackrel{>}{_{\sim}} 0.20\,\mathrm{m}$ for the ``granular soft surface'' that possibly is located ``on top of a more rigid layer''. We are encouraged by the similarity between the layer thickness inferred from the Philae measurements, and the estimate based on our calculations.\\ \noindent Target areas \#1--12 are located in the Ma'at and Serqet regions on the small lobe, extending from the border of Hatmehit towards the Hathor cliff. Collectively, these areas accumulate an average of $0.6\,\mathrm{m}$, with the largest deposits (\#5--6) found at the Hatmehit border. Target areas \#13--20 are located in the eastern Ash region on the large lobe and extend from the border with Aten westward up to the antimeridian. As a group they have the largest average layer thickness of $1.7\,\mathrm{m}$ with the thinnest layers (\#16 and \#17) located at the very center. Target areas \#21--29 are also located in Ash but in its western part stretching from the antimeridian up to the borders to Seth, Anubis, and Aten. The average layer thickness is $0.7\,\mathrm{m}$ for this group with the largest accumulation (\#29) near those borders. Finally, target areas \#30-31 are located in Imhotep, just south of the equatorial plane within the vast smooth region that dominates this morphological unit. Because of efficient self--cleaning the net accumulation here is the lowest, $0.1\,\mathrm{m}$.\\ \begin{table} \begin{center} {\bf } \end{center} \begin{center} \begin{tabular}{||l|r|r|r||l|r|r|r||} \hline \hline Target & Lat & Long & Thickness $\mathrm{[m]}$ & Target & Lat & Long & Thickness $\mathrm{[m]}$\\ \hline \#1 & $40.9^{\circ}$ & $338.2^{\circ}$ & 0.86 & \#17 & $49.6^{\circ}$ & $158.1^{\circ}$ & 0.77\\ \#2 & $34.6^{\circ}$ & $320.2^{\circ}$ & 0.77 & \#18 & $58.1^{\circ}$ & $171.1^{\circ}$ & 2.95\\ \#3 & $30.3^{\circ}$ & $332.9^{\circ}$ & 0.24 & \#19 & $49.2^{\circ}$ & $177.1^{\circ}$ & 1.44\\ \#4 & $27.4^{\circ}$ & $340.2^{\circ}$ & 0.40 & \#20 & $42.5^{\circ}$ & $180.5^{\circ}$ & 1.22\\ \#5 & $19.8^{\circ}$ & $346.0^{\circ}$ & 1.82 & \#21 & $33.9^{\circ}$ & $203.1^{\circ}$ & 0.87\\ \#6 & $9.1^{\circ}$ & $349.2^{\circ}$ & 1.48 & \#22 & $30.3^{\circ}$ & $200.9^{\circ}$ & 0.42\\ \#7 & $19.0^{\circ}$ & $339.9^{\circ}$ & 0.19 & \#23 & $25.4^{\circ}$ & $206.4^{\circ}$ & 0.79\\ \#8 & $16.4^{\circ}$ & $336.0^{\circ}$ & 0.00 & \#24 & $24.4^{\circ}$ & $198.8^{\circ}$ & 0.04\\ \#9 & $21.2^{\circ}$ & $331.7^{\circ}$ & 0.62 & \#25 & $19.1^{\circ}$ & $195.9^{\circ}$ & 0.85\\ \#10 & $13.9^{\circ}$ & $330.6^{\circ}$ & 0.00 & \#26 & $17.6^{\circ}$ & $202.9^{\circ}$ & 0.51\\ \#11 & $6.4^{\circ}$ & $333.5^{\circ}$ & 0.27 & \#27 & $19.2^{\circ}$ & $206.1^{\circ}$ & 0.00\\ \#12 & $15.3^{\circ}$ & $315.9^{\circ}$ & 0.41 & \#28 & $16.8^{\circ}$ & $212.1^{\circ}$ & 0.51\\ \#13 & $48.87^{\circ}$ & $108.0^{\circ}$ & 3.64 & \#29 & $11.3^{\circ}$ & $210.1^{\circ}$ & 2.07\\ \#14 & $56.5^{\circ}$ & $120.9^{\circ}$ & 1.30 & \#30 & $-10.7^{\circ}$ & $118.6^{\circ}$ & 0.11\\ \#15 & $45.7^{\circ}$ & $135.6^{\circ}$ & 1.80 & \#31 & $-7.2^{\circ}$ & $146.1^{\circ}$ & 0.09\\ \#16 & $58.9^{\circ}$ & $142.4^{\circ}$ & 0.41 & & & & \\ \hline \hline \end{tabular} \caption{Target area locations and thickness of accumulated layers formed by fallback of material released from the southern hemisphere.} \label{tab1} \end{center} \end{table} \subsection{Volatile loss during the transfer through the coma} \label{sec_results_iceloss} We now proceed to the question of the ice abundance in airfall deposits. Observations of dust jet switch--off after local sunset shows that the water ice sublimation front is located just a few millimeters below the nucleus surface \shortcite{shietal16}. The process that liberates and ejects chunks in the cm--m class, including but not limited to various forms of cracking (\shortciteNP{elmaarryetal15b}; \shortciteNP{augeretal18}) and cliff collapse \shortcite{pajolaetal17}, combined with gas drag, should produce coma chunks with an initial ice abundance that may be similar to that of the bulk nucleus. However, some of this ice will be lost during the flight through the coma, thereby lessening the initial ice abundance in airfall deposits.\\ \noindent In order to estimate the ice abundance in fresh airfall deposits, we used the code \texttt{NIMBUS} described in Sec.~\ref{sec_method_grainthermo} to study a $D=0.1\,\mathrm{m}$ chunk in the coma, rotating about a fixed axis with coordinates $\{\alpha,\,\delta\}=\{0^{\circ},\,45^{\circ}\}$ in the equatorial system with a period $P=20\,\mathrm{min}$ (individual chunks with similar sizes and rotation periods have been observed in the 67P coma, see \shortciteNP{davidssonetal15b}). These simulations assumed a porosity of $70\%$, micro--meter sized grains (influencing the volume sublimation rate $q$) and pore lengths and radii of $10\,\mathrm{\mu m}$ and $1\,\mathrm{\mu m}$, respectively (influencing the gas diffusivities $\Phi$ and $\Psi$ as well as the magnitude of radiative heat transfer). The chunk was resolved by 18 latitudinal segments and 69 cells in the radial direction using geometric progression such that surface cells were $0.5\,\mathrm{mm}$ thick and core cells were $1\,\mathrm{mm}$ thick. The diurnal skin depth was resolved by $\sim 10$ cells.\\ \noindent After a $12\,\mathrm{h}$ flight this body lost a total of $4.8\%$ of its original ice content. Because parts of the chunk received little radiation due to its spin axis orientation, and because small bodies of this type may not maintain a fixed spin axis, we then introduced a constant precession rate of $\alpha$ from $0^{\circ}$ to $360^{\circ}$ in one hour, combined with a declination nutation with a full oscillation from $\delta=90^{\circ}$ to $-90^{\circ}$ and back again during the same amount of time. Because of the more even distribution of illumination, the volatile loss increased marginally to $6.4\%$ of the initial ice content.\\ \noindent Next, a $D=0.01\,\mathrm{m}$ chunk was considered. The rotation period was reduced to $2\,\mathrm{min}$ but the same precession and nutation periods were applied as before. Because of the smaller size of this chunk its ice loss was more substantial. Yet after $3\,\mathrm{h}$, only $37\%$ of the ice had been lost, increasing to $45\%$ loss after $6\,\mathrm{h}$ and to $56\%$ loss after $12\,\mathrm{h}$. This illustrates that the ice--loss rate slows down rapidly once the sublimation front has moved a few millimeters below the surface, and that chunks as small as $\sim 1\,\mathrm{cm}$ are capable of retaining a substantial fraction of their water ice on time--scales similar to the coma flight time. Coupled with the observation that the majority of mass in smooth region deposits is bound in $D\stackrel{>}{_{\sim}} 1\,\mathrm{cm}$ chunks (with at least half of that mass in $D\stackrel{>}{_{\sim}} 0.1\,\mathrm{m}$ chunks) this means that fresh northern airfall deposits are nearly as icy as the southern source regions that produced them.\\ \noindent Figure~\ref{fig8} shows the internal distributions of porosity, temperature, and $\mathrm{H_2O}$ gas pressure for a $D=1\,\mathrm{cm}$ chunk about $11.4\,\mathrm{h}$ into the simulations. The sublimation front has withdrawn to a depth of $\sim 1\,\mathrm{mm}$ and the depth does not vary significantly with latitude because of the tumbling of the chunk that exposes all parts of the surface to strong solar illumination over time. At the particular snapshot shown in Fig.~\ref{fig8} the subsolar point has been at high northern latitudes sufficiently long for the south pole to cool off, and $\mathrm{H_2O}$ vapor has recondensed within the dust mantle. This is hinted at by a slight increase of porosity in the middle of the southern hemisphere dust mantle, and is seen more clearly in the lower right panel showing the ice abundance (normalized to the initial amounts) versus radial distance at latitude $25^{\circ}\,\mathrm{S}$. At depth the normalized ice abundance is slightly above unity because of downward diffusion and recondensation of vapor. Near the surface the ice abundance has previously gone to zero in the dust mantle, and recently been slightly elevated again, because of the previously mentioned near--surface cooling and associated vapor recondensation. The subsolar point is moving south which means that this frost is starting to sublimate at latitudes near the equator. That gives rise to the local vapor pressure maximum seen in the lower left panel.\\ \noindent In order to explore the robustness of our results, we made three tests where the nominal parameters in the $D=0.1\,\mathrm{m}$ model were changed. First, the radius of constituent grains was changed from $r_{\rm g}=1\,\mathrm{\mu m}$ to $100\,\mathrm{\mu m}$, which decreases the volume sublimation rate by two orders of magnitude ($q\propto r_{\rm g}^{-1}$). The ice loss as well as internal temperatures and vapor pressures changed insignificantly. This can be understood as follows. During one second the part of the difference between absorbed flux and thermally emitted flux that is not used to heat sub--surface material must necessarily be consumed by net sublimation of ice. If an amount of energy $\Delta E$ is available, a mass $\Delta m=\Delta E/\mathcal{L}(T)$ must be converted to vapor (where the latent heat is evaluated at the temperature of the sublimation front). The temperature and pressure gradients on both sides of the sublimation front will evolve in such a way that gas diffusion removes $\Delta m$ from the sublimation front region. This flow gives rise to continuous reductions of the vapor pressure $p_{\rm v}$ below the local saturation pressure $p_{\rm sat}$, which triggers sublimation because $q\propto p_{\rm sat}(T)-p_{\rm v}$. The time--scale for re--establishing saturation conditions after a temporary deviation is extremely short (micro--seconds or less) because $q\gg0$ when $p_{\rm v}<p_{\rm sat}$ (and $q$ rapidly goes to zero when $p_{\rm v}\rightarrow p_{\rm sat}$). Therefore, even if the response time to pressure changes is increased two orders of magnitude because of $r_{\rm g}$, this time is still much shorter than the characteristic time scales of gas diffusion and heat conduction, thereby making the model insensitive to the size of the icy particles.\\ \noindent In another test the dimensions of the cylindrical tubes used to evaluate gas diffusivity were increased by two orders of magnitude from radius $1\,\mathrm{\mu m}$ to $0.1\,\mathrm{mm}$ and from length $10\,\mathrm{\mu m}$ to $1\,\mathrm{mm}$. Note that the pore radius also affects the radiative component of heat conduction, but that parameter was still kept at $1\,\mathrm{\mu m}$ in order to isolate the dependence of the ice loss on the gas diffusion modeling. In this case, the water ice loss increased from $6.4\%$ to $8.9\%$. When the gas diffusivity increases drastically because of the longer and wider pores, the main effect is a reduction of temperature and vapor pressure, in such a way that the resulting gas fluxes inward and outward are essentially the same as before. The same net sublimation $\Delta m$ is required in order to consume the discrepancy between absorbed and emitted radiation that does not change much, and that goal can be met for a cooler and less pressurized gas. For example, a cell that had $T=204.4\,\mathrm{K}$ with the stricter tubes cooled by $15.4\,\mathrm{K}$ to $T=189.0\,\mathrm{K}$ when the tubes widened, and the vapor pressure fell from $p_{\rm v}=0.2234\,\mathrm{Pa}$ to $p_{\rm v}=0.0055\,\mathrm{Pa}$. The sub--surface cooling slightly reduces the surface temperatures as well, meaning that less energy is lost radiatively to space and more energy is available for sublimation. That is why the ice loss becomes somewhat larger, but this increase is still modest.\\ \noindent In the final test, the pore radii used to regulate the radiative contribution to heat conduction was increased by two orders of magnitude as well. This is expected to have a stronger effect on the ice loss because the dust mantle becomes rather warm ($325\,\mathrm{K}$) compared to the icy interior, and wider pores allow for the thermal radiation to penetrate more efficiently to the ice. In fact, the total ice loss increased from $6.4\%$ to $13.7\%$. Yet, this increase is not large enough to alter our overall conclusions -- coma chunks preserve water ice rather efficiently.\\ \noindent We do not expect these numbers to be strongly dependent on the dust--to--ice mass ratio. The water ice sublimation front is expected to withdraw to similar depths regardless of water abundance. At that critical depth, the cooling by net sublimation is reduced by the dust mantle quenching of gas diffusion to the point that the hot dust mantle dissipates the majority of absorbed energy through thermal reradiation. The fractional volatile losses in terms of volume and mass are thus expected to be similar.\\ \noindent We now consider the presence and survival of $\mathrm{CO_2}$ ice in coma chunks. \shortciteN{davidssonetal20} reproduced the observed pre--perihelion $\mathrm{CO_2}$ production rate curve of 67P from $3.5\,\mathrm{AU}$ to $1.24\,\mathrm{AU}$ with a \texttt{NIMBUS} model that had the perihelion $\mathrm{CO_2}$ sublimation front located $\sim 1.9\,\mathrm{m}$ below the surface near the equator, and where that depth diminished with latitude to $\sim 0.2\,\mathrm{m}$ at the south pole. If the suggested presence of $\mathrm{CO_2}$ ice at shallow depths is correct, it is conceivable that regular sublimation--driven activity occasionally can expel extremely cold chunks containing supervolatiles at perihelion. The presence of $\mathrm{CO_2}$ ice patches on the surface of 67P observed spectroscopically by \shortciteN{filacchioneetal16} could be the result of such temporary exposure of near--surface $\mathrm{CO_2}$ ice. There are other forms of activity that might produce $\mathrm{CO_2}$--rich chunks as well. For example, the cliff collapse in Aswan observed by OSIRIS suddenly exposed material located $\sim 12\,\mathrm{m}$ below the previous surface \shortcite{pajolaetal17}, and though such events are rare, they could occasionally inject unusually cold chunks into the coma that still contain $\mathrm{CO_2}$. It is interesting and important to understand to what extent such material could be transported and mixed in with other airfall material.\\ \noindent Therefore, in our last numerical experiment, 5\% condensed $\mathrm{CO_2}$ relative to $\mathrm{H_2O}$ (by number) was added to the chunk, and the initial temperature was lowered from $T(t=0)=150\,\mathrm{K}$ to $T(t=0)=50\,\mathrm{K}$ to avoid an explosive sublimation of the carbon dioxide. Our \texttt{NIMBUS} simulations of a $D=0.1\,\mathrm{m}$ chunk shows that it takes $2\,\mathrm{h}$ for all $\mathrm{CO_2}$ to be lost. A substantially larger chunk could potentially preserve a fraction of its $\mathrm{CO_2}$ ice during the transfer to the northern hemisphere. After a total of $12\,\mathrm{h}$ exposure to the Sun at the perihelion distance of 67P, the amount of water ice lost in the $D=0.1\,\mathrm{m}$ chunk is $5.4\%$, i.e., somewhat lower than in the simulation without $\mathrm{CO_2}$. This reduction happens for two reasons; 1) more energy is required to heat the body which start out $100\,\mathrm{K}$ colder than in the other simulations; 2) energy is needed in order to sublimate the carbon dioxide. As a consequence, the water sublimation is somewhat delayed. \subsection{Ice loss from airfall deposits} \label{sec_results_survivability} To perform the illumination calculations (including shadowing and self heating effects from the irregular nucleus) and the thermophysical simulations for the entire orbit, with rotation resolved (solving the thermophysical equations, Eq.~\ref{eq:06}--\ref{eq:09} requires a time step of $\sim 10\,\mathrm{s}$) is computationally demanding. Therefore, we have not performed these calculations for all 31 target areas, but for a subset of four. These areas were selected to represent each of the major regions under study. Specifically, we consider \#3 in Ma'at on the small lobe, \#16 in eastern Ash, \#24 in western Ash, and \#31 in Imhotep on the large lobe.\\ \noindent The results of these simulations are shown in Fig.~\ref{fig9}. The black curves in those plots show the surface temperature, with the daytime peak as solid curves and the nighttime minimum as dashed curves. The gray areas mark the part of the orbit between the inbound and outbound equinoxes when the level of southern hemisphere activity is highest and most airfall takes place. Target areas \#3, \#16, and \#24 are all located on the northern hemisphere. Because of the spin axis orientation they are illuminated at aphelion and the surface temperatures increases as the comet travels toward perihelion. The temperature peaks around $200\,\mathrm{K}$ at all three locations, and the day/night amplitude is typically $10$--$30\,\mathrm{K}$. However, when the comet approaches the inbound equinox, the importance of the spin axis orientation becomes evident and the temperature plummets, as the regions enter into polar night. During the perihelion passage, the temperatures are as low as $60$--$70\,\mathrm{K}$ (with little to no dependence on nucleus rotational phase) and they are completely inactive. Therefore, airfall material originating from the strongly sublimating southern hemisphere has no problem landing in these target areas and they are flash--frozen after their coma flight. The generally good correspondence between inactivity and the deposition period in gray is the reason why we ignore self--cleaning for these areas.\\ \noindent When target areas \#3, \#16, and \#24 emerge from polar night after the outbound equinox at $2.6\,\mathrm{AU}$, the surface temperatures are lower and (because of their strongly non--linear relation) the gas production rates are much lower than on the inbound branch, again due to the spin obliquity effect. Therefore, it is expected that much of the airfall ice collected at perihelion will survive the journey towards aphelion. Consistent with observations (e.g., \shortciteNP{gulkisetal15}; \shortciteNP{hassigetal15}; \shortciteNP{finketal16}), the northern hemisphere produces water vapor vigorously within $\sim 3\,\mathrm{AU}$ pre--perihelion because the airfall material in these regions reach $T\stackrel{>}{_{\sim}} 190$--$200\,\mathrm{K}$ for the first time since it was expelled from the southern hemisphere six years earlier.\\ \noindent The situation for target area \#31, that is located on the southern hemisphere fairly close to the equator, is quite different. It is poorly illuminated at aphelion, and its peak surface temperature is $\sim 40\,\mathrm{K}$ below the others. However, as the region approaches the Sun it remains fully illuminated and the temperature peaks at $\sim 230\,\mathrm{K}$, which is the highest among the four by far. Although it is bombarded with airfall material from even more active regions closer to the subsolar point at perihelion, it is capable of re--ejecting this material or preventing it from landing in the first place. That is why we considered self--cleaning for target areas \#30--31.\\ \noindent Figure~\ref{fig9} also shows solid and dashed red curves. Those are the maximum and minimum temperatures $0.2\,\mathrm{m}$ below the surface. Because of the low conductivity of the porous comet material, the diurnal surface temperature amplitude is rapidly damped with depth, and at this level of magnification the solid and dashed curves are inseparable. In brief, $0.2\,\mathrm{m}$ below the surface, a thermometer would not be capable of telling whether the Sun is above or below the local horizon. We note that these calculations were performed without accounting for airfall. We ran one simulation for target area \#16 where a $0.5\,\mathrm{m}$ thick layer at temperature $200\,\mathrm{K}$ was deposited instantaneously onto the nucleus at perihelion when this location had a surface temperature of $\sim 70\,\mathrm{K}$. This procedure was intended to simulate the arrival of warm airfall (originating from the fully illuminated southern hemisphere) to the northern hemisphere that had polar night. After $\sim 150\,\mathrm{days}$ the temperature profile in the upper $0.2\,\mathrm{m}$ differed at most a few Kelvin between models with and without airfall. Because this thermal relaxation time is rather short compared to the orbital period we consider Fig.~\ref{fig9} a sufficiently good representation of temperatures at the target sites.\\ \noindent A detailed comparison of outbound temperatures indicates that some regions will lose less of the airfall ice they have accumulated than others. In order to compare the sites with each other, we integrated the gas production rates from the outbound equinox to perihelion and normalized with respect to the site with the highest loss, thereby obtaining an index $L'$ quantifying the relative capacity to get rid of the ice. We then defined $S'=1-L'$ as the relative capacity to retain ice, and renormalized the set to the highest retainer. Thereby we obtained an index $\mathcal{S}$ that we refer to as ``survivability''. We also form an index $\mathcal{R}$ by normalizing the thickness of the accumulated airfall layer to the highest in the set that we call ``reachability''. If a target location simultaneously is characterized by a high reachability (it is reached by much airfall material) and high survivability (it remains comparably cool on the outbound orbital branch), then it is likely to be relatively active when the comet approaches perihelion on the inbound orbital branch.\\ \noindent In this system, target site \#16 has $\{\mathcal{R},\,\mathcal{S}\}=\{1,\,1\}$ which means that eastern Ash has the potential of being more active before pre--perihelion than \#3 in Ma'at with $\{\mathcal{R},\,\mathcal{S}\}=\{0.58,\,0.75\}$, and much more active than \#24 in western Ash ( $\{\mathcal{R},\,\mathcal{S}\}=\{0.11,\,0.31\}$) and \#31 in Imhotep ( $\{\mathcal{R},\,\mathcal{S}\}=\{0.22,\,0.07\}$). \section{Discussion} \label{sec_discussion} A number of authors have discussed the transfer of material from one location on 67P to another, the details of the airfall deposition process, and its effect on observable properties, using different methods. We here recapitulate their work and compare their results with our own. \shortciteN{pajolaetal17} calculated the maximum daily lift pressure (the product between production rate and expansion velocity that is proportional to the maximum size of liftable chunks) for different regions on 67P and used that information to offer an explanation for the differing size distributions at Agilkia (coarse and fine chunks) and Sais (coarse chunks with $D \stackrel{>}{_{\sim}} 3\,\mathrm{cm}$ only). Their discussion provides valuable insight into the dynamics of dust deposition and self--cleaning. The region Bes on the southern hemisphere reaches the highest lift pressure at perihelion and is expected to send a wide range of chunk sizes toward the northern hemisphere. Hapi experiences an extensive polar night period and can collect a substantial amount of material. Once illuminated, its own lift pressure is many orders of magnitude smaller than that of Bes, which, according to \shortciteN{pajolaetal17}, may explain why the coma in August 2014 and in the following months (primarily fed by Hapi contributions) was dominated by rather small ($0.1$--$1\,\mathrm{mm}$) chunks. Hapi is unable to eject most of the large chunks it receives and may experience an unusually large accumulation of airfall material over time, unless fragmentation of dm--m chunks into $0.1$--$1\,\mathrm{mm}$ chunks on the surface is efficient. As a contrast, the lift pressure at Sais (once it emerges from polar night) is just an order of magnitude lower than for Bes, suggesting that all but the largest airfall chunks will be ejected after a brief residence on the nucleus. This may explain why the size distribution at that location (measured at the end of September 2016, $3.8\,\mathrm{AU}$ post--perihelion) was skewed towards coarse chunks. Agilkia, on the other hand, does not experience polar night at perihelion, which prevents it from accumulating chunks smaller than $\sim 0.1\,\mathrm{m}$ at that time. However, it was observed in November 2014 ($3.0\,\mathrm{AU}$ from the Sun, inbound), at a time when its lift pressure had been lower than that of Hapi since aphelion. The fine airfall seen at Agilkia but missing in Sais therefore originated from Hapi while its coarse airfall came from Bes and other regions on the southern hemisphere \shortcite{pajolaetal17}.\\ \noindent In our work we only consider self--cleaning from regions \#30--31 that are strongly illuminated around perihelion. This means that we ignore the self--cleaning observed by \shortciteN{pajolaetal17} in, e.g., Sais at Ma'at, because the level of activity after the outbound equinox is still very low (Fig.~\ref{fig9}). Also, on the inbound leg we ignore the contribution arriving to our target areas from Hapi. In both cases, our assumptions concern very small particles, and we have already demonstrated that the mass carried by chunks smaller than a few centimeters is a relatively small fraction of the total mass (see Sec.~\ref{sec_method_airfall}). Therefore, our results are only slightly affected by this simplification (because we here only consider the total amount of mass, and do not attempt to accurately describe the size distribution of accumulated airfall at the target sites).\\ \noindent \shortciteN{kramerandnoack15} performed numerical integration of dust chunk trajectories under the influence of realistic nucleus gravity, centrifugal and Coriolis forces (as they were working in a non--inertial frame), and gas drag. Their purpose was to explain the orientation of wind tails observed by \shortciteN{mottolaetal15} at Agilkia. They assumed that each point on the nucleus surface emitted the same fixed amount of gas flux. Thus, there was no dependence of local outgassing rate on solar location, the model nucleus had no day/night side, and the model coma was static and highly isotropic compared to the comet coma. However, the irregular nucleus shape created local concentrations of gas in strongly concave areas, like the Hapi neck region. The resulting dust flow had two major characteristics; 1) a flow out of Hapi that diverged into one flow over Seth and Ash on the large lobe and another flow over Hathor, Ma'at, and Hatmehit on the small lobe; and 2) a clockwise flow in the equatorial plane, as seen from the cometary north pole in a body--fixed frame, caused by the counter--clockwise nucleus rotation. Using this model, \shortciteN{kramerandnoack15} obtained dust flow vectors in the Agilkia region that were consistent with the wind tail directions observed by \shortciteN{mottolaetal15}.\\ \noindent \shortciteN{laietal17} performed an investigation of airfall deposition that also considered realistic nucleus gravity, centrifugal and Coriolis forces, and gas drag. In their work, a fixed outgassing rate was applied to each point on the surface. These outgassing rates were taken to be proportional to the average cosine of the local day--time solar zenith angle during nucleus rotation. This anisotropic gas flow roughly captures the dependence of coma number density with nucleus latitude but smears its local time dependence, except for a strong day/night asymmetry. This anisotropic coma was merged at a 0.85 weight with an isotropic component with weight 0.15, and the total gas production rate was normalized to match the observed one. This was done once per month from January--December 2015 in order to account for the changing subsolar latitude and evolving total gas production rate with heliocentric distance. For each of the dozen static outgassing patterns, Direct Simulation Monte Carlo simulations were performed to calculate the gas number density, translational temperature, and expansion velocity as functions of position within the coma, needed for gas drag evaluations. \shortciteN{laietal17} used 14 dust classes with sizes in the $2.8\,\mathrm{\mu m}$--$4.5\,\mathrm{cm}$ range, each represented by $\sim 4\cdot 10^5$ particles launched from random locations on the surface that had different weights depending on local gas production conditions and chunk cross sections. In this manner, the local airfall and self--cleaning rates could be calculated at a given time, and the results were integrated over the considered orbital arc to estimate the local net deposition during one orbit. We note that \shortciteN{thomasetal15b} applied a similar model, however, limited the study to ejection of dust from Hapi to other parts of the northern hemisphere. They found that ejection speeds of $v_{\rm d}\stackrel{>}{_{\sim}} 0.5\,\mathrm{m\,s^{-1}}$ are necessary to leave the Hapi valley and that $\stackrel{>}{_{\sim}} 50\%$ of the particles escape the nucleus if $v_{\rm d}\stackrel{>}{_{\sim}} 1\,\mathrm{m\,s^{-1}}$, thus providing tight speed constraints for that transfer route.\\ \noindent \shortciteN{laietal17} found that the largest liftable chunk on the southern hemisphere has $D=1\,\mathrm{cm}$ and that a net mass loss occurs everywhere except in Hapi where a net airfall deposit of $0.4\,\mathrm{m}$ depth is formed. We note that the $D\leq 1\,\mathrm{cm}$ limit is small compared to the observed numerous coma chunks in the $D=0.1$--$1\,\mathrm{m}$ class (e.g., \shortciteNP{rotundietal15}; \shortciteNP{davidssonetal15b}; \shortciteNP{agarwaletal16}), as well as compared to the size of chunks observed in resolved smooth terrains. Furthermore, their $0.5$--$1\,\mathrm{m}$ net loss on most of the northern hemisphere, combined with their $1$--$1.8\,\mathrm{m}$ net loss on most of the southern hemisphere, translates to a total loss of $\sim 2.5\,\cdot 10^{10}\,\mathrm{kg}$ (if applying an average loss of $1\,\mathrm{m}$, a nucleus surface area of $46.9\,\mathrm{km^2}$ and assuming that the density is that of the bulk nucleus, $530\,\mathrm{kg\,m^{-3}}$, see \shortciteNP{jordaetal16}). This is $2.4$ times higher than the observed nucleus net mass loss of $1.05(\pm 0.34)\cdot 10^{10}\,\mathrm{kg}$ according to the Rosetta/RSI radio science experiment \shortcite{patzoldetal19}. We suspect that the maximum liftable dust size at the perihelion subsolar point becomes significantly underestimated when calculating production rates based on the average cosine of the solar zenith angle. As a consequence, the amount of airfall is underestimated because the size distribution of ejected particles is truncated at a size below which return as airfall is highly uncommon. Simultaneously, the outgassing is rather strong in polar night regions (15\% of the daytime production), which leads to a high self--cleaning rate.\\ \noindent Except for an initial acceleration phase aimed at determining a relation between local gas production rates, dust size, and their velocities, the current work does not consider gas drag effects on the dust orbits. This is admittedly a weakness. However, we do not think it is meaningful to engage in computationally demanding gas coma modeling that enables continuous gas drag, unless the nucleus model source that feeds the coma model is reproducing the strong temporal, latitudinal, and longitudinal variations in gas production rate known to occur on the real nucleus (during nucleus rotation and throughout the orbit). The summary above reminds us that such an elaborate treatment is beyond the current state--of--the--art of the field. It is unlikely that attempts to account for the orbital perturbations due to outgassing from the chunks would be realistic (due to their small size, chunks are likely having an excited spin state, and when combined with thermal inertia effects the time--evolution of the net non--gravitational force vector is unpredictable). Therefore, this ``rocket effect'' is ignored as well. Because of the point--mass assumption we also have not applied a realistic gravity field, but based on the comparison with a more appropriate approach described in Sec.~\ref{sec_method_numorb} we think that this may not have drastic consequences concerning our overall conclusions. Furthermore, we have ignored the effect of solar radiation pressure on the dynamics of coma chunks. Radiation pressure does have an affect on micron--sized chunks (e.g., \shortciteNP{tenishevetal11}), but considering the small contribution of such particles to airfall deposits in terms of mass, this simplification has no important effect on our conclusions. With these limitations in mind, our calculations support the hypothesis (\shortciteNP{kelleretal15}; \citeyearNP{kelleretal17}) of a significant transport route from the southern to the northern hemisphere and demonstrate that both Ash and Ma'at are plausible destinations of airfall, as previously has been suggested (\shortciteNP{thomasetal15a}; \citeyearNP{thomasetal15b}). We find that the amount of airfall varies systematically between the four main regions of study (Ma'at, eastern and western Ash, and Imhotep) and within those regions. About 40\% of the considered target areas receive at most $0.5\,\mathrm{m}$ airfall and about $30\%$ receive at least $1\,\mathrm{m}$ according to our study. Considering the global coverage OSIRIS resolution of $0.2$--$3\,\mathrm{m\,px^{-1}}$ \shortcite{preuskeretal17} it is therefore not surprising that morphological changes because of airfall are difficult or impossible to verify observationally in many locations, but that such changes definitively are detected in others \shortcite{huetal17}.\\ \noindent Our second goal consisted of estimating the amount of water ice loss during the transfer of chunks through the coma. We also wanted to explore the survivability of the substantially more volatile $\mathrm{CO_2}$ because of the chemical north/south dichotomy measured at 67P (\shortciteNP{hassigetal15}; \shortciteNP{fougereetal16}; \shortciteNP{finketal16}) and interpreted in the context of coma transport and airfall \shortcite{kelleretal17}. For a nominal set of model parameters our \texttt{NIMBUS} simulations showed that a dm--sized chunk would retain more than $90\%$ of its water ice when suspended for $12\,\mathrm{h}$ and that a cm--sized chunk would keep almost half its water ice in the same time. This lends substantial credibility to the claim by \shortciteN{kelleretal17} that the airfall is ``wet'' and naturally explains the substantial level of water--driven activity on the northern hemisphere observed by Rosetta. We also found that a dm--sized chunk containing $5\%$ $\mathrm{CO_2}$ relative to water would lose all of it in $2\,\mathrm{h}$. It also means that a meter--sized chunk might be able to transfer supervolatiles from the south to the north. If so, then the low levels of $\mathrm{CO_2}$ outgassing from the northern hemisphere is a consequence of the small number of such bodies (and it may indicate that most $\mathrm{CO_2}$ ice could have been lost already prior to departure). Alternatively, some $\mathrm{CO_2}$ may have been transported from the south to the north, not as an independently condensed form of ice but trapped within water ice. Laboratory experiments (e.g., \shortciteNP{edridgeetal13}) demonstrate that a fraction of $\mathrm{CO_2}$ trapped in amorphous ice survives the release during crystallization as well as the cubic--hexagonal transformation and is released during water sublimation. We did not model that process explicitly. We note that Philae detected some highly volatile species at Agilika, such as $\mathrm{CO}$ and $\mathrm{CO_2}$ with Ptolemy \shortcite{wrightetal15}, and $\mathrm{CH_4}$ with COSAC \shortcite{goesmannetal15}. Those species may have been trapped in water ice within airfall material and released during water sublimation. Alternatively, they may have originated from underneath the airfall layer. \texttt{NIMBUS} simulations by \shortciteN{davidssonetal20} show that the measured pre--perihelion $\mathrm{CO_2}$ production rate curve is reproduced when the depth of the $\mathrm{CO_2}$ sublimation front is located $3.8\,\mathrm{m}$ below the surface on the northern hemisphere, and at $1.9\,\mathrm{m}$ below the surface on the southern hemisphere at aphelion (these depths are modified in the model over time because of dust mantle erosion as well as $\mathrm{CO_2}$ sublimation, see Sec.~\ref{sec_results_iceloss}). Supervolatiles like $\mathrm{CO}$ and $\mathrm{CH_4}$ may be released during sublimation of $\mathrm{CO_2}$ at the front, or from even larger depths by segregation processes, if they are trapped in $\mathrm{CO_2}$, as suggested by \shortciteN{gascetal17}. Although presence of $\mathrm{CO_2}$ in the airfall material itself remains a possibility, it cannot be the major source of $\mathrm{CO_2}$ release from the northern hemisphere before the inbound equinox, because the $\mathrm{CO_2}$ production does not correlate with that of $\mathrm{H_2O}$ (\shortciteNP{luspaykutietal15}; \shortciteNP{gascetal17}).\\ \noindent Our third goal was to quantify the relative capacity of different airfall deposition sites to retain their water ice content after deposition. To this end we introduced a survivability index $\mathcal{S}$ and combined it with a reachability index $\mathcal{R}$ that measures the relative capability to collect airfall material. We performed an investigation limited to four sites, where \#16 in eastern Ash came out on top, with \#3 in Ma'at second, and \#24 (western Ash) and \#31 (Imhotep) on a shared third place. Comparing with the $\mathrm{H_2O}$ potential activity map of \shortciteN{fougereetal16} derived from long measurement series by ROSINA and corrected for illumination biases, it is clear that \#16 is substantially more active than \#24 and \#31, which is consistent with our work. However, \#3 is the most active of them all. Interestingly, the high--activity belt shown by \shortciteN{fougereetal16}, stretching from Ash, Seth, to Ma'at along the (anti)meridian with an epicenter in Hapi, has a strong resemblance with the simulated model flow pattern of airfall material originating from Hapi (\shortciteNP{thomasetal15b}; \shortciteNP{kramerandnoack15}). It is therefore possible that the ice variability introduced by airfall from the southern hemisphere at perihelion is mixed and masked by redistribution of airfall due to Hapi activity at the time ROSINA performed the measurements. It is encouraging that the activity map of \shortciteN{laraetal15}, based on the footprint of dust jets observed early during the Rosetta mission, suggest high levels of activity both at \#3 and \#16.\\ \noindent To what extent are airfall deposits common features on comet nuclei? We expect that three criteria need to be fulfilled in order to create thick and wide--spread airfall deposits. First, the spin pole needs to have such an orientation that a large fraction of the nucleus surface experiences polar night at perihelion. In terms of the spin axis obliquity and argument (for definitions see, e.g., \shortciteNP{sekanina81}; \shortciteNP{davidssongutierrez04}) the most favorable configuration is an obliquity near $90^{\circ}$ and an argument near $90^{\circ}$ or $270^{\circ}$ (i.e., the perihelion subsolar point coincides with one of the comet nucleus poles). Second, the post--perihelion equinox should take place as late as possible. The equinox passage means that the Sun is in the equatorial plane of the comet and that fresh airfall deposits no longer are protected from solar illumination. If airfall is illuminated sufficiently close to perihelion, strong activity and self--cleaning may take place that prevents long--term deposits from forming. Third, the comet needs to produce abundant chunks that are too large to reach escape velocity. The presence of a detectable debris trail is a potential indicator that a specific comet is prone to eject particularly large chunks. A Spitzer survey of 34 comets showed that 27 objects, or $79\%$, had trails consisting of millimeter--sized particles and larger \shortcite{reachetal07}, suggesting that 67P is not unusual.\\ \noindent Figure~\ref{fig11} shows the time evolution of the angle between the solar direction and the comet nucleus pole that is sunlit at perihelion (here referred to as the polar angle), for four comets that were targets of spacecraft missions. The smaller the polar angle, the larger the fraction of the comet surface has polar night at perihelion. If it reaches $90^{\circ}$ (marked by the horizontal line), equinox takes place and the Sun is in the equatorial plane of the comet (i.e., no region has polar night). Among the four, Comets 67P and 19P/Borrelly seem to have the best conditions for airfall deposition. The perihelion polar angles are $42^{\circ}$ for 67P and $29^{\circ}$ for 19P/Borrelly, i.e., the polar night regions are large. Comet 67P reaches equinox later (221 days post--perihelion, outside the figure) than 19P/Borrelly (97 days post--perihelion), but self--cleaning is probably limited in both cases. 67P evidently produces large chunks (as observed by Rosetta, but also inferred from the existence of a trail; \shortciteNP{sykeswalker92}; \shortciteNP{kelleyetal09}; \shortciteNP{ishiguroetal09}), that enables airfall deposit formation, but this may not be the case for 19P/Borrelly. The reports on the Deep~Space~1 flyby of 19P/Borrelly do not mention large coma chunks (\shortciteNP{soderblometal02}; \shortciteNP{boiceetal02b}), and the comet does not have a trail \shortcite{ishiguroetal09}. Despite the favorable geometric conditions, Comet 19P/Borrelly may therefore not have large smooth terrains. Unfortunately, Deep Space 1 images cannot provide an answer because the flyby took place just 8 days after perihelion \shortcite{soderblometal02}, i.e., a potential airfall coverage would have been hidden from view on the dark polar night side.\\ \noindent Comets 81P/Wild~2, and particularly 9P/Tempel~1, are least suitable for vast airfall deposit formation. 9P/Tempel~1 has a perihelion polar angle of $86^{\circ}$ and passes equinox just three weeks later. It is therefore surprising that 9P/Tempel~1 has prominent smooth areas \shortcite{ahearnetal05b}. Particularly the smooth area ``S2'' near the south pole \shortcite{veverkaetal13} would be most consistent with airfall onto the weakly illuminated and least active part of the nucleus, while the origin of localized smooth areas elsewhere is less obvious. It is also interesting to note that observations of a slowly expanding coma arc during the Deep~Impact flyby suggests the presence of fairly large (millimeter--sized) slow--moving chunks \shortcite{farnhametal07}, and a trail associated with Comet 9P/Tempel~1 has been observed both with IRAS \shortcite{sykeswalker92} and with Spitzer \shortcite{reachetal07}. This suggests that the spin axis orientation plays a minor role for the existence of smooth airfall terrain, though a large coverage probably still requires a small perihelion polar angle. If true, this could mean that all comets with an appropriate dust size distribution are capable of building smooth airfall terrains. For that reason, the case of 81P/Wild~2 is somewhat puzzling. With a polar angle of $70^{\circ}$, a rather large part of the nucleus has polar night at perihelion. The equinox passage is just 40 days post--perihelion, but judging from Comet 9P/Tempel~1, that may not prevent airfall debris from remaining on the surface. The Stardust spacecraft collided with a couple of mm--sized dust grains \shortcite{tuzzolinoetal04} and the comet has a trail according to \shortciteN{ishiguroetal09}. Furthermore, the Stardust flyby took place 98 days post--perihelion \shortcite{brownleeetal04}, at a time when the regions that experienced polar night at perihelion had become visible. Yet, the Stardust images \shortcite{brownleeetal04} do not reveal smooth terrains of the type seen on 9P/Tempel~1 or like Imhotep or Hapi on 67P. Potentially, the airfall material on 81P/Wild~2 is hiding in plain view, i.e., as a thin coverage on top of the underlying rugged topography, similar to the Ash and Seth regions on 67P. The nature of those regions only became evident in Rosetta images that had significantly better resolution than the best Stardust images ($14\,\mathrm{m\,px^{-1}}$; \shortciteNP{brownleeetal04}). Alternatively, airfall deposits do not form on Comet 81P/Wild~2.\\ \noindent Two spacecraft targets, Comets 1P/Halley and 103P/Hartley~2, were omitted from Fig.~\ref{fig11} because they are complex rotators (the polar angle is not easily calculated). Large chunks are common around 103P/Hartley~2 \shortcite{kelleyetal13} and it has a trail \shortcite{reachetal07}. EPOXI images revealed a smooth neck region \shortcite{ahearnetal11} reminiscent of Hapi on 67P. With airfall confirmed on 67P, and suspected on 9P/Tempel~1 and 103P/Hartley~2, it indeed appear to be a common process in comets. \section{Conclusions} \label{sec_conclusions} We have modeled the transfer of material from the highly active southern hemisphere of comet 67P to its northern hemisphere during the perihelion passage. We find that such transport routes exist and that the Ash and Ma'at regions receive plenty of airfall, thereby supporting previous hypotheses (\shortciteNP{thomasetal15a}, \citeyearNP{thomasetal15b}; \shortciteNP{kelleretal15}, \citeyearNP{kelleretal17}). The amount of airfall accumulated between the inbound and outbound equinoxes is typically a few times $0.1$--$1\,\mathrm{m}$ (some of which will be removed in other parts of the orbit, i.e., these numbers should not be interpreted as \emph{net} orbital accumulations, but rather as an estimate of the material the northern hemisphere has to its disposal to drive activity pre--perihelion). The distribution of airfall is heterogeneous, with the most substantial accumulation in eastern Ash, followed by western Ash and Ma'at at similar levels, with the least airfall at central Imhotep because of efficient self--cleaning (only these four regions were considered, e.g., Hapi was not included in the study).\\ \noindent We have also modeled the loss of both $\mathrm{H_2O}$ and $\mathrm{CO_2}$ from $0.01$--$0.1\,\mathrm{m}$ diameter chunks in the coma, using the elaborate \texttt{NIMBUS} model. We find that a cm--sized chunk can retain roughly half its water ice during a $12\,\mathrm{h}$ exposure to the Sun at perihelion, while a dm--sized chunk holds on to more than 90\% of the water ice. If there is $\mathrm{CO_2}$ at a 5\% level relative to water by number, a dm--sized chunk loses all carbon dioxide in two hours. We therefore support the scenario described by \shortciteN{kelleretal17} of a ``wet'' airfall deprived of supervolatiles that is responsible for the observed water--dominated comet activity prior to the inbound equinox.\\ \noindent Finally, we studied the longterm loss of ice following the near--perihelion airfall accumulation. We demonstrated that the surface temperatures are kept comparably cool on the northern hemisphere during the outbound orbital branch, which helps explain the high level of activity in the north on the inbound orbital branch when the deposits finally are heated to high temperature. We introduce the reachability and survivability indices $\mathcal{R}$ and $\mathrm{S}$ as a way to classify the potential of a region to have a high level of activity by simultaneously accumulating a high amount of wet airfall and to preserve it to the following perihelion passage. This work also serves as a guide for selecting the most promising volatile rich smooth terrain sites in the northern hemisphere for future comet nucleus sample return missions. \bigskip \bigskip \noindent {\sl ACKNOWLEDGMENTS.} Parts of the research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.\\ \noindent \emph{COPYRIGHT}. \textcopyright\,2020. All rights reserved. \bibliography{Davidsson_etal_airfall_ver04acc.bbl} \clearpage \linespread{1}

Title: The Pristine Inner Galaxy Survey (PIGS) V: a chemo-dynamical investigation of the early assembly of the Milky Way with the most metal-poor stars in the bulge

Abstract: The investigation of the metal-poor tail in the Galactic bulge provides unique information on the early Milky Way assembly and evolution. A chemo-dynamical analysis of 17 very metal-poor stars (VMP, [Fe/H] $<-2.0$) selected from the Pristine Inner Galaxy Survey was carried out based on Gemini/GRACES spectra. The chemical abundances of $\alpha-$elements (Mg, Ca, and Ti), odd-Z (Na, K, and Sc), Fe-peak (Cr and Ni), and neutron-capture (Ba) elements are determined from model atmosphere analyses. The chemistry suggests that the majority of our stars are very similar to metal-poor stars in the Galactic halo. Orbits calculated from {\it Gaia} EDR3 imply these stars are brought into the bulge during the earliest Galactic assembly. Most of our stars have large [Na,Ca/Mg] abundances, and thus show little evidence of enrichment by pair-instability supernovae. Two of our stars (P171457, P184700) have chemical abundances compatible with second-generation globular cluster stars, suggestive of the presence of ancient and now dissolved globular clusters in the inner Galaxy. One of them (P171457) is extremely metal-poor ([Fe/H] $<-3.0$) and well below the metallicity floor of globular clusters, which supports the growing evidence for the existence of lower-metallicity globular clusters in the early Universe. A third star (P180956, [Fe/H] $\sim-2$) has low [Na,Ca/Mg] and very low [Ba/Fe] for its metallicity, which are consistent with formation in a system polluted by only one or a few low-mass supernovae. Interestingly, its orbit is confined to the Galactic plane, like other very metal-poor stars found in the literature, which have been associated with the earliest building blocks of the Milky Way.

https://export.arxiv.org/pdf/2208.13791

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} Galaxy: formation - Galaxy: evolution - Galaxy: bulge - Galaxy: abundances - stars: kinematics and dynamics - stars: Population II \end{keywords} \section{Introduction} The oldest and most chemically pristine stars in the Galaxy are expected to have been enriched by only one or a few individual supernovae or hypernovae events. This means that studies of their chemical abundance patterns and orbital dynamics are invaluable for learning about the lives and deaths of the first stars, and the assembly history of the Galaxy \citep{Freeman02, Tumlinson10, Wise12, Karlsson13}. Successive generations of stars enriched the interstellar medium, while gas inflows dilute it, contributing to a complex star formation history that depends on location within the Milky Way Galaxy. In cosmological simulations, low-metallicity stars (\FeH\footnote{\FeH $= \log(\rm{N_{Fe}/N_{H}})_{\star}-\log(\rm{N_{Fe}/N_{H}})_{\odot} $, in which $\rm{N_X}$ is the number density of element X.}$\leq-2.5$) form in the first 2--3 $\Gyr$ after the Big Bang, and mostly in low-mass systems, the so-called ``building blocks", \citep{Starkenburg17a, ElBadry18, Sestito21}. These building blocks gradually merged to form the proto-Milky Way. These stars are expected to occupy the deepest parts of the gravitational potential, \ie near the bulge, while late accretion of dwarf satellites are expected to deposit metal-poor stars primarily in the halo \citep{Bullock2005, Johnston2008, Tissera12}, or even in the disc for planar accretions \citep[\eg][]{Abadi03,Sestito21,Santistevan21}. While the metal-poor stars in the Galactic bulge are important tracers of the earliest stages in the formation of the Milky Way, they are extremely difficult to find \citep[e.g.,][]{Schlaufman14}. Firstly, the region of the bulge is dominated by a metal-rich population of both young and old stars, disrupted globular clusters, and ongoing star formation \citep{Ness13a, Ness2014, Bensby13,Bensby17,Schiavon17, Schultheis19}. Secondly, the heavy and variable interstellar extinction, extreme stellar crowding, and presence of complex foreground disc stellar populations have made photometric surveys of metal-poor stars extremely challenging. The ARGOS spectroscopic survey found that fewer than 1\% (84) of the stars in their sample have $\FeH<-1.5$ \citep{Ness13b}. The Extremely Metal-poor BuLge stars with AAOmega \citep[EMBLA,][]{Howes14,Howes15,Howes16} survey selected VMP targets with a metallicity-sensitive photometric filter from the SkyMapper Southern Survey \citep{Bessell11,Wolf18} for low-resolution spectroscopy with the Anglo-Australian Telescope. EMBLA analysed with high-resolution spectroscopy 63 stars with [Fe/H] $< -2.0$, where the majority resemble chemically metal-poor stars in the Galactic halo. The only noticeable differences were a lack of carbon-rich stars, and possibly a larger scatter in [$\alpha$/Fe] abundances. A detailed kinematics analysis of their sample also raised questions about what it means to be a ``bulge star", \ie a star that formed in the bulge versus one passing through the bulge on a radial orbit. Reducing their sample to stars with apocentric distances $\le$5 kpc (36 stars), however, did not alter their conclusions \citep{Howes16}. The Pristine Inner Galaxy Survey \citep[PIGS,][]{Arentsen20a,Arentsen20b} is similar to the EMBLA survey in that metal-poor targets have been selected from the narrow-band photomery, in this case the Pristine survey \citep{Starkenburg17b}. The Pristine survey is a narrow-band imaging survey carried out at the Canada-France-Hawaii Telescope (CFHT), where the Ca\ii{} HK filter, in combination with broad band photometry, has been shown to find low-metallicity stars ([Fe/H]$<-2.5$) with $\sim$56\% efficiency in the Galactic halo \citep{Youakim17, Aguado19, Venn20}. The power of the Pristine survey has been demonstrated by the discovery of two new ultra metal-poor stars \citep[``UMP", $\FeH<-4.0$,][]{Starkenburg19,Lardo21}, or $\sim$5\% of the total known UMP stars so far \citep[see the compilation in ][]{Sestito19}. The Pristine-selected metal-poor targets in the bulge were examined with low-/medium-resolution spectroscopic observations obtained with the the AAOmega spectrograph on the Anglo Australian Telescope (AAT), from which stellar parameters, metallicities, and carbon abundances were derived for $\sim$12,000 stars. \citet{Arentsen20b} report an efficiency of $\sim$80\% in finding very metal-poor stars (``VMP"\footnote{This nomenclature, \eg VMP, EMP, UMP, has been introduced in \citet{Beers05}.}, $\FeH<-2.0$) in the bulge avoiding the most highly extincted regions. \citet{Arentsen20b} used the PIGS/AAT observations to study the kinematics of metal-poor stars in the inner Galaxy, finding that the rotation around the Galactic centre decreases with decreasing metallicity, \ie lower metallicity stars are more dispersion-dominated as the Galactic halo. To chemically examine the low-metallicity tail of the Galactic bulge also requires consideration of the contributions from disrupted globular clusters and later accretions of dwarf galaxies. Some studies have suggested that up to $\sim$25\% of the stellar mass of the inner region of the Milky Way is made of dissolved ancient globular clusters \citep[\eg][]{Shapiro10, Kruijssen15, Schiavon17}. One study \citep{Schiavon17} based this claim on the large number of nitrogen-rich stars that resemble the chemistry of second-generation stars in globular clusters \citep[\eg][]{Gratton04,Bastian18}. More recently, a few bulge stars with chemistry similar to second-generation globular cluster stars were found in the Chemical Origins of Metal-poor Bulge Stars (COMBS) survey \citep{Lucey19, Lucey21, Lucey22}. The COMBS survey is based on VLT/UVES+GIRAFFE+FLAMES spectra of red giants in the bulge, and also reported that the number of halo stars passing through the bulge (``interlopers") increases with decreasing metallicity. In this paper, we report on a chemo-dynamical investigation of 17 VMP candidates selected from the PIGS survey and observed with the high-resolution GRACES spectrograph at Gemini North. This work aims to confirm the low metallicity of these targets, investigate the orbits of these stars within the inner Galaxy, and examine their connections with the early assembly of the Milky Way through studies of their detailed chemical abundance ratios. In particular, we explore the chemical signatures of our 17 metal-poor stars ($-3.3 <$ [Fe/H]$< -2.0$) in comparison to other metal-poor systems, such as ultra-faint dwarf galaxies and globular clusters, as well as the dispersed Galactic halo. In Section~\ref{datasec}, we discuss the target selection process, the observations, and the data reduction of our spectra. Section~\ref{kinesec} describes the kinematical analysis, including estimates of the radial velocity, the heliocentric distance, and the orbital parameters using an appropriate Galactic potential. In Section~\ref{stellarparamssec}, a description of the stellar parameters (effective temperature and surface gravity) determinations and the effects of high extinction are reported. In Section~\ref{models}, the model atmospheres, radiative transfer, line lists, and metallicity determinations are discussed. In Section~\ref{chemsec}, the method used to measure the chemical abundances from the observed spectra is described, focusing on iron-group elements, $\alpha$-, odd-Z, and neutron-capture elements. The scientific implications of our chemo-dynamical analyses are described in Section~\ref{discussionsec}, particularly in the context of Galaxy formation and evolution. \section{Data}\label{datasec} \subsection{Target selection} The targets in this work were selected from the larger sample of PIGS \citep{Arentsen20a}. All the PIGS stars were selected from MegaCam photometry observed at the CFHT\footnote{The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the Native Hawaiian community. We are very fortunate to have had the opportunity to conduct observations from this mountain.}. Then, AAT/AAOmega low/medium-resolution spectra were taken and analysed \citep{Arentsen20b} with both \textsc{FERRE} \citep{Allende06} and ULySS \citep{Koleva09} yielding estimates of metallicity, effective temperature, surface gravity, radial velocity, and carbonicity. Throughout this paper, when comparing stellar parameters and metallicities to this sample, the \textsc{FERRE} output are considered and denoted as AAT (see Sections~\ref{gracesferreRV}~and~\ref{metsec}). From the largest dataset of PIGS/AAT, a sample of 20 stars were observed at higher resolution with GRACES as part of the Gemini Large and Long Program LLP$-102$ started in 2019A and ended in 2021A. We selected the targets to have $\FeH_{\mathrm{AAT}}\leq-2.5$, V $\leq15$ mag, $T_{\mathrm{eff,AAT}}\leq 5500$ K. The cut on magnitudes allows us to achieve a SNR$\sim30$ within a reasonable exposure time, see Table~\ref{tab:obs}. The selection in effective temperature permits us to observe spectral lines that are too weak at higher temperatures, \eg Ba lines. While the metallicity limit is necessary to explore the most metal-poor tail of the inner galaxy. Of the full sample we observed only less than a third of the potential targets. At the epoch of the proposal submission, the astrometric {\it Gaia} DR2 data \citep{Gaia16,Gaia18,Lindegren18} poorly constrained the orbits at the distance of the bulge, and hence no selection based on orbital properties was not done. Due to numerous unexpected events, such as the protests at Mauna Kea, the global Covid-19 pandemic, and the bad weather, we were only able to collect a sample composed of $17$ VMP bulge stars and 2 VMP standard stars, HD~122563 and HD~84937 \citep[\eg][]{Amarsi16,Sneden16,Karovicova18}. Figure~\ref{longlat} shows the PIGS footprint in Galactic coordinates, with the GRACES stars highlighted. \subsection{GRACES observations and reduction} Observations were conducted with the Gemini Remote Access to CFHT ESPaDOnS Spectrograph \citep[GRACES,][]{Chene14,Pazder14} in the 2-fibre (object+sky) mode with a resolution of R$\sim40000$. GRACES consists in a 270-m optical fibre that links the Gemini North telescope to the Canada–France–Hawaii Telescope ESPaDOnS spectrograph \citep{Donati06}, which is a cross-dispersed high resolution \'echelle spectrograph. The spectral coverage of GRACES is from 4500 \AA{} to 10000 \AA{} \citep{Chene14}. However, our spectra are dominated by noise below the spectral region of 4900-5000 \AA{}. The GRACES spectra were first reduced using the Open source Pipeline for ESPaDOnS Reduction and Analysis \citep[OPERA,][]{Martioli12} tool, which also corrects for heliocentric motion. Then the reduced spectra were reprocessed following the procedure described in \citet{Kielty21}. The latter pipeline allows us to measure the radial velocity of the observed star, to co-add multiple observations for a given target to check for possible radial velocity variations, to correct for the motion of the star, and to eventually renormalise the flux. This procedure also improves the signal-to-noise ratio in the overlapping spectral order regions. Figure~\ref{spectraex} shows the spectra of three stars (P170438, P180956, and P183229) with the very metal-poor standard star HD122563 in the Mg\ione{} b, the Ca\ione{}, and Ba\ii{} regions. The observed stars with their Pristine name, {\it Gaia} EDR3 ID, their {\it Gaia} photometry, and the log of the observations are reported in Table~\ref{tab:obs}. \begin{table*} \caption{Log of the observations. The short name, the Pristine name, the {\it Gaia} EDR3 source ID, the G and BP-RP from {\it Gaia} EDR3, the reddening from the 3D map of \citet{Green19}, the total exposure time, the number of exposures, and the SNR are reported. The SNR is measured as the ratio between the median flux and its standard deviation in two spectral regions, the 5175$-$5182 \AA{} (@Mg\ione{} b) and the 5950$-$6000 \AA{} (@6000\AA{}) ranges. P180503 is the star for which the 3D extinction map from \citet{Green19} does not provide a value, therefore we report the \textsc{StarHorse} extinction \citep{Anders19}.} \label{tab:obs} \resizebox{\textwidth}{!}{ \begin{tabular}{ccccccccc} \hline Short name & Pristine name & source ID & G & BP-RP & E(B-V) & T$_{\rm exp}$& N$_{\rm exp}$ & SNR \\ & & & (mag) & (mag) & (mag) & (s) & & @Mg\ione{} b,@6000 \AA{} \\ \hline P170438 & Pristine\_170438.40-261742.8 & 4111904146257749504 & 15.52 & 1.33 & 0.41 & 4800 & 2 & 13, 37 \\ P170610 & Pristine\_170610.81-290322.3 & 6029916002325425536 & 14.30 & 1.47 & 0.34 & 1800 & 1 & 22, 51 \\ P171457 & Pristine\_171457.25-232718.6 & 4114176871140428416 & 14.12 & 1.68 & 0.52 & 1800 & 1 & 22, 44 \\ P171458 & Pristine\_171458.77-220807.2 & 4114599427192094592 & 14.83 & 1.63 & 0.46 & 1800 & 1 & 18, 43 \\ P180118 & Pristine\_180118.30-295346.9 & 4050241536946876928 & 14.96 & 1.75 & 0.85 & 3600 & 2 & 11, 37 \\ P180503 & Pristine\_180503.40-272725.4 & 4062947566467257472 & 13.80 & 1.93 & 0.64 & 1800 & 3 & 8, 34 \\ P180956 & Pristine\_180956.78-294759.8 & 4050071013878221696 & 13.50 & 1.55 & 0.53 & 2700 & 3 & 36, 78 \\ P181306 & Pristine\_181306.64-283901.5 & 4050649838085533184 & 14.98 & 1.33 & 0.47 & 7200 & 4 & 19, 53 \\ P182129 & Pristine\_182129.69-245815.3 & 4053226917177000448 & 13.60 & 1.77 & 0.51 & 1800 & 3 & 30, 63 \\ P182221 & Pristine\_182221.12-265025.3 & 4052704649149713536 & 15.78 & 1.22 & 0.44 & 4800 & 2 & 11, 34 \\ P182244 & Pristine\_182244.51-223836.4 & 4089749772871612416 & 14.12 & 1.85 & 0.58 & 1800 & 1 & 25, 47 \\ P182505 & Pristine\_182505.97-261308.2 & 4052845008684897408 & 14.86 & 1.39 & 0.38 & 2400 & 1 & 15, 37 \\ P183229 & Pristine\_183229.69-250729.1 & 4076280038267332736 & 15.13 & 1.41 & 0.35 & 3600 & 2 & 10, 26 \\ P183335 & Pristine\_183335.04-263056.1 & 4075864732108631808 & 14.38 & 1.37 & 0.36 & 2700 & 3 & 9, 31 \\ P184338 & Pristine\_184338.08-241508.3 & 4078080041827258112 & 14.56 & 1.43 & 0.35 & 1800 & 1 & 11, 29 \\ P184700 & Pristine\_184700.56-251720.9 & 4073414191180594944 & 16.19 & 1.55 & 0.33 & 7200 & 3 & 16, 16 \\ P184855 & Pristine\_184855.88-300124.9 & 6761365842153603968 & 13.67 & 1.42 & 0.22 & 3600 & 3 & 38, 70\\ \hline \end{tabular}} \end{table*} \section{Kinematical analysis}\label{kinesec} \subsection{Radial velocity: GRACES vs. medium-resolution spectroscopy}\label{gracesferreRV} A necessary step for the orbital parameters inference is to measure the radial velocity, RV. A first measurement for the RV is performed by the reduction pipeline. The spectra are cross-correlated with well known reference metal-poor star spectra (HD 122563). The cross-correlation is made selecting H$\alpha$, H$\beta$, and the Mg\ione{} b lines. Then the corrected spectra are cross-correlated using the more precise \textsc{fxcor} routine from \textsc{IRAF}\footnote{\url{ https://github.com/iraf-community}} \citep{Tody86,Tody93}. The RV of our targets were previously determined by medium-resolution observations \citep{Arentsen20a}, and Figure~\ref{rvgracesferre} displays a comparison between our measurements and the former, which shows in general good agreement. The difference in velocity between the two sets of spectra at different resolution is always positive which might indicate a possible offset between the two instruments. The median difference in radial velocity between the two measurements is $2.13\kms$ with a dispersion of $1.28\kms$, removing the outlier star (P182221, $\sim33 \kms$). A discussion on P182221 is reported in Section~\ref{cempgc}. Due to the much higher resolution and the SNR of the GRACES spectra, the determination of RV and its uncertainty from our observations are preferred and considered for the orbital parameters inference. \subsection{Distances and spatial distribution} Another essential ingredient for the recipe of the kinematical analysis is the determination of the heliocentric distances. The improvements that the {\it Gaia} early data release 3 \citep[hereafter {\it Gaia} EDR3,][]{GaiaEDR3,Lindegren21} provided for astrometric measurements, \ie parallaxes and proper motions, play a crucial role for stars in our sample. It is now well-known that it is ill advised to simply invert the parallax for inferring the distance \citep[\eg ][]{Bailer15,Bailer18}. This is especially true when the parallax and its uncertainty are poorly constrained, \eg $\varpi\leq 0 \mas$ and/or $\sigma_{\varpi}/\varpi\geq 10$ percent. In this work, we infer the distances with a Bayesian approach similar to that of \citet{Bailer15}, and following Equations 8 to 11 of \citet{Sestito19}. Briefly, this consists in a Gaussian likelihood for the parallax distribution and a prior on the stellar density distribution that takes into account the Galactic disc and halo \citep[for more details see][]{Sestito19}. The zero point offset has been applied to the {\it Gaia} EDR3 parallaxes \citep{Lindegren21} using the python \textsc{gaiadr3\_zeropoint}\footnote{\url{https://gitlab.com/icc-ub/public/gaiadr3\_zeropoint}} package. The distance is provided with a probability distribution function (PDF or posterior), which, in some cases, is far from being a Gaussian-like distribution \citep{Bailer15}. Figure~\ref{spacedist} displays the spatial distribution in Galactic Cartesian coordinates of this PIGS sample. For each star, the median and the standard deviation of the Galactocentric coordinates PDF are shown. \subsection{Orbital parameters}\label{orbsec} The final step for the orbital inference is to feed \textsc{Galpy}\footnote{\url{http://github.com/jobovy/galpy}} \citep{Bovy15} with the inferred distances and RV, and the proper motions and coordinates from \textit{Gaia} EDR3. Since we deal here with objects that are in the inner region of the Milky Way, we need to account for the presence of a rotating bar in the Galactic gravitational potential. Therefore, the potential we use is composed by a Navarro-Frenk-White dark matter halo \citep[][\textsc{NFWPotential}]{NavarroFrenkWhite97}, a Miyamoto-Nagai potential disc \citep[][\textsc{MiyamotoNagaiPotential}]{MiyamotoNagai}, an exponentially cut-off bulge (\textsc{PowerSphericalPotentialwCutoff}), and a rotating bar potential (\textsc{DehnenBarPotential}). All of the aforementioned potentials, with the exclusion of the bar, are usually summoned by the \textsc{MWPotential14} package. However, we adopt a more massive and up-to-date halo \citep{BlandHawthorn16}, with a mass of $1.2\times10^{12}\msun$ (vs. $0.8\times10^{12}\msun$ for \textsc{MWPotential14}). The bar is invoked from the \textsc{DehnenBarPotential} package, which consists in a Dehen bar potential \citep{Dehnen00} and generalised to 3D following \citet{Monari16}. This choice of the Galactic potential, especially the settings of the rotating bar (\eg pattern speed and scale length of the bar), is in line with the recent dynamical analysis of bulge stars by the COMBS survey \citep{Lucey21}. From the distance inference the majority of the stars are placed very close to the inner region of the bulge ($<5\kpc$, see Figure~\ref{spacedist} in Appendix~\ref{apporb}), and due to the uncertainties on the distance, it is hard to discern if they are in front or just beyond the Galactic centre. Depending on their location either in front or behind, the orbital parameters might drastically change, resulting in a change of their Galactic rotation direction (retrograde vs. prograde). Therefore for each star, we create a grid of distances with a step of $0.1\kpc$ within $\pm 1\sigma$ from the maximum of the distance PDF. For each point of the grid (\ie at a fixed distance), we perform a Monte Carlo with 1000 random draws on the other parameters (\eg RV, coordinates) to infer the orbital parameters and their uncertainties. In case of the proper motion components, we consider their correlation given the coefficients from {\it Gaia} EDR3, drawing randomly with a multivariate Gaussian function. The RV and coordinates are treated as a Gaussian. The integration time is set to 1 Gyr. Figure~\ref{kinefig} shows the main median orbital parameters inferred from \textsc{Galpy}\footnote{Note that \textsc{Galpy} cannot infer the energy, the angular momentum, and the action variables for the potential we have adopted, given the presence of a rotating bar.} and considered for this analysis, namely the maximum height from the plane $Z_{\rm max}$, the apocentre $r_{\rm apo}$, the pericentre $r_{\rm peri}$, and the eccentricity $\epsilon$. The variation of the orbital parameters as a function of the distance grid steps is shown in Figure~\ref{kinefig_grid}. The sample is catalogued into 4 groups according to their median $Z_{\rm max}$ and their median $r_{\rm apo}$. In this space, the variation of the orbital parameters does not strongly impact the classification of the stars. When a star could be classified as belonging to more than one group, the final choice mirrors the classification according to the maximum of the distance PDF. We want to emphasise that at this stage of the narrative, we are not discriminating between halo, bulge, or disc stars. This will be discussed in Section~\ref{discussionsec}. The 4 categories are described as follow: \begin{itemize} \item \textbf{Bulge group}. The stars in this group do not venture outside a sphere of 3.5 $\kpc$ from the Galactic centre, \ie $Z_{\rm max}\leq3.5 \kpc$ and $r_{\rm apo}\leq3.5 \kpc$. In this category there are 3 stars, P171458, P180118 and P180503. The targets within this group are marked with a blue star marker in the Figures of this work. \item \textbf{Confined group}. These stars have $Z_{\rm max}\leq3.5 \kpc$ and $r_{\rm apo}>3.5 \kpc$. They are confined close to the Milky Way disc and this group is composed by 9 objects, P170438, P170610, P180956, P181306, P182244, P182505, P183229, P183335 and P184338. \item \textbf{Inner halo group}. This group is composed by 4 stars with $Z_{\rm max}>3.5 \kpc$ and $r_{\rm apo}\leq20 \kpc$. P171457, P182129, P182221, and P184700 belong to this group. \item \textbf{Outer halo group}. Only P184855 is catalogued in this group and it has a $Z_{\rm max}>3.5 \kpc$ and $r_{\rm apo}>20 \kpc$. \end{itemize} As it appears from Figure~\ref{kinefig}, all the stars, independently of their group, display a high eccentricity $\epsilon >0.55$ and their pericentre is located in the inner Galactic region $r_{\rm peri}<2 \kpc$. To be noted, stars in the Confined group have not necessarily a planar orbit. This is because for the majority of them the $Z_{\rm max}$ to $r_{\rm apo}$ ratio is not small ($0.25<Z_{\rm max}/r_{\rm apo}<0.85$). Two of them, P170610 and P180956, have a very small ratio (\ie $Z_{\rm max}/r_{\rm apo}<0.15$), which indicates their planar orbit. Table~\ref{tablekine} contains all the kinematical parameters used in this work. \begin{table*} \caption{Orbital parameters for the stars in this sample. The heliocentric distance, the maximum height from the MW plane, the apocentre and pericentre distances, the eccentricity, the Galactic cartesian coordinates (X, Y, Z), and the dynamical group classification are reported for each star denoted by their short name. For the dynamical group classification, B = bulge, C = confined, I = inner halo, and O = outer halo.} \label{tablekine} \resizebox{\textwidth}{!}{ \begin{tabular}{cccccccccccccccccc} \hline name short & D & $\sigma_{\rm D}$ & $Z_{\rm max}$ & $\sigma_{Z_{\rm max}}$ & $r_{\rm apo}$ & $\sigma_{r_{\rm apo}}$ & $r_{\rm peri}$ & $\sigma_{r_{\rm peri}}$ & $\epsilon$ & $\sigma_{\epsilon}$ & X & $\sigma_X$ & Y & $\sigma_Y$ & Z & $\sigma_Z$ & Group \\ & $(\kpc)$ &$(\kpc)$ &$(\kpc)$&$(\kpc)$&$(\kpc)$&$(\kpc)$&$(\kpc)$&$(\kpc)$& & &$(\kpc)$&$(\kpc)$&$(\kpc)$&$(\kpc)$&$(\kpc)$&$(\kpc)$ & \\ \hline P170438 & 5.40 & 1.74 & 1.66 & 0.56 & 6.77 & 1.05 & 1.57 & 0.21 & 0.64 & 0.04 & 2.65 & 1.63 & -0.27 & 0.08 & 0.86 & 0.25 & C\\ P170610 & 4.09 & 0.48 & 0.58 & 0.01 & 5.38 & 0.49 & 1.43 & 0.11 & 0.58 & 0.01 & 3.97 & 0.48 & -0.35 & 0.04 & 0.52 & 0.06 & C \\ P171457 & 7.38 & 1.07 & 4.34 & 0.31 & 9.67 & 0.55 & 0.75 & 0.15 & 0.85 & 0.02 & 0.72 & 1.01 & 0.11 & 0.01 & 1.12 & 0.15 & I \\ P171458 & 7.59 & 1.38 & 1.79 & 0.17 & 3.19 & 0.30 & 0.49 & 0.05 & 0.74 & 0.03 & 0.65 & 1.38 & 0.25 & 0.05 & 1.23 & 0.23 & B \\ P180118 & 8.24 & 0.37 & 0.94 & 0.03 & 1.72 & 0.22 & 0.11 & 0.01 & 0.85 & 0.01 & -0.20 & 0.38 & 0.13 & 0.01 & -0.49 & 0.02 & B \\ P180503 & 8.22 & 0.47 & 1.49 & 0.12 & 1.83 & 0.28 & 0.31 & 0.07 & 0.71 & 0.09 & -0.18 & 0.45 & 0.49 & 0.03 & -0.42 & 0.03 & B \\ P180956 & 3.30 & 0.27 & 1.81 & 0.04 & 12.91 & 0.19 & 0.69 & 0.08 & 0.90 & 0.01 & 4.71 & 0.27 & 0.11 & 0.01 & -0.27 & 0.02 & C \\ P181306 & 5.24 & 1.22 & 2.87 & 0.08 & 4.20 & 0.19 & 0.28 & 0.06 & 0.87 & 0.03 & 2.84 & 1.18 & 0.29 & 0.07 & -0.45 & 0.11 & C \\ P182129 & 6.12 & 0.98 & 5.73 & 0.41 & 8.20 & 0.16 & 0.86 & 0.14 & 0.81 & 0.03 & 1.91 & 0.96 & 0.79 & 0.12 & -0.53 & 0.09 & I \\ P182221 & 7.94 & 1.43 & 4.07 & 1.09 & 14.98 & 3.09 & 0.99 & 0.16 & 0.87 & 0.03 & 0.06 & 1.39 & 0.81 & 0.14 & -0.84 & 0.15 & I \\ P182244 & 7.99 & 0.92 & 3.03 & 0.12 & 4.17 & 0.18 & 0.64 & 0.17 & 0.73 & 0.06 & 0.08 & 0.95 & 1.34 & 0.16 & -0.59 & 0.07 & C \\ P182505 & 4.80 & 0.86 & 2.65 & 0.03 & 5.32 & 0.51 & 1.20 & 0.36 & 0.65 & 0.05 & 3.31 & 0.86 & 0.55 & 0.10 & -0.51 & 0.10 & C \\ P183229 & 4.23 & 0.61 & 1.23 & 0.12 & 4.52 & 0.22 & 0.65 & 0.13 & 0.75 & 0.03 & 3.86 & 0.59 & 0.61 & 0.09 & -0.52 & 0.08 & C \\ P183335 & 6.82 & 1.13 & 2.72 & 0.17 & 4.23 & 0.41 & 0.18 & 0.02 & 0.88 & 0.01 & 1.32 & 1.16 & 0.85 & 0.15 & -0.96 & 0.17 & C \\ P184338 & 8.00 & 1.07 & 2.61 & 0.19 & 6.28 & 0.12 & 0.98 & 0.05 & 0.74 & 0.01 & 0.19 & 1.04 & 1.42 & 0.19 & -1.28 & 0.17 & C \\ P184700 & 8.25 & 1.39 & 5.60 & 0.16 & 10.88 & 0.25 & 1.03 & 0.27 & 0.82 & 0.04 & -0.04 & 1.40 & 1.37 & 0.24 & -1.49 & 0.26 & I \\ P184855 & 5.71 & 0.67 & 10.62 & 0.89 & 37.06 & 2.15 & 0.22 & 0.03 & 0.99 & 0.00 & 2.46 & 0.63 & 0.53 & 0.06 & -1.25 & 0.14 & O \\ \hline \end{tabular}} \end{table*} \subsection{The kinematical sample from the literature} We compare the orbital parameters of the PIGS/GRACES sample with two datasets from the literature. The first is a compilation made of 36 stars from the EMBLA survey \citep{Howes14,Howes15,Howes16}. Since the orbital parameters in that sample were inferred before {\it Gaia} DR2, we re-calculate their orbit using the most up-to-date {\it Gaia} EDR3 astrometric solutions, the method describe in this Section, and the RV from the \citet{Howes15,Howes16}. In the case of stars from \citet{Howes16}, the RV has been inferred from the Galactocentric velocity inverting their Equation~4. The second dataset is from the recent work of \citet{Lucey22}, which is part of the COMBS survey \citep{Lucey19}. The complete sample is composed of 319 stars, however we restrict the kinematical comparison to the 27 stars with \FeH$<-1.7$. Both datasets are marked with blue circles in Figure~\ref{kinefig}. All the panels in Figure~\ref{kinefig} display that the distribution of the stars in our sample kinematically matches the literature's compilation. All the displayed VMPs towards the inner region of the MW have high eccentricity ($\epsilon>0.5$), small pericentre ($\lesssim 2\kpc$), and a combination of $Z_{\rm max}$ and $r_{\rm apo}$ that makes them inhabit various regions of the Galaxy. \section{The stellar parameters and the effects of the high extinction}\label{stellarparamssec} The effective temperature is measured using the \citet[][hereafter MBM21]{Mucciarelli21} colour-temperature relation which combines the InfraRed flux method from \citet{Gonzalez09} with the photometry from {\it Gaia} EDR3. The input parameters for this inference are the {\it Gaia} EDR3 (BP-RP) colour, the reddening in this colour, a metallicity estimate, and knowledge of whether a star is in the dwarf or giant phase. The 3D extinction map from \citet{Green19} was used to correct the photometry for extinction\footnote{For P180503, the \citet{Green19} 3D extinction map does not provide a value, therefore the \textsc{StarHorse} extinction \citep{Anders19} was adopted for this star.}. This map provides the reddening E(B-V) that has to be converted to {\it Gaia} filters. Then, the {\it Gaia} extinction coefficients were derived using $\rm A_V/E(B-V)= 3.1$ \citep{Schultz75} and the $\rm A_G/A_V = 0.85926$, $\rm A_{BP} /A_V = 1.06794$, $\rm A_{RP} /A_V = 0.65199$ relations \citep{Marigo08,Evans18}. As input metallicities, we adopt the values from the AAT analysis \citep{Arentsen20a}. Since the MBM21 relation needs the knowledge on the nature of the star (\ie dwarf or giant), we infer a first effective temperature that is the average of both the dwarf and giant solutions. With this first guess, a first estimate on the surface gravity using the Stefan-Boltzmann law\footnote{$L_{\star} = 4\pi R_{\star}^2 \sigma T_{\star}^4$; the radius of the star can be calculated from this equation, then the surface gravity is inferred assuming the mass.} is derived. This latter step requires as input the previously inferred effective temperature, the distance of the object, the {\it Gaia} EDR3 G photometry, the extinction, and the bolometric corrections on the flux \citep{Andrae18}. Then, with this first estimate of the surface gravity, we iterate the process to find the effective temperature and subsequently a new inference on the surface gravity. These steps have been iterated 1000 times in order to converge to a final estimate of surface gravity and effective temperature. The final values do not depend on the initial inference, especially on the dwarf-giant average made in the first step. For each step, we perform a Monte Carlo on all the input parameters to estimate the uncertainties on the effective temperature and surface gravity. The input parameters are randomised within $1\sigma$ using a Gaussian distribution, except for the stellar mass and the extinction. The stellar mass is treated with a flat prior from 0.5 to 0.8 $\msun$, which is consistent with the mass of very metal-poor stars. The extinction is also described with a flat prior with a width of $30$ percent from the assumed value from \citet{Green19}. The mean uncertainties on the effective temperature is $\sim113$ K, while on the surface gravity it is $\sim0.14$ dex. This method has been shown to provide reliable stellar parameters that are in agreement, within the uncertainties, with values obtained with spectroscopic methods \citep[\eg][]{Kielty21,Lardo21}. Figure~\ref{kielfig} shows the Kiel Diagram of the inferred stellar parameters colour-coded by $\FeH_{\rm GRACES}$ with a VMP MESA/MIST\footnote{\url{https://waps.cfa.harvard.edu/MIST/}} \citep{Dotter16,Choi16} isochrone as a reference. All the stars appear to be giants. The inferred stellar parameters are reported in Table~\ref{tab:stellarparams}. The photons of these targets travelled across multiple clouds of interstellar medium (ISM) before they got collected by the CCD of the telescope. Since the determination of the effective temperature and surface gravity highly depends on the de-reddened photometry, we investigate how much a different set of extinction coefficients can affect the final values of $\rm A_{G}$, $\rm A_{BP}$, and $\rm A_{RP}$. For this purpose, the extinction coefficients from \citet[][]{Casagrande21} have been tested. Using GALAH data \citep{Buder21}, they calibrate the extinction coefficients for {\it Gaia} filters as a function of the colour, BP$-$RP, \citep[see Figure~1 and Appendix~B in ][]{Casagrande21}. Therefore, $\rm A_{G}$, $\rm A_{BP}- A_{RP}$ have been computed with the new relation and tested against the ones adopted in this work. Considering a 30 percent relative uncertainty on the E(B-V) from the \citet{Green19} 3D dust map towards the bulge region, we find that the $\rm A_{G}$ from \citet{Casagrande21} and the one adopted in this work differs by less than 0.6$\sigma$. The difference between the two methods for $\rm A_{BP}- A_{RP}$ is less than 0.15$\sigma$. Therefore, the two methods agree within the uncertainties. Nonetheless, the derived stellar parameters used for this work have been checked against the excitation potential as described in Section~\ref{chemsec} (see Figure~\ref{fitfig}) before being adopted as the final values. We stressed that, since the {\it Gaia} filters are very broad, the extinction coefficient of these filters depends on the effective temperature of a star, and is not a fixed number. This effect would be very important for photometric calibrations, while is negligible for spectroscopic analyses, as this work. \section{Model Atmospheres Analysis} \label{models} \subsection{Model atmospheres} The first step to measure the chemical abundances in the stellar spectra is to have an ad hoc model atmosphere. The most up-to-date \textsc{MARCS}\footnote{\url{https://marcs.astro.uu.se}} models \citep{Gustafsson08,Plez12} are generated. In particular, for stars with log(g)$<3.5$, \textsc{OSMARCS} spherical models are used. \subsection{The lines list and the atomic data} A lines list very similar to the one adopted by \citet{Kielty21} is employed for this work. It contains Fe lines from \cite{Norris17} and \cite{Monty20}, while the other species are generated with \textsc{linemake}\footnote{\url{https://github.com/vmplacco/linemake}} \citep{Placco21}. In addition, K\ione{} lines are from the National Institute of Standards and Technology \citep[NIST,][]{NIST_ASD}\footnote{NIST database at \url{https://physics.nist.gov/asd}}. \subsection{Lines measurements} Some of the observed spectra suffer from poor flux normalization due to the suboptimal signal-to-noise ratio across the wavelength range. This translates into a failure of automatic line fits and procedures to measure chemical abundances. To obviate to this problem, the automatic procedure described in \citet{Kielty21} is run to identify the spectral lines and to create a common line list between the stars. Then, their equivalent widths (EW) are measured with \textsc{IRAF} \citep{Tody86,Tody93} using the \textsc{splot} routine. Multiple profiles (Gaussian, Voigt, integral of the flux) have been adopted to measure the EW, then the median is taken as final value. We discard strong lines (EW $>140$ m\AA, where differences in our measurement methods exceeded $\sim$15\%) and very weak lines at the level of the noise (EW $<20-25$ m\AA), depending on the location in our spectra. Then, the \textsc{autoMOOG} code is used to infer the chemical abundances from the input EW and atmosphere models. This code is an automated version of the more popular \textsc{MOOG}\footnote{\url{https://www.as.utexas.edu/~chris/moog.html}} code \citep{Sneden73,Sobeck11}. A table containing the EW measurements is provided as a machine readable table in the Supplementary materials. \subsection{Metallicity: GRACES vs. AAT}\label{metsec} Given the SNR of the observed spectra, the A(Fe\ione{}) is measured using from 8 to 63 lines in the $[4871,6678]$ \AA{} spectral range, while A(Fe\ii{}) only from 2 lines ($\lambda\lambda 4923.922, 5018.435$ \AA). The final [Fe/H] is calculated as the mean of A(Fe\ione{}) and A(Fe\ii{}), weighted by the number of lines and then scaled by the solar Fe content. Solar abundances are from \citet{Asplund09}. Figure~\ref{gracesferre} shows the comparison between the LTE $\FeH$ inferred in this work vs. the ones previously determined with FERRE at low/medium-resolution from AAT \citep{Arentsen20a}. The markers are colour-coded by the SNR measured at the Mg\ione{} b region (see also Table~\ref{tab:obs}). All the stars have low/medium $\FeH_{\rm AAT}\leq-2.5$, while only 11 have also $\FeH_{\rm GRACES}\leq-2.5$ from high-resolution. The remaining 6 stars are still VMP. Our method has also been calibrated on two VMP standard stars (HD122563 and HD84937), reproducing the literature values from the stellar parameters to the metallicity. Therefore, the deviation of the two measurements is thought to originate from a difference in the spectral resolution, and the high-resolution results are preferred in this work. Table~\ref{tab:stellarparams} reports the RV of the targets, the inferred stellar parameters (T$_{\rm eff}$, logg, $\xi$), and the \FeH{} from LTE analysis. \begin{table*} \caption{The radial velocities, the effective temperature, surface gravity, microturbulence velocities, and LTE metallicities are reported with their uncertainties.} \label{tab:stellarparams} \begin{tabular}{ccccccccccc} \hline name short & RV & $\sigma_{\rm RV}$ & T$_{\rm eff}$ & $\sigma_{\rm T_{eff}}$& logg & $\sigma_{\rm logg}$ & $\xi$ & $\sigma_{\xi}$ & \FeH{} & $\sigma_{\rm [Fe/H]}$ \\ & $(\kms)$ & $(\kms)$& (K) & (K) & & &$(\kms)$ &$(\kms)$ & & \\ \hline P170438 & -166.93 & 1.00 & 5566 & 132 & 2.49 & 0.25 & 2.33 & 0.10 & -2.35 & 0.04 \\ P170610 & -6.60 & 1.31 & 4994& 102 & 2.06 & 0.12 & 2.24 & 0.10 & -2.71 & 0.03 \\ P171457 & 265.99 & 0.59 & 5040& 98 & 1.24 & 0.13 & 2.29 & 0.10 & -3.26 & 0.05 \\ P171458 & 95.36 & 0.45 & 4990& 101 & 1.57 & 0.14 & 2.27 & 0.10 & -2.9 & 0.04 \\ P180118 & 86.84 & 0.84 & 5978& 154 & 1.45 & 0.09 & 2.61 & 0.10 & -2.18 & 0.04 \\ P180503 & -90.12 & 0.47 & 4895& 97 & 1.13 & 0.05 & 2.18 & 0.10 & -3.26 & 0.13 \\ P180956 & 262.45 & 0.54 & 5391& 133 & 1.87 & 0.10 & 2.50 & 0.10 & -2.03 & 0.02 \\ P181306 & 66.65 & 1.12 & 5752& 129 & 2.28 & 0.19 & 2.30 & 0.10 & -2.48 & 0.06 \\ P182129 & -154.23 & 0.56 & 4868& 107 & 1.11 & 0.13 & 2.44 & 0.10 & -3.11 & 0.02 \\ P182221 & -33.79 & 2.48 & 5962& 148 & 2.32 & 0.16 & 2.42 & 0.10 & -2.61 & 0.06 \\ P182244 & -225.93 & 1.58 & 4886& 92 & 1.02 & 0.12 & 2.48 & 0.10 & -2.97 & 0.04 \\ P182505 & 26.10 & 2.97 & 5287& 135 & 2.26 & 0.16 & 1.91 & 0.10 & -2.79 & 0.04 \\ P183229 & 1.41 & 1.02 & 5160& 119 & 2.48 & 0.12 & 2.14 & 0.10 & -2.56 & 0.03 \\ P183335 & 50.41 & 1.85 & 5304& 113 & 1.74 & 0.13 & 2.52 & 0.10 & -2.23 & 0.07 \\ P184338 & 218.38 & 0.35 & 5121& 103 & 1.56 & 0.12 & 2.63 & 0.10 & -2.39 & 0.04 \\ P184700 & -368.27 & 1.17 & 4842& 100 & 2.17 & 0.14 & 1.94 & 0.10 & -2.65 & 0.06 \\ P184855 & -458.3 & 0.78 & 4811& 95 & 1.51 & 0.12 & 2.41 & 0.10 & -2.85 & 0.03 \\ \hline \end{tabular} \end{table*} \subsection{Checking the stellar parameters} With high-resolution spectroscopy, it is possible to test if the input stellar parameters, \ie effective temperature, surface gravity and microturbulence velocity, are correct. Wrong estimates of the microturbulence velocity will produce a slope in the A(Fe\ione) vs. reduced EW, log(EW/$\lambda$), relation. We adopted the \citet{Mashonkina17} and \citet{Sitnova19} relations as starting values for the microturbulence velocities and then refined to flatten the slope of the A(Fe\ione) $-$ log(EW/$\lambda$) curve. In a similar way, a wrong estimate of the effective temperature would produce a slope in the A(Fe\ione) $-$ Excitation potential (EP) space. The slope from the linear fit has an absolute value $< 0.1$ dex eV$^{-1}$ using the effective temperatures from MBM21. These values are smaller than the dispersion in the measurements of the chemical abundances, therefore there is no need to further tune the effective temperatures. Figure~\ref{fitfig} summarises the aforementioned relations for P182244 using the output from \textsc{autoMOOG}. Historically, the Fe\ione{} $-$ Fe\ii{} ionisation balance has been widely used as a sanity check on the surface gravity. However, the recent work from \citet{Karovicova20} showed that tuning the surface gravity to achieve the balance between Fe\ione{} and Fe\ii{} can lead to wrong chemical abundances and gravity estimations. They used extremely precise interferometric observations of metal-poor stars, producing the most precise and accurate stellar parameters for a set of metal-poor benchmark stars. \citet{Karovicova20} found that the deviation from the Fe\ione{} $-$ Fe\ii{} ionisation balance can reach up to $\sim0.8$ dex. This effect is very important when dealing with (very) metal-poor cold giants \citep[see Figure~7 of][\eg \FeH$<-2.0$, log(g)$<3$, and T$_{\rm eff}\sim5500$ K]{Karovicova20}. This is exactly the same range in stellar parameters of the PIGS/GRACES stars. For this reason, we refrain from tuning the surface gravity to reach this balance. Appendix~\ref{appdr3} and Figure~\ref{gaiadr3_params} discuss the comparison between our set of stellar parameters and the one released from the most recent {\it Gaia} DR3 \citep[][]{Andrae22,Gaia16}. \subsection{Uncertainties on the chemical abundances} \textsc{AutoMOOG} provides estimates of the chemical abundances A(X) and their uncertainties $\sigma_{\rm A(X)}$. The abundance uncertainties are calculated by adding the line-to-line scatter ($\sigma_{\rm EW}$) in quadrature with the uncertainties imposed by the stellar parameter uncertainties ($\sigma_{\rm T_{eff}}$, $\sigma_{\rm logg}$, $\sigma_{\rm \FeH}$, see Table~\ref{tab:stellarparams}). The final uncertainty on element X is given by $\delta_{\rm A(X)}=\sigma_{\rm A(X)}/\sqrt{{\rm N_X}}$ if the number of the measured spectral lines is ${\rm N_X}>5$, or $\delta_{\rm A(X)}=\sigma_{\rm A(Fe\ione)}/\sqrt{{\rm N_X}}$. The uncertainties on the chemical abundances are provided as a machine readable table in the Supplementary material. \section{Chemical abundances analysis}\label{chemsec} The spectral coverage of the GRACES spectrograph enables estimates of the abundances of Fe-peak (Fe, Cr, Ni), $\alpha-$ (Mg, Ca, Ti), odd-Z (Na, K, Sc), and neutron-capture process (Ba) elements. In addition, some intrinsically weak lines (\eg O\ione{} 7770 \AA{} and Eu\ii{} 6645 \AA) are in the spectral region covered by our GRACES spectra; however, these lines are detectable only if the stars are highly enhanced in the element \citep[][]{Kielty21}. Chemical abundances in both LTE and NLTE analysis are provided as a machine-readable table in the Supplementary material. \subsection{The Swan band and the Carbon from low-resolution spectroscopy}\label{cempsec} According to the low/medium resolution PIGS campaign \citep[][]{Arentsen21}, P183229 and P184700 are C-enhanced with [C/Fe]$=2.10\pm 0.22$ and [C/Fe]$=2.85\pm 3.12$, respectively. In general, Carbon is visible in the wavelength range covered by GRACES spectra through the Swan band ($\lambda5100-5200$\AA) but only when the star is C-enhanced and relatively cold \citep[with the exception for SDSS J081554.26$+$472947.5][]{Ganzalez20}. This is the case for P183229 and P184700, for which the Swan band is very pronounced. The large uncertainty on the [C/Fe] for P184700 might originate from a combination of low SNR and such extreme carbon-enhancement that the models don't fit well. The left panel of Figure~\ref{spectraex} displays the Swan band for P183229 in comparison to other Carbon-normal stars. We thus qualitatively confirm the C-enhanced nature of these two stars. A third star, P182221 has [C/Fe] $ =0.64 \pm0.73$ from PIGS/AAT. This value is slightly below the CEMP threshold ([C/Fe]$=0.7$), although its high uncertainty. The GRACES spectrum does not display an evident Swan band as for the other two CEMP stars. \subsection{$\alpha$-elements} $\alpha$-elements are primarily formed in massive stars before being ejected by core-collapse supernovae and during the supernovae event \citep[\eg][]{Timmes95,Kobayashi20}. There are only three $\alpha$-elements which produce lines in the GRACES spectra, Mg, Ca and Ti. The A(Mg\ione{}) is from two lines of the Mg\ione{} Triplet ($\lambda\lambda 5172.684, 5183.604$\AA), the weaker $5528.405$\AA{} line, and the $4702.991$ \AA{} line for which the SNR is high. The A(Ca\ione{}) is inferred from up to 10 spectral lines, from 5588 \AA{} to 6500 \AA. The Ca Triplet has been excluded since it shows strong lines ($>140$ m\AA). In these GRACES spectra, up to 4 and 6 lines of Ti\ione{} and Ti\ii{} are present \citep{Lawler13,Wood13}, respectively. The three left panels of Figure~\ref{chemfig} display the [Mg,Ca,Ti/Fe] ratios as a function of the \FeH{}, corrected for NLTE effects (see Section~\ref{nltesec}). When present both Ti\ione{} and Ti\ii{} lines, the [Ti/Fe] is the average of [Ti\ione{}/Fe] and [Ti\ii{}/Fe]. \subsection{Odd-Z elements} Odd-Z elements are tracers of core-collapse supernovae. In particular, the difference in energy between the neutron capture and the $\alpha-$ particle capture produce the so called odd-even effect in the chemical yields \citep[\eg][]{Heger10,Takahashi18}. Three odd-Z elements are observable in the spectra, Na, K and Sc. The Na abundance is measurable from the Na\ione{} Doublet ($\lambda\lambda 5889.951,5895.924$ \AA). The ISM Na\ione{} D lines are present with multiple components and might have formed from clouds at a similar RV of our targets. Therefore, the ISM and stellar component could be blended. In 5 stars analysed out of 17, it is not possible to measure Na\ione{} D EW due to the blending. The two panels of Figure~\ref{Nafig} show two cases in which the ISM Na\ione{} D is blended (upper panel for P183335) and not blended (bottom panel for P182129). K\ione{} is observable with two lines at $\lambda\lambda7664.899, 7698.965$ \AA{} \citep{Falke06,Trubko17}. These lines are very close to water vapour lines of the Earth's atmosphere. A(K\ione{}) is measurable only when at least one line is not blended with the atmospheric lines and when the SNR is sufficiently high. A(K\ione{}) is measurable in 1 star from the $\lambda\lambda7664.899$\AA{} line and from $\lambda\lambda7698.965$ \AA{} in 6 other stars. Only in one star are both K\ione{} visible and yield the same estimate of [K/Fe]. Therefore, A(K\ione{}) is measured in 8 stars out of 17. Sc is present with one Sc\ii{} line at $\lambda\lambda5526.785$ \AA{} \citep{Lawler19}. For 10 stars, the SNR in the Sc\ii{} region is very low; therefore, it is not possible to fit the line and measure A(Sc\ii). The abundances of K and Sc have been measured with the \textsc{synth} configuration in \textsc{MOOG} and taking the hyperfine structure into account for Sc. Figure~\ref{chemfig} shows only [Na/Fe] (NLTE corrected) ratio among the odd-Z elements, since Sc and K are measurable only for a few stars. \subsection{Fe-peak elements} Fe-peak elements are important tracers of stellar evolution. In the early Universe, when the very metal-poor stars were forming, Fe-peak elements were produced primarily in core collapse supernovae \citep[\eg][]{Tolstoy09,Heger10}. While at much higher metallicities, hence later in cosmic time, Fe-peak elements were formed in supernovae type Ia \citep{Nomoto13}. The Fe-peak elements that are observable in the GRACES spectra are Fe (see Section~\ref{metsec}), Cr and Ni. A(Cr\ione) is measured with up to 5 spectral lines \citep[$\lambda\lambda 5206.023, 5208.409, 5345.796, 5348.314, 5409.783 $\AA,][]{Sobeck07}, while Ni\ione{} is present with up to 4 lines \citep[$\lambda\lambda 5115.389, 5476.904, 5754.656, 6482.796 $ \AA,][]{Wood14}. Figure~\ref{chemfig} shows [Cr/Fe] (NLTE) and [Ni/Fe] (LTE) as a function of \FeH{} for the stars in this sample. \subsection{Neutron-capture process elements} Neutron-capture elements can be formed through two main channels, the rapid and the slow neutron captures. Rapid-process elements are formed if their nuclear production timescale is much shorter than the time needed by the $\beta^{-}$ decay. This is the case for core collapse supernovae and neutron-star mergers. Otherwise, if the timescale for their synthesis is longer as in the stellar atmospheres of AGB stars, then these elements are named slow-process. The neutron-capture process elements present are Ba, with up to three Ba\ii{} lines ($\lambda\lambda 5853.69, 6141.73, 6496.91 $ \AA, e.g., see Fig.~\ref{spectraex}), and Y, with only one Y\ii{} line \citep[$\lambda\lambda 5200.413$ \AA,][]{Hannaford82,Biemont11}. To infer the A(Ba\ii{}), \textsc{MOOG} has been run with the synthetic configuration to take the hyperfine structure and corrections into account. The Y\ii{} line is measurable in only 6 stars. [Ba/H] (LTE) as a function of \FeH{} is shown in Figure~\ref{bafig}, while [Ba/Fe] is displayed in Figure~\ref{badwarf} and discussed in Section~\ref{planarsec}. The GRACES spectra cover only one very weak Eu\ii{} line at $\lambda\lambda 6645.11$\AA{}. This line is visible only when a star is Eu-rich \citep[see Figure~8 in][]{Kielty21}, which is not the case for any of our stars. Upper limits to A(Eu\ii) do not constrain if these stars were polluted by r- or s-processes. Figure~\ref{bafig} shows that two stars, P183229 and P184700, are strongly Ba-rich. This enhancement is explained by their CEMP nature as discussed in Sections~\ref{cempsec}~and~\ref{cempgc}. P182221, seems to be slightly enriched in Ba too ([Ba/H]$\sim-2$ at $\FeH\sim-2.6$). As for the other two Ba-rich stars, the Eu line is not detectable, therefore we can exclude that this star is r-process enhanced. The likely CEMP nature of this star is discussed in Section~\ref{cempgc}. \subsection{NLTE corrections}\label{nltesec} The elemental abundances in the atmospheres of very metal-poor stars are affected by departures from Local Thermodynamic Equilibrium (LTE). Thus, the statistical equilibrium solutions need to correct for radiative effects (non-LTE, or NLTE effects), which can be large for some species. To correct for NLTE effects in Fe\ione{} and Fe\ii{} \citep{Bergemann2012}, Mg\ione{} \citep{Bergemann2017}, Ca\ione{} \citep{Mashonkina17}, Ti\ione{} and Ti\ii{} \citep{Bergemann2011}, and Cr\ione{} \citep{Bergemann2010b} the MPIA webtool database\footnote{\url{http://nlte.mpia.de}} has been used. On the other hand, \textsc{INSPECT}\footnote{\url{http://inspect-stars.com}} has been adopted to correct for NLTE effect in Na\ione{} \citep{Lind2012}. In this sample, NLTE corrections are on order of $0.1-0.3$ dex for lines of Fe\ione{} and $<0.1$ dex for lines of Fe\ii{}. Similarly small NLTE corrections are found for lines of Mg\ione{} ($<0.1$ dex) and Ti\ione{} ($<0.05$ dex). Larger corrections were found for lines of Ca\ione{} ($\sim0.25$), Ti\ii{} ($\sim0.5$ dex), Na\ione{} ($0.2-0.4$ dex for the Na\ione{} D resonance lines), and Cr\ione{} ($\sim0.5$ dex). NLTE chemical abundances are provided as a machine-readable table in the Supplementary material. \section{Discussion} \label{discussionsec} For this discussion, a literature selection of stars in the Milky Way halo and bulge, globular clusters, and dwarf galaxies was put together to compare with the chemical results from this work. Metal-poor MW halo and bulge stars have been selected with $-4<$\FeH$<-1.7$ to match the PIGS/GRACES sample. The compilation includes the high-resolution spectral analysis results from \citet{Aoki13,Yong13}, and particularly stars in \citet{Kielty21}, which have been observed with GRACES at SNR comparable to targets in this work, and analysed with a similar methodology (line lists, stellar parameters, model atmospheres). The bulge compilation is from the high-resolution optical spectral analyses by \citet{Howes14,Howes15,Howes16,Koch16,Reggiani20,Lucey22}. A compilation of stars with high resolution spectral analyses in globular clusters was collected from \citet{Pancino17,Pritzl05,Larsen22,Martin22}, and for stars in a selection of dwarf galaxies, including Hercules \citep{Koch08,Koch13,Francois16} and Segue 1 \citep{Frebel14}. \subsection{The Inner Galaxy}\label{nonpeculiar} According to various cosmological simulations \citep[\eg][]{Salvadori10,Tumlinson10,ElBadry18,Starkenburg17a,Sestito21}, during the first $2-3$\Gyr{} many low-mass systems ($\sim10^8\msun$) merge somewhat chaotically to form the proto-Galaxy. Each of these building blocks brought in the oldest and most pristine stars, the ISM, and the dark matter. During this phase, because the mass ratio of the merging clumps was not extreme, stars from different building blocks were able to populate may regions of the Galaxy, including its very central regions. As the Galaxy grew more massive, later accretions were able to provide their stars mainly to the outer halo and possibly to the disc. Therefore, the inner galaxy should include some of the oldest stars accreted during the early assembly, while the halo would be a mixture of stars brought in across the early and later accretion history. In addition, a VMP star ($\FeH\leq-2.0$) that forms in an evolved dwarf galaxy and is accreted at a later time, will show a different chemical signature than a \textit{normal} halo star. This has been shown for stars in the halo, \eg r-process-weak stars \citep[\eg][]{Kielty21,Lucchesi22}, r-process-enhanced stars \citep[\eg][]{Hansen18}, or stars with low [Ca/Mg] ratio \citep[\eg][]{Sitnova19, Venn20}. In combination with the kinematics, this is the basis for the discovery of accreted structures in the Galaxy, \eg the Gaia-Enceladus/Sausage system \citep[\eg][]{Belokurov18,Helmi18}, Sequoia \citep[\eg][]{Myeong19, Monty20}, the Inner Galactic Structure \citep{Horta21}, and Thamnos \citep{Koppelman19}. In particular, the [$\alpha$/Fe], Fe-peak, and the odd-Z element abundance ratios can differ from those of the metal-poor MW halo stars. Thus, the chemistry of stars in the inner Galaxy can be used to test scenarios for the formation and evolution of the Milky Way. When we examine our chemical abundances for stars that appear constrained to the bulge, we do not find significant differences for most stars from that of the MW halo metal-poor stars. This includes the distribution of the $[\alpha/{\rm Fe}]$ ratio around $\sim0.4$, as in Figure~\ref{chemfig}, and the typical [Ba/H] ratio trend with the increase of the \FeH{}, as in Figure~\ref{bafig}. These findings are in agreement with the high-resolution analysis of bulge VMPs from \citet{Howes14,Howes15,Howes16}. They discuss that the majority of the stars in their sample are indistinguishable from MW halo stars at the same metallicities. Generally, we observe the same and we conclude that the majority of stars with high resolution spectral analyses in the bulge resemble those of the majority of stars in the halo. \subsection{No evidence for PISNe}\label{pisnesec} During the early Universe and amongst the earliest generations of stars, some of the low-metallicity stars are predicted to have formed from gas enriched only in the yields from Pair-Instability Supenovae \citep[PISNe,][]{Ji15}. This is when the highly energetic thermonuclear explosions of very massive stars ($150<{\rm M}_{\mathrel{prog}} < 260 \msun$) are carried by the pair production of electrons and positrons formed in the massive CO cores ($ >65 \msun$). The question of the initial mass function for the first stars is still unresolved; however, many theoretical models suggest that first stars were very massive, up to a few hundreds of $\msun$, given that molecular hydrogen would have been the only available coolant \citep[\eg][]{Omukai01,Bromm02,Stacy10}. PISNe produce a strong odd-even effect \citep[ratios of odd-Z to even-Z elements, such as $\rm{[Al/Mg]}$,][]{Heger02, Aoki14}, however the typical metal-poor halo stars (with [Fe/H] $< -2.5$) have chemical abundances that resemble the predicted yields from lower mass, metal-poor core-collapse supernovae \citep[CCSNe, \eg][]{Joggerst10, Ishigaki18}. As the ejecta from multiple CCSNe mix together, any unique Pop III abundance patterns which may have been preserved in metal-poor stars would be erased. Nevertheless, \citet{Takahashi18} predict PISNe yields as a function of progenitor mass (and with few differences between rotating and non-rotating models), which we compare to the results for the inner Galaxy stars from our GRACES spectral analysis. The chemical abundances for Mg, Ca, and Na, observable in our GRACES spectra, do not vary significantly with PISNe progenitor mass, i.e., a PISN produces a high ratio of Ca to Mg ($0.5<$[Ca/Mg]$<1.3$) and very low amounts of Na ([Na/Mg]$\sim-1.5$). In Figure~\ref{pisnefig}, the [Na/Mg] vs. [Ca/Mg] for our PIGS/GRACES sample are shown, together with bulge stars from the literature. For this latter comparison, stars with $\FeH >-2$ have not been removed to show the dearth of stars with detectable PISNe signatures. Three stars with inner halo-like orbits (P171457, P182221, and P184700) have high [Ca/Mg] compatible with PISNe yields (blue shaded area), yet their observed [Na/Mg] are too high. This is similar to the results from the COMBS survey \citep[Figure~13 of][]{Lucey22}, which made use of Al (not covered by GRACES) instead of Na. Thus, in both studies, no signature of PISNe yields is found. Alternatively, \citet{Salvadori19} investigate the combination of PISNe and SN II yields, which produce different yields (see grey shaded area in Figure~\ref{pisnefig}). Again, none of our stars occupy the overlapping PISNe + SNe II regions. \citet{Salvadori19} suggest that the smoking gun for the detection of PISNe yields would include low chemical abundances of N, Zn, and Cu; unfortunately, none are measurable in our GRACES spectra. They also suggest that PISNe would pollute the ISM up to $\FeH\sim-2.0$, making it unlikely to ever detect PISNe signatures in the most metal-poor stars. \subsection{Comparisons with Globular Clusters} \subsubsection{Second-generation globular cluster stars}\label{secgen} It has been proposed that some of the building blocks of the Galactic bulge could also have been from ancient globular clusters disrupted at early times \citep[\eg][]{Shapiro10,Kruijssen15,Bournaud16}. Quantitatively, \citet{Schiavon17} and \citet{Horta21b} have estimated that up to $\sim25$\% of the stellar mass of the inner halo within 2 \kpc{} from the Galactic centre is made of disrupted GCs, where those clusters were more massive (by $10-100$) than those observed today. Similarly, \citet{Martell11} suggested that a minimum of 17 percent of the present-day mass of the stellar halo was originally formed in globular clusters. Some research has confirmed the presence of stars in the bulge with peculiar chemical signatures similar to those in present-day GCs \citep[\eg][]{Trincado17,Schiavon17,Lucey19,Lucey22}, \eg the Na-O and/or Al-Mg anticorrelation found only in globular cluster red giants \citep[\eg][]{Gratton04, Martell11, Carretta12, Pancino17}. This is typical of the so-called second-generation globular cluster stars (or enriched stars), which their chemical imprint is thought to be governed by CNO cycle processing at high temperatures \citep[\eg][]{Gratton04,Bastian18}. The spectral coverage of GRACES contains four Al\ione{} lines (at $\AA6 696.015,6 698.673,8 772.866,8 773.896$\AA), however, no lines were detected. Upper limit estimates suggest [Al/Fe]$\leq+2$, which does not provide a meaningfully discriminating constraint. Therefore, we substitute Al with Na to study the odd-even effect. \citet{Pancino17} suggests that [Na/Mg] is smaller than [Al/Mg] in displaying the anti-correlation effects. To separate out the first ([Al/Mg] and [Na/Mg] normal) and second-generation stars ([Al/Mg] and [Na/Mg] rich), we cut the sample at [Mg/Fe]$\leq0.1$ vs. [Mg/Fe]$\leq0.4$ to help the second-generation population stand out more clearly (the first generation stars and MW halo-like stars possess [Mg/Fe]$\sim0.4$). The [Na/Mg] vs. [Mg/Fe] abundances are shown in Figure~\ref{gcsecfig}. Chemical abundances from \citet{Pancino17} are also shown, which are not NLTE corrected, hence we report the LTE measurements for our sample as well. Two GRACES stars, P171457 and P184700, clearly populate the second-generation GC region, given their high [Na/Mg] and low [Mg/Fe] ratios. Both have inner halo-like orbits, and one of them (P184700) is a CEMP star (see Section~\ref{cempgc}). Two other stars (P180118 and P182221) have lower [Na/Mg] ratios compatible with the first generation stars. \citet{Lucey22} also reported two stars compatible with the second-generation using the [Al/Fe] vs. [Mg/Fe] space. We do not show their sample in Figure~\ref{gcsecfig} since they do not provide LTE Na abundances and the majority of their stars are not very metal-poor. \subsubsection{Extragalactic globular clusters}\label{extragc} The chemical abundances of 7 GCs in M31 were compared with GCs in the MW, Sagittarius, Fornax, and the Large Magellanic Clouds by \citet{Sakari15}. They found that the [Mg/Ca] distribution of the extragalactic GCs is wider and mainly negative in the M31 GCs, whereas it peaks at [Mg/Ca]$\sim 0$ in the MW GCs. The only exception is NGC~2419, which has stars with a wide range of [Mg/Fe] values, and negative values for [Mg/$\alpha$] \citep[\eg][]{Mucciarelli12}. These features, together with the dispersion in the abundances of K and Sc, and its retrograde orbit, have led to speculation that this GC has an extragalactic origin \citep{Cohen12}. \citet{Pancino17} collected data from various Galactic and extragalactic GCs. They looked at the distribution of the $\alpha$-elements available, removing the second-generation stars (Al-enhanced and Mg-poor). They reinforce the hypothesis that all the extragalactic GCs have statistically lower mean Mg content than MW GCs. In our GRACES dataset, we find four stars with [Mg/Fe]$<0$ (see Figure~\ref{gcsecfig}). Figure~\ref{gc_extra} displays three panels, [Mg/Fe], [$\alpha$/Fe] and [Mg/Ca] ratios as a function of \FeH. The three inner halo-like stars (P171457, P182221 and P184700) have [Mg/Ca] in the range $[-0.85,-0.55]$, compatible with an extragalactic GC origin. The one bulge star (P180118) has a [Mg/Ca] $\sim-0.25$, which is low but also in the overlapping region between the Galactic and extragalactic GCs, thus its origin is less clear. We highlight that if the inner-halo star P171457 truly formed in an accreted extragalactic GC, then its metallicity (\FeH$_{\rm NLTE}=-3.25\pm0.05$) challenges the current estimates for the metallicity floor for GC \citep[\FeH$\sim-2.8$, \eg][and references therein]{Beasley19}. This would not be the first observation that calls into question the metallicity floor threshold; \citet[][]{Martin22} report the discovery of the remnant of the most metal-poor (\FeH$\sim-3.4$) GC known to date, the C$-19$ stellar stream. The chemistry of C$-19$ indicates a GC origin (e.g., range in Na abundances with no dispersion in Fe). However, its dynamics as measured from the radial velocity dispersion of its member stars are hotter than expected for a classical GC stream \citep{Errani22,Yuan22}, suggesting perhaps a new formation mechanism for GCs in ancient dwarf galaxies. If P171457 is confirmed as a star stripped and accreted from a dissolved GC, this would indicate that more extremely metal-poor GCs formed at early times but that tidal interactions with the MW may have dispersed them in the bulge and the halo. \subsection{Three CEMP stars: P182221, P183229, and P184700} \label{cempgc} Carbon-enhanced metal-poor (CEMP) stars are common amongst the metal-poor stars in the Galactic halo, reaching frequencies of $\sim30$\% at \FeH$<-2.0$ and up to $80$\% in the UMP regime \citep[\FeH$<-4.0$, \eg see][]{Placco14,Yoon18}, but see \citet{Arentsen22} for caveats. According to \citet{Beers05} and later confirmed by \citet{Norris13}, there are various classes of CEMP which differentiate by their Eu and Ba content. Stars are r-process-enhanced if [Eu/Fe]$>1.0$ (CEMP-r), s-process-enhanced if [Ba/Fe]$>1.0$ and [Ba/Eu]$>0.5$ (CEMP-s), mixed if $0.0<$[Ba/Eu]$<0.5$ (CEMP-r/s), and with no overabundance of n-capture elements if [Ba/Fe]$<0$ (CEMP-no). In particular, CEMP-s stars are enhanced due to the contribution of an AGB donor \citep[\eg][]{Masseron10,Hansen16}, \ie they are or were in a binary system. While CEMP-no are important tracers of spinstars \citep[\eg][]{Meynet06,Meynet10} or faint supernovae \citep[\eg][]{Umeda03,Umeda05,Tominaga14}. In the PIGS survey, \citet{Arentsen21} discovered 96 new CEMP stars in the inner Galaxy, including 62 with \FeH$<-2.0$. Previously, it was thought that CEMP stars were not common in the bulge, as only one CEMP-s \citet{Koch16} and one CEMP-no \citep{Howes15,Howes16} star had been identified. \citet{Arentsen21} further showed that in the EMP regime (\FeH$<-3.0$) the percentage of CEMP stars is $42_{-13}^{+14}$ percent, in agreement with the Galactic halo populations. There are two evident C-enhanced stars in this sample, P183229 and P184700, which was first noticed in the low-resolution AAT spectra from the PIGS sample \citep{Arentsen21} and the C$_2$ Swan bands can be seen in our GRACES spectra (\eg see Fig.~\ref{spectraex}). Both stars are also enhanced in Ba (see Figure~\ref{bafig}), identifying them as CEMP-s stars \citep[\eg][]{Milone12,Lucatello15,Hansen16}. The chemistry of P184700 is also compatible with second-generation stars in globular clusters (discussed in Sections~\ref{secgen}-\ref{extragc}). P184700 has a high [Na/Fe]$_{\rm LTE}=0.83\pm0.10$, and low [Mg/Fe]$_{\rm LTE}=-0.09\pm0.10$. Some CEMP stars have shown extreme Na enhancements, up to [Na/Fe]$\sim3.0$ on the main sequence, whereas most CEMP stars on the RGB have Na ranging from [Na/Fe]$=0.0-1.5$ \citep{Aoki07,Aoki08}. These enhancements are thought to depend on properties of the AGB donor \citep[e.g.,][]{Stancliffe09}. However, a low [Mg/Fe] value is not expected from the CEMP nucleosynthesis and mass transfer models, but can be lower in second-generation GC stars. Furthermore, lower [Mg/Fe] ratios are typically found in extragalactic systems, in both field stars and GCs \citep[see Section~\ref{extragc} and][]{Venn04, Pritzl05, Sakari15, Pancino17, Hasselquist21}. Given the rarity of binarity in very dense environments as shown in various works \citep[\eg][]{Dorazi10,Milone12,Lucatello15}, the possible association of P184700 with globular clusters would be quite a rare event. Further high-resolution spectroscopic follow-up of this star can provide a better insight into its origin. A third star, P182221, is likely to be (or was) a CEMP-s star. The large uncertainty on the C (see Section~\ref{cempsec}), the slightly enhancement in Ba (see Figure~\ref{bafig}), and the stellar parameters suggest (see Figure~\ref{kielfig}) this star is a horizontal branch star that has experienced maximal carbon-depletion once up to the RGB tip \citep{Placco14}. This depletion can reach up to 0.5 dex in [C/Fe]. If so, this would indicate the CEMP-s nature of this star. In favour of this scenario, the GRACES-AAT RV discrepancy (see Figure~\ref{rvgracesferre}, up to $\sim30\kms$) would also suggest that P182221 is in a binary system. Further measurements of the RV are needed to confirm the RV variability. \subsection{P180956: A very interesting star with planar orbit}\label{planarsec} The majority of the stars in the PIGS/GRACES sample that are confined to the MW plane (Z$_{\mathrm{max}}\leq3.5$\kpc) have small apocentric distances ($<6.5$\kpc), implying that they are confined to the inner region of the MW. The only planar-like star that plunges very far and beyond the Sun is P180956. This star has an apocentre of R$_{\mathrm{apo}} = 12.9 \pm 0.2\kpc$ and a Z$_{\mathrm{max}}=1.81\pm0.04\kpc$. This star at $\FeH\sim-2.0$, also exhibits a peculiar chemistry in the $\alpha$-elements, in the odd-Z elements, and Ba. The $\alpha$-elements are slightly enriched in Mg, yet challenged in Ca and Ti, such that [Ca/Mg]$\sim -0.6$. The odd-Z elements show that this star is extremely Na-poor ([Na/Fe]$\sim-1.0$), but also rich in K and Sc ([K/Fe]$\sim0.9$, [Sc/Fe]$\sim0.3$). The heavy element Ba is quite low, where [Ba/Fe]$\sim-1.6$ is amongst the lowest values of all stars known in this metallicity range (see Figure~\ref{badwarf}). A similar pattern in the $\alpha$-elements has been seen in a few low-metallicity stars \citep[HE~1424-0241, HE~2323-0256, HE~2139-5432, and HE~1327-2326, see][]{Sitnova19}, which have inner halo orbits with eccentricities $\epsilon\geq0.7$ \citep{Sestito19}. A few inner halo stars with similar behaviours in the $\alpha$-elements were also found by \citet{Venn20, Kielty21}, although at slightly lower metallicities ($\FeH\sim-2.5$). The peculiar chemistry, \ie very low [Ca/Mg] and [Ba/Fe] ratios, can be explained by the enrichment of one or very few core collapse supernovae \citep[the ``one-shot" model,][]{Frebel12}. The expectation is that strong feedback effects shut off the star formation after a Population III star explodes. The ejected yields would then produce very little Ba, even at higher metallicities ([Fe/H]$>-2$, as found in some stars in Coma Berenices \citep{Frebel12}, Segue 1 \citep{Frebel14}, and Hercules \citep{Koch08,Koch13,Francois16}. Figure~\ref{badwarf} shows the [Ba/Fe] ratio vs. \FeH{} for the PIGS/GRACES sample in comparison with Segue 1 and Hercules stars. The ``one-shot" model has, however, been questioned, \eg the linear decrease in [$\alpha$/Fe] with increasing [Fe/H] in Hercules and Com Ber suggests contributions from SNe Ia \citep[][]{Koch08,Waller22}. Regardless its origin, this chemical abundance signature is unique to stars that form in UFD galaxies, strongly suggesting that P180956 was captured from an UFD galaxy. It has been proposed by several recent publications that VMPs confined to the MW plane might be the relics of the building blocks that formed the proto-MW \citep[\eg][]{Sestito19,Sestito20,DiMatteo20,Carter21,Cordoni21}. \citet{Sestito21} proposed that a retrograde planar population is an excellent proxy for the early Galactic assembly, while a prograde VMP planar population could trace later accretion events at very low inclination angles. In fact, observations of a planar population at very high eccentricities have been found by \citet{Sestito19,Sestito20}, and proposed to have been brought in by one big merger event at early times. However, these studies used cosmological simulations or kinematical properties only, and lacked evidence from chemical abundances. The few planar stars that have been also observed with both high-resolution spectroscopic abundances and detailed Gaia-based kinematics \citep[][ and this work]{Kielty21,Venn20,DiMatteo20} have all pointed to peculiarities in their chemo-dynamical properties. A thorough high-resolution investigation of a larger sample of VMP planar stars will provide better insight, \eg using the upcoming spectroscopic surveys WEAVE \citep{WEAVE12} and 4MOST \citep{deJong19}. \subsection{Connection with the Inner Galaxy Structure and with Aurora}\label{igs} \citet{Horta21} used SDSS/APOGEE DR16 \citep{Ahumada20} to discover a population of metal-poor stars in the inner region of the MW ($R_{GC}<4\kpc$), which they called the Inner Galaxy Structure (IGS). This population has different chemo-dynamical properties from the more metal-rich bulge (\FeH$>-0.5$). The IGS is composed of high eccentricity stars ($\epsilon>0.6$), with $\FeH >-1.7$ and $0.3<$[Mg/Fe]$<0.4$, but an unusually negative ratio of [Al/Fe]$<0$ and positive [Mg/Mn]$>0.3$. They infer that the IGS host would have had stellar mass of $\sim5\times10^8\msun$, \ie twice the mass of the Gaia-Enceladus/Sausage system \citep[\eg][]{Belokurov18,Helmi18}. As all the stars in this paper are more metal-poor (\FeH$_{\rm LTE}<-2.0$) than the IGS stars discovered so far ($\FeH >-1.7$), then our sample is not sufficient to search for new members of the IGS. Nevertheless, one star P180956 (discussed in Section~\ref{planarsec}) has $\FeH_{\rm NLTE} =-1.8\pm0.1$. An evaluation of its chemical abundances shows that its [Mg/Fe] may be slightly too large for membership in the IGS, but its low Na is consistent with the expectation of low Al. The high eccentricity of P180956 would be compatible with the IGS, given the wide range selected by \citet{Horta21}, $e>0.6$. Many accreted structures have been extended towards the VMP regime \citep{Yuan20} applying self-organizing maps algorithms in the 4D space of orbital energy and angular momentum. However, higher signal-to-noise spectral observations and over a wider wavelength region than available to GRACES are necessary to measure the chemical abundance of Al, Mn, and N, and to test its connection to the IGS. Very recently, \citet{Belokurov22} used SDSS/APOGEE DR 17 \citep{ApogeeDR17} and Gaia data to investigate the metal-poor ([Fe/H]$>-1.5$) in-situ population of the Galaxy. They propose that a metal-poor halo could have formed early in the proto-Galaxy. This chaotic early phase of the Milky Way, dubbed Aurora, would produce stars with a large scatter in their chemistry due to the lumpy ISM. This would affect the elements usually used to discern the multi-populations in globular cluster stars (\eg Na, Al, N). According to \citet{Belokurov22}, high-eccentricity structures in the inner Galaxy, such as the IGS, would be part of Aurora. Our sample is more metal-poor than the Aurora’s stars studied so far. Therefore, a firm conclusion on their association with Aurora is beyond the scope of this work. \section{Conclusions} We present a chemo-dynamical investigation of 17 stars selected from the low/medium resolution spectroscopic campaign of the Pristine Inner Galaxy survey \citep{Arentsen20b}. Spectral observations with the Gemini North/GRACES and {\it Gaia} EDR3 astrometric solutions were used to infer precise chemical abundances, stellar parameters, distances, and orbits. Our sample is divided into four dynamical sub-groups; (i) stars confined to the bulge, (ii) stars confined into the MW plane, and two groups that dive further out to the (iii) inner and (iv) outer halo. The red spectral coverage of GRACES allowed us to determine the chemical abundances for several species, including Fe-peak (Fe, Cr, Ni), $\alpha-$ (Mg, Ca, Ti), odd-Z (Na, K, Sc), and neutron-capture (Ba) elements. By combining the chemistry and kinematics, we have investigated the properties of this sample of stars to find the following: \begin{enumerate} \item The majority of the stars in this sample have a chemical signature indistinguishable from that of Milky Way halo stars. If the bulge and the halo formed at early times from numerous building blocks, that are depositing their stars, their pristine gas, and dark matter into the forming Galaxy, then the chemistry of the bulge stars should resemble those in the halo (with the exception that the outer halo where more recent accretion will preferentially deposit their stars). This is additional confirmation of the hierarchical assembly of the Galaxy. \item We do not detect the signatures of PISNe yields \citep[as in][]{Takahashi18}, nor PISNe + CCSNe yields \citep[as in][]{Salvadori19}. \item Some of the stars in our sample are chemically compatible with second-generation stars in GCs. This is reinforced through examination of the chemistry of stars in extragalactic GCs, mainly their negative [Mg/Ca]. \item One possible second-generation GC star, P171457, is extremely metal-poor ($\FeH\sim-3.2$). If confirmed, this would indicate another dissolved extremely metal-poor GC in the MW, similar to the recently discovered halo stellar stream \citep[C-19,][]{Martin22}. \item We confirm the nature of two C-enhanced stars. We find high Ba, indicating they are CEMP-s stars. One of them, P184700, has the chemical abundances of a second-generation GC star. \item P182221 is likely a CEMP-s star. [C/Fe] measurement from AAT is highly uncertain, while its stellar parameters and and the slightly enhancement in Ba suggest this star past the RGB phase. This would indicate the depletion of C in its atmosphere. Moreover, the high RV discrepancy between GRACES and AAT would also be in favour of the binarity of this star. \item P180956 is a very metal-poor ($\FeH\sim-2.0$) star confined to the MW plane with an apocentric distance of $\sim12\kpc$ and a pericentre $<1\kpc$. Its chemistry includes low [Ba/Fe], [Na/Fe], and [Ca/Mg], which suggest it originated in an UFD galaxy that was polluted by only 1 or a few core collapse supernovae ($\sim20\msun$). Similar stars discovered in the Pristine survey \citep{Sestito20} and followed up with Gemini-North/GRACES and CFHT/ESPaDoNS spectroscopy have been found on planar orbits \citep{Venn20, Kielty21}. These may point to one or more very early accretion events, compatible with the building block merger phase \citep[][]{Sestito21}. These chemo-dynamical studies confirm the importance of very metal-poor stars with planar orbits as tracers of the early MW assembly. \end{enumerate} This chemo-dynamical investigation of the very metal-poor tail of the inner Galaxy opens a window on the early assembly of the Milky Way. It unveils the variety of the building blocks, from systems chemically similar to globular clusters to ultra faint dwarfs galaxies. Further spectroscopic observations towards bluer regions of the spectra are needed to better characterise the properties of the relics of these ancient and dissolved systems. This is a task easily achievable by the forthcoming Gemini High-Resolution Optical SpecTrograph \citep[GHOST,][]{Pazder20}. \section*{Acknowledgements} We acknowledge and respect the l\textschwa\textvbaraccent {k}$^{\rm w}$\textschwa\ng{}\textschwa n peoples on whose traditional territory the University of Victoria stands and the Songhees, Esquimalt and $\ubar{\rm W}$S\'ANE\'C peoples whose historical relationships with the land continue to this day. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the Native Hawaiian community. We are very fortunate to have had the opportunity to conduct observations from this mountain. We acknowledge the traditional owners of the land on which the Anglo Australian Telescope stands, the Gamilaraay people, and pay our respects to elders past and present. FS thanks Tim Beers, Ani Chiti, Anna Frebel, and Ian Roederer for the interesting discussions and feedback about this work at the 2022 JINA-CEE meeting in Notre Dame. The authors thanks Vasily Belokurov for his feedback on Aurora. We thank the anonymous referee for their comments that helped to improve this manuscript. FS thanks the Dr. Margaret "Marmie" Perkins Hess postdoctoral fellowship for funding his work at the University of Victoria. KAV thanks the National Sciences and Engineering Research Council of Canada for funding through the Discovery Grants and CREATE programs. AA, NFM, and ZY gratefully acknowledge support from the French National Research Agency (ANR) funded project ``Pristine'' (ANR-18-CE31-0017) and from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 834148). DA acknowledges support from the ERC Starting Grant NEFERTITI H2020/808240. JIGH acknowledges financial support from the Spanish Ministry of Science and Innovation (MICINN) project PID2020-117493GB-I00. ES acknowledges funding through VIDI grant "Pushing Galactic Archaeology to its limits" (with project number VI.Vidi.193.093) which is funded by the Dutch Research Council (NWO). The authors thanks the International Space Science Institute (ISSI) in Bern, Switzerland, for funding the Team “The Early Milky Way” led by Else Starkenburg. This work is based on observations obtained with GRACES, as part of the Gemini Large and Long Program, GN-X-LP-102 (where X includes semesters 2019A–2021A). Based on observations obtained at the international Gemini Observatory, a program of NSF’s NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. On behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\'{o}n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\'{i}a e Innovaci\'{o}n (Argentina), Minist\'{e}rio da Ci\^{e}ncia, Tecnologia, Inova\c{c}\~{o}es e Comunica\c{c}\~{o}es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). Based on observations obtained through the Gemini Remote Access to CFHT ESPaDOnS Spectrograph (GRACES). ESPaDOnS is located at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawai’i. ESPaDOnS is a collaborative project funded by France (CNRS, MENESR, OMP, LATT), Canada (NSERC), CFHT and ESA. ESPaDOnS was remotely controlled from the international Gemini Observatory, a program of NSF’s NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), Agencia Nacional de Investigaci\'{o}n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\'{i}a e Innovaci\'{o}n (Argentina), Minist\'{e}rio da Ci\^{e}ncia, Tecnologia, Inova\c{c}\~{o}es e Comunica\c{c}\~{o}es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. We thank the Australian Astronomical Observatory, which have made these observations possible. This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France \citep{Wenger00}. This work made extensive use of \textsc{TOPCAT} \citep{Taylor05}, the \textsc{DUSTMAPS} \textsc{Python} package \citep{Green19}, and \textsc{MATLAB} \citep{MATLAB21}. This research was enabled in part by support provided by WestGrid (\url{https://www.westgrid.ca}) and Compute Canada (\url{www.computecanada.ca}). \section*{Data Availability} GRACES spectra are available at the Gemini Archive web page \url{https://archive.gemini.edu/searchform}. The data underlying this article are available in the article and in its online supplementary material. \bibliographystyle{mnras} \bibliography{graces_bulge.bib} \appendix \section{Orbital parameters as a function of the distance grid}\label{apporb} The spatial distribution in Galactic Cartesian coordinates is shown in the three panels of Figure~\ref{spacedist}. As described in Section~\ref{orbsec}, a grid of distances with a step of $0.1\kpc$ within $\pm 1\sigma$ from the maximum of the distance PDF has been created for each star. Then, the orbits has been computed for each step of the grid and varying the other parameters (\eg RV, proper motions, coordinates etc.) with a Monte Carlo. The orbital parameters for each step of the distance grid are reported in Figure~\ref{kinefig_grid}. \section{Comparison with the stellar parameters from {\it Gaia} DR3}\label{appdr3} It is tempting to use the new {\it Gaia} DR3 \citep{Andrae22,Gaia16} catalogue of stellar parameters and metallicities; however, \citet{Andrae22} warn on the quality of the stellar parameters for stars with poor parallax measurements (\ie $\varpi/\sigma_{\varpi}<20$) and farther than 2\kpc. This is exactly the regime of the metal-poor stars towards the bulge. As expected, we find the {\it Gaia} DR3 photometric results for these stars to be quite poor. Figure~\ref{gaiadr3_params} shows the temperatures, gravities, and metallicities from this work and from the COMBS \citep{Lucey22} and EMBLA \citep{Howes14,Howes15,Howes16} surveys compared withe {\it Gaia} DR3 photometric results. {\it Gaia} DR3 temperatures are typically lower than the values used in spectroscopic works (top panel of Figure~\ref{gaiadr3_params}). The {\it Gaia} DR3 surface gravities have a flat distribution around logg$\sim4.5$ dex, while the sample spans a range of 5 dex (central panel of Figure~\ref{gaiadr3_params}). The {\it Gaia} DR3 metallicities are distributed around \FeH$\sim0.0$, while the stars are from super-solar to extremely metal-poor, spanning a range of 5 dex (bottom panel of Figure~\ref{gaiadr3_params}). \bsp % \label{lastpage}

Title: Atmospheric Monitoring and Precise Spectroscopy of the HR 8799 Planets with SCExAO/CHARIS

Abstract: The atmospheres of gas giant planets are thought to be inhomogeneous due to weather and patchy clouds. We present two full nights of coronagraphic observations of the HR 8799 planets using the CHARIS integral field spectrograph behind the SCExAO adaptive optics system on the Subaru Telescope to search for spectrophomometric variability. We did not detect significant variability signals, but placed the lowest variability upper limits for HR 8799 c and d. Based on injection-recovery tests, we expected to have a 50% chance to detect signals down to 10% H-band photometric variability for HR 8799 c and down to 30% H-band variability for HR 8799 d. We also investigated spectral variability and expected a 50% chance to recovery 20% variability in the H/K flux ratio for HR 8799 c. We combined all the data from the two nights to obtain some of the most precise spectra obtained for HR 8799 c, d, and e. Using a grid of cloudy radiative-convective-thermochemical equilibrium models, we found all three planets prefer supersolar metallicity with effective temperatures of ~1100 K. However, our high signal-to-noise spectra show that HR 8799 d has a distinct spectrum from HR 8799 c, possibly preferring more vertically extended and uniform clouds and indicating that the planets are not identical.

https://export.arxiv.org/pdf/2208.05594

\begin{CJK*}{UTF8}{gbsn} \title{Atmospheric Monitoring and Precise Spectroscopy of the HR 8799 Planets with SCExAO/CHARIS\footnote{Based on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.}} \correspondingauthor{Jason Wang} \email{jwang4@caltech.edu} \author[0000-0003-0774-6502]{Jason J. Wang (зЋ‹еЉІйЈћ)} \altaffiliation{51 Pegasi b Fellow} \affiliation{Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA} \affiliation{Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA} \author[0000-0002-8518-9601]{Peter Gao} \affiliation{Earth \& Planets Laboratory, Carnegie Institution for Science, 5241 Broad Branch Rd NW, Washington, DC 20015, USA} \author[0000-0001-6305-7272]{Jeffrey Chilcote} \affiliation{Department of Physics and Astronomy, University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, IN, 46556, USA} \author[0000-0002-3047-1845]{Julien Lozi} \affiliation{Subaru Telescope, NAOJ, 650 North A{'o}hoku Place, Hilo, HI 96720, USA} \author[0000-0002-1097-9908]{Olivier Guyon} \affiliation{Subaru Telescope, NAOJ, 650 North A{'o}hoku Place, Hilo, HI 96720, USA} \affiliation{Steward Observatory, University of Arizona, 933 N Cherry Avenue, Tucson, AZ 85719, USA} \affiliation{Astrobiology Center of NINS, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan} \author[0000-0002-4164-4182]{Christian Marois} \affiliation{National Research Council of Canada Herzberg, 5071 West Saanich Rd, Victoria, BC, V9E 2E7, Canada} \affiliation{University of Victoria, 3800 Finnerty Rd, Victoria, BC, V8P 5C2, Canada} \author[0000-0002-4918-0247]{Robert J. De Rosa} \affiliation{European Southern Observatory, Alonso de C\'{o}rdova 3107, Vitacura, Santiago, Chile} \author[0000-0003-2806-1254]{Ananya Sahoo} \affiliation{Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA} \affiliation{Subaru Telescope, NAOJ, 650 North A{'o}hoku Place, Hilo, HI 96720, USA} \author[0000-0001-5978-3247]{Tyler D. Groff} \affiliation{NASA-Goddard Space Flight Center, Greenbelt, MD, USA} \author[0000-0003-4018-2569]{Sebastien Vievard} \affiliation{Subaru Telescope, NAOJ, 650 North A{'o}hoku Place, Hilo, HI 96720, USA} \author[0000-0001-5213-6207]{Nemanja Jovanovic} \affiliation{Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA} \author[0000-0002-7162-8036]{Alexandra Z. Greenbaum} \affiliation{IPAC, MC 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA} \author[0000-0003-1212-7538]{Bruce Macintosh} \affiliation{Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA} \keywords{Exoplanet atmospheres ({487}), Exoplanet atmospheric variability ({2020}), Coronagraphic imaging ({313}), Spectrophotometry ({1556})} \section{Introduction} \label{sec:intro} Atmospheric spectroscopy provides a key window into understanding the nature of exoplanets. Through the measurement of molecular absorption features, we can learn about the composition of planets to uncover their nature and their formation history. However, molecular absorption is not the only feature in planetary spectra. Clouds are another central aspect of planetary atmospheres. Clouds trace the weather on other planets, and the detailed monitoring of their clouds can help us understand exoplanet atmospheric dynamics \citep{Tan2017}. They can also alter molecular absorption feature strengths, and affect our understanding of the composition of other planets \citep{Burningham2017, Molliere2020}. The atmospheres of directly imaged planets are generally similar to higher mass brown dwarfs of the same effective temperature \citep[e.g.,][]{Chilcote2017}.Planets of the L and T spectral type have redder spectra in the near-infrared than the higher-mass brown dwarfs, which could be due to having thicker clouds \citep[e.g.,][]{DeRosa2016}. For the four super-Jupiters orbiting HR 8799 \citep{Marois2008, Marois2010}, we have accumulated a diverse set of photometry and spectra that point to a complex cloud model \citep{Bowler2010, Barman2011, Konopacky2013, Currie2014, Ingraham2014, Skemer2014, Zurlo2016, Bonnefoy2016, Greenbaum2018, Molliere2020, Ruffio2021, Wang2020, Wang2021}. In particular, the temperature of the four planets lie around the transition between the L and T spectral types, where partly cloudy atmospheres are thought to induce the enhanced photometric variability ($>2$\% in $J$ band) we see in free-floating substellar objects at this temperature range \citep{Radigan2014}. If patchy clouds exist in these planetary atmospheres and the planets are not viewed pole-on, we expect photometric variability in the light curves of these planets as different patches of clouds rotate into and out of view \cite{Vos2017}, although exoplanet weather could induce some variability signal even for planets viewed pole-on \cite{Tan2021}. Rotational variability has been seen in the light curves of wide separation and free-floating planetary mass objects with nearly the same mass and atmospheric properties as the HR 8799 planets \citep{Biller2018,Bowler2020,Zhou2020}. There has also been indirect evidence for this in the HR 8799 planets as patchy cloud models have fit 3 to 5 $\mu$m photometry of the HR 8799 planets better \citep{Skemer2014}. This has motivated monitoring programs for the HR 8799 planets, but no variability has been detected \citep{Apai2016,Biller2021}. One issue is the difficulty in performing precise photometry in these high contrast imaging datasets, stemming from effectively removing the glare of the host star as well as obtaining precise photometric calibration with the host star. Due to this, the current photometric precision obtained on the HR 8799 planets is an order of magnitude away from what has been obtained for free-floating planetary mass objects \citep{Biller2018,Biller2021}. In this paper, we present the results of monitoring the HR 8799 planets from Maunakea, where the system is visible for the entirety of our two full nights, using the Subaru/CHARIS integral field spectrograph. In Sections \ref{sec:obs} and \ref{sec:data}, we describe the observations and data reduction respectively. We discuss the stability of the spectrophotometric calibration in Section \ref{sec:satspots}. We do not see any significant periodic variability signatures and place limits on the photometric and spectral variability of the planets in Section \ref{sec:var}. However, we were able to combine all the spectra we obtained to precisely measure the near-infrared spectra of the inner three planets (HR 8799 c, d, and e) and fit atmospheric models to constrain the properties of these three planets in Section \ref{sec:atm-fits}. \section{Observations} \label{sec:obs} \subsection{HR 8799} HR 8799 was observed at the Subaru telescope on two consecutive nights for nearly the entire time using the CHARIS integral field spectrograph \citep{Groff2015,Groff2017} behind the SCExAO high order adaptive optics system with its Lyot coronagraph \citep{Jovanovic2015_scexao}. We generated ``satellite spots" with the deformable mirror to create calibration point sources with attenuated copies of the stellar spectrum in the image since the star is behind the coronagraph. We quickly modulated the deformable mirror to make the spots incoherent with the speckle field \citep{Jovanovic2015}. We obtained 20~s exposures using the CHARIS low-resolution mode that obtained R$\sim$20 spectra from $J$ through $K$ band. On 2018 September 1 (night 1), we obtained 1,201 usable frames of data from 7:03 to 15:34 UT, resulting in a total on-sky integration time of 24,805~s (6.89~hours). On 2018 September 2 (night 2), we obtained 1,253 usable frames of data from 6:11 to 15:45 UT after discarding 89 frames with unusual detector noise, resulting in a total on-sky integration time of 25,879~s (7.19~hours). Altogether, we obtained 14.08 hours of integration time on HR 8799 over the course of the two nights. We note that a portion of this data was used in \citet{Wang2020} to study HR 8799 c specifically, and here we will use this data to study all three planets in our field of view. \subsection{HD 187003} On each night, we observed the stellar binary HD 187003 beforehand to calibrate the brightness and stability of the satellite spots generated by the deformable mirror. HD 187003 has a secondary located at $\sim$0.6~arcsec away from the primary with a $K$-band flux ratio of $\sim3 \times 10^{-2}$. This stellar companion is favorable for studying the stability of the satellite spots, since it is inside the field of view and not so bright that it saturates the image. On 2018 September 1, we obtained 30 frames with 15~s integration times in the same low-resolution mode with the Lyot coronagraph as the HR 8799 data. On 2018 September 2, we obtained a longer 87 frame sequence with the same configuration. Furthermore, on 2018 September 2, we took three 200~s images where the primary star was not behind the coronagraph, but the whole field of view was attenuated by a neutral density filter to prevent saturation. These images were used to directly measure the flux ratio of the stellar binary at each wavelength. With the binary flux ratio known, we can then calibrate the satellite spot flux ratio in each exposure where the binary is occulted. \section{HR 8799 Data Reduction}\label{sec:data} \subsection{Spectral Datacube Processing} First, we will discuss the processing steps to generate and calibrate the 3D spectral datacubes. The initial steps were performed using the CHARIS data reduction pipeline \citep{Brandt2017}. The raw 2D detector frames consist of an array of microspectra, with each corresponding to one pixel of the final 3D datacube. The 2D images were dark subtracted using dark frames taken at the end of each night. Using a narrow-band tuneable filter, point spread functions (PSFs) of the microspectra at each wavelength were taken at the end of each night and compiled into a wavelength solution for each spectrum. The microspectra were then extracted using the algorithm in \citet{Brandt2017} to create 3D spectral datacubes. The datacubes have 22 spectral channels, with each channel corresponding to an image of HR 8799 at a specific wavelength. Each datacube was then flat fielded using a flat field cube where all of the microspectra were illuminated by a flat lamp. Note that the whole 2D detector was not illuminated by the flat field, so this flat field is a combination of detector response, lenslet response, and cube extraction effects. Following, we located the position of the four satellite spots in each wavelength channel of each datacube to determine the position of the occulted star and to extract off-axis PSFs. We fit each of the four satellite spots in each wavelength channel of each cube with a 2D Airy function. We fit the position, peak flux, and width of the Airy function using Nelder-Mead minimization. We averaged the position of the four satellite spots to calculate the location of the star behind the occulting mask as a function of wavelength. We also used the position of each spot to extract out a 15x15 pixel cutout of each satellite spot. We interpolated each spot so that it is centered on the middle pixel of the cutout, and subtracted the background in the cutout by taking the average background value between 9 and 12 pixels away from the center of the satellite spot. We combined the cutouts of the four spots to create a single satellite spot PSF for each frame. \subsection{Stellar PSF Subtraction} In order to look for temporal variability, we broke up the observing sequences into one- and two-hour chunks for the analysis. We chose these two binning strategies because two-hour chunks gave us better parallactic angle rotation for ADI, but one hour chunks gave us better temporal resolution. Regardless, for each chunk, we performed stellar PSF subtraction using only data from that chunk to remove the glare of the host star. We used the open-source Python package \texttt{pyKLIP} \citep{Wang2015} to model and subtract off the glare of the star using principal component analysis \citep[PCA,][]{Soummer2012}. We employed both angular differential imaging (ADI) and spectral differential imaging (SDI) to select images of the system taken at other times and wavelengths to create reference images of the stellar PSF \citep{Marois2000, Sparks2002, Liu2004, Marois2006}. The PCA modes were constructed using the 200 most correlated reference PSFs from these reference images. For each planet, we varied the minimum number of pixels the planet has to move due to ADI and SDI for that frame to be included in the reference images and the number of PCA modes used. We create a fiducial image using 50 PCA modes and used reference images where the planets moved at least 1 pixel due to ADI and SDI. The two-hour-chunk reduction are plotted in Figure \ref{fig:hr8799_gallery}. HR 8799 c and d are clearly detected in all images, but HR 8799 e is marginally detected in the first and last two hour chunk due to the limited parallactic angle rotation for ADI. This is consistent with previous findings that ADI is more effective than SDI for SCExAO/CHARIS PSF Subtraction \citep{Gerard2019}. \subsection{Spectral-photometric Extraction}\label{sec:spectral_extraction} Next, we measured the flux of each planet at each wavelength in the CHARIS observations. Due to the known effects of overfitting caused by PCA, we used the forward modeling framework presented in \citet{Pueyo2016} and \citet{Greenbaum2018} and implemented in \texttt{pyKLIP} to obtain the spectrum of each planet in each chunk of data. The forward modelling utilizes the fact we know the instrumental PSF (derived from the satellite spot PSFs), how the planet moves in the images due to ADI and SDI, and a linear approximation for how the planet perturbs the PCA fitting to self-consistently recover the ratio of the planet's flux at each wavelength to the instrumental PSF flux (i.e., satellite spot flux). As there is a concern that our linear approximation of each planet's effect on the stellar PSF subtraction may not be valid for precision variability monitoring of bright planets \cite{Pueyo2016}, we injected eight simulated planets at the same separation as each of the three real planets but at different position angles (we only injected one simulated planet at a time to avoid cross talk). Each simulated planet was injected into the data with the spectrum of the real planet we had previously extracted from the data to accurately measure any biases from spectral extraction. We then extracted the spectrum of each simulated planet using the same method. For each real planet at each wavelength in each chunk of data, we computed a bias factor that was the average ratio of the measured simulated planet flux with the input simulated planet flux. We found biases between the injected and measured fluxes of up to 10\%. We used the measured bias factors to correct our HR 8799 planet spectra. We computed uncertainties for the flux of each real planet at each wavelength in each one- and two-hour chunk by computing the standard deviation of the measured simulated planet fluxes and correcting for the bias term. To convert the flux measurement from a ratio of planet flux to satellite spot flux to the planet's flux in physical units, we use the satellite spot flux ratios for each night calculated in Section \ref{sec:satspot-phot} to derive a planet to star flux ratio and a 7330~K PHOENIX stellar model and the star's $K$-band magnitude of 5.24 \citep{Cutri2003} to derive the spectrum of each planet in $\textrm{W}/\textrm{m}^2/\mu\textrm{m}$. The 7330~K PHOENIX spectrum agrees well with the effective temperature of the star found in other recent work \citep{Wang2020}. We applied this process to both the one-hour and two-hour chunking of the data. To obtain the highest SNR spectrum of each planet from each night, we then combined the two-hour-chunk spectra in time using a weighted mean where the weights are the inverse square of the uncertainties. This is nearly identical to running the data analysis on an entire night of data, but saves on computation time. We derived the errors on these stacked spectra using the formal uncertainty propagation for a weighted mean. We plot these spectra for all three planets in Figure \ref{fig:hr8799_spec_fullnight}. For HR 8799 c in $H$- and $K$-band, we reach signal-to-noise ratios (SNRs) $> 50$ per spectral channel. We note that the plotted errors consist only of measurement uncertainty and does not include systematic errors in the spectro-photometric calibration that we will explore in Section \ref{sec:satspots}. We can see in Figure \ref{fig:hr8799_spec_fullnight} that the spectra from night 2 appear to be systematically brighter than the spectra from night 1 if we take the photometric calibration at face value. Interestingly, if we normalize each night's spectra to the $K$-band flux of HR 8799 c that night, then the three spectra have better agreement in Figure \ref{fig:hr8799_spec_fullnight}, indicating that the first order correction is a constant scale factor across all wavelengths between nights. This could be due to a bias in the satellite spot ratio between nights, as it is unlikely all three planets brightened in the same way. In Figure \ref{fig:hr8799_spec_fullnight}, the night 1 spectrum of HR 8799 e has enhanced flux at 2.3~$\mu$m that is not seen in night 2 and the other planets. We believe this is likely spurious, and possibly caused by imperfect speckle subtraction at the location of the planet. We will next investigate the stability of the satellite spots to use for spectro-photometric calibration and whether there are systematics to account for. \section{Satellite Spot Analysis} \label{sec:satspots} Due to variable adaptive optics correction and unknown atmospheric transmission, simultaneously imaged reference stars are needed to photometrically calibrate data on the HR 8799 planets. With a 2" field of view, the only reference star in the CHARIS data is HR 8799 itself, which is blocked by the coronagraph. Satellite spots are essential for spectro-photometric calibration of this kind of coronagraphic data \citep{Marois2006b,Sivaramakrishnan2006}. Unlike other instruments that use a static pupil mask or deformable mirror offset, the spots used here were generated by placing a fast modulating sine pattern on the deformable mirror to make them incoherent with the speckle field \citep{Jovanovic2015}. We characterize the stability of the SCExAO satellite spots using the data on the HD 187003{, which harbors a companion at $\sim$0.6~arcsec}. We focus on this data because it allows us to calibrate out most effects due to changing atmospheric transmission or adaptive optics correction, since they should affect both components in the same way. To our knowledge, HD 187003 is not a known variable star. Even though the primary is a spectroscopic binary consisting of two equal-mass G-type stars, it is not an eclipsing system \citep{Griffin2001}. Typical G-type stars have variability observed at the mmag-level \citep{Ciardi2011}, which is below the 1-10\% variability amplitudes that we can concerned with in this work. {The companion at 0.6~arcsec is an early M dwarf based on our measured $K$-band flux ratio of $\sim$1:30 relative to the unresolved twin G-type stars. Based on \citet{Ciardi2011}, we expect the variability of M-dwarfs to be below 10~mmag, which again is below the 1\% variability amplitudes we are concerned with. For the rest of this paper, we will call this 0.6~arcsec companion the ``binary companion" or ``secondary" for simplicity, since the fact the primary is an unresolved spectroscopic binary can be safely ignored. } We performed a similar data analysis routine as what was done for measuring the photometry of the planets in each image (see Section \ref{sec:data}). We performed dark subtraction of the raw 2D images, extracted the microlens spectra to form a 3D spectral data cube using the CHARIS data reduction pipeline, and then flat fielded the 3D spectral data cube using data from a flat field lamp taken after the night. This results in one spectral cube per exposure. In Figure \ref{fig:satspot_imgs}, we show representative images of the binary used in the calibration analysis after we have reconstructed them into spectral data cubes. We note that for the exposures taken with the neutral density filter and without the coronagraph, we could not flat field that data since the the flat field exposures were taken with the focal plane mask in that blocks the flat field lamp at the center of the focal plane where the primary star happened to lie in the images. This affects the computed flux ratio of the binary companion at the percent-level, which only affects the photometric calibration at the percent-level. We find later in this section that the photometric calibration of the satellite spots is uncertain beyond this level due to other factors, so the effect is negligible. \subsection{Binary for Satellite Spot Characterization}\label{sec:satspots-eqn} To assess the stability of the satellite spots both photometrically and spectrally over time, we computed the flux ratio between the binary companion and the four satellite spots in each individual frame (each wavelength channel of each spectral datacube). We used the following relations to define our stability tests. Let us define the flux density of the primary and secondary star as a function of wavelength as $F_{A}(\lambda)$ and $F_{B}(\lambda)$ respectively. Then the flux ratio of the secondary to the primary is \begin{equation} FR_{B,A}(\lambda) = F_{B}(\lambda)/F_{A}(\lambda). \end{equation} The satellite spots (labeled with a subscript $s$) are at a given flux ratio relative to the primary star at each wavelength ($FR_{s,A}(\lambda)$) so that their flux density is \begin{equation} F_{s}(\lambda) = FR_{s,A}(\lambda) F_{A}(\lambda). \end{equation} Thus, the flux ratio of the secondary relative to the primary can be written as \begin{equation}\label{eqn:fr-basic} FR_{B,A} = \frac{F_{B}}{F_{s}/FR_{s,A}} = FR_{s,A}/FR_{s, B} \end{equation} where each term depends on wavelength but we have removed the explicit wavelength dependence from the equations. Here $FR_{s, B}$ is the flux ratio of the satellite spots relative to the secondary, which we can directly measure. We also make the assumption that the flux ratio of the secondary relative to the primary is constant, since neither star is known to be significantly variable. Thus, our assumption is that if we measure variations in the flux ratio of the secondary relative to the satellite spots, this is linearly proportional to instabilities in $FR_{s,A}$, the amplitude of the satellite spots in our images. From just the binary to satellite spot flux ratio as a function of wavelength, we can assess the spectral stability of the satellite spots. Since we assume $FR_{B,A}$ does not change in time, then $FR_{B,A}(\lambda_1)/FR_{B,A}(\lambda_2) = const$ for any two wavelengths $\lambda_1$ and $\lambda_2$. Expanding this ratio of flux ratios and moving terms around so that $\lambda_1$ is only in the numerator, we get that \begin{equation} \frac{FR_{s,A}(\lambda_1)}{FR_{s,A}(\lambda_2)} = \frac{FR_{B,A}(\lambda_1)}{FR_{B,A}(\lambda_2)}\frac{FR_{s,B}(\lambda_1) }{FR_{s, B}(\lambda_2) }. \end{equation} This means that any changes in $FR_{s,B}(\lambda_2)/FR_{s,B}(\lambda_1)$ is linearly proportional to changes in $FR_{s,A}(\lambda_2)/FR_{s,A}(\lambda_1)$. Changes in the latter should be due to changes in the spectral stability of the satellite spots, so $FR_{s,B}(\lambda_2)/FR_{s,B}(\lambda_1)$ allows us to assess the spectral stability of the satellite spots. We focused on testing the spectral stability in the ratio of photometry in different bands (i.e., colors) so we will take the mean of the flux ratio over each band. Thus, what we are comparing is \begin{equation}\label{eqn:fr-phot} \frac{\sum_i FR_{s,B}(\lambda_i) }{\sum_j FR_{s,B}(\lambda_j)} = \frac{ \sum_i FR_{s,A}(\lambda_i)/FR_{B,A}(\lambda_i)}{ \sum_j FR_{s,A}(\lambda_j) / FR_{B,A}(\lambda_j) } \end{equation} where for each of the bands, we sum over all wavelength indices (either $i$ or $j$) that are defined to be in that band. We also assumed each band has the same number of wavelength channels so that the normalization by number of wavelengths for the mean cancels out. We defined bands from the CHARIS spectra that are roughly the $J$, $H$, and $K$ bands, as well as a custom ``F190" band that is centered on the water absorption band at roughly 1.9~$\mu$m. For this work, we defined the bands as follows: $J$ band is wavelength channels 1 through 4 inclusive (1.20-1.33 $\mu$m); $H$ band is channels 9 through 12 inclusive (1.58-1.74 $\mu$m); the custom F190 band is channels 13 through 16 inclusive (1.80-2.00 $\mu$m); $K$ band is channels 17 through 20 inclusive (2.07-2.29 $\mu$m). All indices are assumed to start from channel 0. We note that these are not the exact definitions for these photometric bands, but are a convenient approximation for this work. \subsection{Satellite Spot Photometry}\label{sec:satspot-phot} We investigated the photometric stability of the satellite spots and measured their flux ratio, $FR_{s,A}(\lambda)$, for photometric calibration. To do this, we need to measure $FR_{B,A}$ amd $FR_{s,B}$ as shown in Equation \ref{eqn:fr-basic}. As we used PSF fitting for the HR 8799 planets, we used PSF fitting here too to measure these relative flux ratios. First for $FR_{s,B}$, since the binary companion is brighter than the satellite spots, we used it as the PSF model that we fit to the satellite spots. To extract the binary companion PSF, we performed centroiding on the companion in each frame by fitting it to a 2-D Airy PSF and then extracting out a 21x21 pixel box based on the centroid. This resulted in a 21x21 PSF of the binary companion for each frame to fit to each of the four satellite spots. We performed the same Airy PSF centroiding to find the approximate location of each satellite spot, and defined a 15x15 pixel fitting region around each satellite spot. We performed a least-squares optimization using the Levenberg-Marquardt algorithm to find the best x and y shifts and the best flux ratio scaling of the companion PSF to match each satellite spot. We took the mean of the four flux ratios obtained from this fit to be the flux ratio between the binary companion and the satellite spots at that given wavelength and time. We averaged over the respective wavelength channels that we have defined in each band to obtain average flux ratios in our $J$, $H$, F190, and $K$ bands for each exposure. We then computed the flux ratio between the primary and secondary star, $FR_{B,A}(\lambda)$. In the images where the primary is behind the neutral density filter rather than the occulting masking, we perform the same PSF fitting routine to measure the binary flux ratio as we had done to measure the flux ratio between the secondary and the satellite spots in the previous spectral stability analysis. With $FR_{B,A}(\lambda)$ measured from the unocculted images, we can use the measured $FR_{s, B}$ in each frame of coronagraphic data to derive $FR_{s,A}$ following the relationship in Equation \ref{eqn:fr-basic}. We averaged the flux ratio of the four satellite spots in each wavelength channel of each exposure. Because the satellite spot flux ratios intrinsically scale $\propto \lambda^{-2}$, we rescaled all of them to a fiducial wavelength $\lambda_0$ by multiplying each flux ratio by $\lambda^2/\lambda_0^2$. We picked $\lambda_0 = 1.55 \mu$m to be consistent with previous work \citep{Currie2018}. We then averaged the rescaled satellite spots across all wavelengths in a single exposure, and plotted the measured satellite spot flux ratio at 1.55 $\mu$m for each exposure in the bottom row of Figure \ref{fig:satspot_time} to assess their photometric stability. We see drifts up to 10\% in the satellite spot flux ratio over the course of half an hour. One possible cause could be due to the fact that SCExAO sees on the wavefront sensor (WFS) the sine pattern from the deformable mirror (DM) that creates the satellite spots. Even though that the SCExAO control loop is designed to filter out this signal, the filtering could be imperfect and could depend on what the signal of atmospheric turbulence looks like on the WFS. Another control-related cause could be imperfect DM calibration, where the voltage to physical displacement conversion is incorrect and the amplitude of the sine wave that creates the satellite spots changes as the shape of the DM changes to correct for atmospheric turbulence. Alternatively, the quasi-static speckles at the location as the satellite spots may be evolving over time as the adaptive optics correction changes. Since speckles are also $\lambda$/D in size, it is difficult to disentangle these two signals when measuring the satellite spot fluxes. We did not alternate turning the spots on and off as has been done in other SCExAO observations to remove quasi-static speckles at the location of the satellite spots \citep{Sahoo2020}. All three processes imply that the flux ratio of the satellite spots could change as conditions and turbulence change. We measured a 2.1\% and 3.2\% scatter in the satellite spot flux ratios as a function of time in night 1 and night 2 respectively. We found an average satellite spot flux ratio at 1.55 $\mu$m of $2.58 \times 10^{-3}$ and $2.65 \times 10^{-3}$ for night 1 and night 2 respectively, a 3\% difference that is much smaller than the drift within night 2. Despite this, the fact the flux ratio is very correlated in time makes it suspect that the flux ratio could stay constant over the course of a whole night. It may be that we observed the binary at similar hour angles on consecutive nights where the atmospheric conditions were both consistent and good. Data taken at other hour angles or in changing conditions may produce a different satellite spot ratio. As seen in Figure \ref{fig:satspot_time}, the 30 minute sequence on night 2 has a possible downwards trend in the satellite spot flux ratio over the full time baseline, indicating there could be longer timescale variations that are not measured in this data. The correlated fluctuations may not average out when taken in more heterogeneous conditions. This may explain why our derived satellite spot flux ratio is also different by 3\% from the $2.72 \times 10^{-3}$ reported by \citet{Currie2018} and why our measured HR 8799 planet spectra are all brighter on the second night (Figure \ref{fig:hr8799_spec_fullnight}). These data only characterize the stability on 30-minute and 24-hour timescales. Future observations that observe binary stars over more varied timescales are necessary to better characterize this instability. \subsection{Satellite Spot Spectral Stability}\label{sec:satspot-spec} We also investigate the spectral stability of the satellite spots by looking at the stability of the color of the flux ratio between the satellite spots and the binary companion. In Figure \ref{fig:satspot_time}. we plot the colors of the satellite spot to secondary flux ratios (i.e., quantity in the left hand side of Equation \ref{eqn:fr-phot}) in the following combinations of bands: $J$/$K$, $H$/$K$, $H$/F190, $K$/F190. In these time-series shown in Figure \ref{fig:satspot_time}, we see some 2\% drifts on the 10-minute timescales as well as on the night-to-night timescales, but they all are much smaller than the photometric drifts. All four ratios show drifts of 1-2\% on the 10- to 30-minute timescales. The $J$/$K$ and $H$/$K$ ratios show negligible ($< 1$\%) variations between the two nights: $J$/$K$ ratio is $0.591 \pm 0.010$ on night 1 and $0.585\pm 0.006$ on night 2 and $H$/$K$ ratio is $0.819 \pm 0.012$ on night 1 and $0.816 \pm 0.010$ on night 2 (the error bars here represent the scatter between exposures within a night). However, the two ratios involving the F190 filter, which is strongly affected by telluric absorption, do see a significant $\sim$2\% shift between the two nights: the $H$/F190 ratio is $0.961 \pm 0.006$ on night 1 and $0.982 \pm 0.009$ on night 2; the $K$/F190 ratio is $1.175 \pm 0.016$ on night 1 and $1.203 \pm 0.013$ on night 2. Given the 2\% variations in the satellites spot colors exist on the minutes to day timescales, we find a systematic floor in measuring the colors of any exoplanet of 2\%. Anisoplanatism could be one possible cause of the drift: the satellite spots probe the atmospheric turbulence of the primary star, which is also used for adaptive optics correction, but the light of the binary (and any planet companions) is separated by 400 to 1000~mas and travels through a slightly different patch of atmosphere and suffers slightly degraded adaptive optics correction. The amount of degradation is time variable depending on the structure of atmospheric turbulence and is also wavelength-dependent as shorter wavelengths are generally more affected. We note that we did not probe drifts on the hour timescales from this data, so there could be additional drifts at those timescales, especially if anisoplanatism changing with changing airmass is a dominant effect. Still, these systematic drifts are only observed at the percent-level. \section{Atmospheric Variability}\label{sec:var} \subsection{Photometric Variability}\label{sec:phot-var} \begin{deluxetable*}{c|c|c|c|c|c} \tablecaption{Peak photometric variability periods from periodogram analysis \label{table:phot_peaks}} \tablehead{ Planet & Band & Period (hours) & Amplitude (W/m$^2$/$\mu$m) & Amplitude (\%) & False Alarm Probability (\%) } \startdata c & J & 4.0 & $2.5 \times 10^{-17}$ & 21 & 50 \\ c & H & 17.1 & $1.8 \times 10^{-17}$ & 11 & 45 \\ c & K & 4.0 & $1.9 \times 10^{-17}$ & 13 & 32 \\ c & F190 & 15.8 & $1.1 \times 10^{-17}$ & 13 & 37 \\ \hline d & J & 9.8 & $1.5 \times 10^{-17}$ & 18 & 34 \\ d & H & 13.9 & $2.4 \times 10^{-17}$ & 17 & 22 \\ d & K & 4.6 & $1.8 \times 10^{-17}$ & 13 & 52 \\ d & F190 & 14.1 & $1.3 \times 10^{-17}$ & 14 & 32 \enddata \end{deluxetable*} Since most brown dwarf variability studies have been looking at photometric bands \citep[e.g.,][]{Radigan2014, Artigau2018}, we will spectrally bin the flux of the planets into various photometric bands and look at the corresponding time series plots to avoid the complexity of the flux changing in 22 spectral channels. We define our own $J$, $H$, $K$ and F190 bands as we detailed in Section \ref{sec:satspots-eqn}. Each band consists of four spectral channels where the flux of the planet is brightest. The uncertainties for each datapoint were estimated based on the scatter in the simulated planet fluxes that were binned into the same spectral bands. We plot the time series for HR 8799 c and d in Figure \ref{fig:cd_phot_time}. The SNR on HR 8799 e was too poor. In Figure \ref{fig:cd_phot_time}, we see that the fluxes on the second night are higher on average compared to the first night, consistent with the global scaling seen in Figure \ref{fig:hr8799_spec_fullnight} which we hypothesize could due to a bias in the satellite spot flux ratios between the two nights. Ignoring this global offset between nights, we see that the photometry of each planet is correlated between bands, but do not appear to be strongly correlated to the other planet. This implies that the variability within a night we see is not due to the instability of the satellite spots, as it should affect both planets in the same way. Instead, this photometric behavior could be due to real changes in the planet's atmosphere or unaccounted-for errors in the spectral extraction. The amplitude of the flux variability is quite large (20\%) and near the limit of what is seen for comparable low mass objects \citep{Bowler2020}. Systematic errors in the spectral extraction process are somewhat accounted for with the simulated planet injection analysis described in Section \ref{sec:spectral_extraction}, which essentially injected planets with a flat light curve and made corrections so that the output spectrophotometry of the simulated planets was flat, on average. However, this process does not account for all possible systematics. One remaining systematic is a mismatch between the true planet PSF and the PSF we derived from the satellite spots, as the satellite spots are radially elongated due to the finite spectral bandwidth of each channel in the CHARIS low-resolution mode. A more detailed study of the accuracy of spectrophotometry in coronagraphic images, which does not exist in the literature to the best of our knowledge, may help determine how relevant this concern is. To formally assess the light curves for any significant periodic variability, we constructed Lomb-Scargle periodograms for the time-series of each planet in each photometric band using the implementation in \texttt{astropy} \citep{Astropy2013,Astropy2018}. We found the results from the one-hour chunks to be more sensitive from a bootstrap analysis of the false alarm probability, so we will focus only on those results. The Lomb-Scargle periodograms are plotted in Figure \ref{fig:periodograms}, and it is clear there is no significant peak in any band. The most significant peak for each band is listed in Table \ref{table:phot_peaks} along with its false alarm probability. All the false alarm probabilities are greater than 5\%, so none of them should be considered significant. We note that this analysis assumes the variability is sinusoidal, and ignores variability in the variability, although this is likely a second-order effect: a primary peak corresponding to the rotation period of the object should still be discernible. To quantify the sensitivity of our data to periodic variability signals, we injected sinusoidal signals into the one-hour-chunk data at a variety of different amplitudes and periods. At each amplitude and period, we performed 20 injection and recovery tests, varying the phase of the sinusoid in each injection. In each test, the signal was injected with noise drawn from Gaussian distributions with widths equal to the quadratic sum of the statistical errors computed from the data and a 3\% systematic error due to time-variable satellite spot calibrations discussed in Section \ref{sec:satspot-phot}. We added the 3\% satellites spot calibration systematic as a random error term as we do not have data to quantify its trend on hour-long timescales. The only measurement with time baselines longer than an hour that we have is the fact that the satellite spot flux ratio changed by 3\% between the two nights. Thus, we caveat the following analysis with the fact our expected sensitivity could be worse due to additional unknown systematics. After the simulated signal was injected, the time-series was run through the same Lomb-Scargle periodogram analysis above and determined to be a detection if the periodogram found the correct period with a false alarm probability $< 5$\%. We computed the fraction of signals at each variability amplitude and period that were detected to determine how complete we are for that amplitude and period (i.e., the chance we would have detected such a signal in the data). These completeness values were evaluated on a grid of variability amplitudes and periods to form completeness maps. Completeness maps for each photometric band and for both planets are shown in Figure \ref{fig:complete-phot}. In the best-case {for rotation periods between 5 and 18 hours}, we maintain $ > 50$\% completeness for $10\%$ variability amplitudes for HR 8799 c in $H$ band. These results are 2-3x better than \citet{Biller2021}, who also did $H$-band photometric monitoring of HR 8799 c with SPHERE. For HR 8799 d, we only reach down to 30\% variability amplitudes in $H$-band, but these are the first significant constraints on photometric variability for this planet. Our $K$-band sensitivities are $> 50$\% complete to variability amplitudes of 20\% for both planets, and are the first in $K$ band for these planets. However, the patchy cloud models from \citet{Skemer2014} do not predict $> 10\%$ variability in $K$-band, so the lack of variability may be expected. We also note that all of the periodogram peaks listed in Table \ref{table:phot_peaks} have amplitudes and periods that correspond to $\leq$~50\% completeness in our completeness maps. This reinforces the non-detection of photometric variability in this HR 8799 dataset. \subsection{Color Variability}\label{sec:color-var} \begin{deluxetable*}{c|c|c|c|c|c} \tablecaption{Peak color variability periods from periodogram analysis \label{table:ratio_peaks}} \tablehead{ Planet & Bands & Period (hours) & Flux Ratio Amplitude & Amplitude (\%) & False Alarm Probability (\%) } \startdata c & J/K & 9.7 & 0.09 & 10.8 & 74 \\ c & H/K & 23.1 & 0.34 & 41.5 & 53 \\ c & H/F190 & 12.5 & 0.16 & 8.3 & 60 \\ c & K/F190 & 8.3 & 0.13 & 7.8 & 32 \\ \hline d & J/K & 6.7 & 0.10 & 15.7 & 61 \\ d & H/K & 8.0 & 0.14 & 13.9 & 30 \\ d & H/F190 & 4.4 & 0.06 & 4.1 & 71 \\ d & K/F190 & 12.1 & 0.24 & 16.0 & 40 \enddata \end{deluxetable*} We also investigated if the broadband colors would be a better diagnostic of variability, as it removes effects such as the global calibration offset between the two nights. There is also scientific motivation for this. The partly cloudy models of the HR 8799 planets presented in \citet{Skemer2014} predict an order of magnitude more variability in $J$ and $H$ band compared to $K$ band. Variability studies of brown dwarfs have also found significantly decreased variability inside the water absorption bands than outside of them \citep{Zhou2020}. Motivated by these previous works, we plot the $J/K$, $H/K$, $H$/F190, and $K$/F190 flux ratios of HR 8799 c and d in Figure \ref{fig:cd_color_time}. We would expect all four to show a similar trend if there is real variability on the planets. The colors evolve much more gradually over time than the photometric light curves presented in the previous section, possibly indicating systematic biases are cancelling out. We find that the scatter between adjacent points is comparable to the formal uncertainties denoted by the error bars, unlike in the case of the photometric light curves where outliers were common. We argue from this that the spectral time series data is likely more robust. In night 1, we find a that the trends in the HR 8799 c colors correlate with trends in the photometry: all four flux ratios increase as the photometry increases for HR 8799 c. However, no clear trend emerges in night 2, and no consistent trend exists for HR 8799 d. The different changes in color between the two planets argues against these trends due to some spectral instability in the satellite spots as that would affect both planets in the same way. To statistically assess the significance of any periodicity in the signal, we run the same Lomb-Scargle periodogram analysis on the 1-hour time series for the flux ratios. This time, we included a 2\% satellite spot color systematic floor as discussed in Section \ref{sec:satspot-spec}. The periodograms are plotted in Figure \ref{fig:periodograms} and the peak periods are reported in Table \ref{table:ratio_peaks}. Again, no significant peaks are found, so we conclude that we should not read too much into any correlations in the HR 8799 c colors. We use the same injection and recovery test as the previous section to assess our sensitivity to color variability at the range of periods and amplitudes in Figure \ref{fig:complete-ratio}. The most sensitivity flux ratio is the $H$/$K$ ratio, for which we should be sensitive to $\sim$20\% variability amplitudes for HR 8799 c. The completeness maps also confirm that none of the peak periods reported in Table \ref{table:ratio_peaks} are significant as they all reside in areas with $\leq 50$\% completeness. \subsection{Comparison with other substellar objects} Most of the near-infrared variability in brown dwarfs and planetary mass companions have been identified with $J$-band photometric monitoring \citep{Radigan2014, Artigau2018}, making it difficult to compare to our constraints due to our poor sensitivity in $J$ band (Figures \ref{fig:complete-phot} and \ref{fig:complete-ratio}). Fortunately, there are a few suitable comparison objects that do have spectral or $H$-band variability, so we will focus on individual comparisons. We can compare these limits to the variability of VHS~J1256-1257~b, a wide separation planetary mass companion with similar mass and temperature as the HR 8799 planets \citep{Gauza2015, Bowler2020, Zhou2020}. Thus we expect similar variability behavior between the HR 8799 planets and VHS~J1256-1257~b, modulo viewing angle. The near-infrared spectra from the Hubble Space Telescope indicates large variability with amplitudes of 10\% in $H$ band 13\% in $J$ band \citep{Bowler2020}. Since their spectrum does not cover $K$ band and we have poor constraints bluer of $H$ band, the comparisons of empirical data is not very constraining. However, their model of variability \citep{Zhou2020} predicts variability amplitudes of $\sim$5\% in the F190 band and $\sim$7\% in the $K$ band (see their Figure 10, noting they are plotting peak-to-valley variability). Given the $\sim$2x smaller variability amplitudes in the F190 and $K$ bands, we expect $\sim$10\% variability amplitudes for the $H$, $H$/F190, and $H$/$K$ time series of the HR 8799 planets if they are viewed edge-on like VHS~J1256-1257~b appears to be \citep{Zhou2020}. While we are not very sensitivity to $H$/F190 or $H$/$K$ spectral variability at these amplitudes, our $H$-band photometric variability sensitivity is $> 50\%$ complete to such an amplitude for periods $< 20$~hours. Thus, it is unlikely that HR 8799 c is viewed equator-on if it is as strongly variable as VHS~J1256-1256~b. Given the current constraint on the line-of-sight inclination of the orbital plane of the four planets is $26.8^\circ \pm 2.3^\circ$ \citep{Wang2018}, the planets would need to have a high obliquity if we were viewing them equator-on. If the HR 8799 planets are not viewed equator-on, then the variability amplitudes would be much lower. The free-floating planetary mass companion PSO~J318.5-22 also has a similar temperature and mass as the HR 8799 planets \citep{Allers2016}, and has a spin axis inclination of $61^\circ \pm 17^\circ$. It only experiences a 2\% variability amplitude in the $H$ band \citep{Biller2018} that is well-below our detection limits. However, our non-detection alone does not rule out the possibility of the HR 8799 planets being equator-on, as they may not have the same high-amplitude variability as VHS~J1256-1256~b. Assuming a simple and naive $\sin(i)$ dependence of variability amplitude on inclination, the $H$-band variability of PSO~J318.5-22 would still be $< 3$\% if viewed equator-on, and an equivalent variability would be undetected in our data for the HR 8799 planets. Hotter temperature objects also are shown to have reduced variability in the $H$ band, but the variability in the water bands are not suppressed, indicating their patchy clouds reside high up in the atmosphere \citep{Yang2015,Lew2016}. Given they are spectrally dissimilar to the HR 8799 planets, it is unlikely that the exact same case is happening for the HR 8799 planets. However, we mention it as a possible reason for spectral variability to be suppressed when comparing $H$ and $K$ band to the F190 band in our analysis. \section{Spectral Modeling}\label{sec:atm-fits} Even though we did not detect significant atmospheric variability, the long time baseline of our observations allowed us to obtain high SNR spectra with which we can constrain atmospheric properties (Figure \ref{fig:hr8799_spec_fullnight}). These spectra achieve higher SNR than previous spectra from low-resolution integral field spectrographs \citep{Oppenheimer2013, Bonnefoy2016, Greenbaum2018}. Given HR 8799 c, d, and e are similar in luminosity and spectral type \citep[e.g.,][]{Greenbaum2018}, and are locked in mean-motion resonance with similar masses \citep{Fabrycky2010,Wang2018}, identifying and characterizing the similarities and differences in these planets' spectra can be insightful. Atmospheric characterization of the HR 8799 planets have been carried out previously using spectra and photometry on numerous occasions (see Section \ref{sec:intro} and references therein), and therefore we will conduct only a limited investigation here focused on our data set and defer detailed global forward model comparisons and retrieval analyses to future work. To aid in our characterization we generate a small custom grid of cloudy radiative-convective-thermochemical equilibrium models using the oft-used thermal structure code EGP \citep[e.g.][]{mckay1989,marley1996,saumon2008,marley2012,Skemer2014,Ingraham2014,Greenbaum2018} with updated opacities from \citet{marley2021}. We assume a solar C/O ratio, as suggested by recent moderate resolution observations \citep{Molliere2020,Ruffio2021}. We vary the effective temperature ($T_\textrm{eff}$) from 950 to 1400 K in steps of 50 K and the log of gravity in cgs units (log(g)) from 3.5 to 4.5 in steps of 0.25; we consider atmospheric metallicities of solar and 10 $\times$ solar; sedimentation efficiencies ($f_\textrm{sed}$) of 1 and 0.5 for silicate and iron clouds; and compare models with uniform clouds and patchy clouds with a fraction of cloud holes ($f_\textrm{hole}$) of 0.05. We base these parameter ranges on previous literature values \citep[e.g.][]{marley2012,Ingraham2014,Bonnefoy2016,Greenbaum2018}. We compare our models to data for each planet by first averaging both nights' data and then optimizing the radius of the model planets to produce a best fit. We compute the reduced $\chi^2$ with 16 degrees of freedom (22 data points minus 6 free parameters: $T_\textrm{eff}$, log(g), atmospheric metallicity, $f_\textrm{sed}$, $f_\textrm{hole}$, and the planet radius). We tested the impact of including literature data at 3-5 $\mu$m in our fits but we found no significant change in our best fit models, only higher $\chi^2$. Systematic shifts in flux calibrations between our data set and previous data sets may complicate joint fits, but when we introduced these shifts as a free parameter we found that the best fit models often tended towards unphysical radii to try to match all the data. We therefore ignored the literature data in our fit, but we do plot them alongside our best fit models to compare them (see below). We find that HR 8799 c and d are distinct from each other, which can also be clearly seen visually in the spectra (Figure \ref{fig:hr8799_spec_fullnight}). \citet{Greenbaum2018} also found differences between the two planets, although not the same differences: they found a fainter K-band spectra for HR 8799 c which they hypothesize could also be due to photometric calibration offsets. On the other hand, we generally find HR 8799 c to be brighter than HR 8799 d in the $J$, $H$, and $K$ bands, but of comparable brightness in the water absorption bands. While the larger uncertainties on HR 8799 e's spectra result in looser constraints, the shape of the spectrum is more similar to planet c than d (Figures \ref{fig:hr8799c_model}, \ref{fig:hr8799d_model}, \ref{fig:hr8799e_model}). However, its flux level is more comparable to HR 8799 d, so we find that, while HR 8799 e has similar atmospheric parameters as HR 8799 c, HR 8799 e has a smaller radius than HR 8799 c. For all three planets we find a preference for supersolar metallicity, $T_\textrm{eff}$'s of $\sim$1100 K, and largely unconstrained log(g) values. Supersolar metallicity has been suggested for all four HR 8799 planets \citep{Barman2011,Bonnefoy2016,lavie2017}, and we confirm this in our analysis. The best fit models for planets c and e do not demonstrate a preference in $f_\textrm{sed}$ or cloud patchiness among our limited parameter search. In contrast, the best fit planet d models consistently favor those with low $f_\textrm{sed}$ (vertically extended) and uniform clouds, suggesting a difference in cloud and/or dynamical processes in the atmosphere of planet d versus planets c and e. A full layer of clouds could be blocking us from seeing deep down into HR 8799 d, causing its $J$, $H$, and $K$ band fluxes to be lower than those of HR 8799 c but keeping its flux in the water absorption band comparable since the water absorption happens above the cloud layer. It should be noted that the model fits to the data for planet c is considerably worse than that for planet d (reduced $\chi^2$ of $>$8 versus $\leq$1.5, respectively); in particular, the best fit planet c model overestimates the J band data by $\sim$30\%, while higher reduced $\chi^2$ models that better fit the J band underestimate the H band flux. In addition, all of our best fit models underestimate the flux between 3 and 4.5 $\mu$m, as compared to literature data taken from \citet{Bonnefoy2016}, by a few to a few tens of percent, while matching the observed $M$ band flux well for planets c and d. Thus, the exact physical process responsible for HR 8799 d being different should be taken with caution, but it may be that HR 8799 d has more extended clouds that cover its entire surface. Discrepancies between our models and the data could arise from a number of sources. As our model grid is limited in size and resolution, especially in atmospheric metallicity, $f_\textrm{sed}$, and $f_\textrm{hole}$, we could have simply missed the best fit solution. Furthermore, there are more dimensions to atmospheric composition and clouds that we did not explore here. A more detailed forward model investigation would be needed to solve this problem. A major assumption of our work is that of thermochemical equilibrium, which many previous works have already shown to be lacking \citep{Barman2011,marley2012,Skemer2014,Ingraham2014,Molliere2020}. Our models all show a deep methane absorption feature centered at 3.3 $\mu$m, which contributes to the underestimation of the flux between 3 and 4.5 $\mu$m compared to previous observations \citep{Bonnefoy2016}. Disequilibrium processes such as vertical mixing would enrich the photospheric region with CO-rich gas thereby reducing methane absorption. The best fit planet radii of our models are $>$1.1R$_J$ for planets c and d and $\sim$1R$_J$ for planet e. Evolution models suggest radii $>$1.1R$_J$ at the ages of the HR 8799 planets, possibly as large as $\sim$1.4R$_J$ \citep{baraffe2003,saumon2008,marley2012}, in agreement with our fits for planets c and d, though we seem to be underestimating the radius of planet e. Underestimation of planet radii by radiative-convective models has been seen previously \citep[e.g.][]{Barman2011,Greenbaum2018} and is likely the result of simplifying assumptions in the model chemistry and physics. \section{Conclusion} In this paper, we present two full nights of spectrophotometric monitoring on HR 8799 c, d, and e using the CHARIS spectrograph behind the SCExAO adaptive optics system on Subaru. We first analyzed the use of satellite spots for precise spectrophotometric calibration of the data, as there are no other reference stars in the images. \begin{itemize} \item The incoherent satellite spots used for photometric calibration have a flux ratio that varies with time with a scatter of at least 3\% and drifts on the 10-minute timescales that can be even larger, so it should be used with caution as a reference for percent-level photometry. \item The spectral (color) stability of the satellite spots is better and experiences much smaller temporal correlations. We find a systematic uncertainty floor of 2\% when measuring the colors of exoplanets using the satellite spots for calibration. \end{itemize} We then extracted the spectra of the HR 8799 planets and performed time series analysis to look for periodic variations. \begin{itemize} \item We did not observe any significant photometric variability, possibly owing to unfavorable viewing geometry, but{, for rotation periods between $\sim$5-18~hours,} achieved $H$-band sensitivity down to 10\% variability amplitudes for HR 8799 c and 30\% variability amplitudes for HR 8799 d. These are the best constraints yet for these two planets. \item We also did not observe significant variability in the planets' colors, but were sensitive to 20\% variability amplitudes in $H$/$K$ flux ratio for HR 8799 c. \item Our limits exclude the highest variability models for HR 8799 c, and support the theory that these planets are not viewed equator-on, leading to lower variability amplitudes. \end{itemize} Our two nights of monitoring data can be stacked together to produce high signal-to-noise spectra of HR 8799 c, d, and e. We compared the planets' spectra to a small grid of radiative-convective-thermochemical equilibrium models with silicate and iron clouds to characterize their atmospheres. \begin{itemize} \item From Figure \ref{fig:hr8799_spec_fullnight}, we can visually see that HR 8799 c is brighter than HR 8799 d in the $J$, $H$, and $K$ bands, while HR 8799 e has a similar shape to HR 8799 c, but with lower flux. \item All three planets are best fit with models with $T_\textrm{eff}$ $\sim$ 1100 K and supersolar metallicity, while log(g) is unconstrained within the bounds of our parameter space. \item Our fits do not constrain cloud properties for planets c and e, as both high (1) and low (0.5) $f_\textrm{sed}$ and uniform and patchy (with $f_\textrm{hole}$ = 0.05) clouds were allowed. In contrast, the best fit models for planet d were consistently those with vertically extended ($f_\textrm{sed}$ = 0.5) and uniform clouds, suggesting that planet d's cloud and/or atmospheric dynamical processes are distinct from those of planets c and e. \item The best fit model planet radii for planets c and d were consistent with evolution models, while that of planet e was underestimated in comparison. \end{itemize} The HR 8799 planets still remain some of the best targets for studying the clouds of exoplanets. Possibly due to an unfavorable viewing geometry, the variability amplitude is not as high as the most photometrically variable planetary mass objects. Future improvements in high-contrast imaging instrumentation can help us reach the necessary precision to detect variability in the light curves of these planets. New ideas to alternate turning the satellite spots on and off to remove the quasi-static speckles at the same location \citep{Sahoo2020} and to generate static, incoherent satellite spots for calibration \citep{Bos2020} are promising avenues to achieve significantly higher photometric calibration precision. Improvements to the SCExAO adaptive optics system \citep{Guyon2021} and new high-contrast imagers coming to Maunakea \citep[e.g. GPI2,][]{Chilcote2020,Marois2020} should provide higher signal-to-noise measurements of the planetary spectra, which is the limiting factor in our analysis. Furthermore, the upgrades to adaptive optics systems should improve signal-to-noise at shorter wavelengths like in the $J$ band where the bulk of substellar object variability has been observed. Obtaining time series data from space using the James Webb Space Telescope would also bypass calibration issues due to observing from the ground. \acknowledgments We thank the anonymous referee for helpful suggestions that improved the manuscript. J.J.W. thanks Jean-Pierre V\'eran for helpful discussions on atmospheric turbulence. J.J.W. and P.G. were supported by the Heising-Simons Foundation 51 Pegasi b postdoctoral fellowship during the bulk of this research project. CHARIS was built at Princeton University under a Grant-in-Aid for Scientific Research on Innovative Areas from MEXT of the Japanese government (\# 23103002) The development of SCExAO was supported by the Japan Society for the Promotion of Science (Grant-in-Aid for Research \#23340051, \#26220704, \#23103002, \#19H00703 \& \#19H00695), the Astrobiology Center of the National Institutes of Natural Sciences, Japan, the Mt Cuba Foundation and the director's contingency fund at Subaru Telescope. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community, and are most fortunate to have the opportunity to conduct observations from this mountain. \facility{Subaru (SCExAO/CHARIS)} \software{{\tt pyKLIP} \citep{Wang2015}, \texttt{astropy} \citep{Astropy2013,Astropy2018} } \clearpage \bibliography{ref}{} \bibliographystyle{aasjournal} \end{CJK*}

Title: On the Edge: the relation between stellar and dark matter haloes of Milky Way-mass galaxies

Abstract: We investigate the build-up of the stellar and dark matter haloes of Milky Way-like galaxies in cosmological hydrodynamics simulations. We show that the stellar halo is made up primarily of stars stripped from a small number of massive dwarfs, most of which are disrupted by the present day. The dark matter halo, on the other hand, is made up primarily of small unresolved subhaloes ($\lesssim 10^6$ M$_{\odot}$) and a ``smooth'' component consisting of particles which were never bound to a subhalo. Despite these differences, the massive dwarfs that make up the majority of the stellar halo also contribute a significant fraction of the dark matter. The stars and dark matter stripped from these dwarfs are related through their kinematics and this leaves imprints in the phase-space structure of the haloes. We examine the relation between the location of features, such as caustics, in the phase space of the stars and dark halo properties. We show that the ``edge'' of the stellar halo is a probe of dark matter halo mass and assembly history. The edges of Milky Way-mass galaxies should be visible at a surface brightness of 31-36 mag arcsec$^{-2}$.

https://export.arxiv.org/pdf/2208.02266

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} Galaxy: halo -- Galaxy: formation -- dark matter \end{keywords} \section{Introduction} \label{intro} Within the Cold Dark Matter (CDM) model, structures form hierarchically, with small haloes forming first and merging to assemble larger objects \citep{fof}. In the early Universe, some of the haloes have potential wells deep enough to allow the gas to become cool and dense, resulting in the formation of the first stars. Smaller haloes, with shallower potential wells may remain completely dark. The formation of stars triggers the reionization of hydrogen and helium atoms. The emitted radiation heats the halo gas above the virial temperature, bringing star formation to a halt in small haloes. More massive haloes will be able to maintain or re-accrete gas and continue star formation over long periods of time \citep{efstathiou,thoul,bullock,benson, benitezfrenk}. These haloes host bright galaxies that we can observe today. The process of hierarchical structure formation leads to the build-up of the stellar and dark matter haloes of galaxies \citep{frenk85}. The dark matter halo of our own Galaxy, the Milky Way, is thus predicted to consist of its primordial dark matter component from the initial peak collapse, smoothly accreted dark matter and the dark matter that came from minor and major mergers with smaller haloes, some of which have hosted stars \citep{white_rees, wang_aq}. The majority of the Milky Way's stars belong to its disk component and the bar; however a non-negligible fraction resides in the Milky Way's extended stellar halo \citep{milkywaymass}. Within the CDM paradigm, these halo stars are the result of tidal stripping of infalling dwarf galaxies\footnote{Some fraction of halo stars could also have come from the heating of the stellar disk stars by supernovae feedback or molecular clouds, or from encounters with dwarf galaxies \citep{bensona,zolotov_halo_insitu, font_insitu, yu_outflow_halo, gomez}. In this work, we exclusively consider the dwarf galaxy origin of stellar haloes.}. The Sagittarius dwarf galaxy \citep{ibata_sag1,ibata_sag2} and its extended tidal tails \citep{majewski_sag_tails} are an example of this process happening at the present day. Past mergers experienced by the Milky Way are expected to have left an imprint in the phase-space structure of our Galaxy. Stars coming from the same progenitor can be seen to ``clump'' in angular momentum -- energy space \citep{helmi_de_zeeuw,gomez_helmi}. Spatially, mergers can leave an imprint in the form of `shells' and `streams' which have been observed in the Milky Way and nearby galaxies \citep{quinn,ibata_m31, mcconnachie_m31,cooper_shell, bernard_panstarss,shipp_des, stellar_stream_legacy_survey}. The identification of these features in the spatial distribution and kinematics of Milky Way's stars, coupled with their chemical abundances, give clues to the accretion history of our Galaxy \citep{johnston_accretion_history, bonaca}. Recently, some of these properties have been used to determine that Milky Way underwent a merger with an object now known as {\it Gaia Enceladus/Sausage} \citep{helmi_enceladus,sausage}. Other progenitors of the present-day Milky Way have been inferred from the chemo-kinematics of the Milky Way's accreted population of globular clusters \citep{kruijssen_gc}. In recent years, the {\it Gaia} satellite \citep{helmiGaia, gaia_edr3} has uncovered a number of disrupted objects within the Milky Way through the coherent kinematics of their stripped stars. Tools such as {\sc streamfinder} \citep{streamfinder} have been used to discover such objects with proper motions and deep photometry. This approach has been effective at identifying a number of globular cluster streams within the Milky Way. Some of these have been associated with dwarf galaxies which now make up the stellar halo \citep{malhan_mw_mergers}. Nevertheless, finding evidence of disrupted dwarf galaxies in stellar motions has proven to be more difficult. This is because the large velocity dispersion in dark matter-dominated dwarf galaxies result in more kinematically hot streams, where the orbits of stripped stars can vary substantially from that of the dwarf \citep{helmi_white}. Moreover, due to their higher mass, dwarf galaxies tend to sink into the centre of the Galaxy by dynamical friction \citep{amorisco_halo}. The stripped stars, particularly near the centre, become phase-mixed over time and the effectiveness of the integrals of motion in identifying coherent structures is then limited in the time-varying asymmetric potential of the Milky Way. For these reasons, it is the stars in the outer halo, with longer dynamical times, that likely hold clues on past mergers of our Galaxy. The CDM paradigm, where the halo assembles largely through tidal stripping of smaller infalling objects, predicts that the majority of the stellar halo and at least a fraction of the dark matter halo have a common origin. It is thus possible that the properties of the Milky Way's stellar halo can be used to investigate those of the dark matter. For instance, the extent of the stellar halo may be directly related to that of the dark matter. Moreover, phase-space features in the stars may also suggest equivalent features in the dark matter. Local dark matter overdensities are important features for direct and indirect searches for the dark matter particle \citep{simpson_aurigaia,necib_gaia, darkshards}. In the spherical collapse model of the formation of virialized structures, overdensities in the early Universe gravitationally attract surrounding material, causing it to collapse and virialize, leading to the formation of `caustic' shells of matter \citep{vogelsberger_caustic1, vogelsberger_caustic2}, corresponding to the apocentres of successively accreted material, with their spacing dependent on the rate of growth of the dark matter halo \citep{gunn_gott,fillmore, bertschinger}. Although this picture is simplified, structures modeled with $N$-body simulations match well analytical predictions \citep{zavala, adhikari, sugiura}. In particular, the outermost shell, corresponding to the first apocentre of the most recently accreted material is related to the ``splashback'' radius of the halo and provides a physical definition of the halo boundary \citep{diemer_and_kravtsov, diemer_2017}. With the inclusion of hydrodynamical processes in cosmological simulations, it has become possible to follow the evolution of the stars and gas after accretion. \cite{edgeofthegalaxy} have shown that together with the outermost `splashback' radius, Milky Way analogues in the APOSTLE \citep{sawalapuzzles,fattahi}, AURIGA \citep{auriga} and ELVIS \citep{elvis} simulations also have a `second caustic' in the dark matter, which roughly coincides with the visual extent of the stellar halo and is located at $\sim 0.6 R_{200,\rm m}$, or near $R_{200, \rm crit}$. This second caustic is directly measurable as the steepest drop in the log-slope of the stellar density distribution or in its radial velocity profile. An observation of the "edge" feature could thus allow us to infer the size of the dark matter halo and its properties would be directly related to the accretion history of the Milky Way. In this work, we investigate the origin of the `second caustic' in the stellar and dark matter haloes of Milky Way~/~M31 analogues in APOSTLE simulations. In Section~\ref{simulations}, we introduce the APOSTLE suite of simulations and provide some definitions that we will use throughout this work. We then split our sample of 10 galaxies into a `quiet' subsample that is more Milky Way-like and an `active' subsample that is more M31-like \citep{deason_mw_m31,pillepich,lachlan}. In Section~\ref{buildup} we examine the historical build-up and present-day composition of the stellar and dark matter haloes of these analogue galaxies. In Section~\ref{phasespace}, we examine the phase-space properties of the `quiet' and `active' galaxies, focusing on the differences between the two and the relation to the phase-space properties of their dark matter haloes. In Section~\ref{formationofcaustics}, we investigate the formation of phase-space features in the stellar and dark matter haloes, looking in particular at the mergers that contributed to their formation, their infall and tidal history. We comment on the halo and galaxy properties which influence the characteristics of the phase-space distribution. In Section~\ref{observations}, we discuss how observations of the luminous stellar component of haloes may be used to uncover the properties of the dark component and observational strategies that the present work suggests. In Section~\ref{conclusions}, we summarise our results. \section{Simulations} \label{simulations} \subsection{APOSTLE simulations} The APOSTLE (A Project Of Simulating The Local Environment) simulations are a suite of $N$-body hydrodynamical zoom simulations of environments resembling the Local Group \citep{fattahi,sawalapuzzles}. Each simulation volume features a Milky Way - M31 analogue pair. The pairs of haloes were selected to match the observational constraints on the Local Group, such as the combined halo mass, galaxy separation, relative radial and tangential velocity and the velocities of nearby galaxies. The M$_{200}$ values of the haloes range between $5\times10^{11}$ and $2.5\times10^{12}$~M$_{\odot}$. The high-resolution zoom region comprises a sphere of $\sim2.5$~Mpc from the barycentre of the halo pair, within a $100^3$~Mpc$^3$ box. The suite consists of 12 volumes simulated at low and medium resolution, while 5 volumes have also been simulated at high resolution. In this work, we analyse these five simulations, giving us a sample of 10 Milky Way-mass galaxies. Their dark matter particle mass is in the range $2.5 - 5\times10^4$M$_{\odot}$ and their gas particles have initial masses in the range $0.5-1\times10^4$M$_{\odot}$; the gravitational softening length, $\epsilon_g = 134$~pc. The APOSTLE suite was run with the {\sc p-gadget-3} code \citep{gadget}, assuming a WMAP-7 cosmology \citep{wmap7}. A {\sc Tree-PM} scheme is used to compute gravitational accelerations. Galaxy formation is modeled using the {\sc eagle} code \citep{eagle1,eagle2}. {\sc Eagle} solves hydrodynamic forces using the smoothed particle hydrodynamics (SPH) {\sc Anarchy} scheme \citep{anarchy,anarchy2} and the pressure-entropy formalism \citep{hopkins}. The {\sc eagle} model was calibrated to reproduce the $z=0.1$ stellar mass function and galaxy sizes above $10^8$~M$_{\odot}$. The model includes cooling, star formation and evolution and feedback from supernovae, stellar mass loss, active galactic nuclei and radiation pressure \citep{gascooling,starformation,starformation2,agnbooth,accretionmergers}. A uniform ionizing background is turned on instantaneously at $z=11.5$ \citep{haardtmadau}. Cooling rates are computed for 11 tracked chemical elements (including hydrogen and iron), assuming ionization equilibrium in the presence of UV and X-ray backgrounds and the cosmic microwave background \citep{enrichment}. Artificial fragmentation of the ISM is prevented by imposing a temperature floor through a polytropic equation of state. Star formation has a metallicity-dependent density threshold that effectively ranges between $n_H$ = $0.1-1$~cm$^{-3}$. Star formation is also pressure-dependent and follows the Kennicutt-Schmidt star formation law \citep{ks1, ks2}. A stellar particle within the simulations represents a simple stellar population following a \cite{chabrier} initial mass function. Feedback from star formation is implemented using the stochastic thermal prescription of \cite{anarchy}. {\sc Eagle} has been shown to reproduce the evolution of the stellar mass function, colours and magnitudes of galaxies, scaling laws of galaxy populations, while APOSTLE provides a good match to the abundances of satellites and dwarf galaxy scaling relations within the Local Group \citep{sawalapuzzles, fattahi, campbell}, including dwarf metallicities \citep{thedistinct}. \subsection{Classification of stellar and dark halo components} In this work we investigate the build-up of the stellar and dark matter haloes of Milky Way-mass galaxies. In order to do so, we must define a redshift-dependent halo boundary. A possible definition of this is the splashback radius; however, the exact definition of this radius is uncertain and complicated by the fact that we are studying group environments. Instead, we opt to use the radius enclosing 200 times the mean matter density of the Universe at each redshift, $R_{\rm 200,m}(z)$, as our definition of the halo boundary. This is also motivated by the finding that the splashback radius tends to be close to this value, and much further our than $R_{200,\rm crit}$ \citep{diemer_and_kravtsov}. We identify substructures in our simulations using the {\sc HBT+} algorithm \citep{hbt}. For each stellar particle within $R_{\rm 200,m}(z)$ of either of the main haloes at $z=0$, excluding those that are identified as bound to satellite haloes, we track the particle back in time until it is identified as being bound to a halo or a subhalo that is not one of the two main haloes. We classify all stellar particles which are bound to the Milky Way~/~M31 analogue at the time of their birth as `disk' particles. A negligible fraction of stellar particles are never identified as bound in the available snapshot outputs and we exclude these particles from the analysis. For the dark matter halo particles, we also track each particle back in time until there is a substructure match. However, unlike the stellar particles, a large fraction of dark matter comes from smooth accretion. This includes unresolved haloes and individual dark matter particles that were never bound to a subhalo. All dark matter particles which are part of one of the main haloes at $z=0$ and for which there is no historical substructure match are classified as the `smooth' dark halo component. Finally, we must decide how to treat mergers of substructures which occur {\it inside} of the Milky Way/M31halo. We opt to count the subhaloes that merged while within the halo as a single object, even if particles had been stripped within the Milky Way/M31 analogue prior to the merger. Additionally, we impose a merging criterion. Namely, we consider a subhalo to have merged with another subhalo when it has lost all of its bound mass, retaining only an `orphan' particle. To be merged with a subhalo, this particle must lie within the maximum subhalo radius, defined by the location of subhalo's furthest bound particle. We further ensure that this particle is gravitationally bound to the subhalo. We establish whether the orphan particle is bound by comparing its relative velocity with respect to the centre of the subhalo to the subhalo's local escape velocity. The latter is computed by performing an NFW profile fit to the subhalo out to the maximum subhalo radius. Finally, at $z=0$, we `climb the tree' and ensure all orphans representing substructures and substructures-of-substructures are assigned to the main progenitor. \section{The build-up of the stellar and the dark matter halo} \label{buildup} \subsection{Halo composition} We begin by studying the build-up of the stellar and dark matter haloes of Milky Way-mass galaxies. Past works have examined the build-up of haloes in dark-matter-only (DMO) simulations, where it was shown that major mergers (mass ratio $> 1:10$) contribute only 20 per~cent of the dark matter halo mass in Milky Way-mass galaxies, with the majority made up of the `diffuse' component, which includes unresolved haloes and minor mergers in roughly equal amounts \citep{wang_aq}. However, this picture need not be reflected in the stellar halo because only the most massive haloes will form a galaxy which can subsequently be deposited into the stellar halo of the Milky Way. The power-law form of the CDM mass function implies that small dwarfs are more abundant than large ones; however it is the less numerous, more massive, haloes that contain the most stellar mass. This opens up the question of whether most of the stellar halo mass is made up of stars gained in mergers with many small dwarfs or few large ones \citep{cooper,deason_mao_wechsler, deason_big_bricks, delucia_helmi, fattahi_two_pops}. Moreover, it is unclear what fraction of the stellar halo is made up of stars stripped from dwarfs that still survive at $z=0$ (e.g. Sagittarius) and what fraction is made up by dwarf galaxies that are fully disrupted by the present day (e.g. Gaia-Enceladus-Sausage). In Fig.~\ref{fig1} we show the make-up of the stellar and dark matter haloes in terms of the peak (that is the maximum value attained) stellar and dark matter masses of infalling dwarfs (upper left) and in terms of the number of dwarfs at a given peak mass (upper right). The peak in the fraction of the stellar halo contributed by objects of a given mass occurs at peak stellar mass of $M_{\rm peak, *}\sim 10^9$M$_{\odot}$, where, on average, dwarfs with this stellar mass make up $\sim 40$~per~cent of the stellar halo. However, it can also be seen that the scatter is rather large and some of the Milky Way analogues either accreted no objects of this stellar mass, or only accreted them at late times, when the amount of stellar stripping was insufficient to contribute any significant mass to the halo (surviving dwarfs are shown by the dotted red line). On the bottom left of Fig.~\ref{fig1}, we show the cumulative version of this figure. Evidently, dwarfs with peak stellar mass of above $10^8$M$_{\odot}$ contribute at least 70~per~cent of the stellar halo. We also note that typically less than 20~per~cent comes from dwarfs surviving at $z=0$. The stellar mass function of the dwarf progenitors (bottom right of Fig.~\ref{fig1}) suggests there are approximately 20 progenitors with stellar mass above $10^8$M$_{\odot}$ (7 surviving) and 6 above $10^9$M$_{\odot}$ (3 surviving) in a typical Milky Way analogue. We now focus on the dark matter. Immediately, we see that the tallest peak in dark matter halo contribution is at $\sim 10^4$~M$_{\odot}$, which is the dark matter particle resolution in our simulations. This is the contribution of the `smooth' and `unresolved' component. Among the resolved bound substructures, it can be seen that significant fractions of the dark matter halo mass come from dwarfs with peak halo mass $\sim 3\times10^{10}M_{\odot}$. The purple histogram highlights subhaloes that have hosted stars in the past. It is evident that all objects above $\sim 10^{10}M_{\odot}$ hosted stars in the past. The cumulative version of this plot (bottom left of Fig.~\ref{fig1}) confirms the previous findings that nearly half of the dark matter halo is in a `smooth' component and the other half in bound structures. It can also be seen that about 40~per~cent of the dark matter halo mass has come from objects that have hosted stars in the past and of those, nearly all had halo mass above $10^{9}M_{\odot}$, consistent with the hydrogen cooling limit arguments, while 30~per~cent of the halo mass comes in subhaloes of peak mass greater than $10^{10}M_{\odot}$. The dark matter haloes of Milky Way-mass galaxies had 100-200 luminous contributors (with stellar masses $\gtrsim 10^3 M_{\odot}$), though only 10-20 of them make up the majority of the stellar halo. \subsection{Active and quiet halo assembly} Before we proceed to examine the radial distributions of various components of the stellar and dark matter haloes, we split our sample of Milky Way and M31 analogues into those with `active' formation histories (haloes which are still rapidly increasing in mass down to $z=0$) and those with `quiet' formation histories (those whose halo growth rate over the past few gigayears is slow). We make this sample separation for two reasons. One is that we expect that the radial distributions of debris from surviving and disrupted dwarfs will be different in these two cases -- the locations of stars coming from surviving dwarfs are expected to peak away from the centre of the galaxy and the disk. Secondly, the sample separation is motivated by the distinction in the inferred formation histories of the Milky Way and M31. The Milky Way is believed to have been relatively quiet \citep{deason_mw_m31, pillepich, lachlan}, while M31 is still actively assembling. In Fig.~\ref{fig2} we show the dark matter halo assembly histories of our Milky Way/M31 analogues; the subsamples are shown in blue (active) and red (quiet). The quiet sample is characterised by analogues that had formed $\sim80$~per~cent of their mass 6~Gyr ago and had a slow growth rate after that. In contrast, the `active' halos formed $\sim80$~per~cent of their mass approximately 3~Gyr ago and their growth thereafter is fast. We find that lower-mass haloes in our sample tend to be `active', while higher-mass haloes are typically `quiet'. This may seem surprising from the hierarchical structure formation considerations. This occurs because our ``zoom'' simulations are constrained to have haloes in the Milky Way / M31 mass range. If we instead define recent accretion history as that within the last 8.5~Gyr \citep{diemer_and_kravtsov}, we recover the expected trend where high-mass haloes have more active assembly histories. We find no correlation between the assembly histories of the two main haloes in each simulation volume. There are two extreme examples in both of these categories. Within the `active' sample there is a galaxy that has built up nearly 50~per~cent of its halo mass in the last 5~Gyr. This is due to a very recent major merger. This galaxy is an outlier in this category, but is, of course, formally `active'. Within the `quiet' sample we see a galaxy which had two large mergers $\sim7$~Gyr ago and has since assembled only about ~10~per~cent of its final mass. This is distinct from the rest of the `quiet' sample, where nearly all of the final halo mass has been built up $\sim9$~Gyr ago. In the following, we keep these galaxies in their respective categories, bearing in mind that the rest of galaxies in each sample have very similar assembly histories, so that our outliers will likely not affect the median radial distributions, but would instead contribute to the scatter. \subsection{Radial distribution of halo components} We have so far shown that the stellar halo of Milky Way-like galaxies is dominated by stars stripped from a few massive dwarfs that are primarily disrupted by $z=0$, while the dark matter halo is dominated by a smooth, unresolved component. We summarize these findings in the top left of Figs.~\ref{fig3} and \ref{fig4} for the active and quiet Milky Way~/~M31 analogues, respectively. It is clear that the smooth component is the dominant contribution to the dark halo. In both active and quiet samples the disrupted dwarfs dominate the stellar halo, but surviving dwarfs contribute more significantly in the active sample. Interestingly, while the mean contributions to the stellar halo of the disrupted and surviving dwarfs are similar in the quiet and active samples, there are clear differences in the contributions of these objects to the dark matter. Specifically, disrupted and surviving dwarfs contribute roughly the same amount of dark matter in the `active' sample, yet they contribute very different fractions to the stellar halo. This suggests that substantial amounts of dark matter have been stripped from surviving dwarfs, but their stellar component has not been significantly affected. We now investigate how these components are distributed within the haloes. We aim to establish, for instance, whether the smooth halo component is dominant at all radii or only in the outskirts of haloes and whether these radial distributions are different in active and quiet Milky Way~/~M31 analogues. In the top-centre panels of Figs.~\ref{fig3} and \ref{fig4}, we show the fraction of the stellar and dark matter haloes that each component contributes at a given radius. In both active and quiet samples, disrupted dwarfs make up almost all of the stellar halo up to 0.1$R_{200, \rm m}$ and remain dominant out to $\sim 0.8 R_{200,m}$, while surviving dwarfs contribute most of the material outside that radius. In the active sample, there is a small radial range between 0.1 and 0.2~$R_{200, \rm m}$ where material stripped from surviving dwarfs dominates on average, providing $\sim65$~per~cent of the visible matter in the halo. This is material stripped recently (within the last 2~Gyr) from surviving dwarfs that wandered close to the centre of the halo. Interestingly, we do not see a corresponding peak in the dark matter contributed by surviving luminous haloes. The reason for this is the outside-in stripping of infalling dwarfs (illustrated in the bottom panels of Figs.~\ref{fig3} and \ref{fig4}), that makes the stripping times of the dark matter and the stars different. For the dark matter, in both active and quiet samples, we see that the smooth component makes up only a small fraction of the inner dark matter halo ($\sim10$~per~cent at 0.01$R_{200, \rm m}$, roughly the Milky Way half-light radius); the majority of the dark matter has come from the dark matter of disrupted dwarf galaxies. The smooth component becomes dominant at $\sim0.2R_{200, \rm m}$, but the dark matter from disrupted dwarf galaxies is still a major contributing component out to $\sim0.3 R_{200, \rm m}$ in the active sample and $\sim0.6 R_{200, \rm m}$ in the quiet sample. \subsection{Radial contribution by mass} We have now established that it is the most massive, luminous dwarfs that make up the majority of the stellar halo. However, it is still unclear whether this is true over the entire radial extent of the halo. It could be the case that the massive dwarfs dominate only in the centre, where most of the stars are expected to be deposited in a merger, while the more numerous small dwarfs with longer dynamical friction sinking timescales deposit their stars in the outer halo, dominating the local stellar content. To address this question, we split the disrupted and surviving dwarfs by mass, such that massive dwarfs above a given threshold mass make up 50~per~cent of the stellar halo. We then show the radial contribution of each component to the stellar halo. In agreement with the upper left panel of Fig~\ref{fig1}, we find that this threshold peak stellar mass is typically $\sim10^9$~M$_{\odot}$ and ranges between $10^{8}-10^{9.5}$~M$_{\odot}$ for our Milky Way~/~M31 analogues, corresponding to LMC/SMC-mass dwarf galaxies. It can be seen that massive dwarfs, surviving and disrupted, contribute over $\sim30$~per~cent of the mass at all radii. The massive disrupted dwarfs clearly dominate the stellar halo content within $0.1R_{200, \rm m}$. The contribution of the most massive dwarfs diminishes between 0.1 and 1$R_{200, \rm m}$. Indeed, the smaller dwarfs become more important in the `intermediate' halo regions. This is likely because their orbits are relatively more tangential due to the reduced effects of dynamical friction. We do find that the debris stripped from lower-mass dwarfs has {\it on average} larger orbital apocentres, however the most massive haloes typically have larger {\it maximum} apocentres due to their larger size and velocity dispersion. It is also clear that the peak in the contribution of surviving dwarfs seen between 0.1 and 0.2 R$_{200,\rm m}$ in active galaxies (top-centre panel) is not caused by the most massive dwarfs that entered the halo. This is not surprising, given that in Fig.~\ref{fig1} we saw that the peak masses of disrupted dwarfs are on average greater than those of surviving ones. In other words, it is likely that the most massive contributor to the Milky Way's stellar halo (in terms of peak stellar mass) has already been disrupted. This picture is somewhat different in the dark matter, where the most massive mergers contribute typically no more than $30-40$~per~cent of the dark matter coming from luminous subhaloes. This is most likely a consequence of the stellar mass--halo mass relation \citep{behroozi, moster}, whereby dark matter dominates the mass of smaller dwarfs (e.g. dwarf spheroidals and the ultra-faints) more so than in the dwarf irregulars (e.g LMC). As a result, smaller dwarf galaxies contribute a larger fraction of their mass in dark matter than the larger dwarfs. Overall, it is remarkable that out to the outermost radius within the halo, the material of 3-4 massive dwarfs can make up nearly half of the stellar halo. \subsection{Radial gradients in accretion and stripping time} We now examine the radial accretion time gradients in the stellar and dark matter haloes, both regarding the time of entry into $R_{200,m}$ and the time when the material was stripped from infalling subhaloes\footnote{ The stripping time is not defined for the smooth halo component, so instead we use the time when these particles entered $R_{200, \rm m}(z)$.} (lower panels in Figs~3 and~4). The radial distribution of the stellar accretion times shows a remarkably flat profile both for disrupted and surviving dwarfs out to large radii, in both subsamples. This either suggests that the majority of dwarfs contributing to the stellar halo were accreted at roughly the same time or that the stellar halo is dominated by one or two objects. In the quiet sample, this may be the case, as few disrupted dwarfs make up nearly half of the stellar halo at all radii. For the active sample, above $\sim 0.4-0.5R_{200, \rm m}$ the stellar halo has an increasing contribution from more recently accreted objects, both surviving and disrupted. When looking at the radial stripping time (bottom centre of Fig.~\ref{fig3}), we begin to see a mild gradient, whereby material near the halo centre was stripped earlier and the material in the outskirts later on, before the slope flattens, such that above 0.2$R_{200, \rm m}$ the halo material was stripped at approximately the same time. This suggests either that the material deposited in the outer parts of the halo from a given dominant object was stripped later on (which seems unlikely as we expect the infalling dwarfs to spiral towards the centre and deposit their stars there), or that the outer regions of the halo are dominated by dwarf galaxies that came on wider orbits, allowing their stars to be stripped later and at larger radii. Overall, for the disrupted dwarfs we observe an offset between the accretion and stripping times of approximately 2~Gyr, while for surviving dwarfs this difference amounts to $\sim5$~Gyr, suggesting that the surviving dwarfs are harder to strip due to the particulars of their mass, the mass of the Milky Way analogue and the orbit. For the dark matter, we see that the `smooth' component in the inner regions is likely of primordial origin, while in the outer regions, where it dominates, the smooth component was deposited into the halo more recently. The latter is, however, also true for other constituents of the dark matter halo, where the outer regions are made up of dark matter accreted recently. Interestingly, the radial distributions suggest that in the outermost regions the dark matter particles had been stripped prior to entering $R_{200, \rm m}$. We also see this odd behaviour for the stars that came from dwarfs that are disrupted. This may be in line with the findings of \cite{wang_aq}, who suggest that some of the "smooth" dark matter component in the halo could have come from previously bound structures that lost dark matter particles during a merger prior to infall. Moreover, considering that we are analysing Local Group-like environments, it is plausible that some of the material could have come from subhaloes that had been previously stripped within the companion halo, then ejected, before infalling into the Milky Way analogue. However, we have found that these objects make up no more than 10~per~cent of the material currently outside $R_{200,\rm m}$. Instead, there are two main sources for the apparent inconsistency between the `stripping' and the `accretion' time. The first contributes at small and large radii and is due to dwarfs that merge with the Milky Way analogue at early times, when R$_{200,\rm m}(z)$ is rather small and thus it takes a while for some of the dark matter particles, which become unbound during the merger, to formally cross R$_{200, \rm m}(z)$ and therefore to be identified as having been ``accreted''. This is particularly likely when the incoming dwarf itself is in the process of assembly and has an extended halo of loosely bound material. The second source is more important in the outermost regions of the halo and consists of massive, LMC-like, dwarfs that have entered the halo recently, so that some of the particles still have not crossed R$_{200,\rm m}(z)$. In both cases, the wide spread in particle binding energies in these massive dwarfs can lead to a sufficiently small tidal radius, and the dark matter can become unbound prior to crossing R$_{200, \rm m}(z)$. It is additionally interesting to note that the discrepancy between the accretion and stripping times is more pronounced in active galaxies than in the quiet sample. This may explain why the disrupted dark haloes contribute more in the active galaxies near $R_{200, \rm m}$ than they do in the quiet sample. As the active sample grows significantly in dark matter in the last 3~Gyr, it is likely that these dark haloes were `pre-processed' by large objects that entered $R_{200, \rm m}$ at late times. It is clear that the central component is made up of stars stripped earlier, while the outer halo is made up of stars stripped later, which is also the case for the dark matter. However, one can also see that all radii, for disrupted and surviving dwarfs, the dark matter is stripped earlier than the stars. In the bottom right of Figs.~\ref{fig2} and \ref{fig3}, we compare the cumulative distributions of stars and dark matter stripped from disrupted and surviving dwarf galaxies. It can be seen that the stars are significantly more centrally concentrated than the dark matter, with a half-mass radius of $\sim0.05 R_{200,\rm m}$ for the disrupted dwarfs, compared to $\sim0.1 R_{200,\rm m}$ for the dark matter. Similarly, half of the mass in stars stripped from surviving objects is at $\sim0.15 R_{200,\rm m}$ for the stars and $\sim0.5 R_{200,\rm m}$ for the dark matter. This reinforces the idea that dark matter in dwarf galaxies typically gets stripped earlier and more efficiently than the stars. \section{Phase-space features in the stellar and dark matter haloes} \label{phasespace} \subsection{The radial velocity -- distance diagram} In this Section we examine some of the orbital properties of the stars and the dark matter in the halo. On the left of Fig.~\ref{fig5} we focus on one example of an active halo. In the top panel, we show the $v_r-D$ diagram for all the stars in the halo (i.e. stars stripped from dwarf galaxies). In the centre we show dark matter that has come from luminous haloes and at the bottom we show all dark matter, including the smooth component. Similarly, in the left panel of Fig.~\ref{fig6}, we show an example of a quiet galaxy. The dashed black lines show the computed log-slope of the density profile. These log-slope profiles are of interest in establishing the ``splashback'' radius of the dark matter and the ``edge'' of the galaxy, corresponding to minima in the log-slope of the profiles. In order to compute the log-slope profiles, we follow the procedure of \cite{edgeofthegalaxy}. Namely, we bin the particles in 75 radial $\log_{10}$-spaced bins and in 11 angular-spaced bins. Since we consider Local Group-like systems, the splashback radii of the two main haloes may overlap. In order to circumvent this, we discard the angles $\cos(\theta) <-0.6$ and $\cos(\theta) > 0.6$ measured from a vector joining the two halo centres. For each radial bin, we then take the median of all bins in angle. Furthermore, we apply the fourth-order Savitzky-Golay filter \citep{savitzky} over the 15 nearest bins to smooth the density profiles and compute the log slope. We use the same number of bins for the stars and for the dark matter. Note that we use a greater number of radial and angular bins than \cite{edgeofthegalaxy}, who analysed a lower-resolution version of APOSTLE\footnote{Since we take a median of the angles, the `edge' of the galaxy we infer is defined by particles that are generally well phase-mixed by the present day, rather than by potentially denser but highly anisotropic particle distributions arising from very recent accretion events \citep{mansfield}. We have also carried out convergence studies, varying the number of radial and angular bins and found that the locations of the steepest log-slope minima do not vary significantly so as to affect the conclusions of this work.}. Examining the two examples visually, it is clear that `quiet' galaxies are more structured in phase space, with clear `shells' of particles moving on similar orbits. The `active' galaxy exhibits less clear structure, with only some shells visible. In the outer regions, the particles stripped from currently surviving objects are also visible. These correspond to `streams' one would observe stemming from dwarfs like Sagittarius. We note that these are less likely to contribute to the fluctuations in the log-slope profile, as in computing these profiles we take the angular average for a given bin in distance. In the log-slope profile of the stars , several features are visible, with log-slope values $<-6$. The dark matter, on the other hand, is smoother than the stars on this diagram. Some shells can be seen, but these are hard to distinguish from the background. In the log-slope profile, one can clearly see the splashback radius just outside $R_{200,\rm m}$ in the quiet example, and slightly inside $R_{200,\rm m}$ in the active example (where the log-slope drops to -4); this is consistent with the results of \cite{diemer_and_kravtsov}. Numerous other minima in the log-slope of the dark matter can also be seen, which seem to coincide roughly with the features in the stars, interestingly including the splashback feature. The similarities in the orbital properties of the dark matter and the stars can be seen more clearly if we only select the dark matter coming from stripped luminous haloes (which, we note, is comparable in amount to the smooth component, though the latter is dominant in the outer regions). This is shown in the middle left panels of Figs.~\ref{fig5} and \ref{fig6}, where it is clear that `shells' of dark matter closely follow those of the stars and the features in the log-slope of the density profile are significantly more pronounced. As in \cite{edgeofthegalaxy}, we see a pronounced feature in the log-slope of the stars at $\sim$0.6R$_{200,m}$, for both active and quiet examples. In fact, several features are seen of comparable log-slope. We note that since \cite{edgeofthegalaxy} analyzed the lower-resolution version of the APOSTLE simulations, some of the drops in the log-slope would likely combine into a single feature in their analysis. Likewise, if we reduce the number of radial bins, nearby log-slope features can merge into one, while varying the number of angular bins can result in different steepness of the log-slope caustics. We now explore whether any of the log-slope features are common among our samples of active and quiet Milky Way analogues. For this, we stack the haloes in each subsample, weighting each particle by the inverse of the total stellar/dark matter halo mass within $R_{200,\rm m}$ of the halo. We additionally normalize the radial velocity by the value of the circular velocity at R$_{200,\rm m}$, to give equal weight to each Milky Way analogue. We then compute the log slope. We find the steepest feature in the log-slope of the stars between 0.5-0.6$R_{200,\rm m}$ for the quiet sample, and at 0.4$R_{200,\rm m}$ for the active sample. Note that this reflects the radii at which the contributions of disrupted luminous haloes are important in the two samples. In Fig.~\ref{fig7} we demonstrate the similarity between the local minima in the log-slope profile of the stars and the dark matter. For each minimum in the stars, we find the nearest minimum in the dark matter (blue squares). We see that the minima in the log-slope of the stars often have a nearby minima in the log-slope of the dark matter. Equivalently, the (over/under)densities in the stellar distribution can be associated with (over/under)densities in the dark matter. We do, however, see some differences at small radii. This is a by-product of the noise in the distribution of the halo stars, compared to a much smoother distribution of the dark matter in the inner regions, given the binning we use to compute the profiles. Red star symbols show the location of the steepest drop in the log-slope of the stars. The black squares show the splashback radius identified in the log-slope profile of the dark matter. The green squares show the nearest minimum in the dark matter to the left of the splashback radius (i.e. ``the second caustic''). One can see that this is often not the steepest drop in the log-slope of the stars (i.e. the "edge" of the galaxy, as defined by the steepest drop in the stellar density log-slope, does not necessarily coincide with the second dark matter caustic). Five out of 10 Milky Way~/~M31 analogues have their steepest density log-slope between 0.5-0.6 $R_{200,\rm m}$, two are located closer to $0.3R_{200,\rm m}$ and 3 above $0.7R_{200,\rm m}$. We have previously shown that some of these discrepancies are due to the accretion histories of these galaxies (i.e. active or quiet) . We will examine this point further in the next section, when we will discuss the origin of the shell features in the $v_r-D$ space. \section{Formation of caustics} \label{formationofcaustics} \subsection{Dark matter and stars stripped from infalling dwarfs} We have so far established that the stars and dark matter stripped from dwarf galaxies get deposited in the halo, forming `shells' in $v_r-D$ space. In particular, stars and dark matter piling up close to their apocentres cause radial overdensities, leading to features in log-slope of the density profile. In this Section, we examine how the stars and the dark matter stripped from dwarf galaxies infalling into the Milky Way~/~M31 analogues are distributed in the halo. We will focus on the particular example of a quiet galaxy, {\sc V1 0}, which we have previously shown on the left panel of Fig.~\ref{fig6}. For this analogue, we identify the biggest contributors to the stellar halo, both disrupted and surviving, and plot the histogram of the stripped particles' locations within the halo. This is shown on the left of Fig.~\ref{fig8}. Different contributors are identified with different colours, ordered by the stellar mass contributed to the halo. Note that this is not necessarily reflected by the peak stellar mass and that the greatest contributors of stars are not necessarily the greatest contributors of dark matter. We see a number of interesting features. First, we are able to determine which dwarfs cause the `overdensities', corresponding to particles piling up at the apocentres, and how these lead to the fluctuations in the log-slope of the stellar density (black solid line compared to black dotted line). Secondly, we see that the coincident minima in the log-slopes of the stars and the dark matter are not necessarily caused by the same dwarf galaxies. For example, the minimum in the stars at $\sim0.8R_{200,\rm m}$ is caused by the stars stripped from dwarf {\sc 837}, whereas the corresponding feature in the dark matter seems to be due to some combination of {\sc 483, 577} and {\sc 837}. We also see that each dwarf contributes substantial amounts of dark matter at each `peak' (likely, subsequent apocentres of stripped particles as the dwarf sinks), with some increase in contribution towards the centre. At the same time, the stars are stripped in small amounts at the outskirts of the halo and substantially more towards the centre. Moreover, `peaks' of stripped stellar and dark matter particles from the same dwarf galaxy do not appear to always align. This may be one of the reasons for the offsets observed between corresponding log-slope minima in the stars and the dark matter: overlapping contributions from various dwarfs and the differences in the stellar and dark matter stripping \citep{libeskind}. In the middle panel of Fig.~\ref{fig8}, we show the radial composition of the stellar and dark matter haloes in cumulative form, focusing on the contributions from the main stellar contributors, but showing in addition the contribution from smaller luminous and dark haloes and the smooth component. For the stellar halo in particular, it is clear that our interpretation of the role of the top 4 mergers in creating the caustics in the log-slope density profile is correct. For the dark matter, the picture is somewhat more complex, as the four main stellar contributors account for no more than 15~per~cent of the dark matter halo out to $R_{200,\rm m}$; this figure primarily captures the radial contribution of each component rather than the overall density. However, one can see that of all components, it is the contributions of dwarfs {\sc 577} and {\sc 483} that show oscillatory behaviour that can be associated with variations in the log-slope. It is remarkable that objects that contribute no more than 10-20~per~cent of the dark matter halo out to $R_{200,\rm m}$ can cause such strong variations in the density slope, which are also observable in the stellar distribution. \subsection{Completed pericentres } It is tempting to associate the consecutive `peaks' in the locations of particles of a given dwarf galaxy to the streams stripped off as the dwarf sinks to the centre of the host galaxy due to dynamical friction. As particles are primarily stripped near pericentre, in this section we explore how the number of pericentres completed by the particles in the halo relates to the particle spatial distribution, and thus to variations in the log-slope of the density profile \citep{diemer_pericentres}. \subsubsection{Counting pericentres} We find the number of pericentres that a particle has completed by counting the number of times that its radial velocity, v$_r$, has changed sign. We also require that an apocentre count can only follow a pericentre and vice-versa. There are, however, some caveats to this method. Firstly, we are largely limited by the frequency of output times in our simulations, which decrease with decreasing redshift. If a particle has completed an orbit and the orbital time is shorter than the time interval between two simulation outputs, a pericentre cannot be counted. This is a major problem of this method. However, as we show in the following, the particles which make up the outer regions of the halo have long orbital time periods and would have completed typically no more than 4 full orbits during the dynamical time of the Milky Way analogue. These particles are immune to the limitation of infrequent simulation time outputs. Another limitation is that subhaloes can interact with each other within a host galaxy, sometimes flipping the sign of $v_r$ with respect of the Milky Way. We avoid this problem issue by requiring that a pericentre or an apocentre of an orbit be counted only if the change in the sign of $v_r$ lasts longer than one simulation output. Again, this can undercount the number of pericentres or apocentres for orbits with short time periods, but has no effect on orbits with long orbital times which are of interest here. \subsubsection{Stellar and dark matter halo split by number of pericentres} On the right panels of Fig.~\ref{fig8}, we show the radial contribution of particles with consecutive numbers of completed pericentres for the stars (top) an dark matter (bottom). It is remarkable to see the differences between the two. The stars have a distinctly clumpy distribution, while the dark matter resembles almost evenly spatially distributed `shells' of matter on consecutive pericentres. In both cases, however, particles with more completed pericentres dominate at smaller radii and particles with fewer pericentres dominate at the outskirts. A clear distinction between stars and dark matter in this case is the presence of dark matter particles that have had one pericentre and one apocentre (lighter blue). These particles effectively define the splashback radius of the halo and there are almost none visible in the stars. This likely reflects the differences in the build-up of the two types of haloes. Dark matter includes the smooth component as well as dark and luminous subhaloes which altogether dominate the outer halo (see centre of Fig.~\ref{fig8}). The stellar halo, on the other hand, is dominated by the debris from a few past mergers in these regions, which have almost no stars with apocentres reaching $\sim R_{200,m}$ (left of Fig.~\ref{fig8}). The `edge' of the galaxy, as defined by the steepest drop in the log-slope of the stellar density distribution, is coincident with the region where the three-pericentre material dominates, while across the two steepest log-slope drops, the 2-pericentre material is dominant overall. The latter could be connected to dwarfs {\sc 577} and {\sc 837} contributing to the stellar halo (see centre of Fig.~\ref{fig8}). \subsection{Examples of past major mergers} We now examine in more detail the processes that lead to the formation of shells in the $v_r-D$ diagram. First, we identify dwarfs the which contributed the most stars to the stellar halo of the Milky Way/M31 analogue. We then follow the history of these objects -- their infall into the Milky Way~/~M31 analogues and the stripping of their stars and dark matter. Figure~\ref{fig9} demonstrates the history of two dwarfs that merged into two different Milky Way~/~M31 analogues. In the upper panel, the infalling dwarf enters the halo on a very radial orbit, reaches its first pericentre at $\sim8$~Gyr and has one further apocentre before merging with the host halo. The orbital apocentre can be seen to be rapidly damped by the effects of dynamical friction \citep{amorisco_halo}. The colours in this plot denote the number of pericentres that the particles have gone through by $z=0$, with blue showing 1 pericentre and red 4 pericentres. We do not include particles with more pericentres in this figure. Firstly, one can see that the material located in the outer stellar halo has been stripped primarily at the first pericentre of the dwarf's orbit. From there, the stellar particles are dumped on a wide range of orbits, showing a spread in orbital energies that leads to some particles having fewer pericentres than others. Those with longer orbital times are the particles making up the outer regions of the stellar halo. In the middle panel of Fig.~\ref{fig9}, it can be seen that the `shells' in the $v_r-D$ diagram, corresponding to particles with varying numbers of pericentres coincide with the minima in the log-slope of the density. The particles that have only had one pericentre (the first pericentre of the merging dwarf) are ejected in some cases beyond the splashback radius of the halo (solid black line) - some can become unbound. In the $v_r-D$ diagram, these particles look as if they are being accreted onto the halo, with generally negative radial velocities, while in the X-Y diagram it can be seen that they form a kind of a `jet' from the centre of the halo. These are akin to Gaia-Enceladus ``arches'' seen in the Toomre diagram \citep{koppelman, naidu_lmm}, which likely originate from stars on prograde orbits within the disk of the infalling dwarf or stars belonging to the dwarf's extended stellar halo. These stellar particles, and their apocentres, could be the best stellar tracer of the splashback radius of the halo (provided the halo has not significantly grown since the merger). The particles which have undergone at least two pericentres, and were originally dumped with lower orbital energies, are the ones that define the steepest drop in the log-slope of the stars. These are clearly on bound orbits within the halo and define the stellar halo `edge', as proposed by \citet{edgeofthegalaxy}. We observe a similar behaviour in the merger shown in the lower panel of Fig.~\ref{fig9}. In this case, however, the merger comes in on a more tangential orbit. The dwarf is able to complete two full orbits before effectively merging with the host halo. The increased circularity of the orbit is also evident in the $v_r-D$ diagram, where the shells of particles are clearly more circular, compared to the `sharper' shells of a merger on a more radial orbit. The X-Y plot on the bottom right of Fig.~\ref{fig9} demonstrates a distinct `umbrella' shape, characteristic of mergers with higher angular momentum \citep{martinez-delgado}. A `jet' of one-pericentre stars is visible once again and extends out to beyond the splashback radius of the halo. Note that the spread in orbital phases means that the `jet' bends over, making a loop. Since the particles spend more time near apocentre, the density is enhanced in the outer radii. In Fig.~\ref{fig10} we show the binding energies of particles that have undergone 1, 2 and 3 pericentres within the Milky Way analogue. In the top panel, we show the binding energy with respect to the dwarf galaxy at the time of infall (left) and with respect to the host halo at $z=0$ (right). We show corresponding properties for the dark matter at the bottom. It can be seen that the particles with successively smaller number of completed pericentres were less bound in the dwarf galaxy at infall. In the dark matter, one can see already a tail of unbound particles at infall, while the stellar component is to a large extent still bound. This confirms that the dark matter is stripped earlier and more effectively than the stars. At the same time, comparing the binding energy distributions within the Milky Way analogue at $z=0$, one can see that the binding energies of the stars and the dark matter with the same number of pericentres are remarkably similar. As the binding energy is, in effect, the total energy of the orbit, it best traces the apocentre of the orbit. This figure thus reinforces the idea that the orbits of stellar particles stripped from a dwarf follow the orbits of dark matter particles of the same energy and have similar phase-space features. This also suggests that semi-analytical dark matter particle tagging techniques can give faithful representations of the stellar distributions \citep{bullock_and_johnston, gomez_aquarius, cooper_tagging}. \subsection{The relation between past mergers and features in the log-slope of the halo density profile} We now examine in more detail the relation between particle apocentres and the ``edge'' of the stellar halo, defined here as the location of the steepest drop in the log-slope of the stars. On the left panel of Fig.~\ref{fig11}, we display the merger histories of the 10 Milky Way~/~M31 analogues (identified with lines of different colours). The vertical location of each point represents the 99$^{\rm th}$ percentile of the apocentres of stars stripped from each dwarf. The dashed line marks the location of the steepest caustic. The sizes of the points reflect the fraction of the halo that each dwarf contributes, while the colours reflect the stellar masses of each dwarf. In this figure we only include objects that contribute at least 1~percent of the accreted stellar halo mass. Several trends are visible. Firstly, the oldest mergers tend to have smaller particle apocentres, and so they do not typically define the ``edge'' of the stellar halo. We can also see that the oldest mergers are typically less massive and thus contribute a smaller fraction of the halo. More recent mergers, on the other hand, have stripped off particles that reach successively larger radii. In each case, we can identify the mergers which most likely define the galaxy ``edge''. These are typically dwarfs that fall in later and are, on average, more massive than those that came in earlier. As such, the stellar halo ``edge'' is caused by the apocentre pile-ups of the stars stripped in the last big merger. We now seek to establish the conditions that determine our ability to use stellar tracers to map dark matter. We have previously seen that log-slope features can be identified in both components at roughly similar locations; however the stripping of the dark matter is more efficient than that of the stars and thus the ability of the stars to trace the dark matter in the halo will depend on the efficiency of the stripping and the similarity of the velocity distributions of stars and dark matter in the infalling dwarf. In the middle panel of Fig.~\ref{fig11}, we show the stellar mass of a dwarf, which is related to the stellar velocity dispersion, and the ratio of maximum apocentres of the stripped stellar and dark matter particles. It can be seen that dwarfs that have more stellar mass -- and thus higher velocity dispersion -- have stripped stars that follow the stripped dark matter orbits more closely. This relation shows scatter, which appears to be related to the time of the merger. For a given stellar mass, the stripped stars trace the dark matter better for early mergers. This suggests that tidal stripping may be more efficient early on in the history of the Milky Way and could, in part, be due to a lower concentration of the dwarfs at higher redshift. Given the differences between the stripping of stellar and dark matter particles, it interesting to ask if there is a relation between the edge of the stellar halo and the location of the splashback radius. On the right of Fig.~\ref{fig11}, we show the ratio of ``edges'' of merger debris to the splashback radius as a function of M$_{200,\rm m}$. The points are coloured by the fraction of total present-day mass within $R_{200,\rm m}$ that was in place at the time of the merger (this includes the merging dwarf). We can see that the more massive Milky Way~/~M31 analogues typically have galaxy ``edges'' that extend closer to the splashback radius, although there is some scatter. We note that the ``edges'' do not necessarily correspond to the furthest apocentres of the merger debris in some cases (see Fig.~\ref{fig7}), though the stellar component beyond the edge is typically rather diffuse. We find that the ``edges'' of our Milky Way analogues range between 0.2-0.65 R$_{\rm splash}$. We can see that the cases where the ``edge'' of the halo is more embedded also corresponds to the cases where the halo has not assembled more than 50~percent of its final mass at the time when the largest contributors to the stellar halo merged with the main galaxy. These objects correspond to our `active' sample of Milky Way~/~M31 analogues. Overall, Fig.~\ref{fig11} suggests that there is no direct conversion between the galaxy ``edge'' and the splashback radius; relating these two properties may require knowledge of the recent growth history of the dark matter halo. We expand on this feature of some of our Milky Way/M31 analogues in the next section. \subsection{What determines the ``edge'' of the galaxy?} We now explore the main factors behind the location of the steepest drop in the log-slope of the density profile of the stars relative to $R_{200,\rm m}$. From past work on the splashback radius of the halo, some of the main factors were identified as the halo mass and the mass accretion rate of the halo, typically defined from a halo dynamical time of $\sim 8.5$~Gyr ago to the present day. While our analogue galaxies were selected to match the constraints for the Milky Way and M31, there is still a noticeable variety in stellar and dark matter halo masses. In Fig.~\ref{fig12}, we show the mass assembly histories of the dark matter halo (left) and the stellar halo (centre). The right panel shows the evolution of the stellar-to-dark matter halo mass ratio. The lines are coloured by the location of the galaxy `edge', defined by the location of the steepest caustic. From the left panel of Fig.~\ref{fig12}, in agreement with the right panel of Fig.~\ref{fig11}, it is clear that the more massive dark matter haloes typically have their stellar halo edge further out relative to $R_{200,m}$, whereas the less massive haloes have a stellar halo edge that is more embedded. This suggests that dark matter halo mass is an important driver of the location of the stellar halo edge. Nevertheless, we also see a number of outliers -- for example, the fourth most massive halo has an edge at $\sim0.5 R_{200,\rm m}$, while the fifth most massive has it at $0.7 R_{200,\rm m}$. Galaxies with bigger stellar haloes also tend to have a higher stellar mass, though it does not appear to be the case that the stellar mass alone can explain the location of the halo edge (middle panel of Fig.~\ref{fig12}). To further investigate the source of this diversity, we look at the historical stellar-to-dark matter halo mass relation (right panel of Fig.~\ref{fig12}). We see that the more massive haloes also tend to have higher $M_*/M_{\rm DM}$, as expected \citep{behroozi, moster}. If the total stellar mass to dark mass in the infalling dwarf is relatively high, this could result in more stars being stripped earlier on after infall, allowing stars to trace the stripped dark matter further out in the halo and simultaneously `pushing' the stellar halo edge further out. However, it does not appear to be the case that the smaller values of $M_*/M_{\rm DM}$ lead to more embedded stellar haloes. For example, the Milky Way~/~M31 analogue with the lowest $M_*/M_{\rm DM}$ has its stellar halo edge at $\sim0.6R_{200,\rm m}$, while a halo with $M_*/M_{\rm DM} \approx 0.003$ has its edge at $\sim0.3R_{200,\rm m}$. However, there is one feature that distinguishes the 'blue' curves from the `green': the rate of change of $M_*/M_{\rm DM}$, whereby haloes with more embedded haloes have $M_*/M_{\rm DM}$ declining more steeply with time. From the middle panel of Fig.~\ref{fig12}, it is clear that the stellar haloes are almost completely assembled $\sim8$~Gyr ago. This suggests that the main driver of the change in $M_*/M_{\rm DM}$ is a faster dark matter assembly. The growth of the dark matter halo in the last few gigayears leads to the increased concentration of the stellar halo within the dark halo, although the total halo mass is also an important factor. The Milky Way~/~M31 analogues where the ``edge'' is more concentrated relative to $R_{200,\rm m}$ or the splashback radius belong to our `active' sample. Looking at the stack of this sample on the right of Fig.~\ref{fig5}, it is clear what causes this accelerated growth in the dark matter mass: these objects tend to have a massive satellite that deposits a large amount of the dark matter into the halo, but not so much in the stars. This is reminiscent of the Large Magellanic Cloud, which could have contributed a significant fraction of the total mass of the Milky Way, but has not experienced significant stripping of stars as it is likely on its first infall \citep{besla_lmc, conroy_lmc,petersen_lmc, lmc_mw_mass}. \section{Observational prospects} \label{observations} In this section, we discuss the prospects for identifying the remnants of the past mergers that contribute to the stellar halo through their kinematic and chemical properties. We further explore whether halo `edges' of external galaxies can be identified with deep photometry and connected to the underlying dark matter halo and its assembly history. \subsection{Chemo-kinematic properties of past mergers} In Fig.~\ref{fig13}, we select 5 important mergers in the history of a quiet Milky Way analogue. These are identified with different colours and the symbol size corresponds to the fraction of stars they contribute to the stellar halo. It can be seen that these different dwarf galaxies are indistinguishable in the energy-angular momentum space at small radii, where most of the data are available (the right panel of Fig.~\ref{fig13} shows the typical galactocentric distance at each energy). Any significant deviations from the mean energy and angular momentum can only be seen in the outer regions of the halo, beyond $\sim 100$~kpc or so, where the stars have not yet phase-mixed. However, the contributing dwarfs have somewhat different metallicities. This suggests a way to distinguish past mergers. Moreover, a metallicity gradient can be seen, whereby the stars in the outer halo are more metal-poor than the stars in the inner halo. The stars which are more bound also tend to have lower metallicities, suggesting that the most bound stars come from the most ancient mergers. While the stars are significantly phase-mixed in the inner regions, making it difficult to distinguish different progenitors through their kinematics, the metallicities can differ sufficiently to tell the separate components apart. If the stellar ages are taken into account, one could use the redshift-dependent mass-metallicity relation to disentangle the different progenitors (note that in this case, mergers that contribute the most stellar mass are more metal-poor, as they are also older). Overall, our results suggest that detailed chemistry and, ideally, stellar ages are required to disentangle the origins of individual stars in the inner regions of the halo (see e.g. \citealt{naidu}), while kinematics are sufficient to identify individual structures in the outer halo. Note, however, that our results also suggest that the stars stripped from the same dwarf can have prograde and retrograge motions and in that case one must employ additional information to avoid classifying these as separate structures \citep{virgo_herc_common, kim_multiple, amarante}. \subsection{The surface brightness of the stellar halo edge} We have so far shown that some of the most important mergers that define the `edge' of the stellar halo of Milky Way-like galaxies leave the most kinematically distinct traces in the outskirts of haloes, beyond $\sim100$~kpc. Aside from kinematics, we have shown that the stellar density profile also shows variations due to particle `pile-ups' at apocentre, which are associated with the dark matter halo mass and assembly history. What do the variations in the log-slope of the stellar density profile mean in terms of surface brightness? To compute the AB V-band magnitudes of the stellar particles in our simulations we use the {\sc fsps} software \citep{fsps1,fsps2}, where we adopt the Padova stellar isochrone library \citep{padova1,padova2} and a \cite{chabrier} initial mass function. For each stellar particle, we provide its age and smoothed metallicity to the code. In the top panel of Fig.~\ref{fig14}, we show the surface brightness profiles of our 10 Milky Way~/~M31 analogues, computed using an arbitrary projection. In order to avoid contributions from the closest galaxy (as our systems form a Local Group), we exclude all particles within $R_{200,\rm m}$ of the companion halo. As for the 3D profiles, we compute the profiles in 75 radial bins and 11 angular bins, taking the median of the angular bins at each radius. We smooth the profiles using the Savitzky-Golay filter \citep{savitzky}. Each analogue is identified with a unique colour. One can see clearly that the surface brightness profiles of the haloes are not smooth, but exhibit variations similar to those found by \cite{edgeofthegalaxy}. These variations are substantially more pronounced in the computed surface brightness slope (bottom panel of Fig.~\ref{fig14}). The vertical dashed lines in Fig.~\ref{fig14} show the location of the steepest drop in the log-slope of the 3D stellar density profile. This is typically further than the nearest drop in the log-slope of the surface brightness, as expected from projection effects. While the location of the halo edge varies, as we have discussed previously, the typical location, 0.5-0.6~$R_{200,\rm m}$, occurs at a V-band surface brightness of between 31-36 mag~arcsec$^{-2}$, marginally achievable with the Euclid Deep Survey \citep{euclid} and the Hubble Ultra Deep Field \citep{hubble_udf}, albeit with typically smaller survey areas. An alternative is to stack images of Milky Way-mass galaxies, as has been done in the past in the search for splashback radius in galaxy clusters \citep{splash_stack}. \section{Summary and Conclusions} \label{conclusions} The Cold Dark Matter model implies hierarchical structure formation in our Universe, where smaller structures form first and accumulate to former larger ones. The model predicts the typical assembly histories, with scatter, for galaxies such as our own Milky Way. Since the presence of dark matter has been observed only indirectly, one has to rely on the visible baryonic component to infer the total matter content and the assembly history of galaxies like the Milky Way. The emergence of cosmological hydrodynamic simulations has allowed us to model the formation and evolution of galaxies within the $\Lambda$CDM paradigm. These simulations make predictions for how the stellar halo of the Milky Way has assembled through past accretion events and how this relates to the assembly of the dark matter halo. In particular, one can look for signatures in the stellar component that would reveal the properties of the dark matter. In this work we have examined the build-up of Milky Way-mass haloes in Local Group-like environments from the APOSTLE suite of simulations. We have examined both stellar and dark matter halo build-up. We find the following: \\ \noindent{\it i)} In the CDM paradigm of hierarchical structure formation, large dark matter haloes are built up through accumulation of smaller clumps \citep{frenk88}. By mass, subhaloes that have hosted stars make up 30-40~per~cent of a galactic dark matter halo, with the `smooth' halo component making up the majority of the mass (35-40~per~cent). The smooth component is itself split into particles that are not in bound structures as well as those in haloes of mass below the resolution limit of our simulations (subhaloes < 10$^6$ M$\odot$, where the power-law form of the CDM mass function breaks down.) The contributions of the dark and the luminous components are quite similar and it is their relative sizes that determine the degree to which the stars in the halo are able spatially to trace the dark matter (the stellar mass -- halo mass relation). \\ \noindent{\it ii)} The accreted stellar halo of Milky Way-like galaxies is primarily built up from disrupted dwarfs ($\sim85$~per~cent). Stars stripped from surviving dwarfs typically make up 10-15~per~cent of the stellar halo. It is typically 5-6 dwarfs with peak stellar mass of $>10^9$M$_{\odot}$ that make up $\sim80$~per~cent of the stellar halo in Milky Way-mass galaxies; of those the majority are disrupted. \\ \noindent{\it iii)} We identify `active' and `quiet' Milky Way~/~M31 analogues in our sample of 10 galaxies, in equal numbers. The main distinction between the two is the relative contribution of surviving subhaloes and the distribution of their debris in the halo. This has to do with the order in which particles are stripped from dwarf galaxies -- dark matter stripping occurs before stripping of stars due to the more extended spatial distribution of dark matter particles and their lower binding energies \citep{libeskind}. We also find that the halo stars in `active' galaxies are more centrally concentrated than in the quiet sample, which overall results in more embedded stellar haloes. Active galaxies also have a more significant dark matter contribution from disrupted dark haloes in the outer regions. Radial accretion and stripping time gradients suggest this is due to subhaloes that began to be stripped prior to crossing $R_{200,\rm m}$ while in a group with larger haloes. \\ \noindent{\it iv)} The disruption of dwarf galaxies as they fall into the Milky Way leaves structural imprints on the phase-space distribution. On a $v_r - D$ diagram, this takes the form of shells of particles following similar orbits in both stars and dark matter. Structures seen in the stellar halo have corresponding structures in the dark matter, although in the latter case they are ``smoothed out" due to the dominance of the smooth dark matter component, particularly beyond $0.1R_{200,\rm m}$ (smaller radii are not better places to look for such structures due to increasingly phase-mixed material in those regions). If the smooth component is removed from the analysis, the $v_r-D$ structure of the dark matter halo in the outer regions follows closely that of the stars. \\ \noindent{\it v)} In agreement with \cite{edgeofthegalaxy}, we find that the log-slope of the stellar halo density has a prominent trough at $\sim0.5 - 0.6$R$_{200,\rm m}$, although the exact location varies, corresponding to apocentre pile-ups of particles that have completed two-three orbital pericentres since infall into the Milky Way~/~M31 analogues. However, in this work we have examined simulations with higher mass resolution than \cite{edgeofthegalaxy}. As such, we were able to detect additional features in the stellar and dark matter halo density profiles, located closer in. Overall, we have found that even out to the splashback radius, variations in the dark matter density profile have corresponding variations in the density profile of the stars. \\ \noindent{\it vi)} We examined the formation of the shells that lead to the formation of the `edge' of the galaxy (i.e. the steepest drop in the log-slope profile of the stars). We found that typically one or two mergers deposit particles in the outer regions that lead to deviations of the halo density profile from smoothness. Contributions from several important mergers can also add up to enhance variations in the density profile. Since the stripping of the dark matter is more efficient and more continuous along an orbit than for the stars, the `peaks' in the radius of stripped dark matter particles can be offset from those of the stars stripped from the same objects, leading to slight differences in the locations of the density log-slope minima. Additionally, the dark matter content at large radii, where we expect to see the stellar halo edge, is dominated by the smooth component which suppresses variations in the density profile of the dark matter compared to corresponding features in the stars.\\ \noindent{\it vii)} We found a common behaviour in mergers that have contributed the most to the stellar halo of Milky Way-like galaxies. In agreement with previous work, we found that these objects enter the halo on very radial orbits due to their large mass and the undergo dynamical friction. The log-slope features that are detected in the stellar halo outskirts correspond to particles that were typically stripped at the first pericentre of the satellite's orbit. The wide range in particle binding energies, particularly in the most massive mergers, leads to a spread of particle apocentres and orbital phases. Particles which were less bound within the infalling dwarf galaxy end up on more energetic orbits within the Milky Way after they are stripped and, consequently, complete fewer orbits by the present day. Depending on the time of the merger, a `jet' of 1-pericentre stellar particles can be seen extending from the halo centre and looping around. If the halo has not grown significantly after the merger, these particles roughly trace the splashback radius of the dark matter halo. The steepest trough in the log-slope of the stellar density is the result of apocentre pile-ups of particles that have completed 2-3 orbits, depending on the halo growth since the merger. However, if the halo growth is particularly fast after the merger, these relations can break down as the stellar halo is dominated by particles that have not completed a full orbit. Although we have not explicitly demonstrated the same features in the dark matter, we have shown that stellar and dark matter particles with the same number of completed pericentres have very similar distributions of binding energies within the Milky Way analogues.\\ \noindent{\it viii)} We have looked into the specific histories of the dark matter and stellar halo growth, and their relation, to establish the main drivers of the location of the halo `edge'. We have found that a large halo mass, which often corresponds to a large mass in the stellar halo, leads to more extended haloes. This reflects the fact that these haloes experienced more massive major mergers that had higher stellar mass contributions, allowing the stars to trace the dark matter out to larger radii. At the same time, halo mass does not seem to be the only important factor. Since the stellar haloes often assembled earlier than the dark matter, the subsequent dark halo growth leads to increased concentration of the stellar halo within the dark halo. This is consistent with previous work exploring the halo splashback radius. In particular, we find stellar halo ``edges'' to be more embedded within the dark matter halo if the halo has recently accreted a massive satellite that contributes roughly 25~per~cent of the dark halo mass, while its stars have generally not been affected by tidal forces. \\ \noindent{\it ix)} We have examined the possibility of uncovering the important mergers contributing to the build-up of the stellar halo. We have found that the debris from these past mergers has a metallicity gradient across the stellar halo, likely stemming from the metallicity gradient within the objects themselves. More massive mergers typically have particles with higher binding energies (lower orbital energies) within the stellar halo and thus contribute more particles at small radii (central $\sim 20$~kpc), making them more likely to be detected. At the same time, across all radii, there is a large spread in the metallicity distribution that may overlap with other contributors. We have also examined the distributions of the orbital energy and angular momentum that may help distinguish different mergers kinematically. We found that at small radii (below $\sim 50$~kpc) the orbits are rather similar, likely due to mixing in a turbulent time-varying gravitational potential; however, at large radii the particle orbits of stars from different progenitor dwarfs become increasingly distinct from each other. This is likely where the particles are still on the first or second pericentre of the orbit, moving still somewhat coherently since the time they were stripped (see, for example, the `jet' features in Fig.~\ref{fig9}). \\ \noindent{\it x)} We find that the `edge' of the stellar halo in Milky Way-like galaxies typically corresponds to a surface brightness of 31-36 mag~arcsec$^{-2}$. Reaching this surface brightness limit is marginally possible with existing, though coverage-limited, ultra deep photometric surveys. Alternatively, one may stack images of Milky Way-mass galaxies in search of the `edge' feature. \\ In this work, we have examined the predictions of the $\Lambda$CDM model, together with the {\sc EAGLE} model of galaxy formation for the assembly of the stellar and dark matter halos of Milky Way~/~M31 analogues in Local Group-like environments. Some aspects of this work may be sensitive to the assumed galaxy formation physics. For example, the details of the stellar-halo mass relation may, to some extent, alter the ability of stripped stars to track the dark matter. This could also be affected by the sizes of the galaxies. We note, however, that the galaxy formation model we have employed has been shown to reproduce these galaxy scaling relations, with the largest uncertainties expected both above the Milky Way mass and in the regime of the classical dwarf galaxies, in which the observed relations are not well constrained. We also do not expect small changes in the halo mass threshold above which galaxy formation occurs to affect our results significantly, since it is the most massive galaxies that contribute the majority of stars in the outer halo. We have shown that the stripping of dwarf galaxy stars leads to the formation of shells in phase space that traces similar shells in the dark matter distribution. It would be interesting to explore whether these features are affected by the nature of the dark matter. For example, warm dark matter leads to the suppression of the matter power spectrum on small scales, such that low-mass haloes and the fraction of the smooth component made up of unresolved haloes would not exist in our simulations, potentially leading to more distinctive caustic features in the dark matter, and perhaps also in the stars, if small subhaloes perturb streams of stripped particles in CDM \citep{lovell_wdm_sfr}. Self-interacting dark matter (SIDM) may also result in a different phase-space picture. Firstly, dark matter haloes may suffer enhanced disruption due to `dark ram pressure' stripping from the host galaxy halo \citep{selfintdm,sirks}. In some SIDM models cores form in dark matter haloes. Core formation is also expected in CDM galaxy formation models in which baryonic feedback is sufficiently impulsive \cite{benitez-llambay}. It has been shown that tidal stripping is more efficient in haloes with cores, enhancing subhalo disruption and producing wider stellar and dark matter streams \citep{errani_streams}. This would lead to fuzzier streams of stripped particles and smoother halo density profiles, in which changes in the smoothly declining density slope are less easily identifiable. \section*{Acknowledgements} AD and CSF are supported by the Science and Technology Facilities Council (STFC) [grant number ST/F001166/1, ST/I00162X/1, ST/P000541/1]. AD is supported by a Royal Society University Research Fellowship. CSF acknowledges aEuropean Research Council (ERC) Advanced Investigator grant DMIDAS (GA 786910). This work used the DiRAC Data Centric system at Durham University, operated by the ICC on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. \bibliographystyle{mnras} \bibliography{edge.bib} % \bsp % \label{lastpage}

Title: Binary open clusters in the Gaia data

Abstract: Context. Observations indicate that the fraction of potential binary star clusters in the Milky Way is either the same or lower than that of the Magellanic Clouds. The unprecedented precision in the parallax measurements by Gaia has allowed for the discovery of a growing number of new binary open clusters (OCs). Aims. We aim to survey the candidates of truly binary open clusters that are formed simultaneously, using information from the Gaia database. Methods. Based on the most recent catalog of open clusters, we investigated the interactions of adjacent binary open clusters in our Galaxy within separations of 50 pc. We compared their coordinates, proper motions, parallaxes, and color-magnitude diagrams(CMDs) via binary plots for all candidate pairs. The candidates of truly binary open clusters are selected on the basis of their common proper motions and consistent behaviors in the CMDs of different clusters that are limited to a separation of 50 pc. Results. About ten pairs of the selected binary open clusters appear to be the same clusters, based on evidence that almost half ofthe cluster members are shared. Fourteen pairs are possibly true binaries, implying that they may come from the same clouds, among which five pairs are newly discovered. In addition, two clusters, UBC 46 and UBC 192, were found to be part of the stellar complex LISCA I. Our results confirm that OCs born in groups are usually composed of young open clusters.

https://export.arxiv.org/pdf/2208.12935

\authorrunning{Fangfang Song} \titlerunning{Binary open clusters in the Gaia data} \title{Binary open clusters in the Gaia data} \author{Fangfang Song\inst{1,2}, Ali Esamdin\inst{1,2},Qingshun Hu\inst{1,2}, Mengfan Zhang\inst{1,2}} \institute{Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, People's Republic of China; \email{aliyi@xao.ac.cn; songfangf@xao.ac.cn} \\ University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China; } \date{Received X XX, XXXX; accepted X XX, XXXX} \abstract {Observations indicate that the fraction of potential binary star clusters in the Milky Way is either the same or lower than that of the Magellanic Clouds. The unprecedented precision in the parallax measurements by Gaia has allowed for the discovery of a growing number of new binary open clusters (OCs).} {We aim to survey the candidates of truly binary open clusters that are formed simultaneously, using information from the Gaia database.} {Based on the most recent catalog of open clusters, we investigated the interactions of adjacent binary open clusters in our Galaxy within separations of 50 pc. We compared their coordinates, proper motions, parallaxes, and color-magnitude diagrams (CMDs) via binary plots for all candidate pairs. The candidates of truly binary open clusters are selected on the basis of their common proper motions and consistent behaviors in the CMDs of different clusters that are limited to a separation of 50 pc.} {About ten pairs of the selected binary open clusters appear to be the same clusters, based on evidence that almost half of the cluster members are shared. Fourteen pairs are possibly true binaries, implying that they may come from the same clouds, among which five pairs are newly discovered. In addition, two clusters, UBC~46 and UBC~192, were found to be part of the stellar complex LISCA I. Our results confirm that OCs born in groups are usually composed of young open clusters. } {} \keywords{open clusters and associations: general -- Astrometry} {} \section{Introduction}\label{section:1} Open clusters (OCs) are born in giant molecular clouds and can go on to form groups such as pairs, triplets, or higher multiplicity systems, according to observational evidence \citep{2016MNRAS.455.3126C}. Contrary to field stars, the distance, reddening, age, and metallicity for member stars of a given open cluster can be determined by the statistical cluster parameters \citep{2021A&A...649A..54P}. Among OCs, binary open clusters play a key role in understanding the formation, evolution, and properties of stars in the Galactic disk \citep{2021ARep...65..755C, 2021A&A...649A..54P}. These structures are also crucial in the study of the mechanisms that lead to cluster formation and evolution \citep{2018MNRAS.474.2277D}. \citet{2009A&A...500L..13D} divided the formation of multiplicity star clusters into five different scenarios, including simultaneous formation, sequential formation, tidal capture, resonant trapping, and optical doubles. The clusters formed from the simultaneous formation are defined as genetic pairs or true binaries because they share common space velocities, ages, and chemical compositions. This means that either they come from the same molecular cloud or they are the final products of multiple mergers of smaller clusters\citep{2009A&A...500L..13D}. These bona fide pairs are ideal laboratories for promoting the development of formation and evolution theories of star clusters that are still poorly understood \citep{2022MNRAS.511L...1P}. The fraction of binary open clusters accounts for roughly 10\% of the known OCs in the Large Magellanic Cloud (LMC), according to the observations \citep{2009A&A...500L..13D}. The estimated fraction of binary clusters in our Galaxy is 12\% \citep{2009A&A...500L..13D}, a proportion that is similar to that of the LMC. Until recently, the binary open cluster of $h$ and $ \chi $ Persei has been the only confirmed genetic pair known in the Milky Way \citep{2010A&A...511A..38V}, which means that more binary open clusters have not been found in the disk. \cite{1989SvA....33....6P} proposed the existence of five possible cluster groups. Later, \cite{1995A&A...302...86S} examined the spatial proximity of open clusters by the catalog of \cite{1995yCat.7092....0L} and suggested 18 probable binary open star clusters and found that about 8\% of open clusters may be genuine binaries. \cite{1997A&AT...14..181L} provided a catalog of 31 probable multiple systems by restricting the spatial vicinities and age coincidences. \citet{2009A&A...500L..13D} found 6 clear candidates of 34 OC pairs by constituted volume-limited samples from WEBDA \citep{2003A&A...410..511M} and NCOVOCC \citep{2002A&A...389..871D}. \cite{2017A&A...600A.106C} detected 19 groupings, including 14 pairs using an adapted version of the friends-of-friends algorithm on the 6D phase-space information, including radial velocities, for the first time, from the Catalogue of Open Cluster Data \citep{2004AN....325..740K, 2005A&A...440..403K} and the Radial Velocity Experiment \citep{2006AJ....132.1645S}. Thanks to the precisison of Gaia data, we have the opportunity to find more new open clusters and new binary open clusters. The second Gaia Data Release (Gaia DR2) provides full astrometric data (positions, parallaxes, and proper motions) and multi-band photometry for about 1.3 billion stars \citep{GaiaDR2}, leading to extensive studies of galactic OCs, namely, exclusions of false clusters, as well as accurate measurements of known star clusters, and searches for new clusters. \cite{Cantat2018} first determined a list of members and cluster parameters for 1229 clusters from the list of known clusters and candidates using Gaia DR2 data. These authors derived fully homogeneous parameters, including parallaxes, proper motions, and the most reasonable distances of the clusters, together with the membership probabilities of the individual stars (based on UPMASK). The number was later expanded to 1481 \citep{Cantat2020} and then updated to 2017 \citep[][CG20]{2020A&A...640A...1C} after a few months. The steadily growing numbers of known open clusters and relatively accurate classification of cluster member stars based on Gaia DR2 has led to enthusiastic attempts to search for binary open clusters. \cite{Soubiran2019} provided 21 cluster pairs differing by less than 100 pc in distance and 5 km/s in velocity. \cite{2019ApJS..245...32L} found 56 candidates for star cluster groups among the Class 1 cluster candidates using the data of Gaia DR2 based on the star clusters’ 3D positions only. In the literature of \cite{2021RAA....21..117C}, a new binary system named Casado 9 and 10 was discovered in the reported 20 new OCs, which have been confirmed manually by Gaia DR2. There are 22 new binary or multiple OCs in a 30 deg sector of the Galactic disc found in \cite{2021ARep...65..755C}, based on the catalogues of OCs \citep{Kharchenko2013, Cantat2018, 2019AJ....157...12B, 2019ApJS..245...32L, 2019JKAS...52..145S, 2020A&A...635A..45C, 2020A&A...640A...1C}, identified by hand between the galactic longitudes of 240$^{\circ}$ and 270$^{\circ}$. Sixty aggregates of clusters are found in \cite{2021A&A...649A..54P} by searching for open clusters that\ share some member stars located at relatively low volumes of the phase space in the catalog of \cite{Cantat2020}. Nevertheless, the possibility for new discoveries of binary open clusters still exists. The aim of this work is to locate more genetic binary open clusters. The present study is organized as follows. In Section 2, we describe the methodology for the selection of candidate OCs belonging to groups and the analysis. Section 3 presents a discussion of the details of each OC binary or multiple system and a comparison with the Two Micron All Sky Survey (2MASS) catalog \citep{2003yCat.2246....0C}. We summarize our results in Section 4. \section{Pair selection and analysis}\label{section:2} The data sets adopted in this work were derived by CG20, which is presented as two tables. The list of clusters (2017 entries) in their Table 1 includes mean positions, mean proper motions, and mean parallaxes, as well as other reliable parameters (ages, extinctions, distance modulus, distances converted from distance modulus, XYZ positions in Galactic cartesian coordinates, and distances from Galactic center) for 1867 clusters. A total of 234\,128 cluster member stars with probability over 0.7 are listed in their other table, which contains the astrometric measurements and photometric observational information for most of the stars by Gaia, the derived membership probabilities, and IDs of the host clusters. They employed uniform analysis methods to ensure unbiased comparisons among clusters.\citet{2019JPhCS1127a2053P} argued that systems located outside the dense region with small separations have a higher probability of being considered real pairs. The linking lengths used to identify pair open clusters are always limited to 100 pc \citep{2017A&A...600A.106C, 2019ApJS..245...32L}. \citet{2009A&A...500L..13D} argued that the pairs' physical separation must be less than 30 pc, while \citet{2022MNRAS.511L...1P} detected pairs with physical separations smaller than 40 pc. The basic criterion adopted in our preliminary selection is arbitrarily set to 50 pc (each pair's spatial separation) based on the 3D positions of the star clusters given in the datasets. The separation between two clusters is calculated using three-dimensional (3D) coordinates that include RA, DEC, and distance based on the Python Astropy package \citep{2013A&A...558A..33A, 2018AJ....156..123A}. The distance to each cluster is computed as $1/\omega$, with $\omega$ as the mean parallax of the cluster given by CG20. We obtained a total of 125 preliminary pair candidates. Sixteen of them are found to contain some member stars, which have also been assigned to another cluster based on the same equatorial coordinates. This means that the host clusters may be physically close to each other -- or even overlapping \citep{2021A&A...649A..54P}. We assume that if the coincidence rate of the member stars in a cluster pair is greater than 50\%, they are considered to be one cluster; these results are listed in Table \ref{binaries1}. Table \ref{binaries1} provides the cluster names, the mean proper motions in RA and Dec, the mean parallaxes, the corresponding numbers of cluster members, the cluster ages, and the extinctions of the pair clusters in proper sequence (Columns 2-13), which are given by CG20. We calculate the cluster spatial separations and the star coincidence rates of cluster members listed in the last two columns of Table \ref{binaries1}. Most of the selected pairs are aggregates of clusters in Table A.1 of \citet{2021A&A...649A..54P}, except for numbers 8-10, which are not included in their paper since clusters UBC 167, UBC 392, and UBC 323 are newly found clusters in the work of \citet{Cantat2020}. Among the ten pairs, the star coincidence rates of four pairs are larger than 0.8, which means they have a greater probability of being the same clusters. The star coincidence rates of the other four pairs are approximately 0.7 and the values of the other two pairs are closer to 0.5. These pairs with star coincidence rates lower than 0.8 require more evidence to verify their characterization. After eliminating those pairs (as shown in Table \ref{binaries1}), details such as coordinates, proper motions, parallaxes, and color-magnitude diagrams (CMDs) can be checked by visual inspection, as shown in Figure \ref{fig:fig1}. If the proper motions of two clusters are similar in panel a and consistent within 3 $\sigma$ error range (as shown in Table \ref{binaries2}), then we considered them as having common proper motions. According to \citet{2020A&A...642L...4K}, if the ages and reddenings of the two clusters cannot be distinguished in the CMD, then the cluster ages and reddenings seem to be equal and the same values can be used for both clusters. Furthermore, the indistinguishable CMDs (panel b) indicates similar ages. The differences in coordinates or parallaxes (panels c, d) indicate that the clusters can be distinguished in the 3D space. The pairs exhibiting similar proper motions and ages can be considered as candidates of truly binary open clusters. In total, we obtained 14 candidate truly pairs from the limited 115 candidate pairs, which are given in Table \ref{binaries2}. The listed parameters are the same as those in Table \ref{binaries1}. Most of the proper motions are consistent in the error range for each pair of the 14 candidates, except pair 5, whose proper motions in the declination direction are consistent within the 2 $\sigma$ error range. The distributions of proper motions, CMDs, parallaxes, and positions of members for the selected 14 open cluster pairs are shown in Figure \ref{fig:fig1}. All the adopted parameters of Figure \ref{fig:fig1} were taken from Gaia DR3 \citep{2022yCat.1355....0G} by cross-matching the equatorial coordinates, because the photometric and astrometric uncertainties are greatly improved compared with those in Gaia DR2. The dark dots and gray histograms represent the former clusters, while the red dots and red histograms are related to the latter clusters. The black and red lines in CMDs are the fitted PARSEC stellar isochrones \citep{2012MNRAS.427..127B}, with the cluster metallicities assumed to be solar. Most isochrones are aptly fit with the parameters of CG20 of these candidates. In contrast, cluster parameters need to change to fit the member distributions, which are the clusters marked with asterisks in Table \ref{binaries2}. The cluster ages and extinctions of CG20 are approximated by artificial neural networks (ANN) with Gaia photometry and parallaxes, and the precision of ANN is affected by the numbers of cluster member stars and the densities of clusters (CG20). Therefore, some cluster isochrone fittings are reconfirmed in this work. Because they are assumed as part of binary open clusters, we can use the same ischrones of their partners. Pairs 2, 3, and 10 are aptly fit with the parameters of latter clusters, whereas Pair 4 and 12 are aptly fit with the parameters of former clusters. The revised parameters can be seen in Table \ref{binaries3}. The clusters marked with an asterisk have some common characteristics, namely, they contain fewer than 100 members and most of them have sparse distributions for their member stars in the CMDs, with their spacial positions shown in Figure \ref{fig:fig1}. In addition, two clusters, UBC~46 and UBC~192, were found to be part of the $h$ and $ \chi $ Persei double clusters in our study (see details in Section 3.1.1). \begin{table*} \centering \fontsize{8} {10pt}\selectfont \tabcolsep 0.10truecm \caption{List of possible one clusters} \begin{tabular}{rccrrrrrrrrrrrrrr} \hline\hline \multicolumn{1}{c}{No. \#} & \multicolumn{1}{c}{Cluster} & \multicolumn{1}{c}{$\mu_{\alpha}\cos\delta$} & \multicolumn{1}{c}{$\mu_{\delta}$} & \multicolumn{1}{c}{$\omega$} & \multicolumn{1}{c}{members} & \multicolumn{1}{c}{$\tau$} & \multicolumn{1}{c}{Av} & \multicolumn{1}{c}{Sep} & \multicolumn{1}{c}{ratio} & \cr & &(mas yr$^{-1}$) & (mas yr$^{-1}$) &(mas) & &(Myr) & (mag) &(pc) & \cr \hline 1 & BH 121 & -6.0(1) & 0.6(1) & 0.39(4) & 174 & 2.63 & 1.11 & 7.78 & 0.92 \\ & IC 2948 & -6.0(1) & 0.65(5) & 0.39(3) & 51 & 6.46 & 1.16 & & \\ 2 & FSR 0686 & -1.1(1) & -2.56(7) & 1.09(4) & 12 & & & 6.83 & 0.91 \\ & UBC 55 & -1.1(1) & -2.6(1) & 1.08(5) & 56 & 33.88 & 1.38 & & \\ 3 & RSG 7 & 4.9(4) & -1.9(7) & 2.3(1) & 43 & 38.90 & 0.57 & 24.78 & 0.72\\ & RSG 8 & 5.3(5) & -1.7(5) & 2.2(2) & 39 & 26.92 & 0.53 & & \\ 4 & Gulliver 56 & 0.53(8) & -3.24(7) & 0.46(4) & 18 & 407.38 & 0.44 & 48.74 & 0.83 \\ & UBC 73 & 0.5(1) & -3.21(9) & 0.45(3) & 50 & 389.05 & 0.56 & & \\ 5 & Gulliver 6 & -0.0(4) & -0.2(4) & 2.4(1) & 318 & 16.60 & 0.25 & 1.78 & 0.72 \\ & UBC 17b & 0.1(1) & -0.2(3) & 2.38(5) & 103 & 11.48 & 0.05 & & \\ 6 & Hogg 10 & -6.20(7)& 1.75(5) & 0.38(2) & 11 & & & 41.99 & 0.72 \\ & NGC 3572 & -6.3(1) & 1.9(2) & 0.38(5) & 75 & 4.79 & 1.53 & & \\ 7 & Kronberger 1 & -0.1(1) & -2.2(2) & 0.4(1) & 32 & 6.03 & 2.05 & 20.29 & 0.53\\ & Stock 8 & 0.1(1) & -2.2(2) & 0.45(5) & 275 & 14.45 & 1.17 & & \\ 8 & UBC 10a & -2.1(1) & -3.0(1) & 1.08(2) & 33 & 14.13 & 0.89 & 9.03 & 0.56 \\ & UBC 167 & -2.08(9)& -3.0(1) & 1.07(3) & 115 & 21.88 & 1.08 & & \\ 9 & UBC 392 & -0.6(1) & -3.3(1) & 1.06(3) & 55 & 70.79 & 1.47 & 8.55 & 0.96\\ & UPK 194 & -0.6(2) & -3.1(3) & 1.05(5) & 267 & 151.36 & 1.67 & & \\ 10& BH 205 & -0.2(1) & -1.1(2) & 0.57(7) & 124 & 6.17 & 1.23 & 18.74 & 0.69\\ & UBC 323 & -0.2(2) & -1.3(2) & 0.58(5) & 170 & 8.91 & 0.72 & & \\ \hline \end{tabular} \begin{list}{}{} {\footnotesize \item[] \textbf{Notes} $\mu_{\alpha}\cos\delta$,$\mu_{\delta}$: average proper motions of cluster members in RA and Dec mas yr$^{-1}$; $\omega$:Mean parallax of cluster members in mas;$members$: Numbers of cluster member stars; $\tau$: cluster age in Myr ; $Av$: Extinction of the cluster in mag; All the above parameters were taken from CG20. Sep: cluster pairs' spatial separation in pc; ratio: the star coincidence rate of cluster members. This Sep and ratio values were calculated in this work. } \end{list} \label{binaries1} \end{table*} \begin{table*} \centering \fontsize{8} {10pt}\selectfont \tabcolsep 0.10truecm \caption{List of candidates of truly binary open clusters} \begin{tabular}{rccrrrrrrrrrrrrrr} \hline\hline \multicolumn{1}{c}{Pairs \#} & \multicolumn{1}{c}{Cluster} & \multicolumn{1}{c}{$\mu_{\alpha}\cos\delta$} & \multicolumn{1}{c}{$\mu_{\delta}$} & \multicolumn{1}{c}{$\omega$} & \multicolumn{1}{c}{members} & \multicolumn{1}{c}{$\tau$} & \multicolumn{1}{c}{Av} & \multicolumn{1}{c}{Sep} & \cr & &(mas yr$^{-1}$) & (mas yr$^{-1}$) &(mas) & &(Myr) & (mag) &(pc) \cr \hline 1 & NGC 869 & -0.7(1) & -1.1(1) & 0.40(4) & 503 & 15.14 & 1.69 & 21.01\\ & NGC 884 & -0.6(1) & -1.1(1) & 0.40(4) & 322 & 17.78 & 1.72 & \\ 2 &$UBC~46^{\star}$ & -0.8(2) & -1.2(2) & 0.40(3) & 65 &112.20 & 1.35 & 29.30\\ & NGC 869 & -0.7(1) & -1.1(1) & 0.40(4) & 503 & 15.14 & 1.69 & \\ 3 &$UBC 192^{\star}$ & -0.4(2) & -1.2(1) & 0.40(3) & 78 & 48.98 & 1.28 & 38.59\\ & NGC 884 & -0.6(1) & -1.1(1) & 0.40(4) & 322 & 17.78 & 1.72 & \\ 4 &UBC 461 & -2.7(1) & 3.11(8) & 0.29(2) & 72 & 47.86 & 0.45 & 27.12 \\ & $UBC~462^{\star}$ & -2.67(8)& 3.04(4) & 0.29(2) & 8 & 39.81 & 0 & \\ 5 &Collinder 135 & -10.0(4)& 6.2(4) & 3.3(1) & 324 & 26.30 & 0.01 & 26.42\\ &UBC 7 & -9.7(2) & 7.0(2) & 3.56(6) & 77 & 31.62 & 0.1 & \\ 6 &NGC 2659 & -5.3(1) & 5.0(1) & 0.45(7) & 82 & 43.65 & 1.21 & 49.45\\ & UBC 482 & -5.2(1) & 5.0(2) & 0.45(3) & 97 & 26.92 & 0.88 & \\ 7 &FSR 0198 & -3.6(2) & -6.6(2) & 0.49(5) & 82 & 4.68 & 2.49 & 28.38\\ & Teutsch 8 & -3.5(1) & -6.7(3) & 0.49(7) & 28 & 3.98 & 2.2 & \\ 8 &Collinder 394 & -1.5(2) & -5.9(2) & 1.39(6) & 156 & 91.20 & 0.54 & 11.49\\ & NGC 6716 & -1.5(2) & -6.0(2) & 1.41(8) & 76 & 97.72 & 0.43 & \\ 9 &Alessi 43 & -5.5(3) & 3.9(3) & 1.03(8) & 284 & 11.48 & 0.99 & 19.76\\ & Collinder 197 & -5.8(3) & 3.9(4) & 1.03(9) & 229 & 14.13 & 1.42 & \\ 10&$FSR 1297^{\star}$ & -2.8(2) & 3.3(2) & 0.75(5) & 25 & & & 41.13\\ & NGC 2362 & -2.8(2) & 3.0(2) & 0.74(7) & 144 & 5.75 & 0.37 & \\ 11&Antalova 2 & 2.0(3) & -1.7(3) & 0.84(7) & 27 & 12.02 & 1.15 & 47.02\\ & NGC 6383 & 2.6(3) & -1.7(2) & 0.9(1) & 245 & 3.98 & 1.3 & \\ 12&Negueruela 1 & -3.0(2) & -1.3(1) & 0.29(6) & 52 & 16.98 & 3.39 & 18.31\\ & $Teutsch~23^{\star}$& -3.0(1) & -1.39(7)& 0.29(4) & 28 & 5.62 & 2.69 & \\ 13& UBC 547 & -0.7(1) & -2.0(1) & 0.31(3) & 175 & 15.14 & 1.63 & 23.54\\ & UBC 549 & -0.81(9)& -1.93(8)& 0.31(2) & 20 & 85.11 & 1.92 & \\ 14& Stock 20 & -3.24(6)& -1.12(6)& 0.34(3) & 38 & 19.05 & 1.09 & 23.74\\ & UBC 410 & -3.1(1) & -1.0(1) & 0.34(3) & 36 & 38.02 & 0.97 & \\ \hline \end{tabular} \begin{list}{}{} {\footnotesize \item[] \textbf{Notes} The given parameters are the same as those in Table \ref{binaries1}. The fitting isochrones of the clusters marked with asterisks are not suitable for the cluster members, which require an adjustment of the parameters to find adequate fittings. } \end{list} \label{binaries2} \end{table*} \begin{table*} \centering \fontsize{8} {10pt}\selectfont \tabcolsep 0.10truecm \caption{Comparation of the fitting parameters: ages and extinctions of candidates for truly binary open clusters} \begin{tabular}{rccrrrrrrrrrrrrrr} \hline\hline \multicolumn{1}{c}{pairs \#} & \multicolumn{1}{c}{Cluster} & \multicolumn{2}{c}{CG20} & \multicolumn{2}{c}{DAML02} & \multicolumn{2}{c}{MWSC} & \multicolumn{2}{c}{This work} \cr\cline{3-10} & &log(t) (yr) & $Av$ (mag) &log(t) (yr) & $Av$ (mag) &log(t) (yr) & $Av$ (mag) &log(t) (yr) & $Av$ (mag) \cr \hline 1-3 &NGC 869 & 7.18 & 1.69 & 7.07 & 1.78 & 7.28 & 1.62 & 7.18 & 1.69 \\ &NGC 884 & 7.25 & 1.72 & 7.10 & 1.74 & 7.20 & 1.74 & 7.25 & 1.72 \\ &$UBC~46^{\star}$ & 8.05 & 1.35 & & & & & 7.18 & 1.69\\ &$UBC 192^{\star}$& 7.69 & 1.28 & & & & & 7.25 & 1.72\\ 4 &UBC 461 & 7.68 & 0.45 & & & & & 7.68 & 0.45\\ &$UBC~462^{\star}$& 7.6 & 0 & & & & & 7.68 & 0.45\\ 5 &Collinder 135 & 7.42 & 0.01 & & & 7.60 & 0.13 & 7.42 & 0.01\\ &UBC 7 & 7.5 & 0.1 & & & & & 7.5 & 0.1\\ 6 &NGC 2659 & 7.64 & 1.21 & 6.89 & 1.59 & 7.36 & 1.58 & 7.64 & 1.21\\ &UBC 482 & 7.43 & 0.88 & & & & & 7.43 & 0.88\\ 7 &FSR 0198 & 6.67 & 2.49 & 7.0 & 2.98 & 6.56 & 2.77 & 6.67 & 2.49\\ &Teutsch 8 & 6.6 & 2.2 & 7.0 & 1.80 & 7.50 & 1.68 & 6.6 & 2.2\\ 8 &Collinder 394 & 7.96 & 0.54 & & & 7.86 & 0.74 & 7.96 & 0.54\\ &NGC 6716 & 7.99 & 0.43 & 7.96 & 0.68 & 7.39 & 0.71 & 7.99 & 0.43\\ 9 &Alessi 43 & 7.06 & 0.99 & & & 7.80 & 0.71 & 7.06 & 0.99\\ &Collinder 197 & 7.15 & 1.42 & & & 7.20 & 1.70 & 7.15 & 1.42\\ 10 &$FSR 1297^{\star}$& & & & & 6.70 & 5.16 & 6.76 & 0.37 \\ &NGC 2362 & 6.76 & 0.37 & 6.70 & 0.31 & 6.64 & 0.52 & 6.76 & 0.37\\ 11 &Antalova~2 & 7.08 & 1.15 & & & 6.0 & 11.55& 7.08 & 1.15\\ &NGC 6383 & 6.6 & 1.3 & 6.96 & 0.92 & 6.60 & 1.29 & 6.6 & 1.3\\ 12 &Negueruela 1 & 7.23 & 3.39 & 6.98 & 3.87 & 6.98 & 3.87 & 7.23 & 3.39\\ &$Teutsch~23^{\star}$& 6.75 & 2.69 & 7.0 & 2.39 & 7.10 & 2.39 & 7.23 & 3.39\\ 13 &UBC 547 & 7.18 & 1.63 & & & & & 7.18 & 1.63\\ &UBC 549 & 7.93 & 1.92 & & & & & 7.93 & 1.92\\ 14 &Stock 20 & 7.28 & 1.09 & 8.34 & 0.62 & 8.34 & 0.62 & 7.28 & 1.09\\ & UBC 410 & 7.58 & 0.97 & & & & & 7.58 & 0.97\\ \hline \end{tabular} \begin{list}{}{} {\footnotesize \item[] \textbf{Notes} Most parameters(ages and extinctions) of the fitted isochrones of the clusters given in this work are from CG20, except the clusters marked with asterisk whose isochrone fittings are reconfirmed in this work. DAML02 refers to the catalog of \citet{Dias2002}, while MWSC is short for Milky Way global survey of star clusters \citep{Kharchenko2013}; The extinction values of DAML02 and MWSC is calculated by the extinction law $Rv = Av/E(B-V)$, with RV=3.1 \citep{1989ApJ...345..245C}. } \end{list} \label{binaries3} \end{table*} \section{Results}\label{section:3} \subsection{Case analysis} In this section, we perform the analysis for the 14 candidates of binary open clusters detected in the above section. The information related to the study is presented in Figure \ref{fig:fig1}. We compare the age and extinction of the clusters with those given in previous literature \citep{Dias2002, Kharchenko2013} to examine the fitted isochrones based on the cluster members, as shown in Table \ref{binaries3}. \subsubsection{$h$ and $ \chi $ Persei aggregate} Pair 1 is the well-known $h$ and $ \chi $ Persei double cluster, which is characterized by an identical mean heliocentric distance of about 2.5 kpc. The position of the two clusters can be clearly distinguished and the distance between them is 21 pc. They may be in the same stellar evolution stage, as illustrated in the first CMD panel in Figure \ref{fig:fig1}. The aggregate is selected on the basis of transitivity, that is to say: if cluster $A$ and cluster $B$ are all friends of cluster $C$, then all the three clusters form an aggregate. We can see in Figure \ref{fig:fig1} that the members of each cluster in Pair 1 and 2 are well mixed with the members of its partner in panels a, b, c. In contrast, the members of each cluster of Pair 3 are clearly different in panel a, but well mixed in panels b, c. Nevertheless, the proper motions of these three pairs are consistent within the error range in Table \ref{binaries2}. This means that the aggregate have almost the same ages, distances, and proper motions. Using the isochrones of $h$ and $ \chi $ Persei based on the ages and extinctions given in CG20, we find that they can be well fitted with the members of clusters in Pairs 2 and 3. The derived ages for UBC46 and UBC 192 in this work are listed in Table \ref{binaries3}, showing that they are clearly younger than the ages given by CG20. Therefore, Pairs 1, 2, and 3 might form a young quadruple aggregate of an age less than 20Myr. The distributions and movement trends of these quadruple open clusters are shown in Figure \ref{fig:fig2}. The red and blue fitted ellipses represent 50\% and 5\% of the peak stellar number density profiles for each cluster, with the red and blue asterisks being their centers, respectively. The blue dashed lines are coincident with the semi-major axes of the outer ellipses, indicating the direction of the cluster elongations \citep{2021ApJ...912....5H}. The red and blue pentagrams of each cluster are linked with a solid green line, which is the distance between the centers of inner and outer fitted ellipses, also regarded as the dislocation of cluster morphology initially defined by \cite{2021A&A...656A..49H}. In their work, the morphological dislocation of an open cluster refers specifically to the Euclidean distance between the centroids of the inner and outer fitted ellipses of the open cluster in the 2D spherical coordinate system. The morphological dislocation measures the stability of the 2D morphology of the cluster; and the larger the morphological dislocation, the less stable the 2D morphology of the cluster. We find that the three clusters of this group (UBC 46, NGC 869, and NGC 884) seem to be elongated as a result of their interaction with each other, since the shaped direction of their outer ellipses intersects with each other. NGC 869 may be the central cluster which are likely to disturb or influence the shapes of UBC 46 and NGC 884; this is because it has a greater number of member stars than the other two clusters. Generally speaking, the higher the number of member stars, the stronger the self-gravity of a cluster and the stronger the cluster’s gravity is expected to be. Moreover, in comparing their dislocations of morphologies, we can note the smallest dislocation of shape in NGC 869, implying the morphological stability of the cluster is higher than that of the rest clusters. This reinforces the idea of NGC 869 as the central cluster of the group. UBC 46 and NGC 884 have dislocations of morphologies about four and eight times that of NGC 869, respectively, which means that their morphological stabilities may be smaller than NGC 869 and more influenced by the central cluster. In addition, although UBC 192 displays an elongated shape that is even larger than the three other clusters and its small dislocation of morphology seems to suggest a high probability with regard to the morphological stability. \citet{2021ApJ...909...90D} found a stellar complex known as LISCA I, which is composed of seven comoving clusters with an extended massive halo distributed within 6 degrees surrounding the well-known $h$ and $ \chi $ Persei double stellar cluster in the Perseus Arm by Gaia-EDR3 data. The stellar complex share the same chemical abundances of half solar metallicity and the same age of $t \sim 20 Myr$, which could place them at a distance of 2.3 kpc and the extinction value set at Av = 1.65 mag. The values of the parameters are consistent with our results. Our findings extend the stellar complex and the new members of UBC 46 and UBC 192 are much closer than the others (the cluster separations are 29.30 pc and 38.59 pc, respectively). \subsubsection{Known binary open clusters} The cluster pair made up of UBC~461 and UBC~462, both open clusters discovered by \citet{2020A&A...635A..45C}, was first mentioned in \citet{2021ARep...65..755C} as a probable physical system of three OC aggregates. Their ages and extinctions are provided by CG20. The PARSEC track fits in well with the cluster parameters of UBC~461. However, for UBC~462, the isochrones based on the adopted cluster parameter cannot fit the color-magnitude diagram constituted by cluster members, due to the fact that only eight member stars are available for this cluster. Their indistinguishable CMDs allow them to share the same track of UBC~461 as shown in Figure \ref{fig:fig1}, which means that the age of UBC~462 may be comparable to that of UBC~461. The cluster pair Collinder~135 and UBC~7 was identified as a physical pair that has since been analyzed in detail by \citet{2020A&A...642L...4K,2021scgr.confE..17K}. In addition, \citet{2020A&A...642L...4K} built a model to show that the two clusters might indeed form a physical pair. Collinder~135 is a well-known open cluster and UBC~7 used to be considered as part of Collinder~135 before Gaia Data was released \citep{2020A&A...642L...4K, Kharchenko2013}. This pair is the only pair in our samples where the clusters are distinctly separated by their parallax histograms, rather than by equatorial coordinates. This indicates that they are obviously separated along the line of sight. NGC~2659 is a young open cluster that is characterized the cluster parameters given by \citet{Kharchenko2013}. UBC~482 was discovered by the work of \citet{2020A&A...635A..45C}. The cluster pair NGC~2659 and UBC~482 are another pair discovered by \citet{2021ARep...65..755C}, which makes up part of a large group containing nine members: Casado 28, NGC 2659, Casado 61, NGC 2645, SAI 92, LP 58, Pismis 8, UBC 482, and Gulliver 5. With the cluster spatial separation of 49.45 pc, this pair has the largest gap among our selected pairs. The cluster pair FSR~0198 and Teutsch~8 can be found in the works of \citet{2021A&A...649A..54P} and \citet{2022Univ....8..113C}, respectively. They are likely members of a triple primordial group of FSR~0198, Teutsch~8, and NGC~6871 \citep{2022Univ....8..113C}; later, Biurakan~2 was added to the group in \citet{2021A&A...649A..54P}. All the relevant parameters of this pair are well-matched except the celestial positions, as illustrated in Figure~\ref{fig:fig1}. Collinder~394 and NGC~6716 were confirmed to be physically interacting binaries by \citet{2022MNRAS.510.5695A}. The pair was discovered by \citet{2019ApJS..245...32L}, and \citet{2020JPhCS1523a2013N} were the first to analyze their morphology, age estimates, photometric mass, and kinematics using Gaia DR2 for the first time. This is the oldest (age > 90 Myr) and closest (Sep = 11.49 pc) pair compared to the other clusters in this work. The cluster pair Alessi~43 (also named ASCC~50) and Collinder~197 was first studied by \citet{2009A&A...500L..13D}. In \citet{2019ApJS..245...32L}, it was assigned to a triple group that included Collinder~197, Alessi~43, and BH~56. However, \citet{2021A&A...649A..54P} argued that it could be a binary cluster. All the relevant parameters (apart from the celestial coordinates) are well matched, as seen in Figure~\ref{fig:fig1}. \subsubsection{New binary open clusters} Pairs 10-14 are newly discovered candidates of binary open clusters in this work. Each pair fits well with the parameters of CG20 (see Figure \ref{fig:fig1} and Table \ref{binaries2}). In particular, UBC~410, UBC~547, and UBC~549 were discovered by \citet{2020A&A...635A..45C}. The other clusters were previously investigated by \citet{Kharchenko2013}. In this section, we present an analysis of the new candidate pairs and compare the cluster parameters with those investigated by the previous literature \citep{Dias2002, Kharchenko2013}. The distance between FSR~1297 and NGC~2362 is 41.13 pc. It is easy to distinguish them from the RA/DEC coordinates. Their CMDs overlap and the clusters share the same isochrone of NGC~2362, which indicates that FSR 1297 probably has the same age and extinction as NGC 2362 as shown in Figure \ref{fig:fig1}. The derived age of FSR~1297 is coincident with \citet{Kharchenko2013}. The ages provided by \citet{Kharchenko2013} for Antalova~2 and NGC~6383 are $logt = 6.0$ yr, and $logt = 6.6$ yr, respectively. The given age for Antalova~2 is quite different from the cluster age of $logt = 7.08$ yr from CG20, as shown in Table \ref{binaries3}. From the CMD in Figure \ref{fig:fig1}, the two isochrones from CG20 both fit well with the cluster members. There are 25 member stars in Antalova~2, which are scattered among the members of NGC~6383. The low number of known member stars makes it difficult to choose a better isochrone for Antalova~2. More member stars are needed to identify the actual age of Antalova~2. However, this does not affect our conclusion that it is a candidate for a truly binary open cluster. The cluster separation of Negueruela~1 and Teutsch~23 is 18.31 pc. The isochrone of Teutsch~23 based on Table \ref{binaries2} does not fit well with the cluster members. The ages given in Table \ref{binaries2} show that Negueruela~1 ($logt = 7.23$ yr) is about 11 Myr older than Teutsch~23 ($logt = 6.75$ yr), which is contradiction to the works of \citet{Kharchenko2013} and \citet{Dias2002}, where the ages in $logt$ for Negueruela 1 and Teutsch 23 are 6.98 yr and 7.0 yr, respectively (as shown in Table \ref{binaries3}). The isochrone of Negueruela~1 fits in well with the member stars of Teutsch~23, indicating that the ages of the two clusters are almost the same, which is consistent with the literature. UBC~547 and UBC~549 were discovered to be open clusters by \citet{2020A&A...635A..45C}. The member stars are mixed in the proper motion space, color-magnitude diagram, and parallax histogram. The derived isochrones by CG20 could potentially fit in with the members, but they are not adequate for a physical binary cluster. The age of UBC~549 (85 Myr) given by CG20 has to be considered with caution since there are no evolved member stars at the upper part of CMD. If this is a true pair, the age of UBC~549 would be comparable to that of UBC~547. The cluster parameters of candidate cluster pair Stock~20 and UBC~410 fit in well to their cluster members, as shown in Figure \ref{fig:fig1}. The age of Stock~20 is specified as $logt = 8.337$ yr, provided by \citet{Kharchenko2013} and \citet{Dias2002}. In CG20, the age of Stock~20 decreases to $logt = 7.28$ yr, which is more suitable for the cluster members. The cluster separation is 23.74 pc. Based on our analysis of the candidates of truly pairs open clusters, the cluster distances in the proper motion space are limited to 0.86 mas/yr, and the parallax differences between the clusters are less than 0.28 mas. If ignoring the disputed pairs, the ages of all candidates of pair OCs are < 100 Myr, with the age gaps less than 20 Myr, and the extinction discrepancies are limited to 0.4 mag. The ages of most binary open clusters can further decrease to 50 Myr, except for the oldest pair, Collinder~394 (91.2 Myr) and NGC~6716 (97.72 Myr). Our results are consistent with the literature \citep{1990PASJ...42..757B, 2022Univ....8..113C}, confirming that the binary open clusters are composed of young star clusters. \subsection{Comparison with 2MASS} The Two Micron All Sky Survey (2MASS) catalog \citep{2003yCat.2246....0C} provides high-quality near-infrared photometry for all-sky stars. We searched for the stars of the 14 analyzed pairs in 2MASS catalog based on the coordinates of the member stars. Figure \ref{fig:fig3} presents the distributions of color-magnitude diagrams (CMDs) and color-color diagrams (CCDs) for members of the 14 open cluster pairs using 2MASS photometric data. As shown in Figure \ref{fig:fig3}, all the pairs do not show a distinction between the two constituent clusters. \section{Summary}\label{section:4} In this study, we identify14 candidates of truly binary open clusters (see Table \ref{binaries2}), including one aggregate (Pairs 1-3, composed of four clusters: the $h$ and $ \chi $ Persei double cluster, UBC 46, and UBC 192), based on the Gaia database and cluster separations limited to 50 pc. By analyzing the distributions and movement trends of these quadruple open clusters, we find that NGC 869 may be the central cluster, with enough potential to disturb or influence the shapes of UBC 46 and NGC 884. Moreover, the morphological stability of NGC 869 is higher than that of the remaining clusters. Pairs 4-9 are known binary open clusters that can be found in the literature, and the parameters we obtained are essentially consistent with the literature. Pairs 10-14 are newly identified candidates of binary open cluster pairs, which are first reported in this work. Incidentally, ten pairs (Table \ref{binaries1}) appear to be the same clusters based on the fact that almost half of the cluster members are shared with one another. Among them, four pairs with ratio values greater than 0.8 have a greater probability of being the same clusters, while the other pairs require further study. \begin{acknowledgements} We are grateful to an anonymous referee for valuable comments which have improved the paper significantly. This work has been financially supported by the Natural Science Foundation of China (NSFC-U2031209). This work has made use of data from the European Space Agency (ESA) mission GAIA processed by Gaia Data processing and Analysis Consortium (DPAC), (https://www.cosmos.esa.int/web/gaia/dpac/consortium). \end{acknowledgements} \bibliographystyle{aa} \bibliography{aanda}

Title: Destruction of Long-Period Comets

Abstract: We identify a sample of 27 long-period comets for which both non-gravitational accelerations and Lyman-alpha based gas production rates are available. Seven of the 27 comets (i.e. 25 percent) did not survive perihelion because of nucleus fragmentation or complete disintegration. Empirically, the latter nuclei have the smallest gas production rates and the largest non-gravitational accelerations, which are both indicators of small size. Specifically, the disintegrating nuclei have a median radius of only 0.41 km, one quarter of the 1.60 km median radius of those surviving perihelion. The disintegrating comets also have a smaller median perihelion distance (0.48 au) than do the survivors (0.99 au). We compare the order of magnitude timescale for outgassing torques to change the nucleus spin, tau, with the time spent by each comet in strong sublimation, Dt, finding that the disrupted comets are those with tau < Dt. The destruction of near-Sun long-period comets is thus naturally explained as a consequence of rotational break-up. We discuss this process as a contributor to Oort's long mysterious ``fading parameter''.

https://export.arxiv.org/pdf/2208.04469

command. \newcommand{\vdag}{(v)^\dagger} \newcommand{\myemail}{skywalker@galaxy.far.far.away} \slugcomment{\textbf{The Astronomical Journal} } \shorttitle{LPCs} \shortauthors{Jewitt} \begin{document} \title{Destruction of Long-Period Comets} \author{David Jewitt$^{1}$\\ } \affil{$^1$Department of Earth, Planetary and Space Sciences, UCLA, \\ 595 Charles Young Drive East, Los Angeles, CA 90095-1567, USA} \email{jewitt@ucla.edu} \keywords{comets: general---comets: } \section{INTRODUCTION} \label{intro} Comets are volatile-rich products of accretion in the Sun's protoplanetary disk. Historically, comets were classified as either short-period or long-period, depending on whether their orbital periods were less or greater than 200 years. With some complications and subtleties, this division into two dynamical groups has survived to the present day. Most short-period (strictly ``Jupiter family'') comets arrive from the Kuiper belt, where they have been stored for the last 4.5 Gyr at temperatures $\sim$40 K. Long-period comets arrive from equally long-term residence in the Oort cloud, where the equilibrium temperature is $\lesssim$10 K. Both short and long-period comets are thought to have formed in the giant planet region of the solar system and were scattered to their respective storage reservoirs in the final stages of the growth of the planets. \cite{Oort50} first examined the distribution of orbital binding energies of long-period comets and found that no purely dynamical model could fit the large observed ratio of first-appearance (``dynamically new'') comets to returning comets. He invoked an ad-hoc ``fading parameter'' by which to decrease the brightness of returning comets and thereby to depress their number in magnitude limited surveys. Subsequent work with a much larger comet sample verified both the dynamics and the need for a fading parameter \citep{Wiegert99, Levison02} but did not identify its physical origin. Observationally, \cite{Bortle91} reported that intrinsically faint long-period comets with small perihelia are less likely to survive perihelion than their brighter counterparts but, again, did not identify the mechanism. Several practical difficulties are inherent in the observational study of long-period comets. Crucially, these objects are generally discovered only a short time before perihelion and they are soon thereafter lost as they recede from the Sun and fade to permanent invisibility. The observational window for long-period comets is consequently short, unlike that of the short-period comets, which can be predictably observed over many orbits. Even worse, long-period comets with perihelia $<$ 1 au must be observed at small solar elongation (Sun-Earth-comet) angles, where ground-based telescopes struggle against low elevation and high sky-brightness constraints. As a result of these practical difficulties, the nuclei of long-period comets are poorly characterized relative to those of short-period comets. Cometary disintegrations, being intrinsically unpredictable and rapidly evolving, are even more difficult to study and rarely reach publication in the refereed literature. \section{THE COMET SAMPLE} We are interested in long-period comets for which there exist both reliable measurements of the total outgassing rates and of the non-gravitational accelerations. For this reason, we focus on long-period comets discovered since the year 2000, to coincide with the period in which high quality physical and dynamical measurements have become routinely available from long-term surveys. Specifically, we rely on systematic measurements of the water production rate obtained from Lyman-$\alpha$ observations with the space-borne SWAN ultraviolet spectrometer aboard SOHO \citep{Bertaux97}, and on all-sky optical surveys (e.g.~the Catalina and, recently, Pan STARRS surveys) which provide precise astrometry over a long timebase. We further restricted the sample to comets with perihelion distances $q \le$ 2 au. \subsection{Non-gravitational Accelerations:} We used orbit solutions from JPL's Horizons\footnote{\url{http://ssd.jpl.nasa.gov/horizons.cgi}} web site to compile a list of LPCs showing evidence for non-gravitational acceleration. The orbital properties of the 27 comets used for this study are listed in Table \ref{list}, where we show the estimated original barycentric orbital elements in order to avoid the effects of planetary perturbations. (To find the latter, we used Horizons to compute the osculating orbital elements on 1900-Jan-01, a time at which all of the comets in the present study were far beyond the planetary region and therefore subject to minimal planetary perturbation). Twelve of the 27 comets are retrograde (inclination $i \ge$ 90\degr), consistent with being drawn from an isotropic distribution, and the eccentricities are all very close to $e$ = 1. By default, non-gravitational parameters are computed when purely gravitational orbits fail to reproduce astrometric data, but otherwise are assumed to be zero. Conventionally, the non-gravitational acceleration is resolved into three, orthogonal components ($A_1$, $A_2$ and $A_3$, expressed in au day$^{-2}$), with $A_1$ being in the radial direction, $A_3$ perpendicular to the plane of the orbit and $A_2$ perpendicular to $A_1$ and $A_3$ (Marsden et al.~1973). The radial component, $A_1$, is normally dominant, because cometary mass loss is concentrated on the heated, sun-facing side of the nucleus producing a recoil force that acts away from the Sun. Gas produced by the sublimation of cometary water ice dominates the instantaneous outflow momentum from the nucleus. For this reason, it is conventional to scale the acceleration by a function representing the instantaneous equilibrium sublimation rate, expressed as $g(r_H)$, such that the total acceleration is \begin{equation} \alpha_{NG} = g(r_H) \left(A_1^2 + A_2^2 + A_3^2\right)^{1/2}. \label{alpha} \end{equation} \noindent Function $g(r_H)$ is defined by \begin{equation} g(r_H) = \alpha_M \left(\frac{r_H}{r_0}\right)^{-m} \left[1 + \left(\frac{r_H}{r_0}\right)^n\right]^{-k} \end{equation} \noindent where $r_0$ = 2.808 au, $m$ = 2.15, $n$ = 5.093, $k$ = 4.6142, and $\alpha_M$ = 0.1113 are constants determined from a fit to a model of sublimation and the normalization is such that $g(1)$ = 1 \citep{Marsden73}. These constants derive from a model of a sublimating isothermal sphere. While this is logically incorrect (since an isothermal sphere would sublimate isotropically and hence experience no net recoil force), the difference from a physically more plausible hemispheric sublimator is minor, at least at distances $r_H \lesssim$ 3 au where the bulk of the absorbed energy is used to break hydrogen bonds in water and the sublimation rate is large. The assumed nucleus temperature distribution over the nucleus surface matters more at larger distances, but the sublimation rate falls exponentially and the resulting recoil force due to distant activity is comparatively small. Values of $A_1$, $A_2$ and $A_3$ and $\alpha_{NG}$ are listed for each comet in Table \ref{properties}. The accelerations are small (the median value at 1 au is $\alpha_{NG} = 4\times10^{-7}$ m s$^{-2}$, about 0.007\% of the solar gravitational acceleration). \subsection{Production Rates:} We searched the literature to find reliable measurements of the mass production rates from those long-period comets having non-zero non-gravitational accelerations. Water molecules dominate the mass flux and outflow momentum from the nucleus. Direct measurements of the water production rate are impractical, but observations of water photodissociation products provide an accurate alternative. We placed the greatest reliance on a long and spectacular series of measurements of Lyman-$\alpha$ emission from cometary hydrogen, a photodissociation product of water, made using the SWAN mapping spectrometer instrument on the SOHO spacecraft (Bertaux et al.~1997). These measurements (Combi et al.~2008, 2009, 2011, 2014, 2018, 2019, 2021, 2021) have the advantage of internal consistency, being made using a single instrument and, in many cases, providing time resolution sufficient to monitor the heliocentric variation of the water production. Where necessary, we used other published production rate data as listed in Table \ref{properties}. Production rates measured as a function of heliocentric distance were interpolated to estimate the production rate at $r_H$ = 1 au, denoted $Q_{H_2O}(1)$, and listed in Table \ref{properties}. Some comets display significant asymmetry, such that $Q_{H_2O}(1)$ differs before and after perihelion. In these cases we list the average value. We estimate that the values of $Q_{H_2O}(1)$ in the table are accurate to no better than a factor of $\sim$2. The mass production rate at 1 au, $\dot{M}(1)$ kg s$^{-1}$, is related to $Q_{H_2O}(1)$ by $\dot{M}(1) = \mu m_H Q_{H_2O}(1)$, where $\mu$ = 18 is the molecular weight of the water molecule, and $m_H = 1.67\times10^{-27}$ kg is the mass of the hydrogen atom. For reference, we note that a production rate $Q_{H_2O}(1) = 10^{29}$ s$^{-1}$ corresponds to $\dot{M}$ = 3000 kg s$^{-1}$. We assume that gas produced on the nucleus surface flows away at the thermal speed $V_{th} = (8 k T/(\pi \mu m_H))^{1/2}$, where $k = 1.38\times10^{-23}$ J K$^{-1}$ is the Boltzmann constant and $T$ is the temperature of the sublimating ice surface. At 1 au, the temperature is depressed by sublimation to about $T$ = 200 K, giving $V_{th}$ = 500 m s$^{-1}$. The temperature and $V_{th}$ change only slightly with heliocentric distance as a result of buffering by sublimation. We ignore solid matter (``dust'') expelled simultaneously from the comets because, although the dust and gas mass production rates may be comparable, the dust mass is dominated by large particles which are poorly coupled to the gas flow, travel at speeds $\ll V_{th}$, and so carry only a small fraction of the outflow momentum. \section{RESULTS} \subsection{Nucleus Radii} \label{nucleus} We use the production rate and non-gravitational acceleration data to estimate the radii of the nuclei in two ways. \textbf{a) Radius from Total Production Rate: } In equilbrium with sunlight, sublimation of cometary ice drives a mass flux, $f_s$ (kg m$^{-2}$ s$^{-1}$), from the surface. Measurements of comets show that sublimation from the nucleus night-side is weak because surface temperatures there are very low. Accordingly, we represent the nucleus as a sphere, sublimating only from the day-side hemisphere, and we use the energy balance equation to calculate $f_s$; \begin{equation} \frac{L_{\odot} (1-A_B)}{4\pi r_H^2} = 2\left[\epsilon \sigma T^4 + f_s(T) H(T)\right]. \label{energy} \end{equation} \noindent Here, the term on the left is the absorbed solar power ($L_{\odot} = 4\times10^{26}$ W is the solar luminosity, $A_B$ = 0.04 the assumed Bond albedo), the first term on the right accounts for radiation cooling ($\epsilon$ = 0.9 is the assumed emissivity, $\sigma = 5.67\times10^{-8}$ W m$^{-2}$ K$^{-4}$ is the Stephan-Boltzmann constant and $T$ is the effective temperature) and the second term represents energy consumed in sublimating ice the latent heat of which is $H = 2\times10^6$ J kg$^{-1}$). Equation \ref{energy} is solved using the Clausius-Clapeyron equation for the pressure vs.~temperature along the sublimation phase change boundary. At $r_H$ = 1 au, for sublimation averaged over the dayside of a spherical ice nucleus, we find $f_s = 2.1\times10^{-4}$ kg m$^{-2}$ s$^{-1}$. The mass loss rate, $\dot{M}$ (kg s$^{-1}$), is then \begin{equation} \dot{M} = 2 \pi r_n^2 f_A f_s \label{emdot} \end{equation} \noindent where $r_n$ is the nucleus radius. The factor $f_A$ represents the fraction of the surface area of the nucleus that contributes to the sublimation flux. Empirically, $f_A$ is a decreasing function of nucleus radius and approaches unity on sub-kilometer short-period comets (Jewitt 2021a). (This trend is probably a result of observational bias favoring the detection of small cometary nuclei having large active fractions over those with less active surfaces). Values $f_A >$ 1 are possible if ice sublimates both from the nucleus and from icy grains ejected from the nucleus. From Equation \ref{emdot} we obtain the nucleus radius \begin{equation} r_1 = \left(\frac{\dot{M}}{2 \pi f_Af_s}\right)^{1/2} \label{rn1} \end{equation} \noindent and the nucleus mass \begin{equation} M_n(1) = \rho_n \left(\frac{2}{9\pi}\right)^{1/2} \left(\frac{\dot{M}}{f_A f_s}\right)^{3/2}. \end{equation} \noindent As a starting point, and in the absence of evidence regarding $f_A$ on the long-period nuclei, we take $f_A$ = 1. Values of $r_1$ are listed for each comet in Table \ref{derived}. If $f_A <$ 1, as in most short-period comets \citep{Ahearn95}, then $r_1$ gives an under-estimate of the true radius. \textbf{b) Radius from Non-Gravitational Acceleration: } Non-gravitational acceleration, $\alpha_{NG}$, is the result of anisotropic mass loss from sublimating ices on the nucleus. We use it to obtain a second estimate of the mass of the nucleus. The force on the nucleus is $k_R \dot{M} V_{th}$, where $V_{th}$ is the outflow speed and $0 \le k_R \le 1$ is a dimensionless constant expressing the fraction of the outflow momentum that goes into accelerating the nucleus (for isotropic outgassing, $k_R$ = 0, while for perfectly collimated outgassing $k_R$ = 1). Then, force balance on a spherical nucleus of density $\rho_n$ gives a second relation for the nucleus radius \begin{equation} r_2 = \left(\frac{3 k_R \dot{M} V_{th}}{4 \pi \rho_n \alpha_{NG}}\right)^{1/3} \label{rn2} \end{equation} \noindent and the nucleus mass is \begin{equation} M_n(2) = \frac{k_R \dot{M} V_{th}}{\alpha_{ng}}. \end{equation} \noindent We take the measured values of $\dot{M}$ and $\alpha_{NG}$ from Table \ref{properties} and, as noted above, we adopt $\rho_n$ = 500 kg m$^{-3}$ and $V_{th}$ = 500 m s$^{-1}$. Measurements from 67P/Churyumov-Gerasimenko give $k_R$ = 0.5 (\cite{Jewitt20}), which we assume to apply to the long-period comets. The resulting values of $r_2$ are also listed for each comet in Table \ref{properties}. The two estimates of the nucleus radii are compared in Figure \ref{radius_radius}. Ideally, we would find $r_1 = r_2$ (indicated in the Figure by the solid diagonal line) but the comets are better described by $r_1 \sim (5/3) r_2$ (shown as a dashed, black line in the figure). \cite{Sosa11} found a similar result and interpreted it to mean that the long-period comets are hyperactive ($f_A >$ 1 in Equation \ref{rn1} acts to reduce $r_1$), allowing substantial sublimation from icy grains in the coma in addition to ice in the nucleus. However, the assumption of hyperactivity is only one of several possible reasons for the difference between $r_1$ and $r_2$. Parameters $k_R$, and $\rho_n$ in Equation \ref{rn2} might also be different from the values assumed and radius estimate $r_2$ can be increased by increasing $k_R$ and/or decreasing $\rho_n$. To consider one example, $r_1$ and $r_2$ could be brought into agreement if the bulk nucleus density in Equation \ref{rn2} were arbitrarily reduced by a factor $(5/3)^3 \sim$ 5 to $\rho_n \sim$ 100 kg m$^{-3}$ instead of 500 kg m$^{-3}$, as assumed. This lower density is by no means ruled out by physics and would still be consistent with values measured in several short-period comets (e.g.~6P/d'Arrest, 19P/Borrelly) \citep{Groussin19}. However, rather than make alternative, weakly justified guesses for some of the parameters in Equations \ref{rn1} and \ref{rn2}, we conservatively choose the average radius, $\overline{r_n} = (r_1 + r_2)/2$, as our best estimate of the nucleus radius, and we take the difference between radii $r_1$ and $r_2$ as a crude measure of the intrinsic radius uncertainty. We feel that this is a good procedure because, if we instead followed \cite{Sosa11} by assuming that $f_A >$ 1, the conclusions to be reached (described in Section \ref{discussion}) would be changed only by becoming stronger. The mean radii are listed in column 4 of Table \ref{derived}. Figure \ref{radius_radius} shows that the disintegrating long-period comets (filled red circles representing C/2001 A2, C/2010 X1, C/2012 S1, C/2017 E4, C/2019 Y4, C/2020 F8 and C/2021 A1) have smaller nuclei, on average, than those that survive perihelion (filled yellow circles). Six of the seven disintegrating comets are sub-kilometer bodies while the seventh (C/2001 A2 (LINEAR)) is only slightly larger at $\overline{r_n} = 1.4\pm0.4$ km (Table \ref{properties}). The median radius of the disintegrating long-period nuclei is 0.41 km (mean value 0.55$\pm$0.15 km, 7 objects) whereas that of surviving nuclei is 1.60 km (mean value 1.96$\pm$0.28 km, 20 objects). The non-parametric KS test was used to assess the likelihood that the two radius distributions are drawn from the same population. This test gave a statistic $D$ = 0.807, with an associated probability $p$ = 0.002, consistent with the visual impression that the surviving and disrupted radius distributions are distinct. We also compared the distributions of perihelion distances of the surviving and disrupted comets. The median perihelion distance of the disrupted comets is 0.48 au (mean value 0.48$\pm$0.09 au, 7 objects) while that of the survivors is 0.99 au (mean value 0.97$\pm$0.10 au, 20 objects). The associated KS statistics are $D$ = 0.697 and $p$ = 0.007, indicating that the hypothesis that the two samples are drawn from the same parent population can again be rejected, although with less confidence. Figure \ref{nga_dmbdt} compares the two measured quantities, the non-gravitational acceleration and the mass loss rate, both at 1 au. The figure shows that $\alpha_{NG}(1)$ and $\dot{M}(1)$ are inversely related, and with a trend that is readily understood. All else being equal, we expect that the mass loss rate should vary as $\dot{M}(1) \propto \overline{r_n}^2$ (Equation \ref{rn1}), while the non-gravitational acceleration should vary as $\alpha_{NG}(1) \propto \dot{M(1)}/\overline{r_n}^3 \propto 1/\overline{r_n}$ (Equation \ref{rn2}), giving $\alpha_{NG}(1) \propto \dot{M}(1)^{-1/2}$. The black line in Figure \ref{nga_dmbdt} has slope -1/2 and evidently matches the data well. A least-squares fit of a power law (dashed red line in the figure) to the data gives $\alpha_{NG}(1) \propto \dot{M}(1)^{B}$ with $B = -0.66\pm0.17$, consistent with this expectation within one standard deviation. The color coding in Figure \ref{nga_dmbdt} also shows that comets with the largest non-gravitational accelerations and the weakest outgassing rates are the most likely to disintegrate, with an approximate separation between disintegrating and surviving comets at $\alpha_{NG}(1) \sim 10^{-6}$ m s$^{-2}$ and $\dot{M}(1) \sim 10^3$ kg s$^{-1}$. By Equations \ref{rn1} and \ref{rn2}, these values correspond to nucleus radii $r_n$ = 0.5 to 0.9 km, consistent with the color-coding in Figure \ref{radius_radius} showing that subkilometer long-period comet nuclei disintegrate. Published examples of well-characterized sub-kilometer long-period comets are few and far between; the compilation by \cite{Lamy04} lists only two. C/1999 S4 (LINEAR) had $r_n$ = 0.45 km \citep{Altenhoff02}, $q$ = 0.765 au and disintegrated spectacularly at perihelion \citep{Weaver01}, consistent with our findings here. C/1983 J1 (Sugano-Saigusa-Fujikawa) had $r_n <$ 0.37 km \citep{Hanner87}, $q$ = 0.47 au but was observed only for a few weeks when near Earth so that its fate is unknown. The paucity of well-studied small nuclei relative to power-law extrapolations from larger sizes \citep{Bauer17} may itself be evidence for the efficient destruction of sub-kilometer long-period comet nuclei. \section{DISCUSSION} \label{discussion} Seven of the 27 LPCs in our sample either fragmented or disintegrated, fates that are indicated in column 9 of Table \ref{derived} by the letters ``F'' and ``D'', respectively. These descriptors are purely morphological; in fragmentation the comet splits into two or more discrete objects typically each retaining a cometary appearance, while in disintegration the comet assumes the appearance of an expanding, diffuse cloud, lacking an obvious source or other embedded structure. The physical relationship between fragmentation and disintegration is unclear. We assume that the former is a mild case of the latter and, for simplicity in the following discussion, we use the term ``disintegration'' to apply to both. \subsection{Disintegration Mechanisms} Given the results in Figures \ref{radius_radius} and \ref{nga_dmbdt}, what mechanisms could be responsible for the disintegration of long-period nuclei? \textbf{Tidal Breakup:} The Roche radius of the Sun for a comet nucleus represented as a fluid body of density $\rho_n$ = 500 kg m$^{-3}$ is $\sim$10$^{-2}$ au. In our sample, only C/2012 S1 (ISON) approached the Sun closely enough ($q$ = 0.012 au) for tidal disruption to be possible. Neither have the remaining comets passed within the Roche spheres of any planet, eliminating tidal breakup as a generally relevant mechanism for this study. \textbf{Sublimation:} Sublimation erosion is typically $\sim$10 m per orbit (computed from Equation \ref{energy}), and thus is too slow to destroy a $\sim$1 km diameter nucleus in the $\sim$1 year spent in strong sublimation while close to the Sun. Moreover, sublimation would naturally produce more steady erosion of the comet, not the catastrophic disintegrations as observed. \textbf{Collisional Disruption:} Interplanetary collisions are very rare. Even in the relatively dense asteroid belt, the collisional disruption timescales of sub-kilometer bodies are measured in 100s of Myr, such that the probability of a destructive collision in the $\sim$1 year spent by each comet near the Sun is negligible. Moreover, the long-period comets have large orbits with random inclinations and disintegrate far above the ecliptic plane, where collisional disruption is even less likely. \textbf{Confined Pressure Explosion:} \cite{Samarasinha01} invoked the build-up of gas pressure in order to explain cometary disruption. The problem with this mechanism is that the effect of heating by the Sun is confined to a thermal skin that is very thin compared to the radius of the nucleus. For example, cometary material has very small diffusivity, resulting in a thermal skin depth ($\sim$10$^{-2}$ m) that is very small compared to the nucleus radius. Unless the full body of the nucleus is permeated by large interconnected voids (but still sealed from the vacuum of surrounding space), any gas pressure build-up must be confined to a thin surface shell incapable of disrupting of the whole nucleus. Likewise, crystallization runaways, although potentially capable of triggering cometary outbursts \citep{Prialnik92}, are necessarily confined by the radial temperature gradient to a surface shell with thickness $\ll r_n$. \textbf{Rotational Instability:} Outgassing exerts a torque on the cometary nucleus capable of substantially changing the spin on a timescale given by \begin{equation} \tau_s = \left(\frac{16\pi^2}{15}\right)\left(\frac{\rho_n r_n^4}{k_T V_{th} P} \right)\left(\frac{1}{\overline{\dot{M}}} \right). \label{tau_s} \end{equation} \noindent Here, $P$ is the instantaneous spin-period, $k_T$ is the dimensionless moment arm, equal to the fraction of the outflow momentum that exerts a torque on the nucleus and $\overline{\dot{M}}$ is the average value of the mass loss rate following the comet in its orbit \citep{Jewitt21a}. For a weakly cohesive nucleus, the end-state of spin-up is rotational disruption into fragments which are themselves subject to rapid disintegration because of the strong dependence of $\tau_s$ on $r_n$ (Equation \ref{tau_s}) and because of the sudden exposure of previously buried volatiles. To calculate $\overline{\dot{M}}$, we use the facts that water is the dominant volatile and that strong sublimation of water ice is restricted to heliocentric distances $r_H \lesssim$ 3 au. Species more volatile than water (notably CO, CO$_2$) can sublimate at lower temperatures and larger distances but their abundances are poorly constrained in our sample of comets and, in the interests of simplicity, we do not consider them here. Their inclusion would only strengthen our conclusions by amplifying the importance of outgassing torques relative to our water ice calculation. Accordingly, we computed the average mass loss rate for each comet using \begin{equation} \overline{\dot{M}} = \frac{\int_{t_0}^{t_0+\Delta t} \dot{M}(r_H(t)) dt}{\Delta t} \end{equation} \noindent where $t_0$ is the pre-perihelion time at which $r_H$ = 3 au and $\Delta t$ is the time spent with $r_H \le$ 3 au. The instantaneous mass loss rate $\dot{M}(r_H(t))$ is computed by solving Kepler's law for $r_H(t)$ and using Equation \ref{energy} to find $\dot{M}(r_H(t)) = 2 \pi f_s(r_H(t)) \overline{r_n}^2$. The average, listed in Table \ref{derived}, is then substituted into Equation \ref{tau_s} in order to calculate the characteristic spin-up time, $\tau_s$. We again used values of the quantities $k_T$, $\rho_n$ and $P$ taken from the cometary literature. The median dimensionless moment arm, $k_T = 0.007$, is determined from measurements of the nuclei of short-period comets \citep{Jewitt21a}. Likewise, the average nucleus density, $\rho_n = 480\pm220$ kg m$^{-3}$, is known only for the nuclei of short-period comets \citep{Groussin19}. We adopt $P$ = 15 hours, equal to the median rotation period measured in short-period comets \citep{Jewitt21a}. While there are no clear reasons to think that the median $k_T$, $\rho_n$ and $P$ should be different in the long-period vs.~short-period comets, we are aware of this possibility and eagerly await direct measurements of these parameters in the former population. Given the uncertainties in these quantities, it is obvious that Equation \ref{tau_s} can provide, at best, a value of $\tau_s$ accurate to no better than order of magnitude. We set a simple criterion for judging the importance of spin-up torques by comparing $\tau_s$ with $\Delta t$, defined above as the time spent by each comet with $r_H \le$ 3 au. If $\tau_s < \Delta t$ then sublimation torques can substantially modify the nucleus spin within a single perihelion passage of the comet, potentially leading to rotational instability and breakup. Otherwise, the outgassing torques are too weak to trigger rotational breakup, at least within a single perihelion passage. Values of $\Delta t$ and $\tau_s$ are listed in Table \ref{derived} and plotted for convenience in Figure \ref{disint}. The table and figure show that six of the seven disintegrated comets in our sample had $\tau_s < \Delta t$, consistent with rotational breakup as the cause of their destruction. The seventh (C/2001 A2) also satisfies this inequality within the error bar on $\tau_s$. Those comets which did not breakup or disintegrate have $\tau_s > \Delta t$, again with some ambiguous cases close to the $\tau_s = \Delta t$ line. Again, while emphasizing the (necessarily) order of magnitude nature of the treatment offered in Section \ref{nucleus} the basic result, that the disintegrating nuclei are those with the shortest spin-up times, is remarkable. Figure \ref{emdot_vs_rn} shows the mass loss rate at $r_H$ = 1 au vs.~the nucleus radius, $\overline{r_n}$. The solid black line in the figure shows $\dot{M}(1) = 1850 \overline{r_n^2}$, with $\overline{r_n}$ in km and $\dot{M}(1)$ in kg s$^{-1}$. Substituting for $\dot{M}(1)$ in Equation \ref{tau_s} and setting $\tau_s = \Delta t$ = 1 year, we calculate the critical radius below which the average long-period comet is susceptible to rotational break-up in a single perihelion passage as $r_n \sim$ 1 km, in agreement with the observation that the disintegrating long-period comets are sub-kilometer objects. \subsection{Relation to the Oort Fading Parameter} Long-period comets with reciprocal semimajor axes $a^{-1} < 10^{-4}$ au$^{-1}$ are known as ``Oort spike'' comets, some but not all of which are dynamically new objects making their first pass through the planetary region. \cite{Oort50} found that a purely dynamical model of comet delivery from a distant reservoir predicts a larger flux of returning objects, relative to dynamically new comets, than is observed. He introduced an ad hoc ``fading parameter'' to bring the dynamical model into agreement with the data. The need for this fading parameter has since been confirmed many times (e.g.~\cite{Wiegert99,Levison02,Neslusan06}) but the physical cause of the fading remains unknown. \cite{Oort50} and others conjecture that it is due to ``surface aging'' in response to insolation. \cite{Levison02} concluded from the small number of detections of returning objects in ground-based surveys that a majority of dynamically new comets are destroyed, not merely faded. The present work shows that disintegration, by removing comets from the observable sample, is a significant fading mechanism. The destruction is size dependent, preferentially afflicting small nuclei, and also perihelion distance dependent, being more probable at small distances than at large. In contrast, published models of the fading parameter instead assume a survival probability that is independent of nucleus size (\cite{Wiegert99,Levison02}). \cite{Wiegert99} examined a sample of comets having median perihelion $q \sim$ 1 au, comparable to that of our sample in Table \ref{list}. As one of several possible solutions, they found a fading law in which the comets are divided into two groups and supposed that some 95\% of long-period comets are destroyed within the first six perihelion passages while the remaining 5\% survive indefinitely. We conjecture that this empirical division into two groups is an artifact of size-dependent rotational disruption; 95\% of the long-period comets are sub-kilometer objects subject to rotational disruption in a few orbits while 5\% are larger and can resist disruption for a much longer time. The implication that only 5\% of long-period comet nuclei have $r_n >$ 1 km at first appears at odds with the radius distribution of the nuclei listed in Table \ref{derived}, where 17 of the 27 nuclei (62\%) are larger than 1 km. However, our sample is highly observationally biased against the inclusion of small nuclei because they produce too little H$_2$O to be detected in the flux-limited Lyman-$\alpha$ data from SWAN. (The largest comets are also excluded because their non-gravitational accelerations are too small to be measured). As a result, the data in Table \ref{derived} severely underestimate the abundance of small LPC nuclei, and cannot be used to assess the intrinsic size distribution of the nuclei. Cometary fading has been reported to extend to at least $r_H \sim$ 10 au \citep{Krolikowska19, Kaib22}, far beyond the region where outgassing torques from sublimating water ice can alter the nucleus spin. Moreover, a growing number of observations show activity in distant comets (e.g.~$\sim$20 - 25 au in the case of C/2014 UN271 Bernardinelli-Bernstein \citep{Farnham21} and even 35 au in the case of C/2017 K2 (PANSTARRS) \citep{Jewitt21b}). Fragmentation has also been inferred at very large distances, for example in comets C/2002 A1 and A2, reported to have split from a common parent when inbound at $r_H \sim$ 22.5 au \citep{Sekanina03}. Water ice is involatile beyond 5 or 6 au and the sublimation of a more volatile material, perhaps carbon monoxide (CO) ice, is a leading candidate for driving this distant cometary activity. Could fading at large distances be due to torques from CO sublimation? A definitive answer to this question cannot be reached given our limited knowledge of the surface properties of distant comets, or calculated from first principles. However, order of magnitude scaling considerations strongly suggest an answer in the negative, as follows. To first order, the ratio of the timescale for spin-up of a given body through sublimation of CO to that for spin-up through sublimation of water ice is (c.f.~Equation \ref{tau_s}) \begin{equation} \frac{\tau_s(\textrm{CO})}{\tau_s(\textrm{H}_2\textrm{O})} \sim \frac{\dot{M}(\textrm{H}_2\textrm{O}) }{\dot{M}(\textrm{CO}) } \frac{ V_{th}(\textrm{H}_2\textrm{O})}{ V_{th}(\textrm{CO})}. \label{ratio} \end{equation} \noindent We compare the sublimation of CO at 10 au to that of H$_2$O at 1 au. Fortunately, measurements of the two ratios on the right hand side of Equation \ref{ratio} are available for the long-period comet C/1995 O1 (Hale-Bopp). There, $\dot{M}(\textrm{H}_2\textrm{O}, r_H = 1 \textrm{ au}) / \dot{M}(\textrm{CO}, r_H = 10 \textrm{ au}) \sim 10^3$ (Figure 5 of \cite{Biver02}) and $V_{th}(\textrm{H}_2\textrm{O}, r_H = 1 \textrm{ au})/ V_{th}(\textrm{CO}, r_H = 10 \textrm{ au}) \sim$ 3 (figure 4a of \cite{Biver02}), giving $\tau_s(\textrm{CO}, r_H = 10 \textrm{ au}) / \tau_s(\textrm{H}_2\textrm{O}, r_H = 1 \textrm{ au}) \sim$ 3000. This factor of 3000 is partly compensated by the longer time spent in the CO sublimation zone. C/1995 O1 had $r_H <$ 30 au for 27 years (compared with $\Delta t \sim$ 1 year for the comets in Table 1) and could have sublimated CO the entire time. Still, based on this scaling argument, spin-up timescales for a given object at 10 au remain two orders of magnitude larger than at 1 au, given C/Hale-Bopp-like outgassing behavior. Substantial spin-up due to CO torques seems unlikely unless the CO/H$_2$O ratios in other LPCs are much larger than measured in C/1995 O1. On this basis we conclude that, while important in the water sublimation zone, spin-up destruction is not an obvious cause of spin-up or fading in any but the tiniest comets beyond it. Oort's fading parameter thus seems likely to have several physical origins of which rotational disruption is only one. For LPCs with $q \lesssim$ 3 au, we offer three predictions that will be observationally testable in the foreseeable future given improved population data. First, LPC nuclei larger than a few km in radius should rarely disrupt or disintegrate, unless by another process (e.g.~tidal disruption, as may be the case for C/2012 S1 (ISON)). Second, the size distribution of LPCs should be flattened at radii $r_n \lesssim$ 1 km, relative to its value at larger radii, owing to the selective loss of small nuclei through rotational instability. Third, accurately bias-corected data should show that the size distributions of long-period comet nuclei vary with $q$, reaching ``primordial'' values only in the outer solar system where mass loss is negligible. Lastly, it is reasonable to expect that the imprints of rotational disruption might be found in the Damocloid population, to the extent that these objects (inactive bodies with Tisserand parameters $T_J \le$ 2) are remnants of formerly active long-period comets. Measurements indeed show a flatter size distribution \citep{Kim14}, consistent with the preferential destruction of smaller Damocloids, but the available sample is small and undoubtedly subject to its own biases. Future work on these objects may also be revealing. \clearpage \section{SUMMARY} We examine a sample of 27 long-period comets for which both non-gravitational accelerations and water production rates are available. Using these two measured quantities we are able to estimate the nucleus sizes, and so to explore the systematics of this population. Seven of the 27 comets ($\sim$25\%) fragmented or disintegrated. \begin{itemize} \item The disintegrating cometary nuclei have systematically smaller radii (median 0.4 km, 7 objects) than those that survive in proximity to the Sun (1.6 km, 20 objects). % \item The disintegrating comets have smaller perihelion distance (median 0.5 au, 7 objects) than those surviving (1.0 au, 20 objects). \item These size and perihelion distance trends are both consistent with nucleus disintegration through rotational instability, triggered by outgassing torques from sublimating water ice. Specifically, the timescale for outgassing torques to change the spin of sub-kilometer nuclei is less than the time spent in strong sublimation. \item Rotational disruption is a cause of the ``fading'' required to fit the orbital semimajor axis distribution of long-period comets. \end{itemize} \clearpage \acknowledgments I thank the anonymous referee for highlighting the importance of fading at large distances, Yoonyoung Kim for additional comments on the manuscript and Man-To Hui for advice about the vagaries of JPL Horizons. Based in part on observations made under GO 16929 with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. \clearpage \clearpage \begin{deluxetable}{lrrrrrrrrr} \tabletypesize{\scriptsize} \tablecaption{Comet Sample \label{list}} \tablewidth{0pt} \tablehead{\colhead{Comet} & \colhead{$a$\tablenotemark{a}} & \colhead{$e$\tablenotemark{b}} & \colhead{$i$\tablenotemark{c}} & \colhead{$q$\tablenotemark{d}} & \colhead{$T_P$\tablenotemark{e}} & \colhead{S/NS\tablenotemark{f}} } \startdata C/2000 WM1 (LINEAR) & 1877 & 0.9997077 & 72.6 &0.549 & 2452297.3 & NS\\ C/2001 A2-A (LINEAR) & 971 & 0.9991978 & 36.5 &0.779 & 2452054.0 & NS \\ C/2001 Q4 (NEAT) & 16725 & 0.9999426 & 99.6 &0.960 & 2453141.5 & S \\ C/2002 T7 (LINEAR) & 47471 & 0.9999870 & 160.6 &0.615 & 2453118.5 & S \\ C/2002 X5 (Kudo-Fujikawa) & 1119 & 0.9998281 & 94.2 & 0.192 & 2452668.5 & NS \\ C/2003 K4 (LINEAR) & 29199 & 0.9999651 & 134.2 & 1.021 & 2453292.3 & S \\ C/2004 Q2 (Machholz) & 2528 & 0.9995221 & 38.6 & 1.208 & 2453395.5 & NS \\ C/2009 P1 (Garradd) & 2384 & 0.9993522 & 106.2 &1.544 & 2455919.3 & NS \\ C/2010 X1 (Elenin) & 48388 & 0.9999900 & 1.8 &0.482 & 2455815.2 & S \\ C/2012 K1 (PANSTARRS) & 26070 & 0.9999597 & 142.4 &1.051 & 2456897.3 & S \\ C/2012 S1 (ISON) & -144820 & 1.0000001 & 62.2 &0.012 & 2456625.3 & S \\ C/2012 X1 (LINEAR) & 145 & 0.9889527 & 44.4 &1.597 & 2456710.3 & NS \\ C/2013 US10 (Catalina) & 19030 & 0.9999569 & 148.9 &0.820 & 2457342.3 & S \\ C/2013 X1 (PANSTARRS) & 3804 & 0.9996529 & 163.2 &1.320 & 2457499.3 & NS \\ C/2014 E2 (Jacques) & 807 & 0.9991826 & 156.4 &0.664 & 2456841.0 & NS \\ C/2014 Q1 (PANSTARRS) & 841 & 0.9996251 & 43.1 &0.315 & 2457210.0 & NS \\ C/2014 Q2 (Lovejoy) & 502 & 0.9974320 & 80.3 &1.290 & 2457052.5 & NS \\ C/2015 ER61 (PANSTARRS) & 718 & 0.9985315 & 6.3 &1.054 & 2457883.5 & NS \\ C/2015 G2 (MASTER) & 6103 & 0.9998724 & 147.6 &0.779 & 2457166.2 & NS \\ C/2015 V2 (Johnson) & 2525 & 0.9999546 & 49.9 &1.631 & 2457916.8 & NS \\ C/2017 E4 (Lovejoy) & 821 & 0.9994044 & 88.2 &0.489 & 2457866.8 & NS \\ C/2017 T2 (PANSTARRS) & 36830 & 0.9999560 & 57.2 &1.619 & 2458974.5 & S \\ C/2019 Y1 (ATLAS) & 209 & 0.9959967 & 73.3 &0.836 & 2458924.0 & NS \\ C/2019 Y4-B (ATLAS) & 501 & 0.9994983 & 45.4 &0.251 & 2459000.5 & NS \\ C/2020 F8 (SWAN) & -4886 & 1.0000874 & 110.8 &0.427 & 2458997.0 & S \\ C/2020 S3 (Erasmus) & 191 & 0.9978918 & 19.9 &0.403 & 2459196.3 & NS \\ C/2021 A1 (Leonard) & 2028 & 0.9996965 & 132.7 &0.616 & 2459582.8 & NS \\ \enddata \tablenotetext{a}{Barycentric semimajor axis, in au} \tablenotetext{b}{Eccentricity} \tablenotetext{c}{Inclination, in degrees } \tablenotetext{d}{Perihelion distance, in au } \tablenotetext{e}{Mean Julian Date of Perihelion } \tablenotetext{f}{S = Spike, NS = Non-Spike } \end{deluxetable} \clearpage \begin{deluxetable}{lccrrrrrrr} \tabletypesize{\scriptsize} \tablecaption{Measured Properties \label{properties}} \tablewidth{0pt} \tablehead{\colhead{Comet} & \colhead{$A_1$\tablenotemark{a}} & \colhead{$A_2$\tablenotemark{a}} & \colhead{$A_3$\tablenotemark{a}} & \colhead{$\alpha_{NG}$\tablenotemark{b}} & \colhead{$Q_{H_2O}(1)$\tablenotemark{c}} & Source\tablenotemark{d} } \startdata C/2000 WM1 (LINEAR) & 5.8e-09 & -8.0e-11 & 0 & 1.2e-07 & 1.7e+29 & C19 \\ C/2001 A2 (LINEAR) & -2.1e-08 & 1.9e-08 & 0 & 5.7e-07 & 1.4e+29 & C08 \\ C/2001 Q4 (NEAT) & 1.6e-08 & 4.8e-10 & 4.3e-10 & 3.2e-07 & 5.3e+29 & C09 \\ C/2002 T7 (LINEAR) & 1.2e-08 & 9.6e-10 & -1.8e-09 & 2.4e-07 & 7.3e+29 & C19, S20 \\ C/2002 X5 (Kudo-Fujikawa) & 2.6e-08 & 5.8e-09 & 0 & 5.4e-07 & 7.5e+28 & C11 \\ C/2003 K4 (LINEAR) & 8.1e-09 & -3.6e-09 & -5.6e-10 & 1.8e-07 & 5.2e+29 & C19 \\ C/2004 Q2 (Machholz) & 1.2e-08 & -1.1e-09 & -2.3e-09 & 2.5e-07 & 6.2e+29 & C19 \\ C/2009 P1 (Garradd) & 2.0e-08 & -1.0e-09 & 0 & 4.0e-07 & 4.8e+29 & C19 \\ C/2010 X1 (Elenin) & -4.8e-08 & 6.3e-08 & 0 & 1.6e-06 & 7.4e+27 & S11 \\ C/2012 K1 (PANSTARRS) & 2.2e-08 & -1.6e-09 & -2.6e-09 & 4.5e-07 & 2.0e+29 & C19 \\ C/2012 S1 (ISON) & 8.5e-08 & 5.8e-09 & 0 & 1.7e-06 & 2.0e+28 & C19 \\ C/2012 X1 (LINEAR) & 3.6e-08 & 2.2e-09 & 5.3e-09 & 7.3e-07 & 1.4e+29 & L14 \\ C/2013 US10 (Catalina) & 7.6e-09 & 6.2e-11 & 1.6e-10 & 1.5e-07 & 2.2e+29 & C19 \\ C/2013 X1 (PANSTARRS) & 2.0e-08 & -4.1e-09 & -7.3e-09 & 4.4e-07 & 5.7e+29 & C19 \\ C/2014 E2 (Jacques) & 2.1e-08 & -2.8e-09 & 0 & 4.3e-07 & 1.3e+29 & C19 \\ C/2014 Q1 (PANSTARRS) & 7.9e-09 & 2.8e-09 & -5.5e-09 & 2.0e-07 & 3.0e+28 & C19 \\ C/2014 Q2 (Lovejoy) & 1.3e-09 & -1.5e-09 & -2.4e-09 & 6.3e-08 & 2.1e+30 & C19 \\ C/2015 ER61 (PANSTARRS) & 5.6e-09 & -3.5e-09 & -4.6e-10 & 1.3e-07 & 8.6e+28 & S20 \\ C/2015 G2 (MASTER) & 1.2e-08 & 5.9e-09 & -4.3e-09 & 2.8e-07 & 5.4e+28 & C19 \\ C/2015 V2 (Johnson) & 2.3e-08 & -4.2e-09 & -4.4e-09 & 4.8e-07 & 1.6e+29 & C21 \\ C/2017 E4 (Lovejoy) & 1.8e-07 & -7.6e-08 & 0 & 3.9e-06 & 1.4e+28 & F18 \\ C/2017 T2 (PANSTARRS) & 3.6e-08 & -7.1e-10 & -3.2e-10 & 7.2e-07 & 3.1e+28 & C21 \\ C/2019 Y1 (ATLAS) & 1.1e-08 & 0 & 0 & 2.2e-07 & 1.4e+28 & C21 \\ C/2019 Y4 (ATLAS) & 2.9e-07 & -9.1e-09 & 0 & 5.8e-06 & 1.0e+28 & C21 \\ C/2020 F8 (SWAN) & 1.5e-07 & -2.6e-08 & 0 & 3.1e-06 & 5.5e+27 & C21 \\ C/2020 S3 (Erasmus) & 1.7e-08 & 6.3e-09 & 0 & 3.6e-07 & 6.0e+28 & C21b \\ C/2021 A1 (Leonard) & 5.8e-08 & -2.0e-08 & 1.1e-08 & 1.3e-06 & 3.0e+28 & C22 \\ \enddata \tablenotetext{a}{Orthogonal components of the non-gravitational acceleration (units au day$^{-2}$) from JPL Horizons} \tablenotetext{b}{Total non-gravitational acceleration at $r_H$ = 1 au (units m s$^{-2}$) from Equation \ref{alpha}} \tablenotetext{c}{Water production rate at $r_H$ = 1 au, molecules s$^{-1}$} \tablenotetext{d}{C08 = \cite{Combi08}, C09 = \cite{Combi09}, C11 = \cite{Combi11}, C19 = \cite{Combi19}, C21 = \cite{Combi21}, C21b = \cite{Combi21b}, C22 = M. Combi (private communication), F18 = \cite{Faggi18}, S11 = D. Schleicher (private communication, cited in \cite{Li15}), S20 = \cite{Saki20} } \end{deluxetable} \clearpage \clearpage \begin{deluxetable}{lccrlrrrrr} \tabletypesize{\scriptsize} \tablecaption{Derived Properties \label{derived}} \tablewidth{0pt} \tablehead{\colhead{Comet} & $r_1$\tablenotemark{a} & $r_2$\tablenotemark{b} & $\overline{r}$\tablenotemark{c} & \colhead{$Q_{H_2O}$\tablenotemark{d}} & $\overline{\dot{M}}$\tablenotemark{e} & \colhead{$\tau_s$\tablenotemark{f}} & \colhead{$\Delta t $\tablenotemark{g}} & \colhead{Note\tablenotemark{h} } } \startdata C/2000 WM1 (LINEAR) & 2.0 & 1.7 & 1.9 & 1.2e+29 & 3628 & 3.13 & 0.96 & \\ C/2001 A2-A (LINEAR) & 1.8 & 1.0 & 1.4 & 7.3e+28 & 2188 & 1.58 & 1.02 & F \\ C/2001 Q4 (NEAT) & 3.6 & 1.8 & 2.7 & 2.2e+29 & 6691 & 7.15 & 1.05 & \\ C/2002 T7 (LINEAR) & 4.2 & 2.2 & 3.2 & 4.7e+29 & 14263 & 6.65 & 0.98 & \\ C/2002 X5 (Kudo-Fujikawa) & 1.3 & 0.8 & 1.1 & 1.2e+29 & 3607 & 0.32 & 0.85 & \\ C/2003 K4 (LINEAR) & 3.5 & 2.2 & 2.8 & 2.1e+29 & 6252 & 9.79 & 1.07 & \\ C/2004 Q2 (Machholz) & 3.9 & 2.1 & 3.0 & 2.3e+29 & 6895 & 10.32 & 1.09 & \\ C/2009 P1 (Garradd) & 3.4 & 1.6 & 2.5 & 1.2e+29 & 3607 & 10.04 & 1.10 & \\ C/2010 X1 (Elenin) & 0.4 & 0.3 & 0.3 & 6.0e+27 & 180 & 0.07 & 0.94 & D \\ C/2012 K1 (PANSTARRS) & 2.2 & 1.2 & 1.7 & 8.8e+28 & 2645 & 2.79 & 1.07 & \\ C/2012 S1 (ISON) & 0.7 & 0.3 & 0.5 & 1.1e+29 & 3306 & 0.02 & 0.78 & D \\ C/2012 X1 (LINEAR) & 1.8 & 0.9 & 1.4 & 3.4e+28 & 1010 & 3.05 & 1.10 & \\ C/2013 US10 (Catalina) & 2.3 & 1.7 & 2.0 & 1.1e+29 & 3240 & 4.67 & 1.02 & \\ C/2013 X1 (PANSTARRS) & 3.7 & 1.7 & 2.7 & 1.7e+29 & 5140 & 9.31 & 1.09 & \\ C/2014 E2 (Jacques) & 1.8 & 1.0 & 1.4 & 8.3e+28 & 2501.0 & 1.38 & 0.99 & \\ C/2014 Q1 (PANSTARRS) & 0.8 & 0.8 & 0.8 & 3.4e+28 & 1010 & 0.43 & 0.89 & \\ C/2014 Q2 (Lovejoy) & 7.1 & 4.9 & 6.0 & 6.5e+29 & 19569 & 61.01 & 1.10 & \\ C/2015 ER61 (PANSTARRS) & 1.4 & 1.3 & 1.4 & 3.4e+28 & 1008 & 3.24 & 1.07 & \\ C/2015 G2 (MASTER) & 1.1 & 0.9 & 1.0 & 3.2e+28 & 973 & 0.99 & 1.02 & \\ C/2015 V2 (Johnson) & 2.0 & 1.1 & 1.5 & 3.8e+28 & 1154 & 4.18 & 1.10 & \\ C/2017 E4 (Lovejoy) & 0.6 & 0.2 & 0.4 & 1.1e+28 & 328 & 0.08 & 0.95 & D \\ C/2017 T2 (PANSTARRS) & 0.9 & 0.5 & 0.7 & 7.4e+27 & 223 & 0.98 & 1.09 & \\ C/2019 Y1 (ATLAS) & 0.6 & 0.6 & 0.6 & 6.7e+27 & 202 & 0.57 & 1.03 & \\ C/2019 Y4 (ATLAS) & 0.5 & 0.2 & 0.3 & 1.3e+28 & 393 & 0.03 & 0.87 & F \\ C/2020 F8 (SWAN) & 0.4 & 0.2 & 0.3 & 4.8e+27 & 143 & 0.04 & 0.92 & D \\ C/2020 S3 (Erasmus) & 1.2 & 0.8 & 1.0 & 5.6e+28 & 1677 & 0.60 & 0.92 & \\ C/2021 A1 (Leonard) & 0.8 & 0.4 & 0.6 & 1.9e+28 & 577 & 0.27 & 0.98 & D \\ \enddata \tablenotetext{a}{Nucleus radius from Equation \ref{rn1}, km} \tablenotetext{b}{Nucleus radius from Equation \ref{rn2}, km} \tablenotetext{c}{Mean nucleus radius, km} \tablenotetext{d}{Average water production rate when $r_H \le$ 3 au} % \tablenotetext{e}{Average mass production rate when $r_H \le $ 3 au, kg s$^{-1}$} \tablenotetext{f}{Spin change timescale, from Equation \ref{tau_s}, in years} \tablenotetext{g}{Elapsed time with $r_H \le $ 3 au, in years} \tablenotetext{h}{D = Disintegrated, F = Fragmented} \end{deluxetable} \clearpage \clearpage \clearpage \clearpage \clearpage

Title: Flash-X, a multiphysics simulation software instrument

Abstract: Flash-X is a highly composable multiphysics software system that can be used to simulate physical phenomena in several scientific domains. It derives some of its solvers from FLASH, which was first released in 2000. Flash-X has a new framework that relies on abstractions and asynchronous communications for performance portability across a range of increasingly heterogeneous hardware platforms. Flash-X is meant primarily for solving Eulerian formulations of applications with compressible and/or incompressible reactive flows. It also has a built-in, versatile Lagrangian framework that can be used in many different ways, including implementing tracers, particle-in-cell simulations, and immersed boundary methods.

https://export.arxiv.org/pdf/2208.11630

\begin{frontmatter} \title{Flash-X, a multiphysics simulation software instrument} \author[1,8]{A. Dubey \corref{cor1}} \ead{adubey@anl.gov} \author[8,1]{K. Weide} \ead{kweide@uchicago.edu} \author[1]{J. O'Neal} \ead{joneal@anl.gov} \author[1,5]{A. Dhruv} \ead{adhruv@anl.gov} \author[3]{S. Couch} \ead{scouch@msu.edu} \author[2]{J.A. Harris} \ead{harrisja@ornl.gov} \author[1]{T. Klosterman} \ead{tklosterman@anl.gov} \author[1]{R. Jain} \ead{jain@anl.gov} \author[1]{J. Rudi} \ead{jrudi@anl.gov} \author[2,12]{O.E.B. Messer} \ead{bronson@ornl.gov} \author[3]{M. Pajkos} \ead{mapajkos@gmail.com} \author[3]{J. Carlson} \ead{jaredc.scholar@gmail.com} \author[12]{R. Chu} \ead{rchu@vols.utk.edu} \author[14]{M. Wahib} \ead{mohamed.attia@riken.jp} \author[1]{S. Chawdhary} \ead{saurabh.chawdhary@gmail.com} \author[4]{P.M. Ricker} \ead{pmricker@illinois.edu} \author[6]{D. Lee} \ead{dlee79@ucsc.edu} \author[7]{K. Antypas} \ead{kantypas@lbl.gov} \author[1]{K.M. Riley} \ead{riley@alcf.anl.gov} \author[7]{C. Daley} \ead{csdaley@lbl.gov} \author[9]{M. Ganapathy} \ead{murali@google.com} \author[10]{F.X. Timmes} \ead{fxt44@mac.com} \author[13]{D.M. Townsley} \ead{dean.m.townsley@ua.edu} \author[11]{M. Vanella} \ead{marcos.vanella@nist.gov} \author[7]{J. Bachan} \ead{john.bachan@gmail.com} \author[1]{P.Rich} \ead{richp@alcf.anl.gov} \author[8]{S. Kumar} \ead{shravan2915@gmail.com} \author[2,12]{E. Endeve} \ead{endevee@ornl.gov} \author[2,12]{W. R. Hix} \ead{raph@ornl.gov} \author[12]{A. Mezzacappa} \ead{mezz@tennessee.edu} \author[2]{T. Papatheodore} \ead{papatheodore@ornl.gov} \address[1]{Argonne National Laboratory, Lemont, IL, 60439, USA} \address[2]{Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA} \address[3]{Michigan State University} \address[4]{University of Illinois, Urbana Champaign} \address[6]{University of California, Santa Cruz} \address[5]{George Washington University} \address[7]{Lawrence Berkeley National Laboratory} \address[8]{University of Chicago, IL, 60637, USA} \address[9]{Google Inc} \address[10]{Arizona State University} \address[11]{National Institute of Standards and Technology} \address[12]{University of Tennessee, Knoxville, TN, 37996, USA} \address[13]{University of Alabama, Tuscaloosa, AL, 35487, USA} \address[14]{RIKEN, Kobe, Japan} \cortext[cor1]{Corresponding author at: Argonne National Laboratory, 9600 S. Cass Ave, Lemont, IL, 60439} \begin{keyword} Multiphysics \sep Simulation software \sep high-performance computing \sep performance portability \end{keyword} \end{frontmatter} \section*{Metadata} \label{} \begin{table}[H] \begin{tabular}{|l|p{6.5cm}|p{6.5cm}|} \hline \textbf{Nr.} & \textbf{Code metadata description} & \\ \hline C1 & Current code version & 1.0\\ \hline C2 & Permanent link to code/repository used for this code version & {https://github.com/\flashx/\flashx} \\ \hline C3& Code capsule& {https://github.com/Flash-X/ Workflows/tree/main/incompFlow/ FlowBoiling} \\\hline C4 & Legal Code License & Apache 2.0 \\ \hline C5 & Code versioning system used & git \\ \hline C6 & Software code languages, tools, and services used & Fortran, C, C++, Python3, MPI, OpenMP, OpenACC, HDF5 \\ \hline C7 & Compilation requirements, operating environments and dependencies & Unix, Linux, OSX based compilers for languages and libraries mentioned above\\ \hline C8 & Developer documentation/manual & {https://{flash-x}.org/ pages/documentation/} \\ \hline C9 & Support email for questions & flash-x-users@lists.cels.anl.gov\\ \hline \end{tabular} \caption{Code metadata. Please note that the code repository is private only because our funding agency requires us to keep a list of people who obtain the code directly from our repo. Anyone can furnish their github id and be added to the list of collaborators.} \label{} \end{table} \section{Motivation and Significance} \label{sec:motive} \flashx \cite{flash-x} is a new incarnation of \flash \cite{flash2009,Fryxell2000}, a multiphysics software system that has been used by multiple science communities. \flashx is meant for use beyond existing \flash science communities. It is designed to be easily adaptable for use by any computational scientists who rely upon differential equations as their primary mathematical model with finite-volume or finite-difference discretization. \flash was designed only for a homogeneous, distributed-memory parallel model with bulk-synchronism, which has rendered it unsuitable for use on many newer system architectures that are heavily reliant on disparate memory spaces (e.g., accelerators). This difficulty is further exacerbated by increasing heterogeneity in hardware as well as solvers within the code. \flashx has a fundamentally redesigned architecture that uses abstractions and asynchronous operations for performance portability across a variety of platforms, both with and without accelerators. Our design is forward-looking in that it makes minimal assumptions about which parallelization or memory models are likely to be prevalent in future platforms. The design relies upon self-describing code components of varying granularity and a tool\-chain that can interpret the metadata of the code components to synthesize application instances. The synthesis is done partly through assembly, partly through code translation, and partly through code generation. Some code assembly features have been imported from \flash, but have been significantly enhanced to discretize components at a finer scope than subroutines or functions. Tools for code translation and runtime management are new and will enable orchestration of computation and data movement between distinct compute devices on a node. In addition to the new architecture, \flashx has newer and higher- fidelity physics solvers. Most notable among these are Spark \cite{couch2021} for magnetohydrodynamics, XNet \cite{xnet,HiTh99} for nuclear burning, thornado \cite{ChEnHa19,LaEnCh21} for neutrino radiation transport, and WeakLib \cite{weaklib,PoBaEn21,Land2018} for tabulated microphysics. Additionally, \flashx can support multiphase flow through a level-set method, which did not exist in \flash releases \cite{DHRUV2021}. \flashx has been exercised on small clusters at Argonne National Laboratory and on leadership-class machines at Oak Ridge National Laboratory and Argonne National Laboratory. Flash-X will showcase the key performance parameters of ExaStar \cite{Exastar}, a project under the Exascale Computing Project\cite{ECP,Kothe2020} (ECP), through a core-collapse supernova (CCSN) simulation on exascale machines to be deployed by the US Department of Energy. To run effectively at scale, \flashx will rely upon the tool\-chain described above. Some components of the tool\-chain are embedded in \flashx, while others are encapsulated into independent libraries that can be used by other codes. Note that compilation and execution of the code do not require using these external libraries; they are used only to orchestrate data movement and computation for better performance. Along with a new architecture, \flashx also adopts a community-based, open development model. The stewardship of the code is guided by a Council representing all the major science communities of \flash/\flashx. More details of our community development model are available at {https://flash-x.org}. \section{Software Description} \label{sec:desc} The \flashx code is a component-based software system for simulation of multiphysics applications that can be formulated largely as a collection of partial and ordinary differential equations (PDEs and ODEs), as well as algebraic equations. The equations are discretized and solved on a domain that can have uniform resolution (UG) or adaptive mesh refinement (AMR). In \flashx, one can select between PARAMESH \cite{MacNeice2000}, an octree-based library written in Fortran, or AMReX \cite{AMReX,zhang2019amrex}, a highly-flexible, patch-based, C++ AMR library. Both AMR frameworks can interface to math libraries such as hypre \cite{falgout2002} and PETSc \cite{petsc}, making those solvers available to \flashx. Physics units are designed to be oblivious of domain decomposition. Bulk of their code is written for block-by-block update, interspersed with invocation of fine-coarse boundary resolution related API functions of the Grid unit as needed. Hyperbolic equations are solved using explicit methods commonly used for compressible flows with strong shocks, described in Section~\ref{sec:func}. For elliptic equations, one can either use an included multipole solver \cite{couch:2013}, AMReX's multigrid solver, or an interface to one of the math libraries. For parabolic equations, one must rely upon library interfaces. The maintained code components are written in a combination of high-level languages such as Fortran, C, and C++, with an embedded domain-specific configuration language (DSCL) that also supports \flashx custom macros. The DSCL permits multiple alternative definitions of macros with a built-in arbitration mechanism to select the appropriate definition for an instance of code assembly. The accompanying configuration tool\-chain can translate and assemble different combinations of the components to configure a diverse set of applications. \flashx has been designed from the outset to be performant with increasing heterogeneity of both the platforms and the solvers within the code. The code uses the Message-Passing Interface (MPI) library for communication between nodes, though more than one MPI rank can also be placed on a node. HDF5 is the default mode for IO. Support for OpenMP, both for threading and for offloading to accelerators, is built into several, though not all, of the solvers. \subsection{Software Architecture} \label{sec:arch} \flashx has composable components with accompanying metadata that can express, for example, inter-component dependency and exclusivity, necessary state variables, etc. The metadata is encapsulated within the code components by accompanying {\it config} files and is parsed and interpreted by the configuration tool, {\em Setup}. Setup parses config files recursively, aggregates requisite components, and assembles a complete application. It also assembles the compilation/make system and runtime parameters for each component included in the application. The Setup tool also implements code inheritance through a combination of keywords in the config files and the Unix directory structure instead of using programming language supported inheritance mechanisms. When Setup parses the source tree, it treats each subdirectory as inheriting all of the files in its parent’s directory. While source files at a given level of the directory hierarchy override files with the same name at higher levels, config files accumulate all definitions encountered. The schematic for inheritance is shown in Figure~\ref{fig:inheritance}. In \flashx parlance, the highest-level code component for a specific type of functionality is called a {\it unit}. Units can have sub\-units. A unit includes an API accessible to the whole code through which it interacts with other units and the driver. While each sub\-unit can have its own sub-components with no restriction on how fine-grained they can become, the general rule of thumb is to keep them as coarse-grained as feasible for ease of maintenance. A unit can have multiple alternative implementations, one of which is required to be a null implementation. If a unit is not needed in a simulation, the null implementation is included. This feature facilitates maintaining very few implementations of the main driver while permitting many combinations of capabilities to be included in an application. Any code component can have multiple alternative implementations, though unlike the unit-level API, lower-level components do not require null implementations. A different mechanism is used when a code component needs to become smaller than a function or a subroutine. Here, we rely on macros to implement alternative definitions of an operation, including the null case. The inheritance mechanism shown in Figure \ref{fig:inheritance} arbitrates on which definition to select. The macros may also have arguments, be inlined, and be recursive. This mechanism serves two purposes. The first is for developer convenience. Certain code patterns repeat often in the code -- for example, invocation of iterators, bounds for loop-nests, and bounds for arrays. We have provided macros for such repeated patterns, and developers can use these at their discretion. Macros make the code compact, reduce cut-and-paste errors, and help to clarify the control flow and semantics of the code. The second, more powerful motivation is that with alternative definitions, we can generate many variants of a code component from the same source. This functionality is particularly useful when different control flow is more suitable for different compute devices. We can keep arithmetic expressions invariant while using macros for the control flow, or vice-versa, thus not only eliminating code duplication but also keeping the maintained code more compact. The schematic for generating variants from a single source where specializations are obtained through alternative macro definitions is shown Figure in \ref{fig:variants}. \subsection{Software Functionalities} \label{sec:func} The \flashx distribution includes solvers for compressible and incompressible fluids, several methods for handling equations of state (EOS), source terms for nuclear burning, several methods for computing effects of gravity, level-set methods for multiphase flow, and several others. The primary formulation for PDEs in \flashx is Eulerian, although a versatile Lagrangian framework is also included that can be configured to do computations such as tracers, particles-in-cell, immersed boundaries, etc. The vast majority of applications using \flashx include some form of hydrodynamics or magnetohydrodynamics in their configuration. However, it is possible to configure applications that completely bypass those solvers. \noindent{\bf Magnetohydrodynamics and Hydrodynamics}: a compressible magnetohydrodynamics/hydrodynamics solver with second- or third-order strong stability preserving (SSP) Runge-Kutta (RK) time integration (Spark) \cite{couch2021}, another compressible hydrodynamics solver with a predictor-corrector formulation \cite{lee2009unsplit,lee2013solution}, and an incompressible hydrodynamics solver with fluid-structure interaction \cite{Vanella2010} are included in the distribution. All of the solvers can be used in 1-, 2-, or 3- dimensional configurations. \noindent{\bf Equations of State}: the code supports several EOS versions suitable for a range of regimes in astrophysical flows. The simplest one is a perfect-gas EOS with a multispecies variant. Another implementation with two variants uses a fast Helmholtz free-energy table interpolation to handle degenerate relativistic electrons and positrons and also includes radiation pressure and ions (via the perfect gas approximation) \cite{TiSw00}. \noindent{\bf Nuclear Burning}: three nuclear reaction networks of varying numbers of species are included in the distribution. Approx-13 and approx-19 \cite{Timmes1999} are inherited from \flash. XNet is a standalone code for evolving astrophysical nuclear burning and is generalizable to arbitrarily large networks as needed for improved physical fidelity of some applications. \noindent{\bf Gravity}: the gravitational potential can be treated very simply as constant, or through a Poisson solve using a multipole or multigrid method depending upon the symmetry of the density field. \noindent{\bf Particles}: this component of the code forms the basis for the Lagrangian framework \cite{Dubey2012}. Particles maintain their own spatial coordinates and are independently integrated in time. They interact with the Eulerian mesh either to obtain physical quantities needed for their advancement or to deposit quantities such as mass, charge, or energy to the mesh, depending on usage. \noindent{\bf Incompressible Fluid Dynamics}: this component of the code solves incompressible Navier-Stokes equations for single and multiphase flow simulations with options for heat transfer and phase transitions \cite{DHRUV2021}. The Navier-Stokes solver is implemented using a fractional-step temporal integration scheme that uses Poisson solver for pressure. Multiphase interfaces are tracked with a level-set function and use ghost-fluid methods to account for forces due to surface tension and mass transfer \cite{DHRUV2019}. The effect of solid bodies on the fluid is modeled using an immersed boundary method that uses Lagrangian particles\cite{Vanella2009}. \noindent{\bf Importable Modules}: \flashx uses GitHub's submodules to import some capabilities that are independently developed and hosted in their own repositories. These include WeakLib for tabulated, nuclear EOS and neutrino-matter interaction rates, and thornado for spectral neutrino radiation transport. \section{Illustrative Examples} \label{sec:example} We describe two example simulations using \flashx from two different science communities. The first is a CCSN simulation that uses compressible hydrodynamics, nuclear EOS, neutrino radiation transport, and self-gravity solvers. The second is a subcooled flow boiling simulation that uses multiphase incompressible Navier-Stokes and heat advection diffusion solver. We perform a CCSN simulation in spherical symmetry, initiated with a low-mass pre-collapse progenitor star previously modeled throughout all stages of stellar evolution \cite{sukhbold_etal_2016}. Electron-type neutrinos and anti-neutrinos are evolved using thornado's two-moment neutrino transport solver and WeakLib's tabulated nuclear EOS \cite{steiner_etal_2010} and neutrino-matter interaction rates \cite{Bruenn_1985}. Compressible hydrodynamics are evolved with Spark, and Newtonian self-gravity is computed using the multipole Poisson solver. For a more detailed description of the physics included, see~\cite{harris2021}. Figure~\ref{fig:s9_ye_1d} shows the evolution of the ratio of electrons to baryons (electron fraction) versus radius during a critical epoch in the simulation that spans the formation of the primary shock-wave during core ``bounce'' --- the phenomena of infalling matter colliding with, and \emph{bouncing} off of, the newly-formed neutron-star. Figure \ref{fig:boiling} provides details for the subcooled flow-boiling simulation which was designed to replicate experiments performed at different gravity levels by Lebon~et~al. \cite{LEBON2019700}. These computations used the multiphase incompressible Navier--Stokes solver along with the phase transition capability, and were preformed at a resolution almost twice the previous state-of-the-art \cite{SATO2018876,DHRUV2019}. Liquid coolant flows over a heater surface with a mean velocity $U_0$, leading to phase-change and formation of vapor bubbles. These vapor bubbles grow, merge, and finally depart the heater surface due to buoyancy which introduces turbulence in the domain. The heat transfer associated with this turbulence is an important parameter in designing cooling systems for automotive and industrial components, but is difficult to quantify through experimental observations/measurements. With Flash-X we are able to address this challenge through targeted high-fidelity simulations to quantify the contribution of turbulent heat flux. \section{Impact} \label{sec:impact} \flash has been an influential code for computing astrophysical flows almost since its inception. \flash's scientific impact is clearly demonstrated by the citation history of the original paper describing the code, as shown in Figure \ref{fig:citations}. Analysis in \cite{Dubey2019,grannan2020understanding} further quantifies the scientific significance and impact of the code on science. \flash has not only been used extensively for science, it has also been among the pioneers in giving due importance to software quality and adopting rigorous auditing and productivity practices \cite{dubey2014evolution}. In recent years, \flash's use has been diminishing in several communities because of its inability to use accelerators effectively. \flashx is designed to fill this gap and become a reliable multiphysics simulation code for the communities that earlier relied on \flash. At least two major communities of \flash users, stellar astrophysics\cite{couch2021,harris2021} and fluid-structure interactions\cite{DHRUV2021}, are already transitioning to \flashx, with some users now exclusively using \flashx. These use cases have also reported on performance gains with the use of GPUs. Not all of \flash's physics capabilities are available in \flashx yet. However, since \flashx is open source, it is expected that interested users will assist in transitioning their capabilities of interest to the \flashx architecture and help grow the \flashx community. Additionally, the new tool-chain for orchestration of data and work movement is still in the early stages of being exercised. Preliminary performance studies of the runtime tool have been very encouraging \cite{oneal2021}. It is expected that full performance gains will have been realized by the next major release. \section{Conclusions} \label{sec:conclusions} Sustained funding under the ECP has permitted modernization of a highly capable community code for current and future platforms. With \flashx, the \flash science communities can embrace heterogeneity and use available hardware effectively. With the move to an open, community-based development model, users are assured of continuity and support for the code without depending on a single funding source. \flash has had a long history of scientific discovery, and \flashx aims to follow in that tradition. With more modern solvers and flexible architecture, \flashx can continue to be a very useful resource for science domains that rely on modeling of partial differential equations. \section{Conflict of Interest} We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. \section*{Acknowledgements} \label{} The authors acknowledge all contributors to the \flashx code, including contributors to the \flash code from whose work \flashx has inherited. This work was supported by the U.S. Department of Energy Office of Science Office of Advanced Scientific Computing Research under contract number DE-AC02-06CH1137. This research was supported by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of two U.S. Department of Energy organizations (Office of Science and the National Nuclear Security Administration) that are responsible for the planning and preparation of a capable exascale ecosystem, including software, applications, hardware, advanced system engineering, and early testbed platforms, in support of the nation's exascale computing imperative. This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725.

Title: Insight-HXMT observation on 4U~1608--52: evidence of interplay between thermonuclear burst and accretion environments

Abstract: A type-I burst could influence the accretion process through radiation pressure and Comptonization both for the accretion disk and the corona/boundary layer of an X-ray binary, and vice versa. We investigate the temporal evolution of a bright photospheric radius expansion (PRE) burst of 4U 1608-52 detected by Insight-HXMT in 1-50 keV, with the aim of studying the interplay between the burst and persistent emission. Apart from the emission from the neutron star (NS) surface, we find the residuals both in the soft (<3 keV) and hard (>10 keV) X-ray band. Time-resolved spectroscopy reveals that the excess can be attributed to an enhanced pre-burst/persistent emission or the Comptonization of the burst emission by the corona/boundary layer. The Comptonization model is a convolution thermal-Comptonization model (thcomp in XSPEC) and the Comptonization parameters are fixed at the values derived from the persistent emission. We find, during the PRE phase, after the enhanced pre-burst/persistent emission or the Comptonization of the burst emission is removed, the NS surface emission shows a plateau, and then a rise until the photosphere touches down to the NS surface, resulting in a flux peak at that moment. We speculate that the findings above correspond to that the obscured lower part of the NS surface by the disk is exposed to the line of sight due to the inner disk evaporation by the burst emission. The consistency between the fa model and convolution thermal-Comptonization model indicates the interplay between thermonuclear bursts and accretion environments. These phenomena did not usually show up by the conventional blackbody model fitting, which may be due to low count rate and narrow energy coverage in previous observations.

https://export.arxiv.org/pdf/2208.13556

\title{Insight-HXMT observation on 4U~1608--52: evidence of interplay between thermonuclear burst and accretion environments} \author{Yu-Peng Chen\textsuperscript{*}} \affil{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China} \author{Shu Zhang\textsuperscript{*}} \affil{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China} \author{Long Ji} \affil{School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai, 519082, China} \author{Shuang-Nan Zhang} \affil{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China} \affil{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China} \author{Ling-Da Kong} \affil{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China} \affil{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China} \author{Peng-Ju Wang} \affil{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China} \affil{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China} \author{Zhi Chang} \affil{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China} \author{Jing-Qiang Peng} \affil{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China} \affil{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China} \author{Jin-Lu Qu} \affil{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China} \affil{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China} \author{Jian Li} \affil{CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei 230026, China} \affil{School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China} \email{chenyp@ihep.ac.cn, szhang@ihep.ac.cn} \keywords{stars: coronae --- stars: neutron --- X-rays: individual (4U~1608--52) --- X-rays: binaries --- X-rays: bursts} \section{Introduction} Type-I X-ray burst, also named thermonuclear bursts, are triggered by unstable thermonuclear burning of the accreted fuel from a low-mass X-ray binary (LMXB) hosting a neutron star (NS) (for reviews, see \citealp{Lewin,Cumming,Strohmayer,Galloway}). Since its first detection in 1975 from 3A~1820--30, so far there are 116 Galactic X-ray binaries observed to produce thermonuclear bursts\footnote{https://personal.sron.nl/$\sim$jeanz/bursterlist.html}, manifesting a sudden increase in the X-ray luminosity followed by an exponential decay and with a typical duration about tens seconds. The most luminous bursts are the photospheric radius expansion (PRE) events, for which the peak flux reaches the Eddington luminosity of the NS. Among some of the thousands of observed bursts \citep{Galloway2020}, observations on bursters by RXTE \citep{int2013,Worpel2013,Ball2004,KeeK2014b}, INTEGRAL \citep{Sanchez2020}, NICER \citep{Keek2018,Keek2018a}, AstroSat \citep{Bhattacharyya2018} and Insight-HXMT \citep{chen2018,chen2019}, revealed interactions between the burst emission and the accretion environment: the continuum spectrum was observed to have an enhancement at soft X-ray and/or a shortage at hard X-rays \citep{Worpel2013,Worpel2015,chen2012,ji2013}. Such spectral deviations are considered as burst-induced and might be relevant to disk reflection, accretion rate increase, and corona cooling \citep{Ball2004,KeeK2014b,Degenaar2018}. Moreover, the reflection spectrum, consisting of discrete lines and a hump peaking at 20--40 keV, is interpreted as disk refection of an illuminant from the corona/boundary-layer. The burst emission could also serve as the illuminant to the disk, and a reflection component is correspondingly observed during the burst. However, so far only iron line is firmly detected during bursts, specifically during the long-duration superbursts \citep{Ball2004,KeeK2014b}. The observations above are the influence of the bursts on the accretion environment; however, there are few observational results reported related to the burst spectral change caused by the accretion disk/corona. 4U~1608--52 is a prolific burster located at the Galactic plane \citep{Belian}. More than 100 type-I X-ray bursts, inhabited its outbursts which have a typical frequency of once per 1--2 years since its discovery, were regularly observed. The distance was estimated as $D\sim$2.9--4.5 kpc based on the peak flux pf PRE bursts $\sim$1.2--1.5 $\times10^{-7}~{\rm erg}~{\rm cm}^{2}~{\rm s}^{-1}$ (e.g. \citealp{Galloway,Poutanen}). Its spin is around $\nu$=619 Hz \citep{Muno,Galloway}, based on the burst oscillation detection. In this present investigation, we provide a broad-band spectral view of 4U~1608--52 during its 2020 outburst observed by NICER and Insight-HXMT, both for its outburst and burst emission. We first describe the data reduction procedure of NICER and Insight-HXMT in Section 2. We then present an in-depth spectral analysis and model parameters of its outburst emission in Section 3.1, burst lightcures and spectral evolution in Section 3.2 and Section 3.3. Finally, we summarize our results and discuss their implications in Section 4. \section{Observations and Data reduction} \subsection{Insight-HXMT} Hard X-ray Modulation Telescope (HXMT, also dubbed as Insight-HXMT, \citealp{Zhang2020}) excels in its broad energy band (1--250 keV) and a large effective area in hard X-rays energy band. \textbf{It carries three collimated telescopes: the High Energy X-ray Telescope (HE; poshwich NaI/CsI, 20–250 keV, $\sim$ 5000 cm$^2$), the Medium Energy X-ray Telescope (ME; Si pin detector, 5–40 keV, 952 cm$^2$) and the Low Energy X-ray telescope (LE; SCD detector, 1–12 keV, 384 cm$^2$).} Under the quick read-out system of Insight-HXMT detectors, there is little pile-up effect event at the PRE burst peak. Insight-HXMT Data Analysis software (HXMTDAS) v2.04 are used to analyze the data. The data are reduced following the recommended procedure of the Insight-HXMT Data Reduction Guide v2.04\footnote{http://hxmtweb.ihep.ac.cn/SoftDoc.jhtml}, which are screened in the standard criterion include in Insight-HXMT pipelines: lepipeline, mepipeline and hepipeline. Two bursts were detected by \textbf{three-payloads of} Insight-HXMT from 4U~1608--52 in its 2020 outburst, as shown in Table \ref{tb}. \textbf{However, the second burst, which is only half bright as the first one, did not fall into the good-time-interval of LE; } thus we only analyzed the first burst occurred at MJD 59069.770768 in this work. The persistent spectra adapted the GTI exclude the time span before the burst peak time 100 s and after the burst peak time 200 s. The persistent spectra of LE are rebinned by ftool grppha with minimum of 100 counts per grouped bin. For the ME, the spetcra are binned up by a factor 20, due to its background is comparable with the source emission. For the burst, we perform the time-resolved spectroscopy with a time resolution of 0.25 s, and define the time of the bolometric flux peak as a time reference (0 point in Fig. \ref{fig_burst_lc} and Fig. \ref{fig_fit}). The burst spectra are rebinned by ftool grppha with minimum of 10 counts per grouped bin. In the calibration experiments on ground and the first two years in orbit, the recommend energy band for spectral fitting of LE is 1-10 keV, except very bright sources with a flux brighter than several Crab. After the midyear of 2019, the recommend band is shrunken to 2-10 keV, which mostly dues to the poor background estimation because of an increase of detector temperatures and something else. However, in the burst spectral analysis, we take the pre-burst emission as background, so we extend the energy band to 1-10 keV in burst spectra fitting, but still adapt 2-10 keV in persistent emission spectral fitting. \textbf{ We notice that, the LE spectral residuals of the persistent emission have rather complex structures in the energy band $<$ 1.5 keV, where the bursts have only a few data points. % } We also use the 2-10 keV to analyze the burst spectra, and get the roughly consistent results within parameter’s error bar. It is a similar consideration that we extend the ME band to 8--30 keV and the HE band to 25--50 keV in the burst spectra fitting. In short, for the persistent emission spectra fitting of LE and ME, the energy bands are limited to 2--10 keV and 10--20 keV; for the burst spectra fitting of LE, ME and HE, the energy bands used are 1--10 keV and 8--30 keV and 25--50 keV respectively. During fittings of the persistent emission, the LE data in 2--10 keV and ME data in 10--20 keV are used, while the ME data $>$ 20 keV and the HE spectra are not used to fit because of faint source flux and strong background. \textbf{As shown in Fig. \ref{sep_nicer}, the persistent emission detected by ME in 20--30 keV and HE in 25--100 keV is very weak compared with the background. Most of these spectral channels are close to or fainter than the systematic uncertainty of the background (1\%); e.g., for the count rate detected by HE in 30--50 keV, the background of $\sim$120 cts/s is comparable to the detected count rate of $\sim$123 cts/s. Thus the persistent emission has a count rate of $\sim$ 3 cts/s which is close to the systematic uncertainty of the estimated background. } In addition, we added a systematic uncertainty of 1\% to the Insight-HXMT spectra in 1--100 keV, to account for systematic uncertainties in the detector calibrations \citep{Li2020}. \subsection{NICER} On 2020 August 8, within the same day when Insight-HXMT detected the burst from 4U 1608--52, NICER also observed the same source. The OBSID is 3657026501, with a good time interval $\sim$ 2 ks and a count rate $\sim$ 900 cts/s in the 0.3–12 keV band. % However, the NICER missed the type-I X-ray burst because of an observation gap. The NICER data are reduced using the pipeline tool nicerl2\footnote{https://heasarc.gsfc.nasa.gov/docs/nicer/nicer\_analysis.html} in NICERDAS v7a with the standard NICER filtering and using ftool XSELECT to extract lightcurves and spcectra. The background is estimated using the tool nibackgen3C50 \citep{Remillard2022}. The Focal Plane Module (FPM) No. 14 and 34 are removed from the analysis because of increased detector noise. The response matrix files (RMFs) and ancillary response files (ARFs) are generated with the ftool nicerrmf and nicerarf. The spectra are rebinned by ftool ftgrouppha \citep{Kaastra2016} optimal binning algorithm and plus minimum of 25 counts per grouped bin. Other rebin method, e.g., minimum of 100 counts per grouped bin by ftool grppha, are adapted. As expected, the fit results are consistent with each other within parameter’s error bar. For the ISM absorption, we use tbabs in the spectral model and wilm abundances \citep{Wilms2000}. To erase the residuals $<$ 1 keV, three absorption edges are added in spectra fitting: 0.56 keV, 0.71 keV, and 0.87 keV. We added a systematic uncertainty of 1\% to the NICER spectra. From NICER and Insight-HXMT lightcurves, the none-burst/persistent emission is stable in our observations. We jointly fit the persistent spectra observed with NICER and Insight-HMXT, as shown in Fig. \ref{sep_nicer}. The joint fit of the spectra covers an energy band of 0.4--10 keV, 2--10 keV and 10--20 keV for NICER, LE and ME, respectively. The spectra are fitted with XSPEC v12.11.1 and the model parameters are estimated with a 68\% confidence level (1 $\sigma$). \section{Analysis and Results} \subsection{None-burst/persistent emission detected by NICER and Insight-HXMT} The jointed NICER and Insight-HXMT data in a broader energy range 0.4--20 keV, give us an opportunity to utilize a more physics meaningful model to fit the persistent emission, rather than the simplified models, i.e., a simple photon power law, a power law with high energy exponential rolloff (cutoffpl in xspec) and a broken power law (bknpow in xspec). We fit the joint of NICER and Insight-HXMT (LE and ME) spectrum with an absorbed convolution thermal Comptonization model (with an input seed photon spectrum diskbb), available as thcomp (a more accurate version of nthcomp) \citep{Zdziarski2021} in XSPEC, which is described by the optical depth $\tau$, electron temperature $kT_{\rm e}$, scattered/covering fraction $f_{\rm sc}$. The hydrogen column (tbabs in XSPEC) % accounts for both the line of sight column density and as well any intrinsic absorption near the source. The seed photons are in a shape of diskbb, since the thcomp model is a convolution model and the fraction of Comptonization photons is also given in the model. Normalization constants are included during fittings to take into account the inter-calibrations of the instruments. We keep the normalization factor of the LE data with respect to the ME and NICER data to unity. Using the model above, we find an acceptable fit:$\chi_{\upsilon}$=0.95 (d.o.f. 846; Fig. \ref{sep_nicer} and Table \ref{persist_fit}), with the inner disc radius $R_{\rm diskbb}$ and scattered/covering fraction $f_{\rm sc}$ are found to be $\sim 15.1_{-0.8}^{+0.9}$ km (with distance 4 kpc and inclination angel 40 degree) and $0.77_{-0.05}^{+0.05}$, respectively. The derived hydrogen column density $N_{\rm H}$ is $\sim$1.3$\times 10^{22}~{\rm cm}^{-2}$, which agrees with values previously reported in a range of 0.9--1.5 $\times 10^{22}~{\rm cm}^{-2}$ \citep{PenninxW1989,ArmasPadilla2017}. The thcomp parameters, $\tau$ and $kT_{\rm e}$ are well consistent with a previous outburst in soft state of 4U 1608--52 \citep{ArmasPadilla2017}, which derived the parameters were derived with the nthcomp model. The inferred bolometric flux in 1--100 keV is $7.33_{-0.03}^{+0.06}\times10^{-9}~{\rm erg}~{\rm cm}^{-2}~{\rm s}^{-1}$. The constant of ME and NICER is 0.93$\pm$0.02 and 1.05$\pm$0.01 respectively. \textbf{Using model of cons*tbabs*(diskbb+nthcomp) to fit the spectra, i.e., non-convolutional Comptonization model, we get a similar results for both the corona and the disk temperatures but with a smaller normalization of the disk. The shortage of the disk normalization than the convolution model is corresponding to the missing part of the disk emission which is supposed to be scattered in the corona. } Another assumption that the seed photons of the Comptonization are from the NS surface, i.e., the diskbb component is substituted by a blackbody component in the aforementioned convolution model, is also attempted. Taking this approach, spectral fits yield a roughly same thcomp parameters but with $\chi_{\upsilon}$=1.12 (the same d.o.f.) and soft residuals $<$ 2 keV. Furthermore, the derived blackbody radius is $34.1\pm1.0$ km, which is far greater than the NS radius. A hybrid model \citep{ArmasPadilla2017}, i.e., three-component model (diskbb+thcomp*bb or bb+thcomp*diskbb) is not attempted, since the above two-component model is able to fit the data. Since there are no iron emission line or reflection bump above 10 keV, no reflection model are used for the spectrum fitting. \subsection{Burst lightcurves by Insight-HXMT} We show the LE/ME/HE lightcurves in Fig. \ref{fig_burst_lc} with a time resolution of 0.1 s. The burst profiles exhibit a typically fast rise and slow (exponential) decay in the soft X-ray band, and manifest a plateau in soft X-ray band (LE) and two peaks in hard X-ray band (ME\&HE), which is a typical characteristic of a PRE burst. In the middle of the PRE phase with a constant luminosity $L_{\rm Edd}$, the burst emission has the lowest blackbody temperature, which could cause a dip in the HE lightcurves. However, interestingly, there is a peak/excess in the HE lightcurves. For the highest 6 points in HE lightcurves in its whole energy band (20--250 keV), the hard excess is 222.3$\pm$39.3 cts/s with 5.6 $\sigma$ detection; meanwhile, the burst emission for HE (for a blackbody with a temperature of 2.0 keV and a bolometric flux of an Eddington luminosity) should be less than 35 cts/s in this energy band. The hard excess in 30--50 keV is 71.5$\pm$18.0 cts/s with 4 $\sigma$ detection; meanwhile, the burst emission should be negligible with $<$ 0.1 cts/s in this energy band. This hard X-ray excess in the lightcurve suggests that there is another provenance except for the burst, which is also visible during the burst spectra analysis below. \subsection{Broad-band spectra of burst emission by Insight-HXMT} When we fit the burst spectra, we estimate the background using the emission before the burst, i.e., assuming the persistent emission is unchanged during the burst. % To account for the effective area calibration deviation, a constant is added to the model. At the first attempt, for LE, the constant is fixed to 1, the others are variable during spectra fitting. The fits indicate that most of the constants of HE and some of the constant of ME are not convergent, owing to the low-significance data. Under this situation, the constant of ME\&HE is fixed at 1 for the combined-spectra fitting. We follow the classical approach to X-ray burst spectroscopy by subtracting the persistent spectrum and fitting the net spectrum with an absorbed blackbody. In the decay phase, such a spectral model generally results in an acceptable goodness-of-fit, with a mean reduced $\chi^{2}_{\upsilon}~\sim$ 1.0 (d.o.f. 20--60). However, we note that a significant residuals are shown below 3 keV and above 10 keV, as shown in the left panel of Fig. \ref{residual}, especially the spectra in the PRE phase, the reduced $\chi^{2}_{\upsilon}$ are above 1.5 (d.o.f. 60--80). To erase the residuals, we first consider the $f_{a}$ model. Following \citet{Worpel2013} we then include an addition component for fitting the variable persistent emission. We assume that during the burst the spectral shape of the persistent emission is unchanged, and only its normalization (known as a $f_{a}$ factor) is changeable. As reported earlier by RXTE and NICER, the $f_{a}$ model provides a better fit than the conventional one (absorbed blackbody). We compare the above two models using the F-test. % In some cases, the $f_{a}$ model significantly improves the fits with a p-value $\sim10^{-5}$. As shown in left panel of Figure \ref{fig_fit}, the spectral fitting results from these two models have differences mainly around the PRE phase. By considering an additional factor $f_{a}$, the burst blackbody flux tends to slightly decrease, and the temperature becomes higher but the radius shrinks. The $f_{a}$ factor reaches a maximum of $6.5_{-1.3}^{+1.3}$ when the radius reaches its peak. % During the PRE phase, the radius is up to $10.8_{-1.0}^{+1.2}$ km, which is two times larger than the radius measured at touch-down time $5.1_{-0.3}^{+0.3}$ km (assuming a distance of 4 kpc). This is typical to a moderate photospheric expansion with a bolometric burst peak flux $F_{\rm bb}$ $15.3_{-0.8}^{+0.8}\times10^{-8}~{\rm erg}~{\rm cm}^{-2}~{\rm s}^{-1}$ in 0.1--100 keV. Since the burst photons could also be affected by the corona/boundary-layer, we thus check if the model used in the persistent emission could be same with the burst emission. By taking the pre-burst emission as background emission, the burst spectra are fitted by the model tbabs*thcomp*bb, in which the thcomp parameters are fixed at the persistent emission fit results. Thus convolution thermal Comptonization model (with an input seed photon spectra blackbody) has the same d.o.f with the canonical blackbody model, and a more d.o.f. than the $f_{a}$ model. The bb and thcomp represents the burst emission from the NS photosphere and a corona/boundary-layer influence on the burst emission. This model allows us to evaluate the contribution from both the up-scattered by the corona/boundayr-layer and direct photons from the NS surface. In the PPE phase, this model provides the best fit and yields physically acceptable spectral parameters; the obtained best-fit parameters are given in the right panel of Fig. \ref{fig_fit}. We find that this convolved thermal-Comptonization model provides an equally good results with $f_{a}$ model but with a more d.o.f., and statistically preferred to the $f_{a}$ model in the middle of PRE phase (with a coolest blackbody temperature). However, in the rising and decay part, such model has a bigger reduced $\chi^{2}_{\upsilon}$ than the $f_{a}$ model and even the canonical blackbody model, which may indicates that the burst emission suffers few Comptonization during this phase. As mentioned above, the free/unfixed parameters include the blackbody temperature $kT_{\rm bb}$ and the normalization $N_{\rm bb}$. The trend of the parameters are similar with the $f_{a}$ model, but with a greater change. Compared to the $f_{a}$ model results, the maximum radius $R_{\rm bb}$ is up to $29.5_{-2.4}^{+2.9}$ km, the minimum temperature $kT_{\rm bb}$ is low to $1.19_{-0.06}^{+0.06}$ keV. Other scenarios, i.e., burst reflection by the disk, NS atmosphere model carbatm/hatm \citep{Suleimanov2011,Suleimanov2012,Suleimanov2018} in Xspec, are also tried to fit the burst spectra, as we did in \citet{chen2019}. However, neither could alleviate the residuals at soft X-ray and hard X-ray bands simultaneously. For the hard X-ray excess detected in the lightcurve during the PRE phase, we calculate and find that the persistent emission has not enough flux to build the enhancement. We fake the HE spectra using the aforementioned model parameters of the persistent emission, % the HE flux of the persistent emission model in 20--250 keV and 30--50 keV is 3 cts/s and 0.7 cts/s. Taking into the factor $f_{a}$ account, this model predicted enhancement flux only equivalent to one tens of the hard excess. The spectra residuals in hard X-ray band are also visible in the middle panel of Fig. \ref{residual} ($f_{a}$ model to fit the burst spectra in the PRE phase). The hard X-ray excess also disfavours the reflection model because of the faint persistent emission in hard X-ray band. \section{Discussion} In this work, we have presented a spectral analysis of a PRE burst and persistent emission from 4U~1608--52 in its 2020 outburst observed by NICER and Insight-HXMT. The persistent emission is well fitted by an absorbed convolution thermal-Comptonization model, in which 77\% disk emission is up-scattered by the corona/boundary-layer. The X-ray burst shows a significant spectral deviation/excess both at $<$ 3 keV and $>$ 10 keV from an absorpted blackbody in the PRE phase. % This excess is consistent with that the burst emission is up-scattered by the corona/boundary-layer, only part of the burst emission without Comptonization is detected, which mimics the Comptonization of the disk emission in the persistent emission. \subsection{X-ray continuum} \textbf{Based on LE\&ME lightcurves and spectral fitting results, the burst locates at the high/soft state (banana state).} Previous works have attempt to fit the spectra with thermal (diskbb or/and blackbody) plus a Comptonization model, rather than a convolution thermal-Comptonization model, which will cause an underestimation of the thermal emission. In this work, adapting thermal-Comptonization model thcomp in XSPEC, the fit results indicate that most of the disk emission is involved in Compton upscattering. Broadly speaking, in an accreting low-magnetic field NS, except for the emission from the NS surface, there are at least two geometrically distinct regions to generate X-ray emission (see the review by \citealp{Done2007}): in the accretion disc and the boundary/spreading layer (BL/SL) (similar to the corona of the case of an accreting black hole). BL is supposed to spread in large radial extent in the disc midplane; whereas SL has a narrower spread but spreads over a considerable height from the equatorial plane toward higher stellar latitudes. There are claimed judgment criteria for BL and SL based on temporal \citep{Gilfanov2003} and spectral \citep{Grebenev2002,Suleimanov2006} characteristics. In the burst review, during the decay part of the burst, the burst emission is well fitted by a blackbody and no strong comptonization/up-scattered emission detected. Thus the hot electrons plasma should not have a significant coverage for the NS surface. Add that into the consideration with a big scattering factor $f_{\rm sc}$ 0.77$_{-0.05}^{+0.05}$ (the hot electrons plasma on the disk in the persistent emission), the corona-like geometry of the BL is favoured. We find that the persistent emission is 4.8\% $L_{\rm Edd}$ and the corona/boundary-layer temperature is 3.02$_{-0.08}^{+0.08}$ keV, which is in the range of Comptonizing temperature expected for NS LMXBs in the soft state \citep{ArmasPadilla2017}. Meanwhile, the scattering factor $f_{\rm sc}$ is 0.77$_{-0.05}^{+0.05}$, which is too large for the corona/boundary-layer with a lamp-post geometry. Given those above, we prefer the corona/boundary-layer with a slab/sandwich geometry, as shown in Figure \ref{fig_illustraction}. Regarding that the temperature and optical depth deviates from the corona canonical value, we also prefer anther corona pattern--a so-called warm layer \citep{Zhang2000} with temperature $\sim$2--3 keV and optical depth $\sim$5--10, which is produced by the magnetic reconnection. The outburst spectral evolution and its understandings will be given in our forthcoming paper. \subsection{Enhanced Persistent Emission up to 50 keV} In several bursters, during the low/hard state, decrease/deficit in the hard X-ray band (30--50 keV) have been observed in short-duration bursts which happened in the low-hard state (\citealp{chen2012} and reference therein). It is expected that the burst emission (2--3 keV), which is relatively cooler than the corona (tens keV), cause the change in the corona structure or temperature. In this work, conversely, an enhancement hard X-ray emission are observed during a short-duration burst, which is first reported in GS~1826--238 by BeppoSAX in 30--60 keV \citep{int1999}. \textbf{However, these two sets of bursts located different spectral state of LMXBs. In both cases, the soft X-ray showers of the burst may manifest as an enhancement of the input seed photons but not a sufficient cooler upon the corona. } \textbf{ Compared with the disk component of the persistent emission, the count rate of the burst at the PRE phase is 4 times more. The emergent photons of the burst could be up-scattered to higher energy by the corona/boundary-layer. For the comptonization of the burst during PRE phase, i.e., a blackbody with temperature 1.22 keV and normalization 1.33, and a hot corona with temperature of 3 keV, optical depth 10.2 and cover-factor 0.76, we fake a spectrum induced by the inverse Compton scattering of the blackbody emission and get a count rate 289 cts/s in the energy band of 30--50 keV. Thus, the up-scattered photons of the burst do cause an enhancement hard X-ray emission.} Based on the burst spectra fit results, the enhancement hard X-ray emission could be related to the up-scattered burst emission by the corona/boundary-layer, just like the situation in the persistent emission, rather than enhancement accretion rate manifesting itself by elevating persistent emission with unaltered spectral shape \citep{Worpel2013,Worpel2015}. \subsection{Dynamical evolution of the disk geometry} As a common sense, the burst emission has an increasing and decreasing area during its rise and decay phase, which corresponding the hot spot spreading in the NS surface. Meanwhile, there are at least two moments when the hot spot covers that whole NS surface, the photoshphere lift-up point and the touch-down point for the PRE burst. \textbf{As shown in Figure 1 of \citet{Shaposhnikov2003}, the hot-spot spreads on the NS surface and then lifts up the photonsphere, i.e., from 'a' stage to 'b' stage in the Figure. There should be a moment that hot spot covers that whole NS surface in the rise phase and vice versa in the decay phase. However, there are some PRE bursts with a short increase time, i.e., that the increase time is too short for telescope to accumulating enough counts in the first moment when the hot spot covers that whole NS surface.} % In practice, the latter is usually used to derive the NS radius. % As shown in Figure \ref{fit_ledd}, at the touch-down point, the burst emission reaches its peak flux, both for the $f_{a}$ model and convolution thermal-Comptonization model. A dynamical evolution of the disk geometry could cause this phenomenon, i.e., the lower NS hemisphere, which is obscured before the burst (the burst PRE phase), appears from the disk after the burst-disk interaction, as shown in Figure 1 of \citet{Shaposhnikov2003} and Figure \ref{fig_illustraction} in this work. In theory, Poynting-Robertson drag could drain the inner-accretion-disk by taking away the momentum of the accretion matter hence enlarging the local accretion rate \citep{int2013,Worpel2013,Worpel2015}, which is faster than it is being refilled \citep{Stahl,Fragile2020}. At this moment, the inner part of the disk is hollowed out by the burst emission. % The flux-temperature diagram of the burst also indicates that the inner disk radius change causes a bigger visible part of the NS surface. If the whole NS surface shows up as a single-temperature blackbody and a constant color correction factor, the burst flux $F$ should scale as $kT_{\rm bb}^{4}$ in the flux-temperature diagram, and the slope represents the emitting area in the double logarithmic coordinates \citep{Guver2012}. As shown in Figure \ref{fig_t_f}, the rising phase and decaying phase obey different $F \propto kT_{\rm bb}^{4}$. We fit the two sets with $F=\frac{R^2}{D^2}\sigma T^4$, the blackbody radius of the rising and decaying phase is 4.3$\pm$0.11 km and 5.3$\pm$0.05 km with $D=4$ kpc. % Assuming the NS radiates at the Eddington limit in the PRE phase and the disk reaches the NS surface before the PRE phase, the blackbody flux ratio detected at the rising phase $F_{\rm rise}$ and decaying phase $F_{\rm decay}$ is positively associated with inclination angel $i$, i.e.,$\frac{F_{\rm rise}}{F_{\rm decay}}=(1+ {\rm cos}~i)/2$ \citep{Shaposhnikov2003,Shaposhnikov2004}. The inclination angel $i$ is estimated as $\sim$ 70$^\circ$. However, this result is bigger than the value $\sim$ 30$^\circ$--40$^\circ$ derived from the spectral fit results on an outburst of 4U~1608--52 by a reflection model \citep{Degenaar2015}. \subsection{Corona/boundary-layer reacting on burst} The interaction between the burst and inhabited persistent emission was first studied from the long-duration, brightest and most vigorous PRE bursts with moderate/super expansions in 4U~1820--30 and 4U~1636--536 \citep{Ball2004,KeeK2014b} (a factor of $\sim$10--10$^{4}$ increases in emission area). Then in short-duration bursts, this interaction was mainly observed as the persistent spectral change, rather than the burst spectral change, i.e., enhancement accretion rate, deficit at hard X-ray band, reflection by the disk and driven outflow. Type-I X-ray burst happens on the NS surface, which is also in the accretion environment. In principle, the burst spectrum may be influenced due to the Comptonization of the burst photons by the surrounding corona/boundary-layer \citep{chen2019}. A Comptt component was reported in bursts of 4U~1608--52 from RXTE observations above 3 keV \citep{Kajava2017}. However, their approach resembled the $f_{a}$ model since the Comptt component is added in the spectra fitting. In this work, the persistent emission is well fitted by a convolution thermal-Comptonization model. As a result, given the similarity, for the burst, adapting the convolution model with the parameters in the persistent emission fitting but with a blackbody emission, this could also fit the short-duration burst spectra in the PRE phase. The goodness of the fit are comparable with the $f_{a}$ model, but with a colder $kT$ and larger $R$ in the middle of the PRE phase. If this is the case, the radius of the photosphere is underestimated with the canonical blackbody model or the $f_{a}$ model. In principle, the interaction between burst and accretion environment, might be expected to have spectral evolution for both of the burst emission and accretion emission during burst, with spectral shape deviation from pure blackbody and the model of the pre-burst emission. The short duration and rapid spectral change limits the accumulated time and photon counts, which in turn requires larger detection area and broad band energy coverage which may be satisfied by the next generation of Chinese mission of so-called eXTP (enhanced X-ray Timing and Polarimetry mission) \citep{Zhang2019} or a contemporary joint observation of the burst by NICER and Insight-HXMT. \acknowledgements We thank the reviewer for the constructive feedback and comments that greatly improved the quality of this paper. This work made use of the data and software from the Insight-HXMT mission, a project funded by China National Space Administration (CNSA) and the Chinese Academy of Sciences (CAS). This research has made use of data and software provided by of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center. This work is supported by the National Key R\&D Program of China (2021YFA0718500) and the National Natural Science Foundation of China under grants 11733009, U1838201, U1838202, U1938101, U2038101. \bibliographystyle{plainnat} \begin{landscape} \begin{table}[ptbptbptb] \begin{center} \caption{The bursts obsid and peak time of 4U~1608--52 detected by Insight/HXMT in 2020 ourbutst } \label{tb} \begin{tabular}{cccccccccccccccccc} \\\hline obsid &Start Time & Elapsed Time (s) & LE GTI (s) & ME GTI (s) & Burst peak time (MJD) \\\hline P030402100401$^{\mathrm{*}}$ & 59069.63746 (2020-08-08T15:16:50) & 14708 & 2480 &3750 &59069.77077 \\\hline P030402100602 & 59079.83751 (2020-08-18T20:04:55) & 11617 & 120 &480 &59079.85655 \\ \hline \end{tabular} \end{center} \begin{list}{}{} \item[${\mathrm{*}}$]{This work.} \item[Note:]{\textbf{Both bursts are detected by the three payloads of Insight-HXMT. However, the second bursts, which is only half bright as the first one, did not located the good-time-interval.} } \end{list} \end{table} \begin{table}[ptbptbptb] \begin{center} \caption{The NICER obsid at the same day when Insight-HXMT detected the burst } \label{table_hxmt} \begin{tabular}{cccccccccccc} \\\hline obsid & Start Time & Elapsed Time (s) & GTI (s) \\\hline 3657026501 & 59069.61551 (2020-08-08T14:43:40)& 3600 &1707\\\hline \hline \end{tabular} \end{center} \end{table} \begin{table} \centering \caption{The results of the spectral fit of the LE, ME and NICER spectra in the 0.4--20 keV range with cons*tbabs*thcomp*diskbb} \label{persist_fit} \vskip -0.4cm \begin{tabular}{ccccccc} \\\hline $N_{\rm H}$ & $\tau$ & $kT_{\rm e}$ & $f_{\rm sc}$ & $kT_{\rm in}$ & $N_{\rm diskbb}$& $\chi_\nu^2$ \\ $10^{22}~{\rm cm}^{-2}$ & & keV & &keV & $10^2$ & \\ \hline $1.33_{-0.01}^{+0.01}$ & $10.2^{+0.3}_{-0.4}$ & $3.02^{+0.08}_{-0.08}$ & $0.76_{-0.05}^{+0.05}$ & $0.68_{-0.02}^{+0.01}$ & $8.38_{-0.96}^{+1.03}$ & 802/846 \\ % \hline \end{tabular} \end{table} \end{landscape} \clearpage \clearpage \clearpage

Title: CUSP: a two cubesats constellation for Space Weather and solar flares X-ray polarimetry

Abstract: The CUbesat Solar Polarimeter (CUSP) project aims to develop a constellation of two CubeSats orbiting the Earth to measure the linear polarisation of solar flares in the hard X-ray band by means of a Compton scattering polarimeter on board of each satellite. CUSP will allow to study the magnetic reconnection and particle acceleration in the flaring magnetic structures. CUSP is a project approved for a Phase A study by the Italian Space Agency in the framework of the Alcor program aimed to develop CubeSat technologies and missions.

https://export.arxiv.org/pdf/2208.06211

\keywords{X-ray polarimetry, solar flares, space weather, detectors, solar physics, CubeSat} \section{INTRODUCTION} \label{sec:intro} % Solar flares (SFs) are violent energetic phenomena taking place on our Sun that can have a strong impact on the human activities both on ground and in space. Solar activity, including SFs, can degrade radio communications, can cause radio blackouts and interference with GPS and satellite communications. Moreover, high-energy particles (protons and electrons) can release their energy in the satellite electronics producing malfunctions and also the loss of the satellite. The occurrence of SFs is also linked to Coronal Mass Ejection (CME) and Solar Energetic Particle (SEPs) events on the ground \cite{Papaioannou2016}. CUSP outcomes are intended to contribute to the present and future networks for Space Weather, including the future ASI SPace weather InfraStructure (ASPIS) \cite{Plainaki2018}. Polarimetry gives instantaneous clues about the flaring event. As soon as photons are detected, they can be downloaded and promptly analysed. CUSP is currently approved for a phase A study by the Italian Space Agency in the framework of the Alcor program aimed to develop CubeSat technologies and missions. CUSP is one of the 20 selected missions among 49 proposals submitted to a call for CubeSat missions that involved 22 participants from Research Institutes and Universities and 78 companies, mainly Small and medium-sized enterprises (SMEs) Over the next years, the Agency plans to deploy them in orbit. INAF-IAPS is the Prime Contractor and the responsible for the scientific payload development which front-end and back-end electronics will be designed and realized by SCAI Connect s.r.l company. The platform, compatible with the CubeSat 6U standard, will be designed and produced by IMT s.r.l. The Interdepartmental Center for Aerospace Industrial Research (CIRI-AERO) of the University of Bologna is responsible for the mission analysis, while the University of Viterbo ``La Tuscia" will take care of the ground segment with the ground station located in the University Campus. \section{CUSP science objective and scientific requirements} In the classical picture of a SF, magnetic reconnection originates the huge release of energy at the top of a magnetic loop. Particles are accelerated along the magnetic field lines towards the lower layers of the solar atmosphere and the interplanetary space. The dominant components of the energy spectrum are: \begin{itemize} \item thermal Bremsstrahlung (expected weakly polarised) \cite{Emslie1980a} and emission lines below 10 keV \item non-thermal Bremsstrahlung from about 10--20 keV, that is expected to be highly polarised \cite{Zharkova2010} \end{itemize} Theoretical models predict high linear polarisation in the X-rays, depending on the particle beaming and magnetic field properties \cite{Zharkova2010,Jeffrey2020}. Moreover, the directivity of accelerated particles can be derived from polarisation measurements. Until nowadays, few measurements with low significance have been performed \cite{Tindo1970,Tindo1972a,Tindo1972b,Tramiel1984,SuarezGarcia2006,Boggs2006}. High significance measurements would allow to overcome degeneracies in particle beaming models resulting from other observables like energy spectra \cite{Jeffrey2020}. To perform a step forward in the understanding of the SF physics, CUSP will measure the linear polarisation in the 20-100 keV energy band with a Minimum Detectable polarisation $<10\%$ at least for SFs most relevant in terms of Space Weather (X class). Moreover, due to the fact that SFs are dynamical events (with time scales from minutes to hours), CUSP will have the capability to study polarisation as a function of time. A detailed assessment of mission requirements will be carried out during phase A and B. CUSP science program also inlcude to perform some ancillary science. CUSP can detect intense X-ray astrophysical sources falling in the Field of View during the year while observing the Sun (F.O.V. $\pm 21^\circ$ at the Sun): \begin{itemize} \item Gamma Ray Bursts: based on the SWIFT/BAT catalog, CUSP is expected to detect 16 GRBs/yrs with a peak flux over the absorber background \item other sources as for example Crab Nebula (PWN), Sco X-1 (LMXB) and A0535+26 (HMXB) \end{itemize} \section{The payload: The hard X-ray polarimeter} The payload hosts a dual-phase Compton scattering polarimeter (operating in the 20-100 keV energy band) that exploits coincidence measurements between plastic and inorganic scintillator rods. Fig.~\ref{fig:views} shows the compact and exploded view of the polarimeter. The low atomic number of the plastic scintillator allows to maximise the scattering probability with respect to the heavier inorganic crystal made of GaGG (Gd3Al2Ga3O12) that maximises the photoelectric absorption of the scattered photon. Polarimetry is performed by measuring the azimuthal angular distribution of plastic/GaGG. If radiation is polarised, the preferential angular direction of polarisation produces a preferential response of the detector in the azimuthal direction (normal to the incident beam axis) as described by the Klain-Nishina cross section for Compton scattering \cite{Heitler1954}: \begin{equation} \frac{d\sigma}{d\Omega}=\frac{{r_0}^2}{2}\frac{{E^\prime}^2}{{E}^2}\Biggr[ \frac{E}{E^\prime}+\frac{E^\prime}{E}-2\sin^2 \theta \cos^2 \phi \Biggl] \label{eq:KN} \end{equation} where \begin{equation} \frac{E'}{E}=\frac{1}{1+\frac{E}{m_e c^2}(1-\cos \theta)}\label{eq:EsuE} \end{equation} The energies of the incident and scattered photons are $E$ and $E^\prime$, respectively. The angle $\theta$ is the scattering angle measured from the incident photon direction and $\phi$ is the azimuthal angle measured from the plane containing both the incoming direction and the electric vector of the incident photon. Linearly polarised photons are preferentially scattered perpendicularly to their polarisation direction. Thus, the response of $\phi$ emission directions for a polarised beam is modulated. The higher the modulated response, the higher the sensitivity of the polarimeter. Therefore, the modulation factor $\mu(\theta)$ (fraction of modulated signal corresponding to 100$\%$ polarised radiation) is: \begin{equation} \mu(\theta)=\frac{N_\mathrm{max}(\theta)-N_\mathrm{min}(\theta)}{N_\mathrm{max}(\theta)+N_\mathrm{min}(\theta)}=\frac{(\frac{d\sigma}{d\Omega})_{\phi=\frac{\pi}{2}}-(\frac{d\sigma}{d\Omega})_{\phi =0}}{(\frac{d\sigma}{d\Omega})_{\phi=\frac{\pi}{2}}+(\frac{d\sigma}{d\Omega})_{\phi =0}}=\frac{\sin^2\theta }{\frac{E}{E^\prime}+\frac{E^\prime}{E}-\sin^2 \theta} \label{eq:Muphi} \end{equation} In the limit of coherent scattering $E = E^\prime$, the maximum modulation factor is for $\theta=90^\circ$ (orthogonally to the incident photons direction). By increasing the energy it occurs at narrower scattering angles (forward folding effect). However, at 100 keV it is still $\theta \simeq 90^\circ$ \cite{Fabiani2012c}. The payload of CUSP comprises a W collimator for limiting the field of view about $\pm 21^\circ$ around the solar direction, the plastic and the inorganic scintillators assemblies which are readout by means of 4 Multi-anodes Photomultiplier Tubes (MAPMTs) for a total of 64 channels and 36 Avalanche Photo-diodes (APDs), respectively. Readout sensors have been selected to have a high heritage due to the very short implementation time required for the project. They are the MAPMT R7600 (rugged version to survive launch vibration) and an SMD version of APD derived from the S8664-55, both by Hamamatsu. MAPMTs and APDs are readout by MAROC 3A and SKIROC 2A ASICs by WEEROC. The payload also comprises A/D conversion, Micro HVs and a payload computer to handle House Keepings. In Fig.~\ref{fig:curves} are shown the current best estimates of the modulation factor $\mu$ (response in terms of modulation for 100$\%$ polarised radiation), efficiency $\epsilon$ (Compton interaction and tagging efficiency) and quality factor $Q$. Tagging efficiency is the probability to detect an event in the scatterer after a detection of an event in the absorber \cite{Fabiani2012c}. The quality factor is: \begin{equation} Q=\mu \sqrt{\epsilon}\label{eq:Q} \end{equation} It identifies the energy range in which the polarimeter is effective in measuring polarisation. This parameter is derived from the Minimum Detectable Popularization (MDP)\cite{Weisskopf2010} by assuming a source dominated observation (if background is negligible). This is true especially for brighter SFs. Table\ref{tab:mdp} reports the current best estimate of the MDP based on benchmark solar flares from Saint-Hilaire et al. (2008)\cite{SaintHilaire2008}. Few minutes of integration time allow to measure the polarisation of solar flares that is expected to be well above the MDP (at a level of some tens of per cent). \begin{table}[ht] \caption{Current best estimate of the MDP based on benchmark solar flares from Saint-Hilaire et al. (2008)\cite{SaintHilaire2008}} \label{tab:mdp} \begin{center} \begin{tabular}{|c|c|c|} \hline \rule[-1ex]{0pt}{3.5ex} Flare Class & Integration Time (s) & MDP ($\%$) \\ \hline \hline \rule[-1ex]{0pt}{3.5ex} M 5.2 & 284 & 9.2 \\ \hline \rule[-1ex]{0pt}{3.5ex} X 1.2 & 240 & 4.8 \\ \hline \rule[-1ex]{0pt}{3.5ex} X 10 & 351 & 0.9 \\ \hline \end{tabular} \end{center} \end{table} \section{The mission concept} We foresee a constellation consisting of 1 orbital plane and 2 CubeSats at 180$^\circ$ of phase difference along the orbit (see Fig.~\ref{fig:orbit}). This configuration allows to have always at least 1 satellite of the constellation in daylight for any $\beta$ angle (orientation of the orbital plane with respect to the Sun) for observing the Sun. Depending on the specific orbit, there is also a partial overlap of different duration between the two observation phases of the two satellites. A preliminary mission analysis identifies as a target a sun-synchronous (SSO) orbit at an altitude between 500 and 600 km with local time ascending node LTAN variable between a Mid-Morning scenario (LTAN = 9:30) and a Noon-Midnight scenario (LTAN = 12:00). This range of orbits offers the largest number of launch opportunities and foresees different orientations of the orbital plane with respect to the Sun with different duration of the eclipse period. Both satellites will be at the same inclination and altitude to have a common orbital period and node precession. The maximum contemporary daylight for both satellites is about 16 minutes per orbit for an SSO Mid-Morning (LTAN = 9:30) at 600 km of altitude. The time scale of the project, until launch, and then the operational phase of the mission, are compatible with the observation of the Sun during the next cycle maximum. Fig.~\ref{fig:solarcycle} reports the prediction of the 25th solar cycle activity by NOAA\footnote{https://www.swpc.noaa.gov/products/solar-cycle-progression} in terms of sunspot number and over-imposed the 3 years nominal operative life of the CUSP mission. The peak of the solar activity and, therefore, the maximum occurrence of solar flares, is expected between 2024 and 2028. The CUSP mission foresees two main operative modes. The observation mode requires that the each satellite rotates around the direction of pointing towards the Sun, a direction aligned with the axis of symmetry of the sensitive elements of the polarimeter. The rotation reduces the systematic effects (spurious modulation) induced by the polarimeter geometry. The duration of peak intensity in the hard X-rays of a solar flare is of the order of a few minutes, thus the rotation speed of 1RPM is sufficient for sampling the modulation curve induced by polarisation on a time scale of 30 seconds (half of the rotation period). The rotation of the polarimeter is obtained with the rotation of the entire satellite. The payload foresees a data storage and pre-processing system on board to archive data before their transmission to the ground. During the downlink phase, the satellite will change its attitude, optimizing the transmission link, thus interrupting the observation phase. The downlink of the data will take place by ensuring the continuity of observation of the solar phenomena of interest, in the context of the constellation of two satellites. \section{The platform} The platform foreseen for the CUSP project (see Fig.~\ref{fig:platform}) is a standard 6U CubeSat architecture developed by IMT s.r.l., that has a consolidated experience in the realization of different CubeSats, also in the institutional framework. The 6U CubeSat is based on the heritage of the HORTA and EOSS platforms (6U CubeSat platforms funded by Italian regional POR / FESR 2014-20 projects of Lazio and Puglia regions, respectively). The preliminary performance of the CUSP platform are reported in Table~\ref{tab:platform} \begin{table}[ht] \caption{Preliminary performance of the CUSP platform.} \label{tab:platform} \begin{center} \begin{tabular}{|c|c|} \hline \rule[-1ex]{0pt}{3.5ex} Peak Power & 30W with deployable panels \\ \hline \hline \rule[-1ex]{0pt}{3.5ex} Battery & $>$75Wh \\ \hline \rule[-1ex]{0pt}{3.5ex} Pointing accuracy & $>$ 1$^\circ$ (down to 0.1$^\circ$ ) \\ \hline \rule[-1ex]{0pt}{3.5ex} Operative frequencies & S / UHF \\ \hline \rule[-1ex]{0pt}{3.5ex} Downlink throughput & Up to 5 Mbps \\ \hline \rule[-1ex]{0pt}{3.5ex} Available interfaces & CAN Bus, I2C, UART, RS422 \\ \hline \rule[-1ex]{0pt}{3.5ex} Regulated bus & 3,3V, 5V e 12V \\ \hline \rule[-1ex]{0pt}{3.5ex} Not regulated bus & 32V (24V-33.6V) or 16V (12V-16.8V) \\ \hline \rule[-1ex]{0pt}{3.5ex} Available volume for the payload & 2.5U \\ \hline \rule[-1ex]{0pt}{3.5ex} Nominal life time & 3 years in LEO \\ \hline \end{tabular} \end{center} \end{table} The platform hosts an On Board Computer that underwent a qualification for radiation hardness above 20 krad and implements mitigation strategies for latch-up errors (SEU / SEL) by architecture, adopting anti-latchup and TMR (Triple Modular Redundancy) circuit. It can also offer memory storage for some Payload data (thanks to NAND memories) with the possibility to manage independently the Payload. During the 1RPM rotation required by the payload, the satellite will provide the determination of the attitude with an accuracy better than 0.5$^\circ$, needed for processing the scientific data. This will be achieved thanks to a high-precision gyroscopes and star trackers. During the non-observational phases of the Payload, the platform returns to the three axes stabilized mode to perform the downlink of the collected data towards the ground station. The telemetry and remote controls are realized through UHF band uplink and downlink communication with omnidirectional turnstile antennas. UHF antenna deployment will be based on the consolidated design by IMT s.r.l. which has already implemented several deployment mechanisms, both of antennas and solar panels. The S-band communication subsystem, in support of the UHF one, provides a higher link speed to allow scientific data transfer. The power needed by the on-board equipment is guaranteed by body mounted and deployable solar panels up to about 30W. The PDU subsystem will provide for the regulation and distribution of on-board energy. Fig.~\ref{fig:platformsub} shows the scheme of a CUSP satellite subsystems. \section{The ground station} The Ground Station (see Fig.~\ref{fig:groundstation}) is located at the “La Tuscia” University of Viterbo, on the building F of the ``Riello" Campus at the following coordinates (see left panel of Fig.~\ref{fig:contacts}): \begin{itemize} \item Latitude = 42.413$^\circ$ N \item Longitude = 12.113$^\circ$ E \item Minimum elevation $\epsilon$ min = 10$^\circ$ \end{itemize} It was built in 2019 as part of the HORTA project (Italian regional funds POR-FESR 2014-2020 of Lazio region). The Station allows for autonomous satellite tracking (using TLE satellite data - Two Lines Elements) and satellite communication. In addition, it can be controlled remotely. Available antennas and bands are: \begin{itemize} \item VHF: Uplink and Downlink \item UHF: Uplink and Downlink \item S-band: Downlink \end{itemize} The UHF/VHF bandwidth are 9.6 kbps as default for downlink (available also 1.2/ 2.4 / 4.8 kbps) and 1.2 kbps as default for uplink (available also 2.4 / 4.8/ 9.6 kbps). The S-band bandwidth is up to 1 Mbps for downlink. The pointing accuracy of the ground station is 0.1° (both azimuth and elevation) with a minimum tracking speed of 2$^\circ$/sec in azimuth, 1.8$^\circ$/sec in elevation. Data received from the ground station is transferred via fibre optics cable to dedicated workstations in the Mission Control Center that allows to schedule the passage of the satellite, the TT\&C and Payload data transceiving activities. Moreover, it provides a Network Server service for data delivery to third parties. From the preliminary Mission Analysis, about 2-4 contacts per day will be possible during mission operations (see right panel of Fig.~\ref{fig:contacts}) \section{The planning} The CUSP project is based on subsystems with high TRL. The Platform and the Ground Station can exploit a large heritage that allow them to guarantee a TRL 7. The Payload is based on elements with a high TRL (MAPMT, APD, scintillators, ASICs, coincidence technique), but the polarimeter as whole needs to be implemented. Thus, a TRL 3 is quoted. The Model Philosophy is based on the production of 1 detector prototype at the end of Phase B. It will be representative of the detector front-end to enhance detector TRL from 3 to 4. Then 1 payload EQM will be designed during phase B to be produced and tested during phase C. it will be representative of the payload to enhance detector TRL from 4 to 7. From the satellite point of view, 2 CubeSats will be produced: \begin{itemize} \item 1 Proto-flight Model (PFM). To be qualified at proto-qualification level \item 1 Flight Model (FM). To be qualified at acceptance level. \end{itemize} The Calibration of the Hard X-ray Polarimeter of each CubeSat will be carried out at INAF-IAPS calibration facility (already employed for calibrating the IXPE Detector Units)\cite{Muleri2021ice}. A 3 months phase A (starting in September 2022) plus possibly a 12 months phase B were approved. \section{Conclusions} Our Sun is still an astrophysical source with many aspects not well understood. Solar Flare mechanisms of production and particle acceleration are still debated. Moreover, solar flares can represent a threat for human technological activities in space and on ground, because they are usually correlated to Solar Energetic Particles Events (SEPs) at the Earth and Coronal Mass Ejections (CMEs). CUSP mission is aimed to measure the linear polarisation of solar flares in the 20-100 keV energy band to probe of particle acceleration and magnetic field behaviour during such energetic events. X-ray polarimetry allows to measure particle directivity and magnetic filed structure of the flaring loop to assess also magnetic reconnection that is thought to be at the origin of solar flares. CUSP will contribute to the understanding of these solar violent phenomena also participating in the present and future networks for Space Weather, including the ASI SPace weather InfraStructure (ASPIS). \bibliographystyle{spiebib} %

Title: Fossil group origins XII. Large-scale environment around fossil systems

Abstract: We analyse the large-scale structure out to 100 Mpc around a sample of 16 confirmed fossil systems using spectroscopic information from the Sloan Digital Sky Survey Data Release 16. We compute the distance between our FGs and the centres of filaments and nodes presented in \citet{Chen2016}. We also study the density of bright galaxies, since they are thought to be good mass tracers, and the projected over densities of galaxies. Finally, we apply a FoF algorithm to detect virialised structures around our FGs, in order to have an estimate of the mass available in their surroundings. FGs are mainly found close to filaments, with a mean distance of $3.7 \pm 1.1$ R$_{200}$ and a minimum distance of 0.05 $R_{200}$. On the other hand, none of our FGs is found close to intersections, with a mean and minimum distance of $19.3 \pm 3.6$ and 6.1 $R_{200}$, respectively. There is a correlation for which FGs at higher redshifts are found in denser regions, when we use bright galaxies as tracers of the mass. At the same time, FGs with the largest magnitude gaps ($\Delta m_{12}$ > 2.5) are found in less dense environments and hosting, on average, smaller central galaxies. Our results suggest that FGs formed in a peculiar position of the cosmic web, close to filaments and far from nodes, in which their interaction with the cosmic web itself can be limited. We deduce that FGs with faint BCGs, large $\Delta m_{12}$, and low redshifts could be systems at the very last stage of their evolution. Moreover, we confirm theoretical predictions that systems with the largest magnitude gap are not massive.

https://export.arxiv.org/pdf/2208.13784

\title{Fossil group origins} \subtitle{XII. Large-scale environment around fossil systems} \authorrunning{S. Zarattini et al.} \titlerunning{Large-scale environment around FGs} \author{S. Zarattini\inst{1,2}, J. A. L. Aguerri\inst{3,4}, R. Calvi\inst{3,4}, M. Girardi\inst{5,6}} \institute{Dipartimento di Fisica e Astronomia ``G. Galilei'', Universit\`a di Padova, vicolo dell'Osservatorio 3, I-35122 Padova, Italy \\ \email{stefano.zarattini@unipd.it} \and INAF - Osservatorio Astronomico di Padova, vicolo dell'Osservatorio 2, I-35122 Padova, Italy \and Instituto de Astrof\'isica de Canarias, calle VГ­a L\'actea s/n, E-38205 La Laguna, Tenerife, Spain \and Departamento de Astrof\'isica, Universidad de La Laguna, Avenida Astrof\'isico Francisco S\'anchez s/n, E-38206 La Laguna, Spain \and INAF-Osservatorio Astronomico di Trieste, via Tiepolo 11, I-34143 Trieste, Italy \and Dipartimento di Fisica, Universit\`{a} degli Studi di Trieste, via Tiepolo 11, I-34143 Trieste, Italy} \date{\today} \abstract{} {We analyse the large-scale structure out to 100 Mpc around a sample of 16 confirmed fossil systems using spectroscopic information from the Sloan Digital Sky Survey Data Release 16.} {We compute the distance between our FGs and the centres of filaments and nodes presented in \citet{Chen2016}. We also study the density of bright galaxies, since they are thought to be good mass tracers, and the projected over densities of galaxies. Finally, we apply a FoF algorithm to detect virialised structures around our FGs, in order to have an estimate of the mass available in their surroundings.} {FGs are mainly found close to filaments, with a mean distance of $3.7 \pm 1.1$ R$_{200}$ and a minimum distance of 0.05 $R_{200}$. On the other hand, none of our FGs is found close to intersections, with a mean and minimum distance of $19.3 \pm 3.6$ and 6.1 $R_{200}$, respectively. There is a correlation for which FGs at higher redshifts are found in denser regions, when we use bright galaxies as tracers of the mass. At the same time, FGs with the largest magnitude gaps ($\Delta m_{12}$ > 2.5) are found in less dense environments and hosting, on average, smaller central galaxies.} {Our results suggest that FGs formed in a peculiar position of the cosmic web, close to filaments and far from nodes, in which their interaction with the cosmic web itself can be limited. We deduce that FGs with faint BCGs, large $\Delta m_{12}$, and low redshifts could be systems at the very last stage of their evolution. Moreover, we confirm theoretical predictions that systems with the largest magnitude gap are not massive.} \keywords{} \section{Introduction} \label{sec:intro} Fossil groups (FGs) were proposed by \citet{Ponman1994}, when they found an apparently isolated giant elliptical galaxy that was surrounded by an X-ray halo, typical of a group of galaxies. Their interpretation of these observations was that this system was the final stage of the evolution of a group of galaxy, in which all the other $M^*$ galaxies (where $M^*$ is the characteristic luminosity of the luminosity function) were merged within the brightest central galaxy (BCG). To accomplish this scenario, FGs were supposed to be older than regular groups and to remain isolated from the cosmic web. In this picture, FGs can be considered fossil relics of the primordial Universe. Only in the last decade the number of known FGs grew enough so to have systematic studies of these objects. Without wanting to be exhaustive, four over-luminous red galaxies were found from \citet{Vikhlinin1999}, five FGs were presented in \citet{Jones2003}, 34 FG candidates were proposed in \citet{Santos2007} \citep[with 15 confirmed in][]{Zarattini2014}, 12 new FGs were presented in \citet{Miller2012}, 18 FG candidates were presented in \citet{Adami2018} and in \citet{Adami2020} a novel probabilistic approach was used to favour statistical studies of FGs. Recently, a list of 36 confirmed FGs (taken from the literature) was presented in the review of in \citet{Aguerri2021}, spreading on the redshift range $0 \le z \le 0.5$. The hierarchical model of structure formation in the Universe is a remarkably successful theory. It predicts that small structures form earlier and that they collapse subsequently into larger structures. This model is proven to be successful at large scales, but at galactic scales it incurs in the so-called ``small-scale crisis'': the number of predicted low-mass sub-halos in simulations around Milky-way like galaxies is larger than the one observed. On the other hand, \citet{D'Onghia2004} found that the small-scale crisis could be affecting the larger halos of FGs and that, in this case, the missing satellites can be as massive as the Milky Way itself. However, \citet{Zibetti2009} and \citet{Lieder2013} found no signs of missing satellites in FGs. The Fossil Group Origins project \citep[FOGO,][]{Aguerri2011} is devoted to the study of one of the largest sample of FGs available in the literature. The previous eleven publications of the FOGO team shed light onto various aspects of FGs. In particular, the formation of their BCGs were studied in \citet{Mendez-Abreu2012}, whereas their stellar populations were analysed in \citet{Corsini2018}. In these works, the authors showed that BCGs in FGs are amongst the most-massive galaxies in the Universe and that their stellar age is compatible with central galaxies in non-fossil systems. At the same time, BCGs are found to be more segregated in the velocity space when compared to non-FGs \citep{Zarattini2019}. The scaling relations between optical and X-ray \citep{Girardi2014,Kundert2015} showed that FGs were not X-ray over luminous systems and that particular attention must be dedicated to the homogeneity of the data in these kind of studies. Also, the luminosity functions (LFs) of FGs were compared with those of non-FGs \citep{Zarattini2015,Aguerri2018}, finding that there is a dependence of the faint-end slope on the magnitude gap. Moreover, FGs were found to host a similar fraction of substructures as non-FGs \citep{Zarattini2016}. Finally, in \citet{Zarattini2021} we showed that one of the main differences between fossil and non-fossil systems can be found in the different orbital shape of their galaxies. In fact, we found that galaxies falling into FGs are found on more radial orbits than in non-FGs, in agreement with theoretical predictions \citep{Sommer-Larsen2005}. Few works were devoted to the study of the large-scale environment around FGs. In particular, \citet{Adami2012} compared the environment of two FGs with one non-FG using photometric data. They found FGs to be more isolated than the control cluster, but the statistic was quite low and prevented them from reaching general conclusion. In a similar way, \citet{Adami2018} used spectroscopic data of RXJ1119.7+2126 (one of the two FGs of their previous paper) and were able to confirm that this system is located in a poor environment. The aim of this work is thus to analyse the large-scale structure around a large sample of spectroscopically-confirmed FGs in order to understand if they are found in any peculiar position of the cosmic web. The cosmology used in this paper, as in the rest of the FOGO publications, is H$_0 = 70$ km s$^{-1}$, $\Omega_\lambda = 0.7$, and $\Omega_M = 0.3$. \section{Sample selection} \label{sec:sample} Our sample is selected from the review of \citet{Aguerri2021}, where a list of spectroscopically-confirmed FGs in presented in table 1. In particular, we chose all the systems which centres are found in the Sloan Digital Sky Survey Data Release 16 (SDSS DR16) footprint and with an upper redshift limit of $z=0.20$. The SDSS spectroscopy is mainly limited at $m_r = 17.77$, that is equivalent to $M_r \sim -22$ at $z=0.2$. The central galaxies of this sample have magnitude ranging between $M_r = -21.3$ and $M_r = -24.1$, the faintest of the sample would not be mapped at $z=0.2$. We thus decided to limit the sample to $z=0.15$, corresponding to a magnitude limit of $M_r \sim -21.5$. The systems selected according to these criteria are 18. We note that we also included SDSS J1045+0420, which redshift is $z=0.154$. We thus looked at the spectroscopic completeness of the various clusters. In fact, it is possible that some cluster is part of the SDSS footprint, but for some reason its spectroscopy is below the standard. We computed the fraction of galaxies with spectroscopy by considering the number of galaxies with spectroscopy and the number of targets (e.g. galaxies with $m_r \le 17.77$). In Fig. \ref{fig:completitud} only two clusters have completeness lower than 65\%: these are UGC 842 and 1RXS J235814.4 + 150524. The former is found in a region of SDSS with very uneven spectroscopic coverage, the latter would also be discarded using the redshift criterium, so we did not check its spectroscopic coverage in details. We remain with 17 FGs after applying this cut. Finally, we will explain in Sect. \ref{sec:FGS28} that we prefer to exclude FGS28 from our sample, due to its doubtful nature. Our final sample is thus composed of 16 FGs which properties are presented in Table \ref{tab:sample}. We note that the coordinates reported in the table are those of the brightest central galaxy (BCG). This was done for having homogeneous centres. In fact, for some systems (like the FOGO ones) the centre reported in the literature is that of the BCG, for some others the centre reported in the literature is obtained from X-ray data. There is a known discrepancy between these two centres, estimated in 13 kpc in \citet{Sanderson2009} using a sample of 65 X-ray-selected clusters and in less than 5\% of $R_{200}$ from \citet{Lin2004}. Moreover, \citet{Aguerri2007} compared the distance between the X-ray peak and the galaxy surface density, finding a mean difference of 150 kpc. We don't expect this difference to impact in the results of our work, since all our tests involve megaparsec scales. Moreover, magnitude gaps (within 0.5 $R_{200}$) and X-ray luminosities are also taken directly from \citet[][]{Aguerri2021} and references therein. However, in Sect. \ref{sec:lx} we will explain how we converted the $L_X$ values to bolometric $L_X$ luminosities and to the same cosmology used in this paper. On the other hand, few masses were available in \citet[][]{Aguerri2021} and references therein, so we decided to estimate them by using their bolometric X-ray luminosities, thus homogenising the computation. The SDSS catalogues were obtained with an SQL query to the CasJobs webpage \footnote{\url{http://skyserver.sdss.org/CasJobs/default.aspx}}. For each system, we looked for all galaxies with available spectroscopy within a radius of 100 Mpc from the centre of the system reported in Table \ref{tab:sample} It is worth noting that for some cluster it was not possible to obtain data for the entire 100-Mpc-radius area. In Appendix \ref{appendix} we present the entire field of view available for each cluster. For 8 clusters, the full area is covered (11 if we consider a circle of 50 Mpc radius). For the remaining clusters the coverage is not full, but we can still use them for the majority of our scopes. \subsection{$L_X$ luminosities} \label{sec:lx} We were able to obtain the X-ray luminosity for all the systems in the sample. All the $L_X$ were taken from the literature and were computed in different bands, cosmologies, and methods. We then converted all the $L_X$ to the same band and cosmology. In particular, we chose to convert to bolometric luminosity and to the cosmology of this paper, when needed. Bolometric luminosities were obtained by multiplying the luminosity in the different bands by a temperature-dependent correction factor. In particular, we found luminosities in the $0.1-2.4$ keV (two FGs) and $0.5-2.0$ keV bands (seven FGs); the remaining seven FGs were found already in bolometric. This factor was computed using the Raymond-Smith code with ICM of 0.3 $Z_\odot$, according to the PIMMS\footnote{\url{https://cxc.harvard.edu/toolkit/pimms.jsp}} webpage. The code needs the value of the X-ray temperature T$_{\rm X}$. The values of T$_{\rm X}$ are available in the literature for seven out of nine clusters. For the remaining two clusters we followed a recursive procedure based on the T$_{\rm X}$-L$_{\rm X}$ relation \citep[see][]{Girardi2014}. The values of the (bolometric) X-ray luminosities are listed in Table 1. \begin{table*} \caption{Global properties of the sample} \label{tab:sample} \begin{center} \tiny \begin{tabular}{lccccccccccc} \hline \noalign{\smallskip} \multicolumn{1}{l}{Cluster name} & \multicolumn{1}{c}{R.A.} & \multicolumn{1}{c}{Dec} & \multicolumn{1}{c}{$\Delta m_{12}$} & \multicolumn{1}{c}{$\Delta m_{14}$} & \multicolumn{1}{c}{M$_{r,BCG}$} & \multicolumn{1}{c}{$z_{\rm \,BCG}$} & \multicolumn{1}{c}{M$_{200}$} & \multicolumn{1}{c}{R$_{200}$} & \multicolumn{1}{c}{L$_{\rm X}$} & \multicolumn{1}{c}{ L$_{\rm X}$ source} \\ \multicolumn{1}{l}{} & \multicolumn{1}{c}{[Deg]} & \multicolumn{1}{c}{[Deg]} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{[Mag]} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{[$10^{14}$ M$_\odot$]} & \multicolumn{1}{c}{[Mpc]} & \multicolumn{1}{c}{[$10^{42}$ erg s$^{-1}$]} & \multicolumn{1}{c}{} \\ \multicolumn{1}{l}{(1)} & \multicolumn{1}{c}{(2)} & \multicolumn{1}{c}{(3)} & \multicolumn{1}{c}{(4)} & \multicolumn{1}{c}{(5)} & \multicolumn{1}{c}{(6)} & \multicolumn{1}{c}{(7)} & \multicolumn{1}{c}{(8)} & \multicolumn{1}{c}{(9)} & \multicolumn{1}{c}{(10)} & \multicolumn{1}{c}{(11)}\\ \noalign{\smallskip} \hline DMM2008 IV & 113.641000 & 26.899000 & 2.4 & 3.0 & -23.43 & 0.08 & 0.34 & 0.67 & 41.8 & [1]\\ FGS03 & 118.184151 & 45.94928 & 2.09 & 2.55 & -22.40 & 0.05 & 1.01 & 0.96 & 29.3 & [2]\\ SDSS J0906+0301 & 136.659489 & 3.0275479 & 3.09 & / & -23.47 & 0.14 & 0.33 & 0.66 & 4.8 & [3] \\ A1068 & 160.182917 & 39.948056 & 2.3 & 3.1 & -23.99 & 0.14 & 2.05 & 1.22 & 2388.5 & [1]\\ SDSS J1045+0420 & 161.452085 & 4.3423831 & 2.00 & / & -23.54 & 0.15 & 1.92 & 1.19 & 43.7 & [3] \\ RXJ1119.7+2126 & 169.899792 & 21.455056 & 2.5 & / & -21.47 & 0.06 & 0.21 & 0.57* & 0.9 & [4] \\ BLOX J1230.6+1113.3 ID & 187.602827 & 11.189658 & 2.1 & 3.5 & -22.51 & 0.12 & 0.18 & 0.54 & 3.3 & [1] \\ XMMXCS J123338.5+374114.9 & 188.410417 & 37.687472 & 2.6 & 3.2 & -22.44 & 0.10 & 0.22 & 0.58 & 5.6 & [1] \\ FG12 & 191.713382 & 0.2970410 & 2.0 & / & -23.62 & 0.09 & 1.73 & 1.15 & 11.0 & [5] \\ RXJ1331.5+1108 & 202.873551& 11.132486& 2.0 & / & -22.48& 0.08& 0.44 & 0.73* & 3.3 & [4] \\ XMMXCS J134825.6+580015.8 & 207.106667 & 58.004389 & 2.0 & 2.6 & -23.23 & 0.13 & 0.54 & 0.78 & 17.8 & [1] \\ FGS20 & 212.517450 & 41.755800 & 2.17 & 2.46 & -23.56 & 0.09 & 0.46 & 0.74 & 9.5 & [2] \\ RXJ1416.4+2315 & 214.112083 & 23.258611 & 2.4 & / & -24.05 & 0.14 & 4.41 & 1.57* & 127.8 & [4] \\ XMMXCS J141657.5+231239.2 & 214.239583 & 23.210889 & 2.8 & 3.1 & -22.94 & 0.12 & 0.20 & 0.56 & 3.2 & [1] \\ RXJ1552.2+2013 & 238.051250 & 20.229167 & 2.3 & / & -23.66 & 0.14 & 1.78 & 1.16 & 36.5 & [4] \\ AWM 4 & 241.237500 & 23.920556 & 2.23 & / & -23.07 & 0.03 & 1.33 & 1.05 & 1.5 & [6] \\ \hline \end{tabular} \end{center} \tablefoot{Columns represent: (1) Cluster name. (2): Right ascension of the BCG. (3): Declination of the BCG. (4): Magnitude gap between the two brightest member galaxies. (5): Magnitude gap between the first and the fourth brightest member galaxies. (6): Absolute magnitude of the BCG, computed using SDSS DR16. (7): Redshift of the BCG. (8): Virial mass of the object estimated from X-ray data. (9): Virial radius, estimated from X-ray data. (10): X-ray bolometric luminosity. (11) Publication from which the original L$_X$ data were taken: [1] \citet{Harrison2012}, [2] \citet{Zarattini2014}, [3] \citet{Proctor2011}, [4] \citet{Jones2003}, [5] \citet{LaBarbera2012}, and [6] \citet{Zibetti2009}. *R$_{200}$ radius computed in this work from the X-ray luminosity given in the original work. We use equation 2 from \citet{Bohringer2007} to compute R$_{500}$ and then convert it to R$_{200}$ using R$_{200} = 1.516 \times$ R$_{500}$ \citep{Arnaud2005,Girardi2014}} \end{table*} \subsection{FGS28} \label{sec:FGS28} FGS28 is found at $z=0.032$. There are four other clusters nearby: NGC 6107 ($z=0.0315$), A2192 ($z=0.0317$), A2197 ($z=0.0301$), and A2199 ($z=0.0299$). The largest velocity difference between all of them is $\sim 600$ km s$^{-1}$. The pair A2197 and A2199 is considered as a supercluster in the literature \citep{Rines2002} and it is known to be connected with a large filament to the Hercules supercluster \citep{Ciardullo1983}. The A2197 mass profile (the closest to the position of FGS28) is better fitted by dividing the cluster in two clumps, that are named East and West in \citet{Rines2002}. Moreover, in \citet{Zarattini2014} we found that FGS28 was peculiar, since it has only one member within R$_{200}$ and there were other 4 members outside this area. Our interpretation is that this is not a real group of galaxy, whereas it is a giant galaxy that is part of the A2197/A2199 supercluster. For these reasons, we prefer to remove FGS28 from our sample of FGs. \section{Fossil systems position in the large-scale structure} \label{sec:lss} In this section we discuss how we define the large-scale structure around our FGs. In particular, in Sect. \ref{sec:filaments} we introduce the catalogue of filaments presented in \citet{Chen2016} and we compute the distance of the FGs in our sample from the centre of filaments and intersections. Then, in Sect. \ref{sec:fof} we explain in details how we apply a friends-of-friends algorithm to our FGs and which are the useful output of the algorithm itself. \subsection{Catalogue of filaments} \label{sec:filaments} The first method that we use to study the large-scale structure of our FG sample is to analyse their position with respect to the catalogue of filaments presented in \citet{Chen2016}. In this catalogue, filaments are found using SDSS data by applying the Subspace Constrain Mean Shift (SCMS) algorithm using galaxy density ridges. In particular, SCMS performs three steps to detect filaments \citep[density estimation, thresholding and gradient ascent, see][and references therein]{Chen2015}. To build the catalogue, the authors used spectroscopically-confirmed galaxies in the redshift range $0.05 < z < 0.7$, divided into 130 redshift bins. As a result, the filament catalogue only covers this redshift range and it is limited in RA and Dec (109 $\lesssim$ RA $\lesssim$ 267 and $-4 \lesssim$ Dec $\lesssim 70$). This approach is similar to that used in \citet{Adami2020}, where they used data from the Canada France Hawaii Telescope Legacy Survey (CFHTLS) to detect FG candidates using photometric redshifts and then used a catalog of filaments and nodes obtained from the same CFHTLS to study the position of their FG candidates with respect to the large-scale structure of the Universe. We are now able to measure the distance of our FGs from the centre of the filaments for most of our systems. In particular, we computed the distance in RA, Dec between our FGs and the closest filaments in the redshift space. The filament catalogue gives the minimum redshift of the filament ($z_{low}$) and all the galaxies in the filament satisfy $z_{low} \le z \le z_{low}+0.005$. This means that, in the velocity space, each filament is $\sim 1500$ km s$^{-1}$ wide. \begin{table} \setlength{\tabcolsep}{15pt} \caption{Distance to filaments and intersections for the FGs in our sample.} \label{tab:filaments} \begin{center} \tiny \begin{tabular}{lcc} \hline \noalign{\smallskip} \multicolumn{1}{l}{Cluster name} & \multicolumn{1}{c}{$D_{fila}$} & \multicolumn{1}{c}{$D_{int}$}\\ \multicolumn{1}{l}{} & \multicolumn{1}{c}{[$R_{200}$]} & \multicolumn{1}{c}{[$R_{200}$]} \\ \noalign{\smallskip} \hline DMM2008 IV & 4.1 & 53.1 \\ FGS03 & 0.7 & 25.6 \\ SDSS J0906+0301 & 11.3 & 21.5 \\ A1068 & 5.4 & 24.7 \\ SDSS J1045+0420 & 14.3 & 14.8 \\ RXJ1119.7+2126 & 4.3 & 9.8 \\ BLOX J1230.6+1113.3 ID & 1.5 & 9.8 \\ XMMXCS J123338.5+374114.9 & 4.6 & 8.3 \\ FG12 & 0.05 & 7.2 \\ RXJ1331.5+1108 & 0.3 & 6.1 \\ XMMXCS J134825.6+580015.8 & 3.4 & 42.1 \\ FGS20 & 0.4 & 7.5 \\ RXJ1416.4+2315 & 2.5 & 9.6 \\ XMMXCS J141657.5+231239.2 & 1.2 & 25.8 \\ RXJ1552.2+2013 & 1.1 & 23.1 \\ AWM 4* & / & / \\ \hline \end{tabular} \end{center} \tablefoot{Column (1): Cluster name. Column (2): Minimum distance to filament in units of $R_{200}$. Column (3): Minimum distance to intersection in units of $R_{200}$} * This system has $z < 0.05$ and it is thus outside the redshift interval of the filament catalogue. For this reason, no distance was computed for it. \end{table} We then measured the distance in RA and Dec from the centre of each FG and the centre of the closest filament. We use this definition because in the catalogue of \citet{Chen2016} the position of the centre of the filament (in both Ra and Dec and in the redshift space) is given. We only use filaments that are found within $z \pm 0.005$ from the target, that is again equivalent to $\pm 1500$ km s$^{-1}$, as above. We repeat the same computation for determining the distance of each FG to the closest intersection, that is also provided in the \citet{Chen2015} catalogue. In Fig. \ref{fig:distances} we show the relation between the magnitude gap ($\Delta m_{12}$), the X-ray luminosity ($L_X$), and the absolute magnitude of the central galaxy (M$_{r,BCG}$) and the distance from the center of the filaments (left panels) or intersections (right panels), as defined in \citet{Chen2016}. It is interesting to note that the majority of the FGs in our sample are found nearby filaments, with a mean distance of 3.7 $R_{200}$ and a minumim distance of 0.05 $R_{200}$. On the other hands, the distance between our FGs and the intersections is larger, with a mean of 19.3 $R_{200}$ and a minimum distance of 6.1 $R_{200}$, as it shown again in Table \ref{tab:filaments}. In the same table the distance between each FG and the center of the closest filament is reported, but in the Appendix \ref{appendix} we show the entire large-scale structure around each FG for an easier visual inspection. It is worth noting that for AWM4 we were not able to compute the distance from filaments and intersections, since this FG has $z=0.0317$, a value below the redshift limit of the \citet{Chen2016} catalogue ($z=0.05$). We are thus able to split our FGs in two categories: systems that are found in a region close to a filament ($D_{fila} \le 5 R_{200}$) and systems that are more isolated. The separation is shown in Fig. \ref{fig:distances} with an horizontal line and in the entire paper FGs close to filaments are shown in blue, whereas red represents FGs that are far from filaments. AWM4, for which the distance to filaments and node was not computed, is eventually shown in black. However, no correlation is found between $\Delta m_{12}$, $L_X$, and M$_{r,BCG}$ and the distance to filaments or intersections. \subsection{Friend-of-friends algorithm} \label{sec:fof} We also run a friends-of-friends algorithm on our data. The algorithm was presented in \citet{Calvi2011} and was applied to all the galaxies within $\pm 3000$ km s$^{-1}$ from the velocity of the parent cluster. Two parameters are used to define friends: a linking length and a linking velocity. For the former, as a first attempt we run the FoF algorithm by using a single linking length of 0.5 Mpc in order to select the core of the clusters and reduce the number of contaminant galaxies. However, the results worsened with $z$: at low redshift ($z < 0.1$) the FoF output follows the visible large-scale structure of each system, but at $z > 0.1$, where data are less sampled, almost nothing was found by the algorithm. Thus, we used a variable linking length depending on redshift: in particular, we used 0.5 Mpc for $z < 0.05$, 1.0 Mpc for $0.05 \le z \ < 0.1$, and 1.5 Mpc for $z \ge 0.1$. This choice was done in order to reflect the smaller number of galaxies found when increasing the redshift. In fact, SDSS spectroscopy is limited in apparent magnitude (down to $r \le 17.77$), which means that increasing redshift will turn in decreasing the number of targets. As a result, the mean distance between galaxies with spectroscopy is expected to grow. On the other hand, the velocity link was chosen to be constant and fixed at $\pm 1500$ km s$^{-1}$. This choice is motivated by the fact that the majority of clusters have velocity dispersions between 300 and 1000 km s$^{-1}$ \citep{Munari2013}. We thus assume that $\pm 1500$ km s$^{-1}$ is enough to include the vast majority of galaxy members, without including too many contaminat galaxies. In the Appendix \ref{appendix}, the results of the FoF algorithm are presented in red. It can be seen that a good agreement between red points and the filamentary structures presented in \ref{sec:filaments} is found. However, an exact measurement of the precision of the match is beyond the scope of this paper and, in the rest of our work, the filament catalogue will be used as the operational definition of the large-scale structure. Once the friends-of-friends have been detected, the code looks for virialised structures (e.g. clusters) by computing the $R_{200}$ radius and the velocity dispersion of the group/cluster, removing the outliers and repeating the process until the number of members converges. For our analysis, we limit the detection of a structure to agglomeration of galaxies that have at least three members and a minimum velocity dispersion of 200 km s$^{-1}$ that is the mean velocity dispersion of galaxy groups found using the same FoF algorithm in \citet{Calvi2011}. The goal of this cut is to remove smaller structures like galaxy pairs or smaller groups \citep[][found that only 11\% of their groups have $\sigma_V < 100$ km s$^{-1}$]{Calvi2011}, that we think are not useful for our work. After convergence, we estimate the mass of each structure (details in Sect. \ref{sec:mass}). The main goal of this procedure is to estimate the mass of groups and clusters around our sample of FGs, to check if there is a relation between the available mass and some of the FG properties. However, sometimes also the FG is detected and in this case we are able to estimate the mass in this alternative way. The FoF-computed mass of each FG is presented in the Appendix \ref{appendix}. \section{General properties} \label{sec:general} In Fig. \ref{fig:general_properties} we show the correlations between some of the main parameters available for our sample. In particular, we focus our attention on the magnitude gap ($\Delta m_{12}$), the X-ray luminosity (L$_X$), the absolute magnitude of the central galaxy (M$_{r,BCG}$), and the virial radius (R$_{200}$). Some correlations are visible, such those between $R_{200}$, $L_X$, and the absolute magnitude of the BCG. The $R_{200} - L_X$ correlations is expected, since the former quantity was computed from the latter. Also the $R_{200} - M_{r,BCG}$ and the $L_X - M_{r,BCG}$ correlations are expected for all clusters, since in general massive clusters host massive BCGs \citep{Lin2004,Brough2008}. For our goals it is more interesting to analyse the correlations involving the magnitude gap: it can be seen that there is no clear correlation when using this parameter. The only interesting result is that the systems with the largest magnitude gap ($\Delta m_{12} \ge 2.5$) are small, with $L_X < 10^{43}$ and $R_{200} < 0.7$. Finally, no specific correlation is found for FGs that are close and far from filaments, although systems with $\Delta m_{12} \ge 2.5$ are mainly close to filaments. However, the one with the largest $\Delta m_{12}$ is far from the closest filament and the statistic is in general very poor for these systems. We then computed the cumulative distribution of $\Delta m_{12}$, $L_X$, $R_{200}$ and M$_{r,BCG}$ in the two subsamples (e.g. FGs close and far from the centre of filaments). The results, presented in Fig. \ref{fig:cumulative}, show that they follow similar relations, with the exception of the absolute magnitude of the BCG. In fact, the three systems that are far from filaments have all BGCs with $M_r < -23.5$, whereas those close to the centre of filaments are more equally distributed in the range $-24 \le M_r \le -21.5$. \section{Large-scale mass distribution} \label{sec:fof_mass} In this section we want to discuss the large-scale environment of FGs with respect to the quantity of mass available in their surroundings and its distribution. Since SDSS spectroscopic data are not homogeneous, the main issue is how to compute precise areas for all the sample, especially for those FGs which mapping is widely incomplete. For this reason, in \ref{sec:pick} we will discuss the Pick theorem and how we use it for our scopes. Once the areas are known, we will compute the local projected over density in Sect. \ref{sec:projected}. Finally, in Sect. \ref{sec:mass} we will analyse the quantity of mass available in the FGs' surroundings by using different indicators: the density and cumulative distribution of bright galaxies and the mass found in groups and clusters from the FoF algorithm. \subsection{The Pick's theorem} \label{sec:pick} The statement of Pick's theorem \citep{Pick1899} claims that, if a regular polygon has integer coordinates for the vertices, the area can be computed as \begin{equation} A = i + \frac{b}{2}-1, \end{equation} where i the number of integer points inside the polygon and b the number of integer points on its boundaries. We used this theorem to compute the area available for each FG in the sample. In particular, we divided our areas in different rectangles and we then applied, to these now regular polygons, the Pick's theorem. To estimate uncertainties, we applied the theorem for those FGs for which the entire 100 Mpc area were available (8 systems). The mean difference between the Pick area and the geometric one is $-2.3$\%, with a maximum of $-3.2$\%. We also compute the difference in the area of 50 Mpc for the 11 systems fully covered out to this radius: in this case, the mean percent error is $-3.2\%$, with a maximum error of $-4.2\%$. The errors are larger in the second case because the number of points used to apply the Pick's theorem is smaller. We highlight that the differences in measurements are always negative. This means that the Pick's theorem is systematically underestimating the geometric areas. Since the errors are connected to the number of available points, we expect to have larger errors for the farthest FGs, since we are using a sample of galaxies that is limited in apparent magnitude. \subsection{Local projected over densities} \label{sec:projected} We were now able to compute the local over densities within circular coronas around our FGs. In particular, we used the Pick's area for the coronas and only galaxies within $\pm 1500$ km s$^{-1}$ and $m_r\le 17.77$ (the SDSS completeness limit for the main spectroscopic sample). Finally, we corrected for spectroscopic completeness. The over density is computed as \begin{equation} OD = 1 + \frac{\Sigma (r)-\overline{\Sigma}}{\overline{\Sigma}} = 1 + \delta \end{equation} where $\Sigma (r)$ is the density at each specific bin radius and $\overline{\Sigma}$ is the mean density computed by using all galaxies between 20 and 50 Mpc. In particular, we firstly analyse the over densities for systems that are closer or farther than 5 $R_{200}$ from the centre of the closest filament. No difference is found within the errors in this case, as it can be seen in the top panel of Fig. \ref{fig:overdensities}. Then, we compute the over densities by dividing our sample in systems with bright and faint central galaxies, using $M_r = -23$ as separation. The result is shown in the central panel of Fig. \ref{fig:overdensities} and in this case a difference is found in the very central distance bin, at least at 1-$\sigma$ level. Finally, in the bottom panel of Fig. \ref{fig:overdensities} we plot the over densities of our systems divided using X-ray luminosity, that is a proxy of the mass of our systems. Again, no difference is found for the over densities of small or massive FGs. \subsection{Available mass in FGs' surroundings} \label{sec:mass} In order to estimate the mass available in the surrounding of our FGs, we use a set of indicators. First of all, we compute the density of bright galaxies ($M_r < -22$) per Mpc$^{2}$ that are found within 50 Mpc from the centre of the FG. The area in degrees was obtained using the Pick's theorem, as explained in Sect. \ref{sec:pick}, to have a proper estimate also for systems that are not fully covered in the 50 Mpc radius. In Fig. \ref{fig:brighter} we show the correlations between the absolute magnitude of the BCGs, the magnitude gap, the redshift, and the X-ray luminosities versus the density of bright galaxies. The most relevant result is obtained for the $\Delta m_{12} -$ density correlation: systems with the largest magnitude gap are always found in low-density regions, whereas systems with the smallest gaps are found in all environments. We repeat the computation using galaxies with $M_r \le -23$, finding exactly the same trend, although with less statistics, and for this reason we prefer to show here the results with $M_r \le -22$. Another useful indicator is the cumulative distribution of bright galaxies, that is shown in Fig. \ref{fig:cumulative_bright}. It can be seen that no differences are found within 50 Mpc between FGs close and far from filaments. A Kolmogorov-Smirnov (KS) test confirms this result by giving a KS probability of 0.99, where differences in the two distributions are expected if this value is smaller than 0.05. However, in the first panel of Fig. \ref{fig:cumulative_bright} small differences seem to appear in the central and external regions: in the former, a smaller number of bright galaxies are found in FGs close to filaments, whereas the opposite is found in the latter. We thus compute the cumulative distribution in a smaller area, namely 20 Mpc, aiming at looking in more details to the differences in the neighborhood of our FGs. Although the difference is visually larger, a KS test confirms that both distributions (FGs close and far from filaments) come form the same parent distribution (KS probability of 0.99 also in this case), so we conclude that no difference is found in the distribution of bright galaxies within 50 Mpc from our FGs and, thus, we did not test the external regions. The FoF algorithm that we used is able to identify virialised objects and to estimate their velocity dispersion. We thus use it to estimate the mass of these virialised systems according to Eq. 1 of \citet{Munari2013}. In Fig. \ref{fig:correlations_mass} we plot the correlation between total mass available within 50 and 100 Mpc and some global quantities of the FG in our sample, namely the magnitude gap, the absolute magnitude of the BCG, the X-ray luminosity, the virial radius, and the redshift. There are 11 FGs well mapped out to 50 Mpc and 8 fully mapped out to 100 Mpc. No particular trend is found, except the one between the available mass and the X-ray luminosity. In fact, it seems that the most luminous FGs have less mass available, but this result can be driven by a couple of points in the most extreme regions of this relation. We run a Spearman correlation test \citep{Spearman1904} that found a significant correlation ($\rho \sim 0.02$) between the X-ray luminosity and the available mass within 50 and 100 Mpc. In both cases, there is a negative correlation (the available mass decreases while the X-ray luminosity increases). Other weak correlations are found, one between the virial radius and the mass found in 100 Mpc ($\rho \sim 0.09$) and of between the redshift and the mass found in 50 Mpc ($\rho \sim 0.06$). The latter trend is apparently the opposite of the one found between redshift and the density of bright galaxies, we will discuss in Sect. \ref{sec:discussion} this contradiction in more details. \section{Discussion} \label{sec:discussion} The results presented in this paper can be interpreted in terms of the formation scenario and evolution of FGs. We show that the majority of the FGs in our sample are found close to the centre of filaments, with a mean distance of $3.7 \pm 1.1$ R$_{200}$, whereas none is found close to intersections (mean distance of $19.3 \pm 3.6$ R$_{200}$). This is surprising, since usually galaxy groups/clusters are though to form in the nodes of the cosmic web \citep[e.g.][]{Cautun2014}. These models predicts that the feeding mechanism for galaxy clusters is to receive new objects falling along filaments into the node. The position of FGs, far from the nodes, could thus be the responsible for the formation of their large magnitude gap. In fact, \citet{Ponman1994} firstly suggested that FGs could have been isolated from the cosmic web, thus having enough time to merge all their bright galaxies with the BCG without receiving more bright galaxies from the surroundings. Our results seem to favour this scenario, in which objects that are moving withing the filaments are not able to leave them to reach the nearby FGs. However, it is difficult to estimate the size of filaments, in particular its width or thickness. For this reason we are not able to definitely claim that FGs are found embedded in filaments or just outside them. It is interesting to note that our results are in good agreement with what was found in \citet{Adami2020}, where the authors analysed a sample of FG identified using a probabilistic approach in the CFHTLS, finding that FGs seem to reside closely to cosmic filaments and do not survive in nodes. In particular, 87\% of their FGs are within 1 Mpc to a filament, whereas 67\% are farther than 1 Mpc from nodes. We are finding a similar trend, but we only have 33\% of our FGs within 1 Mpc from a filament. This fraction rise to 73\% if we consider systems within 2 Mpc from the centre of the filament. This difference can be due to the different cuts used in \citet{Adami2020} to define filaments and nodes close (in the redshift space) to their target FGs: in fact, since they worked with photometric redshift instead of spectroscopic ones, their filament catalogue uses slices that are thicker than ours, possibly leading to projection effects (e.g. more filaments and nodes in the same projected area with respect to \citet{Chen2016}), as they also noted in their work). A similar conclusion can be reached for the distance from the nodes, with the difference that all our FGs (100\%) are farther than 1 Mpc from nodes, whereas in \citet{Adami2020} they found 67\%. We need to use a distance of 10 Mpc to recover a similar fraction for our sample (60\%). Again, differences in the construction of the catalogues of filaments and nodes or different strategy when analysing the data could lead to these differences that, it is worth noticing, are quantitative but not qualitative. Within this scenario, it is interesting to note that FGs with the largest magnitude gaps are not massive (all the four systems with $\Delta m_{12} \ge 2.5$ have $L_X < 10^{43}$ erg s$^{-1}$). This can be a boost for the merging timescale of bright galaxies within this systems, since the available mass is smaller (fewer massive galaxies) and the relative velocity between galaxies is smaller too, due to the shallower gravitational potential that groups have with respect to clusters. These two conditions, together with the presence of more radial orbits expected in FGs \citep[as predicted theoretically and, later, observationally confirmed in][]{Sommer-Larsen2005,Zarattini2021} are the main ingredient that reduce the timescale of dynamical friction \citep[see eq. 4.2 of ][]{Lacey1993}. \citet{Dariush2010} suggested that only FGs with small central galaxies could be real FGs, intended as systems with older formation than regular groups and clusters that passively evolve without many interactions with the cosmic web. Thus, we can now suggest that these old FGs could be those found aside of cosmic filaments, which peculiar position is able to prevent further accretions. Our results support the scenario in which FGs could be a transitional stage in the life of a group/cluster and that the magnitude gap can be reduced when a bright galaxy falls into the FG potential \citep{Aguerri2018,Kim2022}. However, we found that the largest gaps are found in FG with a low density of bright galaxies around. In particular, systems with $\Delta m_{12} \ge 2.5$ has less than one galaxy brighter than $M_r = -22$ within 50 Mpc, whereas five out of ten systems with $\Delta m_{12} < 2.5$ have more than one bright galaxy within their 50 Mpc radius. This could mean that systems with $2.0 \le \Delta m_{12} \le 2.5$ can still be changing their gaps and become non-fossils (because they are closer to $\Delta m_{12} = 2.0$ and they have more bright galaxies nearby), but that FGs with $\Delta m_{12} > 2.5$ are probably less inclined to change the gap to values smaller than $\Delta m_{12} = 2.0$. A similar suggestion can be obtained for the most-isolated FGs, since they presents only very bright BCGs and are, on average, quite massive. In this case, the boost in the merging timescale could have been given by the higher available mass for the satellite galaxies, whereas their isolation from both filaments and nodes could have avoided the subsequent arrival of new massive satellites. However, the paucity of isolated FGs prevents us to reach a robust conclusion on this subsample, since only three FGs are isolated in our sample. We also analysed the galaxy over density within 100 Mpc. We confirmed that at very large radii ($r > 20$ Mpc) no differences can be found, with the over density that oscillates around zero. However, when dividing the sample of FGs into those with bright and faint BCGs, a difference can be found (at $1-\sigma$ level) in the most-central distance bin: systems with brightest galaxies are found in larger over densities. This could be connected with a larger number of bright satellites available for merging nearby those massive BCG. On average, FGs with BCGs brighter than -23 have $1.5\pm 1.1$ bright galaxies within 50 Mpc, whereas FGs with BCGs fainter than -23 have $0.6 \pm 0.4$ of these galaxies available. The result is not statistically significant, however we note that the latter systems have all less than 1 bright galaxy within 50 Mpc, whereas the former systems spread the entire range between 0 and 3.4 bright galaxies within 50 Mpc. However, other correlations appear when comparing the density of bright galaxies with the global properties of our FGs. First of all, a correlation between $L_X$ and the density of bright galaxies is found, where systems with small $L_X$ seem to be more isolated. A Spearman test was run, giving a positive correlation (coefficient of 0.56) and rejecting the null hypothesis (probability of 0.02). Moreover, a correlation between redshift and the density of bright galaxies is found, too (Spearman coefficient and probability of 0.85 and $3\times 10^{-5}$, respectively). The first interpretation could be a sort of selection effect due to observations. However, \citet{Verevkin2011} showed that the redshift distribution in the SDSS-DR7 is peaked at $z\sim 0.08$ and then quickly drops at higher redshift. We did not expect this distribution to change in more recent data releases, since the complete main spectroscopic sample (the one that we are using here, limited to $m_r = 17.77$) was released with DR7. Newer SDSS releases have indeed more redshifts, but the target selection is different and focused on high-redshift galaxies \citep[e.g. the BOSS survey,][]{Dawson2013}. If a selection effect would be present, we would also expect the density to be higher for low-redshift object, that is not the case. In fact, we find the opposite correlation. Our interpretation of this result is that FGs at higher redshift are still found in lively environments, where some major merger can still happen. Indeed, this relation can be seen as an indicator of merger probability, higher for systems at higher redshifts, where more bright galaxies (e.g. more mass) is available. Using numerical simulations, \citet{Kundert2017} showed in the middle-left panel of their fig. 7 that the number of major mergers in FGs is still growing in the redshift range $0.2 \lesssim z \lesssim 0.1$, whereas it stops growing for $z \lesssim 0.1$, a result that can explain what we found in the redshift versus density-of-bright-galaxies correlation. We also studied the mass available around FGs using the FoF algorithm. In this case, we found an opposite trend with $z$: the mass available (within 50 Mpc) is higher for systems at low redshift. However, we believe that this result can be more affected by biases. In fact, we used different (and arbitrary) linking length for systems with different redshift. This was done to have a good visual agreement between the filaments of \citet{Chen2016} and the FoF galaxies. Since the linking length is smaller for systems at low redshift (0.5 Mpc for $z < 0.05$ versus 1.5 Mpc for $z > 0.1$), we can expect an excess of linked galaxies in the lowest-redshift FGs. Moreover, we are also excluding small groups from the FoF computation, thus again favouring the detection of groups at low redshifts, where the density of data is higher. Finally, the density of bright galaxies is more robust when using a survey limited in apparent magnitude, as the SDSS. In fact, $M_r = -22$ is equivalent, at $z=0.15$ to $m_r = 17.2$, that means that the SDSS spectroscopy (e.g. the galaxies used in this work) is complete also at these redshifts. Concluding, we found that the very large scale environment (distances larger than 10 Mpc) is not having any role in the evolution of FGs. Hints are found that some difference can be due the the environment at distances smaller than 10 Mpc. This can be due to the presence of filaments, whose mean distance is 3.7 $R_{200}$ (or 3.0 Mpc). \section{Conclusions} \label{sec:conclusions} We analysed a sample of 16 FGs with $z \le 0.15$, for which the magnitude gap was spectroscopically confirmed to be $\Delta m_{12} \ge 2$ and with spectroscopic compeleteness larger than 65\% in SDSS-DR16. The aim of this work is to test the large-scale environment surrounding FGs and, for this reason, we downloaded all the spectroscopic data available in the SDSS DR16 within a 100 Mpc radius. Our results can be summarised as follows: \begin{itemize} \item The majority of FGs in our sample is found close to the centre of filaments, with a mean distance of 3.7 $\pm$ 1.1 $R_{200}$ (or 3.0 $\pm$ 0.8 Mpc). \item At the same time, all our FGs are found far from nodes, with a mean distance of 19.3 $\pm$ 3.6 $R_{200}$ (or 16.8 $\pm$ 2.6 Mpc). \item FGs with the largest magnitude gap ($\Delta m_{12} > 2.5$) are small and not massive ($R_{200} < 0.7$ Mpc and $L_X < 10^{43}$ erg s$^{-1}$). \item FGs with the largest magnitude gaps ($\Delta m_{12} > 2.5$) are found in low-density environments. \item Only FGs with small magnitude gaps ($\Delta m_{12} < 2.5$) can be candidate to be transitional systems (e.g. systems that can become non-fossil in the near future). In fact, some of them are found in dense environments and new mergers can not be excluded. \item The galaxy over density at large scale ($r > 20$ Mpc) varies around the zero value (e.g. there is no over density at such scales). \item FGs at higher redshift ($z > 0.1$) have a higher probability of suffering other major mergers, since they are found in denser environment than low-redshift FGs. \end{itemize} Our interpretation of these results is that FGs are usually found in a peculiar position with respect to the cosmic web: in fact, they seem to be located close to filaments, whereas galaxy groups and clusters are expected to be found close to nodes. The smaller FGs could be the final end product of group evolution and we do not expect them to evolve anymore. On the other hand, massive FGs at $z>0.1$ could still be evolving, since they are found in denser environment, that we interpreted as having a higher probability to suffer other major mergers, as it is also expected from numerical simulations. Finally, we confirmed that the cosmic web seems to be homogeneous at scales larger than 20 Mpc. We now plan to apply the same techniques presented in this paper to a larger sample of clusters and groups, spanning the $\Delta m_{12}$ range between $0 \le \Delta m_{12} < 2$, that is complementary to the sample analysed in this publication. In this way, as we already did e.g. in \citet{Zarattini2015}, we will look for dependencies between the position of groups/clusters in the cosmic web and their magnitude gaps. \section*{Acknowledgements} We thank the anonymous referee for her/his comments that helps in clarifying the paper and in particular the discussion of the results. SZ is supported by Padova University grant Fondo Dipartimenti di Eccellenza ARPE 1983/2019. JALA was supported by the Spanish Ministerio de Ciencia e Innovaci\'on by the grant PID2020-119342GB-I00. RC acknowledges financial support from the Agencia Estatal de Investigaci\'on del Ministerio de Ciencia e Innovaci\'on (AEI-MCINN) under grant ``La evoluci\'on de los c\'umulos de galaxias desde el amanecer hasta el mediod\'ia cГіsmico'' with reference PID2019-105776GB-I00/DOI:10.13039/501100011033. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrof\'isica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut f\"ur Astrophysik Potsdam (AIP), Max-Planck-Institut f\"ur Astronomie (MPIA Heidelberg), Max-Planck-Institut f\"ur Astrophysik (MPA Garching), Max-Planck-Institut f\"ur Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observat\'ario Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Aut\'onoma de M\'exico, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University. \bibliography{bibliografia} \appendix \section{Plots for visual inspection} \label{appendix} In this appendix we present plots that are useful for having an at-a-glance view of the large scale environment of each FG in our sample. In the following, we will discuss some features of specific FGs that can be deduced from the plots themselves. These are mainly qualitative comments. \subsection*{DMM 2018 IV} This FG is found at $z=0.0796$ and SDSS data only map approximately 50\% of the 100 Mpc radius. In particular, the right side (e.g. R.A. $>$ R.A.$_{{\rm FG}}$) is almost complete, whereas very few data are found in the left side. The FG is detected also by the FoF algorithm and we can estimate a mass of $2.7\times 10^{13} M_\odot$ for it. Interestingly, various groups are found along the closest filament, plus there is a filament on the right that is not perfectly followed by the FoF groups, although the filament is not completely identified. We suggest that this differences could be due to the incompleteness of the data and that DMM 2018 IV could be close to a node of the cosmic web, although not identified in the \citet{Chen2016} catalogue. \subsection*{FGS03} Also for FGS03 the completeness of the 100 Mpc coverage is not 100\%. There is a quadrant, the bottom-right one, that is well mapped and the remaining is less covered in SDSS. The group is identified in the FoF algorithm and we can estimate the mass of $1.9\times 10^{13} M_\odot$. FGS03 seems to be found within a filament. \subsection*{SDSSJ0906} The coverage of SDSSJ0906 is larger than 50\% and it is mainly focused on the upper part (e.g. Dec $>$ Dec$_{{\rm FG}}$). Here the density of galaxies is lower since the redshift is $z=0.1359$, one of the highest of the sample. However, it is interesting to note that the filaments found on the top-right side of SDSSJ0906 are well followed by the groups and clusters found by the FoF algorithm. This FG is one of the closest to a node. \subsection*{Abell 1068} This FG is approximately at the same redshift as the previous one, so similar considerations are valid in terms of number of SDSS galaxies in the region. However, in this case the 100\% Mpc are fully covered by the data. The system is found to be in a sort of a void, although it is not in the middle but closer to one of the walls. Many galaxies are found between the FG and this wall, but they are not virialised for the FoF algorithm. This can be interpreted as the presence of another filament in the region, but this is a tentative interpretation and more data are needed to confirm our hypothesis. \subsection*{SDSSJ1045} This FG in the one at the highest redshift in our sample, so again the density of SDSS galaxies is not very high. The coverage of the 100 Mpc is not complete, but it is complete the one of 50 Mpc, that we used for some of our test along the paper. The FoF algorithm found a small amount of group/clusters, probably due to the low density of point, that reflects in a larger mean distance between point (that is, the linking length of the algorithm). This is why we chose an adapting linking length: in fact, using for example a shorter linking length of 0.5 Mpc nothing was found in this region. Probably, a linking length of 2 Mpc could be better for systems with $0.15 < z < 0.2$, but since this is the only FG in our sample with $z>0.15$ and by a very small amount (in fact, it is at $z=0.154$) we prefer to maintain only three different values for the linking length. \subsection*{RXJ1119} The 100 Mpc radius of this FG is fully covered and there are a lot of galaxies in the field, due to the low redshift of the FG ($z=0.06$). In this case, the effectiveness of the FoF algorithm is very high and a lot of group/systems are found in comparison with other FGs. The identified systems are in good agreement with the position of nodes and filaments from \citet{Chen2016}. \subsection*{BLOXJ1230} For this FG ($z=0.12$), the FoF algorithm is finding few groups/clusters. Indeed, the density of galaxies is quite low, despite a full coverage of the 100 Mpc radius. Most of the FoF detections are on the left side, in a position where a node and 3 or 4 filaments are present. There is still a rather good agreemen between the FoF results and the \citet{Chen2016} catalogue. \subsection*{XMMXCSJ123338} For this FG the coverage of the 100 Mpc radius is complete. There are many visible filamnets, expecially in the bottom-left part of the plot, decently mapped also with the FoF algorithm. However, it is interesting to note that few groups/clusters are found within 50 Mpc from the FG, despite the presence of 6 nodes in the same region, according to \citet{Chen2016}. \subsection*{FG12} The sky coverage for FG12 is approximately 50\%: the upper part is well mapped, whereas poor coverage is found in the lower part of the plot. In this case, the system is also identified from the FoF algorithm, with an estimated mass of $1.1\times 10^{15} M_\odot$. FG12 is massive and it is found along a filament and very close to a double node. The filamentary stucture is well reproduced also by the FoF algorithm. However, only not-virialised galaxies are found in the closest node. On the other hand, other quite massive systems are visible just below our FG, so the area seem to be crowded. \subsection*{RXJ1331} The sky coverage of RXJ1331 is almost complete on the entire 100 Mpc radius, only a small are in the bottom of the plot is missing. Galaxies in the central region are recognised as in a structure, but they are not virialised and so we can not say that RXJ1331 is found by the FoF algorithm. For this reason, we are not able to give an estimation of its mass. This is again a region characterised by a large amount of groups/clusters, especially in the bottom-left part, where a large number of systems are found in a region connecting two nodes. By looking at the FoF results, we also expect a node to be found at approximately R.A. = 188 and Dec = 6, although it is not present in the \citet{Chen2016} catalogue. \subsection*{XMMXCSJ134825} Another relatively-high redshift system, with the 100 Mpc radius fully mapped but with low galaxy density. Something is detected in the centre, but all those galaxies are found to be not virialised, thus we are not able to measure the mass of this FG using the FoF algorithm. Few virialised objects are found, as for the other high-redshift (e.g. $z > 0.1$) systems of our sample. \subsection*{FGS20} In the case of FGS20, the full 100 Mpc area is covered in SDSS and our target FG is also found by the FoF algorithm. We thus estimate for it a mass of $2.3\times 10^{14} M_\odot$ . There are 6 nodes of the cosmic web within 50 Mpc radius, according to \citet{Chen2016}, but in this case the agreement between this catalogue and our FoF algorithm is not very good. However, FGS20 is found embedded in a void, closed by filament at all sides, but very close to one of these walls. \subsection*{RXJ1416} This is another high-redshift FG for our sample ($z=0.137)$, for which few galaxies are available and the FoF algorithm struggles to find objects, as we already explained for SDSSJ0906. However, RXJ1416 is found by the algorithm and we can thus estimate a mass of $2.7\times 10^{14} M_\odot$ for it. This FG is found very close to a filament, but the closest node is at more than 80 Mpc. However, we can not exclude that a node could be found at the position of the FG, since at least four filaments, some of the incomplete, seem to converge to its position. Moreover, we expect a node to be found at 50 Mpc on the left, where a concentration of galaxies are detected as four different systems by the FoF. Also, the absence of nodes on the right and top parts of the plot could be a hint of an incomplete detection in \citet{Chen2016} in this region, as well as of a not-sufficient coverage by SDSS data. \subsection*{XMMXCSJ141657} The sky coverage for this FG is complete out to 100 Mpc. The target FG is found by the FoF algorithm, but as a not-virialised structure. We are thus unable to estimate its mass with this method. A filament is close to our FG, but it seems to be truncated and nothing (filaments or nodes) are found in the central bottom part of the plot. We thus speculate that there could be another undetected filament/node in the region connecting our FGs with the systems found at approximately R.A. = 213 and Dec = 12. We also speculate that a node could be found close to this FG, due to the apparent convergence of various filaments, some of which not detected in \citet{Chen2016}, like the one that the FoF algorithm seems to find on the left side of the target, almost horizontal. \subsection*{RXJ1552} This is another system almost at our redshift limit, with full coverage of the 100 Mpc radius. Some galaxies are found to be part of a structure at the FG position, but they seem to be not virialised, so we are not able to estimate a mass for our FG in this way. Filaments and nodes are mostly found in the lower part of the plot, in good agreement with the FoF detections. The majority of the mass in the central 50 Mpc seems to be destined to the node that is found at the bottom-right of the FG. \subsection*{AWM4} This is the lowest redshift FG in our sample and its redshift is not compatible with the \citet{Chen2016} catalogue. For this reason, we did not highlight filaments and nodes in this plot. However, the FoF algorithm seems to find a series of filaments converging to a point that is at the bottom of our FG. Moreover, there are other two possible nodes in the top-right and bottom-left parts of the plot, giving a sort of s-shape to the galaxies detected with the FoF algorithm. The same algorithm also found a group at the position of our FG, so we can estimate for it a mass of $1.5\times 10^{14} M_\odot$. It is worth noting that the data for this systems only cover $\sim50$ Mpc radius, nothing is found in the SDSS beyond this limit. \newpage \section{The large-scale structure of FGS28} \label{appendixB} In Fig. \ref{fig:FGS28} we present the large scale structure around FGS28. As we mentioned in Sect. \ref{sec:FGS28}, we did not include this FG into our sample for different reasons. However, since we mentioned the supercluster and filament that are found very close to FGS28 in both apparent position and redshift, we include the figure for sake of clarity. \vspace{2cm} \noindent\begin{minipage}{\textwidth} \captionof{figure}{The large scale structure around FGS28. Galaxies with SDSS spectroscopy within $\pm 1500$ km s$^{-1}$ from FGS28 redshift are represented in black. Moreover, in the bottom-left panel, the large red, green, and blue ellipses represent 5, 50, and 100 Mpc respectively (as in Fig. \ref{fig:lss-appendix}). The small violet, brown, green, and blue ellipses are the four clusters that are found close to FGS28. In the top-right panel, a zoom can be seen, were the color code is the same with the exception of the red ellipse, that now simply identify the position of FGS28. In this panel, the legend associate each coloured ellipse to a specific clusters. It can be seen that FGS28 is identified by a single galaxy with spectroscopic redshift in SDSS, supporting the idea that this is a isolated galaxy found in the surroundings of a large supercluster.} \centering \includegraphics[width=0.99\hsize]{images/FGS28_side.png} \label{fig:FGS28} \end{minipage}

Title: Transitions and Origin of the Type-B Quasi-Periodic Oscillation in the Black Hole X-ray Binary MAXI~ J1348--630

Abstract: The fast transitions between different types of quasi-periodic oscillations (QPOs) are generally observed in black hole transient sources (BHTs). We present a detailed study on the timing and spectral properties of the transitions of type-B QPOs in MAXI~J1348--630, observed by \emph{Insight}-HXMT. The fractional rms variability--energy relationship and energy spectra reveal that type-B QPOs probably originate from jet precession. Compared to weak power-law dominated power spectrum, when type-B QPO is present, the corresponding energy spectrum shows an increase in Comptonization component and the need for {\tt\string xillverCp} component, and a slight increase of height of the corona when using {\tt\string relxilllp} model. Therefore, we suggest that a coupled inner disk-jet region is responsible for the observed type-B QPOs transitions. The time scale for the appearance/disappearance of type-B QPOs is either long or short (seconds), which may indicate an instability of disk-jet structure. For these phenomena, we give the hypothesis that the Bardeen-Petterson effect causes disk-jet structure to align with BH spin axis, or that the disappearance of small-scale jets bound by the magnetic flux tubes lead to the disappearance of type-B QPOs. We observed three events regarding the B/C transitions, one of which occurred in a short time from $\sim 9.2$ Hz (C) to $\sim 4.8$ Hz (B). The energy spectral analysis for the other two transitions shows that when type-C QPO is present, the Comptonization flux is higher, the spectrum is harder and the inner radius of disk changes insignificantly. We suggest that type-C QPOs probably originate from relatively stronger jets or corona.

https://export.arxiv.org/pdf/2208.07066

command. \documentclass[twocolumn]{aastex631} \usepackage{graphicx,multirow,hyperref,url,color,xspace} \usepackage{soul} \usepackage{amsmath,amssymb} \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \shorttitle{} \shortauthors{L.H.X et al.} \usepackage{booktabs} \begin{document} \title{Transitions and Origin of the Type-B Quasi-Periodic Oscillation in the Black Hole X-ray Binary MAXI~ J1348--630} \author{H.X.Liu*} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \email{liuhexin@ihep.ac.cn} \correspondingauthor{Y. Huang} \email{huangyue@ihep.ac.cn} \correspondingauthor{Q.C.Bu} \email{bu@astro.uni-tuebingen.de} \author{Y. Huang*} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{Q.C.Bu*} \affiliation{Institut f\"ur Astronomie und Astrophysik, Kepler Center for Astro and Particle Physics, Eberhard Karls Universit\"at, Sand 1, 72076 T\"ubingen, Germany} \author{W. Yu} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{Z.X.Yang} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{L. Zhang} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \author{L.D.Kong} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{G.C.XIAO} \affiliation{Purple Mountain Observatory, Chinese Academy of Sciences,Nanjing 210034, People's Republic of China} \author{J.L.Qu} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{S.N.Zhang} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{S.Zhang} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{L.M.Song} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{S.M.JIA} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{X.MA} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{L.TAO} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{M.Y.GE} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{Q.Z.LIU} \affiliation{Purple Mountain Observatory, Chinese Academy of Sciences,Nanjing 210034, People's Republic of China} \author{J.Z.YAN} \affiliation{Purple Mountain Observatory, Chinese Academy of Sciences,Nanjing 210034, People's Republic of China} \author{R.C.MA} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{X.Q.REN} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{D.K.ZHOU} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{T.M.LI} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{B.Y.WU} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{Y.C.XU} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{Y.F.DU} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{Y.C.FU} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{Y.X.XIAO} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \author{G.Q. Ding} \affiliation{Xinjiang Astronomical Observatory, Chinese Academy of Sciences,150, Science 1-Street, Urumqi, Xinjiang 830011, China} \author{X.X. YU} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, People's Republic of China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People's Republic of China} \begin{abstract} The fast transitions between different types of quasi-periodic oscillations (QPOs) are generally observed in black hole transient sources (BHTs). We present a detailed study on the timing and spectral properties of the transitions of type-B QPOs in MAXI~J1348--630, observed by \emph{Insight}-HXMT. The fractional rms variability--energy relationship and energy spectra reveal that type-B QPOs probably originate from jet precession. Compared to weak power-law dominated power spectrum, when type-B QPO is present, the corresponding energy spectrum shows an increase in Comptonization component and the need for {\tt\string xillverCp} component, and a slight increase of height of the corona when using {\tt\string relxilllp} model. Therefore, we suggest that a coupled inner disk-jet region is responsible for the observed type-B QPOs transitions. The time scale for the appearance/disappearance of type-B QPOs is either long or short (seconds), which may indicate an instability of disk-jet structure. For these phenomena, we give the hypothesis that the Bardeen-Petterson effect causes disk-jet structure to align with BH spin axis, or that the disappearance of small-scale jets bound by the magnetic flux tubes lead to the disappearance of type-B QPOs. We observed three events regarding the B/C transitions, one of which occurred in a short time from $\sim 9.2$ Hz (C) to $\sim 4.8$ Hz (B). The energy spectral analysis for the other two transitions shows that when type-C QPO is present, the Comptonization flux is higher, the spectrum is harder and the inner radius of disk changes insignificantly. We suggest that type-C QPOs probably originate from relatively stronger jets or corona. \end{abstract} \keywords{Black hole binaries, X-Ray, QPO, jet} \section{Introduction} \label{sec:intro} Most of the black hole X-ray binaries (BHXBs) in the Galaxy are discovered as transients whose spectral and timing properties change with time. During a typical outburst, a BHXB goes through a transition from the low/hard (LHS) to the high/soft (HSS) state through relatively short--lived intermediate states, which are known as hard intermediate state (HIMS) and soft intermediate state (SIMS) \citep{2010LNP...794...53B,2006ARA&A..44...49R,2009MNRAS.400.1603M}. The rapid flux variations found in the X-ray light curves of BHXBs are generally considered to be a feature of the inner accretion flow in the vicinity of black holes. Low frequency Quasi Periodic Oscillations (LFQPOs), characterized by discrete peaks in the Power Density Spectra (PDS), are commonly observed in the 0.01--30 Hz range in BHXBs. Based on the properties of PDS in several typical BHXBs, XTE J1550--564, GX 339--4, and XTE~J1859+226 \citep{1999ApJ...526L..33W,2001ApJS..132..377H,2011MNRAS.418.2292M,2005AIPC..797..225C}, LFQPOs are classified into three types, namely type-A, B, and C. Type-A QPOs are rarely detected during the HSS, and they are characterized by a weak and broad peak in the PDS. During the transition state---SIMS, type-B QPOs are usually observed, and typically have a rather high amplitude in the PDS of a power law shape with a centroid frequency around 4--6 Hz. Generally seen in the LHS and HIMS, type-C QPOs are characterized as a strong narrow peak with broadband noise in the PDS, which may be accompanied by harmonics and possible sub-harmonics \citep{2004A&A...426..587C}. Through timing analysis, rapid transitions between two types of QPOs (type-B QPOs transit to other types or disappear), have been observed in a few BHXBs. As reported by \citet{1991ApJ...383..784M}, \citet{2003A&A...412..235N} and \citet{2011MNRAS.418.2292M}, the transitions of type-B QPOs are often observed in GX~339--4. In XTE~J1550--564 \citep{2001ApJS..132..377H,2016ApJ...823...67S}, transition events related to type-B/A QPOs variations were observed along with the change of energy spectra. Similar rapid transitions involving type-B QPOs and spectral variations in XTE~J1859+226 were reported by \citet{2004A&A...426..587C} and \citet{2013ApJ...775...28S}. In XTE~J1817--330, \citet{2012A&A...541A...6S} found that a type-B QPO $\sim$ 6 Hz QPO switches to a type-A QPO with a decrease in flux. Among all the outbursts of H1743-322 observed by Rossi X-ray Timing Explorer (\emph{RXTE}), not only were type B/A QPOs transitions observed, \citep{2005ApJ...623..383H} but also three B/C transitions \citep{2021ApJ...911..127S}. For MAXI~J1535--571, \citet{2018ApJ...866..122H} found type-B QPOs $\sim$ 10 Hz in correspondence with flux changes in X--ray, and \citet{2018ApJ...865L..15S} reported a weak type A/B QPO transition with no sudden flux change. The type C/B QPO transitions were also found in GRS 1915+105 \citep{2008MNRAS.383.1089S} and MAXI J1820+070 \citep{2020ApJ...891L..29H}. Although LFQPOs have been discovered for decades, there is still no consensus for their physical explanations. Some promising models have been proposed based on observational data, which can be divided into two categories: the geometric effects of the hot inner/outer flows and the instability of accretion flows. The relativistic precession model was firstly proposed by \citet{1998ApJ...492L..59S} to explain the origin and the behaviour of the LFQPO and kHz QPOs in neutron star X-ray binaries. This model has been continuously modified \citep{2009MNRAS.397L.101I,2015MNRAS.448.1298P,2017MNRAS.464.2643V,2021NatAs...5...94M} later and applied to BHXBs. In these improved models, the QPOs are from Lense-Thirring (L-T) precession of a hot inner flow (a radially extended corona or jet). These observational evidences support the geometric explanations for LFQPOs \citep{2006ApJ...642..420S,2009MNRAS.397L.101I,2016MNRAS.461.1967I}. In the other kind of model, the QPOs come from intrinsic instabilities in the accretion flow, for example, magneto-acoustic wave propagation \citep{2004ApJ...612..988T,2010MNRAS.404..738C}, or the accretion-ejection instability (AEI)---a product of instability accretion model \citep{1999A&A...349.1003T}. It is thought that the different types of QPOs observed in the BHXBs systems are associated with different physical mechanisms in the inner region of the accretion disk. However, the inner disk variation could not be conclusively demonstrated previously, which may be due to the energy band of \emph{RXTE}. Therefore, the transition mechanism between different types of QPOs has not been clear. Using the data of \emph{Insight}-HXMT, with a wider energy band (1 keV--250 keV), it is now possible to obtain more information and to better understand the changes that occur in the inner disk regions as well as in the high-energy radiation regions where the LFQPOs and their transitions are observed. MAXI J1348--630 was discovered as a X-ray binary by the X-ray telescope \textit{MAXI}/GSC on January 26, 2019 \citep{2019ATel12425....1Y}. \citet{2020ApJ...897....3J} estimated the mass of the BH to be $9.1^{+1.6}_{-1.2}$ $M_{\sun}$. They also found that the viscous timescale in this outburst is $\sim$3.5 days. The distance of the source was estimated as $\sim$2.2 kpc by \citet{2020MNRAS.tmpL.236C}. \citet{2020ApJ...899L..20T} proposed that the black hole is more massive if the disk is inclined and the black hole is spinning. These results suggest that MAXI~J1348--630 may host a relatively massive black hole among the known BHXBs in our Galaxy. \citet{2019ATel12497....1C} reported that a radio source is detected at a position consistent with the location of MAXI~J1348--630, displaying, on 2019-02-09, a preliminary flux density of 520.3 $\pm$ 5.0 mJy. The QPOs transitions between different typical QPOs were also observed in the 2019 outburst of MAXI~J1348--630. For this outburst, \citet{2021MNRAS.505.3823Z} have analyzed four fast appearance/disappearance of type-B QPO using the data from \emph{NICER}, in the energy band of 0.5--10 keV. \emph{Insight}-HXMT has observed several transitions involving the type-B QPOs with its three main detectors: LE, ME, and HE, covering a broad-band energy (1--250 keV). Using the data from \emph{Insight}-HXMT, the detailed timing and spectral analyses for the transitions given in this work may provide more information to understand the underlying physical processes. The paper is organized as follows: the data reduction is introduced in Section \ref{data}, and we present the timing and spectral results in Section \ref{Results}; in Section \ref{discussion}, we present the summary and discussions; finally, we present conclusions in Section \ref{conclusion}. \section{Observations and Data Reduction} \label{data} \emph{Insight}-HXMT, China's first X-ray astronomical satellite which was successfully launched on June 15, 2017, carries three sets of main instruments (LE/ME/HE, short for the Low/Medium/High Energy X-ray Telescope respectively). More details about \emph{Insight}-HXMT can be found in \citet{2020SCPMA..63x9502Z}, \citet{2020SCPMA..63x9505C} (LE), \citet{2020SCPMA..63x9504C} (ME) and \citet{2020SCPMA..63x9503L} (HE). MAXI J1348--630 was observed regularly with \emph{Insight}-HXMT from January 27 2019 to May 15. We analyzed the observations in the SIMS around the peak of this outburst (see the Figure \ref{fig:evolution}) in this work. The \emph{Insight}-HXMT Data Analysis software (HXMTDAS, v2.04) is used to process and filter the data. Since the background and some particle events do not have significant influences on the PDS, the screening criteria of good time-intervals (GTIs) in timing analysis can be loosened. Therefore, the light curves to produce dynamic power spectra (DPS) are generated with the following criteria: the offset for the point position $ < 0.06^\circ$, the elevation angle (ELV) $ > 10^\circ$. The selection of GTIs is more demanding when extracting the energy spectra: ELV $>10^\circ$; the geometric cutoff rigidity (COR) $>6^\circ$; the offset for the point position $<0.04^\circ$; data are used at least 300 s before and after the South Atlantic Anomaly (SAA) passage. The backgrounds are estimated with the official tools: LEBKGMAP \citep{2020JHEAp..27...24L}, MEBKGMAP \citep{2020JHEAp..27...44G} (ME)and HEBKGMAP \citet{2020JHEAp..27...14L} in version 2.0.6. The energy bands, adopted for energy spectral analysis are 1--1.8, 2--10 keV (LE), 10--35 keV (ME) and 35--100 keV (HE) in this work. The XSPEC software package v12.11 \citep{1996ASPC..101...17A} is used to fit the spectra. Uncertainty estimated for each spectral parameter is for 90\% confidence level, and a systematic error of 1\% is added. Detailed discussions of its calibrations are given in \citet{2020JHEAp..27...64L}. \section{Data Analysis and Results} \label{Results} \subsection{Timing Analysis and Results} \label{Timing Results} Using the archived data of \emph{Insight}-HXMT observations of MAXI~J1348--630, we plot the evolution of its light curves and hardness in Figure \ref{fig:evolution}, with each point in the diagram representing an observation. The top panel represents the count rate of LE (1--10 keV), while the bottom panel shows the hardness ratio that is defined as count rates ratio between 4--10 keV and 2--4 keV. According to the the spectral analysis, \citet{2022arXiv220111919Z} have defined four states for this outburst. Huang et al. (2022) (In prepation) studied in detail the timing properties in the 2019 outburst. They found that the observational properties are consistent with BHXBs and the type-C QPOs frequency varied between 0.26 and 7.31 Hz with decreasing hardness. The appearance of the type-B QPO indicates the source was in its SIMS, as shown in the shaded area of Figure \ref{fig:evolution}. In this work, we focus on studying the transitions of QPOs during the SIMS of this outburst, while the detailed information are listed in Table \ref{QPO}. For our timing analysis, we compute the power density spectra (PDS) with a time resolution of 1/128 s using 16 s data segments (corresponding to a Nyquist frequency of 64 Hz) for the observation listed in Table \ref{QPO}. In order to search for the QPOs, the PDS is normalized according to Leahy normalization \citep{1983ApJ...272..256L}. In Figure \ref{fig:Epoch}, we show the light curves from three energy bands (LE:1--10keV; ME:10--35 keV; HE: 35--100 keV), hardness and DPS with a time interval of 16 s for part of SIMS \uppercase\expandafter{\romannumeral1} (from MJD 58523.8 to MJD 58527.8) and SIMS \uppercase\expandafter{\romannumeral2} (MJD 58539.1--MJD 58540.2). The results show that in the bottom panel of Figure \ref{fig:Epoch}, type-B QPOs often disappear or transit into other types of QPOs when the counts rate change. Based on the occurrences of QPOs and variations of rate shown in Figure \ref{fig:Epoch}, we divide the whole SIMS into 7 epochs and mark them by different colors, which are also referred in the last column of Table \ref{QPO}. Epoch 1 is the period from MJD 58522.60 to MJD 58524.25, during which no intermittent QPO and no significant changes in flux are observed. Therefore it is not shown in Figure \ref{fig:Epoch}. During Epoch 2 (MJD 58524.27--58524.41), a fast transition between type-B and type-C QPOs is detected. The fast re-occurrence of type-B QPOs on short time-scale is found during Epoch 3 (MJD 58524.77--58524.94). For Epoch 4 (MJD 58525.06--58525.87) and Epoch 5 (MJD 58526.19--58527.10),there are long time transition intervals between the appearance and disappearance of type-B QPOs. To clarify, the corresponding DPS are plotted in Figures \ref{fig:tr-b} and \ref{fig:tr-c}. In Epoch 6 and Epoch 7, the transitions between type-B and type-C QPOs are observed, accompanied by significant changes in the X-ray flux (also see Figure \ref{fig:tr-c}). Based on the DPS, the average PDS is produced according to Miyamoto normalization \citep{1991ApJ...383..784M} in units of (rms/mean)$^{2}$Hz$^{-1}$ after subtracting the Possion noise, in order to get the characteristics of QPOs for the different ObsIDs. % The PDSs are fitted by a model consisting of multiple Lorentzians \citep{2002ApJ...572..392B}. The background contribution to the QPO fractional rms are corrected according to \citep{1990A&A...227L..33B,2015ApJ...799....2B} \begin{equation} {\rm rms} = \sqrt{R}\times \frac{(S+B)}{S}, \label{eq:quadratic} \end{equation} here \emph{R} is the power calculated with the integration of the QPO Lorentzian function over the frequency range 0.01--64 Hz, \emph{S} is the source count rate, while \emph{B} is the background count rate. We fit the PDS of all QPOs and the results are given in Table \ref{QPO}. The frequency of type-B QPOs varied within a narrow range of 4.3--4.7 Hz and 4.0--4.2 Hz in SIMS \uppercase\expandafter{\romannumeral1} (MJD 58522.7--58527.8) and SIMS \uppercase\expandafter{\romannumeral2} (MJD 58539.1--58540.2), respectively. In order to investigate the energy dependence of the type-B QPOs, type-B QPOs observations are divided into 12 energy bands from 1--100 keV: 1--2.5, 2.5--3.5, 4.5--5.5, 5.5--7.0, 7.0--10.0, 10.0--14.0, 14.0--18.0 18.0--30.0, 30.0--40.0, 40.0--50.0 and 50.0--100.0 keV. The corresponding centroid frequencies and the rms of type-B QPOs from all epochs are plotted in Figure \ref{fig:qpo}. The results show that rms increased with energy from 1 keV and reached a plateau above 10 keV. Meanwhile, the frequency of QPO remains unchanged with the energy. \subsubsection{The temporary characteristics of Type-B QPOs} In this work, we perform spectral analysis for these observations using two different models mentioned above in the energy band 1--100 keV. When source was in its SIMS, the QPOs frequently appeared or disappeared as shown in Figure \ref{fig:Epoch}. When the source entered Epoch 3 from Epoch 2, the count rates of ME, HE and the hardness decrease abruptly, while the DPS show that there are multiple rapid transitions between no QPO (hereafter NO-Q) and type-B QPOs (hereafter B-Q, also see in Table \ref{QPO}). In order to show the temporary nature of the type-B QPOs clearly, the DPS and PDS of partial Epoch 3 observations is shown in the top panel of Figure \ref{fig:tr-b}. We can see that type-B QPOs disappeared suddenly on a shorter time scale (about ten seconds), while no significant changes are observed in the flux. The PDS of NO-Q is characterized by a weak power-law noise. Unfortunately, the LE data is not available during this period, but previous studies using \emph{RXTE} and \emph{NICER}'s data have shown that the disappearance of type-B QPOs are often accompanied by a flux decrease in low energy band \citep{2013ApJ...775...28S,2021ApJ...911..127S,2021MNRAS.505.3823Z}. Type-B QPOs can also be present or disappear for a few hours. In Epoch 1, type-B QPO are observed continuously for more than 3 days. In Epochs 4 and 5, QPOs are more likely to disappear completely at the relatively lower hardness and flux of the high energy bands (i.e. ME, HE), as shown in Figure \ref{fig:Epoch} (MJD 58525.1-58525.7 and MJD 58526-58526.8), which is consistent with the results of \citet{2021MNRAS.505.3823Z}. In the bottom panels of Figure \ref{fig:tr-b}, we show examples of DPS and PDS for a long-time scale transition between B-Q and NO-Q and find that the PDS of NO-Q show a weak power-law noise at low frequencies. We will discuss the possible physical causes of this transition in combination with information on time variation and energy spectrum in section \ref{Spectral Results}. \subsubsection{Transitions between type-B and type-C QPOs} Several obvious transitions between type-B and type-C QPOs are observed in this outburst, as shown in Figure \ref{fig:tr-c}. The first one is observed on $\sim$ MJD 58524.4 (in Epoch 2), which occurred in a short period of $\sim$ 2000 s. During the first $\sim$ 1800 s, the averaged PDS show appearances of a QPO with two peaks of $\sim$ 2.8 Hz (rms $\sim$ 8.6\%) and $\sim$ 9.3 Hz (rms $\sim$ 10.3\%). Depending on the characteristics of type-C QPOs ($\sim$ 7 Hz with small rms $\leq$ 10\% in Huang et al.(In preparation)) at the HIMS of this source, it can be inferred that the QPO with frequency of $\sim$ 9.3 Hz is type-C. A type-B QPO with central frequency of $\sim$ 4.8 Hz (rms $\sim$ 8.1\%) is detected in the last $\sim 200$ seconds. It is likely that a type-C QPO appeared subsequently, though this is hard to confirm because of its short duration. This transition occurred in a particularly short period of time as we can see from the DPS shown in the top panel of Figure \ref{fig:tr-c}. The second transition (denoted as Epoch 6) is accompanied by a significant decrease in the count rate of LE and increase in the count rates of ME and HE (see the Figure \ref{fig:Epoch} and the middle panel of Figure \ref{fig:tr-c}), when a significant evolution of this type-C QPO is observed. The middle panel of Figure \ref{fig:tr-c} demonstrates that this QPO has two peaks and evolves over time; in particular, the lower frequency peak evolved from $\sim$ 2.9 Hz to $\sim$ 3.2 with a slight change in rms from 8.1\% to 8.2\%, while the higher peak evolved from 7.6 Hz to 10.0 Hz (rms $\sim$ 10.5\%-11.2\%). During the SIMS \uppercase\expandafter{\romannumeral2}, the transition of a type-B at frequency of $\sim$ 4.3 Hz to a type-C QPO at frequency of $\sim$ 7.8 Hz is detected (denoted as Epoch 7), accompanied by a smaller low-energy rate (1--10 keV) and a larger high-energy rate ($\geq$ 10 keV), as shown in the bottom panel of Figure \ref{fig:tr-c}. The last two type-C QPOs also exhibited energy-dependent properties, i.e., the PDS exhibit different shapes for different energy bands. \subsection{Spectral Analysis and Results} \label{Spectral Results} Due to the limitation of LE GTIs, only data from Epoch 4 to Epoch 7 are used for the energy spectrum study. In order to study the energy spectral characteristics regarding different types of QPOs, we first combine the energy spectra based on PDS (i.e., NO-QPO, type-B QPO, and type-C QPO) that are observed in Epochs 4, 5, 6, and 7, namely NO-Q, B-Q, and C-Q, to increase the signal-to-noise. Before fitting the spectra, we first compare the spectra between the segments of NO-Q (C-Q) and B-Q qualitatively. The energy spectrum ratio between the B-Q and NO-Q observations is shown in the top panel of Figure \ref{fig:spectra ratio}. The ratios between B-Q and NO-Q indicate differences at both low and high energy bands. At energies below 3 keV, the ratio approaches 1, which indicates that the energy spectrum shape has slightly changed. However, the ratio is more or less constant above 10 keV, and the flux of B-Q is significantly higher than that of NO-Q at energies above 3 keV. These results are consistent with the phenomenons that type-B QPOs appear in the high energy bands. The energy spectra ratio between the type-C QPOs and type-B QPOs is shown in the bottom panel of Figure \ref{fig:spectra ratio}. We find that even after the source re-entered the SIMS \uppercase\expandafter{\romannumeral2} for 12 days, the trend of the energy spectra ratio remains the same when the QPO transitions occurred. The ratio (C-Q/B-Q) is appreciably less than 1 at energies \textless 10 keV (maximum value $\sim$ 0.8 at 4--5 keV), which indicates that the spectral shapes have significantly changed at low energy bands. At energies $>$ 10 keV, the ratio increases with energy with a value $>1$, which indicates that the C-Q flux is larger at higher energies. These agree with the fact that the spectra of type-B QPOs are softer than those of type-C QPOs in a transition. To quantitatively study the evolution of spectral parameters, different models are applied to the observations in Epoch 5. As shown in Figure \ref{fig:example-spectra}, the first trial of fitting is with a model consisting of a multi-color disk component and a Compton component \citep{1996MNRAS.283..193Z, 1999MNRAS.309..561Z}, i.e., {\tt\string tbabs}$\times$({\tt\string diskbb+nthcomp}). The {\tt\string nthcomp} is the thermally comptonized continuum model that may arise from a hot corona associated with accretion disc in the BHTs. Therefore, during the fitting, we link the seed photon temperature ($T_{\rm bb}$) of the {\tt\string nthcomp} model with the inner disc temperature ($kT_{\rm in}$) of the $diskbb$ component. The goodness of fit ($\delta\chi^{2} > 2$) are mostly attributed to an obvious asymmetry of the broadened iron line. Accordingly, we add an extra {\tt\string gaussian} component to fit the spectra and the residuals are shown in the 4-5th panels of Figure \ref{fig:example-spectra}. However, a hint of a narrow iron line seems to be present in the residuals. Therefore, we add another non-relativistic reflection model {\tt\string xillverCp} with the inclination angle fixed at $36.5^{\circ}$, $\Gamma$ and $kT_{\rm e}$ linked to the parameters of {\tt\string nthcomp} and we calculate the significance using $F-{\rm test}$ in XPSEC. The best-fitting parameters of model 1, i.e, {\tt\string tbabs}$\times$({\tt\string diskbb+gaussian+nthcomp+xillverCp}), are shown in Tables \ref{B-nthcomp} and \ref{C-nthcomp}. Although the fitting is significantly improved, the centroid line energy of Gaussian is less than the typical 6.4--7.1 keV iron regime, and the profile is extremely wide (1--2 keV). The values are un-physical, which is similar to the result in Cygnus~X-1 reported by \citet{2014ApJ...780...78T}. The broad emission feature is normally seen in the reflection spectra. An accretion disk with a lamppost geometry corona (the relativistic reflection model), as model 2, {\tt\string tbabs}$\times$({\tt\string diskbb+relxilllp}) is introduced to fit the spectra. For the {\tt\string relxilllp} components of all spectra, an extreme spin value of 0.8 and an inclination angle of $36.5^\circ$ (given in \citealt{2022arXiv220101207J}) are chosen and fixed during spectral fittings. We show the best-fitting parameters in Tables \ref{B-relxilllp} and \ref{C-relxilllp}. In the {\tt\string relxilllp} model, the reflection fraction $R_{\rm f}$ is defined as the ratio of the incident photon intensity that illuminates the accretion disc to that observed directly, $\Gamma$ and $E_{\rm cut}$ denote the initial spectral index and its deviation energy from a simple power-law shape. Other parameters in {\tt\string relxilllp} provide information about the accretion disk: the inner radius of disk $R_{\rm in}$, ionization of the accretion disk log($\xi$) and iron abundance of the material $A_{\rm Fe}$. However, the iron abundance is very high and often pegs at its limit during our fitting, probably because the reflection comes from a high-density disk, which can be referred to \citet{2021MNRAS.508..475C} for specific studies. An example (NO-Q in Epoch 4) of the MCMC (Markov Chain Monte Carlo) analysis for this model is present in Figure \ref{fig:mcmc}. \subsubsection{The spectral results of B-Q and NO-Q} To study the recurrences of type-B QPOs, we compare the segment NO-Q and B-Q energy spectra in Epochs 4 and 5 (also see Table \ref{fig:Epoch}). By using model 1, {\tt\string tbabs}$\times$({\tt\string diskbb+gaussian+nthcomp+xillverCp}), to fit the spectra in Epoch 4 and Epoch 5, we find that the disk temperature remains almost the same at $kT_{\rm in} \sim$ 0.74--0.75 keV along with the power-law index at $\Gamma \sim$ 2.20; the change in the normalization of {\tt\string diskbb} is also negligible within the error range. However, compared to NO-Q, the thermal flux is lower and the Comptonization flux is higher in B-Q. Especially for Epoch 5, the thermal flux is reduced by 5.3\% and the Comptonization flux increases by 58.3\%, which coincides with the direct energy spectrum comparison in Figure \ref{fig:spectra ratio}. To investigate which spectral parameter is responsible for the spectral changes, we perform a simultaneous fit, using NO-Q and B-Q spectra following the method of \citet{2013ApJ...775...28S}. First we fix the adjacent NO-Q and B-Q spectral parameters together and free them one by one to obtain the corresponding residuals and the changes of the parameters. For each epoch, the values of the segment B-Q spectrum are set to the best-fit parameters of segment NO-Q spectrum and the $\chi^{2}/d.o.f$ can be seen in the Table \ref{compare-nthcomp}. For Epochs 4 and 5, we free the {\tt\string nthcomp}, {\tt\string Gaussian}, and {\tt\string diskbb} normalization parameter and find that the fitting are improved significantly. Therefore, it can be assumed that some changes occurred in the corona and disk at this time. In the same way as in model 1, the smallest set of parameters is listed in Table \ref{B-relxilllp} for Epoch 4 and Epoch 5. A relatively small variation in $N_{\rm diskbb}$ is observed in Epoch 4, whereas in Epoch 5 both $N_{\rm diskbb}$ and $kT_{\rm in}$ are found to be variable. Compared to the spectral results using the simple model, little change in photon index $\Gamma$ are observed for NO-Q/B-Q during this fitting. During the fitting of the four spectra, the corona height $h$ seems to be larger, while the change in $R_{\rm in}$ is insignificant $\sim 1 R_{\rm ISCO}$ (Radius of the innermost stable circular orbit). The inner disk radius can also be calculated from disk normalization parameter as outlined by \citet{1984PASJ...36..741M}, whereby $R_{\rm in}^{2} = f^{4}D_{10}^{2}(N_{\rm diskbb}/{\rm cos}\theta)$, where $f$ is the color correction factor \citep{1998PASJ...50..667K}, $D_{10}$ is the distance to the source in units of 10 kpc, and $\theta$ is the disk inclination. We adopt the best-fit $N_{\rm diskbb}$ values with their uncertainties from Table \ref{B-relxilllp}. Assuming a canonical value for color correction of $f=1.5$, a fixed disk inclination of $36.5^{\circ}$ and a fixed distance of 2.2 kpc, $R_{\rm in}$ is found to be $87.4\pm0.3$ km, $85.1^{+0.4}_{-0.8}$ km, for NO-Q and B-Q, respectively, in Epoch 4, and $91.3\pm0.1$ km, $81.5\pm0.3$ km in Epoch 5. The inner disk radius increases slightly in NO-Q if the assumption of $R_{\rm in}^{2} \propto N_{\rm diskbb}$ is considered. \subsubsection{The spectral results of B-Q and C-Q} Intuitively, it can be seen from the bottom panel of Figure \ref{fig:spectra ratio} that the disk component of B-Q is remarkably stronger compared to that of C-Q. In order to understand the change of accretion disk configurations in SIMS (B-Q) and HIMS (C-Q), we fit the energy spectra of Epoch 6 and Epoch 7 with models 1 and 2, respectively. From the best fitting results of the two models (see Table \ref{C-nthcomp} and Table \ref{C-relxilllp}), it can be inferred that for both B-Q and C-Q epochs, the parameters of the disk have changed significantly. For both model 1 and model 2, the C-Q spectra had a relatively lower disk temperature $kT_{\rm in}$ and a larger inner disk radius ($N_{\rm diskbb}$) compared to the B-Q spectral state, meanwhile $\Gamma$ is slightly smaller in C-Q, suggesting a harder spectra. Correspondingly, the thermal component flux is weaker and the Comptonized component is stronger in C-Q. For Epochs 6 and 7, after freeing $kT_{\rm in}$, $\Gamma$ and $N_{\rm diskbb}$, the residuals are improved significantly (\ref{compare-nthcomp}). These results suggest that the accretion disk moved inward when the source entered the SIMS from the HIMS, i.e, B-Q has a smaller inner radius than C-Q. This is consistent with the standard accretion disk model \citep{1997ApJ...489..865E}. Using model 2 (see Table \ref{C-relxilllp}), it is also suggested that the normalization of the {\tt\string diskbb} is larger at C-Q. Using the parameters in Table \ref{C-relxilllp}, $R_{\rm in}$ is found to be $77.1\pm0.7$ km, $83.7\pm0.9$ km, $82.9\pm0.6$ km, for B-Q, C-Q, and B-Q, respectively, in the SIMS \uppercase\expandafter{\romannumeral1}. After 12 days in the SIMS \uppercase\expandafter{\romannumeral2}, $R_{\rm in}$ is found to be $82.9\pm0.3$ km, $87.5\pm0.2$ km, $85.3\pm0.3$ km for B-Q, C-Q, and B-Q. However, $R_{\rm in}$ in {\tt\string relxilllp} of C-Q is found to be the same as that of B-Q: $\sim 1 R_{\rm ISCO}$ for Epochs 6 and 7. We also find that the height of the corona $h$ is slightly larger in B-Q. For both epochs (see Table \ref{compare-relxilllp}), after freeing the parameters $kT_{\rm in}$, the chi-squared value are improved significantly. However, for Epoch 7, all the parameters need to be free except for log($\xi$). Furthermore, an improvement of fitting for Epoch 6 is also present after freeing $h$, which indicates that the height of the corona indeed changed. From the above results, we conclude that after the source MAXI~J1348--630 entered the SIMS (B-Q), it may have returned to the HIMS (C-Q) again, or returned to the SIMS (SIMS \uppercase\expandafter{\romannumeral2}) after HSS, suggested by changes in PDS and energy spectrum. \section{Summary and Discussions} \label{discussion} In this paper, we focus on studying the timing and spectra properties of the QPO transitions observed in MAXI~J1348--630 using data from \emph{Insight}-HXMT during its 2019 outburst. The results can be summarized as follows: \begin{itemize} \item The type-B QPOs ($\sim$ 4.2 -- 4.7 Hz) in this source are recurrent, on timescales from seconds to hours; \item The variation amplitude (rms) of QPOs has a same relation with energy, i.e., rms increases with energy below 10 keV and remains constant above 10 keV to 100 keV. \item When type-B QPOs disappear \begin{itemize} \item the PDS show a weak power-law noise; \item the flux of the Comptonization component decreases, while the spectral index $\Gamma$ remains unchanged; \item the inner disk radius increases slightly when inferred from disk component ($N_{\rm diskbb}$); \item the corona height of type-B QPOs observation is larger than that of NO-Q when including an additional reflection model {\tt\string xillver} by need for the continuum fitting. \end{itemize} \item Three transitions are found between the type-C and type-B QPOs with the following results: \begin{itemize} \item a transient timescale as short as $\sim$ 10 seconds; \item the Comptonization flux is higher and the spectral index is smaller in C-Q; \item the inner radius of the accretion disk is moderately larger for C-Q as inferred from the disk component, while it remains constant $\sim 1 R_{\rm ISCO}$ using reflection model; \item the corona height is larger in B-Q using the reflection model. \end{itemize} \end{itemize} The above results suggest that the type-B and C QPOs are associated with different physical mechanisms within the inner region of the accretion disk. In the next sub-sections, we will discuss those results and possible physical mechanisms of the QPOs. \subsection{LFQPOs and Models} \label{LFQPOs} When the last type-C QPO at $\sim 7.35$ Hz in the HIMS disappeared, a subsequent type-B QPO detection means that MAXI~J1348--630 entered SIMS. The spectral fitting results show that throughout the SIMS, the inner radius of the disc was very close to the ISCO, which is consistent with the results of \citet{2022arXiv220111919Z}. The same results have been found in several other BHTs, such as H1743-322 \citep{2021ApJ...911..127S}, XTE~J1859+226 \citep{2013ApJ...775...28S}, etc. The model proposed for type-B QPOs is not as comprehensive as for type-C, but there is much evidence that they may have originated from a jet-like structure. The most convincing evidence is that type-B QPOs and jet ejections (or radio flares) are observed to occur close in time \citep{2009MNRAS.396.1370F,2019ApJ...883..198R,2020MNRAS.498.5772R,2020ApJ...891L..29H}. Recently, observations on MAXI~J1820+070 \citep{2020ApJ...891L..29H} have given strong evidence that there is a connection between the appearance of type-B QPOs and the launch of discrete jet ejections because the short time delay ($\sim$ 2-2.5 hr) between the transition and a strong radio flare. During the current outburst of MAXI~J1348--630, two radio flares, considered to be associated with the discrete ejection of the same radio knot \citep{2021MNRAS.504..444C}, were detected in MJD 58523.2 and MJD 58527.7, which is reported by \citet{2019ATel12497....1C} and \citet{2020MNRAS.tmpL.236C} respectively. The first flare was about 0.6$\sim$0.8 day from the start of the SIMS. About $\sim 0.1$ day before the start of the second one, we observe a sharp drop in the count rate of LE and a slight rise in the count rate of ME/HE, while the averaged-PDS shows type-C QPOs. A type-B QPO with frequency $\sim 4.5$ Hz appeared almost simultaneously with the second radio flare, which started at MJD 58527.5 and peaked with flux $252 \pm 13$ mJy at MJD 58527.66, suggesting a strong connection between the type-B QPOs and the ejection. For the type-B QPOs observed in this source, using the observations obtained with the \emph{NICER} observatory, \citet{2020MNRAS.496.4366B} computed the energy dependence of the rms and the phase lags at the QPO frequency, sampling for the first time at energies below 2 keV. Concentrating on the phase lags, \citet{2021MNRAS.501.3173G} successfully used a two-component Comptonization model to fit them and explained the radiative properties of the type-B QPOs in 0.8--10 keV energy range by two physically connected Comptonization regions. Our result shows that the fractional rms of type-B QPOs increase with energy up to 10 keV and stay more or less constant above 10 keV in all observations, which implies a Comptonization origin of the type-B QPO. In the jet precession model, the jet velocity can be measured by the QPO amplitude \citep{2021NatAs...5...94M}. Assuming the QPOs rms of $\sim 15\%$ (see Figure \ref{fig:qpo}; B-QPO rms), an observed inclination angle of $\sim 30^{\circ}$, and the jet half-opening angle of $\sim 3^{\circ}$, the jet velocity can be determined as $0.55-0.9c$. Assuming $v = \beta \times c$ to be the escape velocity of the black hole, the height of the ejection knot can be calculated as $\sim 6.6-2.5 r_{\rm g}$ ($\beta = 0.55-0.9 c$) , which is consistent with the emitting scale height of the hard component (Table \ref{B-relxilllp}). Recently \citet{2020A&A...640A..18M} also have found that type-B QPOs require a small transition radius ($r_{J}$, truncation radius due to a jet in the inner region) around 2.5 $r_{\rm g}$. These results indicate that type-B QPOs are produced very close to the BH. Furthermore, the similar rms--energy correlations for different LFQPOs in the 2-30 keV range are found with \emph{RXTE} in XTE~J1859+226 \citep{2004A&A...426..587C}, GRS~1915+015 \citep{2004sf2a.conf..437R,2012Ap&SS.337..137Y,2013ApJ...767...44Y,2016ApJ...833...27Y}, H1743--322 \citep{2013MNRAS.433..412L}, and XTE~J1550--564 \citep{2013MNRAS.428.1704L}, ), in which a corona origin of type-C QPOs is considered. Thanks to the large effective area of \emph{Insight}-HXMT at high energies, the similar rms--energy correlations above 30 keV have been found in MAXI~J1535-571 \citep{2018ApJ...866..122H}, MAXI~J1820+070 \citep{2021NatAs...5...94M} and MAXI~J1631-479 \citep{2021ApJ...919...92B}. Their results all suggest a geometric origin of the type-C QPOs, while \citet{2021NatAs...5...94M} proposed that the type-C QPOs originate from jet rather than corona. Therefore, whether the origin of type-C QPOs lies in the corona or the jet is still an open issue. A recent study in GRS 1915+105 \citep{2022NatAs.tmp...80M} proved that the X-ray corona can morph into a jet. They found that the corona is hot and extended, and covers the inner parts of the disk when the type-C QPO frequency is in the 2--6 Hz range. Then the corona will become a jet as the type-C QPO frequency decreases below $\sim 2$ Hz. A similar conclusion was also proposed by \citet{2022arXiv220511581A} for MAXI~J1348--630, who suggested that there are differences in the geometry of the system at the different phases of the outburst according to the relationship between QPOs frequency and their phase lags. During the the SIMS in MAXI~J1348--630, three transitions from type-B QPO to type-C and then to type-B are observed by \emph{Insight}-HXMT. One of them is only detected by ME and HE, but it is a continuous transition from a type-C QPO at $\sim$ 9.2 Hz to a type-B QPO at $\sim$ 4.8 Hz in a short period of $\sim 10$ s (see the top panel of Figure \ref{fig:tr-c}), which also occurred in MAXI J1820+070 \citep{2020ApJ...891L..29H}. However, the flux at the high energy band changed very slightly in our observation. This may indicate that both the type-C and B QPOs are either generated in different regions located at different radii or in same region with different modulations. For the other two occurrences of type-C QPOs between type-B QPOs, we can present not only the differences in the power spectra but also the differences in the energy spectra. The bottom panel of Figure \ref{fig:spectra ratio} clearly shows that the type-C QPOs have harder spectra than type-B in energy above 10 keV, and less flux at low energies. Furthermore, it can be inferred from Tables \ref{C-nthcomp} and \ref{C-relxilllp} that the energy spectra of C-Q have less disc contribution and low disk temperature ($kT_{\rm in}$). The non-relativistic model seems to be less significant at C-Q in the fit of model 1, probably because of the contraction of the jet structure. This hypothesis is consistent with the $h$ parameter change in model 2. The inner disk radius of C-Q is moderately larger as inferred from the disk normalization parameter: $\sim1.71R_{\rm ISCO}$, $\sim1.86R_{\rm ISCO}$, $\sim1.84R_{\rm ISCO}$ for B-Q, C-Q, and B-Q, respectively, in the SIMS \uppercase\expandafter{\romannumeral1}. However, $R_{\rm in}$ given by the reflection model remains at $R_{\rm ISCO}$. One possible reason for this is that the broad iron line leads to a small radius when applying the reflection model, even in the hard state \citep{2010MNRAS.402..836R,2022ApJ...932...66R}. A recent simulation study reported by \citet{2022arXiv220103526L}, assuming cold, optically thick, clumps of gas falling into the black hole, also suggested that a broader iron line usually appears in the harder spectral states. Another reason could be that the inner radius varied so little during the transitions that the reflection model is insensitive to it. We test this by fixing $R_{\rm in}$ to the radius value derived from $N_{\rm diskbb}$ and find that the effect on the other parameters is insignificant. We consider the L-T precession under the truncated disc/corona (hot inner flow) geometry \citep{2009MNRAS.397L.101I}, assuming that both QPOs $\sim$ 7--9 Hz and $\sim$ 4--5 Hz originate from the precession of an inner flow, a $\sim$ 7--9 Hz QPOs spectrum should have a smaller inner radius than the latter. The observations reveal the difference of quality factor ($Q$), suggesting that the precessions of different morphological structures leads to different $Q$. The type-B QPOs appear when the disk is close to the ISCO, which may indicate that the motion of the disk caused a deformation of the Comptonization region, and thus the flux modulation of different properties is observed in MAXI~J1348--630. \citet{2020A&A...640L..16K} quantitatively explained that the type-B QPOs in GX~339--4 come from the precession of non-solid structures, while type-C QPOs originate from the precession of solid inner flow. In the simulation results of \citet{2018MNRAS.474L..81L}, a solid-body-like precession of the tilted disk-jet system can explain type-C QPOs. Nevertheless, it is also difficult to explain the difference between type-C and B QPOs using solid/non-solid body behavior, which requires better simulations and higher quality observations. Based on the above, we find that both type-B and C QPOs may be related to the so-called jet. Nevertheless, it is clear that types C and B QPOs have significantly different timing and spectral features as summarized in GX~339--4 by \citet{2011MNRAS.418.2292M} as well as in MAXI~J1348--630. Their common Comptonization origin properties and the short transition time scale suggest that a rapid change in physical properties or morphology of the Comptonization results in their differences. The jet/corona in low-luminosity states (hard states) is suggested to be formed in the BP mechanism (\citealt{1982MNRAS.199..883B}. see also e.g. \citealt{1999MNRAS.303L...1B,2001ApJ...548L...9M}). \citet{2001ApJ...548L...9M} further suggested that during very high accretion states, the more relativistic jet rather than the `BP' jet may be associated with the BZ mechanism \citep{1977MNRAS.179..433B}. This jet has a higher Lorentz factor, causing the propagation of an internal shock through the slower-moving outflow in front of it, which may lead the observation of discrete ejection \citet{2004MNRAS.355.1105F}. Therefore, we argue that the compact and steady `BP' jet in low accretion phases is associated with type-C QPOs, while the fast `BZ' jet in SIMS is associated with type-B QPOs. As the accretion rate increases, the truncation radius moves inward, resulting in an increase in the frequency of type-C QPOs \citep{2006A&A...447..813F,2022arXiv220103526L}. Broadband noise in PDS during hard states are assumed to arise from the propagation of magneto-rotational instability (MRI; \citealt{1991ApJ...376..214B}) fluctuations of the inner hot flow \citep{2009MNRAS.397L.101I,2011MNRAS.415.2323I,2022ApJ...932....7Y}. During the SIMS phase, the frequency range of type-B QPOs varies narrowly, which may be related to a particular radius and a detailed discussion is given in sub-section \ref{Type-B QPO}. The noise weakness during SIMS may be due to the magnetic field of `BZ' jet emanating from the black hole, meanwhile the MRI originating with the disk is suppressed. \subsection{The disappearances of Type-B QPOs} \label{Type-B QPO} The disappearance of type-B QPOs (NO-Q/B-Q) has been studied using archival data from the \emph{RXTE}, for example, on XTE~J1817--330, XTE~J1859+226, and H1743--322 \citep{2012A&A...541A...6S,2013ApJ...775...28S,2021ApJ...911..127S}. From the energy spectra comparisons (Figure \ref{fig:spectra ratio}), it can be seen that the difference is very small at energies below 2 keV where the emission is dominated by the thermal component. The most noticeable difference is in the energies above 2 keV, suggesting that the disappearance of type-B QPOs may be related to changes in the Comptonized component. The same result has been found in this source on the results of \citet{2021MNRAS.505.3823Z} and several of the above sources. The observed transient properties of type-B QPOs can occur on timescales of seconds to hours, which implies that the Comptonization radiation region is unstable. When type-B QPOs appeared, the increase in the high-energy flux indicates that there is either an extra Comptonized region contributing to the energy spectra, or the flux in the same region increases for some reasons. Our results give that the height of the corona is greater at B-Q, although the difference in the spectral shape is insignificant (small change in $\Gamma$). For the type-B QPOs observed in this source, using the observations obtained with the \emph{NICER} observatory, \citet{2021MNRAS.505.3823Z} gave a hint that an additional component involving the transitions of type-B QPOs may be related to the base of jet. In section \ref{LFQPOs}, we discussed that the type-B QPOs originating from the jet near the BH and the disk is also very close to the ISCO at this time. Therefore, we suggest that a coupled inner disk-jet region is responsible for the observed type-B QPOs transitions. Although the jets powered by BZ and BP effect are continuous and steady, the magnetic field lines close to the BH can be complicated by the strong gravity. However, the simulation results of different ways of \citet{2019MNRAS.484.4920Y,2019MNRAS.487.4114Y} and \citet{2021MNRAS.507..983L} allow us to understand the precessing jet near the BH and its instability from two perspectives. Theoretical studies show that the jet with transient properties can only exist when the accretion disk is very close to the BH and the magnetic fields on the accretion disk are quite in-homogeneous \citep{2022arXiv220100512Y,2021EGUGA..2316423L,2009RAA.....9..725W,2019MNRAS.484.4920Y,2019MNRAS.487.4114Y,2009MNRAS.395.2183Y}. The small-scale magnetic flux tube model proposed by \citet{2019MNRAS.484.4920Y,2019MNRAS.487.4114Y} argue that a closed zone with a size of a few gravitational radii is formed by the magnetic flux tubes, which may arise due to magnetorotational instability or magnetic buoyancy. This zone connecting the central compact object and the accretion disc may be able to produce strong X-ray emission and can be seen as the small-scale jet, which precesses and modulates X-ray emission (LFQPOs) as the BH rotates. Considering that the type-B QPOs arise from the precession of small-scale jets that are produced only when the disk is close to the BH, it is natural that the frequency of the type-B QPOs varies within a narrow range. Obviously, the field lines in and around this closed structure are actually continuously reconnecting i.e. the field lines connecting a point of the BH to a point on the disk only maintain the connectivity for a short time, and the rotation of the BH increases the toroidal twist of this field line, which eventually breaks. After that, open field lines are formed on the disk. As a result, the field lines on the accretion disk around the closed region alternate between closed and open configurations, leading to the sporadic type-B QPOs observed as well. The Comptonization emission still observed in NO-Q may be due to the remaining symmetric corona or a collimated smaller-scale jet (smaller height in NO-Q). In addition to the small-scale magnetic tube model, another mechanism involving titled disks can explain the rapid transitions of type-B QPOs. Because the accreted material is most likely independent of the BH spin, its angular momentum is expected to be misaligned with respect to the BH spin, resulting in a titled accretion disc. \citet{2019MNRAS.487..550L} performed 3D general relativistic magnetohydrodynamics (GRMHD) simulations of tilted discs around rapidly spinning BHs. They suggested that such disk undergo L-T precession and can launch relativistic jets that propagate along the disc rotation axis. As we discussed above, the flux modulation produced by such jets that precess with discs allows us to observe type-B QPOs. \citet{1975ApJ...195L..65B} predicted that in the presence of realistic magnetized turbulence, the inner part of the disk ($< r_{\rm BP}$, Bardeen-Petterson radius) can align with the BH spin axis, while the outer disc is titled. Recently, \citet{2021MNRAS.507..983L} found a smaller ($\lesssim 5-10 r_{\rm g}$ ) alignment radius than predicted by analytic models through GRMHD simulations, which is consistent with our energy spectral fitting results. When the disk reaches the Bardeen-Petterson transition radius, the jet is parallel to the spin axis of the BH due to the dragging effect, and type-B QPOs generated by the jet precession will not appear. The observed PDS noise at this time is very small, probably due to the collimation of the jet. However, the Comptonization component is observed at this time, indicating that it is different from the HSS. Therefore, it is highly likely that there is a parallel jet generating Comptonization radiation, but without obvious modulations in its flux. The results show that the Comptonization flux decreases when the disk tears, probably because the outer disc cannot maintain a sufficient gas supply \citep{2021MNRAS.507..983L}. Conversely, disk flux less than 2 keV increases slightly, which is also seen in recent simulations of \citet{2022arXiv220103085M}, although their main research is not type-B QPOs. They suggested that the absence of a warp in the aligned portion of the disk reduces the inflow speed of the gas, which increases the density. Furthermore, the obtained rms-spectra are more or less the same regardless of how much the flux of the hard component increases when type-B QPOs are present, suggesting that type-B QPOs are more likely to arise from the overall behavior of one component. It is accepted that the transient jets (discrete ejection) on large-scales could be the result of the ejection of the corona at state transitions \citep{2003ApJ...595.1032R,2003ApJ...597.1023V} and are associated with presence of type-B QPOs. In addition, the radio flares in XRBs are proposed to be produced by the adiabatic expansion of ejected plasma blobs \citep{2004MNRAS.355.1105F,2012MNRAS.421..468M}; hence the ejection components should be launched before strong radio flares are observed. If the ejections of the corona (Comptonization part) are associated with the absence of type-B QPOs, although type-B QPOs occasionally disappear before or after ejections, it is possible that the opened magnetic field lines (parallel to the BH axis) can accelerate the plasma to a very large distance, leading to the in radio band discrete ejections. \subsection{Unified model for type-C and type-B QPOs} More and more observational results in the BHT family indicate that LFQPOs may be a geometrical effect, which is also supported by our findings in MAXI~J1348--630. However, in the geometrical effect models, explaining the transition between different type QPOs, as well as the transient properties of type-B QPOs remain open issues. Based on the above results in BHT MAXI~J1348--630, we propose a scenario illustrated in Figure \ref{fig:model} to explain the timing and spectral evolution as observed in the BHTs. In the LHS and HIMS (panel (a) of Figure \ref{fig:model}), the L-T precession of the corona/jet in the inner regions of the disk (or titled disk) produces type-C QPOs due to the misalignment with BH spin axis. With the increase of accretion rate, the inner edge of the accretion disk gradually approaches the BH, while the corona/jet becomes smaller and its precession frequency increases. Although the jet-structure shown in top panel of Figure \ref{fig:model}, considering frequency ($\sim 7-10$) Hz and the variation in spectra of type-C QPO, we argue that it may be possible that the vertical-jet becomes a corona structure during the QPO transitions, or vice versa (B-C-B), as suggested in GRS 1915+105 \citep{2022NatAs.tmp...80M}. In the SIMS, the inner radius of the accretion disk approaches its ISCO (panel (b) of Figure \ref{fig:model}) and the precessing jets parallel to the titled disc produce type-B QPOs. The innermost parts of the accretion disk align with the BH equator due to the Bardeen-Petterson effect, while the outer parts remain tilted and the disc forms a smooth warp in between (panel (c) of Figure \ref{fig:model}). Since the jets are launched along the direction of the BH spin axis, type-B QPOs disappear due to the absence of precession effect on the modulation of the flux. Alternatively, we can also explain the absence of type-B QPOs by the disappearance of small-scale jets bound by the magnetic flux tubes near the BH. In SIMS, possibly for some reason, inner disc may subsequently recede, when the larger scale Comptonisation component (corona/strong-jet) dominates the radiation and therefore the appearance of type-C QPOs may be observed. But what causes the outward movement of the disk is not yet precisely conclusive. \section{Conclusions} \label{conclusion} In this work, we investigate the spectral and timing properties of the type-B QPOs transitions found in the BHT source MAXI~J1348--630. An additional {\tt\string xillver} component is required in the modelling of the continuum of type-B QPO, while the reflection modelling also suggests a larger height of the corona. Combined these with other results, we suggest that type-B QPOs probably originate from the precession of the weak-jet in a titled disk-jet structure located relatively close to the BH. However, due to the Bardeen-Petterson effect, the misaligned inner-disk in the high spin BH system can tend to be parallel to the direction of the BH spin axis, which can explain the disappearance of type-B QPOs in the BHXBs. We can also explain the absence of type-B QPOs by the disappearance of small-scale jets bound by the magnetic flux tubes. In this source, \emph{Insight}--HMXT observed a type-C/B transition that occurs within a very short time scale. It is still unclear whether type-C QPOs originate from the jet or the corona precession. Based on our results, there is only a slight increase in the inner disk radius when type-C QPO appears. However, due to the limitation of the observation time, it is not clear whether the disc is moving outward or the reasons for the type-C/B QPO transitions in SIMS. \begin{acknowledgements} This work made use of the data from the mission,a project funded by China National Space Administration (CNSA) and the Chinese Academy of Sciences (CAS).This work is supported by the National Key R\&D Program of China (2021YFA0718500). This work is supported by the National Natural Science Foundation of China (Grant No. 11733009, 11673023, U1838202, U1838201, U1938102, U2038104 and U2031205.), the CAS Pioneer Hundred Talent Program (grant No. Y8291130K2) and the Scientific and technological innovation project of IHEP (grant No. Y7515570U1). \end{acknowledgements} \bibliography{maxi}{} \bibliographystyle{aasjournal} \begin{deluxetable*}{cccccccc}[!htbp] \label{QPO} \tablecaption{QPO types classification and variability parameters.} \tablewidth{500pt} \tabletypesize{\scriptsize} \tablehead{ \colhead{ObsID} & \colhead{Start-Time} & \colhead{End-Time} & \colhead{QPO Type$^{a}$} & \colhead{Frequency$^{b}$} & \colhead{${{\rm FWHM}_{\rm QPO}}^{c}$} & \colhead{${{\rm rms}_{\rm QPO}}^{d}$} & \colhead{Epoch$^{e}$}\\ \colhead{} & \colhead{MJD} & \colhead{MJD} & \colhead{} & \colhead{(Hz)} & \colhead{(Hz)} & \colhead{(\%)} } \startdata P021400201101 & 58522.607 & 58522.756 & B & $4.66 \pm 0.02$ & $0.54 \pm 0.05$ & $13.9 \pm 0.1$ & {}\\ P021400201102 & 58522.808 & 58522.889 & B & $4.49 \pm 0.02$ & $0.41 \pm 0.05$ & $14.0 \pm 0.1$ & {}\\ P021400201103 & 58522.943 & 58522.955 & B & $4.46 \pm 0.04$ & $0.38 \pm 0.07$ & $14.6 \pm 0.2$ & {}\\ P021400201201 & 58523.314 & 58523.449 & B & $4.61 \pm 0.09$ & $0.60 \pm 0.14$ & $13.4 \pm 0.2$ & EP 1\\ P021400201501 & 58523.801 & 58523.948 & B & $4.54 \pm 0.02$ & $0.41 \pm 0.05$ & $14.3 \pm 0.1$ & {}\\ P021400201502 & 58524.072 & 58524.114 & B & $4.69 \pm 0.06$ & $0.43 \pm 0.09$ & $13.0 \pm 0.2$ & {}\\ P021400201503 & 58524.136 & 58524.246 & B & $4.57 \pm 0.07$ & $0.36 \pm 0.17$ & $9.7 \pm 0.6$ & {}\\ \hline P021400201504 & 58524.275 & 58524.345 & B & $4.55 \pm 0.04$ & $0.45 \pm 0.13$ & $13.2 \pm 0.3$& {}\\ P021400201504 & 58524.372 & 58524.393 & C & $9.23 \pm 0.32$ & $7.21 \pm 0.93$ & $15.9 \pm 2.1$& EP 2\\ P021400201504 & 58524.393 & 58524.396 & B & $4.81 \pm 0.15$ & $1.24 \pm 0.32$ & $12.3 \pm 3.1$ & {}\\ P021400201505 & 58524.407 & 58524.743 & B & $4.56 \pm 0.02$ & $0.59 \pm 0.05$ & $13.5 \pm 0.1$ {}\\ \hline P021400201506 & 58524.779 & 58524.799 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201506 & 58524.799 & 58524.805 & B & $4.47 \pm 0.13$ & $0.79 \pm 0.32$ & $9.5 \pm 2.3$ & {}\\ P021400201506 & 58524.805 & 58524.808 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201506 & 58524.848 & 58524.856 & B & $4.33 \pm 0.07$ & $0.43 \pm 0.12$ & $9.1 \pm 1.1$ & {}\\ P021400201506 & 58524.856 & 58524.861 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201506 & 58524.861 & 58524.865 & B & $4.46 \pm 0.05$ & $0.43 \pm 0.08$ & $13.1 \pm 1.2$ & EP 3 \\ P021400201506 & 58524.865 & 58524.875 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201507 & 58524.917 & 58524.922 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201507 & 58524.922 & 58524.931 & B & $4.35 \pm 0.04$ & $0.35 \pm 0.11$ & $10.3 \pm 0.4$ &{}\\ P021400201507 & 58524.931 & 58524.933 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201507 & 58524.933 & 58524.941 & B & $4.45 \pm 0.05$ & $0.44 \pm 0.09$ & $13.5 \pm 0.7$ & {}\\ \hline P021400201508 & 58525.064 & 58525.176 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201509 & 58525.198 & 58525.239 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201510 & 58525.386 & 58525.468 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201511 & 58525.500 & 58525.601 & NO-Q & \nodata & \nodata & \nodata & EP 4\\ P021400201512 & 58525.632 & 58525.737 & B & $4.35 \pm 0.04$ & $0.57 \pm 0.09$ & $13.0 \pm 0.1$ & {}\\ P021400201513 & 58525.787 & 58525.869 & B & $4.33 \pm 0.03$ & $0.49 \pm 0.07$ & $13.3 \pm 0.1$ & {}\\ \hline P021400201601 & 58526.195 & 58526.232 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201602 & 58526.379 & 58526.455 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201603 & 58526.495 & 58526.595 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201604 & 58526.625 & 58526.730 & NO-Q & \nodata & \nodata & \nodata & {}\\ P021400201605 & 58526.781 & 58526.815 & NO-Q & \nodata & \nodata & \nodata & EP 5\\ P021400201605 & 58526.815 & 58526.863 & B & $4.45 \pm 0.06$ & $0.44 \pm 0.07$ & $13.7 \pm 0.2$ &{}\\ P021400201606 & 58526.913 & 58526.995 & B & $4.57 \pm 0.04$ & $0.69 \pm 0.09$ & $14.6 \pm 0.1$ & {}\\ P021400201607 & 58527.022 & 58527.096 & B & $4.49 \pm 0.03$ & $0.47 \pm 0.08$ & $13.6 \pm 0.1$ & {}\\ \hline P021400201701 & 58527.372 & 58527.448 & B & $4.56 \pm 0.04$ & $0.51 \pm 0.08$ & $13.2 \pm 0.1$ & {}\\ P021400201702 & 58527.487 & 58527.525 & C & $7.42 \pm 0.07$ & $4.13 \pm 0.32$ & $18.6 \pm 3.1$ & {}\\ P021400201702 & 58527.551 & 58527.588 & C & $10.21 \pm 0.16$ & $6.23 \pm 0.51$ & $19.2 \pm 3.4$ & EP 6\\ P021400201703 & 58527.618 & 58527.723 & B & $4.44 \pm 0.02$ & $0.41 \pm 0.03$ & $14.1 \pm 0.2$ & {}\\ P021400201704 & 58527.774 & 58527.856 & B & $4.34 \pm 0.03$ & $0.52 \pm 0.07$ & $14.3 \pm 0.1$ & {}\\ \hline P021400202501 & 58539.134 & 58539.161 & B & $3.98 \pm 0.08$ & $0.45 \pm 0.15$ & $11.9 \pm 0.2$ & {}\\ P021400202502 & 58539.171 & 58539.534 & B & $4.16 \pm 0.02$ & $0.47 \pm 0.04$ & $12.6 \pm 0.1$ & {}\\ P021400202503 & 58539.559 & 58539.663 & B & $4.15 \pm 0.04$ & $0.55 \pm 0.08$ & $13.1 \pm 0.1$ & {}\\ P021400202504 & 58539.741 & 58539.795 & B & $4.33 \pm 0.31$ & $0.71 \pm 0.86$ & $11.2 \pm 0.3$ & {}\\ P021400202504 & 58539.795 & 58539.804 & C & $7.78 \pm 0.40$ & $9.34 \pm 1.03$ & $20.1 \pm 6.8$ & EP 7\\ P021400202505 & 58539.828 & 58539.955 & C & $7.43 \pm 0.36$ & $11.77 \pm 0.88$ & $22.1 \pm 6.3$ & {}\\ P021400202505 & 58539.955 & 58539.940 & B & $4.08 \pm 0.02$ & $0.52 \pm 0.05$ & $12.7 \pm 0.4$ & {}\\ P021400202506 & 58539.993 & 58540.004 & B & $4.13 \pm 0.08$ & $0.39 \pm 0.18$ & $10.4 \pm 0.2$ & {}\\ P021400202507 & 58540.125 & 58540.463 & B & $3.96 \pm 0.03$ & $0.56 \pm 0.09$ & $11.3 \pm 0.1$ & {}\\ \enddata \tablecomments{$^{a}$ The label `C' means type-C QPO are detected. Similarly, `B' corresponds to type-B QPO, `NO-Q' corresponds to NO-QPO.\\ $^{b}$ Centroid frequency of the Lorentzian component.\\ $^{c}$ Width of the Lorentzian component.\\ $^{d}$ Fractional rms of the LFQPO.\\ $^{e}$ The divided epochs.} \end{deluxetable*} \begin{deluxetable*}{cccccc}[!htbp] \tablecaption{Best-fit spectral parameters of epochs with B-QPO and NO-QPO using model 1 {\tt\string tbabs}$\times$({\tt\string diskbb+gaussian+nthcomp+xillverCp)} \label{B-nthcomp}} \tablewidth{500pt} \tabletypesize{\scriptsize} \tablehead{ \colhead{Component} & \colhead{Parametes} & \multicolumn{2}{c}{EP 4} & \multicolumn{2}{c}{EP 5}\\ \cmidrule(rr){3-4} \cmidrule(rr){5-6} \colhead{} & {} & \colhead{NO-Q} & \colhead{B-Q} & \colhead{NO-Q} & \colhead{B-Q} } \startdata TBabs & $N_{\rm H}(\times10^{22}\rm cm^{-2}$) &$0.913\pm0.009$ &$0.901\pm0.003$ &$0.908\pm0.006$ &$0.893_{-0.007}^{+0.009}$ \\ \hline Diskbb & $kT_{\rm in} ({\rm keV})$ &$0.758\pm0.003$ &$0.752_{-0.005}^{+0.004}$ &$0.743\pm0.003$ &$0.734_{-0.007}^{+0.006}$ \\ { }& norm &$27555_{-571}^{+621}$ &$27987_{-672}^{+421}$ &$30067_{-525}^{+582}$ &$28496_{-1007}^{+1203}$ \\ \hline Gaussian & $E_{\rm line}({\rm keV})$ &$5.0_{-0.3}^{+0.2}$ &$4.3_{-0.4}^{+0.3}$ &$4.8_{-0.3}^{+0.2}$ &$3.7_{-0.6}^{+0.4}$ \\ { } & $\sigma ({\rm keV})$ &$1.47_{-0.17}^{+0.16}$ &$1.87_{-0.15}^{+0.13}$ &$1.66_{-0.12}^{+0.13}$ &$2.03_{-0.19}^{+0.21}$ \\ { } & norm &$0.27_{-0.07}^{+0.09}$ &$0.62_{-0.13}^{+0.16}$ &$0.38_{-0.07}^{+0.08}$ &$1.06_{-0.28}^{+0.31}$ \\ \hline Nthcomp & $\Gamma$ &$2.26\pm0.02$ &$2.22_{-0.02}^{+0.03}$ &$2.21_{-0.02}^{+0.01}$ &$2.23_{-0.02}^{+0.03}$ \\ { } & $kT_{\rm e} (\rm keV)$ &$105.1_{-34.9}^{+59.9}$ &$61.3_{-8.8}^{+20.1}$ &$102.9_{-27.5}^{+94.6}$ &$54.7_{-8.4}^{+20.1}$ \\ { } & norm &$3.47_{-0.26}^{+0.19}$ &$3.86_{-0.13}^{+0.15}$ &$2.64_{-0.10}^{+0.11}$ &$4.80_{-0.14}^{+0.19}$ \\ \hline Xillver & $A_{\rm Fe}$ &$2.40_{-1.02}^{+1.47}$ &$10.00_{-6.28}^{*}$ &$10.00_{-6.29}^{*}$ &$10.00_{-6.15}^{*}$ \\ { } & log($\xi$) &$3.85_{-0.37}^{+0.29}$ &$3.47_{-0.37}^{+0.43}$ &$3.61_{-0.57}^{+0.44}$ &$3.84_{-0.29}^{+0.28}$ \\ { } & norm &$0.017_{-0.008}^{+0.010}$ &$0.025_{-0.001}^{+0.003}$ &$0.002_{-0.001}^{+0.003}$ &$0.007_{-0.002}^{+0.008}$ \\ {} & probability & 3.24e-07 & 7.02e-08 & 0.0065 & 2.351e-07\\ \hline { } & $\chi^2/d.o.f$ &730.3/728 & 726.3/746 & 670.0/733 & 590.6/675 \\ \hline Flux & Total &$16.68$ & $16.66$ & $15.50$ &$16.59$ \\ { } & Diskbb &$12.78 $ & $12.36 $ & $12.51 $ &$11.35 $ \\ { }& Gaussian &$0.29 $ & $0.43 $ & $0.31 $ &$0.64 $ \\ { } & Nthcomp &$3.61$ & $3.87$ & $2.68$ &$4.60$ \\ \enddata \tablecomments{Error bars are of 90\% confidence limit.\\ $^{*}$ means that the parameter pegs at its limit.\\ Flux unit is $10^{-8} \rm ergs/\rm cm^{2}/\rm s$ in the energy band 1–100 keV.\\ } \end{deluxetable*} \begin{deluxetable*}{cccccccc} \tablecaption{Best-fit spectral parameters of epochs with B-QPO and C-QPO using model 1 {\tt\string tbabs}$\times$({\tt\string diskbb+gaussian+nthcomp+xillverCp)} \label{C-nthcomp}} \tablewidth{500pt} \tabletypesize{\scriptsize} \tablehead{ \colhead{Component} & \colhead{Parametes} & \multicolumn{3}{c}{EP 6} & \multicolumn{3}{c}{EP 7}\\ \cmidrule(rr){3-5} \cmidrule(rr){6-8} \colhead{} & {} & \colhead{B-Q} & \colhead{C-Q} & \colhead{B-Q} & \colhead{B-Q} & \colhead{C-Q} &\colhead{B-Q} } \startdata TBabs & $N_{\rm H}(\times10^{22}\rm cm^{-2}$) &$0.899_{-0.017}^{+0.018}$ &$0.870\pm0.011$ &$0.894\pm0.007$ &$0.859\pm0.006$ &$0.856_{-0.011}^{+0.004}$ &$0.867_{-0.007}^{+0.008}$\\ \hline Diskbb & $kT_{\rm in} ({\rm keV})$ &$0.759_{-0.046}^{+0.010}$ &$0.683_{-0.007}^{+0.006}$ &$0.737_{-0.006}^{+0.005}$ &$0.689\pm0.004$ &$0.646_{-0.004}^{+0.007}$ &$0.685\pm0.004$ \\ { }& norm &$23662_{-1425}^{+4510}$ &$32228_{-1289}^{+1376}$&$28206_{-797}^{+923}$ &$28437_{-617}^{+676}$ &$31670_{-734}^{+747}$ &$28920_{-681}^{+757}$ \\ \hline Gaussian & $E_{\rm line}({\rm keV})$ &$4.6_{-3.8}^{+0.4}$ &$3.5_{-0.4}^{+0.4}$ &$3.9_{-0.5}^{+0.4}$ &$3.5_{-0.4}^{+0.3}$ &$3.4_{-0.3}^{+0.3}$ &$3.5_{-0.5}^{+0.3}$ \\ { } & $\sigma ({\rm keV})$ &$1.13_{-0.36}^{+1.77}$ &$2.07_{-0.18}^{+0.18}$ &$1.96_{-0.16}^{+0.17}$ &$2.10_{-0.11}^{+0.13}$ &$2.24_{-0.10}^{+0.10}$ &$2.02_{-0.20}^{+0.16}$ \\ { } & norm &$0.26_{-0.12}^{+1.56}$ &$1.16_{-0.27}^{+0.31}$ &$0.81_{-0.20}^{+0.28}$ &$0.87_{-0.14}^{+0.20}$ &$1.04_{-0.14}^{+0.16}$ &$0.69_{-0.19}^{+0.22}$ \\ \hline Nthcomp & $\Gamma$ &$2.31_{-0.06}^{+0.02}$ &$2.15\pm0.02$ &$2.25_{-0.02}^{+0.03}$ &$2.20\pm0.01$ &$2.10\pm0.01$ &$2.22\pm0.02$ \\ { } & $kT_{\rm e} (\rm keV)$ &$121.2_{-20.4}^{+23.1}$ &$84.3_{-18.1}^{+32.2}$ &$111.3_{-29.2}^{+108.1}$ &$108.5_{-23.9}^{+49.3}$ &$84.1_{-14.7}^{+22.8}$ &$142.2_{-48.8}^{+339.6}$ \\ { } & norm &$4.86_{-4.86}^{+4.86}$ &$5.52_{-0.27}^{+0.30}$ &$4.19_{-0.16}^{+0.17}$ &$3.18_{-0.09}^{+0.09}$ &$2.90_{-0.11}^{+0.12}$ &$2.77_{-0.17}^{+0.13}$ \\ \hline Xillver & $A_{\rm Fe}$ &$4.92_{-3.27}^{+4.88}$ &\nodata &$6.53_{-3.85}^{+-6.53}$ &$4.03_{-1.47}^{+2.81}$ &\nodata &$3.82_{-1.18}^{+1.95}$ \\ { } & log($\xi$) &$4.15_{-1.24}^{+0.49}$ &\nodata &$3.50_{-0.29}^{+0.33}$ &$3.65_{-0.19}^{+0.19}$ &\nodata &$3.95_{-0.18}^{+0.30}$ \\ { } & norm &$0.034_{-0.025}^{+0.012}$ &\nodata &$0.007_{-0.002}^{+0.006}$ &$0.006_{-0.002}^{+0.003}$ &\nodata &$0.011_{-0.004}^{+0.006}$ \\ {} & probability & 0.006882 & \nodata (0.75) & 2.25e-10 & 1.19e-21& \nodata & 4.29e-23\\ \hline { } & $\chi^2/d.o.f$ &307.5/322 & 344/309 & 660.2/736 & 974.4/1017 & 682/615 & 792.0/919 \\ \hline Flux & Total &$16.63$ &$15.35$ &$15.90$ &$12.04$ &$11.02$ &$11.60$ \\ { }& Diskbb &$11.32 $ &$9.03 $ &$11.41 $ &$8.47 $ &$7.34 $ &$8.45 $ \\ { }& Gaussian &$0.60 $ &$0.66 $ &$0.51 $ &$0.49 $ &$0.57 $ &$0.42 $ \\ { } & Nthcomp &$4.71$ &$5.67$ &$3.98$ &$3.08$ &$3.12$ &$2.73$\\ \enddata \tablecomments{Error bars are of 90\% confidence limit.\\ $^{*}$means that the parameter pegs at its limit.\\ Flux unit is $10^{-8} \rm ergs/\rm cm^{2}/\rm s$ in the energy band 1–100 keV. } \end{deluxetable*} \begin{deluxetable*}{lccccrcccc}[!htbp] \tablecaption{Results of simultaneous spectral fits using model {\tt\string tbabs}$\times$({\tt\string diskbb+gaussian+nthcomp)} \label{compare-nthcomp}} \tablewidth{0pt} \tabletypesize{\scriptsize} \tablehead{ \colhead{} & \multicolumn{2}{c}{EP 4} & \multicolumn{2}{c}{EP 5} & \colhead{} &\multicolumn{2}{c}{EP 6} & \multicolumn{2}{c}{EP 7}\\ \cmidrule(rr){2-3} \cmidrule(rr){4-5} \cmidrule(rr){7-8} \cmidrule(rr){9-10} \colhead{} & \colhead{NO-Q} & \colhead{B-Q} & \colhead{NO-Q} & \colhead{B-Q} & \colhead{} & \colhead{B-Q} &\colhead{C-Q} & \colhead{B-Q} &\colhead{C-Q} } \startdata {Parameters} & \nodata & ${\chi^2_{red}/d.o.f}^{d}$ & \nodata &$\chi^2_{red}/d.o.f$ & {Parameters} & \nodata & $\chi^2_{red}/d.o.f$ & \nodata &$\chi^2_{red}/d.o.f$ \\ All tied &\nodata & 6326.3/1502 &\nodata &100662.1/1436 & All tied &\nodata & 32532.22/656 & \nodata & 81580.34/1657\\ ${N_{\rm nthcomp}}^{a}$ & \nodata & 3712.9/1501 & \nodata & 13564/1435 &$N_{\rm diskbb}$ & \nodata & 7403.7/655 & \nodata & 16945.38/1656\\ ${N_{\rm diskbb}}^{b}$ & \nodata & 1867.9/1500 & \nodata & 5265.4/1434 & $\Gamma $ &\nodata & 5312.17/654 & \nodata & 16403.51 /1655\\ ${N_{\rm gaussian}}^{c}$ & \nodata & 1610.7/1499 & \nodata & 1658.7/1433 & $kT_{\rm in}$ &\nodata &1443.28/653 & \nodata &2797.47/1654\\ {} & {} & {} & {} & {} & $N_{\rm nthcomp}$ &\nodata &763.98/652 & \nodata &2604.69/1653\\ \hline $N_{\rm nthcomp}$ &$3.46_{-0.10}^{+0.11}$ & $3.72\pm0.01$ & $2.56_{-0.08}^{+0.09}$ & $4.23\pm0.01$ & $N_{\rm diskbb}$& $28089_{-1922}^{+3099}$ &$34772_{-347}^{+353}$ & $29730_{-815}^{+884}$& $34052_{-154}^{+155}$\\ $N_{\rm diskbb}$ & $27663_{-448}^{+485}$&$26967_{-26}^{+25}$&$29914_{-467}^{+516}$&$28036_{-31}^{+32}$ & $\Gamma$ & $2.22_{-0.02}^{+0.02}$ & $2.15\pm0.01$ & $2.16\pm0.01$ &$2.108\pm0.003$\\ $N_{\rm gaussian} $ & $0.34_{-0.04}^{+0.05}$& $0.41\pm0.01$ & $0.40_{-0.05}^{+0.06}$ & $0.72\pm0.01$ & $kT_{\rm in}$ & $0.747_{-0.020}^{+0.013}$ & $0.684\pm0.002$ &$0.84\pm0.01$ &$0.684\pm0.002$\\ {} & {} & {} & {} & {}& $N_{\rm nthcomp}$ & $1.22_{-0.09}^{+0.11}$ &$1.51\pm0.02$ & $0.87\pm0.02$&$0.931_{-0.007}^{+0.008}$ \enddata \tablecomments{Error bars are of 90\% confidence limit.\\ $^{*}$ means that the parameter pegs at its limit\\ $^{abc}$ Normalization of \emph{nthcomp}, \emph{diskbb} and \emph{gaussian}, respectively.\\ $^{d}$ The chi-squared obtained after freeing the parameters.\\ \emph{Last row} presents results obtained after freeing the above parameters. } \end{deluxetable*} \begin{deluxetable*}{llcccc}[!htbp] \tablecaption{Best-fit spectral parameters of epochs with B-QPO and NO-QPO using model 2 {\tt\string tbabs$\times$({\tt\string diskbb+relxilllp)}} \label{B-relxilllp}} \tablewidth{500pt} \tabletypesize{\scriptsize} \tablehead{ \colhead{Component} & \colhead{Parametes} & \multicolumn{2}{c}{EP 4} & \multicolumn{2}{c}{EP 5}\\ \cmidrule(rr){3-4} \cmidrule(rr){5-6} \colhead{} & {} & \colhead{NO-Q} & \colhead{B-Q} & \colhead{NO-Q} & \colhead{B-Q} } \startdata TBabs & $N_{\rm H}(\times10^{22}\rm cm^{-2}$) &$1.04_{-0.01}^{+0.02}$ &$1.02\pm0.01$ &$1.007\pm0.001$ &$1.03\pm0.01$ \\ \hline Diskbb & $kT_{\rm in} ({\rm keV})$ &$0.762_{-0.002}^{+0.001}$ &$0.763_{-0.002}^{+0.003}$ &$0.746\pm0.001$ &$0.759\pm0.001$ \\ { }& norm &$25078_{-142}^{+234}$ &$23760_{-413}^{+212}$ &$27333_{-58}^{+85}$ &$21770_{-178}^{+166}$ \\ \hline Relxilllp & $h (GM/C^2)$ &$2.95_{-0.12}^{+0.08}$ &$3.18_{-0.12}^{+0.09}$ &$2.86_{-0.20}^{+0.04}$ &$3.21_{-0.07}^{+0.04}$ \\ { } & $R_{\rm in}$ (ISCO) &$1.00_{*}^{+0.16}$ &$1.00_{*}^{+0.13}$ &$1.00_{*}^{+0.09}$ &$1.00_{*}^{+0.07}$ \\ { } & $\Gamma$ &$2.40_{-0.02}^{+0.03}$ &$2.35\pm0.01$ &$2.39\pm0.01$ &$2.36\pm0.01$ \\ { } & $a^*$ ($c{\rm J}/GM^2$) & \multicolumn{4}{c}{0.80 (fixed)}\\ { } & $\theta$ (deg) & \multicolumn{4}{c}{36.5 (fixed)} \\ { } & log($\xi$) &$3.77_{-0.06}^{+0.21}$ &$4.25_{-0.11}^{+0.07}$ &$4.23_{-0.04}^{+0.01}$ &$4.36_{-0.01}^{+0.01}$ \\ { } & $A_{\rm Fe}$ &$6.03_{-1.24}^{+0.94}$ &$10.00_{-1.76}^{*}$ &$9.67_{-0.44}^{+0.03}$ &$10.00_{-0.71}^{*}$ \\ { } & $E_{\rm cut ({\rm keV}})$ &$241.1_{-30.2}^{+61.8}$ &$219.1_{-24.5}^{+14.8}$ &$500.0_{-121.9}^{*}$ &$204.3_{-14.4}^{+12.3}$ \\ { } & $R_{\rm ref}$ &$0.62\pm0.03$ &$0.63\pm0.03$ &$0.74_{-0.03}^{+0.01}$ &$0.67_{-0.02}^{+0.02}$ \\ { } & norm &$1.43\pm0.15$ &$0.98_{-0.09}^{+0.08}$ &$1.06_{-0.04}^{+0.06}$ &$1.40_{-0.01}^{+0.12}$ \\ \hline { } & $\chi^2_{red}/d.o.f$ &786.5/729 & 755.2/747 & 724.6/734 & 617.1/676 \\ \hline Flux & Total &$17.49$ &$17.46$ &$16.14$ &$17.53$ \\ { } & Diskbb &$11.77 $ &$11.22 $ &$11.59 $ &$9.97 $ \\ { }& Relxilllp & $1.41 $ &$1.72 $ &$1.42 $ & $2.22 $ \\ { } & Cutoffpl &$4.32$ &$4.48$ &$3.18$ &$5.32$\\ \enddata \tablecomments{Error bars are of 90\% confidence limit.\\ $^{*}$ means that the parameter pegs at its limit.\\ Flux unit is $10^{-8} \rm ergs/\rm cm^{2}/\rm s$ in the energy band 1–-100 keV.\\ } \end{deluxetable*} \begin{deluxetable*}{llcccccc}[!htbp] \tablecaption{Best-fit spectral parameters of epochs with B-QPO and C-QPO using model 2 {\tt\string tbabs}$\times$({\tt\string diskbb+relxilllp)} \label{C-relxilllp}} \tablewidth{500pt} \tabletypesize{\scriptsize} \tablehead{ \colhead{Component} & \colhead{Parametes} & \multicolumn{3}{c}{EP 6} & \multicolumn{3}{c}{EP 7}\\ \cmidrule(rr){3-5} \cmidrule(rr){6-8} \colhead{} & {} & \colhead{B-Q} & \colhead{C-Q} & \colhead{B-Q} & \colhead{B-Q} & \colhead{C-Q} &\colhead{B-Q} } \startdata TBabs & $N_{\rm H}(\times10^{22}\rm cm^{-2}$) &$1.022_{-0.004}^{+0.007}$ &$1.004_{-0.012}^{+0.009}$ &$1.026_{-0.007}^{+0.008}$ &$0.975_{-0.003}^{+0.001}$ &$0.938_{-0.006}^{+0.003}$ &$0.969_{-0.007}^{+0.003}$\\ \hline Diskbb & $kT_{\rm in} ({\rm keV})$ &$0.777_{-0.003}^{+0.004}$ &$0.710_{-0.003}^{+0.004}$ &$0.756\pm0.002$ &$0.706\pm0.001$ &$0.669_{-0.004}^{+0.002}$ &$0.699_{-0.002}^{+0.001}$ \\ { }& norm &$19467_{-400}^{+336}$ &$22968_{-560}^{+532}$ &$22533_{-323}^{+326}$ &$22565_{-158}^{+139}$ &$25110_{-188}^{+102}$ &$23894_{-210}^{+100}$ \\ \hline Relxilllp & $h (GM/C^2)$ &$4.06_{-0.31}^{+0.23}$ &$2.99\pm0.08$ &$4.54_{-0.11}^{+0.04}$ &$2.61_{-0.04}^{+0.03}$ &$2.51_{-0.03}^{+0.04}$ &$2.68_{-0.05}^{+0.03}$ \\ { } & Rin (ISCO) &$1.0_{*}^{+0.9}$ &$1.0_{*}^{+0.4}$ &$1.0_{*}^{+0.1}$ &$1.00_{*}^{+0.02}$ &$1.00_{*}^{+0.06}$ &$1.00_{*}^{+0.04}$ \\ { } & $\Gamma$ &$2.37_{-0.02}^{+0.01}$ &$2.30_{-0.02}^{+0.01}$ &$2.40\pm0.02$ &$2.38_{-0.01}^{+0.02}$ &$2.30_{-0.01}^{+0.02}$ &$2.38\pm0.01$ \\ { } & $a^*$ ($c{\rm J}/GM^2$) & \multicolumn{6}{c}{0.80 (fixed)}\\ { } & $\theta$ (deg) & \multicolumn{6}{c}{36.5 (fixed)} \\ { } & log($\xi$) &$4.28_{-0.13}^{+0.18}$ &$4.21_{-0.04}^{+0.03}$ &$4.28_{-0.11}^{+0.13}$ &$4.28_{-0.03}^{+0.02}$ &$4.27_{-0.04}^{+0.02}$ &$4.27_{-0.06}^{+0.07}$ \\ { } & $A_{\rm Fe}$ &$6.83_{-1.65}^{+0.96}$ &$5.01_{-0.33}^{+0.02}$ &$8.12_{-0.75}^{+1.42}$ &$8.03_{-0.37}^{+0.29}$ &$10.00_{-0.28}^{*}$ &$7.52_{-0.72}^{+0.82}$ \\ { } & $E_{\rm cut ({\rm keV}})$ &$350_{-60}^{+25}$ &$500_{-51}^{*}$ &$372_{-63}^{+93}$ &$500_{-60}^{*}$ &$500_{-11}^{*}$ &$500_{-62}^{*}$ \\ { } & $R_{\rm ref}$ &$0.64_{-0.12}^{+0.08}$ &$0.73_{-0.01}^{+0.09}$ &$0.63_{-0.04}^{+0.03}$ &$0.88_{-0.06}^{+0.08}$ &$1.05_{-0.01}^{+0.04}$ &$0.96_{-0.06}^{+0.04}$ \\ { } & norm &$0.70_{-0.09}^{+0.08}$ &$0.52_{-0.05}^{+0.01}$ &$1.40_{-0.12}^{+0.11}$ &$1.75_{-0.10}^{+0.12}$ &$1.13_{-0.02}^{+0.01}$ &$1.17_{-0.08}^{+0.07}$ \\ \hline { } & $\chi^2_{red}/d.o.f$ &310.7/323 & 335.1/307 & 711.2/737 & 1120.3/1018 & 737.9/613 & 867.5/920 \\ \hline Flux & Total &$17.51$ &$16.13$ &$16.77$ &$12.61$ &$11.47$ &$12.12$ \\ {} & Diskbb &$10.03 $ &$7.41 $ &$10.21 $ &$7.45 $ &$6.41 $ &$7.51 $ \\ { }& Relxilllp &$2.25 $ &$2.57 $ &$1.81 $ &$1.96 $ &$2.09 $ &$1.88 $\\ { } & Cutoffpl &$5.24$ &$5.79$ &$4.78$ &$3.22$ &$2.96$ &$2.72$ \enddata \tablecomments{Error bars are of 90\% confidence limit.\\ $^{*}$ means that the parameter pegs at its limit.\\ Flux unit is $10^{-8} \rm ergs/\rm cm^{2}/\rm s$ in the energy band 1-–100 keV.\\ } \end{deluxetable*} \begin{deluxetable*}{lccccrcccc}[!htbp] \tablecaption{Results of simultaneous spectral fits using model 2 {\tt\string tbabs}$\times$({\tt\string diskbb+relxilllp)} \label{compare-relxilllp}} \tablewidth{0pt} \tabletypesize{\scriptsize} \tablehead{ \colhead{} & \multicolumn{2}{c}{EP 4} & \multicolumn{2}{c}{EP 5} & \colhead{} &\multicolumn{2}{c}{EP 6} & \multicolumn{2}{c}{EP 7}\\ \cmidrule(rr){2-3} \cmidrule(rr){4-5} \cmidrule(rr){7-8} \cmidrule(rr){9-10} \colhead{} & \colhead{NO-Q} & \colhead{B-Q} & \colhead{NO-Q} & \colhead{B-Q} & \colhead{} & \colhead{B-Q} &\colhead{C-Q} & \colhead{B-Q} &\colhead{C-Q} } \startdata Parameters & \nodata & ${\chi^2_{red}/d.o.f}^{c}$ &\nodata & $\chi^2_{red}/d.o.f$ & Parameters & \nodata & $\chi^2_{red}/d.o.f$ &\nodata & $\chi^2_{red}/d.o.f$ \\ All tied & \nodata & 8460.37/1502 & \nodata & 111803.8/1436 & All tied &\nodata & 29227.48/656 & \nodata & 64119.46/1658\\ $\Gamma$ & \nodata & 4063.71/1501 & \nodata & 14354.44/1435 & $kT_{\rm in} ({\rm keV})$ & \nodata & 3404.34/655 & \nodata & 4316.91/1657\\ ${N_{\rm diskbb}}^{a}$ & \nodata & 1742.65/1500 & \nodata & 8201.09/1434 & $h (GM/C^2)$ &\nodata & 1235.55/654 & \nodata &4022.04/1656 \\ $h (GM/C^2)$ & \nodata & 1628.05/1499 & \nodata & 3552.20/1433 & $N_{\rm diskbb}$ &\nodata & 983.54/653 & \nodata &2569.73/1655\\ $kT_{\rm in} ({\rm keV})$ & \nodata & 1571.34/1498 & \nodata & 2096.51/1432 & $\Gamma$ ;$E_{\rm cut}$ &\nodata &700.66/651 & \nodata &2102.18/1653\\ ${N_{\rm relxilllp}}^{b}$ & \nodata & \nodata & \nodata & 1980.67/1431 & $R_{\rm ref}$ &\nodata &655.98/650 & \nodata & 2084.41/1652\\ $E_{\rm cut}$;$R_{\rm ref}$ & \nodata & \nodata & \nodata & 1378.33/1429 & &\nodata & \nodata & \nodata & \nodata\\ \hline $\Gamma$ & $2.40_{-0.02}^{+0.03}$ & $2.38_{-0.02}^{+0.01}$ & $2.39 \pm 0.01$ & $2.32\pm0.01$ & $kT_{\rm in}$ & $0.777_{-0.003}^{+0.004}$ & $0.707\pm0.002$ & $0.706\pm0.001$ & $0.661\pm0.001$\\ $N_{\rm diskbb}$ & $25078_{-142}^{+234}$ & $23474_{-123}^{+119}$ & $27333_{-58}^{+85}$ & $21383_{-221}^{+135}$ & $h$ & $4.42_{-0.11}^{+0.33}$ &$3.77_{-0.15}^{+0.14}$ &$2.61_{-0.04}^{+0.03}$ &$2.47_{-0.01}^{+0.02}$\\ $h (GM/C^2)$ & $2.95_{-0.12}^{+0.08}$ & $3.33\pm0.01$ & $2.86_{-0.20}^{+0.04}$ & $3.51_{-0.02}^{+0.03}$ & $N_{\rm diskbb}$ & $19467_{-400}^{+336}$ & $23232_{-398}^{+406}$ &$22565_{-158}^{+139}$ &$26296_{-144}^{+152}$\\ $kT_{\rm in} ({\rm keV})$ & $0.762_{-0.002}^{+0.001}$ & $0.765\pm0.001$ & $0.746\pm0.001$ & $0.763\pm0.001$ & $\Gamma$ & $2.37_{-0.02}^{+0.01}$ & $2.33_{-0.01}^{+0.01}$ & $2.38\pm0.01$ &$2.34\pm0.01$\\ $N_{\rm relxilllp}$ & \nodata & \nodata & $1.06_{-0.04}^{+0.06}$ & $1.59_{-0.18}^{+0.11}$ & $E_{\rm cut}$ & $350_{-60}^{+25}$ & $500^{*}$ & $500^{*}$ &$500^{*}$\\ $E_{\rm cut}$ & \nodata & \nodata & $500^{*}$ & $159_{-10}^{+11}$&$R_{\rm ref}$ & $0.64_{-0.12}^{+0.08}$ &$0.92_{-0.11}^{+0.14}$ & $0.88_{-0.06}^{+0.08}$ &$1.12_{-0.05}^{+0.06}$\\ $R_{\rm ref}$ & \nodata & \nodata & $0.74_{-0.03}^{+0.01}$ & $0.62_{-0.02}^{+0.04}$ & &\nodata & \nodata & \nodata & \nodata\\ \enddata \tablecomments{Error bars are of 90\% confidence limit.\\ $^{*}$ means that the parameter pegs at its limit.\\ $^{ab}$ Normalization of \emph{diskbb} and \emph{relxilllp}.\\ $^{c}$ The chi-squared obtained after freeing the parameters.\\ \emph{Last row} presents results obtained after freeing the above parameters. } \end{deluxetable*}

Title: Impact of dust size distribution including large dust grains on magnetic resistivity: an analytical approach

Abstract: This paper investigates the impact of dust size distribution on magnetic resistivity. In particular, we focus on its impact when the maximum dust size significantly increases from sub-micron. The first half of the paper describes our calculation method for magnetic resistivity based on the model of \citet{1987ApJ...320..803D} and shows that the method reproduces the results of a more realistic chemical reaction network calculations reasonably well. Then, we describe the results of the resistivity calculations for dust distributions with large maximum dust grains. Our results show that resistivity tends to decrease with dust growth, which is particularly true when the dust size power exponent $q$ is $q=2.5$. On the other hand, the decrease is less pronounced when the dust size power exponent $q$ is $q=3.5$, i.e., when the small dust is also responsible for the dust cross-section. Our results suggest that detailed dust coagulation and fragmentation processes play a vital role in the magnetic resistivities in protostar formation.

https://export.arxiv.org/pdf/2208.00601

\begin{keywords} star formation -- circum-stellar disk -- methods: magnetohydrodynamics -- protoplanetary disk \end{keywords} \section{Introduction} \label{intro} Non-ideal effects (Ohmic dissipation, Hall effect, and ambipolar diffusion) play a crucial role for formation and evolution of protostars and protoplanetary disks. For example, Ohmic dissipation and ambipolar diffusion enable formation and stable existence of protoplanetary disks without catastrophic magnetic braking \citep[e.g.,][]{2011PASJ...63..555M,2015ApJ...810L..26T,2015MNRAS.452..278T,2015ApJ...801..117T, 2016A&A...587A..32M, 2016MNRAS.457.1037W, 2017ApJ...846....7K, 2017ApJ...835L..11T,2018MNRAS.473.4868Z,2021MNRAS.502.4911X}. The relatively weak magnetic field of the protostar ($\sim 1$ kG) stems from non-ideal MHD effects in the first core and disk \citep{2007ApJ...670.1198M,2015MNRAS.452..278T,2018A&A...615A...5V}. This weak magnetic field around the protostar is the key for protostellar jets formation \citep{2008ApJ...676.1088M,2013ApJ...763....6T,2019ApJ...876..149M}. The degree of impact of the non-ideal effect depends on the magnetic resistivities, which are determined by the amount of charged particles, and hence ionization chemistry. In the ionization chemistry, dust grains absorb the ions and electrons and affect their abundance. The adsorption efficiency depends on the total cross section of dust grains. Furthermore, a large population of charged small dust grains ($\lesssim$10 \nm) can contribute to the conductivities \citep{2016MNRAS.460.2050Z}. The dust size distribution in the previous studies is often assumed to be that of the interstellar medium (ISM) such as MRN size distribution \citep{1977ApJ...217..425M} or sub-micron sized dust grains. However, the distribution may change through dust coagulation during protostar formation, particularly in the protoplanetary disks. By assuming that the relative velocity among the dust is determined by turbulence, the growth timescale $\tgrowth$ of dust grains in the disk is calculated as \citep{2007A&A...466..413O}, \begin{eqnarray} t_{\rm growth} &=& 1.6 \times 10^3 \alpha_{10^{-2}}^{-1/2} \rho_{\rm mat,2 \gcm}^{1/2} a_{\rm d, 1 \mm}^{1/2} f_{0.01} ^{-1} \\ & &n_{\rm g,10^{11} \ccm}^{-1/2}c_{\rm s,190 \ms}^{-1/2} M_{*, 0.1 \msun}^{-1/4} r_{10 {\rm AU}}^ {3/4} {\rm year}. \nonumber \end{eqnarray} where $n_{\rm g}$, $\alpha$, $\rho_{\rm mat}$, $a_{\rm d}$, $f$, $c_{\rm s}$, $M_*$ denotes the gas number density, viscous $\alpha$ value, material density and size of the dust grains, dust-to-gas mass ratio, sound velocity, and mass of the central protostar, respectively. $f_{X}$ means $f_{X}=(\frac{f}{X})$. Here we assume turbulent velocity and timescale of largest eddy to be $\dv_L=\sqrt{\alpha} c_{\rm s}$ and $t_L=\Omega^{-1}$, respectively. Thus, the dust growth timescale is about 100 times smaller than the age of Class 0/I young stellar objects (YSOs), and dust growth may proceed even in the early evolution of circumstellar disks. Actually, recent 3D simulation by \citet{2021ApJ...920L..35T} shows that dust growth in the disk (and reflux of large dust to the envelope). The question we address in this paper is how the change of dust size distribution caused by the dust growth affects resistivities. Care should be taken in applying the usual chemical-reaction-network calculations to calculate the resistivities with large dust grains because the mean grain charge $\langle Z \rangle$ of $\gtrsim 1 \mum$ is typically \citep{1987ApJ...320..803D} \begin{eqnarray} \langle Z\rangle \sim - 20 a_{\rm d,10 \mum} T_{10 \rm K}, \end{eqnarray} when abundances of ions and electron is much larger than that of dust grains. Since dust grains with different charges need to be treated as different chemical species in chemical reaction network calculations, a vast number of charged dust species should be considered, which is computationally demanding. Therefore, analytical models of ionization chemistry such as \citet{1987ApJ...320..803D, 2009ApJ...698.1122O,2021ApJ...913..148T} are more suitable for the calculation of resistivities with large dust grains. Thus, we adopt this approach in this paper. This paper is organized as follows. In \S \ref{sec_method}, we describe our analytical model which is based on \citet{1987ApJ...320..803D}. In \S \ref{sec_validation}, we validate the analytical model by comparing it with chemical reaction calculations. In \S \ref{sec_results}, we investigate the magnetic resistivities with large dust grains. Finally, the results are summarized and discussed in \S \ref{discussion}. \section{Equilibrium charge distribution and magnetic resistivity} \label{sec_method} \subsection{Equilibrium of ionization recombination reaction} We start from equations for chemical equilibrium in the gas phase. \begin{eqnarray} \label{chemical_equilibrium} \zeta n_{\rm g} -s_{\rm i} u_{\rm i} n_{\rm i} \overline{( \sigma_{\rm d}\langle \Jt_{\rm i}(I,Z)\rangle)} n_{\rm d}-\beta n_{\rm i} n_{\rm e}&=&0 \nonumber \\ \zeta n_{\rm g} -s_{\rm e} u_{\rm e} n_{\rm e} \overline{( \sigma_{\rm d}\langle \Jt_{\rm e}(I,Z)\rangle)} n_{\rm d}-\beta n_{\rm i} n_{\rm e}&=&0 \end{eqnarray} where $n_{\rm d}$ is the dust number density. $ u_{\rm i}=n_{\rm i}^{-1}\sum_k n_{\rm i}^{(k)} u_{\rm i}^{(k)}$, $ \beta=n_{\rm i}^{-1}\sum_k n_{\rm i}^{(k)} \beta^{(k)}$, and $ \zeta=n_{\rm g}^{-1}\sum_k n_{\rm g}^{(k)} \zeta^{(k)}$, are average ion velocity, gas-phase recombination rate coefficient, ionization rate. Here $n_{\rm i}=\sum_k n_{\rm i}^{(k)}$ and $n_{\rm g}=\sum_k n_{\rm g}^{(k)}$ are the total number density of ion and neutral respectively. $\Jt_{\rm i(e)}(I,Z)$ is the effective cross sections normalized by $\sigma_{\rm d}(I)=\pi (a_{\rm d}(I))^2$ between dust grains and ion (electron). $a_{\rm d}$ is the dust radius. $\overline{(A)}$ and $\langle A \rangle$ denotes the average over dust size ($I$) and charge ($Z$), respectively. \citet{1987ApJ...320..803D} derive the approximation formula for the effective cross section of charged particles as, \begin{eqnarray} \label{Jtilde_mean} \Jt(\tau, \nu)=\begin{cases} \left(1-\frac{\nu}{\tau}\right)\left[1+\left(\frac{2}{\tau-2\nu}\right)^{1/2}\right] ~(\nu < 0)~ \nonumber \\ 1+(\frac{\pi}{2 \tau})^{1/2}~ (\nu = 0) \\ \left[1+(4\tau+3\nu)^{-1/2}\right]^2\exp(-\frac{\nu}{\tau(1+\nu^{-1/2})}) ~(\nu > 0), \end{cases} \end{eqnarray} where $\nu=\nu(Z,q_{\rm i(e)})=Ze/q_{\rm i(e)}$ where $q_{\rm i(e)}$ are charge of ion or electron. $\tau=\tau(I)= a_{\rm d}(I) k_{\rm B} T/e^2$ is the normalized temperature and $e$ is the elementary charge. These formula are correct within few \% for $\tau>10^{-3}$ (thus, the particle size of $a_{\rm d}>1.67 {\rm nm} (T/(10 K))^{-1}$ which is enough for our purpose). Using $\Jt(\tau, \nu)$, we obtain, \begin{eqnarray} \label{Jtilde_ion} \Jt_{\rm i}(I, Z)=\begin{cases} \left(1-\frac{Z}{\tau(I)}\right)\left[1+\left(\frac{2}{\tau(I)-2Z}\right)^{1/2}\right] ( Z < 0) \\ 1+(\frac{\pi}{2 \tau(I)})^{1/2} (Z = 0) \\ \left[1+(4\tau(I)+3Z)^{-1/2}\right]^2\exp(-\frac{Z}{\tau(I)(1+Z^{-1/2})}) (Z > 0). \end{cases} \end{eqnarray} for singly charged ions and \begin{eqnarray} \label{Jtilde_electron} \Jt_{\rm e}(I, Z)=\begin{cases} \left(1+\frac{Z}{\tau(I)}\right)\left[1+\left(\frac{2}{\tau(I)+2Z}\right)^{1/2}\right] ~{\rm for} ~ Z > 0 \\ 1+(\frac{\pi}{2 \tau(I)})^{1/2} ~{\rm for} ~ Z = 0 \\ \left[1+(4\tau(I)-3Z)^{-1/2}\right]^2\exp(\frac{Z}{\tau(I)(1+(-Z)^{-1/2})}) ~{\rm for} ~ Z < 0. \end{cases} \end{eqnarray} for electrons. The charge neutrality condition is given as \begin{eqnarray} \label{charge_neutrality} n_{\rm i}-n_{\rm e}+\overline{\langle Z \rangle}n_{\rm d}=0. \end{eqnarray} The governing equations for dust charging is the detailed balance equation for dust grains which is given as \begin{align} \label{detailed_balance} &n_{\rm i}s_{\rm i}u_{\rm i} n_{\rm d}(I,Z) \sigma_{\rm d}(I) \Jt_{\rm i}(I,Z)= \nonumber \\ &n_{\rm e}s_{\rm e}u_{\rm e} n_{\rm d}(I,Z+1) \sigma_{\rm d}(I) \Jt_{\rm e}(I,Z+1) \nonumber \\ &\therefore \epsilon \frac{n_{\rm d}(I,Z) \Jt_{\rm i}(I,Z)}{n_{\rm d}(I,Z+1) \Jt_{\rm e}(I,Z+1)}=1, \end{align} where we define \begin{eqnarray} \epsilon \equiv \frac{n_{\rm i}s_{\rm i}u_{\rm i}}{n_{\rm e}s_{\rm e}u_{\rm e}}. \end{eqnarray} The final governing equation is number density conservation for each $I$ \begin{eqnarray} \label{number_density_conservation_Z} n_{\rm d}(I)=\sum_Z n_{\rm d}(I,Z). \end{eqnarray} We assume that $n_{\rm g}$, $\zeta$, $s_{\rm i(e)}$, $a_{\rm d}(I)$, $\beta$, and $n_{\rm d}(I)$ are known. Our purpose is to obtain $n_{\rm i}$, $n_{\rm e}$, and $n_{\rm d}(I,Z)$. In the following three subsections, we describe the procedure for computing these quantities. \subsection{Low $\tau$ case} For $\tau\ll1$, the dust charge concentrates to $Z=-1,0,1$ \citep{1987ApJ...320..803D}. Therefore, equations (\ref{detailed_balance}) and (\ref{number_density_conservation_Z}) are reduced to be \begin{eqnarray} \label{low_temp_case_detailed_balance1} \epsilon \frac{n_{\rm d}(I,-1) \Jt_{\rm i}(I,-1)}{n_{\rm d}(I,0) \Jt_{\rm e}(I,0)}&=&1, \\ \label{low_temp_case_detailed_balance2} \epsilon \frac{n_{\rm d}(I,0) \Jt_{\rm i}(I,0)}{n_{\rm d}(I,1) \Jt_{\rm e}(I,1)}&=&1, \end{eqnarray} and \begin{eqnarray} \label{low_temp_number_density_conservation_Z} n_{\rm d}(I)=n_{\rm d}(I,-1)&+&n_{\rm d}(I,0)+n_{\rm d}(I,1). \end{eqnarray} Furthermore, we prohibit the transition to $|Z|>1$, \begin{eqnarray} \label{low_temp_case_detailed_balance3} n_{\rm d}(I,-1) \Jt_{\rm e}(I,-1)=0, \nonumber \\ n_{\rm d}(I,1) \Jt_{\rm i}(I,1)=0. \end{eqnarray} By using $\Jt_{\rm i}(I,Z=-1)=\Jt_{\rm e}(I,Z=1)=\Jt(\tau(I),\nu=-1)$, we obtain \begin{eqnarray} \label{low_temp_case_number_density} n_{\rm d}(I,-1) &=& \frac{1}{\epsilon} \Omega(I) n_{\rm d}(I,0) \nonumber\\ n_{\rm d}(I,1)&=& \epsilon \Omega(I) n_{\rm d}(I,0) \nonumber\\\nonumber\\ n_{\rm d}(I,0)&=& \frac{n_{\rm d}(I)}{\Xi(I, \epsilon)}, \nonumber\\ \end{eqnarray} from equations (\ref{low_temp_case_detailed_balance1}), (\ref{low_temp_case_detailed_balance2}) and (\ref{low_temp_number_density_conservation_Z}). Here we have introduced \begin{eqnarray} \Omega(I) = \frac{\Jt(\tau(I),0)}{\Jt(\tau(I),-1)}, \\ \Xi(I, \epsilon) = \left[\frac{\Omega(I)}{\epsilon}+1+\epsilon \Omega(I) \right]. \end{eqnarray} By knowing $n_{\rm d}(I,Z)$, we can calculate $\langle Z \rangle$, $\langle \Jt_{\rm i}(\tau, Z) \rangle$, and $\langle \Jt_{\rm e}(\tau, Z) \rangle$ for low $\tau$ case as \begin{eqnarray} \label{low_temp_case_Z_J} \langle Z \rangle_{Z, {\rm low}}&=&\frac{\Omega(I)}{\Xi(I,\epsilon)}\left[\epsilon- \epsilon^{-1} \right], \\ \langle \Jt_{\rm i}(\tau(I)) \rangle_{Z, {\rm low}}&=&\frac{1}{\Xi(I,\epsilon)}\left[\epsilon^{-1} \Omega(I) \Jt_{\rm i}(I,-1) \right. \nonumber \\ &+& \left. \Jt_{\rm i}(I,0)+ \epsilon \Omega(I) \Jt_{\rm i}(I,1) \right] \nonumber \\ &=&\frac{\Jt(I,0) }{\Xi(I,\epsilon)}\left[\epsilon^{-1}+1 \right], \\ \langle \Jt_{\rm e}(\tau(I)) \rangle_{Z, {\rm low}}&=&\frac{1}{\Xi(I,\epsilon)}\left[\epsilon^{-1} \Omega(I) \Jt_{\rm e}(I,-1) \right. \nonumber \\ &+& \left. \Jt_{\rm e}(I,0)+ \epsilon \Omega(I) \Jt_{\rm e}(I,1) \right] \nonumber \\ &=&\frac{\Jt(I,0) }{\Xi(I,\epsilon)}\left[\epsilon +1 \right], \end{eqnarray} where $\Jt_{\rm i}(I,1)=0$ and $\Jt_{\rm e}(I,-1)=0$ have been used. The results above generalize the low $\tau$ case of \citet{1987ApJ...320..803D}. If we assume \begin{eqnarray} \label{low_temp_case_detailed_balance_draine} \epsilon \frac{n_{\rm d}(I,-1) \Jt_{\rm i}(I,-1)}{n_{\rm d}(I,0) \Jt_{\rm e}(I,0)}&=&1, \nonumber\\ n_{\rm d}(I)&=&n_{\rm d}(I,-1)+n_{\rm d}(I,0), \nonumber\\ n_{\rm d}(I,-1) \Jt_{\rm e}(I,-1)&=&0, \nonumber\\ n_{\rm d}(I,0) \Jt_{\rm i}(I,0)&=&0, \end{eqnarray} (i.e., we only consider $Z=0, -1$), instead of equations (\ref{low_temp_case_detailed_balance2}), (\ref{low_temp_number_density_conservation_Z}), and (\ref{low_temp_case_detailed_balance3}), and $\Jt(\tau, 0)\sim \sqrt{\pi/(2\tau)}$, $\Jt(\tau, -1)\sim 2/\tau$, we recover equations (4.11) to (4.13) of \citet{1987ApJ...320..803D}. \subsection{High $\tau$ case} The equations in the previous subsection hold as long as the dust charge remains at $Z=\pm 1, 0$. However, as $\tau$ becomes large (i.e., the temperature of the gas increases or the size of the dust increases), the typical dust charge becomes $Z \sim -k_{\rm B} T a_{\rm d}/e^2 =-\tau \ll -1 $ and very small. Thus, the strategy in the previous section of solving the detailed balancing equations in sequence is not useful for $Z \ll 1 $ (or $\tau \gg1$) because we have to consider a large number of detailed balance equations. On the other hand, for large $\tau$, the dust charge distribution can be treated as a continuous distribution, known to become Gaussian distribution. Then $\langle Z \rangle , \langle \Jt_{\rm i} \rangle , \langle \Jt_{\rm e} \rangle$ can be obtained analytically \citep{1987ApJ...320..803D,2009ApJ...698.1122O}. For $\tau \gg 1$, we can approximate $\Jt_{\rm i}(I, Z)$ and $\Jt_{\rm e}(I, Z)$ of equation (\ref{Jtilde_ion}) and (\ref{Jtilde_electron}) assuming $Z<0$, \begin{eqnarray} \label{Jtilde_ion_high} \Jt_{\rm i}(I, Z)= \left(1-\frac{Z}{\tau(I)}\right) \end{eqnarray} for singly charged ion and \begin{eqnarray} \label{Jtilde_ion_high} \Jt_{\rm e}(I, Z)= \exp \left(\frac{Z}{\tau(I)}\right) \end{eqnarray} for electron. For $\tau \gg 1$, the solution of detailed balance equation (i.e., equation (\ref{detailed_balance})) is given as \citep{1987ApJ...320..803D,2009ApJ...698.1122O}, \begin{eqnarray} \label{Gaussian} n_{\rm d}(I,Z)=\frac{n_{\rm d}(I)}{\sqrt{2 \pi \langle \Delta Z^2\rangle}} \exp\left[-\frac{(Z-\langle Z \rangle)^2}{2 \langle \Delta Z^2 \rangle}\right], \end{eqnarray} where \begin{eqnarray} \label{Gaussian_Z} \langle Z \rangle_{Z, {\rm high}}&=&\psi \tau, \\ \label{Gaussian_dZ} \langle \Delta Z^2 \rangle&=&\frac{1-\psi}{2-\psi} \tau. \\ \end{eqnarray} The dimensionless parameter $\psi$ is the solution of the equation of \begin{eqnarray} s_{\rm i} n_{\rm i} u_{\rm i} \Jt_{\rm i}(I, \langle Z \rangle )&=& s_{\rm e} n_{\rm e} u_{\rm e} \Jt_{\rm e}(I,\langle Z \rangle) \nonumber,\\ \label{psi_eq} \therefore \epsilon (1-\psi)&=&\exp \left( \psi \right), \end{eqnarray} and hence, $\psi$ is a function of $\epsilon$. Equations (\ref{Gaussian_Z}) to (\ref{psi_eq}) are derived from the detailed balance equation (\ref{detailed_balance}) with the assumption of $n_{\rm d}(I,Z+1)\sim n_{\rm d}(I,Z)+\partial n_{\rm d}(I,Z)/\partial Z $ and $\Jt_{\rm e}(I,\langle Z \rangle +1) \sim\Jt_{\rm e}(I,\langle Z \rangle)$ \citep[for the detail, see][]{2009ApJ...698.1122O}. Using these equations, we can calculate $\langle Z \rangle$, $\langle \Jt_{\rm i}(\tau, Z) \rangle$, and $\langle \Jt_{\rm e}(\tau, Z) \rangle$ for high temperature case as, \begin{eqnarray} \label{high_temp_case_Z_J} \langle \Jt_{\rm i}\rangle_{Z, {\rm high}} &=&(1-\psi(\epsilon)) \\ \langle \Jt_{\rm e}\rangle_{Z, {\rm high}} &=&\exp[\psi(\epsilon)]. \end{eqnarray} \subsection{Number density of ions and electrons} In the previous two sections we have obtained $\langle Z \rangle$, $ \langle \Jt_{\rm i}\rangle$, and $ \langle \Jt_{\rm i}\rangle$ for each $I$ in the high and low $\tau$ limits. Following the approach of \citet{1987ApJ...320..803D}, we approximate these values for general $\tau(I)$ as, \begin{eqnarray} \label{general_approx1} \langle Z \rangle(I, \epsilon)&=&\langle Z \rangle_{{\rm high}}(I, \psi(\epsilon))+\langle Z \rangle_{\rm {\rm low}}(I, \epsilon), \\ \langle \Jt_{\rm i} \rangle(I, \epsilon) &=& \langle \Jt_{\rm i}\rangle_{Z, {\rm high}}(\psi(\epsilon))+ \langle \Jt_{\rm i} \rangle_{Z, {\rm low}}(I, \epsilon), \\ \label{general_approx2} \langle \Jt_{\rm e} \rangle(I, \epsilon)&=& \langle \Jt_{\rm e}\rangle_{Z, {\rm high}}(\psi(\epsilon))+\langle \Jt_{\rm e} \rangle_{Z, {\rm low}} (I, \epsilon). \end{eqnarray} Here we explicitly write the variables of these quantities. By summing these up for $I$, we can calculate $\overline{\langle Z\rangle}$, $ \overline{\langle \Jt_{\rm i}\rangle}$, and $ \overline{\langle \Jt_{\rm i}\rangle}$ as a function of $\epsilon$. Then $n_{\rm i}$ and $n_{\rm e}$ are obtained from equation (\ref{chemical_equilibrium}) as a function of $\epsilon$, \begin{align} n_{\rm i}&\equiv n_{\rm i}(\epsilon)=\frac{u_{\rm e} s_{\rm e} \overline{\sigma_{\rm d} \langle \Jt_{\rm e}\rangle}n_{\rm d}}{2 \beta} \\ &\left(\sqrt{1+\frac{4 \beta \zeta n_{\rm g}}{s_{\rm i} u_{\rm i} s_{\rm e} u_{\rm e} \overline{\sigma_{\rm d} \langle \Jt_{\rm i}\rangle}~ \overline{ \sigma_{\rm d}\langle \Jt_{\rm e}\rangle}n_{\rm d}^2}}-1 \right), \nonumber \\ n_{\rm e}&\equiv n_{\rm e}(\epsilon)=\frac{u_{\rm i} s_{\rm i} \overline{\sigma_{\rm d} \langle \Jt_{\rm i}\rangle}n_{\rm d}}{2 \beta} \\ &\left(\sqrt{1+\frac{4 \beta \zeta n_{\rm g}}{s_{\rm i} u_{\rm i} s_{\rm e} u_{\rm e} \overline{\sigma_{\rm d} \langle \Jt_{\rm i}\rangle}~ \overline{ \sigma_{\rm d}\langle \Jt_{\rm e}\rangle}n_{\rm d}^2}}-1\right). \nonumber \end{align} The charge neutrality condition becomes, \begin{eqnarray} \label{gx} n_{\rm i}(\epsilon)-n_{\rm e}(\epsilon)+n_{\rm d}\overline{\langle Z \rangle}(\epsilon)=0. \end{eqnarray} Equations (\ref{gx}) is a nonlinear algebraic equation for $\epsilon$, and we solve this equation with Newton-Raphson method. \subsection{Conductivity and magnetic resistivity} Using the $n_{\rm i}$, $n_{\rm e}$, and $n_{\rm d}(I, Z)$ obtained in the previous sections, the conductivity is calculated as follows \citep{2007Ap&SS.311...35W}, \begin{eqnarray} \sigma_{\rm O}&=&\sum_{\rm s} \frac{c}{B} n_{\rm s}q_{\rm s}\beta_{\rm s},\\ \sigma_{\rm H}&=&-\sum_{\rm s} \frac{c}{B} \frac{n_{\rm s} q_{\rm s}\beta_{\rm s}^2}{1+\beta_{\rm s}^2},\\ \sigma_{\rm P}&=&\sum_{\rm s} \frac{c}{B} \frac{n_{\rm s} q_{\rm s} \beta_{\rm s}}{1+\beta_{\rm s}^2} . \end{eqnarray} where $\sigma_{\rm O, H, P}$ are the Ohmic, Hall, and Pedersen conductivities, respectively, of the charged species. \begin{eqnarray} \beta_{\rm s}=\frac{q_{\rm s} B}{m_{\rm s} c \gamma_{\rm s} m_{\rm g} n_{\rm g}}, \end{eqnarray} is the product of the cyclotron frequency and the collision frequency with the neutral gas. The subscript ${\rm s}$ denotes the charged species. Here $n_{\rm s}$ and $q_{\rm s}$ are the number density and charge of the species ${\rm s}$. $B$ and $c$ are the magnetic field strength and speed of light, respectively. $\gamma_{\rm s} =\langle \sigma v\rangle_{\rm s}/(m_{\rm s}+m_{\rm g})$ and $\langle \sigma v\rangle_{\rm s}$ is the collisional momentum transfer rate between species ${\rm s}$ and the neutrals. $m_{\rm g}$ is the mean mass of the gas. The momentum transfer rate between neutral and charged species was calculated using the equations described in \citet{2008A&A...484...17P}. The conductivities of dust grains are separately calculated from low temperature and high temperature dust size distribution (equations (\ref{low_temp_case_number_density}) and (\ref{Gaussian})) and then summed up. This treatment is necessary to correctly calculate the Pedersen conductivity to which both the positively and negatively charged dust grains positively contribute. This method adds the conductivity of the dust in duplicate, but we confirmed that this does not cause an error because the contribution of dust at higher temperature is small (see \S \ref{sec_validation}). The Ohmic, Hall, and ambipolar resistivities are calculated as \begin{eqnarray} \eta_{\rm O}&=&\frac{c^2}{4 \pi}\frac{1}{\sigma_{\rm O}},\\ \eta_{\rm H}&=&\frac{c^2}{4 \pi}\frac{\sigma_{\rm H}}{(\sigma_{\rm H}^2+\sigma_{\rm P}^2)},\\ \eta_{\rm A}&=&\frac{c^2}{4 \pi}\frac{\sigma_{\rm P}}{(\sigma_{\rm H}^2+\sigma_{\rm P}^2)}-\eta_{\rm O}. \end{eqnarray} \subsection{Chemical reaction network calculation} We perform chemical reaction network calculations to compare with the analytical model above. In the chemical reaction network calculations, we consider ion species ${\rm H^+,H_2^+,H_3^+,HCO^+,Mg^+}$ ${\rm He^+,C^+,O^+,O_2^+,H_3O^+,OH^+,H_2O^+}$ and neutral species ${\rm H,H_2, He, CO, O_2, Mg, O, C, HCO, H_2O, OH, N, Fe}$. We also consider neutral and singly charged dust grains, G$^0$, G$^-$, G$^+$. We consider cosmic-ray ionization, gas-phase and dust-surface recombination, and ion-neutral reactions. We also considered the indirect ionization by high-energy photons emitted by direct cosmic-ray ionization (described as CRPHOT in the UMIST database). The initial abundance and reaction rates are taken from the UMIST2012 database \citep{2013A&A...550A..36M}. We neglect grain-grain collisional neutralization so that the chemical network calculations are consistent with the analytical model. The chemical reaction network is solved using the CVODE package \citep{hindmarsh2005sundials}. We calculate the conductivities using the abundances of charged species in the equilibrium state. \section{Results} In this section, we compare the analytical calculation with chemical network calculations and previous studies to justify the analytical calculations in \S 3.1. In \S 3.2, we investigate the impact of the dust size distribution with large dust grains on magnetic resistivity using the analytical calculation. \subsection{Validation of the analytic model} \label{sec_validation} In this subsection, we compare the analytic model with the chemical reaction network calculation. Here, we assume that the temperature is $T=10(1+\gamma_T (n_{\rm g}/n_c)^{(\gamma-1)}) ~{\rm K}$, where $\gamma=7/5$ and $n_c=2.6\times 10^{10} \ccm$, the magnetic field is $0.2 n_{\rm g,\ccm}^{1/2} \mu G$ \citep[i.e., assuming flux freezing; see e.g.,][]{2002ApJ...573..199N}, the dust internal density is $\rho_{\rm mat}=2 \gcm$, the dust-to-gas mass ratio is $f=0.01$, and the cosmic ray ionization rate of $\xi_{\rm CR}=10^{-17} {\rm s^{-1}}$ except for the calculations presented in figure \ref{abundance_marchand}. We assume that the dominant ion is HCO$^+$ and adopt its recombination rate of $\beta=2.4 \times 10^{-7}(T/300)^{-0.69}$ and its mean molecular weight of $\mu_{\rm I}=29$ for ion in the analytic model. \subsubsection{Fractional abundances} Figure \ref{abundance} shows the fractional abundance of ions, electrons and dust grains for mono-sized dust of $a_{\rm d}=0.1 \mum$ and MRN dust size distribution \citep{1977ApJ...217..425M}, in which the dust size is assumed to be \begin{eqnarray} \label{power_ad} \frac{d n_{\rm d}}{d a_{\rm d}} = A a_{\rm d}^{-q} (a_{\rm min}<a_{\rm d}<a_{\rm max}), \end{eqnarray} where $q=3.5$, $a_{\rm min}=5 \nm$, and $a_{\rm max}=250 \nm$. $A=(4-q)\rho_{\rm d}/((4/3\pi)\rho_{\rm mat} (\mu_{\rm g}/\mu_{\rm H}) n_{\rm g} |a_{\rm max}^{4-q}-a_{\rm min}^{4-q}|)$ is a constant for normalization. $\mu_{\rm g}=2.34, \mu_{\rm H}=1.4$, and $\rho_{\rm d}$ is the dust mass density. $d n_{\rm d}~ d a_{\rm d}$ is the number of dust grains whose sizes are between $a_{\rm d}$ and $a_{\rm d}+da_{\rm d}$ per hydrogen nucleus. The top panel of figure \ref{abundance} shows the fractional abundance with $a_{\rm d}=0.1 \mum$. The ion abundance of the analytic model is in good agreement with the chemical reaction network. On the other hand, our model tends to overestimate the electron abundance in high density region. The figure shows that ion and electron abundance difference between analytic calculation and chemical reaction network is within a factor of three. The negatively charged dust grains G$^-$ and neutral dust grans G$^0$ are dominant in low ($\lesssim 10^{10} \ccm$) and high-density regions ($\gtrsim 10^{10} \ccm$), respectively, and are in a good agreement between the analytic model and chemical reaction network. Although the abundance of G$^+$ and G$^0$ slightly different between the two calculations around $10^8 \ccm$, this does not cause errors for conductivities. Bottom panel of figure \ref{abundance} shows the fractional abundance with MRN size distribution. Even with the size distribution, the similar trend is seen as in the case with $a_{\rm d}=0.1 \mum$, and the analytic model and chemical reaction network are in a good agreement. \subsubsection{Conductivity and resistivity} Figure \ref{sigma_01mum} shows the conductivities from the chemical network calculation and the analytic calculation of this work with $a_{\rm d}=0.1 \mum$. The figure shows that electrons dominate Ohmic conductivity, so there is about a factor of three discrepancy over the entire region due to differences in the electron abundance. On the other hand, for the Hall and Pedersen conductivities, the deviation is much smaller than for the Ohmic conductivity because ions dominate them. As a result, the resulting error for Ohmic resistivity is also about a factor of three, and for Hall and ambipolar resistivity, the error is smaller than that apart from $\eta_{\rm H}$ of the very low and high density region as shown in figure \ref{eta_01mum}. In this figure, we also plot the conductivities obtained by the method of \citet{2009ApJ...698.1122O} which is valid in $\tau \gg 1$. Since $\tau<1$ almost everywhere in this plot, we can see that the difference becomes larger when the dust charge affects the ion/electron abundance. Figure \ref{sigma_MRN} shows the conductivities from the chemical network calculation and the analytic model with MRN size distribution. In the low-density region of $n_{\rm g} \lesssim 10^{11} \ccm$, Ohmic conductivity is determined by that of electrons as with $a_{\rm d}=0.1 \mum$, so there is up to about a factor of three discrepancy due to differences in the abundance of electrons. In the high-density region of $n_{\rm g} \gtrsim10^{11} \ccm$, Ohmic conductivity is determined by that of dust grains, and the discrepancy becomes much smaller. For Hall conductivity, the deviation is sufficiently small apart from the low-density region of $n_{\rm g}\lesssim10^{5}$ which is due to simplified ion treatment and high-density region of $n_{\rm g}\gtrsim10^{14}$ which is due to the difference of electron abundance. The deviation is sufficiently small for Pedersen conductivity because they are mainly determined by small dust grains. As a result, the resulting error for resistivities is also within a factor of three for MRN size distribution apart from the low density region of $n_{\rm g}<10^{-6} \ccm$ where non-ideal MHD effect is not important as shown in figure \ref{eta_MRN}. \subsubsection{Average dust charge from low to high temperature} The results in the previous subsection only confirm that the analytical model is consistent with chemical reaction network calculations for small $\tau$ (i.e., small dust and low temperature). In this subsection, we will further check that our model in the high-temperature regime is consistent with \citet{1987ApJ...320..803D}. Figure \ref{Zmean_tau} shows the mean dust grain charge $\langle Z \rangle$ as a function of normalized temperature $\tau$. The cyan dashed line shows the analytic formula of \citet{1987ApJ...320..803D} which is given as \begin{eqnarray} \label{DS87} \langle Z \rangle=\frac{-1}{1+\sqrt{\tau_0/\tau}}+\psi\tau, \end{eqnarray} where $\tau_0=8/(\pi \mu)(m_{\rm e}/m_p)$ and $\mu=(s_{\rm e} n_{\rm e}/n_{\rm i})^2(m_{\rm i}/m_p)$. The figure shows $\langle Z \rangle$ obeys $\langle Z \rangle\propto \tau$ and our analytic model well reproduces the analytic formula of \citet{1987ApJ...320..803D}. This means that our model can correctly calculate the ionization state of dust grains not only for small $\tau$ but also large $\tau$. \subsection{Impact of dust size on magnetic resistivity} \label{sec_results} In this subsection, we investigate the impact of dust growth on the magnetic resistivities. We assume power law dust size distribution (equation (\ref{power_ad})) and vary $a_{\rm min}$, $a_{\rm max}$, and $q$ as parameters. The important quantity is total cross-section $S_{\rm tot}$ of the dust grains, \begin{align} S_{\rm tot} &= \mu_{\rm H}/\mu_{\rm g} n_{\rm g} A \int_{a_{\rm min}}^{a_{\rm max}} \pi a_{\rm d}^2 a_{\rm d}^{-q} da_{\rm d} \nonumber \\ &\propto \frac{|{a_{\rm min}}^{-q+3}-{a_{\rm max}}^{-q+3}|}{|{a_{\rm min}}^{-q+4}-{a_{\rm max}}^{-q+4}|}, \end{align} which determines the absorption efficiency of ions and electrons. If $4>q>3$ and $a_{\rm max} \gg a_{\rm min}$, $ S_{\rm tot} \propto {a_{\rm min}}^{-q+3}/{a_{\rm max}}^{-q+4}$ and both small and large dust grains affects the total cross section. In this case, it is expected that the impact of dust growth would be less significantly on the absorption efficiency of ions and electrons, and hence resistivities. On the other hand, if $q<3$ and $a_{\rm max} \gg a_{\rm min}$, $ S_{\rm tot} \propto {a_{\rm max}}^{-1}$ and only large dust grains determines the total surface area. Thus, it is expected that dust growth would significantly change the resistivities for $q<3$. It is pointed out that $q\sim 2.5$ when the coagulation process dominates while $q\sim 3.5$ when the disruption process dominates \citep{1993Icar..106...20M}. Thus, we investigate the resistivities with $q=2.5$ and $q=3.5$. On the other hand, a large population of charged small grains ($\lesssim$10 \nm) can be responsible for the conductivities \citep[][see also figure \ref{sigma_MRN}]{2016MNRAS.460.2050Z}, and decreases the resistivities at envelope and disk \citep{2018MNRAS.478.2723Z,2019MNRAS.484.2119K,2020ApJ...900..180M, 2020ApJ...896..158T}. Thus, whether such small grains exist or not would also be important. In this subsection, we investigate the resistivities with $a_{\rm min}=5 \nm$ and $a_{\rm min}=0.1 \mum$. Figure \ref{eta_amin5_q25} shows the resistivities with $a_{\rm min}=5 \nm$ and $q=2.5$ for different $a_{\rm max}$. The figure shows that $\eta_{\rm O}$ tends to decrease with increasing $a_{\rm max}$ aside from $a_{\rm max}=0.25\mum$ to $a_{\rm max}=2.5\mum$ in high density region. The increase of $\eta_{\rm O}$ and $\eta_{\rm A}$ from $a_{\rm max}=0.25\mum$ to $a_{\rm max}=2.5\mum$ in the high-density region is due to the decrease of dust conductivity. Then, it can be seen that resistivities converge to the power law of the form of $\eta_{\rm O, H, A} \propto n_{\rm g}^{p}$ with respective constant power exponent ${\rm p}$ as the dust size increases to $a_{\rm max}=2.5\times 10^3\mum$. They are power laws determined by the chemical equilibrium of cosmic-ray ionization and gas-phase recombination. $\eta_{\rm H}$ becomes positive (shown with dashed lines) in $a_{\rm max}> 250 \mum$ almost entire density region. This is because the relative velocity between ions and electrons determines the Hall current. Figure \ref{eta_amin5_q35} shows the resistivities with $a_{\rm min}=5 \nm$ and $q=3.5$ for different $a_{\rm max}$. The figure shows that $\eta_{\rm O}$ tends to decrease with increasing $a_{\rm max}$ in the low-density region, which is due to the decrease of the total cross-section. On the other hand, $\eta_{\rm O}$ increases in the high-density region, which is due to the decrease of dust conductivity. $\eta_{\rm H}$ and $\eta_{\rm A}$ also increase in the high-density region, which is also due to the decrease of dust conductivity there. The difference between $q=2.5$ (dotted lines) and $q=3.5$ (solid lines) is striking. If we compare $\eta_{\rm O}$ and $\eta_{\rm A}$ at $10^{15}\ccm$ between figure \ref{eta_amin5_q25} and \ref{eta_amin5_q35} , the difference is more than $10^3$ times greater, which may significantly affect the disk evolution. Figure \ref{eta_amin100_q25} shows the resistivities with $a_{\rm min}=0.1 \mum$ and $q=2.5$ for different $a_{\rm max}$. By removing small dust grains, resistivity behavior is simplified because the dust grains themselves are no longer responsible for conductivity and serve only as adsorber of ions and electrons. The figure shows that $\eta_{\rm O}$ tends to decrease with increasing $a_{\rm max}$ in the entire density region. It can be seen that resistivities converge to the single power law as the dust size increases, which is the same as the figure \ref{eta_amin5_q25}. Again $\eta_{\rm H}$ becomes positive in $a_{\rm max}> 250 \mum$ almost entire density region, which is also consistent with the figure \ref{eta_amin5_q25}. Figure \ref{eta_amin100_q35} shows the resistivities with $a_{\rm min}=0.1 \mum$ and $q=3.5$ for different $a_{\rm max}$. Similar to the result of figure \ref{eta_amin100_q25}, $\eta_{\rm O}$ tends to decrease with increasing $a_{\rm max}$. However, the decrease is less pronounced because of the contribution of the small dust to the cross-section. $\eta_{\rm A}$ increases in the low-density region of $n_{\rm g} \lesssim 10^{10} \ccm$ and converges to the single power law as the dust size increases. On the other hand, it decreases with increasing $a_{\rm max}$ in the high density region of $n_{\rm g} \gtrsim 10^{10} \ccm$. However, again the decrease is less pronounced. $\eta_{\rm H}$ tends to decrease with increasing $a_{\rm max}$. \section{Discussion} \label{discussion} In this paper, we investigate the impact of dust size distribution with large dust grains on the magnetic resistivities using the analytic method based on \citet{1987ApJ...320..803D}. Our test results show that the analytic model can correctly calculate the ionization state from small $\tau$ (low temperature or small dust) to large $\tau$ (high temperature or large dust). Therefore, the method is applicable to a dust size distribution that simultaneously contains small dust with $\tau<1$ and large dust with $\tau>1$, and can be used over a broader class of dust size distribution than previous study which uses the Gaussian charge distribution such as \citet{2009ApJ...698.1122O,2021ApJ...913..148T}. The calculation results with large dust grains show that the resistivity tends to decrease with dust growth. This is particularly true when the dust size power exponent $q$ is $q=2.5$ (i.e., in the case the coagulation process dominates in the dust size evolution, and only large dust grains are responsible for the dust cross-section). On the other hand, the decrease is less pronounced when the dust size power exponent $q$ is $q=3.5$, (i.e., in the case the disruption process dominates in the dust size evolution, and the small dust grains are also responsible for the dust cross-section). Our results suggest that detailed dust coagulation and fragmentation processes play a crucial role to investigate the impact of non-ideal effects, in particular in the high density region of $10^{10} \ccm$. Recently, \citet{2021A&A...649A..50M} proposed a similar method that also can be used to calculate magnetic resistivity analytically. Our numerical tests showed that our method seems to be more robust and applicable over a wide parameter range (e.g., when the dust grains are highly depleted). When we implemented and tested their algorithm, we found that it did not converge in the limit where the total dust charge goes zero. This is because their method uses $\psi$ as the basic variable and use equation (A.3) and (A.4) of \citet{2021A&A...649A..50M} to determine $\epsilon$ and $n_{\rm i}$. However their equation (A.4) becomes singular when $\epsilon=1$ and $\langle \bar{Z} \rangle=0$ i.e., gas phase recombination determines the ionization state. That would be a reason why the solution does not converge. Our model does not include charge neutralization due to grain-grain collisions which many previous studies have included \citep[e.g.,][]{1990MNRAS.243..103U,2015MNRAS.452..278T,2016A&A...592A..18M}. One might find this to be a flaw in our model. However, we would argue here that inclusion of charge neutralization by grain-grain collisions is debatable and may not necessarily describe a realistic dust charge state in dense region. This is because, when (sub-)micron-sized small dust particles collide, the dust particles tends to coalesce and grow rather than bounce because the collisional velocity of small dust grains tends to be much smaller than their bouncing velocity \citep{1997ApJ...480..647D,2000PhRvL..85.2426B,2012Icar..218..688W,2015ApJ...798...34G}. More quantitatively, bouncing threshold velocity for dust grains composed of SiO$_2$ bellow which the collsion results in 50 \% sticking is given as \begin{align} \Delta v_{\rm stick}=\left(\frac{m_{\rm d}}{m_{\rm th}}\right)^{-5/18} \sim 3.9 \times 10^2 \left(\frac{a_{\rm d}}{10 \nm}\right)^{-15/18} \cms, \end{align} where $m_{\rm th}=1.1 \times 10^{-15} ~{\rm g}$ \citep{2012Icar..218..688W}. On the other hand, the relative velocity of the dust (assuming Brownian motion) is given as \begin{align} \Delta v_{\rm Brown}=\sqrt{\frac{16 k_{\rm B} T}{\pi m_{\rm d}}} \sim 91 \left(\frac{a_d}{10 \nm}\right)^{-3/2} \left(\frac{T}{100 {\rm K}}\right)^{1/2} \cms \end{align} By solving the inequality of $\Delta v_{\rm stick}>\Delta v_{\rm Brown}$, we can conclude that the silicate dust grain with $a_{\rm d}>1.2 (T/100 {\rm K})^{3/4} \nm$ tends to stick rather than bounce, which is much smaller than the minimum size of MRN size distribution ($a_{\rm min}=5 \nm$). Note also icy or porous dust grains are even more sticky \citep{2009ApJ...702.1490W}. Hence, grain-grain collisions lead to dust growth and a change in the dust size distribution rather than bounce. Therefore, the approximation that ignores charge neutralization due to grain-grain collisions is not necessarily a flaw in our model, but rather a difference in the approximation of how we view the dust collision process. Note also that neglecting grain-grain neutralization does not cause the change on the resistivities when we consider the large dust (e.g., $\gtrsim 1\mum$), which is the main subject of current and our subsequent studies. This is because the charged dust grains determines the resistivities only when there are sufficient sub-micron dust grains ($\lesssim 100\nm$). Several studies have investigated the effect of dust size distribution on magnetic resistivities. \citep[e.g.,][]{2018MNRAS.478.2723Z,2020A&A...643A..17G}. Our results seems to be consistent with these studies. For example, in comparison with \citet{2018MNRAS.478.2723Z}, figure \ref{eta_amin5_q35} shows that $\eta_{\rm O}$ increases from $a_{\rm max}=2.5 \times 10^{-1} \mum$ to $a_{\rm max}=2.5 \mum$ at $n_g \sim 10^{14} \ccm$, and decreases at $n_g \sim 10^{12} \ccm$. $\eta_{\rm A}$ increases in $n_g<10^{10} \ccm$ and converged to the power law of $n_g^{-1/2}$ \citep{1983ApJ...273..202S} as dust size increases. $\eta_{\rm H}$ becomes positive and almost constant as dust size increases. Although \citet{2018MNRAS.478.2723Z} treats chemical reactions in more detail and the dust size distribution considered is different (they changed $a_{\rm min}$ instead of $a_{\rm max}$), these trends are largely consistent with their results (see their figure 5). On the other hand, the comparison with \citet{2020A&A...643A..17G} is difficult because their dust size distribution changes as density increases and they also include non-thermal dust drift due to ambipolar diffusion. However, the following consistent trends can be observed. Our figure \ref{eta_amin100_q25} shows that smaller power exponent $q$ (meaning that small dust aggregates are less abundant) causes significant decreases of $\eta_{\rm O}$ and increase of $\eta_{\rm A}$ around intermediate density ($n_g \sim 10^9 \ccm$). On the other hand, figure 9 of \citet{2020A&A...643A..17G} shows that larger $V_{\rm AD}$ (causing the removal of small dust) results in the decrease of $\eta_{\rm O}$ and the increase of $\eta_{\rm A}$ around the intermediate density. These points are consistent. The method described in this paper is less computationally expensive than conventional chemical reaction network calculations and easily converges to the solutions because it only requires to perform one-dimensional Newton-Raphson method twice (in determining $\psi$ and $\epsilon$). Our method typically requires only a few (typically 1-4 times) iterations for each Newton-Raphson calculation to obtain a result. Therefore, it can be easily used in 3D simulations with negligible computational costs. Upon request, we will provide a sample implementation of our method to the readers. We plan to use the analytic model in this paper in 3D MHD simulations of disk formation and evolution which incorporates dust growth. \section*{Acknowledgments} This work is supported by JSPS KAKENHI grant number 18H05437, 18K13581, 18K03703. \appendix \section{Comparison with \citet{2021A&A...649A..50M}} For comparison with the previous study by \citet{2021A&A...649A..50M}, we plot the number density of ions and electrons and $-\langle Z\rangle$ calculated from our analytic model in figure \ref{abundance_marchand}. In this figure, we assume the parameters of \citet{2021A&A...649A..50M}. This figure can be directly compared to figure 2 of \citet{2021A&A...649A..50M}. The results in this figure are in good agreement with their results. Quantitatively, we confirmed that $n_{\rm i}/n_{\rm e} $ converges to $\Theta \equiv s_{\rm e} (m_{\rm i}/m_{\rm e})^{1/2}=107$ in $n_{\rm g} \gtrsim 10^{12} \ccm$, which is also in good agreement with their results. \bibliography{article}

Title: Improving Black Hole Accretion Treatment in Hydrodynamical Simulations

Abstract: The large galactic scales are connected to the many orders of magnitude smaller supermassive black hole (SMBH) scales by an episodic cycle of feeding and feedback. Active galactic nuclei (AGN) are powered by accretion onto SMBH and the majority of AGN energy, in near-Eddington regime, is produced in thin sub-pc accretion discs. Currently, it is very difficult to model processes that occur on vastly different scales, ranging from the circumnuclear gas reservoirs at tens to hundreds of parsecs, down to the accretion disc scales at <0.01 pc. While sub-grid prescriptions used in large-scale or cosmological simulations are able to reproduce large-scale feedback, we propose using a more realistic model in parsec-scale simulations, where it is important to get accurate timescales to understand how feedback affects gas dynamics and star formation in the vicinity of the AGN. To test our approach we use a sub-resolution thin accretion disc model, coupled to the SMBH, in a set of hydrodynamical simulations of a retrograde collision between a gas ring and a molecular cloud in an environment similar to the Galactic centre using the SPH code Gadget-3. The disc-mediated feeding of the SMBH is relatively smooth and delayed compared to an instantaneous feeding prescription. While the reduction of accretion due to feedback is present in both accretion disc and instantaneous feeding simulations, a clear central cavity appears only in accretion disc runs - hinting that a less volatile accretion phase could have a greater impact on the surrounding gas.

https://export.arxiv.org/pdf/2208.12692

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} accretion, accretion discs -- galaxies: active -- Galaxy: centre -- Galaxy: evolution \end{keywords} \section{Introduction} \label{intro} Various correlations between properties of the supermassive black holes (SMBH) found at the centres of galaxies with the properties of their respective host galaxies imply a close link between galaxy and SMBH evolution \citep[eg.][]{1998Rees, King2003, Ciotti2007a, Ciotti2007b, 2009Cattaneo, Novak2011, 2013Kormendy, Ciotti2017}. Understanding how this co-evolution occurs is an important challenge that requires both observation and increasingly detailed modeling. Much of the difficulty in understanding these systems comes from the fact that SMBH and relevant scales (sub-parsec to a few parsecs) are extremely small when compared with the galactic (kilo-parsec) scales, yet we see that the large and the small galactic scales are connected by an episodic cycle of feeding and feedback \citep{Gaspari2020}. Kilo-parsec scale outflows are detected in many galaxies \citep{McKinley2021,Laha_rew2021} and observations at pc-scale resolution reveal an intricate picture of feeding and feedback in currently active galaxies \citep[eg.][]{Burillo2005,NGC10682019, NGC6132019, NGC10972019, General_circinus_NGC1068, NGC1275_2019, Burillo2021}. Some of these features are also detected in our own Galaxy, which is currently inactive \citep{gcrev}. Large, kpc-scale, outflows \citep{fermior, eRosita2020} are linked to the Galactic Centre by a $\sim100$~pc bridge of X-ray chimneys and radio bubbles \citep{XRAYCHIMNEY, 430RADIOBUBBLES}. Cosmological and galactic-scale simulations are capable of recreating the observed scaling relations when feedback is included \citep[eg.][]{Di_Matteo_2005,Filloux_2010,iliustris2014, Steinborn_2015, Eagle22015, Alcazar2016, Romulus2017,TNG12019, TNG22019}. In addition, the inclusion of feedback allows the simulations to recreate and explore the observed features such as Fermi bubble-like outflows \citep{TNG50_bubbles2021} or the impact of active galactic nuclei (AGN) on the interstellar medium (ISM) \citep{SIMBA_2019, ANG_wind_impact2020}. While these models are relatively successful at recreating the large scale features and observed relations, they are not capable, nor try to, accurately model the parsec-scale gas dynamics. In order to study SMBH feeding, AGN feedback and their impact on the local environment in more detail we still need to refer to smaller-scale simulations \citep[eg.][]{StarFormationGC1, Alig2, LUCASMCINFALL,Hopkins2016, CMZMOD, Tartenas2020, Tress2020} where gas dynamics in central parts of a galaxy can be resolved or utilize sub-resolution prescriptions to track unresolved physical processes \citep{Negri_2017}. The usual way to circumvent the limitation of finite resolution in large numerical simulations is to use a sub-grid prescription for an unresolved physical process. A go-to sub-grid prescription for the process of accretion was proposed by \cite{Springel_BONDi_2005} and uses the Bondi-Hoyle-Lyttleton accretion flow solution \citep[Bondi method;][]{Bondi_Lyttleton_1939, Bondi_hoyle_1944, Bondi_1952}. This method is very convenient, especially in large scale simulations, as the SMBH feeding depends entirely on gas properties ``far away'' from the black hole. In this case, gas particles are not captured and instantly removed, but are only used to determine said ambient properties on which accretion depends. Once the SMBH accretion rate is determined, its mass is increased and particles are removed stochastically from the surrounding medium ensuring approximate conservation of mass. However, the classical Bondi approach does not provide a unique way to evaluate the ambient gas properties. It also requires the unrealistic assumption of absence of angular momentum which may lead to an over- or underestimation of SMBH accretion in certain situations \citep{Hobbs_2012,Negri_2017}. These issues are somewhat mitigated by introducing a numerical correction factor $\alpha$ \citep{Springel_BONDi_2005} and/or using other modifications to the Bondi method \citep[e.g.,][]{Booth2009,AngMomInBondi, Eagle1}. An alternative to the Bondi method that also depends on the ambient properties of gas uses gravitational torques within a certain radius of the SMBH to determine its rate of accretion \citep{Alcazar2016}. Another approach is to calculate the amount of SMBH accretion from the amount of matter added to the accreting sink particle. This amount is determined not by the ambient properties of gas surrounding the sink particle (or at least not directly), but only by the particles that get swallowed after coming "close enough" and/or fulfilling some other accretion criteria (eg. low angular momentum, low internal energy, etc.). This method can be extended by coupling the SMBH to a gas reservoir that stores the accreted material and `drip-feeds' it to the SMBH \citep{Power2011, GizmoMethod_2015}. These ``two-stage'' prescriptions allow the simulators to delay and keep the black hole accretion rate close to some chosen accretion disc model. The main disadvantage of this sort of a two-stage prescription is that the properties of the accretion flow are largely dependent on a set of freely chosen parameters - viscosity timescale, accretion efficiency, etc. In particular, the viscous timescale varies dramatically even in an $\alpha-$prescription accretion disc depending on the assumed accretion radius ($t_{\rm visc}\propto R^{3/2}$). This means that this single parameter may artificially increase or reduce the amount and significantly impact the timing of feedback injections in a given simulation, since $t_{\rm visc}$ might vary between a few yr and a few Myr inside a given accretion disc \citep{Tartenas2020}, although in practice some `reasonable number' in between the two extremes is often chosen. This may not be as important in large scale simulations, but it may be critical if we want to, for example, isolate the impact that black hole wind has on star formation in the central $\sim100$~pc. Results from small scale simulations also help to improve existing sub-grid prescriptions used in large scale simulations. In this paper, we aim to improve the tracking of SMBH accretion in small-scale simulations. To do so, we extend the usual two-stage SMBH accretion model for hydrodynamical simulations by including a simple $\alpha$-accretion disc \citep{ShakuraSunyaev} prescription coupled to the SMBH sink particle. This allows us to reduce the number of free parameters with a relatively low impact on computational cost. The disc is evolved viscously with a separate timestep criterion from the simulation as a whole, but is synchronized at SMBH timesteps. Feedback is directly determined from the parameters of the accretion disc, with radiation efficiency naturally aproaching $\eta\approx0.0625$, which is correct for our chosen Paczy\'nsky-Wiita potential \citep{Paczy1980}. We test the prescription in a smoothed-particle hydrodynamics (SPH) \texttt{Gadget-3} simulation \citep{GADGET2005} of a cloud impacting a gas ring around the SMBH. We show that feedback has a significant effect in regulating the growth of the SMBH. Models with the accretion disc method produce a clear cavity in the centre, abruptly stopping any further accretion; this is not reproduced by models with instantaneous accretion. We argue that our result is more realistic than the alternative. The layout of this paper is as follows. First, we describe the the general setup of our simulations in \cref{sec:PaN}. Section \cref{sec:AccPartSetup} shows the key assumptions and equations determining the evolution the accretion disc particle. Results of the main set of hydrodynamical simulations are shown in \cref{results}, with possible implications, limitations and further research directions discussed in \cref{Discussion}. Conclusions follow in \cref{sec:Conclusions}. \section{Numerical setup}\label{sec:PaN} As our accretion disc method is intended to improve simulations of the vicinity of an AGN, we test our approach with a set of simulations of retrograde collisions between a gas ring and a molecular cloud in an environment similar to that of the Milky Way centre. In a previous paper \citep{Tartenas2020} we showed that similar configurations, without feedback, result in an AGN phase lasting $\gtrsim100$~kyr. We use the N-body/SPH code \texttt{Gadget-3} \citep{GADGET2005} with the SPHS formulation \citep{SPHS2012} and the appropriate Wendland kernel function C$^2$ \citep{kernel} with neighbour number $N_{\rm neigh} = 100$\footnote{In the making of this paper we extensively used \texttt{matplolib} \citep{matplotlib}, \texttt{pygadgetreader} \citep{pygadgetreader}, \texttt{numba} \citep{numba} and \texttt{numpy} \citep{numpy}}. The gas ring and cloud are each composed of $N_{\rm part} \approx 5\times10^5$ particles of mass $m_{\rm SPH} \approx 0.4 \, \msun$. The resolved mass is $M_{\rm res} = N_{\rm neigh} m_{\rm SPH} \approx 40 \textrm{M}_{\odot}$. The initial velocities are determined using a potential given by: \begin{equation} \phi = -\frac{GM_{\rm BH}}{r} + 2\sigma^2 \log \frac{r}{r_0}, \label{pot} \end{equation} where the first term is the gravitational potential of a point mass (the SMBH) and the second is an isothermal potential with velocity dispersion $\sigma = 100$~km$\,$~s$^{-1}$; r$_{0}$ is an arbitrary large constant. Our chosen potential corresponds to an enclosed mass $M_{\rm enc} = M_{\rm bh}$ at $R_{\rm enc} = 0.8$~pc. A more detailed description of the initial conditions and the physics included in our simulations is given below. \subsection{Initial Conditions \label{sec:IC} } \textbf{The supermassive black hole}: Our model contains a SMBH with an initial mass of $M_{\rm BH} = 4\times10^6\,\msun$. The chosen mass is similar to the SMBH mass at the centre of the Milky Way, determined from the orbits of S stars - $4.02\,\pm0.16\pm\,0.04 \times10^6\,\msun$ \citep{BHMASS2016}. The SMBH is fixed at the centre of the system. The SMBH is coupled with a sub-resolution accretion disc. Particles are accreted by this combined SMBH-accretion disc particle if they fall inside $r_{\rm sink}=0.01$~pc, which is approximately the minimal volume spatial resolution in the model, and have orbits with circularization radius $r_{\rm circ} < r_{\rm sink}$. This excludes the accretion of particles with high angular momentum. Further evolution of this gas is followed using the accretion disc particle method (see sec. \ref{sec:AccPartSetup}). \textbf{The circumnuclear ring}: The toroidal gas ring is similar in size and mass to the Circumnuclear Ring (CNR) found at the centre of the Milky Way \citep{minmass}. It has an inner radius $R_{\rm in} = 1.5$~pc, an outer radius $R_{\rm out} = 4$~pc. The mass estimates of the current CNR vary by almost two orders of magnitude; for simplicity we set the initial mass of the ring to $M_{\rm ring} = 10^5\, \msun$ \citep{gas} and set its density to a constant value. Ring particles move in circular orbits with speeds $v_{R1.5} \sim 181$ km s$^{-1}$ at the inner edge and $v_{R4} \sim 160$ km s$^{-1}$ at the outer edge. \textbf{The molecular cloud}: An infalling molecular cloud is placed $12$~pc away from the target (Fig. \ref{fig:TorCloud}). The cloud contains the same amount of gas as the CNR $M_{\rm cloud} = 10^5\, \msun\,$ and is also of constant density. We set the radius $r_{\rm cloud} = 3$~pc based on observational estimates of typical sizes of clouds of this mass in the Galaxy \citep{cloudSurvey}. The cloud is set on a parabolic orbit that crosses the ring at a point $3$~pc away from the origin along the Y axis. The initial velocity of the centre of the cloud is $v_{\rm cl} = 220$km s$^{-1}$; this corresponds to a parabolic velocity at the position of the cloud's centre. While the molecular cloud in our simulations does not correspond to any particular feature currently observed in the Galactic Centre, an appearance of such a cloud over several million years seems likely given that both observations and simulations show an inflow of matter into the central few parsecs \citep{Sormani2019, Tress2020}. In fact, the CNR itself could have been formed by a capture of a similar cloud \citep{CNRVirialmass, CNRform2015, CNRform2018}. \textbf{Background gas}: The primary intent in including the diffuse background is to facilitate the tracking of the feedback energy input, in particular in directions perpendicular to the rotation plane of the CNR. As a result, we set up only a strongly idealized spherical gas distribution. It contains $M_{\rm bg} = 1.2\times10^3 \msun$ extending out to $r_{\rm bg} = 25$~pc and following an isothermal density profile. The background gas consists of $N_{\rm bg} \approx 3\times10^5$ particles; the mass of each particle is 100 times lower than that of particles in the cloud and the ring. The initial velocity of the background gas is set to zero. \textbf{Turbulence}: In addition to orbital velocities, MC and CNR particles are given velocities from a turbulent velocity field. The velocity field is generated based on the example of \cite{Turbulencija}, with velocity amplitude $\sigma_{\rm turb} = 37.5$ km s$^{-1}$. Turbulence is only present initially and dissipates over time, since currently there is no further turbulence driving. Nevertheless, it results in the formation of a turbulent density field, which is more realistic than smooth gaseous structures. \subsection{Cooling} \label{sec:Cooling} The thermal properties of the gas particles are determined using two methods depending on the temperature of the gas. For gas with temperatures between $20$~K and $10^4$~K an empirical function by \cite{cool_to_20K} is used: \begin{equation} \log(\Lambda/n^2_{\rm H})= -24.81 + 2.92x - 0.6982x^2+\log(Z/Z_{\odot}), \end{equation} here $x \equiv \log(\log(\log(T)))$; $\Lambda/n^2_{\rm H}$ is in units of erg$\,$s$^{-1}$cm$^{3}$; $n_{\rm H}$ is the number density of H (in cm$^{-3}$). $\log(Z/Z_{\odot}$ is the metallicity in solar units; we assume $Z = Z_\odot$, therefore the metallicity term disappears. This prescription is based on the assumption that radiative cooling occurs via fine structure and metastable lines of C, N, O, Fe, S, and Si and that the above elements are in ionization equilibrium maintained by locally produced cosmic rays. For temperatures above $T \geq 10^4$~K we follow an approach from \cite{QuasarHEATING} where the change in energy $\dot{E}$ is approximated by: \begin{equation} \dot{E} = n_{\rm H}^2 (S_1 + S_2 + S_3). \label{eq:QH} \end{equation} Here, $S_1, S_2$ and $S_3$ correspond to the effect of different physical processes: \begin{equation} S_1 = -3.8\times10^{-27} T^{1/2}, \end{equation} is the bremsstrahlung cooling term, \begin{equation} S_2 = 4.1\times10^{-35}(1.9\times10^7 - T)\xi \end{equation} is the Compton heating or cooling and \begin{equation} S_3 =10^{-23} \frac{a + b ( \xi / \xi_0)^c}{1 + ( \xi / \xi_0)^c} \end{equation} is a combination of photoionization heating and line and recombination continuum cooling. The parameters here are: \begin{align} a &= -\frac{48}{\exp({25\log{T} -4.35)^2 }} -\frac{80}{\exp({5.5\log{T} - 5.2)^2 }}-\\ &-\frac{17}{\exp({3.6\log{T} -6.5)^2 }},\\ b &= \frac{1.7\times10^4}{T^{0.7}},\\ c &= 1.1 - \frac{1.1}{\exp(T/1.8\times10^5)} + \frac{4\times10^{15}}{T^4},\\ \xi_0 &= \frac{1}{1.5 T^{-1/2} 1.5\times10^{12} T^{-5/2}} + \\ &+ \frac{4\times10^{10}}{T^2} \left[ 1 + \frac{80}{\exp( (T-10^4)/1.5\times10^3 )} \right]. \\ \end{align} To determine the ionization parameter $\xi$ we use: \begin{equation} \xi = \frac{L_{\rm{disc}}}{n r^2}, \end{equation} where $L_{\rm{disc}}$ is the luminosity of the accretion disc. For temperatures $T>3\times10^7$~K, the $S_3$ term is dropped from equation (\ref{eq:QH}) \citep{QuasarHEATING}. In addition to the above (mainly cooling) processes, a background photoelectric heating from grains is taken into account. Using functions from \cite{UVBGHEATING}, the photoelectric heating rate is given by: \begin{equation} \Gamma_{\rm pe} = 10^{-24} \epsilon \chi \rm{n}_{\rm{H}}\,\rm{erg}\,\rm{s}^{-1}\,\rm{cm}^{-3}, \end{equation} where $\epsilon$ is the heating efficiency and $\chi$ is the far-UV flux normalized to the Habing field appropriate for the Solar neighborhood; $\chi=100$ is appropriate for the Galactic Centre. An approximate expression for the heating efficiency is: \begin{equation} \epsilon = \frac{4.87\times10^{-2}}{1 + 4\times10^{-3}(\chi T^{1/2}/n_e)^{0.73}} + \frac{3.65\times10^{-2}(T/10^4\rm{K})}{1+2\times10^{-4}(\chi T^{1/2}/n_e)}, \end{equation} where $n_{\rm e}$ is the electron number density defined in terms of a ionization fraction $f_{\rm ion} = n_{\rm e} / n_{\rm gas} = 10^{-3}$. The precise value of $n_{\rm e}$ has a negligible effect on our results. The balance of heating and cooling depends on the distance from the SMBH, the AGN luminosity and the gas density. At the spatial scales relevant for our simulations, with $L_{\rm AGN} = 0.5\le{}$, the equilibrium temperature of background gas is about $10^6$~K. The much denser torus and cloud have equilibrium temperatures of order $\sim10^5$~K. This produces a dichotomy which is useful for our purposes: the background gas is easily heated to temperatures higher than the virial temperature ($T_{\rm vir} \sim 5\times10^5$~K in our chosen potential), so the background does not accrete rapidly on to the SMBH even in the absence of feedback. The cloud and ring, on the other hand, are not heated to virial temperatures and so behave more-or-less balistically. \subsection{Star formation} \label{sec:star_formation} Our current model setup lacks resolution to accurately simulate star formation in the vicinity of the SMBH (star formation near the SMBH is discussed further in section \cref{sec:lost_gas}), but the removal of overly dense particles from hydrodynamical calculations helps speed up the simulation without sacrificing realism. Therefore, we introduce the following star formation prescription. Gas particles are probabilistically transformed to collisionless star particles if their density is higher than the tidal density and their Jeans mass is lower than the resolved mass. Star formation is related to the dynamical time of the particle: \begin{equation} t_{\rm{dyn}, i} = \sqrt{ \frac{3\pi}{32\rm{G}\rho_i}} \end{equation} which allows us to define the probability of the transformation, $P_{\rm{sf},i}$, as: \begin{equation} P_{\rm{sf},i} = 1 - {\rm exp}{ \left ( -\frac{\eta_{\rm sf} \Delta t_i} {t_{\rm dyn,i}}\right) }, \end{equation} here $\eta_{\rm sf} = 0.1$ is the star formation efficiency; $\Delta t_i$ is the timestep of the particle. \subsection{Feedback} \label{sec:Feedback} The physical basis of our feedback implementation is the AGN wind feedback model \citep[for a review, see][]{King2015ARAA}. Numerically, we implement it with an extended version of the Monte Carlo radiation transfer method from \cite{MCRadTrans}. Here, the AGN wind is represented by a number of isotropically distributed discrete energy-momentum packets generated at the source location. The number of these is determined by the luminosity of the accretion disc particle. The momentum of a single packet is defined in relation to the SPH particle mass, $p_{\rm{\gamma}}=p_{\rm{sph}}=m_{\rm{sph}}\times \sigma \approx 8\times10^{39} \rm{g}\,\rm{cm}\,\rm{s}^{-1}$. Since the momentum is defined based on the higher mass of cloud/ring rather than background particles, our approach may introduce artificial shocks in the background gas. However, since the primary purpose of the background gas is as a tracer of energy imparted by the AGN, this has little impact on our results. An emitted packet moves steadily outwards with the velocity of $v_{\rm{\gamma}}=0.1c$. Once it comes close enough to some SPH particles it transfers its momentum and and energy to them over several steps. The SPH density field is directly used to determine the amount of momentum passed to each particle: \begin{equation} \Delta \textbf{p}_{\rm{\gamma},i}= \frac{\rho_{i}(\textbf{r})}{\rho(\textbf{r})}\Delta \textbf{p}_{\rm{\gamma}} \end{equation} here $\Delta \textbf{p}_{\rm{\gamma}}$ and $\Delta \textbf{p}_{\rm{\gamma}, i}$ are the total amount of momentum transferred to the gas in a given step and the momentum transferred to the $i$-th particle, while $\rho(\textbf{r})$ and $\rho_i(\textbf{r})$ are the density field value at that point and the contribution of the $i$-th particle to that field. The radiation pressure force $\textbf{f}_{\rm{rad}, i}$ on each particle then is: \begin{equation} \textbf{f}_{\rm{rad}, i} = \frac{\sum_\gamma \Delta \textbf{p}_{\rm{\gamma},i}}{\Delta t_i}. \end{equation} The same principle is applied when calculating the energy transfer, except that we replace $\Delta \textbf{p}_{\rm{\gamma}}$ with $\Delta E_{\rm{\gamma}}$, where $E_{\rm{\gamma}} = \eta p_{\rm{\gamma}}c /2$, where the radiative efficiency $\eta= L_{\rm disc} / \dot{M}_{BH} c^2 $ is determined directly from the accretion disc model. The total energy injected to the gas approaches \citep{Ring_Eta} \begin{equation} E_{\rm wind} = \frac{\eta^2}{2} M_{\rm acc} c^2. \label{eq:Eabs} \end{equation} \section{Accretion disc particle} \label{sec:AccPartSetup} The core of our prescription is a one-dimensional thin accretion disc model which we couple to the SMBH sink particle in the hydrodynamical simulation. Here, we present the details of this model. The geometry of the accretion disc is defined by a number of concentric rings of logarithmically increasing radius as shown in Fig. \ref{fig:GeoFeed}. The innermost and outermost annuli are used as outflow-type boundaries. The inner lies inwards of the the innermost stable orbit and the outermost outwards of the sink radius of the particle. We make a simplifying assumption that the geometry of the accretion disc is fixed, that is, we do not need to adjust the radii of the annuli as the mass of the black hole grows, since the growth is negligible when compared to the initial SMBH mass. As is usual for a two-stage approach, the mass of particles that cross some predetermined $r_{\rm{sink}}$ and fulfil accretion criteria is added to the accretion disc. As the smoothing length of the accreted SPH particle can be comparable to the size of the disc, some care needs to be taken when adding the accreted material on to the disc. Our solution is to smoothly distribute the accreted material over a number of rings spanned by the particle, as illustrated in fig. \ref{fig:GeoFeed}. We centre the distribution defined by the kernel of the SPH particle around the circularization radius of the accreted particle given by $R_{\rm circ} = J_{\rm part}^2 / (G M_{\rm BH}$), where $J_{\rm part}$ is the angular momentum per unit mass of the accreted particle. The portion of the distribution that would end up outside the disc is put on the outer boundary. At the inner boundary, the distribution is reflected. This is a compromise between a more rigorous calculation and the simple mass injection to a single ring. We evolve the disc using a pseudo-Newtonian Paczy\'nsky-Wiita (PW) potential \cite{Paczy1980}: \begin{equation} \phi = \frac{-G M_{\rm BH}}{R-R_{\rm g}}, \label{eq:PWpot} \end{equation} where $R$ is the distance from the SMBH and $R_{\rm g}$ is the Schwarzschild radius. The equation for the viscous evolution of the disc is then: \begin{equation} \frac{\partial \Sigma}{\partial t} = \frac{3}{R}\frac{\partial }{\partial R} \left [ \frac{(R - R_{\rm g})^2}{R^{1/2}(R-3R_{\rm g})}\ \frac{\partial}{\partial R} \left ( \nu \Sigma R^{3/2} \frac{R-\frac{1}{3}R_{\rm g}}{(R-R_{\rm g})^2} \right ) \right ]; \label{eq:Diff} \end{equation} with viscosity defined as $\nu = \alpha c_{\rm s} H$, where $c_{\rm s}$ is the speed of sound, $H$ is the height of the disc and $\alpha=0.1$ \citep{ShakuraSunyaev} \footnote{See Appendix \ref{App:Derivations} for a detailed derivation of this and other relevant disc equations.}. We solve the diffusion equation numerically, updating the surface density in each annulus comprising the accretion disc $\Sigma(R_i, t)$ after each timestep: \begin{equation} \Sigma(R_i, t+\Delta t) = \Sigma(R_i, t) + \frac{\partial \Sigma(R_i, t)}{\partial t} \Delta t; \label{eq:finiteDiff} \end{equation} Accretion disc evolution occurs after each sink particle step. The dynamical timescale of the accretion disc can be much smaller than that of the simulation as a whole so we use a different criterion to determine a timestep appropriate for the disc. An initial timestep estimate is calculated by: \begin{equation} \Delta t_{\rm est} = \rm{C}\, \rm{min}\left[\Delta t_i\right] = \rm{C}\, \rm{min}\left[\frac{\Delta R_i^2}{ \nu_i}\right], \end{equation} with an arbitrary Courant factor, in our case $\rm{C} = 0.01$ works well. The actual $\Delta t$ is then defined in reference to the sink particle's timestep $\Delta t_{\rm sink}$: \begin{align} n_{\rm steps} &= \textrm{ceil}\left[ \frac{\Delta t_{\rm sink}}{\Delta t_{\rm est}} \right] + 1, \\ \Delta t &= \frac{\Delta t_{\rm sink} }{n_{\rm steps}}, \end{align} to ensure that both $\Delta t n_{\rm steps} = \Delta t_{\rm sink}$ and $\Delta t < \Delta t_{\rm est}$ conditions are met. Here $\textrm{ceil}\left[x\right]$ is the ceiling function. After a few steps, the properties of the accretion disc change and it may turn out that $\Delta t$ is too large to produce a stable result. In this case the timestep is halved and the system is restored to a configuration at the end of the sink particle step, repeating the previous steps and continuing with the corrected timestep. Note that only the viscous evolution of the accretion disc is performed using these, generally smaller, timesteps. Both the accretion on to the disc and the ejection of feedback energy-momentum packets occur at the main simulation timesteps. \begin{table*} \begin{tabular}{@{}LLS[table-format=1.1]S[table-format=2.1]S[table-format=3.0]S[table-format=3.0]S[table-format=1.2]S[table-format=1.2]S[table-format=1.2]S[table-format=1.2]@{}} \toprule \textnormal{Run} & \textnormal{FB} & \multicolumn{1}{c}{$E_{\rm tot.o}$} & \multicolumn{1}{c}{$E_{\rm tot.i}$} & \multicolumn{1}{l}{$t_{\rm Edd}$} & \multicolumn{1}{r}{$t_{\rm stop}$} & \multicolumn{1}{l}{$M_{\rm acc.tot.}$} & \multicolumn{1}{l}{ $M_{\rm acc.bh.}$} & \multicolumn{1}{l}{$M_{\rm acc.esc.}$} & \multicolumn{1}{r}{$M_{\rm peak.disc.} $ }\\ \cmidrule(ll){5-6} \cmidrule(ll){7-10} & & \multicolumn{1}{l}{$10^{57}$ erg} & \multicolumn{1}{l}{ $10^{55}$ erg} & \multicolumn{2}{c}{kyr} & \multicolumn{4}{c}{$10^5\msun{}$} \\ \midrule \multicolumn{1}{l}{\texttt{nFBr0}} & \textnormal{off} & 5.3 & \multicolumn{1}{c}{\text{-}} & 98 & \multicolumn{1}{c}{\text{-}} & 1.17 & 0.44 & 0.60 & 0.42 \\ \multicolumn{1}{l}{\texttt{nFBr1}} & \textnormal{off} & 5.3 & \multicolumn{1}{c}{\text{-}} & 163 & \multicolumn{1}{c}{\text{-}} & 1.18 & 0.45 & 0.62 & 0.41 \\ \multicolumn{1}{l}{\texttt{nFBr2}} & \textnormal{off} & 5.0 & \multicolumn{1}{c}{\text{-}} & 137 & \multicolumn{1}{c}{\text{-}} & 1.10 & 0.41 & 0.58 & 0.46 \\ \multicolumn{1}{l}{\texttt{nFBr3}} & \textnormal{off} & 5.9 & \multicolumn{1}{c}{\text{-}} & 145 & \multicolumn{1}{c}{\text{-}} & 1.29 & 0.48 & 0.68 & 0.46 \\ \multicolumn{1}{l}{\texttt{FBr0}} & \textnormal{on} & 2.5 & 7.3 & 0 & 181 & 0.55 & 0.21 & 0.28 & 0.32 \\ \multicolumn{1}{l}{\texttt{FBr1}} & \textnormal{on} & 3.2 & 9.2 & 3 & 276 & 0.69 & 0.26 & 0.35 & 0.32 \\ \multicolumn{1}{l}{\texttt{FBr2}} & \textnormal{on} & 3.3 & 9.7 & 66 & 207 & 0.71 & 0.28 & 0.37 & 0.39 \\ \multicolumn{1}{l}{\texttt{FBr3}} & \textnormal{on} & 3.2 & 9.1 & 32 & 219 & 0.69 & 0.26 & 0.35 & 0.37 \\ \bottomrule \multicolumn{1}{l}{\texttt{INSTr0}} & \textnormal{on} & 3.3 & 15.9 & 107 & \multicolumn{1}{c}{\text{-}} & 0.44 & 0.44 & \multicolumn{1}{c}{\text{-}} & \multicolumn{1}{c}{\text{-}} \\ \multicolumn{1}{l}{\texttt{INSTr1}} & \textnormal{on} & 3.4 & 16.3 & 117 & \multicolumn{1}{c}{\text{-}} & 0.63 & 0.63 & \multicolumn{1}{c}{\text{-}} & \multicolumn{1}{c}{\text{-}} \\ \multicolumn{1}{l}{\texttt{INSTr2}} & \textnormal{on} & 3.5 & 16.5 & 140 & \multicolumn{1}{c}{\text{-}} & 0.68 & 0.68 & \multicolumn{1}{c}{\text{-}} & \multicolumn{1}{c}{\text{-}}\\ \multicolumn{1}{l}{\texttt{INSTr3}} & \textnormal{on} & 3.5 & 16.8 & 142 & \multicolumn{1}{c}{\text{-}} & 0.63 & 0.63 & \multicolumn{1}{c}{\text{-}} & \multicolumn{1}{c}{\text{-}} \\ \bottomrule\bottomrule \end{tabular} \caption{ Main results of all simulations. All values calculated at $t = 500$~kyr. FB column shows whether feedback is turned on in the run. The total energy generated by the accretion disc or sink particle is $E_{\rm tot.o}$ and the total energy absorbed by the surrounding gas is $E_{\rm tot.i}$ (note that $\eta=0.1$ for the INST runs). $t_{\rm Edd}$ is the time spent at or over the Eddington luminosity and $t_{\rm stop}$ is the time at which the central cavity appears and the disc accretion fully stops. The total accreted mass is given by $M_{\rm acc.tot.}$, total mass fed to the SMBH is $M_{\rm acc.bh.}$, the total amount of mass escaping via the outer disc boundary is $M_{\rm acc.esc.}$ and the maximum amount of matter contained in the accretion disc is $M_{\rm peak.esc.}$. } \label{tab:ResultSummary} \end{table*} After each disc timestep, the parameters of the accretion disc are also updated. The PW potential yields the following expression for the viscous dissipation per unit disc face area $D(R)$: \begin{equation} D(R) = \frac{3}{8} \frac{\dot{M}}{\pi} \frac{G M_{\rm BH} }{ R} \left( \frac{1}{R-R_{\rm g}} - \frac{3^{3/2} R_{\rm g}^{1/2}}{2 R^{3/2}} \right) \frac{\left( R-R_{\rm g} \right )^2}{ R-\frac{1}{3}R_{\rm g}}, \end{equation} which we use to calculate the luminosity of each annulus: \begin{equation} \label{eq:AccDiscLum} L(R) = 2 \pi \left[D\left(R_{\rm out}\right)R_{\rm out} + D\left(R_{\rm in}\right)R_{\rm in}\right] \left(R_{\rm out} - R_{\rm in}\right) \end{equation} here $R_{\rm in}$ and $R_{\rm out}$ are, respectively, the inner and outer radii of the $i$-th annulus. Summing over all annuli gives the total luminosity of the accretion disc which we use to determine the amount of energy and/or momentum to be injected into the surrounding gas. The radiative efficiency of accretion stays close to the expected value for the PW potential $\eta \sim 6.25\%$ \citep{Paczy1980} throughout our simulation runs. This is somewhat higher than $\eta = 5.7\%$ appropriate for a Schwarzschild black hole; on the other hand, any real black hole is expected to be at least somewhat rotating, which increases the radiative efficiency. \section{Results }\label{results} We ran 12 simulations in total: four with (FB) and four without feedback (nFB) using our accretion disc method, plus four runs with feedback using an instantaneous accretion prescription (INST). In the latter group, we set the radiative efficiency to the usual value $\eta=0.1$, calculate the luminosity using $L = \eta \dot{M}_{\rm BH} c^2$ and limit it by the Eddington limit. Note that since the matter fed to the disc is assumed to be instantaneously transported to the SMBH in the INST case, the nFB disc feeding rate would correspond to the SMBH accretion rate in an INST run without feedback. This makes the additional four runs of INST without feedback redundant. The differences among the four simulations in each group are only stochastic: the cloud and ring are realised from different initial distributions of particles. We performed these stochastically differing simulations in order to understand which of our results are robust and represent actual differences in the accretion methods. The key results are summarized in Table \ref{tab:ResultSummary}. All runs produce qualitatively similar behaviour, with a single clear outlier \texttt{FBr2}. In general, the system evolves as follows \footnote{Density maps representative of the general evolution are presented in Appendix \ref{app:grids}}. It takes about $40$~kyr for the cloud to reach the torus. Due to the initial turbulence seeding, at this time there are already clumps and filaments present and some of portion of the cloud misses the ring; this material returns to the system later via an elongated stream. The cloud impacts the torus in the direction opposite to its rotation, thus a significant portion of gas loses angular momentum and is quickly transported to the accretion disc, with the disc accretion rate exceeding the Eddington mass accretion rate of the SMBH by almost an order of magnitude. Later on, a central disc/ring system forms within approximately the central parsec, surrounded by a larger ring that is still in the process of settling by $t = 500$~kyr. However, some identifiable differences between simulation sets begin to appear after a few tens of kyr following the initial collision. In Fig. \ref{fig:3Diff} we compare a representative run from each set at $t=200$~kyr; around this time a central cavity becomes clearly apparent in the FB runs. At this point the nFB run has a more compact ($r_{\rm out}\sim0.1$~pc) central structure which is slightly warped in the centre. In contrast, both FB and INST produce a more extended central structure ($r_{\rm out} \gtrsim 0.2$~pc), with a central cavity appearing only in FB runs. Over time the differences in the extent of the central structure seem to mostly disappear (eg. Fig. (\ref{fig:grid_00}-\ref{fig:grid_00i}) but the nFB central discs remain more warped and central cavities in the FB runs become even more pronounced. A comparison between accretion rates in all runs is shown in Fig. \ref{fig:AccRates_var}. The blue in nFB and FB runs corresponds to the accretion disc feeding rate, while red shows the SMBH feeding rate; in the INST case, green shows the feeding rate on to the SMBH sink particle. In all cases the dotted lines are mean values of the four simulations in the set, and the filled regions encompass the range of variations in the four simulations. We can see that accretion on to the disc/sink particle is relatively highly variable with differences by a factor two or more common on timescales $<10$~kyr, which is due to the chaotic nature of clumpy matter infall. In contrast, the disc-mediated feeding of the SMBH is relatively smooth and slightly delayed in time. In FB runs, the feeding of the disc comes to an abrupt stop between $\sim 180-280$~kyr, which can be tied to the appearance of central cavities, while in the nFB runs the disc particle is still being fed after this time at a gradually decreasing rate. We examine the nFB and FB runs further in Fig. \ref{fig:MassGrowth} where we show the amount of matter contained within each reservoir of our sink particle. Here, the total amount of gas fed to the disc is shown in black, while the amount of matter feeding the SMBH and escaping via the outer boundary is shown in red and green respectively. We can see the disc itself (blue lines) is far from empty at the end of the rapid accretion stage, so the feeding of the SMBH continues for $\sim 200$~kyr more even after the central region is cleared of gas. The feeding of the SMBH sink particle continues also in the INST runs, which did not produce the central cavity, but the rate of accretion is strongly diminished, generally to lower values than the SMBH accretion rate in the FB runs. In total, between $\sim 1.1\times10^5 \msun{}$ and $\sim 1.3\times10^5\msun{}$ feeds the disc particle in the nFB runs; this is a significant fraction of the initial gas mass ($\sim 2\times10^5\msun{}$). More than a third of this mass ends up feeding the SMBH and about a half escapes the disc via the outer boundary (Fig. \ref{fig:MassGrowth}). Similar proportions are found in the FB runs, with the total amount of accretion reduced by about a half. The commonality between these proportions in all runs suggests that the stochasticity and the relatively chaotic dynamics of the surrounding gas have little effect on the accretion disc itself. We explore how our choice of minimum smoothing length effect the disc(sec. \cref{sec:Param_tune}) and how the escaping gas might affect the system (sec. \cref{sec:lost_gas}) in the Discussion. The evolution of luminosity in the three simulation sets is shown in Fig. (\ref{fig:Luminosity_var}). The stalled accretion also results in a similarly decreased luminosity in both sets with feedback when compared to the nFB runs\footnote{Note that in the INST simulations, we compute the luminosity of the accretion disc, but do not generate feedback particles, thus it has no effect on the surrounding gas.}. Luminosity in INST simulations is highly variable as it follows the variations of accretion on to the sink particle. Conversely, the evolution of disc luminosity in both FB and nFB is relatively smooth, since it follows the feeding of the SMBH; the radiative efficiency stays nearly constant at $\eta\approx6.3$~\% during the period of activity. Accretion disc luminosities reach super-Eddington values in all except one FB run. While feedback significantly reduces gas infall, thus the super-Eddington phases are not sustained for long and the peak luminosity is $< 30\%$ (60\% in nFB) higher than $\le{}$ , we note that our simple model does include any mass loss from the disc itself that occurs in the super-Eddington regime. More tests are required to determine if some artificial limit on luminosity in our sub-resolution prescription is warranted. The disc in the centre of the system has a collimating effect on the isotropically emitted SMBH wind; this leads to a conical outflow (Fig. \ref{fig:outflow}). The total energy produced by the accretion disc in the FB runs is between $2.5\times10^{57}$~erg and $3\times10^{57}$~erg, but only a portion of this energy is absorbed by the surrounding ISM in accordance with (\ref{eq:Eabs}), so the actual energy injected in gas particles in our simulation is about $7.8\times10^{55}$~erg to $10.3\times10^{55}$~erg. This value is still a few orders of magnitude higher than the estimate for progenitor event of the 430-pc radio bubbles (\citep[$7\times10^{52}$~erg]{430RADIOBUBBLES}). It also exceeds the energy required to form the Fermi Bubbles by at least a factor of a few \citep[$\sim10^{54-55}$~erg]{fermior}. In all cases except one, SMBH winds are not strong enough to completely disrupt the system, leaving an intact central disc and a surrounding ring system, just with a pronounced central cavity. As the system evolves, more and more of the background gas is removed and the opening angle of the cone increases. The same general behaviour describes the INST runs, but they produce approximately twice the amount of energy - this is largely due to the larger radiative efficiency value $\eta_{\rm INST}/ \eta_{\rm FB} \simeq 1.6$. Also, as there is less of a delay between accretion and luminosity peaks in INST runs, the effect of feedback is near immediate. But, as in the case with the central cavity, there is less of a contrast between the outflow cone and the background; this suggests that in the INST runs, AGN feedback is less efficient in pushing the gas out. By $t=500$~kyr the outer portion of the central disc and the surrounding inner filaments/rings are aligned with the $xy$-plane within a few degrees in all runs, while the inner portion of the central disc is somewhat warped - the disc rotational plane shifts up to $50^{\deg}$ in nFB and up to $20^{\deg}$ in INST. A transient misaligned ($\sim20^{\deg}$) disc is present in two FB runs. It is possible that the SMBH wind pressure on the central disc helps to keep it within the same rotational plane. In the run \texttt{FBr2} feedback pushes out all the gas (Fig. \ref{fig:grid_12}) except for a part of the inner disc. Interestingly, the total energy ejected as wind (table \ref{tab:ResultSummary}) is similar to that in the other runs, but a longer period of time was spent in the super-Eddington regime as shown by the luminosity functions of all simulations in Fig. (\ref{fig:time_over}). Overall, at least for luminosities $L>0.5 L_{\rm Edd}$, the variation of the initial particle distribution is less important in determining the luminosity function than the differences between the accretion and/or feedback prescriptions. Another interesting aspect is that the total energy radiated as feedback is considerably larger in the INST simulations; curiously, this does not result in a complete disruption of the initial system as it does in \texttt{FBr2} or even in the creation of central cavities. It appears that sporadic bursty feedback is far less efficient at ejecting the gas and stopping further SMBH accretion than continuous energy injection, even at a milder rate. \subsection{A comment on performance} An important aspect of any numerical method is its computational cost. Although we did not do rigorous benchmarking, we compared the time that our different simulations take to run. On average it takes about 160 wall-clock hours to calculate each nFB run and about 140 wall-clock hours to calculate each FB run up to $t=500$~kyr on 32 CPUs (for comparison, INST runs took slightly longer, 148 wall-clock hours, on the same system and setup). In both cases the time spent on accretion disc calculations is almost negligible: on average, the fraction of time spent on tasks directly related to the disc is $\sim0.6\%$. Interestingly, the significant amount of time spent on calculating the interactions between feedback packets and gas particles in the FB runs ($\sim32\%$ on average) is more than offset by the time saved when a large number particles that would otherwise require very small time-steps are pushed out of the centre, resulting in quicker calculations. \section{Discussion }\label{Discussion} \subsection{Chaotic nature of model evolution} All our simulations produce qualitatively similar results, especially as far as the large-scale morphology of the system is concerned. The stochastic variations between runs seem to mostly affect the smaller-scale central features, e.g. the central cavity, the precise shape and size of the central disc/ring structures, etc. However, a more detailed analysis reveals more evidence of chaotic behavior. For example, the \texttt{FBr2} run shows very different morphology when compared to all other FB runs, characterized by an almost complete absence of gas in the central $5$ parsecs of the system by the end of the AGN phase. The case is especially intriguing as there are no significant differences in the total accreted mass or total injected energy, which are both only $\sim3\%$ larger than in other FB runs (Table \ref{tab:ResultSummary}). The main difference seems to be that the \texttt{FBr2} feeds the disc at a relatively consistent rate for over $100$~kyr. This can be seen in Fig. \ref{fig:MassGrowth} - the black curves show the total mass injected into the disc. \texttt{FBr0} and \texttt{FBr1} runs both have a significant dip in disc accretion rate at $\sim100$~kyr while \texttt{FBr3} somewhat lags behind the other runs. The consistent rate of feeding allows \texttt{FBr2} to maintain $L_{\rm disc} \geq \le{}$ for a longer period of time (Table \ref{tab:ResultSummary}). While not as extreme as the case of \texttt{FBr2}, the run \texttt{FBr3} also has significantly less gas remaining in the central $5$~pc at $t=500$~kyr when compared with the remaining two FB runs. \texttt{FBr3} is slightly less energetic than two of its counterparts but sustains a super-Eddington luminosity for $\sim33$~kyr. This suggests that even a brief period of super-Eddington feedback is critical in the shaping of resultant system's morphology. The results of INST simulations support this argument - there we put an artificial cap at $L_{\rm AGN} = \le{}$ and no run resulted in significant removal of gas from the centre. A counterexample to this argument can be seen when comparing the runs \texttt{FBr0} and \texttt{FBr1} - even though \texttt{FBr1} has a higher luminosity for the majority of the activity phase and the luminosity in \texttt{FBr0} never exceeds the Eddington limit, the central cavity in \texttt{FBr0} is more pronounced. In an attempt to understand the detailed reasons behind these differences, we check whether the more continuous high luminosity keeps the surrounding gas hot making it easier for feedback to push it out. Unfortunately, the results are inconclusive. There are no clear temperature differences between the four FB runs, where the majority of gas in the inner parsec stays at about $10^4$~K throughout the simulation. Meanwhile, in both nFB and INST runs, after the peak of the AGN phase, the temperature drops significantly down to $\gtrsim10^1$~K. Nonetheless, the inner $0.1$~pc remains continuously populated by gas with temperatures over $10^4$~K in INST runs, while all FB runs had this population pushed out at the time of the appearance of the central cavity. To summarize, there are no clear trends in gas temperature that explain the difference in kinematics. Additionally, we measure the weight of the disc and compare it with the outward force of the AGN wind pressure in each simulation with feedback \citep{WEIGHTS}. The weight of the disc is given by: \begin{equation} W_{\rm disc} \sim g(R)M_{\rm disc}(<R), \end{equation} where $M_{\rm disc}(<R)$ is the mass contained within $R$ and $g(R)$ is the $g(R)$ is the gravitational acceleration at $R$: \begin{equation} g\left(R\right) = \frac{\textrm{G}M_{\rm{bh}}}{R^2} + \frac{2\sigma^2}{R}. \label{gacc} \end{equation} The outward force is: \begin{equation} F_{\rm out} \sim \frac{\alpha(R)}{4\pi} \frac{L_{\rm disc}}{c}, \end{equation} here the factor $(\alpha(R))/ 4\pi$ is the ratio between the solid angle subtended by the disc from the point of view of the SMBH and the total solid angle of a sphere, which gives the fraction of AGN wind flux absorbed by the disc. We estimate $\alpha(R)$ using a wedge cut of the density field of the central disc. We take the disc's height to be the full width at half maximum in the density field perpendicular to the disc's radius. Again, we see no significant differences between FB runs - the outward force acting on the disc always outgrows the disc's weight, initially at the centre, and gradually pushes out the gas, producing the central cavity. Conversely, in INST runs the outward force remains smaller than the weight of the disc, explaining the lack of the central cavity; the artificially imposed Eddington limit is probably responsible for this. It is difficult determine what precisely causes \texttt{FBr2} to behave so differently. The results still suggest that the morphology of the resultant system may be very sensitive to the interaction between the infalling gas and the SMBH feedback. This implies that it might be dangerous to rely on a preset and constant $t_{\rm visc}$ parameter as a slightly different choice might result in a completely different evolution of the system. This also hints that the problem of dynamical perturbations in the central few parsecs of galaxies is truly chaotic - relatively small stochastic differences may result in drastically different outcomes. A more comprehensive study varying feedback parameters and prescriptions is required to determine if this result is not an aberration. \subsection{Parameter Tuning}\label{sec:Param_tune} One of the main advantages of our approach is that it gives consistent results that are less reliant on free parameters, but this comes at the cost of simplicity and a small increase in computational cost. For example, we do not define a specific viscous timescale $t_{\rm visc}$ or select a specific radius of gas accretion (although we do need to specify the outer radius of the disc). The choice of these parameters can sometimes in a large part determine the outcome of a simulation as even a small increase in luminosity may result in significant changes in chaotic systems. In our accretion disc model, the disc evolution is set to tend towards the standard quasi-steady thin $\alpha$-accretion disc, but at each step some mass distribution is added on to the disc, depending on the properties of the accreted gas particles. Given this, it is necessary to understand how the evolution of the disc depends on this perturbation, where the distribution is determined by the parameters of the infalling matter. The angular momentum of the accreted particle is used to determine the circularization radius $R_{\rm circ}$, i.e. the radius at which its contribution is centered, and the numerical smoothing length parameter $h$ is chosen to represent the spatial extent of a particle's contribution, similar to its role in the SPH formulation of hydrodynamics. We performed a set of disc simulations with different fixed values of $R_{\rm circ}$ and $h$ \footnote{An implementation of this in \texttt{Python} is freely available in \href{https://github.com/Caradryan/accretiondisc}{\texttt{https://github.com/Caradryan/accretiondisc}}} . The rate of disc feeding is similar to that seen in the nFB set of simulations, with $\sim10^5\msun{}$ of gas added to the accretion disc over a period of $200$~kyr at a constant rate. The resulting growth curves of the accretion disc (blue), the black hole (red) and the escaping matter (green) are shown in Fig. \ref{fig:hsml_rcirc}. Each of the four subplots at the top shows results of simulations with a different fixed $R_{\rm circ}$ value, while different line-styles correspond to different values of $h$. Here we clearly see several trends, which are all quite intuitive. First, the closer to the center a particle is fed to the disc, the more material ends up feeding the SMBH, and conversely, the further out the particle is inserted, the more mass escapes the disc via the outer boundary. The smoothing parameter $h$ in this case acts to soften this effect, as higher values allow for more matter to be placed in the middle annuli of the disc (Note the dash-double-dot line in Fig. \ref{fig:hsml_rcirc}). Interestingly, the nFB set of SPH simulations show results somewhat similar to those of steady accretion between $R_{\rm circ}=0.003$~pc and $R_{\rm circ}=0.005$~pc (Fig. \ref{fig:MassGrowth}) meaning that few accreted particles had very small or very large angular momenta. This being the case, we can also infer from Fig. \ref{fig:hsml_rcirc} that our chosen value for the minimum smoothing length $h_{\rm min} = 0.01$~pc had limited effect on the results: while some of the matter was put directly into the SMBH and the outer boundary, the amount was not overwhelming. In fact, even a significantly larger value $h = 0.015$~pc in the idealised tests did not have an impact larger than the stochastic variability between SPH simulations. In addition to the spatial resolution that depends on smoothing length, the mass resolution may also impact our accretion disc. To test this, we again performed a set of simulations with \texttt{Python} version of our code. Both the feeding radius $R_{\rm circ}=0.003$~pc and the smoothing length $h = 0.01$~pc remain constant, while the mass of a single mass portion is varied. The portion masses for these runs are chosen in proportion to the mass of the SMBH and are incremented by an order of magnitude in each run, from $M_{\rm part}/M_{\rm BH} = 10^{-7}$, up to $M_{\rm part}/M_{\rm BH} = 10^{-2}$ (the latter corresponding to the ratio in the SPH simulations). We set a target for total mass to be fed in a quater of the total run time of $500$~kyr. The target mass is the same as the mass of one portion with the ration $M_{\rm part}/M_{\rm BH} = 10^{-2}$ - $4\times10^4\msun{}$. The amount of mass contained in each component is shown in Fig. (\ref{fig:MassGrowth_mpart_diff}), where colors correspond to the component and thinner lines correspond to the simulations with less massive feeding particles. The more massive mass portions resulted in a more step-like total growth curve (black), and a spikier changes in accretion disc mass (blue), but even so, over the longer period of time the mass in each component seems to converge. In Fig. (\ref{fig:L_mpart_diff}) we show that the very sudden increases in of disc mass result in large luminosity spikes, which, while short, are also highly super-Eddington. This suggests that a limit on luminosity is necessary if very massive particles are used. At the same time, the accretion disc of the scale of used in these runs ($\sim 0.01$~pc) is probably too small for simulations that would also make use of particles as massive as used in the tests - the fact that parameters converge even in this limiting case suggests that over the longer timescales relevant to larger simulations the undesired short bursts of accretion may average out. We discuss the applicability of our model to large-scale simulations in Section \cref{sec:appl}. In general, care should be taken when choosing the spatial resolution of the main simulation, as it could result in an artificially lower accretion disc mass and/or excessive instantaneous feeding/escape. Choosing a sink radius much larger than approximately the intended accretion disc size may also result in accretion of matter with larger than intended angular momentum and matter accumulation in the furthest parts of an unphysically large accretion disc. Our prescription does not account for the possible fragmentation and star formation at the outskirts of the disc; we leave this topic for further research. In addition to exploring our own model, we can compare its results to the simpler two-stage accretion disc particle method (DP) using accretion data from the nFB simulation \texttt{nFBr0} as a baseline. We cannot use \texttt{FB} simulations for this because feedback has an effect on the dynamical evolution of the simulation. A convenient prescription for the rate of SMBH accretion that depends only on parameters readily available in \texttt{nFBr0} data is given by \citep{AccDiscParticle}: \begin{equation} \dot{M}_{\rm BH} = \rm{min}\left[ M_{\rm disc} / t_{\rm visc}, \dot{M}_{\rm Edd}\right], \label{eq:DP_main} \end{equation} where $M_{\rm disc}$ is the accretion disc mass, $t_{\rm visc}$ is an arbitrarily chosen constant viscous timescale and $\dot{M}_{\rm Edd}$ is the Eddington mass accretion rate. Note that $M_{\rm disc}$ used here is not equal to the disc mass in nFB run; instead, $M_{\rm disc}$ is given by the rate of sink accretion, subtracting the mass added to the SMBH using the prescription (\ref{eq:DP_main}). We use a set of four \texttt{Python} scripts that describe the evolution of the black hole and the gas reservoir coupled to it. In addition to the Eddington-limited model, we run scripts with no limits and a an additional limit on the maximum allowed accretion disc mass according to the gravitational stability criterion from \citep{Pringle1981}: \begin{equation} M_{\rm disc,max} \approx \frac{H}{R} M_{\rm BH}, \label{eq:Stable} \end{equation} where the disc height-radius ratio is taken to be a constant $H/R = 0.002$, typical for a thin disc. We show the results of this procedure in Fig. \ref{fig:DP_post}. The blue and red curves represent the rate of disc and SMBH feeding from the simulation \texttt{nFBr0}, respectively, and green ones represent the DP calculations with different choices of $t_{\rm visc}$. A disc evolution that most resembles the SMBH feeding in the \texttt{nFBr0} simulation is marked in black; when calculating it, we weigh the whole $500$~kyr duration evenly. The bottom panel shows how the closest DP calculations compare with the variable $t_{\rm visc} = M_{\rm disc} / M_{\rm \dot{M}_{\rm BH}}$ calculated from \texttt{nFBr0}. The \texttt{No Limit} approach seems to work best for the $\sim200$~kyr period where the variable $t_{\rm visc}$ remains approximately constant, while overestimating the feeding in later stages. Applying the luminosity and/or disc mass limits substantially reduces the best-fitting $t_{\rm visc}$ which also reduces the time lag between disc and SMBH feeding. We note that even if the evolution seen in the main simulation could be approximated, the `correct' $t_{\rm visc}$ is not known {\em a priori} and could be difficult to deduce as it corresponds to relatively small feeding radii $R_{\rm feed}\sim 2-5 \times10^{-4}$~pc. These values are approximately an order of magnitude lower than the $R_{\rm circ} \sim 1e-3$~pc of the majority of the accreted particles in the hydrodynamical simulations. This means that correcting only for angular momentum would not be enough to reproduce accretion disc results with AD prescription. \subsection{Star formation on the disc outskirts}\label{sec:lost_gas} Over the induced activity period about half of matter added to the accretion disc particle escapes through the outer boundary (Fig. \ref{fig:MassGrowth} - green) in all of our simulations. At the moment, simulations just track the amount of matter escaping over time; it is not taken into account when calculating the evolution of the accretion disc or returned to the main hydrodynamical simulation. Neglecting this `lost mass' allows us not to complicate the calculations, but it is worthwhile to consider the possible interaction between this gas and the rest of the gas close to the SMBH as we plan to do when improving the prescription in the future. In particular, accounting for this gas in the hydrodynamical simulation may lead to either additional SMBH accretion, additional star formation in the centre, or both. We expect that interactions between the gas escaping the accretion disc and the surrounding gas can produce additional accretion. Gas escaping via the outer boundary has $R_{\rm circ}\sim0.01$~pc, therefore needs only a small reduction in angular momentum to get accreted back, so adding this gas to the hydro simulation can result in additional accretion as it collides with infalling material.Although the majority of mass escapes the disc in the later stages of the AGN episode, the peak escape rate occurs at about the same time as central cavities appear in the FB runs, when the central disc structure is somewhat settled. Additional gas might even prevent central cavities from forming, as the total mass of gas escaping via the outer boundary is comparable to the total mass contained within the central few parsecs ($\sim10^4\,\msun{}$). We can also envision a scenario, similar to the one described in \cite{Hobbs2011}, in which this escaping gas provides an additional barrier to accretion as the mixing of different angular momentum gas creates a peaked angular momentum distribution and a pronounced ring with $r>0.01$~pc. So the escaping matter might extend the AGN episode by providing some additional material to the accretion disc, but it might also create a more dense environment around the SMBH - maybe leading to star formation. The main obstacle to star formation in the vicinity of the SMBH is the tidal force in the region where SMBH potential dominates ($r<0.8$~pc in our case). As a first approximation, to allow for star formation, the density in a given region has to exceed the tidal density $\rho_{\rm tidal}$: \begin{equation} \rho_{\rm esc} \gtrsim \rho_{\rm tidal} \simeq \frac{3 M_{\rm BH}}{4 \pi R^3}, \label{eq:tidal} \end{equation} where $\rho_{\rm esc}=M_{\rm esc} / V$ is the density of the escaping gas, while $V$ is the volume of the region containing that gas. We can very roughly estimate where star formation might occur due to the escaped gas if we assume that all the escaped matter stays within this volume. The upper density estimate can be found by assuming that all the gas stays within a wedge-shaped disc of angle: \begin{equation} \alpha=\arctan\left( \frac{H_{\rm max}}{2R_{\rm max}}\right), \end{equation} here $R_{\rm max}$is the radius where the accretion disc is at its maximum height $H_{\rm max}$. We can use $\pi/2 - \alpha$ to define a cone sector of a sphere; then the angle subtended by the disc is $4\pi$ minus the angle of the conical sector. Thus the tidal density is given by: \begin{equation} \rho_{\rm esc} = \frac{M_{\rm esc}}{V_{\rm sphere} - 2 V_{\rm sector}}=\frac{3M_{\rm esc}}{4\pi R^3 \cos\left(\pi/2 - \alpha\right) }. \end{equation} In our simulations, the value of $\alpha$ is very small, of order $10^{-3}$. In this case, $\rho_{\rm esc}$ is larger than $\rho_{\rm tidal}$ at $R<1$~pc after enough matter escapes the disc ($t=180$~kyr in FB simulations). Admittedly, this is an extreme case. Density is greatly reduced if we allow for even a small variation of the accretion disc plane. Consider a small angle $\beta$ by which the accretion disc tilts. If we assume that the disc oscillates quickly enough, we expect that this effectively results in matter distributing over a larger volume defined by a larger wedge angle $\bar{\alpha} = \alpha + \beta$. Angles as small as $\beta\gtrsim0.25\deg$ result in density low enough that star formation is no longer possible, even if all of the matter escaping the disc throughout the $500$~kyr duration of the FB simulations accumulates within the specified volume. Since discs that allow for star formation appear to be very thin, an interesting aspect to consider is the required spatial resolution in order to be able to track star formation around the SMBH in an SPH simulation. For simplicity, consider, that the minimum resolvable height of the disc is the minimum smoothing length of an SPH particle in a given simulation: $H(R) \sim h_{\rm min}$. At radii below some threshold, where the disc is unresolved in the $z$-direction, its height remains constant and the volume can be approximated by a cylinder, giving a modified expression for $\rho_{\rm esc}$: \begin{equation} \rho_{\rm esc} = \frac{M_{\rm esc}}{ \pi R^2 H}. \end{equation} Equating this with the tidal density we get we get the maximum height required to reach the tidal density at a given radius: \begin{equation} H_{\rm max} =\frac{4}{3} \frac{M_{\rm esc}}{ M_{\rm BH}}R. \end{equation} In our simulations, this value is \begin{equation} H_{\rm max, FB} \sim 0.013 R; \qquad H_{\rm max, nFB} \simeq 0.02 R, \end{equation} where the differences arises due to differences in $M_{\rm esc}$. Given that our simulations have $h_{\rm min} = 0.01$~pc, we see that they would not be able to resolve star formation closer than $R \sim 0.5-0.8$~pc from the SMBH. \subsection{Applicability to larger scales} \label{sec:appl} So far, we only apply our accretion disc in a system where the the spatial resolution is comparable to the size of the accretion disc, but it should be possible to apply the same approach to larger scales. \cite{Salas2021} performed simulations of the central molecular zone (CMZ) with turbulence driving. They suggest that turbulence results in a quasi-continuous inflow of matter towards the centre; they estimate the effective viscosity $\nu$ using the $\alpha$-accretion disc theory and find it consistent with \cite{Sormani2018} models of nuclear rings present in the central $\sim1$~kpc of galaxies, where viscosity due to turbulence is included as a free parameter. This suggests that the accretion disc grid can be extended outward in a straightforward manner up to the scales of at least tens of parsecs, where circumnuclear discs and/or rings are found. But extending the disc this far may be unnecessary, since parsec-scale resolutions are common in modern galaxy simulations; therefore, extending the sub-resolution grid out to $\sim1$~pc should be sufficient. However, this would still result in a temporal and spatial resolution and would require assuming a continuous disc in the centre, which is clearly not the case in many circumstances (eg. current GC, cf. \cite{gas}, or the results of our FB simulations). In addition, an overly large sink radius would result in the smoothing out of even relatively large perturbations, leading to relatively smooth and long but weak periods of AGN activity. Some of the downsides of this extension may be circumvented by applying more stringent accretion criteria or, possibly, modifying the evolution equation itself, by accounting for disc instabilities and turbulence. Given that the current prescription requires very little computational power to process, some complication to the underlying model will not reduce its applicability from a processing power standpoint. Another interesting possibility is a combination our accretion disc method with a method applicable to larger scales. For example, \cite{Alcazar2016} applies and extends an analytical model of mass transport due to gravitational torques developed by \cite{Hopkins2011} to determine BH accretion in cosmological simulations: \begin{equation} \dot{M}_{\rm BH} = \left(1 - \eta\right)\times\dot{M}_{\rm Torque}, \end{equation} where $\dot{M}_{\rm Torque}$ is the mass transport due to the local properties of the unresolved central region. The model is intended to be used in cosmological and other large scale simulations; it describes the inflow of matter from scales of $0.1-1$~kpc down to the SMBH (sub-parsec) scales. It would be interesting to estimate $\dot{M}_{\rm disc}$ following the same formalism, especially in large-scale simulations with temporal resolution fine enough for the delay of feedback to have a meaningful impact. \section{Conclusions}\label{sec:Conclusions} We developed a simple 1D accretion disc prescription coupled to the SMBH sink particle in \texttt{Gadget-3} in order to increase the realism of black hole accretion and feedback. The prescription is based on the $\alpha$-thin accretion disc model of \cite{ShakuraSunyaev}, but uses the Paczy\'{n}ski-Wiita potential \citep{Paczy1980}. We assume that the disc is stable and quickly returns to a quasi-steady state after each mass injection. We test the prescription by simulating a retrograde collision between a torus-shaped ring surrounding a SMBH and an infalling cloud in an environment similar to the Galactic Centre. We run three sets of four simulations: with feedback from our accretion disc (FB), without feedback (nFB) and with instantaneous accretion and feedback (INST). The disruption of the initial system results in an AGN phase lasting a couple hundred kyr. Feedback reduces the total accreted mass in both sets of feedback simulations ($M_{\rm acc.tot} \sim 6\times10^4 \, \msun{}$) when compared with runs without feedback ($M_{\rm acc.tot} \sim 1.2\times10^5 \, \msun{}$) by about a half. The major differences between simulations with instantaneous accretion and those with our accretion disc prescription are: \begin{itemize} \itemsep0em \item The growth rate of the SMBH, $\dot{M}_{\rm BH}$, is reduced and spread more evenly over time in the accretion disc prescription simulations; the change in luminosity $L_{\rm disc}$ closely follows $\dot{M}_{\rm BH}$. \item Radiation from the disc carries away $\eta\sim 6.25\%$ of the rest mass energy of infalling matter, which is expected in the Paczy\'{n}ski-Wiita potential and within $10\%$ of the expected value from the relativistic Schwarzschild solution. \item Feedback in the FB simulations expels gas from the central $0.1-1$~pc region, producing a central cavity. This is not reproduced in INST runs, although there the aggregate energy input into the gas is higher by a factor $\sim 1.5-2$. \item A significant amount of matter escapes via the outer boundary of the accretion disc; we neglect this in our current simulations. \end{itemize} While improvements are necessary, we show that the current implementation of the accretion disc sub-grid prescription works consistently while requiring negligible additional computational power. It provides robust results that differ significantly from instantaneous feeding prescription. Our approach is less reliant on free parameters, most importantly the viscous timescale used to artificially delay SMBH feedback. Thus, our accretion disc prescription should be especially useful in simulations of galactic nuclei on scales of tens of parsecs, where a lot of questions about the interplay between feeding and feedback and their link to star formation remain unanswered. In future, we plan to improve the model by tracking the direction and warping of the disc plane and by consistently tracing the gas that is removed form the accretion disc via its outer boundary. Inclusion of these effects may help us to better understand how collimated feedback affects the surrounding gas. \section*{Acknowledgements} This research was funded by the Research Council Lithuania grant no. S-MIP-20-43. The simulations were performed on the supercomputer GALAX of the Center for Physical Sciences and Technology, Lithuania. We thank Jonas BialopetraviДЌius for his helpful insights concerning the \texttt{Python} implementation of the accretion disc code. \section*{Data availability} A \texttt{Python} implementation of the accretion disc particle is available at \href{https://github.com/Caradryan/accretiondisc}{\texttt{https://github.com/Caradryan/accretiondisc}}. Simulation results and \texttt{Gadget-3} implementation are available upon reasonable request. \bibliographystyle{mnras} \bibliography{literatura} % \appendix \section{Thin disc evolution equations in the P-W potential} \label{App:Derivations} For completness we provide a more detailed derivation of thin accretion disc equations, following \cite{frank_king_raine_2002} but using the \cite{Paczy1980} potential (PW). Equations (\ref{eq:mass_cons})-(\ref{eq:insert}) and (\ref{eq:mdot_par})-(\ref{eq:solve_for_c_G}) are included for posterity, as they are identical to the Keplerian case. For a more detailed description of the Keplerian case consult \cite{Pringle1981} or \cite{frank_king_raine_2002}. The disc is characterized by its surface density $\Sigma(R, t)$, which is given by integrating the gas density $\rho$ in the $z$~direction. the amount of matter contained in a single annulus between $R$ and $R + \Delta R$ is $2\pi R\Delta R\Sigma$; similarly, the total angular momentum is $2\pi R\Delta R\Sigma R^2 \Omega$. The rate of change of these quantities is determined by the net flow from neighbouring annuli: \begin{equation} \begin{split} \frac{\partial}{\partial t}(2 \pi R \Delta R \Sigma) &= v_R(R, t) 2 \pi R\Sigma (R, t) \\ &- v_R(R+\Delta R, t) 2 \pi (R + \Delta R) \Sigma(R+\Delta R, t) \\ &\approx -2 \pi \Delta R \frac{\partial}{\partial R} (R \Sigma v_{R}). \label{eq:mass_cons} \end{split} \end{equation} As $\Delta R \rightarrow{} 0$, we get the mass conservation equation: \begin{equation} R \frac{\partial \Sigma}{\partial t} + \frac{\partial}{\partial R} \left( R \Sigma v_{R} \right) = 0. \end{equation} The conservation of angular momentum is constructed in the same way from the rate of change of angular momentum, but an additional transport term due to the viscous torques $G(R, t)$ is included: \begin{equation} \begin{split} \frac{\partial}{\partial t}(2 \pi R \Delta R \Sigma R^2 \Omega) &= v_R(R, t) 2 \pi R\Sigma (R, t) R^2 \Omega(R) \\ &- v_R(R+\Delta R, t) 2 \pi (R + \Delta R) \Sigma(R+\Delta R, t) \\ &\times(R+\Delta)^2\Omega(R+\Delta R) +\frac{\partial G}{\partial R}\Delta R \\ &\approx -2 \pi \Delta R \frac{\partial}{\partial R} (R \Sigma v_{R} R^2 \Omega)+\frac{\partial G}{\partial R}\Delta R. \label{eq:mom_cons} \end{split} \end{equation} again taking $\Delta R \rightarrow{} 0$ we arrive at the angular momentum conservation equation: \begin{equation} R \frac{\partial}{\partial t}(\Sigma R^2 \Omega) + \frac{\partial}{\partial R}(R \Sigma v_R R^2\Omega) = \frac{1}{2\pi}\frac{\partial G}{\partial R}. \end{equation} The expression for the torque includes the viscosity term $\nu=\alpha c_{\rm{s}} H$: \begin{equation} G(R) = 2\pi R \nu \Sigma R^2 \frac{\partial \Omega}{\partial R}. \label{eq:torq} \end{equation} Using (\ref{eq:mass_cons}) we can simplify (\ref{eq:mom_cons}): \begin{equation} R \Sigma v_R \frac{\partial}{\partial R}\left(R^2 \Omega\right) = \frac{1}{2\pi}\frac{\partial G}{\partial R}, \label{eq:simpl} \end{equation} note that we assume that the $\partial \Omega / \partial t$ term can be safely neglected as the change in potential due to the increase in SMBH mass is negligible. Combining (\ref{eq:mass_cons}) and (\ref{eq:simpl}) allows us to eliminate $v_R$. \begin{equation} R \frac{\partial \Sigma}{\partial t} = -\frac{\partial}{\partial R}\left(R \Sigma v_R\right) = -\frac{\partial}{\partial R} \left[\frac{1}{2\pi \frac{\partial}{\partial R}\left(R^2 \Omega\right)} \frac{\partial G}{\partial R} \right]. \label{eq:insert} \end{equation} We now introduce the PW potential \begin{equation} \phi = -\frac{\rm{G}M_{\rm BH}}{R-R_g}, \end{equation} from which a modified expression of angular velocity $\Omega$ follows: \begin{equation} \Omega = \left(\frac{1}{R}\frac{{\rm d}\phi}{{\rm d}R}\right)^{1/2} = \left( \frac{\rm{G}M_{\rm BH}}{R^3} \right)^{1/2} \left( \frac{R}{R-R_g} \right). \end{equation} Here $R_g$ is the Schwarzschild radius. Inserting expressions for \begin{equation} \frac{\partial}{\partial R}\left(R^2 \Omega\right) = \frac{R}{2}\frac{R - 3R_g}{ \left(R-R_{\rm g}\right)^2}\left( \frac{G M}{R^3} \right)^{1/2} \end{equation} and \begin{equation} \begin{split} \frac{\partial G}{\partial R} &=\frac{\partial }{\partial R}2\pi R \nu \Sigma R^2 \Omega' \\ &= 2 \frac{3}{2} \pi R \nu \Sigma R^2 \left( \frac{G M}{R^3} \right)^{2} \left( \frac{R - R_{\rm g}/3}{\left(R-R_{\rm g}\right)^{1/2}} \right) \end{split} \end{equation} to (\ref{eq:insert}) and simpolifying we arrive at the main diffusion equation: \begin{equation} \frac{\partial \Sigma}{\partial t} = \frac{3}{R}\frac{\partial }{\partial R} \left[ \frac{\left(R-R_{\rm g}\right)^2}{R^{1/2}\left(R-3R_{\rm g}\right)}\ \frac{\partial}{\partial R} \left ( \nu \Sigma R^{3/2} \frac{R-R_{\rm g}/3}{\left(R-R_{\rm g}\right)^2} \right ) \right]; \label{eq:Diff_again} \end{equation} This is the main viscous evolution equation with the PW potential that our prescription solves by finite differences. Equation (\ref{eq:Diff_again}) reduces to the standard Keplerian form if we set $R_{\rm g}=0$. So see whether it behaves as expected we perform a diffusion evolution test with constant viscosity. Results for both PW (blue) and standard Keplerian (black) potentials are shown in Fig. (\ref{fig:DiffTest}). We see a noticeable effect of the boundary condition $\Sigma_[0] = 0$ and a slight difference between the PW and Keplerian distributions, in that the PW potential results in somewhat faster diffusion. This is expected, because the PW potential effectively brings the material `closer' to the origin of the potential, so its evolution is faster. We assume that the disc is stable, thin and its parameters should tend to solutions for the steadily accreting disc. In a sense, each feeding cycle is a perturbation, while viscous diffusion distributes the matter to ever more closely resemble the ideal quasi-steady thin accretion disc. This assumption greatly simplifies the calculation of the disc parameters, allowing us to disregard the time derivative in the conservation equations (\ref{eq:mass_cons}) and (\ref{eq:mom_cons}). From these we get: \begin{equation} \begin{split} \dot{M} &= 2\pi R \Sigma (-v_R), \label{eq:mdot_par} \end{split} \end{equation} where $\dot{M}$ is the accretion rate, and \begin{equation} \begin{split} R \Sigma v_R R^2 \Omega &= \frac{G}{2\pi} + \frac{C}{2\pi}. \label{eq:solve_for_c} \end{split} \end{equation} Here C is some constant. Inserting the expression (\ref{eq:torq}) for $G$ we get \begin{equation} \begin{split} -\nu \Sigma \frac{\partial \Omega}{\partial R} &= \Sigma (-v_R)\Omega + \frac{C}{2 \pi R^3}. \label{eq:solve_for_c_G} \end{split} \end{equation} Equation (\ref{eq:solve_for_c_G}) can be solved for $C$; applying $\partial \Omega/\partial R=0$ at the innermost stable orbit ($R=3R_{\rm g}$) we get: \begin{equation} \begin{split} C &= -\frac{3}{2}\dot{M}(GM\cdot3 R_g)^{1/2}. \end{split} \end{equation} This allows us to get a useful expression: \begin{equation} \begin{split} \nu \Sigma &= \frac{\dot{M}}{3\pi} \left( \frac{1}{R-R_{\rm g}} - \frac{3^{3/2}R_g^{1/2}}{2 R^{3/2}} \right) \left( \frac{ \left(R - R_{\rm g}\right)^2 }{R - R_{\rm g}/3} \right) \end{split} \end{equation} Using this we can get the expression for energy dissipation $D(R)$ \begin{equation} \begin{split} D\left(R\right) &= \frac{R^2}{2} \nu \Sigma \left(\frac{\partial \Omega}{\partial R}\right)^2 \\ &= \frac{3}{8}\frac{\dot{M}}{\pi}\frac{GM}{R}\left( \frac{1}{R-R_{\rm g}} - \frac{3^{3/2} R_{\rm g}^{1/2} }{2 R^{3/2} } \right) \left( \frac{ (R - R_{\rm g})^2 }{R - R_{\rm g}/3} \right) \end{split} \end{equation} Using this we can get then expression for central temperature \begin{equation} \begin{split} T_c^4 &= \frac{3\tau}{4\sigma}D(R)\\ &=\frac{27}{32}\frac{\tau}{\sigma}\nu \Sigma \frac{GM}{R} \frac{(R-R_{\rm g}/3)^2}{(R-R_{\rm g})^4}, \end{split} \end{equation} where $\tau = \kappa\Sigma/2$ and $\kappa = 0.348$~cm$^2\,$~g$^{-1}$. To get the height of the accretion disc, we again repeat the considerations outlined in \citep{frank_king_raine_2002}. Taking the Euler equation: \begin{equation} \rho \frac{\partial \textbf{v}}{\partial t} + \rho \textbf{v}\cdot \Delta \textbf{v} = -\Delta P + \textbf{f}_g, \end{equation} where $\textbf{f}_g,$ is the force density due to gravitation, we assume hydrostatic equilibrium in the $z$ direction and neglect the velocity terms arriving at: \begin{equation} \begin{split} \frac{\partial}{\partial z} P = \rho \frac{\partial}{\partial z} (\phi) = \rho \frac{\partial}{\partial z} \left (- \frac{G M}{(R^2 + z^2)^{1/2} - R_{\rm g}} \right) \end{split} \end{equation} completing the partial derivative on the right hand side and moving $\rho$ to the left we get \begin{equation} \begin{split} \frac{1}{\rho} \frac{\partial}{\partial z} P = \frac{G M z}{(R^2 + z^2)^{1/2}\left[ (R^2 + z^2)^{1/2} - R_{\rm g} \right]^2}. \end{split} \end{equation} Following the argumentation from \cite{frank_king_raine_2002}: if the scaleheight in $z$ direction is $H$, then $\frac{\partial P}{\partial z}\sim \frac{P}{H}$ and $z\sim H$. The thin disc assumption gives $H \ll R$; using $P \sim \rho c_{s}^2$ we get our final expression for $H$: \begin{equation} \begin{split} H &= c_s \left(R-R_{\rm g}\right)\left( \frac{R}{GM} \right)^{1/2}, \label{eq:H} \end{split} \end{equation} where the speed of sound $c_{s}$ is given by: \begin{equation} \begin{split} c_s = T_{\rm c}^4 \left( \frac{\rm{k}\Gamma}{m_{\rm p} \mu} \right)^{1/2}, \end{split} \end{equation} with the mean molecular weight $\mu = 0.63$ and $\Gamma = 5/3$. To check if our equations give us the expected results we perform a test, where an accretion disc is fed at a constant rate. After some time, the disc approaches a steady state, that is, the disc's mass remains constant as does the rate of SMBH feeding and matter escaping via outer boundary. To find the analytically expected result for comparison, we solve equations for the surface density $\Sigma$: \begin{multline} \Sigma^5 = \frac{64}{729}\frac{GM\sigma_{SB}}{R\kappa} \left(\frac{\dot{M}}{\pi} \right)^3 \left( \frac{1}{R-R_{\rm g}} - \frac{3^{3/2} R_{\rm g}^{1/2}}{2 R^{3/2}}\right)^3 \cdot \\ \cdot \left( \frac{3 m_{\rm p} \mu }{5 \alpha k (R-R_{\rm g})} \right)^4 \left( \frac{(R-R_{\rm g})^2}{R - R_{\rm g}/3} \right)^5, \label{eq:SA} \end{multline} where $m_{\rm p}$ is the proton mass and $k$ is the Boltzmann constant, and the central temperature $T_{\rm c}$: \begin{equation} T_{\rm c}^4 = \frac{9}{64} \frac{\kappa \Sigma}{\sigma_{\rm SB}} \frac{\dot{M}}{\pi} \frac{G M_{\rm BH}}{R} \left( \frac{1}{R- R_{\rm g}} - \frac{3^{3/2} R_{\rm g}^{1/2} }{2 R^{3/2}} \right) \left( \frac{R - R_{\rm g}/3}{ (R - R_{\rm g})^2} \right). \label{eq:TcA} \end{equation} We input the SMBH mass $M_{\rm BH}$ and SMBH accretion rate $\dot{M}$ from the model into equations (\ref{eq:SA}) and (\ref{eq:TcA}) to get the radial structure of a steadily accreting disc as expected for the given parameters and plot it together with the radial structure obtained via model calculations in Fig. \ref{fig:CompareAnal}. The upper two panels show surface density $\Sigma$ and central temperature $T_{\rm c}$, respectively. The red curves represent the analytical results for the given disc parameters, while the dotted line shows results taken from a \texttt{Gadget} simulation. The bottom panel shows the relative deviation from the analytical solution for the both surface density (blue) and the central temperature (green). We can see, that both the central temperature and the surface density deviate more from the analytical solution the closer they get to the disc boundaries, but the agreement is generally very good; even the relatively large deviation at the outer boundary is relatively unimportant as it generates a negligible portion of the whole luminosity. \section{Representative surface density maps} \label{app:grids} The following figures show density maps of four simulations at a few different stages in their evolution. In Fig. \ref{fig:grid_00} and Fig. \ref{fig:grid_10} density maps of runs nFBr0 and FBr0 are shown. We can see a very similar initial evolution. From the third panel on, we can see that the FBr0 has a distinct central cavity that increases in size as the simulation progresses. In Fig. \ref{fig:grid_00i} we see that in a run with an instantaneous feedback prescription, INSTr0, similar initial evolution occurs, but despite generally stronger feedback the run fails to produce a central cavity. Three of the runs shown, nFBr0, FBr0 and INSTr0 have the same initial particle distribution. Fig. \ref{fig:grid_12} shows the evolution of the outlier run FBr2 in which almost all of the initial gas is pushed out during the AGN phase. \bsp % \label{lastpage}

Title: The black hole X-ray binary MAXI J1348$-$630 in quiescence

Abstract: The properties of the disk/jet coupling in quiescent black hole low mass X-ray binaries (BH LMXBs) are still largely unknown. In this paper we present the first quasi-simultaneous radio and X-ray detection in quiescence of the BH LMXB MAXI J1348$-$630, which is known to display a hybrid disk/jet connection that depends on the accretion rate. We performed deep X-ray and radio observations using the Chandra X-ray Observatory and the Australia Telescope Compact Array. MAXI J1348$-$630 is detected for the first time in quiescence at an X-ray luminosity $L_{\rm X} = (7.5 \pm 2.9) \times 10^{30} (D/2.2 \ {\rm kpc})^2$ erg s$^{-1}$: one of the lowest X-ray luminosities observed for a quiescent BH LMXB, possibly implying a short orbital period for the system. MAXI J1348$-$630 is also detected in radio at $L_{\rm R} = (4.3 \pm 0.9) \times 10^{26} (D/2.2 \ {\rm kpc})^2$ erg s$^{-1}$. These detections allow us to constrain the location of MAXI J1348$-$630 on the radio/X-ray diagram in quiescence, finding that the source belongs to the standard (radio-loud) track in this phase. This provides a strong confirmation that hybrid-correlation sources follow the standard track at low luminosities and down to quiescence, thus improving our knowledge of the disk/jet connection in BH LMXBs.

https://export.arxiv.org/pdf/2208.00100

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} accretion, accretion discs -- black holes physics -- stars: individual:~\maxithirt{} -- ISM: jets and outflows -- radio continuum: stars -- X-rays: binaries \end{keywords} \section{Introduction} \label{sec:Introduction} Black holes (BH) low mass X-ray binaries (LMXBs) are generally observed when they enter into bright outbursts. During such phases, the X-ray luminosity from the accretion disk increases by several orders of magnitude, and the systems transition between different accretion states with well-defined spectral and timing properties \citep{Homan_belloni, Remillard_xrb, Tetarenko_2016, Belloni_Motta_2016}. In addition, synchrotron-emitting compact jets and moving plasma bubbles, which carry away a significant fraction of the accretion power, are observed in the radio band during, respectively, the hard state and the hard-to-soft state transition (e.g.\ \citealt{Corbel_2000, Fender_2001, Fender2006}). However, BH LMXBs are for most of their time in a quiescent state, with low accretion rates and X-ray luminosities in the range between $10^{30}$ and $10^{34}$ erg s$^{-1}$, corresponding to Eddington fractions of $10^{-9} \sim 10^{-5}$ for a 10 $M_{\odot}$ BH (e.g.\ \citealt{Plotkin2013, Reynolds_2014, Plotkin_2021}). While quiescence is the most common state for a BH LMXB, the properties of the accretion flow and of the jets at low luminosities still need to be properly characterized. When in quiescence, the infrared and optical emission from these systems is dominated by the companion star, while compact jets, albeit faint, are still present and can be detected in the radio band (e.g.\ \citealt{Plotkin_2017_v404, Gallo2019}). It appears that the jet properties are essentially the same between outburst and quiescence, with the radio spectrum that might change from flat-to-inverted to moderately steep \citep{Corbel2013_corr, Tremou2020}. This supports the current picture of the quiescence state as a sort of low-luminosity version of the hard state, with very similar spectral and physical properties (e.g.\ \citealt{Gallo2019}). Moreover, the compact jets in quiescence might channel a larger fraction (with respect to the outburst phase) of the radiative power of the accretion flow, strongly motivating radio observations of quiescent accreting compact objects (e.g.\ \citealt{Plotkin_2019}). The dominance of the jets at these low luminosities is also believed to be responsible for an observed progressive X-ray spectral softening (e.g.\ \citealt{Corbel2006, Plotkin2013}). In analogy with the hard state, the properties of the accretion flow should also be similar, and, according to one of the main proposed scenarios, the BH could be fed by a hot radiatively inefficient accretion flow (e.g.\ \citealt{Narayan_1994, Esin}). Compact jets from BH LMXBs are fundamentally connected to the accretion flow, as the levels of jet emission strongly depends on the accretion rate, both in outburst and in quiescence. This is observed in the hard state as a non-linear correlation between the radio and X-ray luminosities taking the form of a power law ($L_{\rm R} \propto L_{\rm X}^{\beta}$). Such correlation is observed over several orders of magnitude in luminosity on the so-called radio/X-ray diagram (e.g.\ \citealt{Hannikainen_1998, Corbel_2000, Corbel_2003}), and it is today an important diagnostic of the accretion/ejection coupling in BH LMXBs, which can also be used to constrain the emission mechanisms in these sources. Two main groups of BHs are currently observed on the radio/X-ray diagram, displaying two distinct correlations with different power law indices $\beta$. The upper track, historically called \textit{standard}, but often referred to as \textit{radio-loud}, is characterised by $\beta \simeq 0.6$ and includes well-known sources as \gx{}, V404~Cyg and \maxieight{} \citep{Corbel_2000, Corbel_2003, Corbel2013_corr, Gallo_2003, Bright, Shaw_2021}. On the other hand, sources on the lower track are labeled as \textit{outliers} (or \textit{radio-quiet}) and display indices $\beta \simeq 1-1.4$ \citep{Coriat, Jonker_MAXI, Gallo_2014, Monageng_2021, Carotenuto_lrlx}. The majority of BH LMXBs appears now to belong to this second group. Interestingly, thanks to dense monitoring campaigns, some of the outliers were found to display a \textit{hybrid} correlation, transitioning between the two tracks at different X-ray luminosities, between $10^{-4}$ and $10^{-3} L_{\rm Edd}$ \citep{Coriat, Carotenuto_lrlx}. While the size of the available data sets of BH LMXBs on the radio/X-ray diagram increased considerably in the recent years, the reasons for the observed distribution of sources on the diagram are still unclear. Possible explanations include geometric effects, different jet physical properties and different accretion flow properties (e.g.\ \citealt{Peer_Casella_2009, Coriat, Xie_2016, Motta_2018}). Much less is known about the properties of the two groups of sources in quiescence, since the quiescent regions of the radio/X-ray diagram are still greatly unexplored, due to the extreme faintness of quiescent BH LMXBs. So far, only a handful of BH LMXBs have been detected in quiescence, mostly appearing to be compatible with the standard track (e.g.\ \citealt{Gallo_2006, Miller-Jones_2011, Corbel2013_corr, Ribo_2017, Tremou2020}), while the behaviour of outliers is completely uncertain. Outlier (or hybrid) sources could re-join the standard track in quiescence \citep{Coriat}, but this has still to be confirmed with additional observations. In this Letter we present the first radio and X-ray detection in quiescence of \maxithirt{}, constraining its location on the diagram at the lowest possible luminosities. \maxithirt{} is a BH LMXB discovered in 2019, when it entered into a bright outburst \citep{Tominaga_1348}. The source appears to be located at a close distance, between 2.2 and 3.4 kpc \citep{Chauhan2021, Lamer_2021}, which is favourable for a study of the system in quiescence. In \cite{Carotenuto2021} we have presented the full radio and X-ray monitoring of \maxithirt{} during its 2019/2020 outburst, and we refer to that paper for details on the various outburst phases. Our radio observations detected and covered the entire evolution of compact jets in the outburst phase, during which \maxithirt{} also displayed some extremely energetic large-scale discrete ejecta \citep{Carotenuto2021, Carotenuto_2022}. Combining the detections of compact jets with the quasi-simultaneous X-ray observations, we were able to cover the whole evolution of \maxithirt{} on the radio/X-ray diagram, finding that this source belongs to the numerous group of outliers. In fact, \maxithirt{} displays a clear hybrid radio/X-ray correlation, re-joining the standard track at $L_{\rm X} \simeq 10^{33}$ erg s$^{-1}$, and it is today the hybrid-correlation source with the most detailed coverage on the diagram, and such coverage was obtained during a single outburst \citep{Carotenuto_lrlx}. Therefore, constraining its behaviour in quiescence is critical for completing the full track of the source on the radio/X-ray diagram, which is something unprecedented an hybrid source. \section{Observations} \label{sec:Observations} To constrain the quiescent level of \maxithirt{} in radio and X-rays, we performed a quasi-simultaneous observation of the source with the NASA \textit{Chandra} X-ray observatory \citep{Weisskopf_2000} and with the Australia Telescope Compact Array (ATCA, \citealt{Frater_1992}). \subsection{\textit{Chandra} observation} \label{sec:Chandra_observation} The \textit{Chandra} observations of \maxithirt{} were performed with the Advanced CCD Imaging Spectrometer (ACIS-S) on 2021 June 28 (30 ks; ObsId 25054) under Director's Discretionary Time (PI: F.\ Carotenuto). We restricted the observation to the back-illuminated chip S3, providing the best low-energy response. The X-ray data analysis was performed using the \textit{Chandra} Interactive Analysis of Observation (CIAO) software 4.14.1 \citep{Fruscione2006}, with the calibration files CALDB version 4.9.6. We filtered to only keep the events in the energy range 0.3--8 keV. No background flare was detected. The \texttt{fluximage} script was used to create the X-ray images, keeping the bin size to 1 (1 pixel = 0.492\arcsec) (see Fig. 1). A single X-ray source is detected at a position consistent with \maxithirt{}. We extracted an energy spectrum in the 0.3--8 keV energy range from a circular source region with a 2\arcsec \ radius, while the background was extracted from an annulus with an inner radius of 10\arcsec \ and an outer radius of 20\arcsec. Spectral analysis was carried out with XSPEC \citep{Arnaud_xspec}. Our main goal with the X-ray spectral analysis was to estimate the unabsorbed flux in the 1--10 keV band. Hence, we fitted the spectrum with a simple absorbed power law model ({\tt tbabs$\times$powerlaw}), for which the {\tt tbabs} component accounts for interstellar absorption, modelled with an equivalent hydrogen column density ($N_{\rm H}$) by using {\tt wilm} abundances \citep{Wilms} and {\tt vern} cross-sections \citep{Verner1996}. Concerning the spectral parameters, the power law photon index was fixed to $\Gamma = 2.2$, in accordance with what is generally measured for BH LMXBs in quiescence \citep{Corbel2006, Corbel2008, Plotkin2013}, and the column density was fixed to $ N_{\rm H} = 0.86 \times 10^{22}$ cm$^{-2}$, measured from X-ray observations during the outburst \citep{Carotenuto2021, Carotenuto_lrlx}. As generally prescribed for low-counts observations, Cash statistics {\tt cstat} \citep{cstat} was used in the fitting process. We then calculated the 1--10 keV X-ray unabsorbed flux using the XSPEC convolution model {\tt cflux}. \subsection{ATCA observation} \label{sec:ATCA observation} A long ATCA observation was performed between 3 and 5 July 2021, less than a week after the \textit{Chandra} observation, for a total of 24 hours on source (proposal ID C3416, PI Carotenuto). During this interval, the array was in the 6B extended configuration. The observation was taken simultaneously in the C and X bands, at central frequencies of 5.5 and 9 GHz, respectively. For each central frequency, the total bandwidth was 2 GHz. PKS~1934--638 was used for bandpass and flux calibration, while PKS~1352--63 was used for the complex gain calibration. The data were first flagged and then calibrated using standard procedures with the Common Astronomy Software Application (\textsc{casa}, \citealt{CASA}). Imaging was carried out with the standard {\tt tclean} algorithm in \textsc{casa} with a natural weighting scheme chosen in order to maximize sensitivity. To obtain the lowest possible rms, we stacked the two bands into a single image, adopting the Multi-term (Multi Scale) Multi-Frequency Synthesis (MTMFS, or MS-MFS, \citealt{Rau_2011}) as a deconvolution algorithm in order to take into account the large fractional bandwidth, using two Taylor terms. In this configuration, the rms reached 3 $\mu$Jy beam$^{-1}$. To obtain the radio flux density $S_{\nu}$, we fitted a point source in the image plane with the \textsc{casa} task {\tt imfit}. \section{The quiescent level of \maxithirt{}} \label{sec:The quiescent level of MAXIJ1348} We obtained a significant detection of \maxithirt{} in quiescence, both in radio and X-rays. The detection is shown in Figure \ref{fig:quiescence_detection}. In the X-rays, 21 counts from \maxithirt{} were collected by \textit{Chandra} (0.3--8 keV) in the 30 ks of exposure, implying an unambiguous detection at the lowest level of activity of the source. After spectral fitting (Section \ref{sec:Chandra_observation}), we obtain an integrated unabsorbed flux $F_{\rm X} = (1.3 \pm 0.5) \times10^{-14}$ erg cm$^{-2}$ s$^{-1}$ in the 1$-$10 keV energy range. This can be then converted to the integrated luminosity $L_{\rm X} = 4\pi D^2 S_{\rm X }$. For consistency with \cite{Carotenuto_lrlx}, in this paper we assume distance $D = 2.2$ kpc \citep{Chauhan2021} and we quote the luminosities with a factor $(D/2.2 \ {\rm kpc})^2$ in order to include the distance estimation by \cite{Lamer_2021}. We obtain $L_{\rm X} = (7.5 \pm 2.9) \times 10^{30} (D/2.2 \ {\rm kpc})^2$ erg s$^{-1}$. \maxithirt{} has also been detected in radio during its quiescence phase. As can be seen from Figure \ref{fig:quiescence_detection}, with 24 h of ATCA exposure time, \maxithirt{} is detected as a point source at its known location, at a significance level of $5\sigma$ and with a peak flux of $15.3 \pm 2.5$ $\mu$Jy. The faint radio emission coming from the core location can be safely attributed to the presence of compact jets in quiescence, as observed for other BH LMXBs \citep{Corbel2013_corr, Gallo2019, Tremou2020}. We then converted the measured radio flux density $S_{\nu}$ to the 5 GHz monochromatic luminosity $L_{\rm R} = 4\pi D^2 \nu S_{\nu}$, conservatively assuming a flat radio spectrum. This leads to a monochromatic 5 GHz radio luminosity $L_{\rm R} = (4.3 \pm 0.9) \times 10^{26} (D/2.2 \ {\rm kpc})^2$ erg s$^{-1}$. With the quasi-simultaneous radio and X-ray detection in quiescence, we obtain a new, critical measurement for \maxithirt{}, completing the coverage of this source on the radio/X-ray diagram. The updated radio/X-ray diagram is shown in Figure \ref{fig:x_radio_corr_quiescence}. One can immediately notice that \maxithirt{} aligns very well with the standard track defined by \gx{}, V404~Cyg and \maxieight{} (e.g.\ \citealt{Corbel2013_corr}). Along the detection in quiescence, the clear confirmation that \maxithirt{} lies on the standard track in quiescence is one of the main results of this work. We performed a a simple power law fit $L_{\rm R} \propto L_{\rm X}^{\beta}$ to the three points on the diagram with $L_{\rm X} < L_{\rm stand} = 3.2^{+61.3}_{-3.0} \times 10^{32} (D/2.2 \ \ {\rm kpc})^2$ erg s$^{-1}$ (see Figure \ref{fig:x_radio_corr_quiescence} and \citealt{Carotenuto_lrlx}), including the detection in quiescence and the two lowest detection in the outburst lying on the standard track. The fit, relying on {\tt curve$\_$fit} from the SciPy package, yields $\beta = 0.57 \pm 0.08$, which is fully consistent with the standard track (e.g.\ \citealt{Corbel2013_corr}). \section{Discussion} \label{Discussion} \maxithirt{} is one of the few BH LMXBs detected in quiescence both in the radio and X-ray bands. Due to the need of quasi-simultaneous deep observations, these detections are exceedingly rare at present. Even if a model-independent mass estimation of the BH is not yet available, the quiescence X-ray luminosity of \maxithirt{} corresponds to an Eddington fraction $\sim$10$^{-8.3} L_{\rm Edd}$, assuming a $10 M_{\odot}$ BH. Currently, the BH LMXB with the lowest known X-ray luminosity in quiescence is GS~2000+25, detected with \textit{Chandra} at $\sim$10$^{30}$ erg s$^{-1}$, namely $\sim$10$^{-9} L_{\rm Edd}$ \citep{Rodriguez_2020}. The detections at such low Eddington fractions are among the lowest level of activity ever explored for an accreting compact object. However, for GS~2000+25 a radio counterpart was not detected \citep{Rodriguez_2020}. The most interesting result of this work revolves around the location of \maxithirt{} on the radio/X-ray diagram in quiescence. In \cite{Carotenuto_lrlx}, we demonstrated that \maxithirt{} belongs to the less-explored group of the so-called \textit{hybrid-correlation} sources. With this detection, we can confirm for the first time that these sources follow the standard track down to quiescence. Moreover, after this work and the coverage presented in \cite{Carotenuto_lrlx}, \maxithirt{} becomes the first BH LMXB to be characterised on the radio/X-ray diagram for over seven orders of magnitude in X-ray luminosity and almost four orders of magnitude in radio luminosity, during a single outburst and in quiescence. The continuity of evolution on the standard track from the end of the outburst (see Figure \ref{fig:x_radio_corr_quiescence} and \citealt{Carotenuto_lrlx}), and the displayed power law X-ray spectrum, are consistent with the current picture of the quiescence state as a low-luminosity analogue of the hard state (e.g.\ \citealt{Gallo2019}). It is then possible to compare the location of \maxithirt{} on the diagram with the handful of other BH LMXBs for which radio and X-ray detections in quiescence are available, shown in Figure \ref{fig:x_radio_corr_quiescence}. \maxithirt{} appears to have a quiescence luminosity that is comparable in radio and X-rays with A0620--00 \citep{Gallo_2006}, MWC~656 \citep{Ribo_2017} and XTE~J1118+480 \citep{Gallo_2014}. Stringent radio upper limits are also available for Swift~J1357.2--0933 \citep{Plotkin_2016} and GS~2000+25 \citep{Rodriguez_2020}. On the other hand, \maxithirt{} displays a quiescent luminosity which is at least one order of magnitude lower than other very relevant quiescent BH LMXBs on the diagram, such as \gx{} \citep{Corbel2013_corr, Tremou2020} and V404~Cyg \citep{Plotkin_2017_v404}. A relation between the quiescent X-ray luminosity $L_{\rm X}$ and the orbital period $P_{\rm orb} $ exists for X-ray binaries \citep{Menou_1999, Garcia_2001}, where a higher $L_{\rm X}$ in quiescence is expected for higher $P_{\rm orb}$, due to the larger size of the accretion disk. While the orbital period of \maxithirt{} is currently unknown, from its quiescence X-ray luminosity it is possible to speculate that its $P_{\rm orb}$ is likely shorter than $10 \sim 20$ h, as can be seen from Figure \ref{fig:p_orb_quiescence}, at both possible distances. This is also in line with the peak luminosity in outburst ($\sim$ $10^{-1} L_{\rm Edd}$, \citealt{Carotenuto2021}), which might suggest a similar order of magnitude for $P_{\rm orb}$ \citep{Wu_2010}. A detection of \maxithirt{} in quiescence on the standard track also supports the findings obtained for V404~Cyg and \maxieight{} \citep{Plotkin_2017_v404, Shaw_2021}, ruling out the possibility that the X-ray emission in quiescence is dominated by the synchrotron radiation produced by the jet. Such scenario, in fact, would result in a steepening of the observed correlation at low luminosities \citep{yuan_cui_2005}, which is instead not observed. The X-ray spectrum of \maxithirt{} in quiescence can be adequately described with an absorbed power law with $\Gamma = 2.2$. The spectral softening when approaching quiescence is a well-known phenomenon \citep{Corbel2006, Corbel2008, Plotkin2013, Liu_index}, observed also in \maxithirt{} \citep{Carotenuto2021}. Such softening could be explained by a lack of hard X-ray photons resulting from the decrease of the inverse Compton scattering efficiency at low luminosities \citep{Esin, Qiao_2013, Liu_index}, or by a dominance at low accretion rate of Comptonization from thermal particles at the base of the jet, and not in the corona \citep{Markoff_corona, Poutanen_2014, Plotkin2015}. Thanks to this new detection in quiescence, we obtained the complete coverage of the evolution of \maxithirt{} on the radio/X-ray diagram, from the quiescent state to the peak luminosity during the outburst. We believe that the full data set of \maxithirt{} on the diagram will turn out to be particularly important for testing different models describing the disk/jet connection in accreting BHs, leading to major improvements in our understanding of hybrid-correlation sources and, more in general, of the behaviour of BH LMXBs on the diagram. \section*{Acknowledgements} We thank the anonymous referee for the careful reading of the manuscript and for the valuable comments. This research has made use of data obtained from the \textit{Chandra} X-ray Observatory and software provided by the Chandra X-ray Center (CXC) in the application package CIAO. FC, SC and AT thank Jamie Stevens and staff from the Australia Telescope National Facility (ATNF) for scheduling the ATCA radio observations. ATCA is part of the ATNF which is funded by the Australian Government for operation as a National Facility managed by CSIRO. We acknowledge the Gomeroi people as the traditional owners of the ATCA observatory site. FC acknowledges support from the Royal Society through the Newton International Fellowship programme (NIF/R1/211296) and from the project Initiative d’Excellence (IdEx) of Universit\'{e} de Paris (ANR-18-IDEX-0001). We acknowledge the use of the Nan\c cay Data Center, hosted by the Nan\c cay Radio Observatory (Observatoire de Paris-PSL, CNRS, Universit\'{e} d'Orl\'{e}ans), and supported by Region Centre-Val de Loire. This project also made use of \textsc{matplotlib} \citep{matplotlib}, \textsc{numpy} \citep{harris2020array} and Overleaf (\url{http://www.overleaf.com}). \section*{Data availability} The un-calibrated ATCA visibility data are publicly available at the ATNF archive at \url{https://atoa.atnf.csiro.au}. The \textit{Chandra} data are instead available from the Chandra Data Archive at \url{https://cxc.harvard.edu/cda}. \bibliographystyle{mnras} \bibliography{maxi1348_paper4} \bsp % \label{lastpage}

Title: Astrochemical model to study the abundances of branched carbon-chain molecules in a hot molecular core with realistic binding energies

Abstract: Straight-chain (normal-propyl cyanide, n - C3H7CN) and branched-chain (iso-propyl cyanide, i - C3H7CN) alkyl cyanides are recently identified in the massive star-forming regions (Sgr B2(N) and Orion). These branched-chain molecules indicate that the key amino acids (side-chain structures) may also be present in a similar region. The process by which this branching could propagate towards the higher-order (butyl cyanide, C4H9CN) is an active field of research. Since the grain catalysis process could have formed a major portion of these species, considering a realistic set of binding energies are indeed essential. We employ quantum chemical calculations to estimate the binding energy of these species considering water as a substrate because water is the principal constituent of this interstellar ice. We find significantly lower binding energy values for these species than were previously used. It is noticed that the use of realistic binding energy values can significantly change the abundance of these species. The branching is more favorable for the higher-order alkyl cyanides with the new binding energies. With the inclusion of our new binding energy values and one essential destruction reaction (i - C3H7CN + H -> CH3C(CH3)CN + H2, having an activation barrier of 947 K), abundances of t - C4H9CN dramatically increased.

https://export.arxiv.org/pdf/2208.03531

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} astrochemistry -- ISM: Molecules -- molecular processes -- ISM: abundances -- ISM: evolution \end{keywords} \section{Introduction} \label{sec:sample1} About 270 molecules comprising 17 different elements have been detected in the interstellar medium (ISM) and circumstellar shells\footnote{\url{https://cdms.astro.uni-koeln.de/classic/molecules}}. H$_2$ is the most abundant molecule in the ISM, whereas CO is second \citep{wils70}. The protonated form of H$_2$, H$_3^+$ is also very plentiful. Despite this, numerous positively (e.g., HCO$^+$, CF$^+$) and negatively (e.g., CN$^-$, $\rm{C_3N^-}$, $\rm{C_4H^-}$) charged radicals, isomers (e.g., HNC, HCN), and isotopes (e.g., deuterium, $^{13}$C, $^{18}$O, $^{29}$Si) were also found. The condensed regions of molecular clouds carry grains, which uphold some simple and complex molecules like H$_2$O, CO, CO$_2$, CH$_3$OH, H$_2$CO, NH$_3$, HCOOH, CH$_4$, etc. \citep{boog15,gora20a}. In the slightly warmer region, radicals become mobile and could produce various complex organic molecules \citep[COMs;][]{das08,das10,das11,das16}. The major portion of the observed complex molecules in the ISM is organic, showing the active participation of the C atom in large molecules detected in space. The presence of larger PAHs ($\rm{C_nH_m}$) \citep{tiel08} and fullerenes (C$_{60}$, C$_{70}$) \citep{cami10} in space was obtained. The presence of complex organics indicates the existence of amino acids in the ISM \citep{sil18,gora20b}. Amino acids (building blocks of protein) are found in carbon chondrites and comets. For example, a carbonaceous meteorite, Murchison, provides more than 70 extraterrestrial amino acids \citep{bott02}. Detected organic molecules carry the aliphatic nature in a simple straight-chain structure (with less than three C atoms). In contrast, a branched carbon-chain structure also exists with four or more C atoms. Several complex species of the prebiotic interest were recently been identified. Some of them are benzonitrile \citep{mcgu18}, propargylimine \citep{bizz20}, ethanolamine \citep{rivi21}, glycolonitrile \citep{zeng19}, ethynyl cyclopropenylidene, cyclopentadiene \citep{cern21a}, indene \citep{burk21,cern21a}, ethynyl cyclopentadiene \citep{cern21b} 1/2 - cyanocylopentadiene \citep{lee21,mcca21}, 1/2 - cyanonaphthalene \citep{mcgu21}, propylene oxide \citep[chiral molecule]{mcgu16}. Iso-butyronitrile or iso-propyl cyanide ($\rm{i-C_3H_7CN}$) is % is the first branched carbon-chain molecule (BCM), which has been observed in the high-mass star-forming regions (HMSFRs), Sagittarius B2 \citep[Sgr B2;][]{bell14}. \cite{paga17} also detected $\rm{n/i-C_3H_7CN}$ for the first time in Orion. The first detection of this kind of molecule has increased the curiosity of other BCMs in the HMSFRs. \cite{bell14} differentiated the mechanism of the formation of both conformers of $\rm{C_3H_7CN}$ and proposed their reaction pathways. They suggested the formation of $\rm{i-C_3H_7CN}$ via the addition of cyanide radical ($\rm{-CN}$) at the secondary carbon in the chain. \cite{etim17} considered various isomers from the C$_5$H$_9$N isomeric group. Butyl cyanide ($\rm{C_4H_9CN}$) belongs to this isomeric group having four forms: the straight-chain normal-butyl cyanide ($\rm{n-C_4H_9CN}$), iso-butyl cyanide ($\rm{i-C_4H_9CN}$), sec-butyl cyanide ($\rm{s-C_4H_9CN}$), and tert-butyl cyanide ($\rm{t-C_4H_9CN}$). According to the expected rotational intensity ratio and considering equal abundances of these species, \cite{etim17} found that $\rm{t-C_4H_9CN}$ is the most favourable candidate among the $\rm{C_5H_9N}$ isomeric group for the future astronomical detection. However, interstellar chemistry is far from equilibrium, and reaction pathways for the formation of these species are very different. \cite{garr17} performed astrochemical modeling to decode how gas and grain are responsible for producing these species. They considered a transformation of species from the surface to the mantle and vice versa. They found $\rm{s-C_4H_9CN}$ with high abundance while $\rm{t-C_4H_9CN}$ with lesser abundance and proposed $\rm{s-C_4H_9CN}$ as future detectable BCM. They implemented some educated guesses of binding energies (BE) to estimate their abundances. Here, in this work, our primary motivation is to study the fate of these species with realistic BEs. Since in the denser region, $\sim 70\%$ of the grain mantle is covered with water, we consider water as a substrate for the computation of the BEs of these species. Computed BEs are then used in our astrochemical model to refrain the abundances of these species. This paper is organized as follows. Firstly, in Section \ref{sec:comp}, computational details, methodology, and reaction pathways are presented. Then, results and discussions are presented in Section \ref{sec:results}. Finally, in Section \ref{sec:conclusions}, we conclude. \section{Computational details and methodology} \label{sec:comp} \subsection{Binding Energy} Merely a hundred years ago, the vast existence of anything but atoms and obscuring tiny dust grains in the ISM was unimaginable. The existence of interstellar dust grains confirmed by \cite{trum30} directed to exploring the presence of organic species on dust grains. Species on grain surfaces generally undergo four different mechanisms: a) Accretion (adsorption) onto the surface, b) desorption from the surface, c) diffusion across the surface or on/within the ice-mantle, and d) reaction. Ice mantles are further processed when it is exposed to various interstellar radiations. The adsorption energy or BE is the surface energy due to electrostatic interaction. It is the energy with which different particles or surfaces incline to attach. For example, it creates a film between the surface of species (adsorbate) and dust grain (adsorbent). We calculate the BE of a species as follows: \[BE = (E_{\rm surface} + E_{\rm species}) - E_{\rm ss},\] where $E_{ss}$ is the optimized energy of a species placed on the grain surface by a weak van der Waals interaction. $E_{\rm surface}$ and $E_{\rm species}$ are the optimized energies of the substrate and target species, respectively. Quantum chemical calculations are used to estimate the optimized energies of adsorbate and adsorbent. Since water molecules dominate the dense part of the interstellar ices \citep{kean01,das10,das11}, we consider it as a substrate. All the quantum chemical calculations are performed using the Gaussian 09 suite of programs \citep{fris13}. We employ the second-order M\o ller-Plesset (MP2) method to optimize the geometries with the aug-cc-pVDZ basis set. \cite{das18} carried out some benchmarking calculations for 16 stable species by considering ZPVE and BSSE corrections. Their computed values better agree with the experiments when ZPVE and BSSE corrections were not included. Here also, we do not consider ZPVE and BSSE corrections into account. The fully optimized structure of the species is further verified by checking the real harmonic vibrational frequencies. Table \ref{tab:be} reports the computed BE of some species related to the formation of BCMs. The ground-state spin multiplicity of these species is also noted for clarity. \cite{das18} estimated BE of $\sim 100$ interstellar species. They proposed a scaling factor of 1.416 and 1.188 for water monomer and tetramer structure. Here, we use water monomer as a binding substrate for all species noted in Table \ref{tab:be}. But, sometimes, the size of the water monomer is minimal compared to the target species acting as an adsorbate. It may lead to some misleading estimations of BEs. For the betterment, we further use water tetramer as a substrate to estimate BEs of some species reported here. \cite{das18,das21} noted that computed BEs are dependent on the chosen adsorption site. Here, for some key species, we calculate BEs at multiple binding sites and noted in Table \ref{tab:be}. For the modeling purpose, we use the average scaled value. The BE values used by \cite{garr17} are also noted for comparison. \begin{table*} \scriptsize {\centering \caption{Calculated binding energy and enthalpy of formation of some key species and comparison with the previous estimation.} \label{tab:be} \begin{tabular}{cccccccccc} \hline Serial & & Ground & \multicolumn{5}{c}{Binding Energies (K)} & \multicolumn{2}{c}{Enthalpy of formation} \\ No. & & State Spin & \multicolumn{5}{c}{ } & \multicolumn{2}{c}{$\rm{\Delta H_f}$ (298 K) (kcal/mole)}\\ & Species & Multiplicity & \multicolumn{4}{c}{This Work (MP2-aug-cc-pVDZ)} & \cite{garr17}& This Work & \cite{garr17}\\ & & & with water monomer & Scaled by 1.416 & with water tetramer & Scaled by 1.188 & & G4 composite method & \\ \hline 1 & $\rm{CH_3CN}$ & Singlet & 2614 & 3702 & & & 4680 &+15.84&+17.70 \\ 2 & $\rm{C_2H_2CN}$ & Doublet & 3263* (2826 / 3700) & 4620* (4001 / 5240) & & & 4187 & +103.49& +105.84\\ 3 & $\rm{CH_2CHCN}$ & Singlet & 2788* (2990 / 2587) & 3948* (4234 / 3663) & 2980 & 3540 & 4637 & +43.31 & +42.95 \\ 4 & $\rm{\dot CH_2CH_2CN}$ & Doublet & 2900 & 4107 & & & 5087 & +61.54 & +55.13 \\ 5 & $\rm{CH_3 \dot CHCN}$ & Doublet & 2730 & 3865 & & & 5087 & +47.14 & +53.23 \\ 6 & $\rm{C_2H_5CN}$ & Singlet & 2867 & 4059 & 4113 & 4886 & 5537 & +10.96 & +12.71 \\ 7 & $\rm{\dot CH_2CH_2CH_2CN}$ & Doublet & 3070 & 4347 & & & 6787&+54.61&+56.38\\ 8 & $\rm{CH_3\dot CHCH_2CN}$ & Doublet & 3000* (3171 / 2830) & 4249* (4491 / 4007) & & & 6787 & +51.69 & +54.48 \\ 9 & $\rm{CH_3CH_2\dot CHCN}$ & Doublet & 2738 & 3877 & & & 6787 & +42.23&+54.48\\ 10 & $\rm{n-C_3H_7CN}$ & Singlet & 2112* (1307 / 2918) & 2991* (1850 / 4132) & 4686 & 5567 & 7237 & +5.56&+7.46 \\ 11 & $\rm{\dot CH_2CH(CH_3)CN}$ & Doublet & 2958 &4188 & & & 6787&+55.26&+54.36\\ 12 & $\rm{\dot CH_3\dot C(CH_3)CN}$ & Doublet & 3290* (3253 / 3328) & 4659* (4607 / 4711) & & & 6787 & +37.35 & +52.46 \\ 13 & $\rm{i-C_3H_7CN}$ & Singlet & 2316* (1743 / 2888) & 3279* (2469 / 4090) & 4184 & 4970 & 7237 & +5.18 & +5.44 \\ 14 & $\rm{CH_3CH_2\dot CHCH_2CN}$ & Doublet & 3224* (3423 / 3024) & 4564* (4847 / 4281) & & & 8487&+46.74& --- \\ 15 & $\rm{n-C_4H_9CN}$ & Singlet & 4464 & 6320 & 3694 & 4388 & 8937&+0.50&+2.65 \\ 16 & $\rm{i-C_4H_9CN}$ & Singlet & 3123 & 4422 & 4336 & 5151 & 8937&$-0.49$&+0.58\\ 17 & $\rm{s-C_4H_9CN}$ & Singlet & 1861 & 2635 & 4472 & 5313 & 8937 &$-0.13$ & $+0.58$ \\ 18 & $\rm{t-C_4H_9CN}$ & Singlet & 3230 & 4574 & 4333 & 5148 & 8937&$-0.83$&$-0.79$\\ 19 & $\rm{\dot CH_2CH_2CH_3}$ & Doublet & 1866 & 2643 & & & 5637&+18.65&+23.90\\ 20 & $\rm{CH_3\dot CHCH_3}$ & Doublet & 1532* (974 / 2091) & 2170* (1379 / 2961) & & & 5637&+14.56&+22.00\\ 21 & $\rm{C_3H_8}$ & Singlet & 1028 & 1456 & & & 6087&$-29.93$&$-25.02$\\ 22 & $\rm{\dot CH_2CH_2CH_2CH_3}$ & Doublet & 1464* (1883 / 1045) & 2073* (2666 / 1480) & & & 7337 & +13.82 & +17.90 \\ 23 & $\rm{CH_3\dot CHCH_2CH_3}$ & Doublet & 2315 & 3278 & & & 7337 & +9.66&+16.00\\ 24 & $\rm{n-C_4H_{10}}$ & Singlet & 1113 & 1576 & & & 7787 & $-35.05$ & $-30.03$ \\ 25 & $\rm{\dot CH_2CH(CH_3)CH_3}$ & Doublet & 1057 & 1496 & & & 7337&+12.96&+17.00\\ 26 & $\rm{CH_3\dot C(CH_3)CH_3}$ & Doublet & 2370 & 3357 & & & 7337 & +5.44 & +15.10 \\ 27 & $\rm{i-C_4H_{10}}$ & Singlet & 1103* (1111 / 1095) & 1562* (1573 / 1551) & & & 7787 & $-35.90$ & $-32.07$ \\ 28 & $\rm{n-C_5H_{12}}$ & Singlet & 3957 &5604 & & & 9487&$-40.05$&$-35.08$\\ 29 & $\rm{i-C_5H_{12}}$ & Singlet & 992* (731 / 1254) & 1405* (1035 / 1775) & & & 9487 & $-40.05$ & $-36.73$ \\ 30 & $\rm{neo-C_5H_{12}}$ & Singlet & 1028 & 1455 & & & 9487 & $-41.83$ &$-40.14$ \\ \hline \end{tabular}} \\ *average of the BE values obtained from multiple binding sites. \end{table*} \subsection{Other chemical parameters} The enthalpies of formation are calculated at 298 K. The calculated enthalpies of formation are subsequently noted in Table \ref{tab:be} and compared with those indicated in \cite{garr17}. They noted these values from the NIST WebBook database and estimated where not available in the literature. The polarizability and dipole moment of these species are also calculated and compared (if available at the NIST WebBook database) in Table \ref{tab:polar_dipole}. These polarizabilities and dipole moments are further used in our model to obtain the destruction of these species by ion-neutral reactions \citep{su82,woon09}. \subsection{Reaction rates} Here, we prepare a reaction network to study the abundance of some BCMs. Most of the reaction rates of the formation BCMs are taken from \cite{garr17}. Here, we carry out transition state (TS) calculations with the Density Functional Theory (DFT) for some specific reactions. The ice-phase geometries of products, reactants, and TSs are optimized with the QST2 method and DFT-B3LYP/6-31+G(d) level of theory. Similarly, the gas-phase geometries of products, reactants, and TSs are optimized using the Berny algorithm and DFT-B3LYP/6-31+G(d,p) and 6-31++G(2d,p) level of theories. All TSs have a single imaginary frequency. The legitimacy of each calculated TS is verified by visually examining the vibrational mode corresponding to the single imaginary vibrational frequency and applying the criterion that it correctly connects the reactants and products through intrinsic reaction coordinate (IRC) paths. Finally, energy barriers are calculated using the TS theory. Formation pathways of the target molecules (vinyl, ethyl, i/n-propyl, i/n/s/t-butyl cyanide) are discussed in Section \ref{sec:EV}-\ref{sec:BC}. \subsubsection{Vinyl and ethyl cyanide \label{sec:EV}} Around the low-temperature regime, ice-phase formation of vinyl cyanide ($\rm{CH_2CHCN}$) could process by the successive hydrogen additions of $\rm{HC_3N}$ (see reactions 1 and 2 in Table \ref{tab:EV}). The first step of this H-addition reaction has an activation barrier of 1710 K (KIDA\footnote{\url{http://kida.astrophy.u-bordeaux.fr}}), whereas the second step is barrierless. In the little warmer region, $\rm{CH_2CHCN}$ could also produce in the ice phase by the radical-radical reaction between CN and CHCH$_2$ (reaction 3). The gas-phase reaction between CN and $\rm{C_2H_4}$ radicals (reaction 4) could contribute to the formation of $\rm{CH_2CHCN}$ beyond 100 K. In the UMIST database \citep{mcel13}, this reaction is allowed only beyond 300 K with the three constants of the reaction: $\alpha=1.25 \times 10^{-10}$, $\beta=0.7$, and $\gamma=30.0$. However, the KIDA databse \citep{wake12} consider this reaction at the low temperature as well. For the lower limit of the temperature 10 K, they noted $\alpha=2.67 \times 10^{-10}$, whereas for 50 K, it is $5.31 \times 10^{-11}$ along with $\beta=-0.69$, and $\gamma= 31.0$. \begin{table} \scriptsize {\centering \caption{Ice-phase reactions considered for $\rm{CH_2CHCN}$ and $\rm{C_2H_5CN}$.} \label{tab:EV} \begin{tabular}{ccc} \hline Reaction & Reactions & Activation Barrier (K) \\ Number (Type) & & \\ \hline \multicolumn{3}{c}{$\rm{CH_2CHCN}$} \\ \hline 1 (NR) & $\rm{H+ HC_3N \rightarrow C_3H_2N}$ & $1710^a$ \\ 2 (RR) & $\rm{H+ C_3H_2N \rightarrow CH_2CHCN}$ & --- \\ 3 (RR) & $\rm{CHCH_2 + CN \rightarrow CH_2CHCN}$ & --- \\ 4 (RR) & $\rm{C_2H_4+CN \rightarrow CH_2CHCN+H}$ & --- \\ \hline \multicolumn{3}{c}{$\rm{C_2H_5CN}$} \\ \hline 5 (NR) & $\rm{H + CH_2CHCN \rightarrow CH_3CHCN}$ & $619^b / 158^c$ \\ 6 (NR) & $\rm{H + CH_2CHCN \rightarrow CH_2CH_2CN}$ & $1320^b / 1603^c$ \\ 7 (RR) & $\rm{H+CH_3CHCN \rightarrow C_2H_5CN}$ & --- \\ 8 (RR) & $\rm{H+CH_2CH_2CN \rightarrow C_2H_5CN}$ & --- \\ 9 (RR) & $\rm{CH_3CH_2 +CN \rightarrow C_2H_5CN}$ & --- \\ 10 (NR) & $\rm{C_2H_5CN+H \rightarrow C_2H_5CNH}$ & $2712^d$ \\ 11 (NR) & $\rm{C_2H_5CN + H \rightarrow CH_2CH_2CN + H_2}$ & $3472^c$ \\ \hline \end{tabular}} \\ Notes: NR and RR refer to neutral-radical and barrierless radical-radical reactions, respectively. \\ $^a$ KIDA (\url{https://kida.astrochem-tools.org}) \\ $^b$ \cite{garr17} \\ $^c$ This work (gas-phase) \\ $^d$ \cite{sil18} \end{table} $\rm{CH_2CHCN}$ is further channelized to form ethyl cyanide ($\rm{C_2H_5CN}$) by successive hydrogenations in the ice phase (reactions $5-8$). \cite{garr17} used an activation barrier of 619 K for the H-addition to the first carbon atom of $\rm{CH_2CHCN}$ (reaction 5) and 1320 K for the H-addition to the second carbon atom of $\rm{CH_2CHCN}$ (reaction 6). \cite{garr17} adopted these activation barriers based on equivalent hydrogenation of $\rm{C_3H_6}$. Our TS calculations find an activation barrier of 158 K and 1603 K for reactions 5 and 6, respectively (see Fig. \ref{fig:reaction_5}). For our chemical model, we use our calculated values for reactions 5 and 6. At the same time, reactions 7 and 8 are considered barrierless. In the warmer region, ice-phase formation of $\rm{C_2H_5CN}$ can follow the radical-radical reaction (reaction 9). $\rm{C_2H_5CN}$ further hydrogenates to form ${\rm C_2H_5CNH}$ (reaction 10, with an activation barrier of 2712 K) which was considered in \cite{sil18}. Furthermore, we explore one hydrogenation abstraction reaction of $\rm{C_2H_5CN}$ (reaction 11). The reaction enthalpy of this reaction is $-3.38$ kcal/mol (DFT). We obtain an activation barrier of 3472 K for this reaction in the gas phase. However, the potential energy surface diagram of gas-phase reaction 11 shown in Fig. \ref{fig:ch2ch2cn} depicts that the energy of the products is less than that of reactants and TS does not converge in the ice phase. Due to these reasons, we do not include this reaction in our network. \subsubsection{Propyl cyanide \label{sec:prop}} Two isomeric forms of propyl cyanide ($\rm{n-C_3H_7CN}$ and $\rm{i-C_3H_7CN}$) are considered in our model. The formation pathways of these two isomeric forms are taken from \cite{garr17}. $\rm{n-C_3H_7CN}$ and $\rm{i-C_3H_7CN}$ formation in the ice phase could be processed by reactions $12-17$ and $18-21$, respectively, noted in Table \ref{tab:prop}. For the destruction of ice-phase $\rm{i-C_3H_7CN}$, we consider a hydrogen abstraction reaction (reaction 22). However, we could not converge the TS of reaction 22 in the ice phase, so here we use it in the gas phase. The potential energy surface diagram of this reaction is shown in Fig. \ref{fig:ch3cch3cn}. We obtain an activation energy barrier of $947$ K for this abstraction reaction. The product of reaction 22 is again utilized in reaction 19 to form $\rm{i-C_3H_7CN}$ again. Reaction 22 is not considered as default in our network unless otherwise stated. \begin{table} \scriptsize {\centering \caption{Ice-phase reactions considered for $\rm{C_3H_7CN}$ isomers.} \label{tab:prop} \begin{tabular}{ccc} \hline Reaction & Reactions & Activation Barrier (K) \\ Number (Type) & & \\ \hline \multicolumn{3}{c}{$\rm{n-C_3H_7CN}$} \\ \hline 12 (RR) & $\rm{H + CH_2CH_2CH_2CN \rightarrow n-C_3H_7CN}$ & --- \\ 13 (RR) & $\rm{H + CH_3CHCH_2CN \rightarrow n-C_3H_7CN}$ & --- \\ 14 (RR) & $\rm{H + CH_3CH_2CHCN \rightarrow n-C_3H_7CN}$ & --- \\ 15 (RR) & $\rm{CH_3 + CH_2CH_2CN \rightarrow n-C_3H_7CN}$ & --- \\ 16 (RR) & $\rm{CH_3CH_2 + CH_2CN \rightarrow n-C_3H_7CN}$ & --- \\ 17 (RR) & $\rm{CH_2CH_2CH_3 + CN \rightarrow n-C_3H_7CN}$ & --- \\ \hline \multicolumn{3}{c}{$\rm{i-C_3H_7CN}$} \\ \hline 18 (RR) & $\rm{H + CH_2CH(CH_3)CN \rightarrow i-C_3H_7CN}$ & --- \\ 19 (RR) & $\rm{H + CH_3C(CH_3)CN \rightarrow i-C_3H_7CN}$ & --- \\ 20 (RR) & $\rm{CH_3 + CH_3CHCN \rightarrow i-C_3H_7CN}$ & --- \\ 21 (RR) & $\rm{CN + CH_3CHCH_3 \rightarrow i-C_3H_7CN}$ & --- \\ 22 (NR) & $\rm{i-C_3H_7CN + H \rightarrow CH_3C(CH_3)CN + H_2}$ & $947^a$ \\ \hline \end{tabular}} \\ Notes: NR and RR refer to neutral-radical and barrierless radical-radical reactions, respectively. \\ $^a$ This work (gas-phase). \end{table} \subsubsection{Butyl cyanide \label{sec:BC}} Here, for the formation of the BCMs belonging to the $\rm{C_5H_9N}$ isomeric group, we consider the reaction pathways adopted in \cite{bell14,garr17}. They found that the radicals take a decisive part in their formation. These radicals were either produced by hydrogenations with carbon double bond or by hydrogen abstraction of a saturated carbon chain by the radicals like OH, NH$_2$, CH$_3$O, CH$_2$OH, etc. Normally, for the computation of the ice-phase reaction rates, the method proposed by \cite{hase92} is used. However, for the hydrogen abstraction reaction, this would underestimate the rate. To avoid this issue, in the context of hydrogen abstraction reactions, we use the mass of the hydrogen atom instead of the reduced mass of the reactants \citep{bell14,garr17}. Following \cite{gann07,garr17}, here also we consider low and high both the activation barriers for the reaction of $\rm{C_2H_2}$, ${\rm C_2H_4}$, $\rm{C_3H_6}$, and $\rm{C_4H_8}$ with the CN radical. Unless otherwise stated, we always use low barriers. For the formation of four isomeric forms of $\rm{C_4H_9CN}$, our considered reactions are noted in Table \ref{tab:BC}. Interestingly, reaction 36 uses $\rm{CH_3C(CH_3)CN}$ as a reactant, which can be produced by our newly proposed hydrogen abstraction reaction of $\rm{i-C_3H_7CN}$ by reaction 22. \begin{table} \scriptsize {\centering \caption{Ice-phase reactions considered for $\rm{C_4H_9CN}$ isomers.} \label{tab:BC} \begin{tabular}{ccc} \hline Reaction & Reactions & Activation Barrier (K) \\ Number (Type) & & \\ \hline \multicolumn{3}{c}{$\rm{n-C_4H_9CN}$} \\ \hline 23 (RR) & $\rm{CH_3 + CH_2CH_2CH_2CN \rightarrow n-C_4H_9CN}$ & --- \\ 24 (RR) & $\rm{CH_2CH_3 + CH_2CH_2CN \rightarrow n-C_4H_9CN}$ & --- \\ 25 (RR) & $\rm{CH_2CH_2CH_3 + CH_2CN \rightarrow n-C_4H_9CN}$ & --- \\ 26 (RR) & $\rm{CH_2CH_2CH_2CH_3 + CN \rightarrow n-C_4H_9CN}$ & --- \\ 27 (RR) & $\rm {H + CH_3CH_2CHCH_2CN \rightarrow n-C_4H_9CN}$ & --- \\ \hline \multicolumn{3}{c}{$\rm{i-C_4H_9CN}$} \\ \hline 28 (RR) & $\rm{CH_3 + CH_3CHCH_2CN \rightarrow i-C_4H_9CN}$ & --- \\ 29 (RR) & $\rm{CH_3CHCH_3 + CH_2CN \rightarrow i-C_4H_9CN}$ & --- \\ 30 (RR) & $\rm{CN + CH_2CH(CH_3)CH_3 \rightarrow i-C_4H_9CN}$ & --- \\ \hline \multicolumn{3}{c}{$\rm{s-C_4H_9CN}$} \\ \hline 31 (RR) & $\rm{CH_3 + CH_3CH_2CHCN \rightarrow s-C_4H_9CN}$ & --- \\ 32 (RR) & $\rm{CH_3 + CH_2CH(CH_3)CN \rightarrow s-C_4H_9CN}$ & --- \\ 33 (RR) & $\rm{CH_2CH_3 + CH_3CHCN \rightarrow s-C_4H_9CN}$ & --- \\ 34 (RR) & $\rm{CN + CH_3CHCH_2CH_3 \rightarrow s-C_4H_9CN}$ & --- \\ \hline \multicolumn{3}{c}{$\rm{t-C_4H_9CN}$} \\ \hline 35 (RR) & $\rm{CH_3C(CH_3)CH_3 + CN \rightarrow t-C_4H_9CN}$ & --- \\ 36 (RR) & $\rm{CH_3C(CH_3)CN + CH_3 \rightarrow t-C_4H_9CN}$ & --- \\ \hline \end{tabular}} \\ Notes: RR refers to barrierless radical-radical reactions. \\ \end{table} \subsection{Physical condition} We consider a free-fall collapsing cloud followed by a warm-up and post-warm-up phases for our physical model \citep{garr13,gora20b,das21,sil21}. During the collapsing phase ($\rm{t_{coll}}$), total hydrogen density (n$_{\rm H}$) can evolve from a low density ($3 \times 10^3$ cm$^{-3}$) to a higher density ($2 \times 10^8$ cm$^{-3}$). The highest density attained at the collapsing phase is kept constant throughout the warm-up and post-warm-up phases. The choice of our highest density is consistent with the density derived from the $\rm{C_3H_7CN}$ emission from the core N2 of Sgr B2 \citep{bell14}. Following \cite{garr08}, we consider that the density and visual extinction parameter ($A_{\rm V}$) is coupled by $\rm{A_{\rm V}=A_{V0}(n_H/n_{H0})^{2/3}}$. Here, $\rm{A_{V0}}$ ($=2$) and n$_{\rm H0}$ ($=3 \times 10^3$ cm$^{-3}$) is the minimum visual extinction and total hydrogen density considered in our model. Using the highest density (n$_{\rm H}=2 \times 10^8$ cm$^{-3}$), in our case, A$_{\rm V}$ could reach a value as high as $3288$. The dust temperature (T$_{\rm dust}$) is derived by the relation provided by \cite{zucc01} and modified by \cite{garr08}. $$\rm{ T_{dust}=18.67-1.637 A_V+0.07518 {A_V}^2 -0.001492 {A_V}^3}. $$ The above relation holds for A$_{\rm V}=2-10$. For A$_{\rm V}=2$, it yields $\rm{T_{dust}} \sim 16 $ K. The dust temperature further decreases as the visual extinction increases. Here, we restrict $\rm{T_{dust}}$ to fall below 8 K. At this phase, the gas temperature ($\rm{T_{gas}}$) is kept constant at 10 K. In the warm-up phase, $\rm{T_{dust}}$ is allowed to increase to 200 K in $\rm{t_{warm1}}$ years. Furthermore, to follow the further evolution, the dust temperature is allowed to increase up to 400 K in another $\rm{t_{warm2}}$ years. Once the dust temperature crosses the gas temperature, gas temperature follows the dust temperature because of the good coupling between the gas and dust at a higher density. Very similar warm-up time scales were considered in \cite{garr13,garr17}. In the post-warm-up phase (for $\rm{t_{pw}}$ years), all the physical parameters are kept constant at their respective highest values. The collapsing and warm-up time would differ between the high mass and low-mass stars. A shorter collapsing time is expected for a high-mass star, whereas relatively longer for a low-mass star. Here, we construct models to explain the abundances observed in Sgr B2. Based on these time scales, we consider two models (Model A and Model B) to explain the observed results. The time scales considered in each model are shown in Table \ref{tab:models}. In Model A, the first warm-up time scale ($t_{warm1}$) is varied, whereas, in Model B, the collapsing time is varied by keeping all other time scales at the fixed value. Fig. \ref{fig:phys} represents the time evolution of all the physical parameters considered in this simulation for Model A only. For all the models, we consider a standard cosmic ray ionization rate of $1.3 \times 10^{-17}$ s$^{-1}$. \begin{table} \centering \caption{Adopted models based on various time scale.} \label{tab:models} \begin{tabular}{c|c|c} \hline Time &Model A& Model B \\ \hline $t_{coll}$& $10^6$ & $10^5-10^6$\\ $t_{warm1}$& $10^5-10^6$ &$5 \times 10^5$\\ $t_{warm2}$& $4.3 \times 10^5$ &$2.12 \times 10^4$\\ $t_{pw}$& $10^5$&$10^5$\\ \hline Total Time &$(1.63-2.53) \times 10^6$&$(0.72 -1.62) \times 10^6$\\ \hline \end{tabular} \end{table} \section{Chemical model results and discussion} \label{sec:results} Here, we use our CMMC (Chemical Model of Molecular Cloud) code \citep{das15a,das15b,das19,das21,gora17a,gora17b,gora20b,sil18,sil21,ghos22} to study the formation of BCMs in Sgr B2(N). Three sets of BE values are used. For all sets, we consider the ratio between the energy for diffusion and energy for desorption ($E_b/E_D$) at 0.5. The set 1 is constructed with the BE values used in \cite{garr17,bell14}. The enthalpies of formation of $\rm{CH_2CHCN}$, $\rm{C_2H_5CN}$, $\rm{C_3H_7CN}$, $\rm{C_4H_9CN}$, and their related precursors are also used from \cite{garr17,bell14}. Set 2 is constructed with the same BE values used in set 1 except those are reported in Table \ref{tab:be}. Whenever we use set 2 BE, we also use our calculated enthalpy of formation values reported in Table \ref{tab:be}. Table \ref{tab:be} shows the BE of 30 relevant species with the monomer water configuration. However, for the eight species ($\rm{CH_2CHCN}$, $\rm{C_2H_5CN}$, $\rm{n/i-C_3H_7CN}$, $\rm{n/i/s/t-C_4H_9CN}$), we note the BE values with the tetramer configuration of water. Where monomer structure is used, we use a scaling factor of 1.416, and for the tetramer structure, a scale factor of 1.188 is used \citep{das18}. Finally, in set 3, we keep the BEs and enthalpies of formation of these 30 species (noted in Table \ref{tab:be}) same as set 2, but for the rest of the species, we use the BEs from the KIDA database. Several differences exist between the BE used in set 2 and set 3. But the significant difference which could alter the abundances of the saturated species on the grain surface is the usage of the slow diffusion rate of the H atom in set 3 ($E_D=650$ K) compared to set 2 ($E_D=450$ K). \begin{table*} \caption{Peak abundances (with respect to H$_2$) and corresponding temperature (in K) obtained from our simulation for Model A with $\rm{t_{warm1}}=3.5 \times 10^5$ years. \label{tab:pk-abn}} { \centering \begin{tabular}{|c|c|c|c|} \hline &set 1&set 2&set 3\\ &&low/high&low/high/low(reaction 22)\\ \hline $\rm{CH_2CHCN}$ & $1.8 \times 10^{-9}, 7.5 \times 10^{-8o}, 7.4 \times 10^{-9g}$ (156, 200$^o$, 167$^g$) & $6.0/6.3 \times 10^{-10} (126)$ & $2.12/2.20/2.06 \times 10^{-9} (143)$ \\ $\rm{C_2H_5CN}$ & $1.3 \times 10^{-8}, 1.1 \times 10^{6o}, 3.5 \times 10^{-8g}$ (116, 150$^o$, 129$^g)$ & $6.4/6.7 \times 10^{-9} (100)$ & $2.52/2.46/3.18 \times 10^{-9} (102)$ \\ $\rm{n-C_3H_7CN}$ & $3.8 \times 10^{-9}, 1.3 \times 10^{-8b}, 1.9 \times 10^{-9g}$ (146, 153$^b$, 156$^g)$ & $5.5/6.1 \times 10^{-10} (106)$ & $1.64/1.94/1.33 \times 10^{-9} (109)$ \\ $\rm{i-C_3H_7CN}$ & $4.5 \times 10^{-9}, 3.2 \times 10^{-8b}, 3.4 \times 10^{-9g} (146, 153^b, 154^g)$ & $10.4/11.4 \times 10^{-10} (108)$ & ${2.01/2.34/1.61} \times 10^{-9}$ (110) \\ $\rm{n-C_4H_9CN}$ & $2.3 \times 10^{-10}, 2.0 \times 10^{-9g} (175,190^g)$ & $1.88/1.94 \times 10^{-10} (103)$ & $7.01/6.96/8.2 \times 10^{-11} (105)$\\ $\rm{i-C_4H_9CN}$ & $2.2 \times 10^{-10}, 3.8 \times 10^{-9g} (175,190^g)$ & $8.39/8.63 \times 10^{-11} (101)$ & $1.04/1.09/0.91 \times 10^{-9} (101)$\\ $\rm{s-C_4H_9CN}$&$6.5 \times 10^{-10}, 3.7 \times 10^{-9g} (175,190^g)$ &$2.93/2.97 \times 10^{-10} (105)$ & $6.78/6.57/7.3 \times 10^{-10}(106)$ \\ $\rm{t-C_4H_9CN}$& $1.3 \times 10^{-10}, 2.0 \times 10^{-10g} (175,190^g)$ & $1.81/1.53 \times 10^{-11} (101)$ & $7.37/1.22/129 \times 10^{-11} (101)$\\ \hline \end{tabular}} \\ $^g$ slow warm-up model with set 1 BEs and low activation \citep{garr17}\\ $^o$\cite{bell16}\\ $^b$\cite{bell14}\\ \end{table*} \begin{table*} {\scriptsize \caption{Ranges of peak abundance ratio obtained from our Model A and Model B with various BEs and their comparison with literature. \label{tab:ratios}} \centering \begin{tabular}{|c|c|c|c|c|} \hline & Set 1 & Set 2 & Set 3 & Observed / other results \\ &low/high&low/high&low/high&\\ \hline ${\rm C_2H_5CN/CH_2CHCN}$ &$5.9-23.4/0.033-22.5$&$13.7-32.5/0.00014-72$&$0.1-9.27/0.1-8.52$&15$^b$, $3.8-9.5^g$\\ ${\rm i-C_3H_7CN/n-C_3H_7CN}$&${1.0-1.2/ 0.78-1.36}$&${1.8-3.5/1.09-3.4}$&${0.7-1.22/0.7-1.21}$&$0.4\pm 0.06^a$, $0.17-3.0^g$, $0.17-0.5^p$\\ ${\rm n-C_3H_7CN/C_2H_5CN}$&${0.04-0.18/0.046-70.3}$&${0.005-0.059/0.0065-3399}$&${0.036-7.74/0.04-7.9}$&$0.029^{a,b}$, $0.024-0.67^g$\\ ${\rm tot-C_3H_7CN/C_2H_5CN}$& ${0.078-0.37/0.086-166}$ &${0.023-0.17/0.0289-7114}$&${0.07-16.4/0.073-16.71}$&$0.041^{a,b}$, $0.98-0.21^g$\\ ${\rm n-C_4H_9CN/n-C_3H_7CN}$&${0.033-0.052/0.03-0.3}$ &${0.34-1.0/0.015-0.77}$&${0.015-0.29/0.018-0.25}$&$<0.59$,$0.1-1.1^g$\\ ${\rm i-C_4H_9CN/n-C_4H_9CN}$&${0.69-2.28/0.70-2.37}$&${0.12-0.44/0.16-14.3}$&${0.46-67/0.57-48.91}$&$0.6-2.2^g$\\ ${\rm s-C_4H_9CN/n-C_4H_9CN}$&${2.45-3.0/2.23-3.03}$&${0.98-1.92/1.02-9.64}$&${3.36-24.8/3.46-17.73}$&$1.7-4.3^g$\\ ${\rm t-C_4H_9CN/n-C_4H_9CN}$&${0.23-1.04/0.19-1.52}$&${0.014-0.12/0.014-5.34}$&${0.025-5.92/0.007-1.91}$&$0.015-0.1^g$\\ ${\rm (i+s+t)-C_4H_9CN/n-C_4H_9CN}$&${3.45-6.38/3.2-5.9}$&${1.12-2.37/1.21-29.3}$&${3.84-97.64/4.04-66.7}$&$3.0-5.8^g$\\ \hline \end{tabular}} \\ $^a$\cite{bell14}, $^b$\cite{bell16}, $^g$\cite{garr17}, $^p$\cite[Orion KL]{paga17}, $^h$High activation \end{table*} \subsection{Vinyl and ethyl cyanide} Table \ref{tab:EV} shows the reactions leading to the formation of $\rm{CH_2CHCN}$ and $\rm{C_2H_5CN}$ in the interstellar condition. Set 1 is constructed with the BE used in \cite{garr17,bell16}. They used higher BE of $\rm{C_2H_5CN}$ (5537 K) than $\rm{CH_2CHCN}$ (4637 K). We also obtain a higher BE of $\rm{C_2H_5CN}$ (4059 K) with the water monomer than $\rm{CH_2CHCN}$ (3948 K). However, the water monomer configuration is relatively smaller (3 atoms) than the adsorbed species (7 atoms for $\rm{CH_2CHCN}$ and 9 atoms for $\rm{C_2H_5CN}$), which could induce some errors on the estimated BEs. \cite{das18} reported that the BE estimation tends to more realistic values as the size of the computed substrate is increased. To check the effect of BEs on the larger substrate, we further use a tetramer water structure (consisting of 12 atoms). As like the monomer configuration, here also, we found that the BE of $\rm{C_2H_5CN}$ (scaled value $\sim 4886$ K) is greater than $\rm{CH_2CHCN}$ (scaled value $\sim 3540$ K). We notice our estimated BE values with the tetramer configuration of water are lower by several hundreds of Kelvin than that of \cite{garr17} for both the species. Set 2 and 3 consider these BE values and enthalpy of formation noted in Table \ref{tab:be} for our simulation. Fig. \ref{fig:EV} shows the time evolution of $\rm{CH_2CHCN}$ and $\rm{C_2H_5CN}$ in the warm-up and post-warm-up phase for set 1 and set 3 BEs. The solid lines represent the gas-phase abundances, whereas the dashed lines represent the ice-phase abundances. Model A with $\rm{t_{warm1}}=3.5 \times 10^5$ years is used for Fig. \ref{fig:EV}. The gas-phase peak abundances (beyond the collapsing time) obtained with various sets of BEs are noted in Table \ref{tab:pk-abn}. We obtain a higher peak abundance of $\rm{C_2H_5CN}$ than $\rm{CH_2CHCN}$ for all the cases. $\rm{C_2H_5CN}$ was mainly produced in the ice phase by the successive hydrogenations of $\rm{CH_2CHCN}$ (reactions $5-8$) and by the radical-radical reaction between $\rm{CH_3CH_2}$ and CN (reaction 9). With the set 1 BE, a peak abundance of $\rm{C_2H_5CN}$ is obtained at 116 K, whereas, with similar BEs, \cite{garr17} got the peak abundance at 129 K (they used $\rm{t_{warm1}= 10^6}$ years). For the set 2 and set 3, it varies in the range $100-102$ K for $\rm{t_{warm1}= 3.5 \times 10^5}$ years. The peak abundance of $\rm{CH_2CHCN}$ for set 1 is obtained at 156 K \citep[][got this at 167 K]{garr17}. We notice a substantial effect of reaction 4 on $\rm{CH_2CHCN}$. According to the UMIST Database for Astrochemistry 2012 \citep{mcel13}, this reaction is valid beyond 300 K. However, following KIDA database \citep{wake12}, we consider that this reaction may process beyond 10 K. The $\alpha$, $\beta$, and $\gamma$ noted in the UMIST database is used as the rate constants of this reaction. The gas-phase abundance profile of $\rm{CH_2CHCN}$ by utilizing the temperature restriction is shown in Fig. \ref{fig:EV} with the solid brown curve. It clearly shows that with the temperature restriction of this reaction, gas-phase peak $\rm{CH_2CHCN}$ abundance drops from $1.8 \times 10^{-9}$ (not considering the temperature limit) to $4.12 \times 10^{-11}$ (considering the temperature limit). The abundance of $\rm{CH_2CHCN}$ and $\rm{C_2H_5CN}$ noted in Table \ref{tab:pk-abn} shows a very minor change between the usage of low/high barrier for the CN addition to $\rm{C_2H_2}$, $\rm{C_2H_4}$, $\rm{C_3H_6}$, and $\rm{C_4H_8}$). \cite{bell14} derived a peak H$_2$ column density of $4.2 \times 10^{24}$ cm$^{-2}$. Furthermore, they extrapolated their obtained column density to a more compact region ($\sim 100$) where the $\rm{C_3H_7CN}$ emission originated. With this consideration, they estimated an average H$_2$ column density of $5.6 \times 10^{24}$ cm$^{-2}$. They identified $154$ transitions of $\rm{C_2H_5CN}$ and 44 transitions of $\rm{CH_2CHCN}$. They estimated a column density of $\rm{C_2H_5CN}$ and $\rm{CH_2CHCN}$ of $6.2 \times 10^{18}$ and $4.2 \times 10^{17}$, respectively. Transforming into the abundances, it yields the abundances of $1.1 \times 10^{-6}$ and $7.5 \times 10^{-8}$ for $\rm{C_2H_5CN}$ and $\rm{CH_2CHCN}$, respectively. Fig. \ref{fig:EV-coll} shows the variation of gas-phase peak abundances (obtained beyond the collapsing time) of $\rm{CH_2CHCN}$ and $\rm{C_2H_5CN}$ with the changes in warm-up time scale (Model A, left panel) and collapsing time (Model B, right panel). Interestingly, in most cases, the peak abundance of $\rm{C_2H_5CN}$ is greater than that of the $\rm{CH_2CHCN}$ with Model A and Model B with various BEs. With the set 1 BEs (red solid curve in the right panel of Fig. \ref{fig:EV-coll}), the gas-phase peak abundance of $\rm{CH_2CHCN}$ and $\rm{C_2H_5CN}$ varies in the range $1.42 \times 10^{-9} - 1.48 \times 10^{-8}$ and $5.19 \times 10^{-11} - 1.19 \times 10^{-7}$, respectively. Only for the lowest warm-up time ($t_{warm1} = 10^5$ years) and set 1, we obtain VC>EC. None of our models (with set 2 and set 3 BEs) can reproduce the observed abundance of $\rm{C_2H_5CN}$ ($1.1 \times 10^{-6}$) and $\rm{CH_2CHCN}$ ($7.5 \times 10^{-8}$) in Sgr B2(N2). From the various models (fast, medium, and slow warm-up along with the low and high activation barriers), \cite{garr17} also obtained a comparatively lower abundance of $\rm{C_2H_5CN}$ ($1.2 \times 10^{-8} - 1.1 \times 10^{-7}$) and $\rm{CH_2CHCN}$ ($1.6 \times 10^{-9} - 1.7 \times 10^{-8}$) than observations. An exciting trend for $\rm{CH_2CHCN}$ is obtained when we vary the warm-up time of Model A. We notice that the peak abundance value of $\rm{CH_2CHCN}$ is gradually shifted towards the higher temperature with the decrease in the warm-up time scale. For example, for set 1, with $\rm{t_{warm1}}=10^6$ years, we get a peak abundance at 146 K (117 K for set 2 and 145 K for set 3), which is shifted to 197 K (143 K for set 2 and 161 K for set 3) with $\rm{t_{warm1}}=10^5$ years. Fig. \ref{fig:EV} shows that the ice-phase abundance of $\rm{CH_2CHCN}$ declines around 70 K. However, its peak abundance appears at a much higher temperature. It is because of the involvement of the gas phase pathways for the formation of $\rm{CH_2CHCN}$. In the $\rm{C_2H_5CN}$, the peak abundance obtained remains roughly invariant with the variation of the warm-up time scale (varies in the range $108-117$ K for set 1, $99-101$ K for set 2, and $95-103$ for set 3). \cite{garr17} obtained $129-131$ K for the different warm-up time scales. \cite{bell16} obtained an abundance ratio between $\rm{C_2H_5CN}$ and $\rm{CH_2CHCN}$ of $\sim 15$ in Sgr B2(N2). The obtained ratio from our various models and various sets of BEs are noted in Table \ref{tab:ratios}. The peak abundance of $\rm{C_2H_5CN}$ and $\rm{CH_2CHCN}$ does not appear simultaneously, but we consider their peak values in deriving this ratio. So, there should be some uncertainty in the derived molecular ratio. For set 1 BE, the values noted in Table \ref{tab:pk-abn}, show a ratio between the peak abundance of $\rm{C_2H_5CN}$ to $\rm{CH_2CHCN}$ as $7.2$. For more higher warm-up time scale ($\rm{t_{warm1}}=5 \times 10^5$ years), it can goes up to $\sim$ 9 (see Fig. \ref{fig:rat_EV}). The right panel of Fig. \ref{fig:rat_EV} shows the ratio obtained by varying the collapsing time (Model B). Overall, with the variation of warm-up ($t_{warm1}=2 \times 10^5 - 10^6$ years) and collapsing time ($t_{coll}=10^5-10^6$ years) shown in Fig. \ref{fig:rat_EV}, we obtain a ratio of $3.8-23.4$ for set 1. For the similar time scales, this ratio varies in the range of $1.56-72$ and $0.8-9.3$ for set 2 and set 3, respectively. We notice a abrupt decrease in ratio for set 1 and set 2 for a shorter warm-up time scale ($t_{warm1}=10^5$ years). It is 0.033 and 0.00014 for set 1 and 2, respectively. A vast difference between the ratio of sets 2 and 3 is observed in both panels of Fig. \ref{fig:rat_EV}. It happens mainly because of the changes in the adsorption energies of the H atom. For set 3, due to the higher adsorption energy of the H atom, it has a longer residence time on the grain, which eventually might helps in the formation of more saturated species. \subsection{Propyl cyanides} Formation pathways of $\rm{i/n-C_3H_7CN}$ are discussed in section \ref{sec:prop}. \cite{garr17} considered the BE of both the isomers the same (7237 K). We obtain relatively lower BE values of these two species with the monomer and tetramer configuration. With the monomer substrate, a higher BE for $\rm{i-C_3H_7CN}$ (3279 K) than $\rm{n-C_3H_7CN}$ (2991 K) is obtained. On the contrary, with the tetramer configuration, we obtain an opposite trend (4970 K for $\rm{i-C_3H_7CN}$ and 5567 K for $\rm{n-C_3H_7CN}$), which are much lower than the values used in \cite{garr17}. Fig. \ref{fig:c3h7cn} shows the time evolution of the two forms of $\rm{C_3H_7CN}$ (i and n) by considering $\rm{t_{warm1}=2 \times 10^5}$ years and $\rm{t_{warm1}=3.5 \times 10^5}$ years, respectively with Model A (with set 1 and set 2 BEs). With the warm-up time scales, we notice a significant difference in the abundance of $\rm{C_3H_7CN}$. In general, a shorter warm-up time yields a larger peak abundance (see the left panel of Fig. \ref{fig:PC-warm-coll}). For $\rm{t_{warm1}}>6 \times 10^5$ years, we obtain i$<$n, whereas the opposite is valid for a shorter warm-up time scale for set 1. For the set 2 BE, we always see i$>$n. For set 3 also, we obtain i$>$n except for the shortest warm-up time ($10^5$ years). The right panel of Fig. \ref{fig:PC-warm-coll} shows the peak abundances of n and i $\rm{C_3H_7CN}$ for the variation in the collapsing time for set 1, set 2, set 3. It shows that for set 1, peak gas-phase abundance of $\rm{i-C_3H_7CN}$ is always greater ($2.97 \times 10^{-9}-1.65 \times 10^{-8}$) than $\rm{n-C_3H_7CN}$ ($2.86 \times 10^{-9 }-1.36 \times 10^{-8}$). For set 2, we always have obtained a high abundance of $\rm{i-C_3H_7CN}$ than $\rm{n-C_3H_7CN}$. For set 3, i$>$n is obtained when ${\rm t_{coll}> 5 \times 10^5}$ years and i$<$n for the shorter ${\rm{t_{coll}}}$. From various models (high/low) barriers and with various warm-up time scales), \cite{garr17} obtained the abundance $1.9 \times 10^{-10}-46.0 \times 10^{-10}$, and $7.6 \times 10^{-10}-34 \times 10^{-10}$ for n and i $\rm{C_3H_7CN}$, respectively. \cite{bell14} calculated the abundance of $3.6 \times 10^{-9}$ and $8.0 \times 10^{-9}$ for n and i conformers of $\rm{C_3H_7CN}$, respectively, whereas from the observations, they obtained abundance of $3.2 \times 10^{-8}$ and $1.3 \times 10^{-8}$, respectively. These observed values are shown in Fig. \ref{fig:PC-warm-coll} with horizontal lines for better understanding. However, as like \cite{bell14,garr17}, none of our models were successful in obtaining such a high abundance with the present reaction network. It might be due to the exceptional environment in the Galactic center region than the other not core as discussed in \cite{bonf19,will20}. The i/n abundance ratio for set 1 with low activation barriers vary between 1 and 1.2 (see Fig. \ref{fig:PC_rat}). With the slow and low barrier model, \cite{garr17} obtained a peak abundance of $1.9 \times 10^{-9}$ and $3.4 \times 10^{-9}$, respectively for $\rm{n-C_3H_7CN}$ and $\rm{i-C_3H_7CN}$, which yields an i/n abundance ratio of 1.79. Fig. \ref{fig:PC_rat} represents the obtained peak ratio between i and n $\rm{C_3H_7CN}$ with the warm-up time scale (left) and collapsing time scale (right). The ratio obtained with other BEs are noted in Table \ref{tab:ratios}. \cite{bell14,bell16} observed the i/n ratio of $\sim 0.4 \pm0.06$. None of our models with the low activation barriers (for the reaction of CN with C$_2$H$_2$, C$_2$H$_4$, C$_3$H$_6$, and C$_4$H$_8$) were able to achieve the observed value (either Model A or Model B). With the high activation barrier, we found that this ratio varies between 0.8 and 1.4 for set 1. In the case of set 2 and set 3, mostly we have i$>$n. With set 3, we obtain $i/n \sim 0.7$ when a shorter warm-up ($t_{warm1}=10^5$ years for Model A) and shorter collapsing time ($t_{coll}=1-3 \times 10^5$ years for Model B) are used. One point to be remembered that the ratio itself is time-dependent, and we consider the ratio with their respective peak values. \cite{bell16} also observed $\rm{n-C_3H_7CN/C_2H_5CN}=0.029$ and $\rm{total \ C_3H_7CN/C_2H_5CN}=0.041$. Figs. \ref{fig:PC_Eth-rat} and \ref{fig:tot-PC_Eth-rat} represent the simulated peak ratio of $\rm{n-C_3H_7CN/C_2H_5CN}$ and $\rm{total \ C_3H_7CN/C_2H_5CN}$, respectively. These ratios are falling in between our simulated range. In section \ref{sec:prop}, reactions related to the formation of $\rm{C_3H_7CN}$ are discussed. The TS calculation of reaction 22 reveals an activation barrier of $947$ K. So far, we do not consider the destruction of $\rm{i-C_3H_7CN}$ by any hydrogenation reaction in our network. By considering reaction 22, we observe a few changes in the abundances of $\rm{i-C_3H_7CN}$ because of the formation by reaction 19 again. However, its effect on $\rm{C_4H_9CN}$ is vital and discussed in the latter portion. The dot-dashed curve represents the peak abundance ratio between i and n $\rm{C_3H_7CN}$ (low barrier) in Fig. \ref{fig:PC_rat}. No significant changes in the abundance of $\rm{i-C_3H_7CN}$ are seen with the inclusion of reaction 22. It is because a substantial part of the product of reaction 22 ($\rm{CH_3(C)CH_3CN}$) hydrogenate to form $\rm{i-C_3H_7CN}$ by reaction 19 again. \subsection{Butyl cyanide} For the formation of various forms of $\rm{C_4H_9CN}$, we consider ice phase reactions $23-36$ of section \ref{sec:BC}. Fig. \ref{fig:c4h9cn} shows the time evolution of n/i/s/t-C$_4$H$_9$CN for various BEs. \cite{garr17} used the BE of these four species as 8937 K. We obtain significantly lower BE values with our calculations. With the monomer configuration, we obtain a higher BE (scaled by 1.416) for n (6320 K) followed by t (4574 K), i (4422 K), and s (2635 K). However, with the larger substrate (tetramer), we obtain the highest BE (scaled by 1.188) for s (5313 K), followed by i (5151 K), t (5148 K), and n (4388 K). With the set 1 BE, we obtain the peak values of these species at 175 K. With the tetramer structure (set 2 and set 3), due to lower BEs, the peak value of these species appears to be around $100-105$ K. Fig. \ref{fig:collapse-butyl} shows the effect on the peak abundances of various forms of $\rm{C_4H_9CN}$ for the variation of warm-up time (top) and collapsing time (bottom). In general, it seems that the shorter collapsing time and shorter warm-up time increase the production of all these species. \cite{garr17} used the set 1 BE values with $\rm{t_{coll}}=10^6$ years. With the shortest warm-up time ($\sim 5 \times 10^4$ years with low and high barriers), they obtained the highest peak abundance of s, followed by n, i, and t, and with the largest warm-up time ($\sim 10^6$ years), they obtained the highest peak abundance of i, followed by s, n, and t. So, in all the cases, they got the lowest abundance of t. In our set 1 model (with a low barrier and ${\rm t_{coll}=10^6}$ years) for most of the warm-up time scale, the sequence is s, n, i, and t. We obtain a dramatic change in the abundances with set 2 and set 3 BE values. For set 3, where the most updated BE values are used, we obtain a peak abundance sequence (with $\rm{t_{coll}=10^6}$ years) of i, s, t, and n for most of the warm-up time (see at the top of Fig. \ref{fig:collapse-butyl}). For the shorter warm-up time ($\sim 10^5$ years), t and n interchange their position. This model has considered $\rm{t_{coll}}=10^6$ years. For the shorter collapsing time, s has the maximum abundance, whereas t has the minimum abundance. With the consideration of the reaction 22, scenario has changed dramatically. With ${\rm t_{coll}=10^6}$ years, when we consider reaction 22, for low barrier case, we have an abrupt increase in the abundance of the $\rm{t-C_4H_9CN}$ (last panel of the top of Fig. \ref{fig:collapse-butyl}). For example, Table \ref{tab:pk-abn} shows the abundance of $\rm{t-C_4H_9CN}$ (with $\rm{t_{coll}}=10^6$ years and $\rm{t_{warm1}=3.5 \times 10^5}$ years) is $7.37 \times 10^{-11}$ when reaction 22 is not considered. On the contrary, with reaction 22, we have its abundance $1.29 \times 10^{-9}$. So, an increase of two orders of magnitude is obtained. It happens due to the production of $\rm{CH_3(C)CH_3CN}$ by reaction 22, which is further used in reaction 36 for the formation of $\rm{t-C_4H_9CN}$. For the shorter collapsing time, we always get a lower abundance of t, the same as that obtained by avoiding reaction 22 with set 3. \cite{garr17} derived an upper limit of $0.59$ for the peak abundance ratio between $\rm{n-C_4H_9CN}$ and $\rm{n-C_3H_7CN}$. Fig. \ref{fig:but_pc-rat} shows this ratio with the warm-up time scale (left) and collapsing time scale (right). The upper limit is shown with the red horizontal line. Modelled peak ratio is less than the derived upper limit for most of the cases, except for set 2 and shorter collapsing time ($\rm{t_{coll}<5 \times 10^5}$ years). It was interesting to see whether the observed ratio between the i and n $\rm{C_3H_7CN}$ is also sustained between the i and n of $\rm{C_4H_9CN}$ or gets amplified or reduced. Fig. \ref{fig:in-butyl} shows the i/n ratio of $\rm{C_4H_9CN}$ for different warm-up (left) and collapsing time (right) scale. For set 1, the i/n ratio for ${\rm C_3H_7CN}$ varies in the range $0.8-1.4$, whereas for $\rm{C_4H_9CN}$, it is $0.69-2.37$. With the same BE value, various warming times, and two types of activation barriers, \cite{garr17} obtained an i/n ratio in the range of $0.6-2.2$, which is in excellent agreement with our model. With the set 2 and 3 BEs, the i/n ratio of $\rm{C_4H_9CN}$ is significantly changed. Comparing Fig. \ref{fig:in-butyl} (i/n ratio of $\rm{C_3H_7CN}$) and Fig. \ref{fig:PC_rat} (i/n ratio of ${\rm C_4H_9CN}$), it is clear that the branching is more favourable for the higher-order alkyl cyanides with the realistic BE sets. \cite{garr17} obtained a s/n, t/n, and (i+s+t)/n ratio of $\rm{C_4H_9CN}$ in the range $1.7-4.3$, $0.015-0.1$, and $3.0-5.8$, respectively. With the set 1, we obtain these ratios $2.23-3.03$, $0.19-1.52$, and $3.2-6.38$, respectively (see Table \ref{tab:ratios}). All these ratios obtained with the set 2 and set 3 BEs are also noted in Table \ref{tab:ratios} and compared with previous observations and modeling results. In general, there are no systematic differences between the results predicted by our models and \cite{garr17}. However, with our more realistic BE values, set 3 \citep[low BEs compared to that used in][]{garr17}, we find that the branching is more favourable. Furthermore, a major difference is obtained when one hydrogen abstraction reaction (reaction 22) of $\rm{i-C_3H_7CN}$ is included. It yields a high abundance of $\rm{t-C_4H_9CN}$. Due to the exceptional environments of the galactic centre region, none of our models and models of \cite{garr17} could possibly explain the observed high abundance of some of the BCMs \citep{bonf19,will20}. The BEs used by \cite{garr17} were some educated estimations. So, we would refer to using our set 3 BEs obtained with the tetramer water substrate for future modeling. We notice a huge impact of collapsing (Model A) and warm-up (Model B) time scales in the abundances of these species. Based on the obtained results, we recommend using relatively moderate collapsing and warm-up time scales ($t_{\rm coll}$=$t_{\rm warm1} \sim 3-5 \times 10^5$ years). \section{Conclusions} \label{sec:conclusions} We carry out an extensive study on the formation of various BCMs. Some of our significant findings from this work are: \begin{itemize} \item One of the critical parameters for astrochemical modeling is the BE of species. Here, we provide a realistic estimation of the BEs for some BCM-related species for the first time. Noticeably lower BE values \citep[as compared to previously used][]{garr17} for $\rm{CH_2CHCN}$ (3540 K), $\rm{C_2H_5CN}$ (4886 K), $\rm{C_3H_7CN}$ (5567 K, 4970 K for n and i $\rm{C_3H_7CN}$, respectively), and $\rm{C_4H_9CN}$ (4388 K, 5151 K, 5313 K, and 5148 K for n, i, s, and $\rm{t-C_4H_9CN}$, respectively) are obtained. \\ \item The enthalpies of formation, polarizabilities, dipole moments, and activation barriers through TS calculations are calculated quantum chemically to better estimate the modeling results. \\ \item \cite{bell14} observed i-PC/n-PC ratio of $\sim 0.4 \pm 0.06$. With set 3, we obtain i-PC/n-PC $\sim$ 0.7 when a shorter warmup ($t_{warm1}=10^5$ years for Model A) and shorter collapsing time ($t_{coll}=1 - 3 \times 10^5$ years for Model B) are used. With the set 2 and set 3, it is observed that the abundances of these species are greatly affected. Compared to the modeled i/n ratio of $\rm{C_3H_7CN}$ (varies in the range $0.7-3.4$ for set 2 and 3 noted in Table \ref{tab:ratios}), the i/n ratio of $\rm{C_4H_9CN}$ is greatly enhanced ($0.12-67$ shown in Table \ref{tab:ratios}). Thus, the branching is more favourable for the higher-order alkyl cyanides. It is also noticed that with the increase in the warm-up and collapsing time, in general, for set 3 (consisting of most updated BEs) ratio of the i/n increases. \\ \item Here, for the destruction of $\rm{i-C_3H_7CN}$, we propose one hydrogen abstraction reaction (reaction 22). The TS calculation of this reaction yields an activation barrier of $947$ K. Inclusion of this reaction drastically increases the abundance of $\rm{t-C_4H_9CN}$ by the CH$_3$ addition of $\rm{CH_3(C)CH_3CN}$ (reaction 36). Furthermore, we found that the formation of $\rm{t-C_4H_9CN}$ is favourable when a longer collapsing time is used along with reaction 22. \end{itemize} \section*{Acknowledgements} S.S. acknowledges Banaras Hindu University and UGC, New Delhi, India, for providing a fellowship. M.S. would like to acknowledge the financial support from S. N. Bose National Centre for Basic Sciences, Salt Lake, Kolkata under the Department of Science and Technology (DST), Government of India. P.G. acknowledges support from a Chalmers Cosmic Origins postdoctoral fellowship. A.P. acknowledges financial support from the IoE grant of Banaras Hindu University (R/Dev/D/IoE/Incentive/2021-22/32439) and financial support through the Core Research Grant of SERB, New Delhi (CRG/2021/000907). AD would like to acknowledge ICSP for support. \section*{Data Availability} The data underlying this article will be shared on reasonable request to the corresponding author. \bibliographystyle{mnras} \bibliography{BCM_mnras.bbl} \clearpage \appendix \section{Potential energy surfaces} Potential energy surface diagrams of the ice-phase chemical reactions 5, 6, 11, and 22 noted in Tables \ref{tab:EV} and \ref{tab:prop} are shown in Figs. \ref{fig:reaction_5}, \ref{fig:ch2ch2cn}, and \ref{fig:ch3cch3cn}, respectively. \clearpage \section{Polarizability and dipole moment} We quantum chemically calculate the polarizability and total dipole moment of BCM-related species noted in Table \ref{tab:polar_dipole} using the Gaussian 09 suite. For these calculations, we use the DFT-B3LYP/6-31G(d,p) level of theory. Our calculated values are compared with the existing experimental values (if available in the NIST WebBook database). \begin{table*} {\centering \caption{Calculated polarizability and total dipole moment of some BCM-related species.} \label{tab:polar_dipole} \begin{tabular}{cccccc} \hline Serial & & \multicolumn{2}{c}{Polarizability (\AA$^3$)} & \multicolumn{2}{c}{Total dipole moment (D)} \\ \cline{3-6} & Species & This Work & Experimental$^a$ & This Work & Experimental$^b$ \\ & & DFT-B3LYP 6-31G(d,p) & & DFT-B3LYP 6-31G(d,p) & \\ \hline 1 & $\rm{CH_3CN}$ & 3.4378 & 4.280 & 3.8279 & 3.92 \\ 2 & $\rm{C_2H_2CN}$ & 4.6441 & --- & 3.2203 & --- \\ 3 & $\rm{CH_2CHCN}$ & 4.9597 & --- & 3.8772 & 3.87 \\ 4 & $\rm{\dot CH_2CH_2CN}$ & 4.7300 & --- & 3.6998 & --- \\ 5 & $\rm{CH_3 \dot CHCN}$ & 5.1242 & --- & 3.8985 & --- \\ 6 & $\rm{C_2H_5CN}$ & 5.0442 & 6.240 & 3.9198 & 4.02 \\ 7 & $\rm{\dot CH_2CH_2CH_2CN}$ & 6.3156 & --- & 3.8816 & --- \\ 8 & $\rm{CH_3\dot CHCH_2CN}$ & 6.4000 & --- & 3.7363 & --- \\ 9 & $\rm{CH_3CH_2\dot CHCN}$ & 6.8268 & --- & 4.0005 & --- \\ 10 & $\rm{n-C_3H_7CN}$ & 6.6846 & 8.400 & 4.0636 & 4.07 \\ 11 & $\rm{\dot CH_2CH(CH_3)CN}$ & 6.3615 & --- & 3.9399 & --- \\ 12 & $\rm{\dot CH_3\dot C(CH_3)CN}$ & 6.8268 & --- & 4.1409 & --- \\ 13 & $\rm{i-C_3H_7CN}$ & 6.6327 & 8.049 & 3.9515 & --- \\ 14 & $\rm{CH_3CH_2\dot CHCH_2CN}$ & 8.0019 & --- & 3.8328 & --- \\ 15 & $\rm{n-C_4H_9CN}$ & 8.3264 & --- & 4.1606 & 4.12 \\ 16 & $\rm{i-C_4H_9CN}$ & 8.2272 & --- & 3.9850 & --- \\ 17 & $\rm{s-C_4H_9CN}$ & 8.2123 & --- & 3.9019 & --- \\ 18 & $\rm{t-C_4H_9CN}$ & 8.1946 & 9.591 & 3.9600 & 3.95 \\ 19 & $\rm{\dot CH_2CH_2CH_3}$ & 4.7330 & --- & 0.2533 & --- \\ 20 & $\rm{CH_3\dot CHCH_3}$ & 4.8263 & --- & 0.2042 & --- \\ 21 & $\rm{C_3H_8}$ & 5.0471 & 5.921 & 0.0491 & 0.08 \\ 22 & $\rm{\dot CH_2CH_2CH_2CH_3}$ & 6.3082 & --- & 0.2525 & --- \\ 23 & $\rm{CH_3\dot CHCH_2CH_3}$ & 6.4519 & --- & 0.2163 & --- \\ 24 & $\rm{n-C_4H_{10}}$ & 6.6446 & 8.020 & 0.0000 & 0.00 \\ 25 & $\rm{\dot CH_2CH(CH_3)CH_3}$ & 6.333 & --- & 0.1989 & --- \\ 26 & $\rm{CH_3\dot C(CH_3)CH_3}$ & 6.5201 & --- & 0.1990 & --- \\ 27 & $\rm{i-C_4H_{10}}$ & 6.6105 & 8.009 & 0.0765 & 0.13 \\ 28 & $\rm{n-C_5H_{12}}$ & 8.2598 & 9.879 & 0.0477 & --- \\ 29 & $\rm{i-C_5H_{12}}$ & 8.1768 & 8.770 & 0.0542 & 0.13 \\ 30 & $\rm{neo-C_5H_{12}}$ & 8.1427 & 10.240 & 0.0001 & 0.00 \\ \hline \end{tabular}} \\ $^a$ \url{https://cccbdb.nist.gov/xp1.asp?prop=9} \\ $^b$ \url{https://cccbdb.nist.gov/xp1.asp?prop=7} \end{table*} \bsp % \label{lastpage}

Title: Interferometric imaging, and beam-formed study of a moving Type IV Radio burst with LOFAR

Abstract: Type IV radio burst has been studied for over 50 years. However, the specifics of the radio emission mechanisms is still an open question. In order to provide more information about the emission mechanisms, we studied a moving type IV radio burst with fine structures (spike group) by using the high resolution capability of Low-Frequency Array (LOFAR) on Aug 25, 2014\textbf{ (SOLA-D-21-00188)}. We present a comparison of Nan\c{c}ay RadioHeliograph (NRH) and the first LOFAR imaging data of type IV radio burst. The degree of circular polarization (DCP) is calculated at frequencies in the range 20$\sim$180 MHz using LOFAR data, and it was found that the value of DCP gradually increased during the event, with values of 10\%$\sim$20\%. LOFAR interferometric data were combined with white light observations in order to track the propagation of this type IV. The kinematics shows a westward motion of the radio sources, slower than the CME leading edge. The dynamic spectrum of LOFAR shows a large number of fine structures with duration of less than 1s and high brightness temperature ($T_\mathrm{B}$), i.e. $10^{12}$$\sim$$10^{13}$ K. The gradual increase of DCP supports gyrosynchrotron emission as the most plausible mechanism for the type IV. However, coherent emissions such as Electron Cyclotron Maser (ECM) instability can be responsible for small scale fine structures. Countless fine structures altogether were responsible for such high $T_\mathrm{B}$.

https://export.arxiv.org/pdf/2208.13670

\begin{article} \begin{opening} \title{Interferometric imaging, and beam-formed study of a moving Type IV Radio burst with LOFAR.} \author[addressref={aff1,aff2}]{\inits{Hongyu}\fnm{Hongyu}~\lnm{Liu}}% \author[addressref=aff3]{\inits{Pietro}\fnm{Pietro}~\lnm{Zucca}}% \author[corref,addressref={aff1,aff4},email={kscho@kasi.re.kr}]{\inits{Kyung-suk}\fnm{Kyung-Suk}~\lnm{Cho}}% \author[addressref=aff5]{\fnm{Anshu}~\lnm{Kumari}}% \author[addressref={aff3,aff6}]{\fnm{Peijin}~\lnm{Zhang}}% \author[addressref={aff7,aff8}]{\fnm{Jasmina}~\lnm{Magdalenić}}% \author[addressref={aff1,aff4}]{\fnm{Rok-Soon}~\lnm{Kim}}% \author[addressref=aff1]{\fnm{Sujin}~\lnm{Kim}}% \author[addressref={aff1,aff9}]{\fnm{Juhyung}~\lnm{Kang}}% \address[id=aff1]{Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Korea} \address[id=aff2]{Center for Magnetic Materials and Devices, College of Physics and Electronic Engineering, Qujing Normal University, Qujing 655011, China} \address[id=aff3]{ASTRON, the Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, Dwingeloo 7991PD, the Netherlands} \address[id=aff4]{University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Korea} \address[id=aff5]{Department of Physics, University of Helsinki, Pietari Kalmin katu 5, 00560 Helsinki, Finland} \address[id=aff6]{Institute of Astronomy and National Astronomical Observatory, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria} \address[id=aff7]{Solar-Terrestrial Center of Excellence–SIDC, Royal Observatory of Belgium, Av. Circulaire 3, B-1180 Brussels, Belgium} \address[id=aff8]{Center for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium} \address[id=aff9]{Astronomy Program, Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea} \runningauthor{H.\ Liu \emph{et al.}} \runningtitle{Moving Type IV Radio burst with LOFAR observation} \keywords{Radio Bursts, Type IV; Radio Emission, Theory} \end{opening} \section{Introduction} \label{S-Introduction} {The Sun is an active star, and its energetic phenomena} is frequently accompanied by electromagnetic emissions over a wide spectral range including radio wavelengths. {Solar radio bursts can show up in dynamic spectrum (a plot of brightness temperature on a frequency-time plane) as the features of increased intensity in comparison to the background emission.} They are classified at meter wavelengths into five major types, Type I to Type V named after Roman numbers \citep{1957CRAS..244.1326B}. Solar radio bursts are generally related to coronal mass ejections (CMEs) and solar flares \citep{2021A&A...647L..12M}. {In this paper we focus on a type IV radio burst, which is a }long-lasting broadband continuum emission in radio wavelengths that usually appears approximately 10 minutes after solar flare \citep{1963IAUS...16..247P}. They are often considered to be generated by a process involving energetic particles trapped in a post flare loop {or outward propagating magnetic structures, such as the CME magnetic fields \citep{1985srph.book..361S,Bastian2001,2019ApJ...870...30V,2021SoPh..296...38L}}. Depending on whether the radio source propagates, we can distinguish moving type IV radio burst (m-TypeIV) from stationary type IV radio burst \citep{1965ASSL....1..408K,1986SoPh..104...19P}. Due to its close relation with CMEs, m-TypeIV has been frequently addressed in solar radio physics since its discovery. {The emission }mechanism of type IV radio bursts is still highly controversial \cite[see e.g.][]{1974SoPh...36..157K,2019A&A...623A..63M,2021SoPh..296...38L}. \cite{1957CRAS..244.1326B} first observed type IV radio emission and proposed synchrotron emission as the mechanism. However, since high degree of circular polarization (DCP) ($>$20\%) is observed during some type IV events, \cite{1969PASA....1..189K} suggested gyrosynchrotron emission instead. \cite{1976ApJ...204..597B} concluded that type IV bursts of narrow bandwidth are generated by plasma emission instead of gyrosynchrotron emission. Later on, \cite{1981SoPh...73..191D} considered plasma emission to explain the observed high brightness temperature ($T_\mathrm{B}$). In addition to already mentioned emission mechanisms, \cite{1986ApJ...307..808W} discussed the possibility of electron cyclotron maser (ECM) emission { alongside the possibility of plasma emission (Upper Hybrid mode)}. Recent studies have featured gyrosynchrotron radiation \citep{2017A&A...608A.137C}, plasma emission \citep{2019ApJ...870...30V}, and ECM emission \citep{2018SoPh..293...58L} as a possible emission mechanisms of the type IV continuum. One important characteristic of type IV continuum is its fine structures \citep{1987SoPh..112..347A,2011ASSL..375.....C}. Recent studies have reported fine structures of very short durations (4 to 60 ms at half power of the burst) \citep{2005ASSL..320..259M,2006ApJ...642L..77M,2008SoSyR..42..434C,2014ApJ...787...45K}. Fine structures may also have different appearances on radio dynamic spectrum. The most frequently observed fine structures of type IV continuum are zebra patterns \citep{2010Ap&SS.325..251T,2018ApJ...855L..29K}, fiber bursts \citep{1983ApJ...264..677B,2014RRPRA..19..295A}, spikes \citep{1986SoPh..104...99B}, and slowly drifting narrowband structures \citep{2014ApJ...787...45K}. Since fine structures usually have higher brightness temperature than background type IV emission, they are more likely to be generated by a different emission process. Early studies of solar radio bursts were focused on understanding their basic physical characteristics employing single frequency observations and dynamic spectra. With the development of radio imaging instruments, more studies have been conducted to track the source of radio bursts on the Sun. Since the Nancay Radio-heliograph (NRH) started its observations in May, 1956 \citep{1957AnAp...20..155B}, metric-wavelength (i.e. $<$500MHz) imaging study of the radio Sun has a history of more than 60 years. Culgoora radio-heliograph was the second one of the type, starting observations from Feb 1968 \citep{1970PASA....1..365W}, but imaging observations stopped working in 1980s. In 1997, Gauribidanur Radio-heliograph \cite[GRAPH;][]{1998SoPh..181..439R} in India came into use, as an addition to the existing NRH instrument that can observe Stokes I and V. In January, 2015, NRH imaging observations stopped, to undergo extensive maintenance. Observations were resumed in November 2020, and currently only Stokes I is observed, as well as the radio image of the Sun at ten separate frequencies within the range of 150 MHz-445 MHz \citep{1997LNP...483..192K}. With the help of NRH, {a number of} studies have been carried out addressing type IV continuum emission \citep{2016A&A...586A..29B,2016ApJ...826..125K}. The Low-Frequency Array (LOFAR) \citep{2013A&A...556A...2V} in the Netherlands is not a solar dedicated instrument but during regular solar dedicated campaigns, the Sun is observed in a similar frequency range as by NRH. There have been several studies on solar type III \citep[e.g., ][]{2014A&A...568A..67M,2018A&A...614A..69R, 2020A&A...639A.115Z, morosan2022}, type II radio bursts \citep[e.g., ][]{2018ApJ...868...79C,2018A&A...615A..89Z,2020ApJ...897L..15M,2021ApJ...909....2M}, and short duration bursts \citep[e.g., ][]{2015A&A...580A..65M,2020ApJ...891...89Z} using LOFAR data. Type IV radio bursts on the other hand, have been reported less often with LOFAR observations. \cite{2019ApJ...873...48G} performed a statistical study of { the positions} of solar radio bursts, among which two were type IV events. The authors found a discrepancy between the obtained radio source locations and the expected radial height as mapped by the Newkirk density model. This discrepancy was attributed to the strong wave scattering due to plasma turbulence in the active corona. In our study, we {examine} a moving type IV radio burst with its fine structures observed by LOFAR on August 25, 2014. Despite of its long history, the emission mechanism of Type IV bursts is not fully understood. {This study will contribute to a better understanding of type IV bursts based on the high resolution observation of LOFAR.} In Section \ref{S2}, we showed an overview of the type IV radio burst and associated solar activity. A comparison of NRH and LOFAR imaging data is then presented. In Section \ref{S3}, we calculated the degree of circular polarization of the type IV. In addition, we combined the LOFAR imaging data with SDO-AIA and Solar and Heliospheric Observatory/Large Angle and Spectrometric Coronagraph (SOHO/LASCO)-C2 to track the type IV radio sources and understand their association with the ambient coronal structures. Finally, we presented details of fine structures and the estimated brightness temperature, followed by discussion on the radiation mechanism. This is the first detailed analysis of a type IV radio burst with LOFAR interferometric imaging data. \section{Observations and Data Analysis} \label{S2} \subsection{ The Low-Frequency Array and Data Preparation} \label{S21} {Low-Frequency Array (LOFAR) utilizes omni-directional antennas to form a phased array}. LOFAR operates in the 10 MHz to 240 MHz frequency range with a frequency resolution of 12.5kHz. There are two types of antennas: Low Band Antenna (LBA) and High Band Antenna (HBA), observing two different frequency ranges 10-90 MHz and 110-240 MHz, respectively. The frequency range is subject to change according to the LOFAR stations used to carry out the observation. While conventional radioheliographs like NRH can only provide Stokes I and V { data}, LOFAR simultaneously observes Stokes I, Q, U and V \citep{2013A&A...556A...2V}. As one of the newest radio telescopes to date, LOFAR greatly improved the angular resolutions of solar observations \citep{2011pre7.conf..507M}. We carried out imaging spectroscopy observations of the Sun with LOFAR on 25 August 2014. For LOFAR observations of this event, the frequency range of LBA is 10 MHz to 90 MHz, and that of HBA is 110 MHz to 190 MHz. % The dynamic spectrum in this work is obtained from core station. The flux value in the dynamic spectrum is un-calibrated relative flux intensity with median background subtracted for each frequency channel, the combined dynamic spectrum of LBA and HBA is shown in Figure \ref{fig1}. The interferometric imaging used the data from both LOFAR core and remote stations, imaging data is available only for HBA in this event. The measurement set is pre-processed with the Default Pre-Processing Pipeline (DPPP; \citealp{2018ascl.soft04003V}). The pre-process includes three steps, to: (1) derive the amplitude and the phase solutions from the analysis of the observation of Virgo-A with a Virgo-A model; {(2) apply the solutions to the solar observations to obtain the calibrated visibilities;} (3) apply the LOFAR-beam corrections to the calibrated visibility. After the pre-processing, we used the WS-Clean \citep{2014MNRAS.444..606O,2017MNRAS.471..301O} for the Fourier transform and deconvolution to obtain the radio flux image. % The calibration in SFU is a straightforward calculation inferred by knowing the beam size of LOFAR and by observing at the same time a calibrator. For more details, see e.g. \citealp{2009IEEEP..97.1431D}. {The imaging and calibration process to obtain the physical unit (brightness temperature in [K]) uses the LOFAR-Sun toolkit with similar procedures as recent solar radio imaging works \citep{2020A&A...639A.115Z,2022ApJ..932.17} .} The coordinate system of the imaging result is transformed from Equinox with J2000 epoch to helioprojective coordinate system. On the lower-right corner in the radio image of this day, there are some projected fake sources. So we have also applied a fade-out fit to the calibrated data on the lower-right corner to erase the fake sources. This doesn't affect the actual radio sources of the event. All the processes mentioned above were done with the ASTRON CEP3 server, and visualization was done using the Solar and Space Weather KSP routines (https://git.astron.nl/ssw-ksp/lofar-sun-tools) using SunPy codes \citep{2015CS&D....8a4009S}. \subsection{Event overview} \label{S22} On Aug 25, 2014, an M2.0 class flare {occurred in NOAA 12146}, starting at 14:46 UT\footnote{\url{https://www.solarmonitor.org/index.php?date=20140825&region=12146}} (SOL2014-08-25T15:10). Meanwhile, SOHO/LASCO CME catalog provided by NASA Goddard Space Flight Center \citep{2009EM&P..104..295G}, reports a halo CME first seen in the SOHO/LASCO C2 field of view at 15:36 UT\footnote{\url{https://cdaw.gsfc.nasa.gov/movie/make_javamovie.php?stime=20140825_1401&etime=20140825_1907&img1=lasc2rdf&title=20140825.153605.p270g;V=555km/s}}. The CME speed is reported to be 555 km/s. The eruptive CME/flare event was {associated with a complex type III/II/IV radio burst shown} by LOFAR combined dynamic spectrum (HBA and LBA) in Figure \ref{fig1}. In Figure \ref{fig1}, a group of type III radio bursts associated with the impulsive phase of the long duration flare is visible on both HBA and LBA at 15:00 UT, followed by a type II radio burst on LBA at 15:09 UT \citep{2020ApJ...897L..15M}. At 15:10 UT, a type IV radio burst was observed, first in HBA and later in LBA. Despite the observation gap between 90 MHz and 110 MHz, it is clear that the type IV radio continuum spans from the HBA to the LBA range. The frequency drift of the continuum emission implies that this is a moving type IV radio burst. Another group of type III bursts was observed at about 15:48 UT in HBA. In this study, we focus only on the type IV emission. \subsection{LOFAR imaging data of the Sun and comparison with NRH} \label{S23} {We performed a detailed comparison of LOFAR and NRH imaging data. The comparison of NRH and LOFAR spatial resolution is important because we employ LOFAR observation with the remote baselines to demonstrate the advantages in beam size and resolution of a long baseline. } For this event, LOFAR HBA has observations from 110 to 190 MHz. Two NRH frequency channels (150.9 MHz and 173 MHz), overlap with LOFAR observations. On Aug 25, 2014, LOFAR had data from 12:12 UT until 16:12 UT, while NRH observed the Sun until 15:23 UT. Since LOFAR has a frequency resolution of 12.5 kHz, radio imaging data can be produced to match the exact NRH frequency. After acquiring the cleaned LOFAR imaging data of the Sun, we re-scaled LOFAR data to make the field of view identical to that of NRH, and plotted LOFAR 150 MHz and NRH 150 MHz observations together. A movie is available in the online material. Some of the frames are shown in Figure \ref{fig2}. Both movie and images show that the positions of LOFAR and NRH radio sources are in agreement. In Figure \ref{fig2} (a), we can see multiple separate sources on LOFAR while in Figure \ref{fig2} (b), we can only see one faint source on NRH. Similarly, in Figure \ref{fig2} (c), at least 3 separate structures are visible on LOFAR images, while in Figure \ref{fig2} (d), we can only see a blurry radio source. The beam size ($\sim$ 200 arcsec along major axes) of LOFAR is shown on the lower-left corner in Figure \ref{fig2} (a) and (c), while beam size ($\sim$ 650 arcsec along major axes) of NRH is shown on the lower-left corner in Figure \ref{fig2} (b) and (d). Since the beam size of LOFAR is smaller than the displacement of the separate radio sources ($\sim$ 500 arcsec), we clearly observe two sources that are separated and resolved. We are able to resolve the multiple sources at the onset of the Type IV which are appearing as a single large source in NRH with ~1.8 km baseline. This indicates the importance of medium baselines 3 to 10 Km to resolve complex radio sources and understand better the evolution and kinematics of the event. \section{Results} \label{S3} \subsection{Degree of circular polarization} \label{S31} A solar radio burst has many physical parameters, and they are crucial not only to determine the emission mechanism, but also to understand the ambient coronal conditions \citep{1975SoPh...43..211M}. One of the important parameters of the radio bursts is the degree of circular polarization (DCP). The degree of circular polarization of solar emission is given by \citep{1992plfa.book.....C}: \begin{equation} \label{eq1} DCP = \frac{V}{I} , \end{equation} where V and I are the corresponding Stokes parameters. LOFAR provides full Stokes parameters, i.e. Stokes I, Q, U, and V. As often observed for radio instrumentation, a small amount of Stokes V on the LOFAR HBA band has leaked into Stokes U. In fact, the polarization leakage (linear spill of intensity from Stoke V to U) is a common instrumental issue in the majority of radio instruments. We simply correct the leakage by using the following process since LOFAR provide four Stokes parameters. In order to obtain the real Stokes V data in HBA, we introduced some corrections. Thus, the DCP of HBA is calculated as follows: \begin{equation} \label{eq2} \vert DCP_\mathrm{corr}\vert = \frac{\sqrt{V^2+U^2}}{I} . \end{equation} The signal leakage to Stokes U is proportional to the Stokes V intensity, making it a clear instrumental effect. Moreover, this correction can be confidently done for the Sun as we can assume that linear component of the incoming radio waves are removed by Faraday rotation (FR) in the corona. Figure \ref{fig3} shows DCP diagrams from 14:40 UT to 16:00 UT at 50 MHz and 70 MHz calculated using Eq. (\ref{eq1}). The presented absolute value of DCP at 120 MHz and 160 MHz was estimated using Eq. (\ref{eq2}). Figure \ref{fig3} shows that the DCP of this type IV continuum {amounts 10\%$\sim$20\%}, and the DCP value rises as type IV develops in time. The increasing DCP after the end of the Type IV burst as shown in Figure 3 (c) and (d), is due to the presence of other types of radio bursts superposed at higher frequencies. The Type I and Type III like sources result in the increase of DCP visible. This is a good indicator that the instrument is capable of measuring DCP well. The increase at 15:40 UT at 120 MHz is a clear sign of the superposition of the type III and Type I sources. We can still show well that the type IV (especially where well isolated in the profiles at 50 and 70 MHz) show a clear trend of decreasing DCP. It can be done thanks to the full Stokes observations of LOFAR. From \ref{fig3} (a) and (b), we conclude that the DCP of this type IV is negative, which means that left hand circular polarization dominates. To preserve the sign of the DCP we use the following equation: \begin{equation} \label{eq3} DCP_\mathrm{corr} =\frac{\vert V\vert}{V} \frac{\sqrt{V^2+U^2}}{I} . \end{equation} We also plot the DCP along vertical lines on the spectrum in Figure \ref{fig1}. For this, we have selected 6 time instances from the spectra, which includes the DCP during no bursts observed and during the type II, III and IV events. The DCP is calculated by averaging the 1 min spectral data and implementing the correction as shown in Eq. (3) Figure \ref{fig35} (a) shows the DCP variation along the full LOFAR frequency range at 12:00 UT when no bursts are present. The flat profile with very little of modulation shows that in the absence of radio burst, the instrument records almost no DCP. Panels (b) to (g) of Figure \ref{fig35} show the DCPs (again along the full LOFAR frequency range) at six different times when there are radio bursts recorded. The time intervals during which different types of radio bursts were recorded are marked with arrows of different colors. Starting from about 15:00 UT a group of type III radio bursts is visible on the spectrum, while from 15:10 UT to 15:50 UT, the type IV which we study is present (see also Fig.1). A gradual change of the DCP towards a negative value is observed as the type IV radio burst propagates away from the Sun (Figure \ref{fig35} panels (d) to (g)). We note that at 63 MHz, there is a spike which can be seen at all considered times. This spike is due to radio frequency interference (RFI). Although the observed gradual rise of DCP value favors gyrosynchrotron radiation as the generation mechanism \citep{1985ARA&A..23..169D}, other physical parameters such as $T_\mathrm{B}$, also need to be considered in order to determine the emission mechanism of this type IV continuum. \subsection{Propagation of radio sources} \label{S32} In order to investigate the properties of the type IV continuum and its association with the ambient coronal structures, we plotted LOFAR 70\% maximum contours of different frequencies on SDO-AIA 171 \AA~ images. The movie is provided as the online material and few frames are shown in Figure \ref{fig4} (b) $\sim$ (f). The LOFAR channels at 125, 160 and 165 MHz suffer from the calibration issues, { and we couldn't identify the radio sources, }so we have excluded them from the study. At 15:15 UT, all radio sources appeared to be situated above the active region (NOAA AR 12146) that was the source of the associated CME/flare event. In Figure \ref{fig4} (c), almost all frequencies seems to propagate toward the west limb of the Sun. At 15:27 UT, radio sources at higher frequencies (170$\sim$185 MHz) started to be observed again closer to the active region. This behavior of the radio sources is expected because the continuum emission decreases in intensity and is no longer observed at higher LOFAR frequencies (see Figure \ref{fig1}). We therefore, start to observe again the radio sources associated with the active region. At the same time, type IV emission at lower frequencies of the HBA band can still be observed until about 15:48~UT. In Figure \ref{fig4} (e) and (f), only 3 low frequency (115$\sim$130 MHz) contours kept propagating westward while all remaining frequencies map the faint radio source, associated with the active region. An expanding post-flare loop was not clearly seen in AIA 171\AA~ running differential images. It might be due to the low emissivity of such loops in EUV wavelengths. Figure \ref{fig5} shows this event in a larger scale, with LOFAR, AIA and LASCO-C2 on the same panel. We can notice the CME eruption in the direction of the propagation of type IV radio source. This CME eruption continued after the type IV disappeared in LOFAR HBA frequency range. Unfortunately, LBA radio imaging was not active for this observation and we cannot confirm the association of the lower frequencies radio sources of the type IV continuum with the CME core. \subsection{Fine structures and emission mechanism} \label{S33} {As already mentioned in Section 1}, a type IV continuum can have superposed fine structures which have duration much shorter than the type IV itself. We analyzed the type IV fine structure in order to obtain more knowledge on the possible origin. We observe different fine structures, such as spikes, drifting spike and patchy, irregular fine structures. To isolate short duration fine structures, we have plotted the dynamic spectrum in a shorter time range for selected areas within this type IV radio burst. Figure \ref{fig6} (a), (b), (c) and (d) {are details} of the dynamic spectrum within time and frequency ranges indicated by red boxes A, B, C and D in Figure \ref{fig1}. Time ranges of left, middle and right columns are 1 minute, 12 seconds and 4 seconds respectively. Despite the short time range, we can see plenty of fine structures that last for about 1s and less. These narrow-band bursts have the characteristics of spikes, some of them show positive frequency drifts, while others show negative frequency drifts \citep{2016A&A...586A..29B}. We measure the drift rate of the fine structure as denoted with a dotted line in the right columns of Figure 7. We find that the mean draft rate is about 20 MHz/s. Brightness temperature plays an important role in determining the emission mechanism. It can be obtained according to Rayleigh-Jeans law: \begin{equation} \label{eq3} T_\mathrm{B} = \frac{\lambda^2}{2k\Omega}S , \end{equation} where $T_\mathrm{B}$ is the brightness temperature, $\lambda$ is the wavelength, k is the Boltzmann constant, $\Omega$ is the beam solid angle, and S is the flux density. We plotted the $T_\mathrm{B}$ profiles at 162 MHz and 122 MHz from 15:20 UT to 15:21 UT in Figure \ref{fig7} (a) and (b). $T_\mathrm{B}$ data provided by LOFAR have a time resolution of 1 s. Note that these plots are along 2 blue parallel lines in left panels of Figure \ref{fig6} (a) and (b). As is shown in Figure \ref{fig6}, there are plenty of fine structures that are brighter than the background type IV. However, only the most intense fine structures have the brightest temperature noticeably different with the type IV continuum. The intense bursts at 162 MHz reach about $6.5\times10^{11}$ K, while the brightness temperature of the type IV background and the weaker fine structures fluctuates around $4\times10^{11}$ K. Situation is similar at 122 MHz but, with in general higher brightness temperatures. The intense burst has brightness temperature of about $10^{13}$ K, while background and weaker fine structures have value of about $2.5\times10^{12}$ K. These examples show that in a case of faint fine structures the brightness temperatures will be on a similar {order} as for the type IV continuum. \section{Discussion and Conclusions} \label{S4} A moving type IV radio burst was observed at LOFAR dynamic spectrum on August 25, 2014. We performed a detailed comparison of NRH and LOFAR imaging. Having a higher spatial resolution LOFAR observed two separate sources at the onset of the m-TypeIV, compared to one unique larger source on NRH. Using the full stokes parameters from the LOFAR dynamic spectra, we calculated the degree of circular polarization during the propagation of this moving type IV. The DCP of this type IV lies in the range 10\%$\sim$20\% and it is left hand polarized. The DCP value showed a gradual increase with the development of the type IV. In addition, the combined LOFAR interferometric data with SDO-AIA 171 and LASCO-C2 indicate that this type IV is generated on expanding flare loops located in the CME core. {The close-in details} of the LOFAR dynamic spectrum showed the existence of countless fine structures. Majority of them look like narrow-band drifting spikes and broadband pulses, and they generally last for less than one second. $T_\mathrm{B}$ analysis showed that the most intense fine structures can exhibit a $T_\mathrm{B}$ as high as $10^{13}$ K. The type IV continuum also showed higher than expected $T_\mathrm{B}$ (up to $10^{12}$ K). We can consider plasma emission, gyrosynchrotron, and ECM as generation mechanisms of type IV background and the fine structure. Our observation presented extremely high brightness temperature and low polarization degree with gradual increase. Regarding type IV background, the observed gradual rise of DCP value favors gyrosynchrotron radiation as the generation mechanism \citep{1985ARA&A..23..169D} because the gradual increase of DCP as shown in Figure \ref{fig3} can't be explained by original plasma emission models \citep{1985srph.book..361S}. One possible argument against the gyrosynchrotron emission is that the emission mechanism has difficulty in explaining high $T_\mathrm{B}$. { \cite{1978SoPh...60..383R} considered the local electron density and claimed that $T_\mathrm{B}$ of gyrosynchrotron emission can't exceed $10^{9}$ K if the source is moderately polarized ($>$40$\%$)}. We note that DCP is not very high (10\%$\sim$20\%), and if the local magnetic field allows a very large number of energetic electron, gyrosynchrotron emission can still be possible. Other physical parameters such as spectral index also need to be inspected in near future to confirm this. Fine structures with even higher $T_\mathrm{B}$ ($\sim10^{13}$ K) than the background type IV may be generated by coherent ECM emission. This speculation needs some justification in the low-frequency regime because the ECM requires a relatively high ambient magnetic field common in the lower corona. \cite{2016A&A...589L...8M} used a PFSS model and calculated the ECM condition to be $\sim$ 500 MHz. However, \cite{2015A&A...581A...9R} used NLFF models and demonstrated that ECM condition can still satisfy at higher corona (1.2 Rs), which supports our conclusion. Another possible interpretation of the results is that a faint gyrosynchrotron background is superposed by high $T_\mathrm{B}$ plasma fine structures. However, we were not able to isolate this faint $T_\mathrm{B}$ background in the spectrum due to numerous fine structures superimposed on the continuum We conclude that the background type IV is most likely generated by gyrosynchrotron emission, while the fine structures may still be generated by coherent ECM emission. However, future observations will be necessary to confirm our conclusion on the emission mechanism of moving type IV radio bursts and fine structure therein. { For example, statistical studies of a number of similar type IV bursts will certainly contribute to our understanding of emission mechanism \citep[][]{salas2020polarisation, Kumari2021}}. \begin{acknowledgements} This research received fund from UST overseas training program (South Korea), the Netherlands institute for Radio Astronomy (ASTRON), and Korea Astronomy and Space Science Institute under the R\&D program Development of a Solar Coronagraph on International Space Station (Project No. 2020-1-850-07) supervised by the Ministry of Science, ICT and Future Planning (South Korea). J.M. acknowledges funding by the BRAIN-be (Belgian Research Action through Interdisciplinary Networks) project CCSOM (Constraining CMEs and Shocks by Observations and Modeling throughout the inner heliosphere). A.K. acknowledges the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme Project SolMAG 724391. We would like to thank SDO, LASCO, NRH and LOFAR operational team for providing open-access data, and SunPy team for the codes. In addition, we acknowledge Eduard Kontar and Nicolina Chrysaphi from University of Glasgow, and Jaeok Lee from KASI for their constructive suggestions. We also thank Sarrvesh Sridhar, Maaijke Mevius and Michiel Brentjens at ASTRON, Netherlands for their beneficial discussions and support. LOFAR, the Low Frequency Array designed and constructed by ASTRON, that has facilities in several countries, that are owned by various parties (each with their own funding sources), and that are collectively operated by the International LOFAR Telescope (ILT) foundation under a joint scientific policy. \end{acknowledgements} \bibliographystyle{spr-mp-sola} \bibliography{reference} \iffalse \bibliographystyle{spr-mp-sola} \fi \end{article}

Title: MagAO-X: current status and plans for Phase II

Abstract: We present a status update for MagAO-X, a 2000 actuator, 3.6 kHz adaptive optics and coronagraph system for the Magellan Clay 6.5 m telescope. MagAO-X is optimized for high contrast imaging at visible wavelengths. Our primary science goals are detection and characterization of Solar System-like exoplanets, ranging from very young, still-accreting planets detected at H-alpha, to older temperate planets which will be characterized using reflected starlight. First light was in Dec, 2019, but subsequent commissioning runs were canceled due to COVID-19. In the interim, MagAO-X has served as a lab testbed. Highlights include implementation of several focal plane and low-order wavefront sensing algorithms, development of a new predictive control algorithm, and the addition of an IFU module. MagAO-X also serves as the AO system for the Giant Magellan Telescope High Contrast Adaptive Optics Testbed. We will provide an overview of these projects, and report the results of our commissioning and science run in April, 2022. Finally, we will present the status of a comprehensive upgrade to MagAO-X to enable extreme-contrast characterization of exoplanets in reflected light. These upgrades include a new post-AO 1000-actuator deformable mirror inside the coronagraph, latest generation sCMOS detectors for wavefront sensing, optimized PIAACMC coronagraphs, and computing system upgrades. When these Phase II upgrades are complete we plan to conduct a survey of nearby exoplanets in reflected light.

https://export.arxiv.org/pdf/2208.07299

\keywords{adaptive optics} \section{INTRODUCTION} \label{sec:intro} MagAO-X\cite{2018SPIE10703E..09M,2020SPIE11448E..4LM} is an ``extreme'' adaptive optics (ExAO) system on the Magellan Clay 6.5 m telescope at Las Campanas Observatory (LCO), in Chile. It is optimized for high-contrast science at visible-to-near-IR wavelengths. MagAO-X is also employed as a testbed for adaptive optics (AO) and segment phasing technologies for the Giant Magellan Telescope (GMT), and as a pathfinder for an ExAO instrument planned for GMT called GMagAO-X. We have successfully completed the first two of several commissioning runs. First-light was in Dec, 2019. Runs in 2020 and 2021 were postponed due to the pandemic. The second commissioning run, as well as initial science operations, occurred in April, 2022. The main science goal of this initial phase is a search for and characterization of young accreting planets in H$\alpha$ orbiting nearby T Tauri and Herbig Ae/Be stars \cite{2020AJ....160..221C}. This project uses the simultaneous differential imaging (SDI) mode of MagAO-X to study where and how this population of low-mass outer extrasolar giant planets (EGPs) form. Additional near-term science cases for MagAO-X include: circumstellar disk characterization; young EGP characterization in the red-optical/near-IR; spatially resolved stellar surface imaging at high spectral resolution; characterization of tight binary star systems; \textit{Kepler} and \textit{TESS} followup; and high spatial resolution imaging of asteroid surfaces and asteroid companion searches. The ultimate goal for MagAO-X is the characterization of nearby temperate exoplanets in reflected light. Achieving the most demanding wavefront control precision required to perform such observations will require continued improvements and upgrades as MagAO-X commissioning proceeds. With a 2040 actuator high order deformable mirror (DM) being controlled at up to 3.6 kHz by a pyramid wavefront sensor (PWFS), MagAO-X is optimized for very high Strehl at short wavelengths. This excellent wavefront quality enables narrow-angle high contrast imaging with coronagraphs. Wavefront quality is further augmented using coronagraphic low-order and non-common path wavefront sensing and control (WFS\&C). In what follows we present a brief overview of the instrument, including the design, specifications, and concept of operations. Following this we report results from our successful second commissioning period, as well as its ongoing use as a laboratory testbed\cite{2020SPIE11448E..2XH, 10.1117/1.JATIS.8.2.021513, 10.1117/1.JATIS.8.2.021515, Demers_2022,Hedglen_2022,Kautz_2022}. Finally we discuss future plans for MagAO-X and in-progress upgrades. \section{INSTRUMENT OVERVIEW} \label{sec:design} Here we present a brief overview of the design and specifications of MagAO-X. See our previous SPIE contributions for more detailed treatments \cite{2018SPIE10703E..09M,2018SPIE10703E..5AV,2018SPIE10703E..55H,2018SPIE10703E..4ZL,2018SPIE10703E..4YC,2018SPIE10703E..2QK,2018SPIE10703E..2NR,2018SPIE10703E..21S,2018SPIE10703E..1TM,2018SPIE10706E..5OK,2018SPIE10703E..1EG,2020SPIE11448E..4LM,2020SPIE11448E..0UC} In addition, the complete preliminary design review (PDR) documentation is available at \url{https://magao-x.org/docs/handbook/appendices/pdr/}, and results of laboratory integration and preparation for shipment can be bound in the pre-ship review (PSR) documentation: \url{https://magao-x.org/docs/handbook/appendices/psr/index.html}. MagAO-X operates on the Nasmyth platform of the 6.5 Magellan Clay telescope (Figure \ref{fig:platform_inset}). The instrument consists of a two-level optical bench (Figure \ref{fig:labeled}). It is a woofer-tweeter system, with a 97-actuator woofer and 2040-actuator tweeter. The pupil illuminates 48 actuators across in the short direction, giving a 13.5 cm actuator pitch. With Clay's 29\% central obscuration, roughly 1640 actuators are illuminated. The top-level contains the k-mirror derotator and atmospheric dispersion compensator. The bottom level houses the pyramid wavefront sensor (PWFS) as well as a Lyot-coronagraph system feeding a dual-EMCCD simultaneous differential imaging (SDI) system. The coronagraph contains a third deformable mirror with 97 actuators downstream of the beamsplitter, which enables non-common path offloading without offsetting the PWFS. A visitor station is currently occupied by the VIS-X spectrograph\cite{2021SPIE11823E..06H}. All of these components and subsystems are described in great detail in the references and documents noted above. \section{LAB TESTBED} MagAO-X is designed for routine shipment between LCO and the University of Arizona in Tucson. Due to the COVID-19 Pandemic, it underwent an extended period of laboratory testing and upgrades between December 2019 and April 2022. Notable examples of the work done during this time are the implementation of Focus Diversity Phase Retrieval (FDPR)\cite{2021JATIS...7c9001V}. FDPR is now used at the telescope to flatten the DMs to maximize Strehl at the focal plane, and the AO is then calibrated around this point. This procedure significantly improves on-sky Strehl. Lab development of advanced control algorithms has also been a major emphasis of MagAO-X development. The Data Driven Subspace Predictive Control (DDSPC) algorithm was implemented on the instrument and demonstrated in the lab.\cite{2021JATIS...7b9001H}. We began on-sky testing in the 2022 run (Haffert et al., in prep). In these proceedings we present new focal plane WFS techniques developed in the lab.\cite{Haffert_2022}. MagAO-X can also be used for remote research and development, as demonstrated by the testing of a reinforcement learning algorithm.\cite{2022arXiv220507554N} MagAO-X is also part of the High-Contrast Adaptive Optics Testbed (HCAT) for the Giant Magellan Telescope (GMT), part of the overall program to develop segment phase sensing and control strategies for the GMT\cite{Demers_2022}. As part of HCAT MagAO-X serves as the AO system and houses phase sensing and control experiments.\cite{2020SPIE11448E..2XH,Hedglen_2022,Kautz_2022,10.1117/1.JATIS.8.2.021513,10.1117/1.JATIS.8.2.021515} \section{ON-SKY RESULTS} MagAO-X returned to LCO in April, 2022, following an extended laboratory upgrade and testing period (see below). The telescope run from 09 to 23 April included 5 nights of commissioning, and 10 nights of shared-risk science verification allocated to members of the Magellan Consortium. During this 2nd commissioning run we demonstrated significant improvements in instrument performance and operations compared to our first-light campaign. These included procedures for flattening the instrument to maximize focal plane Strehl using our internal source each night, closed-loop pupil alignment control, the implementation of modal offloading to the woofer (instead of the actuator basis). Together these improvements resulted in a more stable instrument compared to 2019, and enabled us to routinely close the loop on 1376 modes (compare to $\sim900$ at first light). The result is shown in Figure \ref{fig:z-band-dark-hole}. We also commissioned several new observing modes. Our Lyot coronagraph system was demonstrated. We currently support classical Lyot coronagraphs with opaque focal plane masks (details here: \url{https://magao-x.org/docs/handbook/observers/coronagraphs.html}). See Figure \ref{fig:coronagraph}, which included closed-loop feedback to the non-common path (NCP) DM inside the coronagraph using light rejected by the reflected focal plane masks and the low-order WFS camera. The coronagraph was used extensively, for instance to observe the low-mass companion PZ Tel B in H$\alpha$ (Figure \ref{fig:pztel}) and the well-known HR 4796 A debris disk (Figure \ref{fig:hr4796A}). The VIS-X spectrograph\cite{2021SPIE11823E..06H} also began on-sky commissioning (Figure \ref{fig:visx}); The Strehl ratio of the image shown in Figure \ref{fig:z-band-dark-hole} is $\sim60\%$, a large improvement over the $\sim46\%$ measured at First-light thanks to the better system flat and higher number of modes in use. However, it is still below the design capability of nearly $90\%$ at \textit{z'}. The main limitation is currently vibrations, where we sometimes experience a residual of up to 10 mas jitter. The most likely culprit is wind shake, as similar behavior and dynamics have been observed with MagAO+VisAO. We are investigating whether this is exacerbated by our pointing offloads to the telescope mount, which occur as often as once per second. \section{UPGRADES AND PHASE II} MagAO-X was returned to Tucson for additional lab testing, HCAT testbed support, and further upgrades. The final major item from our original plan is to test the DARKNESS MKID IFU behind MagAO-X\cite{Swimmer_2022}. We anticipate conducting an on-sky demo with DARKNESS at LCO in early 2023. We have also begun a significant upgrade program called ``MagAO-X Phase II''. We plan to upgrade the real-time and instrument control computers and GPUs to reduce compute latency and enable additional real-time processing. We are also procuring new CMOS cameras for low-order WFS, including a new Lyot-LOWFS capability. The Lyot-LOWFS is key for the phase induced amplitude apodization complex mask coronagraph (PIAACMC) being installed as part of this upgrade. Finally, we are upgrading the NCP DM to a 1000 actuator MEMS device to enable FPWFS\cite{Haffert_2022} without the need for PWFS offloading. \section{CONCLUSION} MagAO-X has now undergone two commissioning runs at LCO, and has begun supporting science observations. Significant improvements in performance were realized between first-light in Dec, 2019 and the 2nd (COVID-delayed) commissioning run in April, 2022. MagAO-X is capable of many exciting science cases, especially utilizing its coronagraphic modes of observation. The instrument will continue to serve as an invaluable laboratory testbed, supporting ongoing high contrast imaging technology development as well as supporting phase sensing and control development for GMT. \clearpage \acknowledgments % We are very grateful for support from the NSF MRI Award \#1625441 (MagAO-X). The Phase II upgrade program is made possible by the generous support of the Heising-Simons Foundation. \bibliography{report} % \bibliographystyle{spiebib} %

Title: Red Spiral Galaxies in the Cosmic Noon Unveiled in the First JWST Image

Abstract: In the first image of the James Webb Space Telescope (JWST) of SMACS J0723.3-7327, one of the most outstanding features is the emergence of a large number of red spiral galaxies, because such red spiral galaxies are only a few percent in the number fraction among nearby spiral galaxies. While these apparently red galaxies were already detected with the Spitzer Space Telescope at $\sim3-4{\rm \mu m}$, the revolutionized view from JWST's unprecedented spatial resolution has unveiled their hidden spiral morphology for the first time. Within the red spiral galaxies, we focus on the three most highly red galaxies that are very faint in the $<0.9\,{\rm \mu m}$ bands and show red colors in the $2-4\,{\rm \mu m}$ bands. Our study finds that the three extremely red spiral galaxies are likely to be in the Cosmic Noon (i.e., $1 < z < 3$) and could be consistent with passive (i.e., $\sim$ zero star-formation rates) galaxies having moderate dust reddening (i.e., $A_{\rm V}\sim1\,{\rm mag}$). These "red spiral" galaxies would be interesting, potentially new population of galaxies, as we start to see their detailed morphology using JWST, for the first time. Finally, we note that the spectral energy distribution of these red $z\sim2.5$ galaxies could mimic $z>10$ Lyman break galaxies and contaminate to $z>10$ galaxy samples, especially when they were faint and small.

https://export.arxiv.org/pdf/2208.00132

command. \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \newcommand\red[1]{\textbf{\color{red}#1}} \usepackage{threeparttable} \begin{document} \title{Red Spiral Galaxies at Cosmic Noon Unveiled in the First JWST Image} \author[0000-0001-7440-8832]{Yoshinobu Fudamoto} % \affiliation{Waseda Research Institute for Science and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan} \affiliation{National Astronomical Observatory of Japan, 2-21-1, Osawa, Mitaka, Tokyo, Japan} \author[0000-0002-7779-8677]{Akio K. Inoue} % \affiliation{Waseda Research Institute for Science and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan} \affiliation{Department of Physics, School of Advanced Science and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo 169-8555, Japan} \author[0000-0001-6958-7856]{Yuma Sugahara} % \affiliation{Waseda Research Institute for Science and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan} \affiliation{National Astronomical Observatory of Japan, 2-21-1, Osawa, Mitaka, Tokyo, Japan} \keywords{Spiral galaxies (1560) --- Galaxy structure (622) --- Galaxy formation (595) --- Galaxy evolution (594) --- Galaxy stellar disks (1594) } \section{Introduction} \label{sec:intro} The spiral structure of galaxies is not only one of the most spectacular features of the Universe but also provides us with important information of galaxy formation and evolution. Since the first systematic classification of the morphology of ``extragalactic nebulae'' \citep{Hubble26}, large efforts have been devoted to studying the morphology of galaxies across cosmic time and to understanding their formation mechanisms \citep[see][for a review]{Conselice14}. However, when and how the galaxy morphology emerged in the early Universe is still largely unknown. Spiral galaxies typically show blue colors in their rest-frame optical wavelength and are, in general, classified as ``normal'' star-forming galaxies (e.g., \citealt{2014MNRAS.440..889S}). Red or passive (i.e. non star-forming or `anemic'; \citealt{1976ApJ...206..883V}) spiral galaxies are, on the other hand, a very minor population in the nearby Universe. In the latest study, \cite{2022PASJ...74..612S} identified nearly a thousand red, passive spiral galaxies among $\sim55,000$ galaxies at $0.01<z<0.3$ from $1000\,{\rm deg^2}$ imaging data obtained with the Subaru/Hyper Suprime-Cam. Hence, the fraction of the red spiral galaxy is only $\sim2\%$ in the local Universe. It was, for the first look, surprising to find many apparently red spiral galaxies in the James Webb Space Telescope (JWST) image of the galaxy cluster, SMACS J0723.3-7327, that was released on July 11th 2022 as part of the early release observations \citep[ERO;][]{Pontoppidan22}. Their redness could indicate several important properties of these spiral galaxies: their dominant stellar ages, dust reddenings, or a combination of these features. Because the released image was taken by the Near Infrared Camera (NIRCam) on JWST over the wavelength range of $0.9-4.4\,{\rm \mu m}$, the redness in NIR may indicate that these spiral galaxies are at high redshift. Spiral galaxies in the distant universe are a very important population to examine the emergence of the spiral structure in galaxies. The most distant spiral galaxy known so far is a gas-rich galaxy, BRI 1335-0417, at $z=4.41$ \citep{2021Sci...372.1201T}. A grand-design spiral structure traced by the [C~{\sc ii}] line was revealed with the Atacama Large Millimeter/submillimeter Array (ALMA); however, the stellar structure of BRI 1335-0417 is still unknown. The most distant, spiral stellar disk galaxies are reported at $z=2$--3 \citep{2003AJ....125.1236D,2012Natur.487..338L,2022MNRAS.511.1502M,Wu2022}. Searching for such galaxies at even higher redshift is the key to knowing when the stellar spiral disks emerged. To characterize red spiral galaxies found in the JWST's ERO data, we carefully select the most extreme red spiral galaxies and examine their spectral energy distribution (SED), as a first case study. This Letter is organized as follows: in \S2 we describe data and the sample used in this study. In \S3, we present our analysis. \S4 shows the results and discussion on the red spiral galaxies found in the JWST images. We conclude in \S5. Throughout this Letter, we assume a cosmology with $(\Omega_m,\Omega_{\Lambda},h)=(0.3,0.7,0.7)$ \section{Data} \label{sec:data} The following images of the Hubble Space Telescope (HST) and JWST are based on a ``first-pass'' reduction of the HST and JWST images of the SMACS-0723 lensing cluster field. All images have been processed with the grizli\footnote{\url{https://github.com/gbrammer/grizli}} software pipeline. Further documentation will be provided by G. Brammer et al. (in prep). \subsection{HST data} As part of the Reionization Lensing Cluster Survey: RELICS \citep{Coe19}, optical and NIR images were obtained using HST. These HST images include data from ACS (F435W, F606W, F814W filters) and WFC3/IR (F105W, F125W, F140W, F160W filters) instruments. All HST data were calibrated and mosaiced in a standard manner. Final images have a pixel scale of $0.04^{\prime\prime}/{\rm pixel}$. \subsection{JWST data} We used JWST NIRCam images of six filters in total (F090W, F150W, F200W, F277W, F356W, and F444W). After the re-processing of the JWST's public data available in the MAST data archive, the final mosaic has $0.02^{\prime\prime}/{\rm pixel}$ for the short wavelength filters (F090W, F150W, F200W), and $0.04^{\prime\prime}/{\rm pixel}$ for the long wavelength filters (F277W, F356W, F444W). A small astrometric offset ($\sim 0.2^{\prime\prime}$) exists between HST and JWST images. The offset, however, did not affect our analysis as we manually corrected positions of apertures during our photometry. \section{Analysis} \label{sec:analysis} \subsection{Selecting Red Spiral Galaxies} We first visually selected bright red spiral galaxies from the false-color image presented within the ERO release\footnote{\url{https://stsci-opo.org/STScI-01G7DDBW5NNXTJV8PGHB0465QP.png}}. Two authors (Y.F. and A.K.I.) visually selected red spiral galaxies from the image independently. We selected 21 galaxies that both agreed as having spiral structures with apparently red colors. While quantitative evaluation of galaxy types would also be preferred \citep[e.g.,][]{Dominguez18,Ferreira22}, visual selections are also known to be effective and widely used, especially when finding apparent structures, such as spiral shape \citep[e.g., Galaxy Zoo Project:][]{Lintott08}. To find the reddest spiral galaxies, we further down-selected samples based on their faintness in $\leq0.9\,{\rm \mu m}$ images (see \S\ref{sec:photometry} below for the photometry of our sample here). In particular, we selected galaxies that are non-detected in the HST F814W image and, at the same time, galaxies that have an extremely red color in the F090W vs. F150W image (Fig. \ref{fig:colors}). The selected red spiral galaxies (RS13 and RS14) were already detected by HST WFC3 and Spitzer IRAC images; however, the angular resolution of these previous instruments did not allow us to study detailed morphology. We can now access the resolved morphology of the red spiral galaxies by JWST's unprecedented spatial resolution and sensitivity in these NIR wavelengths (Fig. \ref{fig:nasaimage}). \subsection{Photometry} \subsubsection{HST and JWST photometry} \label{sec:photometry} To measure galaxy-wide integrated fluxes, we performed aperture photometry of both HST and JWST images for the red spiral galaxies. We used python package \texttt{photoutils} \citep{larry_bradley_2020_4044744} using elliptical apertures. We applied apertures large enough to enclose entire galaxies in the JWST F444W image to obtain integrated fluxes. Measured flux errors incorporated background noises by scaling pixel-by-pixel root mean squares (RMSs) to aperture sizes, and Poisson noises of measured fluxes. \subsubsection{JWST photometric uncertainty} As calibrations of NIRCam data progressed, studies reported that the NIRCam's photometric zero points have uncertainties, which could be up to $\sim20\,\%$ different from the pre-flight measurements \citep{Adams2022,Morishita22,Rigby22} and/or could be time-variable \citep{Nardiello2022}. In this study, we use the photometric zero-point of JWST NIRCam data based on \citet{Adams2022}, which is derived using updated JWST's calibration references for the SMACS 0723 field used using in-flight data. These updated zero-points were, however, based on preliminary calibrations of JWST. Therefore, further uncertainties could still be expected. To incorporate such uncertainties, we additionally applied $10\%$ of flux uncertainties for all measured fluxes from JWST. These uncertainties were added quadratically to the measured aperture flux errors. Additionally, we tested the effect of changing photometric zero points in the following discussions, including values derived from in-flight throughput in \citep{Rigby22} and the original pre-flight values. \subsubsection{RS14} Within the extremely red spiral galaxies, RS14 has already been studied in several papers. In particular, \citet{Sun22} reported ALMA continuum detection in $1.1\,{\rm mm}$ image obtained among the ALMA large program ALCS: 2018.1.00035.L (PI: K. Kohno). Most recently, \citet{Cheng22} presented a joint JWST-ALMA study of this source. Also, \citet{Carnall22} reported spectroscopic redshift of $z_{\rm spec}=2.463$ determined by multiple emission lines such as H$\alpha$, [NII], and [SII] emission lines (ID 9239 of NIRSpec spectroscopy as part of the same JWST ERO). In the following analysis, we used the ALMA continuum flux and the NIRSpec redshift for RS14. \subsection{Lensing Magnification} We estimated gravitational lensing magnification based on the recent work by \citet{Golubchik22}. Their lens model is constructed with \texttt{Light-Traces-Mass} method \citep[e.g.,][]{Broadhurst05}, which infers the lens mass contributions from the cluster's light distributions, using HST and VLT MUSE data. We estimated lensing magnification factors by assuming redshifts from the following SED fittings. \subsection{Spectral Energy Distribution fittings} \label{sec:SEDfit_results} To constrain the physical properties and redshift of RS13, one of our extremely red spiral galaxies, we used SED fitting code \texttt{PANHIT} \citep{Mawatari20}. We assumed a Chabrier initial mass function in a range of 0.1--100 M$_\odot$ \citep{2003PASP..115..763C}, the BC03 stellar population model \citep{2003MNRAS.344.1000B}, a nebular continuum and line emission model \citep{2011MNRAS.415.2920I}, and a standard dust attenuation curve \citep{Calzetti00}. The star formation history was assumed to be the ``delayed-$\tau$'' model \citep{2014ApJS..214...15S} to describe a variety of star formation histories, and the star-formation time-scale $\tau_{\rm SF}$ is a free parameter between 0.01 and 10 Gyr. We allowed for metallicities ranging from $0.005\times{Z_{\odot}}$ to $2.5\times{Z_{\odot}}$, where the Solar metallicity of $Z_\odot=0.02$. The dust attenuation in the $V$-band ($A_{V}$) is also a free parameter. The age of the stellar population is another free parameter between 1 Myr and 15 Gyr, and the cosmic age limits it at the redshift of interest. For RS13, the redshift is another free parameter and we examined redshifts from 0.1 to 13.0 with a step of 0.1. The stellar mass and star formation rate (SFR) were determined by scaling the amplitude of the SED. The SED fits performed $\chi^{2}$ minimization algorithm for the photometry data, including non-detection bands \citep[i.e., upper limits;][]{Sawicki12}. Additionally, to test robustness of our SED fittings, we also used SMC dust extinction curve\citep{Gordon2003} as well as assuming constant and then truncate star formation history. For RS14, we run the fitting with the spectroscopic redshift of $z=2.463$ \citep[][]{Carnall22} and including the ALMA continuum flux \citep{Cheng22}. {\tt PANHIT} assumes energy conservation between the stellar radiation absorbed by dust and the far-infrared (FIR) emission from dust. We assumed modified black-body functions with an emissivity index $\beta=2$ and the dust temperature, $T_{\rm d}$, between 10 and 50 K with a step of 5 K. Note that the cosmic microwave background temperature at $z=2.463$ is 9.5 K, a strict lower limit of $T_{\rm d}$. \section{Results and Discussion} \label{sec:results} \subsection{SED fitting reuslts} We found that the redshifts of extremely red spirals are in a range of $z_{\rm ph}=1$--3. Especially, RS14 ($z_{\rm spec}=2.463$) is one of the highest redshift spiral galaxies identified so far \citep[][]{2003AJ....125.1236D,2012Natur.487..338L,2022MNRAS.511.1502M,Wu2022}. Our SED fits showed that two types of integrated properties exist for the extremely red spiral galaxies at $1<z<3$: (1) Old stellar population that is consistent with almost completely passive, non star-forming galaxies, or (2) Dust-obscured highly elevated star-formation activities. In our SED fits, these properties are indicated from the JWST's coverage of $\lambda_{\rm rest}\sim4300 - 12800\,\text{\AA}$ photometry. The red colors of these bands can be explained by old stellar populations with moderate dust attenuation, or dust obscured star-formation activity. Especially, the possibility of a population of red and dead spiral galaxies at $z>1$ in our extremely red spiral sample is very interesting, as they are very rare in the local Universe \citep{2022PASJ...74..612S}. We note that our SED fit results could have some caveats, such as uncertain calibration of JWST photometry, which would change observed colors of these galaxies \citep[see also][for different photometric zero-point corrections]{Morishita22}. Also, the physical properties of these red spiral galaxies are not well studied yet. Thus, other assumptions, that are not included in our SED fits, could be allowed; such as extremely steeper dust attenuation curve. However, we test several reasonable cases to incorporate these possible caveats. As a result, we always find that the extremely red spiral galaxies show photometric redshift $z\sim1-3$ and have passive stellar population or dust obscured star formation activities. In particular, RS14 always show passive stellar population even if we use several photometric zero points or assumptions for SED fits. Table~\ref{tab:sedresults} is a summary of the fitting results as well as the lensing magnification factors. Following, we describe results of our SED fitting in detail: \renewcommand{\arraystretch}{0.85} \begin{table}[h] \centering \caption{A summary of SED fitting results and lensing magnification factors.} \label{tab:sedresults} \begin{tabular}{ccccc} \hline RS13 & best fit\\ \hline $z_{\rm ph}$ & $2.2\pm0.1$ & $2.8^{+0.1}_{-0.2}$ \\ $\chi^2_\nu$ & 1.25 & 1.54 \\ $\log_{10}(M_* \times \mu ~[{\rm M_\odot}])$ & $9.95^{+0.07}_{-0.01}$ & $10.45_{-0.01}^{+0.04}$\\ $SFR \times \mu$ (M$_\odot$ yr$^{-1}$) & $450\pm170$ & $0.17_{-0.17}^{+0.01}$\\ $A_V$ (mag) & $3.07_{-0.07}^{+0.06}$ & $0.59_{-0.20}^{+0.13}$\\ Age (Gyr) & $0.020_{-0.002}^{+0.025}$ & $0.36_{-0.09}^{+0.13}$ \\ $\tau_{\rm SF}$ (Gyr) & 0.01--10 & $0.03_{-0.02}^{+0.01}$ \\ $Z$ & 0.03--0.06 & 0.03--0.05 \\ $\mu$ & $1.67\pm0.02$ & $1.75\pm0.03$ \\ \hline RS14\\ \hline $z_{\rm sp}$ & 2.463 & \\ $\chi^2_\nu$ & 3.84 \\ $\log_{10}(M_* \times \mu ~[{\rm M_\odot}])$ & $11.43\pm0.03$\\ $SFR \times \mu$ (M$_\odot$ yr$^{-1}$) & 0.00 \\ $A_V$ (mag) & $0.84_{-0.03}^{+0.12}$ \\ Age (Gyr) & $2.4_{-0.2}^{+0.1}$ \\ $\tau_{\rm SF}$ (Gyr) & 0.03--0.1 \\ $Z$ & 0.007--0.012 \\ $T_{\rm dust}$ (K) & $\sim15$ \\ $\mu$ & $2.16\pm0.06$ & \\ \hline \end{tabular} \end{table} \subsubsection{RS14} We found that the $T_{\rm d}=15$ K case delivered the minimum $\chi^2$ value compared to other $T_{\rm d}$ cases. Higher $T_{\rm d}$ with fixing 1.1 mm flux leads to larger IR luminosity, and thus, larger dust attenuation; however, it makes the rest-frame optical SED traced by NIRCam too red. As such, the $T_{\rm d}=15$ K, rather low temperature \citep[e.g.,][]{2018A&A...609A..30S}, seems the best representation for RS14, that is shown in Fig.~\ref{fig:SED_RS14}. The best-fit model for RS14 is passive, old stellar population, and moderate dust attenuation ($A_{\rm V}=0.84^{+0.12}_{-0.03}$). The age is much longer than the star-formation time-scale, $\tau_{\rm SF}\lesssim0.1\,{\rm Gyr}$, and the current SFR is almost zero. This feature support the very low $T_{\rm d}$ ($15\,{\rm K}$) that is heated by old stars rather than massive young stars. The stellar mass is relatively high, $1.0\times10^{11}\,{\rm M_{\odot}}$ after correcting a magnification of $\mu=2.16$ at $z_{\rm spec}=2.463$. The best-fit passive solution of the RS14 does not change if we use different photometric zero point, including pre-flight value, or if we use different assumptions for the SED fitting. In particular, we also applied a steeper dust attenuation (i.e., the one from SMC dust extinction curve; \citealt{Gordon2003}). However the results do not change. While the galaxy-wide integrated property of RS14 is consistent with non-star-forming passive galaxy, JWST's F090W image (the rest-frame wavelength of $\sim2600\,\text{\AA}$) shows a few faint clumps in the bulge and some parts of the spiral arm region of RS14, which is smoothed out by our photometry using a large aperture. As these clumps can be seen in the rest-frame UV wavelength, a small amount of star-formation activity is still on-going in RS14. This is also consistent with the H$\alpha$ emission line detection \citep{Carnall22}. Hence, RS14 may have a small but substantial star-formation activity. \subsubsection{RS13} For RS13, we found a global $\chi^{2}$ minimum in the redshift range of $1\lesssim z\lesssim3$ (lower right panel of Fig. \ref{fig:SED_RS1213}), showing two local minima at $z=2.2$ and $z=2.8$. The best redshift solution of RS13 is $z_{\rm ph}=2.2\pm0.1$, having a large dust reddening of $A_{\rm V}=3.07^{+0.06}_{-0.07}$ with a high SFR of $\sim450\,\rm{M_{\odot}\,yr^{-1}}$. This solution explains some of the $>1\,\rm{\mu m}$ fluxes seen in NIRCam bands are produced by strong optical-to-NIR nebular lines, instead of the stellar continuum. With the young age of $0.02\,\rm{Gyr}$, this solution suggests that RS13 is a dusty starburst spiral galaxy. Another redshift solution is $z_{\rm ph}=2.8^{+0.1}_{-0.2}$. For this solution, RS13 is, similar to RS14, a passive galaxy (SFR consistent with zero) with the stellar age of $0.4\,{\rm Gyr}$, showing the clear Balmer break around F140W and F150W filter wavelengths. The stellar mass is $\sim2\times10^{10}\,{\rm M_{\odot}}$ after correcting a magnification. Although these two solutions show largely different properties for RS13, both solution shows that the redshift of RS13 to be $1<z<3$. This general results do not change if we use different photometric zero points or different assumptions for SED fittings. Further disentangling these two possible properties (i.e., a dusty starburst spiral or a passive spiral) requires spectroscopic redshift. \section{Conclusion} In this paper, we studied the properties of two extremely red spiral galaxies (RS13 and RS14) found from the JWST ERO image data release of SMACS 0723. These extremely red spiral galaxies are among a sample of red spiral galaxies visually selected from the ERO data. By performing SED fitting to these extremely red spiral galaxies, we found following results: \noindent$\bullet$ Most likely redshifts of the extremely red spirals are $1 < z < 3$, i.e., in the Cosmic Noon. The results show that these red spiral galaxies in JWST images contain the most distant spiral galaxies known to date. Within them, RS14 is currently one of the most distant stellar-spiral galaxy, having $z_{\rm spec}=2.463$. \noindent$\bullet$ Our SED fits suggest that the extremely red spiral galaxies are passive galaxies or heavily dust-obscured galaxies. These properties are indicated from the red colors of the rest-frame wavelength of $\sim4300 - 12800\,\text{\AA}$ in the JWST photometry. \noindent$\bullet$ One of the extremely red spiral galaxies, RS14, is found to be passive spiral galaxy. Finding passive spiral galaxies in the early Universe is surprising, as most of the spiral galaxies found in the local Universe is young star forming galaxies. Further detailed studies would be required to understand the formation mechanisms and evolutionary path of the red spiral galaxies. Follow-up observations using integral field spectroscopy, including high-resolution ALMA observations, will provide kinematics, molecular gas, and dust distribution of high-redshift spiral galaxies. \begin{acknowledgments} We thank Fengwu Sun, Ryosuke Uematsu, and Marc Postman for very helpful discussions. Y.F., A.K.I., and Y.S. acknowledge support from NAOJ ALMA Scientific Research Grant number 2020-16B. The Early Release Observations and associated materials were developed, executed, and compiled by the ERO production team: Hannah Braun, Claire Blome, Matthew Brown, Margaret Carruthers, Dan Coe, Joseph DePasquale, Nestor Espinoza, Macarena Garcia Marin, Karl Gordon, Alaina Henry, Leah Hustak, Andi James, Ann Jenkins, Anton Koekemoer, Stephanie LaMassa, David Law, Alexandra Lockwood, Amaya Moro-Martin, Susan Mullally, Alyssa Pagan, Dani Player, Klaus Pontoppidan, Charles Proffitt, Christine Pulliam, Leah Ramsay, Swara Ravindranath, Neill Reid, Massimo Robberto, Elena Sabbi, Leonardo Ubeda. The EROs were also made possible by the foundational efforts and support from the JWST instruments, STScI planning and scheduling, and Data Management teams. \end{acknowledgments} \vspace{5mm} \facilities{HST, JWST (STIS)} \software{astropy \citep{astropy:2013,astropy:2018}, photoutils \citep{larry_bradley_2020_4044744}, PANHIT \citep{Mawatari20} } \bibliography{RedSpirals}{} \bibliographystyle{aasjournal}

Title: Non-Gaussianity effects on the primordial black hole abundance for sharply-peaked primordial spectrum

Abstract: We perturbatively study the effect of non-Gaussianities on the mass fraction of primordial black holes (PBHs) at the time of formation by systematically taking its effect into account in the one-point probability distribution function of the primordial curvature perturbation. We focus on the bispectrum and trispectrum and derive formulas that describe their effects on the skewness and kurtosis of the distribution function. Then considering the case of narrowly peaked spectra, we obtain simple formulas that concisely express the effect of the bi- and trispectra. In particular, together with the $g_{\rm NL}$ and $\tau_{\rm NL}$ parameters of the trispectrum, we find that non-Gaussianity parameters for various types of the bispectrum are linearly combined to give an effective parameter, $f_{\rm NL}^{\rm eff}$, that determines the PBH mass fraction in the narrow spectral shape limit.

https://export.arxiv.org/pdf/2208.02941

\flushbottom \section{Introduction} The possibility that black holes may be formed in the very early universe was suggested about half a century ago \cite{Hawking:1971ei,Carr:1974nx,Carr:1975qj}. Since then it has been a topic of constant interest in cosmology, but it has never been explored in depth. However, thanks to the rapidly growing interest in gravitational wave cosmology in recent years, as well as to the theoretical progress and technical developments, the primordial black holes (PBHs) have become one of the hottest topics in cosmology today (for recent reviews, see e.g., \cite{Carr:2020xqk,Carr:2021bzv}). One of the most studied PBH formation mechanisms is the collapse of a region with a sufficiently large curvature perturbation, presumably produced from inflation, during the radiation-dominated early universe. While such high peaks in the curvature perturbation hardly exist in the conventional models of inflation, since there are virtually no stringent observational constraint on cosmologically very small scales that are relevant for the PBH formation, various models that can produce sufficiently abundant PBHs have been discussed in the literature. The abundance of PBHs produced in the early Universe plays a crucial role in the present Universe. Depending on the typical mass of PBHs, various effects which are observationally detectable are anticipated if the abundance is sufficiently large. There is no compelling evidence for the existence of PBHs at any mass scale up to the present time. Hence, what we have so far is the upper limit in the PBH abundance in broad ranges of mass scale (For a recent review, see Ref.~\cite{Carr:2020gox}). If PBHs are formed from high peaks of the curvature perturbation, an upper bound on the PBH abundance places an upper bound on the amplitude of the primordial power spectrum. The observed amplitude of the power spectrum on scales of the cosmic microwave background (CMB) and the large-scale structure (LSS) is too small to produce PBHs. However, scales relevant to the PBH formation are typically much smaller than those scales. Thus, various mechanisms that give rise to the amplitude of the power spectrum on small scales much larger than that on the CMB scale have been proposed (see, e.g., Ref.~\cite{Green:2014faa}). In many of those models, however, the non-Gaussianities in the curvature perturbation are negligible or simply ignored. Nevertheless, even if they are small, they may considerably affect the PBH formation, as it is acutely sensitive to non-Gaussian features in the tail of the probability distribution function \cite{Byrnes:2012yx}. The effects of non-Gaussianity upon the abundance of PBHs have been investigated by many authors using various methods \cite{Bullock:1996at,PinaAvelino:2005rm,Seery:2006wk,Shandera:2012ke,Young:2013oia,Young:2015cyn,Atal:2019cdz,Yoo:2019pma,Kitajima:2021fpq,Taoso:2021uvl,Riccardi:2021rlf,Pi:2021dft,Young:2022phe,Kehagias:2019eil}. However, in most of the previous work, only the local-type of non-Gaussianity is considered. Even for the bispectrum, which is the lowest non-trivial order in perturbation where the non-Gaussianity appears, only a limited number of authors consider the other types, such as equilateral- and orthogonal-types of non-Gaussianity~\cite{Shandera:2012ke,Young:2015cyn}. The effect of the trispectrum of non-Gaussianity on the abundance have not been discussed much, while the clustering of PBHs in the presence of local-type trispectra are considered in, e.g., Refs.~\cite{Tada:2015noa,Suyama:2019cst,Matsubara:2019qzv}. Thus, it is desirable to understand the effect of various types of non-Gaussianity on the PBH formation more systematically. In this paper, we analytically study the effect of non-Gaussianity up through the trispectrum in a model-independent way as much as possible. To analytically address the problem, we use a set of approximations that are valid in many situations. In the presence of skewness and kurtosis in the primordial fluctuations with hierarchical orders, we derive a formula for the non-Gaussian corrections to the abundance of PBHs in a high-peaks limit, generalizing the known formula for the threshold. We also derive integral formulas to calculate the values of skewness and kurtosis in popular models of non-Gaussianity, including local-, equilateral-, folded- and orthogonal-type models for bispectrum, and generalized local-type models for trispectrum. These integral formulas are further reduced to asymptotic formulas of analytic forms by taking the limit of a sharply-peaked shape of the primordial spectrum. The derived formulas provide useful apparatus for predicting the abundance of PBHs in a variety of models. This paper is organized as follows: In section~\ref{sec:Abundance}, after the basic notations and definitions we use in this paper are given, high-peaks formulas for the PBH formation are introduced. In section~\ref{sec:Cumulants}, we derive integral formulas for skewness and kurtosis parameters for a class of non-Gaussian models mentioned above, for an arbitrary shape of the primordial spectrum. In section~\ref{sec:Narrow}, the derived integral formulas are reduced to analytic forms by taking the sharply-peaked limit of the spectrum. In section~\ref{sec:Numrical}, we numerically evaluate the behavior of derived formulas and demonstrate it in several cases. \section{\label{sec:Abundance} The abundance of PBHs from non-Gaussian initial conditions} \subsection{Definitions} First, we define the fundamental quantities used in this paper. The PBHs are considered to be formed from large positive perturbations of the 3-dimensional curvature $\delta R^{(3)}$ at an early stage of the Universe, well before the astrophysical structure formation takes place. The 3-dimensional curvature perturbation on comoving slices is characterized by the curvature perturbation in Fourier space, denoted by $\calR(\bm{k})$. In linear order, we have $\delta R^{(3)}=4(k^2/a^2)\calR$, where $a$ is the scale factor. Since the Einstein equations imply $\delta R^{(3)}\approx 6H^2\varDelta$ on or above Hubble scales at linear order, where $H=\dot{a}/a$ is the Hubble parameter and $\varDelta$ is the energy density contrast on comoving slices, large positive 3-dimensional curvature perturbations are equivalent to large positive energy density fluctuations. If its probability distribution is Gaussian, the statistical properties of the comoving curvature perturbation are completely characterized by the power spectrum $P_\calR(k)$. The power spectrum is the two-point correlation of the fluctuations in Fourier space, and is defined by \begin{equation} \langle\calR(\bm{k}_1)\calR(\bm{k}_2)\rangle_\mathrm{c} = (2\pi)^3 \delta_\mathrm{D}^3(\bm{k}_1 + \bm{k}_2) P_\calR(k_1), \label{eq:2-01a} \end{equation} where $\delta_\mathrm{D}^3(\bm{k})$ is the Dirac's delta function and $\langle\cdots\rangle_\mathrm{c}$ denotes the cumulants or connected part of the statistical average. For the two-point cumulant above, it can be replaced by a simple average, assuming the mean value of curvature fluctuations is zero, $\langle\calR\rangle = 0$. The presence of non-Gaussianity gives rise to higher-order correlations, such as the bispectrum, trispectrum, and so forth. The bispectrum $B_\calR(\bm{k}_1,\bm{k}_2,\bm{k}_3)$ and the trispectrum $T_\calR(\bm{k}_1,\bm{k}_2,\bm{k}_3,\bm{k}_4)$ are three- and four-point functions in Fourier space, defined by \begin{align} \langle\calR(\bm{k}_1)\calR(\bm{k}_2)\calR(\bm{k}_3)\rangle_\mathrm{c} &= (2\pi)^3 \delta_\mathrm{D}^3(\bm{k}_1 + \bm{k}_2 + \bm{k}_3) B_\calR(\bm{k}_1,\bm{k}_2,\bm{k}_3), \label{eq:2-01b}\\ \langle\calR(\bm{k}_1)\calR(\bm{k}_2)\calR(\bm{k}_3) \calR(\bm{k}_4)\rangle_\mathrm{c} &= (2\pi)^3 \delta_\mathrm{D}^3(\bm{k}_1 + \bm{k}_2 + \bm{k}_3 + \bm{k}_4) T_\calR(\bm{k}_1,\bm{k}_2,\bm{k}_3,\bm{k}_4). \label{eq:2-01c} \end{align} The three-point cumulant in Eq.~(\ref{eq:2-01b}) again can be replaced by simple average because the mean value is zero. However, the cumulant in Eq.~(\ref{eq:2-01c}) cannot be replaced by simple average, and we have $\langle\calR_1\calR_2\calR_3\calR_4\rangle_\mathrm{c} = \langle\calR_1\calR_2\calR_3\calR_4\rangle - \langle\calR_1\calR_2\rangle\langle \calR_3\calR_4\rangle - \langle\calR_1\calR_3\rangle\langle \calR_2\calR_4\rangle - \langle\calR_1\calR_4\rangle\langle \calR_2\calR_3\rangle$. The appearance of Dirac's delta functions in the above definition is due to the assumed statistical homogeneity in 3-space with translational symmetry. In general, in linear order, the relation between the comoving curvature perturbation $\calR$ and the density contrast $\varDelta$ on comoving slices is given by \cite{Liddle:2000cg,Green:2004wb,Young:2014ana} \begin{equation} \label{eq:2-02} \varDelta(\bm{k};t) = \mathcal{M}(k) \calR(\bm{k})\,, \end{equation} where \begin{equation} \label{eq:2-03} \mathcal{M}(k;t) = \frac{2+2w}{5+3w} \left(\frac{k}{aH}\right)^2, \end{equation} and $w=p/\rho$ is the equation of state parameter. The PBH formation criteria are typically described by smoothed density fields with a smoothing radius of the horizon scale, $R=(aH)^{-1}$. In the following, we assume the formation of PBHs takes place in a radiation-dominated epoch, $w=1/3$, and thus we use \begin{equation} \mathcal{M}(k) = \frac{4}{9} k^2R^2\,, \label{eq:2-04} \end{equation} for the coefficient of Eq.~(\ref{eq:2-02}). The smoothed density field in configuration space is given by \begin{equation} \label{eq:2-05} \varDelta_R(\bm{x}) = \int \frac{d^3k}{(2\pi)^3} e^{i\bm{k}\cdot\bm{x}} \varDelta(\bm{k}) W(kR) \end{equation} where $W(kR)$ is the window function for the smoothing. In this paper, we adopt the Gaussian window function, \begin{equation} \label{eq:2-06} W(kR) = \exp\left(-\frac{k^2R^2}{2}\right)\,. \end{equation} \subsection{The abundance of PBHs} The abundance of PBHs is frequently modeled by the initial mass fraction $\beta$ of the universe that turns into PBHs at the time of formation. One of the simplest and most commonly used criteria for the PBH formation is to set a threshold density contrast $\varDelta_\mathrm{c}$. Then $\beta$ may be estimated by computing the fraction of space with $\varDelta_R(\bm{x}) > \varDelta_\mathrm{c}$ in the initial density field. This gives \begin{equation} \beta_\mathrm{th} = \int_{\varDelta_\mathrm{c}}^\infty P(\varDelta_R;R)\,d\varDelta_R, \label{eq:3-01} \end{equation} where $P(\varDelta_R;R)$ is the probability distribution function of the initial density field smoothed over a horizon scale $R$. In the analogy to the Press-Schechter theory of structure formation \cite{Press:1973iz}, a fudge factor 2 is sometimes put in front of the right-hand side (RHS) of the above equation. Our discussion below does not depend on whether this fudge factor is present or not. Another criteria of the PBH formation is to use the peak theory \cite{Bardeen:1985tr,Green:2004wb}. Given the number density of peaks $n_\mathrm{pk}(\varDelta_\mathrm{c};R)$ of the smoothed density field above the threshold, $\varDelta_\mathrm{c}$, the mass fraction is given by \begin{equation} \beta_\mathrm{pk} = (2\pi)^{3/2}R^3 n_\mathrm{pk}(\varDelta_\mathrm{c};R), \label{eq:3-02} \end{equation} where the prefactor $(2\pi)^{3/2} R^3$ corresponds to the effective volume of the Gaussian filter. There is yet another formation for the threshold based on the so-called Compaction function \cite{Musco:2020jjb,Musco:2018rwt,Kehagias:2019eil}. But as its relation to the probability distribution function seems rather non-trivial, we leave this case for future studies. When the initial density field is given by a random Gaussian field, the one-point probability distribution function is given by $P = (2\pi {\sigma}^2)^{-1/2}e^{-{\varDelta_R}^2/(2{\sigma}^2)}$, where ${\sigma}^2 = \langle{\varDelta_R}^2\rangle$ is the variance of the smoothed density field. In this case, Eq.~(\ref{eq:3-01}) is given by \begin{equation} \beta_\mathrm{th}^\mathrm{G} = \frac{1}{2} \mathrm{erfc}\left(\frac{\nu}{\sqrt{2}}\right) \approx \frac{1}{\sqrt{2\pi}} \frac{e^{-\nu^2/2}}{\nu}, \label{eq:3-03} \end{equation} where $\nu \equiv \varDelta_\mathrm{c}/\sigma$ is the normalized threshold. The last expression of the above equation is an asymptotic form for a high threshold of $\nu\gg 1$. The number density of peaks in a random Gaussian field is calculated from the joint probability distribution of field derivatives and is analytically given by an integral form \cite{Bardeen:1985tr}. An asymptotic form of the result for high peaks ($\nu\gg 1$) is given by \begin{equation} n_\mathrm{pk} = \frac{1}{(2\pi)^2} \left(\frac{{\sigma_1}^2}{3{\sigma}^2}\right)^{3/2} (\nu^2-1) e^{-\nu^2/2}, \label{eq:3-04} \end{equation} where \begin{equation} {\sigma_j}^2 = \int \frac{k^2dk}{2\pi^2} k^{2j}P_\varDelta(k) W^2(kR) \label{eq:3-05} \end{equation} and $P_\varDelta(k)$ is the power spectrum of the density field $\varDelta$. The corresponding expression for the mass fraction is given by \begin{equation} \beta_\mathrm{pk}^\mathrm{G} = \frac{1}{\sqrt{2\pi}} \left(\frac{R\sigma_1}{\sqrt{3}\sigma}\right)^3 (\nu^2-1) e^{-\nu^2/2}. \label{eq:3-06} \end{equation} There are differences in the predictions of the two different models, Eqs.~(\ref{eq:3-03}) and (\ref{eq:3-06}) with the Gaussian initial conditions. However, the overall shape is relatively close to each other if we take a higher threshold value $\nu$ for the peak theory and the threshold theory for reasonable ranges in the PBH mass \cite{Green:2004wb,Young:2014ana}. \subsection{Non-Gaussian distributions} The one-point distribution function of non-Gaussian fields are characterized by higher-order cumulants $\langle {\varDelta_R}^n \rangle_\mathrm{c}$. We assume a hierarchical scaling of higher-order cumulants, $\langle {\varDelta_R}^n \rangle_\mathrm{c} \propto {\sigma}^{2n-2}$, and define reduced cumulants \begin{equation} S_n \equiv \frac{\langle {\varDelta_R}^n \rangle_\mathrm{c}} {{\sigma}^{2n-2}}, \label{eq:4-01} \end{equation} which are considered to be of order unity or less. In general, the probability distribution function in the integrand of Eq.~(\ref{eq:3-01}) can be expanded in an Edgeworth series, \begin{multline} P(\varDelta_R;R) = \frac{e^{-\nu^2/2}}{\sqrt{2\pi}\sigma} \Biggl\{ 1 + \frac{S_3}{3!} H_3(\nu) \sigma + \left[\frac{S_4}{4!} H_4(\nu) + \frac{1}{2!}\left(\frac{S_3}{3!}\right)^2 H_6(\nu) \right] \sigma^2 \\ + \left[\frac{S_5}{5!} H_4(\nu) + \frac{S_3S_4}{3!4!} H_7(\nu) + \frac{1}{3!}\left(\frac{S_3}{3!}\right)^3 H_9(\nu) \right] \sigma^3 + \cdots \Biggr\}, \label{eq:4-02} \end{multline} where $H_n(\nu) = e^{\nu^2/2}(-d/d\nu)^ne^{-\nu^2/2}$ is the Hermite polynomial. In practice, one can truncate the series in some order when the inequality $\sigma \ll \nu^{-3}$ is satisfied. The Edgeworth expansion has been used to investigate the abundance of PBHs \cite{Shandera:2012ke}. For high peaks, $\nu \gg 1$, the truncated Edgeworth expansion is only applicable for sufficiently small $\sigma$. Integrating the Edgeworth series above, the mass fraction in the threshold model, Eq.~(\ref{eq:3-01}) reduces to \begin{multline} \beta_\mathrm{th} = \beta_\mathrm{th}^\mathrm{G} + \frac{e^{-\nu^2/2}}{\sqrt{2\pi}} \Biggl\{ \frac{S_3}{3!} H_2(\nu) \sigma + \left[\frac{S_4}{4!} H_3(\nu) + \frac{1}{2!}\left(\frac{S_3}{3!}\right)^2 H_5(\nu) \right] \sigma^2 \\ + \left[\frac{S_5}{5!} H_3(\nu) + \frac{S_3S_4}{3!4!} H_6(\nu) + \frac{1}{3!}\left(\frac{S_3}{3!}\right)^3 H_8(\nu) \right] \sigma^3 + \cdots \Biggr\}. \label{eq:4-03} \end{multline} In the high-peaks limit, the mass fraction in the threshold model has been derived in the context of biased structure formation, and the result is given by \cite{Matarrese:1986et,Franciolini:2018vbk} \begin{equation} \beta_\mathrm{th} \approx \frac{1}{\sqrt{2\pi}} \frac{e^{-\nu^2/2}}{\nu} \exp\left( \sum_{n=3}^\infty \frac{\nu^n}{n!} \frac{\langle {\varDelta_R}^n\rangle}{\sigma^n} \right) = \frac{\nu^{-1}}{\sqrt{2\pi}} \exp\left[ -\frac{\nu^2}{2} \left(1- 2\sum_{n=3}^\infty \frac{{\varDelta_\mathrm{c}}^{n-2}}{n!}S_n\right) \right]. \label{eq:4-04} \end{equation} This result can also be obtained by summing up all the infinite series of the leading contributions in the high-peaks limit $H_n(\nu) \rightarrow \nu^n$ in the Eq.~(\ref{eq:4-03}). In this way, the abundance of PBHs for non-Gaussian initial conditions in the high-peaks limit is characterized by the series of reduced cumulants $S_n$. The above formula for the non-Gaussianity in the high-peaks limit can be generalized to the peaks model of Eq.~(\ref{eq:3-06}). The details of the derivation are given in Appendix~\ref{app:A01}. One finds that the non-Gaussian contributions are the same as those in the high-peaks limit of the threshold model. Namely, \begin{equation} \beta_\mathrm{pk} \approx \frac{1}{\sqrt{2\pi}} \left(\frac{R\sigma_1}{\sqrt{3}\sigma}\right)^3 (\nu^2-1) \exp\left[ -\frac{\nu^2}{2} \left(1- 2\sum_{n=3}^\infty \frac{{\varDelta_\mathrm{c}}^{n-2}}{n!}S_n\right) \right]. \label{eq:4-05} \end{equation} Therefore, irrespective of the formation models of PBHs, the effect of non-Gaussianity in the high-peaks limit may be expressed in the generic form, \begin{equation} \beta \approx \beta^\mathrm{G} \exp\left(\nu^2 \sum_{n=3}^\infty \frac{{\varDelta_\mathrm{c}}^{n-2}}{n!}S_n \right) = A(\nu) \exp\left[ -\frac{\nu^2}{2} \left(1- 2\sum_{n=3}^\infty \frac{{\varDelta_\mathrm{c}}^{n-2}}{n!}S_n\right) \right], \label{eq:4-06} \end{equation} where \begin{equation} \beta^\mathrm{G} = A(\nu) e^{-\nu^2/2} \label{eq:4-07} \end{equation} is the mass fraction for the Gaussian initial condition, and $A(\nu)$ is the prefactor that depends on the formation models of PBHs, i.e., \begin{equation} A(\nu) \equiv \frac{1}{\sqrt{2\pi}} \times \begin{cases} \displaystyle \nu^{-1} & \mathrm{(threshold \ model)}, \\ \displaystyle \left(\frac{R\sigma_1}{\sqrt{3}\sigma}\right)^3(\nu^2-1) & \mathrm{(peaks\ model)}. \end{cases} \label{eq:4-08} \end{equation} We note that the asymptotic formula~(\ref{eq:4-06}) is consistent only when \begin{equation} 2\sum_{n=3}^\infty \frac{{\varDelta_\mathrm{c}}^{n-2}}{n!}S_n < 1\,, \label{eq:4-09} \end{equation} otherwise, the mass fraction $\beta$ exceeds unity in the limit $\nu\gg1$. In this paper, we assume the above condition is satisfied. When the values of reduced cumulants $S_n$ are of order unity, the above condition is safely satisfied. For example, when all the reduced cumulants $S_n$ have the same value, the condition~(\ref{eq:4-09}) with $\varDelta_\mathrm{c} \simeq 1/3$ implies $S_n \lesssim 8$. Thus it is not too restrictive. On the other hand, if the condition~(\ref{eq:4-09}) is not met, the asymptotic formula~(\ref{eq:4-06}) breaks down. This is because the resummation of leading contributions in the expansion in Eq.~(\ref{eq:4-03}) for the threshold model is not justified as $S_n$ are no longer of $\mathcal{O}(1)$. The same applies to the resummation in the peaks model, Eq.~(\ref{eq:a-15}). For example, when higher-order cumulants of a non-Gaussian model satisfy $\langle{\varDelta_R}^n\rangle_\mathrm{c} \sim \mathcal{O}(\sigma^n)$ [instead of $ \sim \mathcal{O}(\sigma^{2n-2})$, c.f., Eq.~(\ref{eq:4-01})], the resummation does not work \cite{Riccardi:2021rlf}. Comparing Eqs.~(\ref{eq:4-06}) and (\ref{eq:4-07}), and noting $\nu = \varDelta_\mathrm{c}/\sigma$, we find that the effects of non-Gaussianity may be conveniently taken into account by replacing the threshold value for the PBH formation in the exponent of the Gaussian prediction, Eq.~(\ref{eq:4-07}), as \begin{equation} \varDelta_\mathrm{c} \rightarrow \varDelta_\mathrm{c}^\mathrm{eff} \equiv \varDelta_\mathrm{c} \sqrt{ 1- S }\,;\quad S\equiv2\sum_{n=3}^\infty \frac{{\varDelta_\mathrm{c}}^{n-2}}{n!}S_n\,, \label{eq:4-10} \end{equation} apart from the prefactor $A(\nu)$. When $S$ is positive, the tail of the distribution function increases, which reduces the effective threshold in comparison with the Gaussian prediction, and the number of PBHs increases. On the contrary, if $S$ is negative, the number of PBHs decreases. Thus, the expected number density of PBHs is exponentially sensitive to non-Gaussianity. We note that the effective threshold above, $\varDelta_\mathrm{c}^\mathrm{eff}$, does not apply to the prefactor $A(\nu)$. Hence the effects of non-Gaussianity are not completely degenerate with the Gaussian case, though the change in the exponent dominates the effect on the number density of PBHs. When the observational constraint is given by $\beta < \beta_0$, where $\beta_0\ll 1$ and thus $\ln(1/\beta_0)>0$, Eq.~(\ref{eq:4-06}) implies \begin{equation} \sigma^2 < \frac{{\varDelta_\mathrm{c}}^2}{2\ln(1/\beta_0)} \left( 1- S \right), \label{eq:4-20} \end{equation} where the logarithm of the prefactor $\ln A(\nu)$ is ignored, assuming $\ln(1/\beta_0) \gg \ln A(\nu)$. Therefore, an observational constraint on the upper limit of the amplitude of primordial spectrum $\sqrt{\mathcal{P}_\calR}$ is tighter for $S>0$. If we only keep the leading term in $S$, $S\propto S_3$, this is in qualitative agreement with the results in Refs.~\cite{Byrnes:2012yx,Young:2015cyn} at linear order in $S_3$ (or in $f_{\rm NL}$ as discussed in the next section). Nonlinear behaviors discussed in these references might be explained by the effect of ${f_\mathrm{NL}}^2$ in the variance, $\sigma^2 \sim {\sigma_\mathrm{G}^2} + \mathrm{(const.)}\times {f_\mathrm{NL}}^2{\sigma_\mathrm{G}^4}$. However, for a narrowly peaked power spectrum, the prefactor (const.) of this relation is suppressed by the width of the spectrum, $\epsilon=\Delta k/k_0\ll1$, where $k_0$ is the peak of the spectrum. As seen from Eq.~(\ref{eq:4-06}), the parameters of reduced cumulants, $S_n$, with arbitrary higher orders may equally contribute if there are of the same order of magnitude. In the following, however, we mainly focus on the effects of bispectrum and trispectrum of the primordial fluctuations, which are responsible to the skewness parameter $S_3$ and the kurtosis parameter $S_4$. When the higher-order cumulants $S_n$ with $n\geq 5$ are absent, Eq.~(\ref{eq:4-06}) reduces to \begin{equation} \beta \approx \beta^\mathrm{G} \exp\left[ \nu^2 \varDelta_\mathrm{c} \left(\frac{S_3}{6} + \frac{{\varDelta_\mathrm{c}}S_4}{24} \right) \right]. \label{eq:4-21} \end{equation} The condition of Eq.~(\ref{eq:4-09}) in this case is given by \begin{equation} S_3 + \frac{S_4}{12} \lesssim 9, \label{eq:4-22} \end{equation} for $\varDelta_\mathrm{c} \simeq 1/3$. The skewness and kurtosis parameters, $S_3$ and $S_4$, are determined by non-Gaussian initial conditions. In the next section, we derive useful formulas to calculate these parameters in typical models of primordial non-Gaussianity. \section{\label{sec:Cumulants} Skewness and kurtosis in models of primordial non-Gaussianity } \subsection{Definitions} The reduced cumulants of third and fourth orders, $S_3$ and $S_4$, are called skewness and kurtosis parameters, respectively. They are related to the bispectrum $B(\bm{k}_1,\bm{k}_2,\bm{k}_3)$ and the trispectrum $T(\bm{k}_1,\bm{k}_2,\bm{k}_3,\bm{k}_4)$ of density field $\varDelta$ by \begin{align} S_3 &= \frac{1}{\sigma^4} \int_{\bm{k}_{123}=\bm{0}} B(\bm{k}_1,\bm{k}_2,\bm{k}_3) W(k_1R)W(k_2R)W(k_3R), \label{eq:5-01a}\\ S_4 &= \frac{1}{\sigma^6} \int_{\bm{k}_{1234}=\bm{0}} T(\bm{k}_1,\bm{k}_2,\bm{k}_3,\bm{k}_4) W(k_1R)W(k_2R)W(k_3R)W(k_4R), \label{eq:5-01b} \end{align} where we use an abbreviated notation, $\bm{k}_{1\cdots n} \equiv \bm{k}_1 + \cdots + \bm{k}_n$, and \begin{equation} \int_{\bm{k}_{1\cdots n}=\bm{0}}\cdots \equiv \int \frac{d^3k_1}{(2\pi)^3}\cdots \frac{d^3k_n}{(2\pi)^3} (2\pi)^3 \delta_\mathrm{D}^3(\bm{k}_1 + \cdots + \bm{k}_n)\cdots. \label{eq:5-02} \end{equation} The bispectrum and trispectrum of the density field are related to those of the curvature perturbation by \begin{align} B(\bm{k}_1,\bm{k}_2,\bm{k}_3) &= \mathcal{M}(k_1) \mathcal{M}(k_2) \mathcal{M}(k_3) B_\calR(\bm{k}_1,\bm{k}_2,\bm{k}_3), \label{eq:5-03a}\\ T(\bm{k}_1,\bm{k}_2,\bm{k}_3,\bm{k}_4) &= \mathcal{M}(k_1) \mathcal{M}(k_2) \mathcal{M}(k_3) \mathcal{M}(k_4) T_\calR(\bm{k}_1,\bm{k}_2,\bm{k}_3,\bm{k}_4). \label{eq:5-03b} \end{align} Substituting Eqs.~(\ref{eq:2-04}), (\ref{eq:2-06}), (\ref{eq:5-03a}) and (\ref{eq:5-03b}) into Eqs.~(\ref{eq:5-01a}) and (\ref{eq:5-01b}), we have \begin{align} S_3 &= \frac{1}{\sigma^4} \left(\frac{4}{9}\right)^3 \int_{\bm{k}_{123}=\bm{0}} (k_1R)^2(k_2R)^2(k_3R)^2 e^{-({k_1}^2+{k_2}^2+{k_3}^2)R^2/2} B_\calR(\bm{k}_1,\bm{k}_2,\bm{k}_3), \label{eq:5-04a}\\ S_4 &= \frac{1}{\sigma^6} \left(\frac{4}{9}\right)^4 \int_{\bm{k}_{1234}=\bm{0}} (k_1R)^2(k_2R)^2(k_3R)^2(k_4R)^2 e^{-({k_1}^2+{k_2}^2+{k_3}^2+{k_4}^2)R^2/2} T_\calR(\bm{k}_1,\bm{k}_2,\bm{k}_3,\bm{k}_4). \label{eq:5-04b} \end{align} The variance $\sigma^2$ is the same as ${\sigma_0}^2$ defined by Eq.~(\ref{eq:3-05}). That is \begin{equation} \sigma^2 = \left(\frac{4}{9}\right)^2 \int \frac{k^2dk}{2\pi^2} (kR)^4 e^{-k^2R^2} P_\calR(k)\,, \label{eq:5-05} \end{equation} where $P_\calR(k)$ is the power spectrum of the comoving curvature perturbation. \subsection{Skewness} There are many models of primordial non-Gaussianity. One of the most commonly assumed model for the bispectrum is the local-type \cite{Gangui:1993tt,Verde:1999ij,Komatsu:2001rj}, which is given by \begin{equation} B_\calR(\bm{k}_1,\bm{k}_2,\bm{k}_3) = \frac{6}{5} f_\mathrm{NL} \left[ P_\calR(k_1) P_\calR(k_2) + \mathrm{cyc.} \right], \label{eq:5-14} \end{equation} where $f_\mathrm{NL}$ is a parameter of non-Gaussianity amplitude, $+\,\mathrm{cyc.}$ represents the two terms obtained by cyclic permutations of the preceding term with respect to $k_1$, $k_2$, $k_3$. Among other types of non-Gaussianity for the bispectrum, popular alternatives are the equilateral \cite{Creminelli:2005hu}, folded \cite{Meerburg:2009ys}, and orthogonal \cite{Senatore:2009gt} types. These may be constructed from the following elements of the bispectrum: \begin{equation} B^\mathrm{I}_{123} \equiv P_1 P_2 + \mathrm{cyc.},\quad B^\mathrm{II}_{123} \equiv (P_1 P_2 P_3)^{2/3},\quad B^\mathrm{III}_{123} \equiv {P_1}^{1/3} {P_2}^{2/3} P_3 + \mathrm{5\ perm.}, \label{eq:5-15} \end{equation} where we denote $P_1 = P_\calR(k_1)$, $P_2 = P_\calR(k_2)$, $P_3 = P_\calR(k_3)$ for simplicity, and $+\,\mathrm{5\ perm.}$ represents the 5 terms obtained by permutations of the preceding term. In terms of these elements, the bispectrum for each type of non-Gaussianity is given by \begin{align} B^\mathrm{loc}_\calR &=\frac{6}{5}f_\mathrm{NL}^\mathrm{loc}B^\mathrm{I}_{123}\,, \label{eq:5-16-0} \\ B^\mathrm{eql}_\calR &= \frac{18}{5} f_\mathrm{NL}^\mathrm{eql} \left( - B^\mathrm{I}_{123} - 2B^\mathrm{II}_{123} + B^\mathrm{III}_{123} \right), \label{eq:5-16a}\\ B^\mathrm{fol}_\calR &= \frac{18}{5} f_\mathrm{NL}^\mathrm{fol} \left( B^\mathrm{I}_{123} + 3B^\mathrm{II}_{123} - B^\mathrm{III}_{123} \right), \label{eq:5-16b}\\ B^\mathrm{ort}_\calR &= \frac{18}{5} f_\mathrm{NL}^\mathrm{ort} \left( - 3B^\mathrm{I}_{123} - 8B^\mathrm{II}_{123} + 3B^\mathrm{III}_{123} \right). \label{eq:5-16c} \end{align} We introduce the skewness parameter elements corresponding to the above three elements $B^A_{123}$ ($A={\rm I},{\rm II},{\rm III}$) of the bispectrum as \begin{equation} S^A_3 \equiv \frac{1}{\sigma^4} \left(\frac{4}{9}\right)^3 \int_{\bm{k}_{123}=\bm{0}} (k_1R)^2 (k_2R)^2 (k_3R)^2 W(k_1R)W(k_2R)W(k_3R)B^A_{123}\,. \label{eq:5-17} \end{equation} The skewness parameters for local, equilateral, folded, and orthogonal types, which we denote by $S^\mathrm{loc}_3$, $S^\mathrm{eql}_3$, $S^\mathrm{fol}_3$, $S^\mathrm{ort}_3$, respectively, are given by linear superpositions of $S^A_3$ ($A={\rm I},{\rm II},{\rm III}$) in exactly the same forms as Eqs.~(\ref{eq:5-16-0}) -- (\ref{eq:5-16c}), \begin{align} S^\mathrm{loc}_3 &= \frac{6}{5} f_\mathrm{NL}^\mathrm{loc} S^\mathrm{I}_3, \label{eq:5-18a}\\ S^\mathrm{eql}_3 &= \frac{18}{5} f_\mathrm{NL}^\mathrm{eql} \left( - S^\mathrm{I}_3 - 2S^\mathrm{II}_3 + S^\mathrm{III}_3 \right), \label{eq:5-18b}\\ S^\mathrm{fol}_3 &= \frac{18}{5} f_\mathrm{NL}^\mathrm{fol} \left( S^\mathrm{I}_3 + 3S^\mathrm{II}_3 - S^\mathrm{III}_3 \right), \label{eq:5-18c}\\ S^\mathrm{ort}_3 &= \frac{18}{5} f_\mathrm{NL}^\mathrm{ort} \left( - 3S^\mathrm{I}_3 - 8S^\mathrm{II}_3 + 3S^\mathrm{III}_3 \right). \label{eq:5-18d} \end{align} We note that, excluding the signs of $f_{\rm NL}^X$ ($X=\mathrm{loc}, \mathrm{eql}, \mathrm{fol}, \mathrm{ort}$) in the coefficients, $S^\mathrm{\rm loc}_3$ is positive definite, while the sign of the other two is indeterminate. Given the values of $f_\mathrm{NL}^{X}$, these skewness parameters are uniquely determined once the primordial power spectrum $P_\calR(k)$ is known. In the high-peaks limit, the effects of skewness in each non-Gaussian type on the abundance of PBHs are given by substituting the results into Eq.~(\ref{eq:4-21}). The skewness parameter elements are given by substituting Eqs.~(\ref{eq:5-15}) into Eq.~(\ref{eq:5-17}). To represent the results in a convenient, compact form, we introduce the dimensionless curvature perturbation power spectrum, \begin{equation} \mathcal{P}_\calR(k) \equiv \frac{k^3P_\calR(k)}{2\pi^2}\,. \label{eq:5-19} \end{equation} Changing the integration variables as $\bm{p}=\bm{k}_1R$, $\bm{q}=\bm{k}_2R$ and $r=|\bm{p}+\bm{q}|$, where the variable $r$ describes the angular degrees of freedom, $\mu = \bm{p}\cdot\bm{q}/(pq) = (r^2 - p^2 - q^2)/(2pq)$, some of the angular integrations may be analytically calculated to give \begin{align} S^\mathrm{I}_3 &= \frac{3}{\sigma^4} \left(\frac{4}{9}\right)^3 \int_0^\infty dp\,dq\,e^{-p^2-q^2} pq \left[(p^2+q^2+2)\frac{\sinh(pq)}{pq} - 2\cosh(pq)\right] \mathcal{P}_\calR\left(\frac{p}{R}\right) \mathcal{P}_\calR\left(\frac{q}{R}\right), \label{eq:5-20a}\\ S^\mathrm{II}_3 &= \frac{1}{\sigma^4} \left(\frac{4}{9}\right)^3 \int_0^\infty dp\,dq\,e^{-(p^2+q^2)/2} pq \left[ \mathcal{P}_\calR\left(\frac{p}{R}\right) \mathcal{P}_\calR\left(\frac{q}{R}\right) \right]^{2/3} \int_{|p-q|}^{p+q} \frac{dr}{2}\,r\,e^{-r^2/2} \left[ \mathcal{P}_\calR\left(\frac{r}{R}\right) \right]^{2/3}, \label{eq:5-20b}\\ S^\mathrm{III}_3 &= \frac{6}{\sigma^4} \left(\frac{4}{9}\right)^3 \int_0^\infty dp\,dq\,p^2q\,e^{-(p^2+q^2)/2} \left[\mathcal{P}_\calR\left(\frac{p}{R}\right)\right]^{1/3} \left[\mathcal{P}_\calR\left(\frac{q}{R}\right) \right]^{2/3} \int_{|p-q|}^{p+q} \frac{dr}{2}\,e^{-r^2/2}\, \mathcal{P}_\calR\left(\frac{r}{R}\right). \label{eq:5-20c} \end{align} Thus, all the skewness parameters in the three types of non-Gaussianity can be computed from the above equations for an arbitrary power spectrum $P_\calR(k)$. The variance $\sigma^2$ of Eq.~(\ref{eq:5-05}) is similarly given by a one-dimensional integral: \begin{equation} \sigma^2 = \left(\frac{4}{9}\right)^2 \int dp\,p^3 e^{-p^2} \mathcal{P}_\calR\left(\frac{p}{R}\right). \label{eq:5-21} \end{equation} \subsection{Kurtosis} In contrast to the case of the bispectrum, not so many variations in the types of the primordial trispectrum have been proposed. Lacking considerations on the general types of the trispectrum, here we focus on the most popular type, that is the generalized local-type \cite{Byrnes:2006vq}, \begin{equation} T_\calR(\bm{k}_1,\bm{k}_2,\bm{k}_3,\bm{k}_4) = \frac{54}{25} g_\mathrm{NL} \left[P_1 P_2 P_3 + \mathrm{3\ perm.}\right] + \tau_\mathrm{NL} \left[P_1 P_2 P_{23} + \mathrm{11\ perm.}\right], \label{eq:5-30} \end{equation} where $P_{23}=P_\calR(|\bm{k}_2+\bm{k}_3|)$, with $g_\mathrm{NL}$ and $\tau_\mathrm{NL}$ being the parameters. If the primordial perturbations emerge from the quantum fluctuations of a single scalar field, the last parameter is related to the parameter of the local-type bispectrum by $\tau_\mathrm{NL} = (36/25){f_\mathrm{NL}}^2$ \cite{Boubekeur:2005fj}. If multiple scalar fields are involved, there is an inequality, $\tau_\mathrm{NL} > (36/25){f_\mathrm{NL}}^2$ \cite{Suyama:2007bg}. In this model, we define elements of the trispectrum by \begin{equation} T^\mathrm{I}_{1234} \equiv P_1 P_2 P_3 + \mathrm{3\ perm.},\quad T^\mathrm{II}_{1234} \equiv P_1 P_2 P_{23} + \mathrm{11\ perm.}. \label{eq:5-31} \end{equation} The corresponding elements of kurtosis are given by \begin{equation} S^A_4 \equiv \frac{1}{\sigma^6} \left(\frac{4}{9}\right)^4 \int_{\bm{k}_{1234}=\bm{0}} (k_1R)^2 (k_2R)^2 (k_3R)^2 (k_4R)^2 W(k_1R)W(k_2R)W(k_3R)W(k_4R)T^A_{1234}, \label{eq:5-32} \end{equation} where $A=\mathrm{I}, \mathrm{II}$. We note that both elements are positive definite, as clear from their definitions. The resulting kurtosis for the local-type is given by \begin{equation} S_4 = \frac{54}{25} g_\mathrm{NL} S^\mathrm{I}_4 + \tau_\mathrm{NL} S^\mathrm{II}_4. \label{eq:5-33} \end{equation} Once the primordial power spectrum is given, one can evaluate the kurtosis. The effect on the abundance of PBHs in the high-peaks limit is given by substituting the result into Eq.~(\ref{eq:4-21}). The kurtosis parameter elements are given by substituting Eqs.~(\ref{eq:5-31}) into Eq.~(\ref{eq:5-32}). Changing integration variables as $\bm{p}=\bm{k}_1R$, $\bm{q}=\bm{k}_2R$, $\bm{r} =(\bm{k}_2+\bm{k}_3)R$, $\mu = \bm{p}\cdot\bm{r}/(pr)$, $\mu' = - \bm{q}\cdot\bm{r}/(qr)$, and expressing $\mu'$ in terms of $s=|\bm{q}-\bm{r}|$, some of the angular integrations can be analytically calculated (see Ref.~\cite{Matsubara:2020knr} for the same type of calculation). The results are \begin{align} S^\mathrm{I}_4 &= \frac{4}{\sigma^6} \left(\frac{4}{9}\right)^4 \int_0^\infty dp\,dq\,dr\, e^{-p^2}e^{-(q^2+r^2)/2} pr \nonumber\\ &\hspace{3.5pc} \times \left[(p^2 + r^2 + 2)\frac{\sinh(pr)}{pr} - 2\cosh(pr)\right] \mathcal{P}_\calR\left(\frac{p}{R}\right) \mathcal{P}_\calR\left(\frac{q}{R}\right) \int_{|q-r|}^{q+r}\frac{ds}{2} e^{-s^2/2} \mathcal{P}_\calR\left(\frac{s}{R}\right), \label{eq:5-34a}\\ S^\mathrm{II}_4 &= \frac{12}{\sigma^6} \left(\frac{4}{9}\right)^4 \int_0^\infty dp\,dq\,dr\,e^{-p^2-q^2-r^2}\frac{pq}{r} \left[(p^2+r^2+2)\frac{\sinh(pr)}{pr} - 2\cosh(pr)\right] \nonumber\\ &\hspace{9pc} \times \left[(q^2+r^2+2)\frac{\sinh(qr)}{qr} - 2\cosh(qr)\right] \mathcal{P}_\calR\left(\frac{p}{R}\right) \mathcal{P}_\calR\left(\frac{q}{R}\right) \mathcal{P}_\calR\left(\frac{r}{R}\right). \label{eq:5-34b} \end{align} The above formulas enable us to compute all the kurtosis parameters of the local type for an arbitrary primordial power spectrum $P_\calR(k)$. \section{\label{sec:Narrow} Power spectrum with a narrow peak} In the previous section, we derived formulas for the skewness and kurtosis that can be used to evaluate their effects on the abundance of PBHs for a general primordial power spectrum. In this section, we focus on the case of a narrowly peaked spectrum, which would lead to a nearly monochromatic mass function of PBHs. We consider the case that the primordial power spectrum peaked at a wavenumber $k_0$. When the sharpness of the peak is extreme, we can approximately substitute \begin{equation} f(k) \mathcal{P}_\calR(k) \rightarrow f(k_0) \mathcal{P}_\calR(k), \label{eq:6-01} \end{equation} where $f(k)$ is an arbitrary function of $k$. Applying this substitution, the variance of Eq.~(\ref{eq:5-21}) reduces to \begin{equation} \sigma^2 \simeq \left(\frac{4}{9}\right)^2 e^{-{k_0}^2R^2}(k_0R)^4 \int \frac{dk}{k}\, \mathcal{P}_\calR(k)\,, \label{eq:6-02} \end{equation} and the skewness parameter elements in Eqs.~(\ref{eq:5-20a}) -- (\ref{eq:5-20c}) reduce to \begin{align} S^\mathrm{I}_3 &\simeq \frac{27}{2}\, \frac{1}{(k_0R)^4} \left[({k_0}^2R^2+1) \frac{\sinh({k_0}^2R^2)}{{k_0}^2R^2} - \cosh({k_0}^2R^2)\right]\,, \label{eq:6-02a}\\ S^\mathrm{II}_3 &\simeq \frac{9}{8}\, \frac{e^{{k_0}^2R^2/2}}{{k_0}^2R^2} \frac{{C_{2/3}}^3}{{C_1}^2}, \label{eq:6-02b}\\ S^\mathrm{III}_3 &\simeq \frac{27}{4}\, \frac{e^{{k_0}^2R^2/2}}{{k_0}^2R^2} \frac{C_{1/3}C_{2/3}}{C_1}, \label{eq:6-02c} \end{align} where we have introduced \begin{equation} C_\alpha \equiv \int \frac{dk}{k} \left[\mathcal{P}_\calR(k)\right]^\alpha. \label{eq:6-03} \end{equation} The integral range of the above is localized in the vicinity of $k_0$. We note that $S^\mathrm{I}_3$ takes the minimum value $S^\mathrm{I}_{3}|_{\rm min} \approx4.93$ at $k_0R\approx1.7$. We also note that $S^\mathrm{II}_3$ and $S^\mathrm{III}_3$ depend on the detailed shape of the peak in the power spectrum through the integrals $C_\alpha$. Due to the assumption of a narrow peak, the values of integrals $C_\alpha$ are small. If we characterize the narrowness of the power spectrum by $\epsilon$ in the space of wavenumber, we have $C_\alpha \sim \mathcal{O}(\epsilon)$ for $\alpha > 0$. Thus the parameters $S^\mathrm{II}_3$ and $S^\mathrm{III}_3$ are of order $\sim \epsilon\,S^\mathrm{I}_3$, \begin{equation} S^\mathrm{I}_3 \gg S^\mathrm{II}_3, S^\mathrm{III}_3. \label{eq:6-04} \end{equation} This is a remarkable property of a narrow power spectrum that simplifies the bispectrum in Eqs.~(\ref{eq:5-16-0}) -- (\ref{eq:5-16c}). For example, if the narrow shape of the power spectrum is given by a rectangular function of width $k_0 \epsilon$, \begin{equation} \mathcal{P}_\calR(k) = \begin{cases} A_0, & \displaystyle \mathrm{if\ \ } 1 - \frac{\epsilon}{2} \leq \frac{k}{k_0} \leq 1 + \frac{\epsilon}{2}, \\ 0 & \mathrm{otherwise}, \end{cases} \label{eq:6-05} \end{equation} where $A_0$ is a normalization constant, then we have $C_\alpha = {A_0}^\alpha \epsilon$, which gives \begin{equation} S^\mathrm{II}_3 \simeq \frac{9\epsilon}{8} \frac{e^{{k_0}^2R^2/2}}{(k_0R)^2}, \qquad S^\mathrm{III}_3 \simeq \frac{27\epsilon}{4} \frac{e^{{k_0}^2R^2/2}}{(k_0R)^2}. \label{eq:6-06} \end{equation} If the narrow shape is given by a Gaussian function of the full width at half maximum (FWHM)\footnote{FWHM for a normal distribution of standard deviation $\sigma$ is given by $2\sqrt{2\ln 2}\,\sigma \simeq 2.35482 \sigma$.} $k_0\epsilon$, \begin{equation} \mathcal{P}_\calR(k) = A_0 \exp\left[-\frac{4\ln 2}{k_0^2\epsilon^2} (k-k_0)^2\right], \label{eq:6-07} \end{equation} then we have $C_\alpha = \sqrt{\pi/\ln 2}\,{\alpha}^{-1/2}{A_0}^\alpha\epsilon$, which gives \begin{equation} S^\mathrm{II}_3 \simeq \frac{81\pi^{1/2}\epsilon}{32\sqrt{6\ln 2}} \frac{e^{{k_0}^2R^2/2}}{(k_0R)^2}, \qquad S^\mathrm{III}_3 \simeq \frac{81\pi^{1/2}\epsilon}{8\sqrt{2\ln 2}} \frac{e^{{k_0}^2R^2/2}}{(k_0R)^2}. \label{eq:6-08} \end{equation} In any case, these parameters are much smaller than $S^\mathrm{I}_3$ by a factor of $\epsilon$. Due to this property, the skewness parameters in Eqs.~(\ref{eq:5-18a}) -- (\ref{eq:5-18d}) are solely determined by a single element $S^\mathrm{I}_3$. Taking account of contributions from all types, the skewness of the density field is given by \begin{equation} S_3 \simeq \frac{6}{5} \left( f_\mathrm{NL}^\mathrm{loc} - 3 f_\mathrm{NL}^\mathrm{eql} + 3 f_\mathrm{NL}^\mathrm{fol} -9 f_\mathrm{NL}^\mathrm{ort} \right) S^\mathrm{I}_3. \label{eq:6-10} \end{equation} Thus different types of non-Gaussianity give the same form of skewness with different amplitudes. This means that one cannot distinguish non-Gaussian types only with the amplitude of the skewness. Apart from the numerical coefficients and the signs in (\ref{eq:6-10}), this fact may be considered rather trivial because the abundance of PBHs is just a single number. Different non-Gaussian types cannot be distinguished by a single number. Similarly, the substitution of Eq.~(\ref{eq:6-01}) in Eqs.~(\ref{eq:5-34a}) and (\ref{eq:5-34b}), the kurtosis parameter elements are given by \begin{align} S^\mathrm{I}_4 &\simeq \frac{243}{8} \frac{1}{(k_0R)^8} \Biggl\{ - e^{-{k_0}^2R^2} \sinh\left(2{k_0}^2R^2\right) + \frac{e^{3{k_0}^2R^2/2}}{4k_0R} \sqrt{2\pi} \left[ 2 -3\, \mathrm{erfc} \left(\frac{k_0R}{\sqrt{2}}\right) + \mathrm{erfc} \left(\frac{3k_0R}{\sqrt{2}}\right) \right] \Biggr\}, \label{eq:6-11a}\\ S^\mathrm{II}_4 &\simeq \frac{243}{(k_0R)^8} \left[({k_0}^2R^2+1) \frac{\sinh({k_0}^2R^2)}{{k_0}^2R^2} - \cosh({k_0}^2R^2)\right]^2 \simeq \frac{4}{3} \left(S^\mathrm{I}_3\right)^2. \label{eq:6-11b} \end{align} The narrowness parameter $\epsilon$ discussed above cancels in this case, and therefore $S^\mathrm{I}_4$ and $S^\mathrm{II}_4$ are equally of order unity in $\epsilon$. Hence both parameters do not depend on the detailed shape of the peak, as in the case of $S^\mathrm{I}_3$. Taking into account all the leading contributions to the skewness and kurtosis, the mass fraction with non-Gaussianity given by Eq.~(\ref{eq:4-21}) in the high peaks limit with a narrow spectrum may be expressed in a single formula, \begin{equation} \beta \simeq \beta^\mathrm{G} \exp\left\{ \nu^2 \varDelta_\mathrm{c} \left[ \frac{1}{5} f_\mathrm{NL}^\mathrm{eff} S^\mathrm{I}_3 + \frac{{\varDelta_\mathrm{c}}}{12} \left( \frac{27}{25} g_\mathrm{NL} S^\mathrm{I}_4 + \frac{2}{3} \tau_\mathrm{NL} (S^\mathrm{I}_3)^2 \right) \right] \right\}, \label{eq:6-12} \end{equation} where \begin{equation} f_\mathrm{NL}^\mathrm{eff} \equiv f_\mathrm{NL}^\mathrm{loc} - 3 f_\mathrm{NL}^\mathrm{eql} + 3 f_\mathrm{NL}^\mathrm{fol} -9 f_\mathrm{NL}^\mathrm{ort}\,, \label{eq:6-13} \end{equation} irrespectively of the detailed shape of the narrow peak. The evaluation of the above formula requires only two parameters of skewness and kurtosis, $S^\mathrm{I}_3$ and $S^\mathrm{I}_4$. For the Gaussian part of the mass fraction, $\beta^\mathrm{G}$, the peaks model, Eq.~(\ref{eq:4-08}), contains a factor $R\sigma_1/\sigma$. Taking the limit of a narrow spectrum in Eq.~(\ref{eq:3-05}), we obtain $\sigma_j = {k_0}^j \sigma$. Therefore, we have \begin{equation} \beta^\mathrm{G} = \frac{e^{-\nu^2/2}}{\sqrt{2\pi}} \times \begin{cases} \displaystyle \nu^{-1}, & \mathrm{(threshold \ model)}, \\ \displaystyle \left(\frac{k_0R}{\sqrt{3}}\right)^3(\nu^2-1), & \mathrm{(peaks\ model)}, \end{cases} \label{eq:6-14} \end{equation} in the limit of a narrow spectrum. \section{\label{sec:Numrical} Numerical demonstrations} In this section, to obtain an intuitive sense of our results, and to find the range of validity of the narrow-peak limit approximation of the spectrum, we numerically evaluate the formulas we derived in the previous section. In the left panel of Fig.~\ref{fig:01}, the skewness parameter elements, $S_3^\mathrm{I}$, $S_3^\mathrm{II}$ and $S_3^\mathrm{III}$ computed from Eqs.~(\ref{eq:6-02a}) -- (\ref{eq:6-02c}) in the narrow peak limit are shown as functions of $k_0R$. The element $S_3^\mathrm{I}$ does not depend on the precise shape of the spectrum in the narrow limit, while the other two elements, $S_3^\mathrm{II}$ and $S_3^\mathrm{III}$, depend on the shape of the narrow peak, hence the rectangular case, Eq.~(\ref{eq:6-05}) and the Gaussian case, Eq.~(\ref{eq:6-07}) are presented. The width of the peak characterized by the narrowness parameter $\epsilon$ is taken to be $\epsilon = 0.03$ in the plot just for illustration. We recall that the elements $S_3^\mathrm{II}$ and $S_3^\mathrm{III}$ are simply proportional to $\epsilon$. As seen from Fig.~\ref{fig:01}, $S_3^\mathrm{I}$ dominates the others when $\epsilon$ is small. For the value of $\epsilon = 0.03$, $S_3^\mathrm{I}$ is approximately an order of magnitude larger than the other two. In the right panel of Fig.~\ref{fig:01}, the kurtosis parameter elements, $S_4^\mathrm{I}$ and $S_4^\mathrm{II}$ computed by using Eqs.~(\ref{eq:5-34a}) and (\ref{eq:5-34b}) are shown in the narrow peak limit. Both of these two do not depend on the precise shape of the narrowness parameter $\epsilon$, and both contribute to the kurtosis parameter, irrespective of the narrowness of the spectrum. Since the values of the two elements $S_4^\mathrm{I}$ and $S_4^\mathrm{II}$ are roughly of $\mathcal{O}(10^1)$ - $\mathcal{O}(10^2)$, Eq.~(\ref{eq:4-22}) requires that the parameters $g_\mathrm{NL}$ and $\tau_\mathrm{NL}$ should be somewhat smaller than unity to satisfy the condition for the consistency of the approximation. In the left panel of Fig.~\ref{fig:02}, comparisons of the narrow limit approximation with the exact numerical results are made for the skewness parameter elements, $S_3^\mathrm{I}$, $S_3^\mathrm{II}$ and $S_3^\mathrm{III}$, and for the kurtosis parameter elements, $S_4^\mathrm{I}$ and $S_4^\mathrm{II}$. The right panel shows the comparison of the exact integrations of Eqs.~(\ref{eq:5-20a}) -- (\ref{eq:5-20c}) with the narrow limit approximation for the elements of skewness. The narrow limit of $S_3^\mathrm{I}$ is independent of $\epsilon$, while those of $S_3^\mathrm{II}$ and $S_3^\mathrm{III}$ are linearly proportional to $\epsilon$. The curved lines represent the exact numerical results, and they converge to the corresponding results in the narrow limit approximation as $\epsilon\to0$. As can be seen from the plot, the approximation of the narrow limit is accurate for $\epsilon \lesssim 0.3$. For larger values of $\epsilon$, the approximations are still fairly good, although the value of $S_3^\mathrm{III}$ is non-negligible in comparison to that of $S_3^\mathrm{I}$. In the right panel of Fig.~\ref{fig:02}, the kurtosis parameter elements are similarly compared. The narrow limits are given by the constant lines, while the exact numerical results are given by the curved lines. The narrow limit approximation is also valid for sufficiently small $\epsilon$, though the range of validity seems slightly smaller, $\epsilon\lesssim0.2$ in comparison with the case of the skewness. In Fig.~\ref{fig:03}, the same comparisons as Fig.~\ref{fig:02} are made, but for the Gaussian shape of the spectrum. The narrow limits of $S_3^\mathrm{I}$, $S_4^\mathrm{I}$ and $S_4^\mathrm{II}$ are the same as those in Fig.~\ref{fig:02} because they do not depend on the precise shape of the narrow spectrum. However, the narrow limits of the elements $S_3^\mathrm{II}$ and $S_3^\mathrm{III}$, as well as the exact numerical results of all the elements do depend on the precise shape of the spectrum. As seen from the figures, the narrow limit approximation is slightly worse than that in the rectangular case for the same value of $\epsilon$. The gradient of $S_3^\mathrm{III}$ as a function of $\epsilon$ is larger in the Gaussian case than in the rectangular case. This results in the fact that $S_3^\mathrm{III}$ is non-negligible in comparison with $S_3^\mathrm{I}$ already for mildly small values of $\epsilon$. In Fig.~\ref{fig:04}, we show the PBH mass fraction $\beta$ for the threshold model (left panel) and for the peaks model (right panel) in the narrow spectral shape limit, given by Eq.~(\ref{eq:6-12}). In this limit, the mass fraction does not depend on the precise shape of the spectrum, nor the precise value of the narrowness parameter. The adopted values of the parameters in the plot are $\varDelta_\mathrm{c}=1/3$, $f_\mathrm{NL}^\mathrm{eff} = g_\mathrm{NL} = \tau_\mathrm{NL} = 0.2$, and $k_0R = \sqrt{2}$. As can be seen in the figure, the PBH mass fraction is significantly enhanced for positive values of the non-Gaussian parameters $f_\mathrm{NL}^\mathrm{eff}$, $g_\mathrm{NL}$, and $\tau_\mathrm{NL}$. The peaks model is known to predict a larger mass fraction than the threshold model for Gaussian initial conditions \cite{Yoo:2018kvb}. In the narrow spectral shape limit, however, the non-Gaussian factor in the exponent of Eq.~(\ref{eq:6-12}) does not depend on whether the formation criterion is determined by the threshold or peaks model, because it contains only $S_3^\mathrm{I}$ and $S_4^\mathrm{I}$, given by Eqs.~(\ref{eq:6-02a}) and (\ref{eq:6-11a}), respectively, which are independent of formation criteria. Therefore, the enhancement factor for the non-Gaussian case in comparison with the Gaussian case is the same for both panels. \section{Conclusions} In this paper, we considered the effect of primordial non-Gaussianities on the abundance of PBHs in the conventional PBH formation scenario in which a PBH is formed from a rare, large positive curvature perturbation at a radiation-dominated stage in the early universe. We first presented a general series form of the one-point distribution function with arbitrary non-Gaussianities. Then we derived an asymptotic formula for the PBH mass fraction at the time of formation in the high-peaks limit. The asymptotic formula for the threshold model for the PBH formation has been known. In this paper, we specifically derived the asymptotic formula for the peaks model and found that the enhancement factor is identical in both models if the non-Gaussianities are the same. Next, we focused on the effect of skewness and kurtosis on the abundance for a few specific models of non-Gaussianity. The skewness and kurtosis are calculated from the bispectrum and trispectrum. For the bispectrum, we considered non-Gaussian models which are described by superpositions of specific forms of the bispectrum given by Eq.~(\ref{eq:5-15}), which include popular models such as the local-type, equilateral-type, folded-type and orthogonal-type non-Gaussianities. The integral formulas to calculate the skewness parameter in this series of models were derived in Eq.~(\ref{eq:5-20a}) -- (\ref{eq:5-21}). For the trispectrum, we considered a local-type model given by Eq.~(\ref{eq:5-30}). The integral formulas of kurtosis parameter were derived in Eqs.~(\ref{eq:5-34a}) and (\ref{eq:5-34b}). The integral formulas to obtain skewness and kurtosis parameters are numerically not so difficult to evaluate. However, to obtain a clear, intuitive understanding of the results, we considered the case when the primordial curvature perturbation power spectrum is sharply peaked at a scale $k_0$ with the width $\Delta k=\epsilon k_0$ where $\epsilon$ ($<1$) is the narrowness parameter. As a result, the precise shape of the spectrum gives only subdominant contributions to the resulting skewness and kurtosis parameters, as shown in Eqs.~(\ref{eq:6-02a}), (\ref{eq:6-04}), (\ref{eq:6-11a}) and (\ref{eq:6-11b}). In the limit $\epsilon\to0$, we obtained an asymptotic formula for the PBH mass fraction, Eq.~(\ref{eq:6-12}) for various types of non-Gaussianities of the bispectrum and trispectrum. In particular, we found that the non-Gaussianity parameters for various types of the bispectrum are linearly combined to give an effective $f_{\rm NL}^{\rm eff}$ that determines the PBH mass fraction. In this paper, we simply assume a linear relation between the curvature perturbation and the density contrast in Eq.~(\ref{eq:2-02}), and also assume formation models of PBHs with threshold and peaks criteria with a fixed value of the threshold. These assumptions are employed in order to make it possible for us to analytically treat the problem. It is not obvious that these simplistic assumptions hold in realistic situations \cite{Riccardi:2021rlf,Young:2022phe,Musco:2008hv}. However, since our non-Gaussian corrections in the high-peaks limit change the abundance only through a multiplicative factor, as given by Eq.~(\ref{eq:4-06}), whose form is common to the two different formation models, namely, the threshold model and the peaks model. Therefore, one may expect that this non-Gaussian factor in the high-peaks limit is universal, not depending on the details of the formation models that can be more complicated than those we assume in this paper. Unfortunately, however, it is difficult to prove this expectation within the context of this paper, as the derivation of our formula in the case of the peaks model, for instance, is already complicated enough. Therefore, we leave this issue for future studies. We systematically studied the effect of non-Gaussianities on the mass fraction of PBHs at the time of formation at the level of bispectrum and trispectrum. We only considered the one-point distribution function of the curvature perturbation, we have no clue about non-Gaussianity effects on the spatial distribution of PBHs. The effects of bispectrum and trispectrum on the PBH clustering are investigated and discussed in Refs.~\cite{Tada:2015noa,Suyama:2019cst,Matsubara:2019qzv}. It seems there has been no further systematic analysis of non-Gaussianity effects on the spatial distribution of PBHs in general situations. This issue certainly deserves further studies. \acknowledgments We thank Chris Byrnes for useful conversations at the early stage of this work. This work was supported by JSPS KAKENHI Grants Nos.~JP19K03835 (TM), 21H03403 (TM), 19H01895 (MS), 20H04727(MS), and 20H05853(MS). \appendix \section{High-peaks limit of the peaks model with non-Gaussianity \label{app:A01} } In this Appendix, we derive the formula of Eq.~(\ref{eq:4-05}) for the mass fraction in the high-peaks limit of the peaks model with primordial non-Gaussianity. We rely on the fact that the number of peaks above a threshold $\nu$ asymptotically approach to the Euler characteristic of the three-dimensional body of the regions where $\varDelta_R > \nu\sigma$ is satisfied. The expected value of Euler characteristic in non-Gaussian fields above a threshold $\nu$ is formally given by \cite{Matsubara:1994wn,Matsubara:1995wj,Matsubara:2020fet} \begin{equation} n_\chi(\nu) = \left\langle \exp\left( \sum_{n=3}^\infty \frac{1}{n!} \sum_{\mu_1,\cdots,\mu_n} M^{(n)}_{\mu_1\cdots\mu_n} \frac{\partial^n}{\partial A_{\mu_1}\cdots\partial A_{\mu_n}} \right) F_\chi(\bm{A},\nu) \right\rangle_\mathrm{G}. \label{eq:a-01} \end{equation} Various quantities in this equation are defined below in order. The 10-dimensional vector of variables $\bm{A}$ is composed of normalized variables, \begin{equation} \bm{A} = \left( \alpha, y_i, Z_{ij} \right), \label{eq:a-02} \end{equation} with $1\leq i \leq j \leq 3$, where \begin{equation} \alpha = \frac{\varDelta_R}{\sigma_0},\quad y_i = \sqrt{3}\,\frac{\partial_i\varDelta_R}{\sigma_1},\quad Z_{ij} = \frac{3}{\gamma} \frac{\partial_i\partial_j\varDelta_R}{\sigma_2} + \delta_{ij} \frac{\varDelta_R}{\sigma_0}, \label{eq:a-02-1} \end{equation} $\partial_i=\partial/\partial x_i$ is a coordinate derivative, and \begin{equation} \gamma = \frac{{\sigma_1}^2}{\sigma_0\sigma_2} \label{eq:a-03} \end{equation} is a spectral parameter, and $\sigma_j$ is defined by Eq.~(\ref{eq:3-05}). The $n$-point cumulants of the variables $\bm{A}$ are denoted by \begin{equation} M^{(n)}_{\mu_1\cdots\mu_n} = \left\langle A_{\mu_1}\cdots A_{\mu_n} \right\rangle_\mathrm{c}. \label{eq:a-04} \end{equation} A local function for the number density of the Euler characteristic is given by \begin{equation} F_\chi(\bm{A},\nu) = \left(\frac{\sigma_1}{\sqrt{3}\sigma_0}\right)^3 \Theta(\alpha-\nu) \delta_\mathrm{D}^3(\bm{y}) \det(\alpha I - Z), \label{eq:a-05} \end{equation} where $\Theta(x)$ is a step function and $\delta_\mathrm{D}^3(\bm{y})$ is the 3-dimensional Dirac's delta function, and $I$ is the $3\times 3$ unit matrix. Finally, we define a Gaussian average by \begin{equation} \langle\cdots\rangle_\mathrm{G} = \frac{1}{(2\pi)^5\sqrt{\det M}} \int d^{10}A \cdots \exp\left(-\frac{1}{2}\bm{A}^\mathrm{T}{M}^{-1}\bm{A}\right), \label{eq:a-06} \end{equation} where $M = (M^{(2)}_{\mu_1\mu_2})$ is a $10\times 10$ matrix of two-point cumulants. The two-point cumulants are explicitly given by \begin{align} & \left\langle \alpha^2 \right\rangle = 1, \quad \left\langle \alpha \eta_i \right\rangle = 0, \quad \left\langle \alpha Z_{ij} \right\rangle = 0, \quad \left\langle y_i y_j \right\rangle = \delta_{ij}, \quad \left\langle \eta_i Z_{jk} \right\rangle = 0, \nonumber\\ & \left\langle Z_{ij} Z_{kl} \right\rangle = \frac{3}{5\gamma^2} \left( \delta_{ij}\delta_{kl} + \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk} \right) - \delta_{ij}\delta_{kl}. \label{eq:a-07} \end{align} Expanding the operators in the exponent of Eq.~(\ref{eq:a-01}), we have the following type of factor in the expansion: \begin{equation} \left\langle \frac{\partial^{m_0}}{\partial \alpha^{m_0}} \frac{\partial^{m_1}} {\partial y_{i_1'}\cdots\partial y_{i_{m_1}'}} \frac{\partial^{m_2}} {\partial Z_{i_1j_1}\cdots\partial Z_{i_{m_2}j_{m_2}}} F_\chi \right\rangle_\mathrm{G}. \label{eq:a-08} \end{equation} This factor is equivalent to \cite{Matsubara:2020fet} \begin{equation} \left(\frac{\sigma_1}{\sqrt{3}\sigma_0}\right)^3 \int_\nu^\infty\!d\nu\, \left(-\frac{d}{d\nu}\right)^{m_0} \left[ \left\langle \delta_\mathrm{D}(\alpha-\nu) \right\rangle_\mathrm{G} \left\langle \frac{\partial^{m_1} \delta_\mathrm{D}^3(\bm{y})} {\partial y_{i_1'}\cdots\partial y_{i_{m_1}'}} \right\rangle_\mathrm{G} \left\langle \frac{\partial^{m_2} \det(\nu I - Z)} {\partial Z_{i_1j_1}\cdots\partial Z_{i_{m_2}j_{m_2}}} \right\rangle_\mathrm{G} \right], \label{eq:a-09} \end{equation} because the three kinds of variables $\alpha$, $y_i$ and $Z_{ij}$ are mutually independent in the Gaussian averages with two-point cumulants. The Gaussian distribution functions of the variables $\alpha$ and $y_i$ are given by \begin{equation} P^{(0)}_\mathrm{G}(\alpha) = \frac{e^{-\alpha^2/2}}{\sqrt{2\pi}}, \quad P^{(1)}_\mathrm{G}(\bm{y}) = \frac{e^{-|\bm{y}|^2/2}}{(2\pi)^{3/2}}, \label{eq:a-10} \end{equation} and the first two factors in the square bracket reduce to \begin{equation} \left\langle \delta_\mathrm{D}(\alpha-\nu) \right\rangle_\mathrm{G} = \frac{e^{-\nu^2/2}}{\sqrt{2\pi}},\quad \left\langle \frac{\partial^{m_1} \delta_\mathrm{D}^3(\bm{y})} {\partial y_{i_1'}\cdots\partial y_{i_{m_1}'}} \right\rangle_\mathrm{G} = \begin{cases} \displaystyle \frac{(-1)^{m_1/2} (m_1-1)!!}{(2\pi)^{3/2}} \delta_{(i_1'i_2'}\cdots\delta_{i_{m_1-1}'i_{m_1}')}, & (m_1: \mathrm{even}), \\ 0, & (m_1: \mathrm{odd}), \end{cases} \label{eq:a-11} \end{equation} where parentheses in the subscript of Kronecker's delta indicate the symmetrization of the indices inside them. For the last factor, we have an identity for the $3\times 3$ matrices, \begin{equation} \det(\nu I - Z) =\nu^3 - \nu^2\,\mathrm{tr}\,Z + \frac{\nu}{2} \left[ \left(\mathrm{tr}\,Z\right)^2 - \mathrm{tr}\,\left(Z^2\right) \right] - \det Z. \label{eq:a-12} \end{equation} For a fixed number of derivatives in Eq.~(\ref{eq:a-08}), $m\equiv m_0+m_1+m_2$, one can see that the dominant contribution in the high-peaks limit, $\nu\rightarrow\infty$, is given by a case of $m_0=m$, $m_1=m_2=0$. In this case, we have $\langle\det(\nu I-Z)\rangle_\mathrm{G} = H_3(\nu)$ due to Eqs.~(\ref{eq:a-07}) and (\ref{eq:a-12}), and Eq.~(\ref{eq:a-09}) reduces to \begin{equation} \frac{1}{(2\pi)^2} \left(\frac{\sigma_1}{\sqrt{3}\sigma_0}\right)^3 e^{-\nu^2/2} H_{m+2}(\nu) \label{eq:a-13} \end{equation} in the high-peaks limit. When the replacement $H_{m+2}(\nu) \rightarrow H_2(\nu) \nu^m$ in the same limit is applied for each term of the expanded series of Eq.~(\ref{eq:a-01}), the resulting infinite series is resummed again. Thus we have \begin{equation} n_\chi(\nu) \approx \frac{1}{(2\pi)^2} \left(\frac{\sigma_1}{\sqrt{3}\sigma_0}\right)^3 e^{-\nu^2/2} H_2(\nu) \exp\left( \sum_{n=3}^\infty \frac{\nu^n}{n!} \left\langle\alpha^n\right\rangle_\mathrm{c} \right). \label{eq:a-14} \end{equation} Identifying the number density of peaks with the Euler number density in the high-peaks limit, the mass fraction of peaks is given by \begin{equation} \beta_\mathrm{pk} \approx (2\pi)^{3/2}R^3 n_\chi(\nu) \approx \frac{1}{\sqrt{2\pi}} \left(\frac{R\sigma_1}{\sqrt{3}\sigma}\right)^3 (\nu^2-1) e^{-\nu^2/2} \exp\left( \sum_{n=3}^\infty \frac{\nu^n}{n!} \left\langle\alpha^n\right\rangle_\mathrm{c} \right). \label{eq:a-15} \end{equation} The cumulants of $\alpha$ are given by $\langle \alpha^n \rangle_\mathrm{c} = \sigma^{n-2}S_n$ where $S_n$ is the reduced cumulants defined by Eq.~(\ref{eq:4-01}), and the above equation is equivalent to Eq.~(\ref{eq:4-06}).

Title: A Comprehensive Perturbative Formalism for Phase-Mixing in Perturbed Disks. I. Phase spirals in an Infinite, Isothermal Slab

Abstract: Galactic disks are highly responsive systems that often undergo external perturbations and subsequent collisionless equilibration, predominantly via phase-mixing. We use linear perturbation theory to study the response of infinite isothermal slab analogues of disks to perturbations with diverse spatio-temporal characteristics. Without self-gravity of the response, the dominant Fourier modes that get excited in a disk are the bending and breathing modes, which, due to vertical phase-mixing, trigger local phase-space spirals that are one- and two-armed, respectively. We demonstrate how the lateral streaming motion of slab stars causes phase spirals to damp out over time. The ratio of the perturbation timescale ($\tau_{\mathrm{P}}$) to the local, vertical oscillation time ($\tau_z$) ultimately decides which of the two modes is excited. Faster, more impulsive ($\tau_{\mathrm{P}} < \tau_z$) and slower, more adiabatic ($\tau_{\mathrm{P}} > \tau_z$) perturbations excite stronger breathing and bending modes, respectively, although the response to very slow perturbations is exponentially suppressed. For encounters with satellite galaxies, this translates to more distant and more perpendicular encounters triggering stronger bending modes. We compute the direct response of the Milky Way disk to several of its satellite galaxies, and find that recent encounters with all of them excite bending modes in the Solar neighborhood. The encounter with Sagittarius triggers a response that is at least $1-2$ orders of magnitude larger than that due to any other satellite, including the Large Magellanic Cloud. We briefly discuss how ignoring the presence of a dark matter halo and the self-gravity of the response might impact our conclusions.

https://export.arxiv.org/pdf/2208.05038

\vspace{1mm} \shorttitle{Phase-space spirals} \shortauthors{} \title{A Comprehensive Perturbative Formalism for Phase-Mixing in Perturbed Disks. I. Phase spirals in an Infinite, Isothermal Slab} \correspondingauthor{Uddipan Banik} \email{uddipan.banik@yale.edu} \author[0000-0002-9059-381X]{Uddipan Banik} \affiliation{Department of Astronomy, Yale University, PO. Box 208101, New Haven, CT 06520, USA} \author[0000-0003-2660-2889]{Martin~D.~Weinberg} \affiliation{Department of Astronomy, University of Massachusetts at Amherst, 710 N. Pleasant St., Amherst, MA 01003} \author[0000-0003-3236-2068]{Frank~C.~van den Bosch} \affiliation{Department of Astronomy, Yale University, PO. Box 208101, New Haven, CT 06520, USA} \label{firstpage} \keywords{ methods: analytical --- Perturbation methods --- Gravitational interaction --- Galaxy: disk --- Galaxy: kinematics and dynamics --- Galaxy stellar disks --- galaxies: interactions --- Milky Way dynamics --- Milky Way disk} \section{Introduction} \label{sec:intro} The relaxation or equilibration of self-gravitating systems is a ubiquitous astrophysical phenomenon that drives the formation and evolution of star-clusters, galaxies and cold dark matter halos. In quasi-equilibrium, the phase-space density of such collisionless systems can be well characterized by a distribution function (DF) which, according to the strong Jeans theorem, is a function of the conserved quantities or actions of the system. When such a system is perturbed out of equilibrium by a time-dependent gravitational perturbation, either external (e.g., encounter with another galaxy) or internal (e.g., bars or spiral arms), the original actions of the stars are modified, and the system has to re-establish a new (quasi-)equilibrium. Since disk galaxies are highly ordered, low-entropy (i.e., cold) systems, they are extremely responsive. Even small gravitational perturbations can induce oscillations in the disk, which manifest as either standing or propagating waves \citep[see][for a detailed review]{Sellwood.13}. Such oscillations consist of an initially coherent response of stars to a gravitational perturbation. This coherent response is called {\it collective} if its self-gravity is included. Over time, though, the coherence {\it dissipates}, which manifests as relaxation or equilibration and drives the system towards a new quasi-equilibrium, free of large scale oscillations. Equilibration in galactic disks is dominated by collisionless effects, including purely kinematic processes like phase-mixing (loss of coherence in the response due to different orbital frequencies of stars), and self-gravitating or collective processes like Landau damping \citep[loss of coherence due to non-dissipative damping of waves by wave-particle interactions,][]{LyndenBell.62} and violent relaxation \citep[loss of coherence due to scrambling of orbital energies in a time-varying potential,][]{LyndenBell.67}. It is noteworthy to point out that without phase-mixing neither Landau damping \citep[][]{Maoz.91} nor violent relaxation \citep[see][]{Sridhar.89} would result in equilibration. A final equilibration mechanism is chaotic mixing, the loss of coherence resulting from the exponential divergence of neighboring stars on chaotic orbits \citep[e.g.,][]{Merritt.Valluri.96,Daniel.Wyse.15,Banik.vdBosch.22}. As long as most of the phase-space is foliated with regular orbits (i.e., the Hamiltonian is near-integrable), chaotic mixing should not make a significant contribution, and phase-mixing may thus be considered the dominant equilibration mechanism. Disk galaxies typically reveal out-of-equilibrium features due to incomplete equilibration. These may appear in the form of bars and spiral arms, which are large-scale perturbations in the radial and azimuthal directions, responsible for a slow, secular evolution of the disk. In the vertical direction, disks often reveal warps \citep[][]{Binney.92}. In the case of the Milky Way (hereafter MW) disk, which can be studied in much greater detail than any other system, recent data from astrometric and radial-velocity surveys such as SEGUE \citep[][]{Yanny.etal.09}, RAVE \citep[][]{Steinmetz.etal.06}, GALAH \citep[][]{Bland-Hawthorn.etal.19}, LAMOST \citep[][]{Cui.etal.12} and above all Gaia \citep[][]{Gaia_collab.16, Gaia_collab.18a, Gaia_collab.18b} has revealed a variety of additional vertical distortions. At large galacto-centric radii ($>10 \kpc$) this includes, among others, oscillations and corrugations \citep[][]{Xu.etal.15,Schonrich.Dehnen.18}, and streams of stars kicked up from the disk that undergo phase-mixing, sometimes referred to as `feathers' \citep[e.g.,][]{Price-Whelan.etal.15, Thomas.etal.19, Laporte.etal.22}. Similar oscillations and vertical asymmetries have also been reported in the Solar vicinity \citep[e.g.,][]{Widrow.etal.12, Williams.etal.13, Yanny.Gardner.13, Quillen.etal.18, Gaia_collab.18b, Bennett.Bovy.19, Carrillo.etal.19}. One of the most intriguing structures is the phase-space spiral discovered by \cite{Antoja.etal.18}, and studied in more detail in subsequent studies \citep[e.g.,][]{Bland-Hawthorn.etal.19,Li.Widrow.21,Li.21,Gandhi.etal.22}. Using data from Gaia DR2 \citep[][]{Gaia_collab.18a}, \cite{Antoja.etal.18} selected $\sim 900$k stars within a narrow range of galacto-centric radius and azimuthal angle centered around the Sun. When plotting the density of stars in the $(z,v_z)$-plane of vertical position, $z$, and vertical velocity, $v_z$, they noticed a faint, unexpected spiral pattern, which became more enhanced when colour-coding the $(z,v_z)$-`pixels' by the median radial or azimuthal velocities. The one-armed spiral makes 2-3 complete wraps, resembling a snail shell, and is interpreted as a signature of phase-mixing in the vertical direction following a perturbation, which \cite{Antoja.etal.18} estimate to have occurred between 300 and 900 Myr ago. More careful analyses in later studies \citep[e.g.,][etc.]{Bland-Hawthorn.etal.19,Li.21} have nailed down the interaction time to $\sim 500\Myr$ ago. The discovery of all these oscillations in the MW disk has ushered in a new, emerging field of astrophysics, known as galactoseismology \citep[][]{Widrow.etal.12, Johnston.etal.17}. Similar to how the timbre of musical notes reveals characteristics of the instrument that produced the sound, the `ringing' of a galactic disk can (in principle) reveal its structure (both stellar disk plus dark matter halo). And similar to how the timbre can tell us whether the string of a violin was plucked (pizzicato) or bowed (arco), the ringing of a galactic disk can reveal information about the perturbation that set the disk ringing. Phase spirals are especially promising in this regard: their structure holds information about the gravitational potential in the vertical direction \citep[in particular, the vertical frequency as a function of the vertical action,][]{Antoja.etal.18} and about the type of perturbation that triggered the phase spiral \citep[e.g., bending mode vs. breathing mode, see][and Section~\ref{sec:impulsive_kick} below]{Widrow.etal.14, Darling.Widrow.19a}. In addition, by unwinding the phase spiral one can in principle determine how long ago the vertical oscillations were triggered. By studying phase spirals at multiple locations in the disk, one may even hope to use some form of triangulation to infer the direction or location from which the perturbation emerged (assuming, of course, that the phase spirals at different locations were all triggered by the same perturbation). However promising galactoseismology may seem, many questions remain: what kind of perturbation can trigger a phase spiral? how long do phase spirals remain detectable, and what equilibration mechanism(s) causes their demise? Can we really constrain the vertical potential of the disk, or does self-gravity of the perturbation make it difficult to achieve? What kind of constraints can we infer regarding the perturber that triggered the phase spiral? Is galactoseismology likely to be confusion limited, i.e., should we expect that each location in the disk experiences oscillations arising from multiple, independent perturbations? If so, how does this impact our ability to extract useful information? Answering these questions necessitates a deep understanding of how the MW disk, and disk galaxies in general, respond to perturbations. To date, these questions have mainly been addressed using numerical $N$-body simulations or fairly simplified analytical approaches. In particular, numerous studies have investigated how the MW disk responds to interactions with the Sagittarius (Sgr) dwarf galaxy \citep[e.g.,][]{Gomez.etal.13, Donghia.etal.16, Laporte.etal.18, Khanna.etal.19,Hunt.etal.21}. While simulations likes these have demonstrated that the interaction with Sgr can indeed spawn phase spirals in the Solar vicinity \citep[][]{Antoja.etal.18,Binney.Schonrich.18, Darling.Widrow.19b, Laporte.etal.19, Bland-Hawthorn.etal.19, Hunt.etal.21, Bennett.etal.21}, none of them have been able to produce phase spirals that match those observed in the Gaia data. As discussed in detail in \cite{Bennett.etal.21} and \cite{Bennett.Bovy.21}, this seems to suggest that the amplitude and shape of the ``Gaia snail" cannot be produced by Sgr alone. An alternative explanation, explored by \citet{Khoperskov.etal.19}, is that the Gaia snail was created by buckling of the MW's bar. However, this explanation faces its own challenges \citep[see e.g.,][]{Laporte.etal.19, Bennett.Bovy.21}. Triggering the Gaia snail with a spiral arm \citep[][]{Faure.etal.14} is also problematic, in that it requires the spiral arms to have unusually large amplitude \citep[][]{Quillen.etal.18}. Clearly then, despite a large number of studies, pinpointing the origin of the phase spiral in the Solar vicinity still remains an unsolved problem. Although simulations have the obvious advantage that they can probe the complicated response of a perturbed disk to a realistic perturbation, which often is analytically intractable, especially if the response is large (non-linear), there are also clear disadvantages. Foremost, reaching sufficient resolution to resolve the kind of fine-structure that we can observe with data sets like Gaia requires extremely large simulations with $N > 10^8-10^9$ particles \citep[][]{Weinberg.Katz.07a, Binney.Schonrich.18, Hunt.etal.21}. Although such simulations are no longer beyond our reach \citep[see e.g.,][]{Bedorf.etal.14, Fujii.etal.19, Hunt.etal.21, Peterson.etal.22}, it is clear that using such simulations to explore large areas of parameter space remains a formidable challenge. To overcome this problem, a semi-analytical approach called the {\it backward-integrating restricted N-body method} was developed originally in the context of perturbation by bars \citep[e.g.,][]{Leeuwin.etal.93,Vauterin.Dejonghe.97,Dehnen.00}, and later on used by \cite{Hunt.Bovy.18} and \cite{Hunt.etal.19} to study non-equilibrium features in the MW caused by transient spiral arms. This method is effectively a Lagrangian formalism to solve the collisionless Boltzmann equation (hereafter CBE) by integrating test particles in the perturbed potential in a restricted N-body framework, i.e., without self-consistently developing the potential perturbation from the DF perturbation. Although appropriate for studying the local kinematic distribution of particles, this approach becomes too expensive to study the global equilibration of a system. Hence, it is important to consider alternative analytical methods that can be used to investigate the global response of a disk. In this vein, this paper presents a rigorous, perturbative, Eulerian formalism to compute the response of a disk to perturbations. In order to gain valuable insight into the physical mechanism of phase-mixing, without resorting to the computational complexity involved in modelling a realistic disk, which we postpone to Paper~II (Banik et al., in preparation), in this first paper in the series we consider perturbations of an infinite slab with a vertical profile, but homogeneous in the lateral directions. Although a poor representation of a realistic galactic disk, this treatment captures most of the essential features of how disks respond to gravitational perturbations. We study the response of the slab to perturbers of various spatial and temporal scales, with a focus on the formation and dissolution of phase spirals resulting from the vertical oscillations and phase-mixing of stars. This paper is organized as follows. Section~\ref{sec:linear_theory} describes the application of perturbation theory to our infinite, isothermal slab. Section~\ref{sec:impulsive_kick} then uses these results to work out the response to an impulsive, single-mode perturbation, which nicely illustrates how phase spirals originate from vertical oscillations and how they damp out due to lateral mixing. Sections~\ref{sec:localized} and~\ref{sec:non-impulsive} generalize this to responses to localized (wave packet) and non-impulsive perturbations, respectively. In Section~\ref{sec:sat_encounter} we investigate the response to satellite encounters and examine which satellite galaxies in the halo of the MW can trigger bending and/or breathing modes strong enough to trigger phase spirals at the Solar radius (still approximating the MW disk as an infinite, isothermal slab). We summarize our findings in Section~\ref{sec:concl}. \newpage \section{Linear perturbation theory for collisionless systems} \label{sec:linear_theory} \subsection{Linear perturbative formalism} Let the unperturbed steady state distribution function (DF) of a collisionless stellar system be given by $f_0$ and the corresponding Hamiltonian be $H_0$. $f_0$ satisfies the CBE, \begin{align} [f_0,H_0]=0, \end{align} where the square brackets correspond to the Poisson bracket. Now let us introduce a small time-dependent perturbation in the potential, $\Phi_\rmP(t)$, such that the perturbed Hamiltonian becomes \begin{align} H=H_0+\Phi_\rmP(t)+\Phi_1(t), \end{align} where $\Phi_1$ is the gravitational potential sourced by the response density, $\rho_1 = \int f_1 d^3\bv$, via the Poisson equation, \begin{align} \nabla^2\Phi_1=4\pi G\rho_1. \end{align} Here $f_1$ is the linear order perturbation in the DF, i.e., the linear response of the system to the perturbation in the potential. The perturbed DF can thus be written as \begin{align} f=f_0+f_1. \end{align} Assuming that the perturbations are small such that linear perturbation theory holds, the time-evolution of $f_1$ is governed by the following linearized version of the CBE \begin{align} \frac{\partial f_1}{\partial t}+[f_1,H_0]+[f_0,\Phi_\rmP]+[f_0,\Phi_1]=0. \label{CBE_perturb} \end{align} In this paper we shall neglect the self-gravity of the disk, i.e., neglect the polarization term, $[f_0,\Phi_1]$, in the lhs of the linearized CBE. We briefly discuss the impact of self-gravity in Section~\ref{sec::caveats}, leaving a more detailed analysis including self-gravity to a forthcoming publication. \subsection{Hybrid perturbative formalism for an infinite slab} \label{sec:slab} We consider the simplified case of perturbations in an infinitely extended slab, uniform in $(x,y)$, but characterized by a vertical density profile $\rho(z)$. Although a rather poor approximation of a realistic galactic disk, this idealized case serves to highlight some of the main characteristics of disk response. We consider perturbations that can be described by a profile in the vertical $z$-direction and by a superposition of plane waves along the $x$-direction, such that $\Phi_\rmP$ and $f_1$ are both independent of $y$. After making a canonical transformation from the phase-space variables $(z,v_z)$ to the corresponding action angle variables $(I_z,w_z)$, Equation~(\ref{CBE_perturb}) becomes \begin{align} \frac{\partial f_1}{\partial t}+\frac{\partial H_0}{\partial I_z}\frac{\partial f_1}{\partial w_z}+\frac{\partial H_0}{\partial v_x}\frac{\partial f_1}{\partial x}-\frac{\partial \Phi_\rmP}{\partial w_z}\frac{\partial f_0}{\partial I_z}-\frac{\partial \Phi_\rmP}{\partial x}\frac{\partial f_0}{\partial v_x}=0. \label{CBE_perturb1} \end{align} The unperturbed Hamiltonian $H_0$ can be written as \begin{align} H_0 = \frac{v^2_x+v^2_y}{2} + \frac{v^2_z}{2} + \Phi_z(z), \end{align} where $v_x$, $v_y$ and $v_z$ are the unperturbed velocities of stars along $x$, $y$ and $z$ respectively, and $\Phi_z(z)$ is the unperturbed potential that dictates the oscillatory vertical motion of the stars. We expand $\Phi_\rmP$ and $f_1$ as Fourier series that are discrete along $z$ but continuous along $x$: \begin{align} \Phi_\rmP(z,x,t)&=\sum_{n=-\infty}^{\infty}\int d k\, \exp{\left[i (n w_z + k x)\right]}\, \Phi_{nk}(I_z,t),\nonumber \\ f_1(z,v_z,x,v_x,v_y,t)&=\sum_{n=-\infty}^{\infty}\int d k\, \exp{\left[i (n w_z + k x)\right]}\, f_{1nk}(I_z,v_x,v_y,t). \end{align} Here $z$ can be expressed as the following implicit function of $w_z$ and $I_z$, \begin{align} w_z = \Omega_z \int_0^{z} \frac{d z'}{\sqrt{2\left[E_z(I_z)-\Phi_z(z')\right]}}. \label{z_wz_Iz} \end{align} where $\Omega_z=\Omega_z(I_z)$ is the vertical frequency of stars with vertical action $I_z$, given in equation~(\ref{omzvx}) below. Here and throughout this paper we express any dependence on the continuous wave number $k$ with an index rather than an argument, i.e., $\Phi_{nk}(I_z,t)$ rather than $\Phi_n(k,I_z,t)$. This implies that any function that carries $k$ as an index is in Fourier space. We express the perturber potential and the DF perturbation or response as linear superpositions of Fourier modes. Since we do not take into account the self-gravity of the response itself, i.e., do not self-consistently solve the Poisson equation along with the CBE, these are not dynamical or normal modes of the system. In other words, the oscillation frequencies of the Fourier modes are just the unperturbed frequencies, $\Omega_z$, and do not follow a dispersion relation as in the self-gravitating case. To aid the visualization of the various Fourier modes, Fig.~\ref{fig:modes} illustrates what the $n=0$, $n=1$ and $n=2$ modes for one particular value of the wavenumber $k$ look like. The figure also indicates the direction of the velocity impulses resulting from an instantaneous perturbation of each mode. Substitution of the above expressions in equation~(\ref{CBE_perturb1}) yields the following evolution equation for $f_{1nk}$ \begin{align} \frac{\partial f_{1nk}}{\partial t}+i(n\Omega_z+k v_x)f_{1nk}=i\left(n\frac{\partial f_0}{\partial I_z}+k\frac{\partial f_0}{\partial v_x}\right)\Phi_{nk}, \label{f1nk_de} \end{align} where we have used that \begin{align} \Omega_z = \frac{\partial H_0}{\partial I_z}\,, \;\;\; v_x=\frac{\partial H_0}{\partial v_x}. \label{omzvx} \end{align} The above first order differential equation in time is easily solved using the Green's function technique. With the initial condition, $f_{1nk}(t_\rmi)=0$, we obtain the following integral form for $f_{1nk}$ for a given time dependence of the perturber potential, \begin{align} f_{1nk}(I_z,v_x,v_y,t)&=i\left(n\frac{\partial f_0}{\partial I_z}+k\frac{\partial f_0}{\partial v_x}\right)\int_{t_i}^{t}d\tau \exp{\left[-i(n\Omega_z+k v_x)(t-\tau)\right]}\, \Phi_{nk}(I_z,\tau). \label{f1nk_slabsol} \end{align} This solution is analogous to the particular solution for a forced oscillator with natural frequencies, $n\Omega_z$ and $k v_x$, which is being forced by an external perturber potential, $\Phi_{nk}$. The time-dependence of this external perturbation ultimately dictates the temporal evolution of the perturbation in the DF, $f_{1nk}$. A {\it net} response requires gradients in the (unperturbed) DF with respect to the actions and/or velocities. Similar solutions for the response of perturbed, collisionless systems have been derived in a number of previous studies \citep[e.g.,][]{LyndenBell.Kalnajs.72, Tremaine.Weinberg.84, Carlberg.Sellwood.85, Weinberg.89, Weinberg.91, Weinberg.04, Kaur.Sridhar.18, Banik.vdBosch.21a, Kaur.Stone.21, Chiba.Schonrich.21}, often in the context of phenomena like angular momentum transport, radial migration or dynamical friction. \subsection{Perturbation in an isothermal slab} \label{sec:iso_slab} The infinite slab has a non-uniform (uniform) density profile along the vertical (horizontal) direction. Therefore the unperturbed motion of the stars is only vertically bounded by a potential but is unbounded horizontally. This implies that the unperturbed DF, $f_0$, involves a potential $\Phi_z$ only along $z$. For simplicity, we assume it to be isothermal but with different velocity dispersions in the vertical direction, $\sigma_z$, and the in-plane directions, $\sigma_x = \sigma_y \equiv \sigma$, i.e., \begin{align} f_0(v_x,v_y,E_z)= \frac{\rho_c}{{(2\pi)}^{3/2}\sigma_z\,\sigma^2} \, \exp\left[-\frac{E_z}{\sigma^2_z}\right] \, \exp\left[-\frac{v^2_x+v^2_y}{2\sigma^2}\right], \label{f_iso} \end{align} where \begin{align} E_z = \frac{1}{2} v^2_z +\Phi_z(z) \label{E_z} \end{align} is the energy involving the $z$-motion. The corresponding density and potential profiles in the vertical direction are given by \begin{align} \rho_z(z) = \rho_c \, {\sech}^2(z/h_z),\;\;\;\;\;\;\;\;\;\;\;\Phi_z(z) = 2 \sigma^2_z \,\ln\left[\cosh(z/h_z)\right], \end{align} where $h_z$ is the vertical scale height \citep[][]{Spitzer.42,Camm.50}. The vertical action, $I_z$, can be obtained from the unperturbed Hamiltonian, $E_z$, as follows \begin{align} I_z = \frac{1}{2\pi} \oint v_z \, d z= \frac{2}{\pi} \int_0^{z_{\rm max}} \sqrt{2[E_z - \Phi_z(z)]} \, d z, \end{align} where $\Phi_z(z_{\rm max})=E_z$, i.e., $z_{\rm max} = h_z \cosh^{-1}\left(\exp{\left[E_z/2\sigma^2_z\right]}\right)$. The time period of vertical oscillation is given by \begin{align} T_z = \oint \frac{d z}{v_z} = 4\int_0^{z_{\rm max}} \frac{d z} {\sqrt{2\left[E_z-\Phi_z(z)\right]}}, \label{T_z} \end{align} and the vertical frequency is $\Omega_z = 2\pi/T_z$. Throughout this paper, to compute the perturbative response of the slab, we shall use typical MW parameter values, i.e., $h_z=0.4$ kpc, $\sigma_z=23$ km/s, and $\sigma=1.5\,\sigma_z=35$ km/s \citep[][]{McMillan.11}. Substituting the above form for $f_0$ (Equation~[\ref{f_iso}]) in Equation~(\ref{f1nk_slabsol}) and using that $\Omega_z = \Omega_z(I_z) = \partial E_z/\partial I_z$ yields the following closed integral form for $f_{1nk}$: \begin{align} f_{1nk}(I_z,v_x,v_y,t) &= -i\left(\frac{n\Omega_z}{\sigma^2_z} + \frac{k v_x}{\sigma^2}\right) \, f_0(v_x,v_y,E_z) \, \int_{t_i}^{t}d \tau \, \exp{\left[-i(n\Omega_z+k v_x)(t-\tau)\right]} \, \Phi_{nk}(I_z,\tau). \label{f1nk_isosol} \end{align} \subsection{Perturber potential} \label{sec:per_pot} The slab response depends on the spatio-temporal nature of the perturber. In this paper we consider two different functional forms of the perturber potential described below. \subsubsection{Separable potential} \label{sec:sep_per_pot} In order to capture the essential physics of perturbative collisionless dynamics without much computational complexity, we shall consider the following separable form for the perturber potential: \begin{align} \Phi_\rmP(z,x,t) = \Phi_\rmN\, \calZ(z) \calX(x) \calT(t), \label{Phip_sep} \end{align} where $\Phi_\rmN$ has the units of potential, and $\calZ$, $\calX$ and $\calT$ are dimensionless functions of $z$, $x$ and $t$ respectively that specify the spatio-temporal profile of $\Phi_\rmP$. Thus, the Fourier transform of $\Phi_\rmP$ can also be written in the following separable form, \begin{align} \Phi_{nk}(I_z,t) = \Phi_\rmN\, \calZ_n(I_z) \calX_k\, \calT(t). \label{Phip_sep_fourier} \end{align} Here $\calZ_n(I_z)$ is the $n^{\rm th}$ Fourier coefficient in the discrete Fourier series expansion of $\calZ(z)$ in the vertical angle, $w_z$, given by \begin{align} \calZ_n(I_z) = \frac{1}{2\pi} \int_0^{2\pi} d w_z \, \calZ(z) \, \exp{\left[-in w_z\right]}, \label{Phip_sep_fourier_z} \end{align} where we have used the implicit expression for $z$ in terms of $w_z$ and $I_z$ given in equation~(\ref{z_wz_Iz}). $\calX_k$ is the Fourier transform of $\calX(x)$, given by \begin{align} \calX_k = \frac{1}{2\pi} \int_{-\infty}^{\infty} d x\, \calX(x) \, \exp{\left[-ikx\right]}. \label{Phip_sep_fourier_x} \end{align} In the following sections, we investigate the slab response to perturbers with various functional forms for $\calX(x)$ and $\calT(t)$, while keeping the form for $\calZ(z)$ arbitrary. We start in Section~\ref{sec:impulsive_kick} with an impulsive ($\calT(t)=\delta(t)$) single-mode ($\calX(x) = \exp[ikx]$) perturbation, followed in Section~\ref{sec:localized} by a perturbation that is temporally impulsive but spatially localized ($\calX(x)=\exp{\left[-x^2/\Delta^2_x\right]}$). In Section~\ref{sec:non-impulsive} we consider the same spatially localized perturbation, but this time temporally extended ($\calT(t)=\exp{\left[-\omega^2_0 t^2\right]}$). \subsubsection{Satellite galaxy along straight orbit} \label{sec:sat_per_pot} As a practical astrophysical application of our perturbative formalism, we also study the response of an isothermal slab to a satellite galaxy or DM subhalo undergoing an impact along a straight orbit with a uniform velocity $\vp$ at an angle $\thetap$ (with respect to the disk normal). We model the impacting satellite as a point perturber, whose potential is given by \begin{align} \Phi_\rmP(z,x,t) = -\frac{GM_\rmP}{\sqrt{{\left(z-\vp\cos{\thetap} t\right)}^2+{\left(x-\vp\sin{\thetap} t\right)}^2}}. \label{Phip_sat} \end{align} In this case the spatial and temporal parts are coupled and thus the slab response needs to be evaluated by performing the $\tau$ integral before the $w_z$ and $x$ integrals (to find $\Phi_{nk}$), as shown in Appendix~\ref{App:sat_disk_resp}. \section{Response to an Impulsive Perturbation} \label{sec:impulsive_kick} In order to gain some insight into the perturbative response of the slab, we start by solving equation~(\ref{f1nk_isosol}) for an instantaneous impulse at $t=0$; i.e., $\calT(t) = \delta(t)$. Here the normalization factor $\Phi_\rmN$ has the units of potential times time. With the initial time $t_i<0$, the integral over $\tau$ yields $\exp{\left[-i(n\Omega_z+k v_x)t\right]}$. Further integrating $f_{1nk}$ over $v_x$ and $v_y$ and summing over all $n$ modes, yields the following form for any given $k$ mode of the perturbed DF for a given action $I_z$ and angle $w_z$, \begin{align} f_{1k}(I_z,w_z,t)&=\sum_{n=-\infty}^{\infty}\exp{\left[in w_z\right]}\int_{-\infty}^{\infty}d v_y\int_{-\infty}^{\infty}d v_x\, f_{1nk}(I_z,v_x,v_y,t)\nonumber \\ &=A_{\rm norm}\, D_k(t)\, R_k(I_z,w_z,t), \label{f1_delta} \end{align} where \begin{align} A_{\rm norm}=\frac{\rho_c}{\sqrt{2\pi}\sigma_z} \exp{\left[-E_z/\sigma^2_z\right]} \end{align} is a normalization factor reflecting the vertical structure of the unperturbed disk, \begin{align} D_k(t)=\exp{\left[-\frac{k^2\sigma^2t^2}{2}\right]} \end{align} is a damping term that describes the temporally Gaussian decay of the response by lateral mixing, and \begin{align} R_k(I_z,w_z,t)=-\Phi_\rmN\calX_k\sum_{n=-\infty}^{\infty}\calZ_n(I_z)\left(k^2t+i\frac{n\Omega_z}{\sigma^2_z}\right)\exp{\left[i n \left(w_z - \Omega_z\,t\right)\right]} \label{Rk_impulse} \end{align} is a (linear) response function that includes vertical phase-mixing. Equation~(\ref{f1_delta}) is the basic `building block' for computing the response of our infinite isothermal slab to a perturbation mode $k$ in the impulsive limit. Using the canonical transformation from $(w_z,I_z)$ to $(z,v_z)$, i.e., using equations~(\ref{z_wz_Iz}) and~(\ref{E_z}), we can transform $f_{1k}(I_z,w_z,t)$ to $f_{1k}(v_z,z,t)$. Upon multiplying this by $\exp{\left[ikx\right]}$ and integrating over $k$, and then integrating further over $v_z$ at a fixed $z$, one obtains the response density as a function of both time and position: \begin{align} \rho_1(z,x,t) &=-\frac{\rho_c \Phi_\rmN}{\sqrt{2\pi}\sigma_z} \sum_{n=-\infty}^{\infty} \int_0^{\Tilde{I}_z} d I_z\, \frac{\Omega_z}{\sqrt{2\left[E_z-\Phi_z(z)\right]}} \exp{\left[-E_z/\sigma^2_z\right]} \exp{\left[i n \left(\Tilde{w}_z - \Omega_z\,t\right)\right]} \,\calZ_n(I_z) \nonumber \\ &\times \int d k\,\exp{\left[ikx\right]}\, \exp{\left[-\frac{k^2\sigma^2t^2}{2}\right]}\left(k^2t+i\frac{n\Omega_z}{\sigma^2_z}\right)\calX_k, \end{align} where $\Tilde{I}_z$ is the solution of $E_z(I_z)=\Phi_z(z)$, and $\Tilde{w}_z$ is the solution for $w_z(z,I_z)$ from equation~(\ref{z_wz_Iz}). In order to gain insight into the slab response for a particular $I_z$ and $w_z$, let us start by analyzing equation~(\ref{f1_delta}) for the $n=0$ mode, an in-plane density wave, for which the perturbation causes an in-plane velocity impulse as depicted in Fig.~\ref{fig:modes}. The response is a standing, longitudinal oscillation in density. The response function for this mode is $R_k(I_z,w_z,t) = \Phi_\rmN \calZ_0(I_z)\calX_k\, k^2 t$, indicating that the amplitude of oscillation initially grows linearly with time. However, this growth is inhibited by the Gaussian damping function $D_k(t) = \exp[-\half k^2 \sigma^2 t^2]$, which describes lateral mixing due to the non-zero velocity dispersion of stars in the $k$-direction. The Gaussian form of this temporal damping term owes its origin to the assumed Gaussian/Maxwellian form of the unperturbed velocity distribution along the plane. Hence, following the perturbation, the $n=0$ mode starts to grow linearly with time, but then rapidly damps away; the response loses its coherence due to mixing in the direction of the wave-vector. In the cold slab limit $(\sigma\to 0)$, without any lateral streaming motion to damp it out, the response will grow linearly in time until it eventually becomes non-linear. This is because in an infinite, laterally homogeneous slab there is no restoring force in the lateral directions, causing the stars to stream uninhibited towards (away from) the minima (maxima) of $\Phi_\rmP$ due to the initial velocity impulse induced. This leads to over- and under-density spikes which cannot be treated using linear theory. Hence, Equation~(\ref{f1_delta}) can only adequately describe the response to an instantaneous $n=0$ mode at early times, or if the damping time $\tau_\rmD \sim (k \sigma)^{-1}$ is shorter than the time-scale of formation of density spikes. The latter is roughly the time needed to cross one quarter of the perturbation's wavelength with the velocity impulse triggered at the zeroes of $\Phi_\rmP$. Therefore, in order for linear theory to be valid, we require that $\sigma > (2/\pi) \, |\Delta v|_{\rm max}$, where $|\Delta v|_{\rm max}=k\, \Phi_\rmN \calZ_0(I_z)\calX_k$. Moreover, upon including self-gravity, it can be found that the $n=0$ mode is linearly stable only for $k>k_J\approx \sqrt{4\pi G \rho_c}/\sigma$ \citep[][]{Binney.Tremaine.08}, or in other words $\lambda<\lambda_J\approx \sigma\sqrt{\pi/G \rho_c}$, where $k_J$ and $\lambda_J=2\pi/k_J$ refer to the Jeans wave-number and Jeans wavelength respectively. In the $\sigma\to 0$ limit, the Jeans wave-length, $\lambda_J\to 0$, and thus the $n=0$ mode becomes globally unstable. Hence, the condition of Jeans stability requires an additional constraint on $\sigma$: $\sigma>\sqrt{4\pi G\rho_c}/k$. The validity of linear perturbation theory thus requires that for each $k$, \begin{align} \sigma > \max{\left[\frac{\sqrt{4\pi G \rho_c}}{k},\frac{2k}{\pi}\Phi_\rmN \calZ_0(I_z)\calX_k\right]}. \end{align} For $n=1$, the perturbation is a standing, transverse wave on the slab, formally known as the bending wave. The perturbation induces velocity impulses in the direction perpendicular to the slab, as indicated in Fig.~\ref{fig:modes}. At the locations marked A and B, separated by a lateral distance of $\pi/k$, these velocity impulses point in the positive and negative $z$-directions, respectively. The top panels of Fig.~\ref{fig:impulsive_slab_n1} illustrate the impact this has at location A. The left-hand panels indicate the velocity impulses (cyan arrows) in the $(z,v_z)$-plane. Prior to the perturbation, due to the vertical restoring force from the slab, each star executes a periodic oscillation in this plane. The black and yellow contours indicate the corresponding phase-space trajectories for two values of $I_z$, while the heat-map indicates phase-space density (bluer colors indicate higher density). The top-middle panel shows that immediately following the impulse, the phase-space density is boosted (reduced) where $v_z>0$ ($v_z<0$), resulting in a dipole pattern in the phase-space distribution of stars. After the impulse, the stars continue to execute periodic motion in the $(z,v_z)$-plane, but starting from their new position (corresponding to a modified action $I_z$). The angular frequency of this periodic motion is $\Omega_z$, which is a function of the (modified) action, and hence, stars with different actions oscillate in the $(z,v_z)$-plane at different frequencies. As a consequence, the perturbed phase-space density shown in the middle panels is wound-up into a {\it phase spiral} of over- and under-densities as depicted in the right-most panels of Fig.~\ref{fig:impulsive_slab_n1}. The bottom panels of Fig.~\ref{fig:impulsive_slab_n1} show what happens following the impulsive perturbation at location B. Since the velocity impulses are now reversed in direction, the phase spiral that emerges is exactly the opposite of that at location A. The creation of phase spirals is an outcome of phase-mixing in the $z$-direction and is described by the oscillatory factor, $\exp[i\,n(w_z -\Omega_z t)]$, that is part of the response function $R_k(I_z,w_z,t)$. It consists of two terms: a term that scales as $k^2 t$, which describes the lateral streaming motion of stars due to the non-zero velocity impulses in the lateral directions (see Fig.~\ref{fig:modes}), and a term that scales as $n\Omega_z/\sigma^2_z$ which purely describes the vertical oscillations. As in the case of the $n=0$ mode, the lack of a restoring force in the lateral directions\footnote{If accounting for self-gravity of the response density, there will be non-zero forces in the lateral direction, but these will promote growth rather than act as a restoring force. This ultimately leads to exponential growth (according to linear theory) of unstable modes and Landau damping of stable modes, which occurs exponentially, i.e., more slowly than the Gaussian lateral mixing in the absence of self-gravity.} causes the perturbation to grow linearly with time in the absence of lateral streaming (for a cold disk with $\sigma \approx 0$). Meanwhile, the phase spirals continue to wind-up, which implies that the vertical bending loses its coherence. Over time, phase-mixing in the vertical direction will ensure that the disk regains mirror-symmetry with respect to the midplane, but with a scale-height, $h_z$, that would be a periodic function of $x$, with a wavelength equal to $\pi /k$ (i.e., half the wave-length of the original perturbation). However, all this ignores lateral mixing due to the unconstrained motion with non-zero velocity dispersion in the $x$ direction. Stars that received an impulse $\Delta v_z>0$ create phase spirals that are exactly the inverse of those created by neighboring stars for which the impulse was negative. Thus lateral mixing between neighboring points on the slab causes a damping of the phase spiral amplitude at any location, a process that is captured by the damping function $D_k(t)$. The lateral mixing timescale is $\tau_\rmD \sim 1/k\sigma$, indicating, as expected, that small scale perturbations (larger $k$) mix faster, and that mixing is more efficient for larger velocity dispersion in the lateral direction. After a few mixing time-scales, the slab will once again be completely homogeneous (laterally), with a scale-height $h_z$ that is independent of location. In addition, the density of stars in the $(z,v_z)$-plane will once again be perfectly symmetric without any trace of a phase spiral. The slab has completely equilibrated, and the only impact that remains of the impulsive perturbation is that the new scale-height is somewhat larger than it was originally, i.e., the impulsive perturbation has injected energy into the disk, which causes it to puff-up in the vertical direction. Hence, the final outcome is as envisioned in the impulsive-heating scenario discussed in the seminal study of \citet{Toth.Ostriker.92}. This persistent effect in the vertical density profile is however only captured in perturbation theory at second order \citep[e.g.,][]{Carlberg.Sellwood.85}; to first order the perturbation simply phase-mixes away in the impulsive limit considered here. For $n=2$, the perturbation triggers a breathing mode, as depicted in Fig.~\ref{fig:modes}, i.e., at a given location A on the slab, the velocity impulses for this mode are positive (negative) for positive (negative) $z$. As evident from Fig.~\ref{fig:impulsive_slab_n2}, this leads to a quadrupole pattern for the initial perturbed phase-space distribution of stars, which becomes a two-armed phase spiral over time, as opposed to the one-armed phase spiral resulting from the $n=1$ mode. This reveals an important lesson: the structure of a phase spiral depends, among others, on which perturbation mode(s) are triggered. The phase spirals in regions A and B are each other's additive inverse. Hence, once again lateral mixing will cause damping of the phase spiral's amplitude, as described by the damping function $D_k(t)$. \cite{Hunt.etal.21} have shown using N-body simulations that two-armed phase spirals can indeed arise from breathing mode oscillations and that both bending and breathing modes can be excited at different locations on the MW disk by satellite-induced perturbations such as the passage of Sagittarius (see section~\ref{sec::MW_satellites} for detailed discussion). To summarize, we see that, in case of our infinite slab, equilibration after an impulsive perturbation is driven by a combination of phase-mixing in the vertical direction and free-streaming damping in the horizontal direction. While the former gives rise to phase spirals in the $(\sqrt{I_z} \cos w_z, \sqrt{I_z} \sin w_z)$ or equivalently the $(z,v_z)$ plane, the latter causes them to damp away by lateral mixing. Due to vertical phase-mixing the phase spiral will continue to wrap itself up into a more and more tightly wound pattern, until its structure can no longer be discerned observationally due to finite-$N$ noise and measurement errors in the actions and angles of individual stars (this is an example of coarse-grain mixing). Hence, even without lateral mixing phase spirals are only detectable for a finite duration. \section{Response to a localized perturbation} \label{sec:localized} In the previous section we investigated the slab response to an external disturbance with a single wavenumber $k$. Realistic perturbations are however localized in space and thus consist of many wavenumbers. In this section we shall look into what happens when the slab is hit by an impulsive perturbation that is spatially localized. For simplicity, we assume that the external perturber behaves as a Gaussian packet with half-width $\Delta_x$ along the $x$ direction, i.e., $\Phi_\rmP$ is given by equation~(\ref{Phip_sep}) with \begin{align} \calX(x) = \exp{\left[-x^2/2\Delta_x^2\right]}. \end{align} The $\calZ(z)$ term in equation~(\ref{Phip_sep}) denotes the vertical structure of the perturber potential, which is part of what dictates the relative strength of bending and breathing mode oscillations. We shall see in the next section, though, that the relative strength of the modes is mostly dictated by the form of $\calT(t)$. For simplicity, we only consider localization along the $x$ and $z$-directions; along the $y$-direction the perturbation is assumed to extend out to infinity. We emphasize, though, that this assumption does not impact the essential physics of phase-mixing and lateral mixing discussed below. The Fourier transform of the perturber potential, $\Phi_{nk}$, is given by equation~(\ref{Phip_sep_fourier}), with \begin{align} \calX_k = \frac{\Delta_x}{\sqrt{2\pi}}\, \exp{\left[-k^2\Delta_x^2/2\right]}. \label{Phink_gaussian} \end{align} Upon substituting the above expression for $\calX_k$ in equation~(\ref{f1_delta}) we obtain the response for a single $k$ mode, $f_{1k}$. After multiplying this by $\exp{\left[ikx\right]}$, integrating over all $k$ and summing over all $n$ modes, we obtain the following final form for the slab response density in the case of a (laterally) Gaussian perturber, \begin{align} f_1(I_z,w_z,x,t) &= \sum_{n=-\infty}^{\infty} \exp{\left[i n w_z\right]} \int_{-\infty}^{\infty} d k\, \exp{\left[ikx\right]}\, f_{1k}(I_z,w_z,t) \nonumber \\ &=A_{\rm norm}\, D(x,t)\, R(I_z,w_z,x,t), \label{f1_local_impulsive} \end{align} where \begin{align} A_{\rm norm}=\frac{\rho_c}{\sqrt{2\pi}\sigma_z} \exp{\left[-E_z/\sigma^2_z\right]} \end{align} is the same normalization factor as in equation~(\ref{f1_delta}), \begin{align} D(x,t)=\frac{\Delta_x}{\sqrt{\Delta_x^2+\sigma^2 t^2}} \exp{\left[-\frac{x^2}{2\left(\Delta_x^2+\sigma^2 t^2\right)}\right]} \end{align} is a factor that captures the decay of the response by lateral mixing, and \begin{align} R(I_z,w_z,x,t)=-\Phi_\rmN \sum_{n=-\infty}^{\infty}\calZ_n(I_z)\left[\frac{t}{\Delta_x^2+\sigma^2 t^2}\left(1-\frac{x^2}{\Delta_x^2+\sigma^2 t^2}\right)+i\frac{n\Omega_z}{\sigma^2_z}\right]\exp{\left[i n \left(w_z - \Omega_z\,t\right)\right]} \label{R_local_impulsive} \end{align} with $\calZ_n(I_z)$ given by equation~(\ref{Phip_sep_fourier_z}), corresponds to the remaining part of the response that includes vertical phase-mixing. The above expression (equation~[\ref{f1_local_impulsive}]) for the slab response to a localized disturbance has several important features. Firstly, the profile of the slab response is nearly Gaussian in $x$ since we assumed a Gaussian form (along $x$) for the perturber potential. Secondly, the $D(x,t)$ factor describes the decay of the response amplitude and widening of the response profile due to mixing by lateral streaming. The mixing in this case occurs as a power law in time rather than like a Gaussian as for a single $k$ mode (see equation~[\ref{f1_delta}]), since the power spectrum of the Gaussian perturber is dominated by small $k$ which mix very slowly, at a timescale, $\tau_\rmD\sim 1/k\sigma$. Thirdly, the $R$ factor captures two important effects: (i) a transient response reflecting an initial linear growth due to the perturber-induced velocity impulse, followed by a subsequent decay by lateral mixing, and (ii) vertical oscillations of stars (for $n\neq 0$) at different frequencies resulting in phase-mixing over time and the formation of phase spirals as described in detail in Section~\ref{sec:impulsive_kick}. The $n=0$ modes, i.e., perturbations confined to the slab, damp out faster than the non-zero $n$ modes that manifest the vertical oscillations of stars. Since the perturber was introduced impulsively by means of a Dirac delta function in time, the higher order oscillations are stronger for the same value of $\calZ_n(I_z)$ as the corresponding changes in the vertical actions have larger amplitude. Typically, for $n\geq 2$, $\calZ_n(I_z)$ gets smaller with larger $n$; hence the $n=2$ breathing mode turns out to be the dominant mode of oscillation for impulsive disturbances. The response characteristics however change as we move to non-impulsive or more temporally extended perturbers in the next section. It takes time for the local response to propagate along the slab by lateral streaming. Initially the perturber's gravity draws in stars towards the center of impact, $x=0$. Thus, immediately following the impulse, the region near the center of impact has a larger concentration of stars, which laterally stream outwards due to non-zero velocity dispersion. This leads to a damping of the response amplitude at small $x$ and growth at large $x$, or equivalently damping and widening of the response profile, which occurs at the rate, \begin{align} \calD_x(t) = \frac{d}{d t} \sqrt{\Delta^2_x + \sigma^2 t^2} = \frac{\sigma^2 t}{\sqrt{\Delta^2_x + \sigma^2 t^2}}. \end{align} This rate of outward streaming of slab material is initially equal to \begin{align} \lim_{t\to 0} \calD_x(t) = \frac{\sigma^2 t}{\Delta_x}, \end{align} but at later times asymptotes to a constant value, \begin{align} \lim_{t\to \infty} \calD_x(t) = \sigma. \end{align} To summarize, the response to a spatially localized perturbation can be understood in the context of that to a single mode plane wave perturbation discussed in the previous section, as follows. In both cases, the response involves vertical oscillations that phase-mix away, thus giving rise to phase spirals. However, whereas the plane wave response maintains its sinusoidal profile in the lateral direction with an overall Gaussian decay of the amplitude due to lateral mixing, the response profile in the case of localized perturbation changes its shape and undergoes both decay and widening. This is because in the latter case the response is a linear superposition of responses to many plane wave perturbations with different $k$, each decaying in amplitude over a time-scale, $\tau_\rmD\sim 1/k\sigma$, due to lateral mixing by free-streaming. Since the spatially Gaussian profile considered here has a Gaussian power spectrum and thus more power on large scales (small $k$) that mix more slowly, the combined response from all $k$ modes undergoes much slower lateral mixing (as a power law) than that from a single $k$ mode. The typical timescale of coarse-grained survival (against free-streaming damping) of the phase spiral in this case turns out to be $\sim (f_{\rm max}/f_{\rm res})\,\Delta_x/\sigma$. Here $f_{\rm max}$ is the maximum amplitude of the phase spiral, which is attained at $t=0$, and $f_{\rm res}$ is the resolution limit. The power law nature of free-streaming damping implies that the response to spatially and temporally localized perturbations (e.g., encounters with satellite galaxies) can be sustained in the disk for a long time. \section{Response to a non-impulsive perturbation} \label{sec:non-impulsive} Thus far we have only considered impulsive perturbations of our slab, with $\calT(t)=\delta(t)$. However, a realistic disturbance would not only have a spatial structure, the effects of which we studied in the previous section, but also be extended in time. In this section we investigate the effect of non-impulsive or temporally extended disturbances on the slab oscillations. In particular, we broaden the Dirac delta pulse from the previous section into a Gaussian in time, i.e., $\Phi_\rmP$ is given by equation~(\ref{Phip_sep}) with $\calT(t) = \frac{1}{\sqrt{\pi}}\,\exp{\left[-\omega^2_0 t^2\right]}$, where $\omega_0$ is the pulse frequency. We define the pulse-width or pulse-time as $\tau_\rmP=\sqrt{2}/\omega_0$. We also assume that the pulse is localized and follows a Gaussian profile in $x$ as in the previous section, i.e., $\calX(x)=\exp{\left[-x^2/2\Delta^2_x\right]}$. As before, $\calZ(z)$ in equation~(\ref{Phip_sep}) denotes some generic vertical profile. The (spatial) Fourier transform of this potential, $\Phi_{nk}$, is provided in equation~(\ref{Phip_sep_fourier}) with $\calX_k$ given by equation~(\ref{Phink_gaussian}) and $\calZ_n$ given by equation~(\ref{Phip_sep_fourier_z}). We can substitute this in equation~(\ref{f1nk_slabsol}) and perform the integration over $\tau$ and $v_x$ to obtain the following expression for the response for a single $k$ mode, \begin{align} &f_{1k}(I_z,w_z,t)=A_{\rm norm}\, D_k(t)\,R_k(I_z,w_z,t), \label{f1k_gaussian} \end{align} where \begin{align} A_{\rm norm}=\frac{\rho_c}{\sqrt{2\pi}\sigma_z} \exp{\left[-E_z/\sigma^2_z\right]} \end{align} is the same normalization factor as in equation~(\ref{f1_delta}), \begin{align} D_k(t) = \frac{\calQ^3}{2\omega_0}\, \exp{\left[-\calQ^2\frac{k^2\sigma^2t^2}{2}\right]} \end{align} is a factor that describes the damping of the response due to lateral mixing, and \begin{align} &R_{k}(I_z,w_z,t)=-\Phi_\rmN\calX_k \sum_{n=-\infty}^{\infty} \calZ_n(I_z) \left\{\, S_{nk} \, \Upsilon_{nk}(t) \, \left(k^2 t + i \frac{n\Omega_z}{\sigma^2_z}\right) \exp{\left[i\, n(w_z-\calQ\,\Omega_z t)\right]} - \calG_{nk}(w_z,t)\right\}, \label{Rk_gaussian} \end{align} with $\calZ_n(I_z)$ given by equation~(\ref{Phip_sep_fourier_z}), includes the vertical phase-mixing of the response. Here $\calQ$ is a factor that depends on the pulse frequency, $\omega_0$, and the wavenumber $k$, and is given by \begin{align} \calQ=\calQ(\omega_0,k\sigma)=\frac{\omega_0}{\sqrt{\omega^2_0+\frac{k^2\sigma^2}{2}}}. \end{align} The mode-strength, \begin{align} S_{nk} = \exp{\left[-\frac{1}{\omega^2_0+\frac{k^2\sigma^2}{2}}\frac{n^2\Omega^2_z}{4}\right]} \label{mode_strength_gaussian} \end{align} is a function that indicates the strength of each $n$ mode, \begin{align} \Upsilon_{nk}(t) &= 1+\erf\left\{\calQ \left(\omega_0 t-i\frac{n\Omega_z}{2\omega_0}\right)\right\} \label{growth_gaussian} \end{align} describes the temporal build-up of the response and the decay of transient oscillations, and \begin{align} \calG_{nk}(w_z,t) = \frac{k^2}{\sqrt{\pi}\,\omega_0\calQ} \exp{\left[-\calQ^2\omega^2_0 t^2\right]} \exp{\left[in w_z\right]} \label{transient_gaussian} \end{align} is another rapidly decaying transient feature. In the $\omega_0 \to \infty$ limit, both $\Upsilon_{n}(t)$ and the mode strength $S_{nk}$ become unity, and $\calG_{nk}(w_z,t) \to 0$, such that we recover the response to impulsive perturbations given in equation~(\ref{f1_delta}) as required. It is interesting to contrast this response to an extended pulse to that in the impulsive limit. First of all, the damping factor, $D_k(t)$, which still owes its origin to lateral mixing due to non-zero velocity dispersion, now depends not only on $k$ and $\sigma$ but also on the pulse frequency $\omega_0$. The damping time is given by \begin{align} \tau_{\rmD} = \frac{1}{k\sigma} \sqrt{1+\frac{k^2\sigma^2}{2\omega^2_0}}, \end{align} which scales as $\sim 1/k\sigma$ in the impulsive/short pulse ($\omega^2_0 \gg k^2\sigma^2/2$) limit indicating that the response mixes away laterally with small scale perturbations mixing faster. In the adiabatic/long pulse ($\omega^2_0 \ll k^2\sigma^2/2$) limit, though, $\tau_\rmD \to 1/\sqrt{2}\omega_0$, i.e., the damping of the response follows the temporal behaviour of the perturbing pulse itself, independent of $k$. The mode-strength reveals several important trends: it exponentially damps away with $n^2$, implying that the lower order modes are much stronger for perturbations that are slower \citep[see also][]{Widrow.etal.14} and/or have larger wavelength (smaller $k$). Therefore the $n=1$ bending modes dominate over the $n=2$ breathing modes for a sufficiently slow pulse. Note, though, that if the pulse is too slow ($\omega_0 \to 0$) the mode strength is super-exponentially suppressed, especially at large scales (small $k$), or if the slab has a small lateral velocity dispersion, $\sigma$, compared to that along the vertical direction, $\sigma_z$ (recall that $\Omega_z\sim \sigma_z/h_z$). This is a classic signature of adiabatic shielding of the slab due to the averaging out of the net response to zero by many oscillations of stars within the (very long) perturbation timescale \citep[cf.][]{Weinberg.94a,Weinberg.94b,Gnedin.Ostriker.99}. Finally, if the perturbation is not impulsive the frequency with which the slab stars oscillate in the vertical direction is modified with respect to their natural frequency according to \begin{align} \Omega_z \to \frac{\omega^2_0}{\omega^2_0+\frac{k^2\sigma^2}{2}} \Omega_z, \end{align} which goes to $\Omega_z$ in the impulsive limit, as expected. For slower pulses however, the vertical motion of the stars couples to the lateral motion \citep[see also][]{Binney.Schonrich.18}, resulting in a reduced oscillation frequency, especially for smaller wavelengths (larger $k$). In the extremely slow/adiabatic limit, $\Omega_z \to 0$, signalling a lack of vertical phase-mixing. This is easy to understand; a forced oscillator remains in phase with the perturber if the frequency of the latter is much lower than the natural frequency. In fact, in the adiabatic limit, the response only consists of resonant stars, for which $n\Omega_z+k v_x=0$ (see Appendix~\ref{App:ad_lim_resp}), and thus no phase spiral emerges. The above response corresponds to a temporally Gaussian pulse for a fixed wavenumber $k$. To get the full response to a localized perturber, we substitute the expression for $\calX_k$ given in equation~(\ref{Phink_gaussian}), in the $k$-response of equation~(\ref{f1k_gaussian}), multiply it by $\exp{\left[ikx\right]}$ and integrate over all $k$. The resultant response is an oscillating function of $w_z$ and has a profile along $x$ which varies with time. For the short pulse/impulsive case, we recover the expression given in equation~(\ref{f1_local_impulsive}). In Fig.~\ref{fig:gaussian_slab_x} we plot the amplitude (relative to the unperturbed DF) of this oscillating response (normalized by the $z$ Fourier component of the perturber potential, $\calZ_n$) as a function of $x$. The columns correspond to four different times since the time of maximum pulse strength, and the rows correspond to two different pulse-times, as indicated. The solid and dashed lines indicate the bending ($n=1$) and breathing ($n=2$) modes, respectively. The short pulse response shown in the upper panels has a Gaussian profile centered on the point of impact at $x=0$ with the initial width very similar to that of the $\Phi_\rmP$ profile (see equations~[\ref{f1_local_impulsive}]-[\ref{R_local_impulsive}]). Over time, this response profile gets weaker and wider like a power law, as the unconstrained lateral motion of the stars causes an outward streaming, and thus decay, of the response. The long pulse response in the lower panels has a different, more extended profile than in the short pulse case; it exhibits some ripples along $x$ besides having an overall smooth behaviour (see Appendix~\ref{App:ad_lim_resp} for the response derived in the adiabatic limit). As time goes on, the response decays away and widens out due to lateral mixing. Unlike the short pulse case, here the response initially decays like $\sim \exp{\left[-\omega^2_0 t^2\right]}$ over a timescale of the pulse-time, $\tau_\rmP=\sqrt{2}/\omega_0$, before attaining a power law decay at large time. The temporal behaviour of the response becomes even clearer in Fig.~\ref{fig:gaussian_slab_t}, where we plot the amplitude of the response as a function of time at two different positions on the slab (different rows), and for three different pulse-times (different columns). As before, the solid and dashed lines indicate the $n=1$ and $n=2$ modes, respectively. Initially the slab response grows nearly hand in hand with the perturbing pulse. This is captured by the $\Upsilon_{nk}(t)$ term (equation~[\ref{growth_gaussian}]) in the expression for $R_k(I_z,w_z,t)$, which scales as $\exp{\left[-\calQ^2 \omega^2_0 t^2\right]}$ at small $t$, but asymptotes to a constant value of $2$ at late times. As the perturber strength falls off, the response decays as a Gaussian for each $k$, as described by the damping factor, $D_k(t) \propto \exp[-{\calQ}^2k^2\sigma^2 t^2/2]$. The combined response from all $k$ however decays at a different rate. For the shortest pulse, for which the response asymptotes to that given by equation~(\ref{f1_local_impulsive}), the damping factor, $D(x,t) \propto 1/t$ at late times. In the intermediate and long pulse cases, the response initially tends to follow the same $\sim \exp{\left[-\omega^2_0 t^2\right]}$ decay as the perturbing pulse, before finally transitioning to a power law fall-off, which typically occurs as $\sim 1/t$, just as in the short pulse case. Importantly, this transition sets in later for longer lasting pulses, such that the late-time response for slower perturbations is drastically suppressed with respect to faster perturbations. From the bottom panels, it is evident that the region ($x=10 h_z$) farther away from the center of impact responds later, with a time-lag of $\Delta t= 10\, h_z/\sigma$ (timescale of lateral streaming), which is $\sim 115$ Myr for the typical MW parameter values adopted here. The breathing mode is the dominant mode in the short pulse case ($\tau_\rmP = 10 \Myr$) while in both the intermediate ($\tau_\rmP = 50 \Myr$) and long ($\tau_\rmP = 100 \Myr$) pulse scenarios the bending mode eventually dominates. Note, though, that if the pulse becomes too long, the long-term response is adiabatically suppressed. Hence, there is only a narrow window in pulse-widths for which bending modes dominate and cause a detectable response. In the next section we examine whether any of the MW satellites have encounters with the disk over timescales that fall in this regime. The response formalism for localized, non-impulsive perturbations developed so far can be used to model the response to transient bars and spiral arms. Encounters with such features can cause transient vertical perturbations in the potential over timescales comparable to the vertical oscillation periods of stars, thereby creating phase-spirals. We discuss this in detail in Paper~II for realistic disk galaxies. \section{Encounters with satellite galaxies} \label{sec:sat_encounter} In all cases considered above we have made the simplifying assumption that the perturber potential is separable, i.e., can be written in the form of equation~(\ref{Phip_sep}). However a realistic perturber is seldom of such simple form. For example, the potential due to an impacting satellite galaxy or DM subhalo (approximated as a point perturber) cannot be written in separable form, thereby making the analysis significantly more challenging. In this section, as an astrophysical application of the perturbative formalism developed in this paper, we compute the response of the infinite slab to a satellite encounter. We relegate the far more involved computation of the response of a realistic disk to an impacting satellite to Paper~II. As shown in Appendix~\ref{App:sat_disk_resp}, the $n\neq 0$ response to a satellite impacting the slab with a uniform velocity $\vp$ along a straight orbit at an angle $\thetap$, at a distance $x$ from the point of impact, can be approximated as \begin{align} f_1(I_z,w_z,x,t) = \frac{\rho_c}{\sqrt{2\pi}\sigma_z} \exp{\left[-E_z/\sigma^2_z\right]}\times i\frac{2G\Mp}{\vp} \sum_{n=-\infty}^{\infty} \frac{n\Omega_z}{\sigma^2_z}\, \Psi_n(x,I_z)\, \exp{\left[i\,\frac{n\Omega_z \sin{\thetap}}{\vp}x\right]} \exp{\left[i n\left(w_z-\Omega_z t\right)\right]}, \label{f1_sat} \end{align} where \begin{align} \Psi_n(x,I_z)&= \frac{1}{2\pi} \int_0^{2\pi} d w_z\, \exp{\left[-i n \left(w_z - \frac{\Omega_z \cos{\thetap} z}{\vp}\right)\right]} K_0\left[\,\left|\frac{n\Omega_z \left(x\cos{\thetap}-z\sin{\thetap}\right)}{\vp}\right|\,\right], \label{Psi_n} \end{align} with $K_0$ the zero-th order modified Bessel function of the second kind. This expression for the response is only valid far away from the point of impact ($x\gtrsim \sigma t$), such that the response can be approximated as a plane wave along $x$, and at late times, after the perturber has moved far enough away from the disk, i.e., for $t \gg (x\sin{\thetap}+z\cos{\thetap})/\vp$). There are several salient features of this response that deserve special attention. The strength of the response is dictated by the $K_0$ function whose argument depends on $\Omega_z \cos{\thetap}\, x/\vp$ (for small $I_z$), which is basically the ratio of the encounter timescale, \begin{align} \tau_{\rm enc} = \frac{x\cos{\thetap}}{\vp}, \label{tau_enc} \end{align} and the vertical dynamical time of the stars, \begin{align} \tau_z = \frac{1}{\Omega_z} \sim \frac{h_z}{\sigma_z}. \label{tau_z} \end{align} From the asymptotic limits of $K_0$ it follows that the response scales as a power law ($\sim \vp^{-1}$) in the impulsive ($\tau_{\rm enc}\ll \tau_z$) limit and as $\sim\exp{\left[-\left|n\Omega_z\cos{\thetap}\right|x/\vp\right]}$ in the adiabatic ($\tau_{\rm enc}\gg \tau_z$) limit. The response peaks roughly at the maximum of the $K_0$ function, which occurs when the encounter timescale is comparable to the vertical dynamical time of the stars, i.e., when $\tau_{\rm enc}\approx 0.6\,\tau_z$, or in other words the `resonance' condition, \begin{align} \frac{x\cos{\thetap}}{\vp} \approx \frac{0.6}{\Omega_z}, \end{align} is satisfied. For encounters faster than this, the response is suppressed like a power law, while for slower encounters it is exponentially suppressed. The $\vp^{-1}$ scaling of the response in the impulsive limit is a well known feature of impulsive perturbations \citep[e.g.,][]{Spitzer.58, Aguilar.White.85, Weinberg.94a, Weinberg.94b, Gnedin.etal.99, Banik.vdBosch.21b}, and the exponential suppression is a telltale signature of adiabatic shielding\footnote{While the adiabatic response in one degree-of-freedom cases, e.g., the vertical phase spiral in the isothermal slab, is exponentially suppressed, that in multiple degree-of-freedom systems such as inhomogeneous disks is usually not because of resonances \citep[][]{Weinberg.94a,Weinberg.94b}.}, similar to the adiabatic suppression of the mode-strength factor in the response to slow Gaussian pulses discussed in section~\ref{sec:non-impulsive}. While the response is heavily damped for very slow encounters, something interesting happens in the mildly slow regime, $\tau_{\rm enc}=x\cos{\thetap}/\vp \gtrsim \tau_z$. In this regime, the ratio of the $n=2$ breathing to the $n=1$ bending mode response scales as \begin{align} f_{21} \equiv {f_{1,n=2} \over f_{1,n=1}} \sim \sqrt{2}\,\exp{\left[-\frac{\Omega_z\cos{\thetap}\,x}{\vp}\right]}. \label{f21} \end{align} Thus the bending mode response exponentially dominates over that of the breathing mode for slower (smaller $\vp$), more distant (large $x$), and more perpendicular ($\thetap\approx 0$) encounters. The bending mode is also more pronounced for stars with larger $\Omega_z$ or equivalently smaller $I_z$. On the other hand, for encounters with $\tau_{\rm enc}=x\cos{\thetap}/\vp < \tau_z$, the breathing modes dominate. Finally, the slab response to the impacting satellite, given in equation~(\ref{f1_sat}), consists of oscillating functions of time, lateral distance $x$, and the vertical oscillation amplitude, $\sqrt{2I_z/\nu}$ (see equations~[\ref{Psi_n_app_epi}] and [\ref{Phin_sat_app}]). This implies that the satellite not only induces temporal oscillations, which give rise to phase-mixing and thus phase spirals due to different oscillation frequencies of the stars (see section~\ref{sec:impulsive_kick}), but also spatial corrugations. These vertical and lateral waveforms have wavenumbers given by \begin{align} k_z = \frac{n\Omega_z\cos{\thetap}}{\vp},\;\;\;\;\;\;\;{\rm and}\;\;\;\;\;\;\;\; k_x = \frac{n\Omega_z\sin{\thetap}}{\vp}, \end{align} respectively. Thus, perpendicular impacts induce only vertical corrugations while planar ones excite waves only laterally. An inclined encounter, on the other hand, spawns corrugations along both directions. Both wavelengths get longer with decreasing mode order, increasing impact velocity, and decreasing vertical frequencies, i.e., increasing actions. \subsection{Impact of satellite galaxies on the Milky Way disk} \label{sec::MW_satellites} The MW halo harbors many satellite galaxies. Some of these are quite massive, with DM halo mass comparable to the disk mass, and either underwent or are about to undergo an encounter with the MW disk within a few hundred Myr from the present day. Hence we expect at least some of them to perturb the disk significantly. Here we use existing data on MW satellites to obtain a rough estimate of the disk response to their encounters with the MW stellar disk. Our formalism provides physical insight into the trends and scalings of the disk response as a function of impact parameters and velocities of the MW satellites. We emphasize upfront, though, that the precise numerical estimates of the responses are to be taken with a grain of salt. These estimates only serve as a crude, order-of-magnitude attempt to compare the relative disk responses to different satellite galaxies. As discussed in more detail in section~\ref{sec::caveats}, these estimates are subject to a number of oversimplifications and caveats. First of all, the MW disk is modelled as an isothermal slab, and we only consider the {\it direct} impact of the satellites. We ignore indirect effects due to the self-gravity of the response. Our approach also ignores the presence of a dark matter halo, which can impact the disk response in several ways (see section~\ref{sec::caveats}). Because of all these shortcomings, we caution against using the following response estimates for comparison with actual data and/or detailed numerical simulations. We consider the MW satellites with parallax and proper motion measurements from Gaia DR2 \citep[][]{Gaia_collab_sat.18b} and the corresponding galactocentric coordinates and velocities computed and documented by \cite{Riley.etal.19} \citep[table A.2, see also][]{Li.etal.20} and \cite{Vasiliev.Belokurov.20}. Of these, we only consider the satellites with known dynamical mass estimates \citep[][]{Simon.Geha.07,Bekki.Stanimirovic.09,Lokas.09,Erkal.etal.19}. Adopting the initial conditions for galactocentric positions ($R,z,\phi$) and velocities ($v_R,v_z,v_\phi$) as the median values quoted by \cite{Riley.etal.19} and \cite{Vasiliev.Belokurov.20}, we simulate the orbits of the galaxies in the combined gravitational potential of the MW halo, disk and bulge, which are respectively modelled by a spherical NFW \citep[][]{Navarro.etal.97} profile (virial mass $M_h=9.78\times 10^{11}\Msun$, scale radius $r_h=16$ kpc, and concentration $c=15.3$), a Miyamoto-Nagai \citep[][]{Miyamoto.Nagai.75} profile (mass $M_d=9.5\times 10^{10} \Msun$, scale radius $a=4$ kpc, and scale-height $b=0.3$ kpc), and a spherical \cite{Hernquist.90} profile (mass $M_b=6.5\times 10^9\Msun$ and scale radius $r_b=0.6$ kpc)\footnote{Our MW potential is similar to {\tt GALPY MWPOTENTIAL2014} \citep[][]{Bovy.15} except for the power-law bulge which has been replaced by an equivalent Hernquist bulge.}. The total mass of our fiducial MW model is thus $1.08\times 10^{12}\Msun$. We evolve the positions and velocities of the satellites both forwards and backwards in time from the present day, using a second order leap-frog integrator. For simplicity, we ignore the effect of dynamical friction\footnote{Dynamical friction might play an important role in the orbital evolution of massive satellites like the Large Magellanic Cloud (LMC) and Sgr, pushing their orbital radius farther out in the past.}. From each individual orbit, we note the time, $t_{\rm cross}$, when the satellite crosses the disk (i.e., crosses $z=0$), and record the corresponding distance, $\xp$, from the Sun, which we integrate backwards/forwards in time using a purely circular orbit up to $t_{\rm cross}$. We also record the velocity, $\vp=\sqrt{v^2_R+v^2_z+v^2_\phi}$, and the angle of impact with respect to the disk normal, $\thetap=\cos^{-1}{(v_z/\vp)}$. Finally, we compute the disk response to the satellite encounter using equation~(\ref{f1_sat}). Results are summarized in Table~\ref{tab:MW_sat_resp} and Figs.~\ref{fig:satellite_constraints1} and~\ref{fig:satellite_constraints2}. In Fig.~\ref{fig:satellite_constraints1}, we plot the impact parameter, $\xp\cos{\thetap}$ (with respect to the Sun), as a function of the encounter velocity, $\vp$, of the satellites, for the penultimate (left-hand panel), last (middle panel), and next (right-hand panel) disk crossings. The red (grey) symbols denote the satellites that induce a strong (weak) amplitude of the bending mode response, $f_{1,n=1}/f_0$, for $I_z=h_z\sigma_z = 9.2 \kpc\kms$. As shown in Appendix~\ref{App:detect_crit}, we consider $f_{1,n=1}/f_0=\delta=10^{-4}$ as a rough estimate for the minimum detectable relative response, i.e., the boundary between strong and weak responses to satellite passage. The solid black line indicates the boundary between bending and breathing modes, i.e., where the breathing-to-bending ratio, $f_{21}$ (equation~[\ref{f21}]), is equal to unity. Hence, the blue and red shaded regions indicate where the response is dominated by bending and breathing modes, respectively. The magenta, dashed line roughly denotes the boundary between a strong bending response (blue shaded region) and a response that is adiabatically suppressed (grey shaded region). The latter is defined by the condition $\exp{\left[-\Omega_z \xp \cos{\thetap}/\vp\right]} < \delta = 10^{-4}$. In Fig.~\ref{fig:satellite_constraints2}, we plot the amplitude of the bending mode response, $f_{1,n=1}/f_0$ (upper panel), and the breathing-to-bending ratio, $f_{21}=f_{1,n=2}/f_{1,n=1}$ (lower panel), in the Solar neighborhood, as a function of the time $t_{\rm cross}$ (in Gyr) when the satellite crosses the plane of the disk, assuming the fiducial MW parameters. Negative and positive $t_{\rm cross}$ correspond to disk crossings in the past and future, respectively, and we once again consider stars with $I_z = h_z \sigma_z = 9.2 \kpc\kms$. Both Fig.~\ref{fig:satellite_constraints1} and the lower panel of Fig.~\ref{fig:satellite_constraints2} make it clear that {\it all} the disk crossings considered here preferentially excite bending rather than breathing modes in the Solar neighborhood. As shown in Section~\ref{sec:impulsive_kick} these trigger one-armed phase spirals in the Solar neighborhood, in qualitative agreement with the MW snail observed in the Gaia data. However, as is evident from the upper panel of Fig.~\ref{fig:satellite_constraints2}, most satellites only trigger a minuscule response in the disk, with $f_{1,n=1}/f_0 < \delta = 10^{-4}$, either because the satellite has too low mass, or because the encounter, from the perspective of the Sun, is too slow such that the local response is adiabatically suppressed. The strongest response by far is triggered by encounters with Sgr, for which the bending mode response, $f_{1,n=1}/f_0$, is at least $1-2$ orders of magnitude larger than that for any other satellite. Based on our orbit-integration, it had its penultimate disk crossing, which closely coincides with its last pericentric passage, about $900\Myr$ ago, triggering a strong response of $f_{1,n=1}/f_0 \sim 0.04$ in the Solar neighborhood. The last disk crossing, which nearly corresponds to the last apocentric passage, occurred about $300\Myr$ ago, triggering a very weak (adiabatically suppressed) response. Sgr is currently near its pericenter and will undergo the next disk crossing in about $30\Myr$, which we estimate to only trigger a moderately strong response with $f_{1,n=1}/f_0\sim 0.001$. We caution, though, that in addition to the caveats listed above and in Section~\ref{sec::caveats} these estimates ignore dynamical friction and are sensitive to the MW potential and the current phase-space coordinates of the satellites. We have checked that a heavier MW model with a total mass of $1.5\times 10^{12}\Msun$ does not change the relative amplitudes of the satellite responses significantly, but brings most of the disk crossing times closer to the present day since the satellites are more bound in a heavier MW. For example, the previous pericentric and apocentric passages of Sgr occur at $\sim 600$ and $200\Myr$ ago in the heavier case. The only satellite apart from Sgr that triggers a response $f_{1,n=1}/f_0 > \delta = 10^{-4}$ is Hercules, whose disk crossing $\sim 500\Myr$ ago caused a bending-mode response, $f_{1,n=1}/f_0 = 1.2\times 10^{-4}$. Segue 2 induces a response that is marginally below the detection threshold. Disk crossings of LMC and Leo II trigger responses that are comparable in strength to that of Hercules, but the crossing times are too far in the past or future for them to be considered as candidates for triggering the Gaia snail. All in all, it is clear then that Sgr is by far the most likely candidate among the MW satellite galaxies considered here to have triggered the one-armed phase spiral in the Solar neighborhood discovered in Gaia DR2 by \citet{Antoja.etal.18}. \begin{table*} \centering \hspace{-3cm} \tabcolsep=0.2 cm \begin{tabular}{c|c|cc|cc|cc} \hline MW satellite & Mass & $f_{1,n=1}/f_0$ & $t_{\rm cross}$ & $f_{1,n=1}/f_0$ & $t_{\rm cross}$ & $f_{1,n=1}/f_0$ & $t_{\rm cross}$ \\ name & $(\Msun)$ & & $(\Gyr)$ & & $(\Gyr)$ & & $(\Gyr)$ \\ & & Penultimate & Penultimate & Last & Last & Next & Next \\ (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) \\ \hline \highlight{Sagittarius} & $10^9$ & \highlight{$4.3\times 10^{-2}$} & \highlight{$-0.92$} & $1.4\times 10^{-10}$ & $-0.3$ & $8.3\times 10^{-4}$ & $0.03$ \\ Hercules & $7.1\times 10^6$ & -- & $-3.57$ & $1.2\times 10^{-4}$ & $-0.51$ & $6.4\times 10^{-5}$ & $3.16$ \\ Leo II & $8.2\times 10^6$ & -- & $-3.61$ & $3.5\times 10^{-5}$ & $-1.81$ & $9.3\times 10^{-5}$ & $2.34$ \\ Segue 2 & $5.5\times 10^5$ & $5\times 10^{-5}$ & $-0.84$ & $3.4\times 10^{-5}$ & $-0.25$ & $1.8\times 10^{-6}$ & $0.27$ \\ LMC & $1.4\times 10^{11}$ & $1.4\times 10^{-4}$ & $-6.97$ & -- & $-2.37$ & $7.2\times 10^{-5}$ & $0.12$ \\ SMC & $6.5\times 10^9$ & $3.6\times 10^{-8}$ & $-3.22$ & -- & $-1.39$ & $1.2\times 10^{-9}$ & $0.22$ \\ Draco I & $2.2\times 10^7$ & -- & $-2.43$ & $5\times 10^{-7}$ & $-1.23$ & $1\times 10^{-7}$ & $0.24$ \\ Bootes I & $10^7$ & -- & $-1.65$ & $4.1\times 10^{-7}$ & $-0.35$ & -- & $0.87$ \\ Willman I & $4\times 10^5$ & -- & $-0.63$ & $1.4\times 10^{-7}$ & $-0.21$ & $2.5\times 10^{-8}$ & $0.4$ \\ Ursa Minor & $2\times 10^7$ & -- & $-2.26$ & $5.5\times 10^{-8}$ & $-1.16$ & $8.6\times 10^{-9}$ & $0.29$ \\ Ursa Major II & $4.9\times 10^6$ & $4.5\times 10^{-8}$ & $-2$ & $6.2\times 10^{-10}$ & $-0.1$ & -- & $0.9$ \\ Coma Berenices I & $1.2\times 10^6$ & $7\times 10^{-10}$ & $-2.47$ & -- & $-0.25$ & -- & $0.69$ \\ Sculptor & $3.1\times 10^7$ & -- & $-2.7$ & $2\times 10^{-10}$ & $-0.46$ & -- & $1.47$\\ \hline \end{tabular} \caption{MW disk response to satellites for stars with $I_z=h_z\sigma_z$ in the Solar neighborhood. Column (1) indicates the name of the MW satellite and Column (2) indicates its dynamical mass estimate from literature \citep[][]{Simon.Geha.07,Bekki.Stanimirovic.09,Lokas.09,Erkal.etal.19,Vasiliev.Belokurov.20}. We assume $10^9\Msun$ for the Sagittarius mass; note that there is a discrepancy between its measured mass of $\sim 4\times 10^8\Msun$ \citep[][]{Vasiliev.Belokurov.20} and the required mass of $10^9-10^{10}\Msun$ for observable phase spiral signatures in N-body simulations \citep[see for example][]{Bennett.etal.21}. Columns (3) and (4) respectively indicate the bending mode response assuming fiducial MW parameters and the crossing time for the penultimate disk-crossing. Columns (5) and (6) show the same for the last disk-crossing, while columns (7) and (8) indicate it for the next one. Only the satellites that trigger a bending mode response, $f_{1,n=1}/f_0\geq 10^{-10}$, in at least one of the three cases are shown. The responses smaller than $10^{-10}$ are considered far too adiabatic and negligible and are marked by dashes. The case most relevant for the Gaia phase spiral is highlighted in red.} \label{tab:MW_sat_resp} \end{table*} We emphasize that the results shown in Figs.~\ref{fig:satellite_constraints1} and~\ref{fig:satellite_constraints2} correspond to stars with a vertical action $I_z=h_z\sigma_z=9.2\, \kpc\kms$. As mentioned above, the strength of the response depends on the ratio of the encounter time scale, $\tau_{\rm enc}$ (equation~[\ref{tau_enc}]) and the vertical oscillation period of stars in the Solar neighborhood, $\tau_z$ (equation~[\ref{tau_z}]). The latter is longer for stars with larger vertical action, and from the perspective of such stars the encounter is more impulsive, resulting in a stronger response. Since the response does not scale linearly with $\tau_{\rm enc}/\tau_z$, the relative response strength of different satellites depends somewhat on the vertical action. We have verified that for $I_z/(h_z\sigma_z) < 3$, which is roughly the range covered by the Gaia phase spiral, the direct response from the encounter with Sgr remains larger than that of any other satellite considered here by at least $1-2$ orders of magnitude. However, for stars with larger actions (larger vertical excursions), the LMC can dominate the response. In particular, for stars with $I_z/(h_z\sigma_z) \gtrsim 6.5$ ($z_{\rm max}\gtrsim 4\, h_z$), which make up the thick disk, the LMC is expected to trigger a stronger response than Sgr during its upcoming disk crossing. To summarize, our analysis suggests that the MW satellites during their most recent and forthcoming disk crossings preferentially excite bending modes in the Solar neighborhood. This is because satellite encounters are fairly distant from the Sun and thus the encounter time exceeds the vertical oscillation time of the stars. However, as previously discussed in section~\ref{sec:sat_encounter} and as evident from the N-body simulation of MW-Sgr encounter by \cite{Hunt.etal.21} (especially the earlier disk passages of Sgr), a satellite passage can trigger breathing modes closer to the point of impact, where the encounter is more impulsive. Since almost all the MW satellites undergo their disk-crossings at $R\gg 8\kpc$, future observations of the outskirts of the disk might reveal breathing instead of bending mode oscillations if they are excited by any of the satellites considered here. \subsection{Caveats} \label{sec::caveats} The above calculation of the response of the MW disk to perturbations is subject to a number of oversimplifications and caveats discussed below. The MW disk is modelled as an isothermal slab, which lacks the axisymmetric density profile and velocity structure that characterize a realistic disk. In particular, whereas the lateral motion in our slab is uninhibited, the in-plane motion in a realistic disk consists of an azimuthal rotation combined with a radial epicyclic motion. Among others, this will have important implications for the global disk response and the rate at which phase spirals damp out due to lateral mixing. In Paper~II (Banik et al., in preparation) we apply our perturbative formalism to a realistic self-gravitating disk galaxy with a pseudo-isothermal distribution function \citep[][]{Binney.10}, and consider both external perturbations (encounters with satellites) and internal perturbations (bars and spiral arms). All responses calculated in this paper only account for the direct response to a perturbing potential. In general, though, the response also has an indirect component that arises from the fact that neighboring regions in the disk interact with each other gravitationally. This self-gravity of the response, which we have ignored, triggers long-lived normal mode oscillations of the slab that are not accounted for in our treatment. Several simulation-based studies have argued that including self-gravity is important for a realistic treatment of phase spirals \citep[e.g.,][]{Darling.Widrow.19a, Bennett.Bovy.21}. Using the Kalnajs matrix method \citep[][]{Kalnajs.77, Binney.Tremaine.08}, we have made some initial attempts to include the self-gravity of the response in our perturbative analysis, along the lines of \citet{Weinberg.91}. Our preliminary analysis shows that the self-gravitating response is a linear superposition of two terms: (i) a continuum of modes given in equation~(\ref{f1nk_slabsol}), dressed by self-gravity, that undergo phase-mixing and give rise to the phase spiral, and (ii) a discrete set of modes called point modes or normal modes \citep[c.f.][]{Mathur.90,Weinberg.91} that follow a dispersion relation. The continuum response can be amplified by self-gravity when the continuum frequencies, $n\Omega_z+k v_x$, are close to the point mode frequencies, $\nu$. Depending on the value of $k$, the normal modes can be either stable or unstable. \cite{Araki.85} find that in an isothermal slab the bending normal mode undergoes fire hose instability below a certain critical wavelength if $\sigma_z/\sigma \lesssim 0.3$ while the breathing normal mode becomes unstable above the Jeans scale. In the regime of stability, the normal modes are undamped oscillation modes in absence of lateral streaming \citep[][]{Mathur.90} but are Landau damped otherwise \citep[][]{Weinberg.91}. For an isothermal slab with typical MW-like parameter values, the point modes are strongly damped since their damping timescale (inverse of the imaginary part of $\nu$) is of order their oscillation period (inverse of the real part of $\nu$), which turns out to be of order the vertical dynamical time, $h_z/\sigma_z$. Moreover, the normal mode oscillations are coherent oscillations of the entire system, independent of the vertical actions of the stars, and are decoupled from the phase spiral in linear theory since the full response is a linear superposition of the two. Based on the above arguments, we conclude that self-gravity has little impact on the evolution of phase spirals in the isothermal slab, at least in the linear regime. We emphasize that \citet{Darling.Widrow.19a}, who found their phase spirals to be significantly affected by the inclusion of self-gravity, assumed a perturber-induced velocity impulse with magnitude comparable to the local velocity dispersion in the Solar neighborhood; hence their results are likely to have been impacted by non-linear effects. Moreover, the self-gravitating response of an inhomogeneous disk embedded in a dark matter halo, as in the simulations of \citet{Darling.Widrow.19a}, can be substantially different from that of the isothermal slab. We intend to include a formal treatment of self-gravity along the lines of \citet{Weinberg.91} in future work. The disk of our MW is believed to be embedded in an extensive dark matter halo, something we have not taken into account. The presence of such a halo has several effects. First of all, the satellite not only perturbs the disk, but also the halo. In particular, it induces both a local wake and a global modal response\footnote{The torque from the local as well as global halo response is responsible for dynamical friction acting on the satellite.} \citep[e.g.,][]{Weinberg.89, Tamfal.etal.21}. The former typically trails the satellite galaxy, and boosts its effective mass by about a factor of two \cite[][]{Binney.Tremaine.08}, which might boost the (direct) disk response by about the same factor. The global halo response is typically dominated by a strong $l=1$ dipolar mode followed by an $l=2$ quadrupolar mode \citep[][]{Tamfal.etal.21}, which might have a significant impact on the disk. The presence of a halo also modifies the total potential. At large disk radii and vertical heights, the halo dominates the potential and will therefore significantly modify the actions and frequencies of the stars, and consequently the shape of the phase spirals. Finally, since the disk experiences the gravitational force of the halo, a (sufficiently massive) satellite galaxy can excite normal mode oscillations of the disk in the halo \citep[see for example][]{Hunt.etal.21}. We intend to incorporate some of these effects of the MW halo in Paper~II. \section{Conclusion} \label{sec:concl} In this paper we have used linear perturbation theory to compute the response of an infinite, isothermal slab to various kinds of external perturbations with diverse spatio-temporal characteristics. Although a poor description of a realistic disk galaxy, the infinite, isothermal slab model captures the essential physics of perturbative response and collisionless equilibration via phase-mixing in the disk, and thus serves as a simple yet insightful case for investigation. We use a hybrid (action-angle variables in the vertical direction and position-momentum variables in the lateral direction) linear perturbative formalism to perturb and linearize the collisionless Boltzmann equation and compute the response in the distribution function of the disk to a gravitational perturbation. We have considered external perturbations of increasing complexity, ranging from an instantaneous (laterally) plane-wave perturbation (Section~\ref{sec:impulsive_kick}), an instantaneous localized perturbation, represented as a wave-packet (Section~\ref{sec:localized}), a non-impulsive, temporally extended, localized perturbation (Section~\ref{sec:non-impulsive}), and ultimately an encounter with a satellite galaxy moving along a straight-line orbit (Section~\ref{sec:sat_encounter}). This multi-tiered approach is ideal for developing the necessary insight into the complicated response that is expected from a realistic disk galaxy exposed to a realistic perturbation. We summarize our conclusions below. \begin{itemize} \item The two primary Fourier modes of slab oscillation are the $n=1$ bending mode and the $n=2$ breathing mode, which correspond to anti-symmetric and symmetric oscillations about the mid-plane, respectively. For a sufficiently impulsive perturbation, the dominant mode is the breathing mode, which initially causes a quadrupolar distortion in the $(z,v_z)$-phase space, that evolves into a two-armed phase spiral as the stars with different vertical actions oscillate with different vertical frequencies. If the perturbation is temporally more extended (less impulsive), the dominant mode is the bending mode. This causes a dipolar distortion in $(z,v_z)$-phase space that evolves into a one-armed phase spiral \citep[see also][]{Hunt.etal.21, Widrow.etal.14}. Due to vertical phase-mixing, the phase spiral wraps up tighter and tighter until it becomes indistinguishable from an equilibrium distribution in the coarse-grained sense. \item Besides vertical phase-mixing the survivability of the phase spiral is also dictated by the lateral streaming motion of stars. The initial lateral velocity impulse towards the minima of $\Phi_\rmP$ tends to linearly boost the contrast of the phase spiral. This is however quickly taken over by lateral streaming (with velocity dispersion $\sigma$), which causes mixing between the over- and under-densities, and damps out the phase spiral amplitude. For an impulsive, laterally sinusoidal perturbation, the disk response is also sinusoidal and damps out like a Gaussian (due to the Maxwellian/Gaussian distribution of the unconstrained lateral velocities) over a timescale of $\tau_\rmD \sim 1/k\sigma$, i.e., small scale perturbations damp out faster, as expected. \item Lateral mixing operates differently for a spatially localized perturbation which can be expressed as a superposition of many plane waves. The response to each of them damps out like a Gaussian (if the perturber is impulsive). Since the power spectrum of a spatially localized perturber with a lateral Gaussian profile is dominated by its largest scales (small $k$) that mix and damp out slower, the net response from all $k$ damps away as $\sim t^{-1}$ (the response profile spreads out as $\sim t$), much slower than the Gaussian damping in case of a sinusoidal perturber. \item The disk response to a non-impulsive perturbation is substantially different from that to an impulsive one. If the temporal strength of the perturber follows a Gaussian pulse with pulse frequency, $\omega_0$ (e.g., a transient bar or spiral arm), the response grows and decays following the temporal profile of the pulse before eventually attaining a $\sim 1/t$ power law fall-off. The response peaks when the pulse frequency, $\omega_0$, is comparable to the vertical oscillation frequency, $\Omega_z$. The response to more impulsive perturbations ($\omega_0 \gg \Omega_z$) is suppressed as $\sim 1/\omega_0$, whereas much slower ($\omega_0 \ll \Omega_z$) perturbations trigger a super-exponentially ($\sim \exp{\left[-n^2\Omega^2_z/4\omega^2_0\right]}$ at small $k$) suppressed response. In this adiabatic limit, the stars tend to remain in phase with the perturber, oscillating at frequencies much smaller than $\Omega_z$, which inhibits the formation of a phase spiral. \item The timescale of perturbation dictates the excitability of different modes, with slower (faster) pulses triggering stronger bending (breathing) modes. An encounter with a satellite galaxy that hits the disk with a uniform velocity $\vp$ and an angle $\thetap$ with respect to the normal at a distance $\xp$ away from an observer in the disk, perturbs the potential at an observer's location with a characteristic time scale $\tau_{\rm enc} \sim \xp\cos{\thetap}/\vp$. If $\tau_{\rm enc}$ is long (short) compared to the typical vertical oscillation time, $\tau_z \sim h_z / \sigma_z$, at the observer's location, the dominant perturbation mode experienced is a bending (breathing) mode. Thus, bending modes are preferentially excited not only by low velocity encounters, but also by more distant and more perpendicular ones. Since the velocities of all MW satellites are much larger than $\sigma_z$, the decisive factor for bending {\it vs.} breathing modes is the distance from the point of impact. This is in qualitative agreement with the results from $N$-body simulations of the MW-Sgr encounter performed by \cite{Hunt.etal.21}, which show more pronounced bending (breathing) modes further from (closer to) the location where Sgr impacts the disk. Moreover, for a given encounter, stars with larger actions undergo stronger breathing mode oscillations since they oscillate slower. \item Besides phase spirals satellite encounters also induce spatial corrugations in the disk response, with vertical and lateral wave-numbers given by $k_z=n\Omega_z\cos{\thetap}/\vp$ and $k_x=n\Omega_z\sin{\thetap}/\vp$, respectively. \end{itemize} As an astrophysical application of our formalism, we have investigated the direct response of the MW disk (approximated as an isothermal slab) to several of the satellite galaxies in the halo for which dynamical mass estimates and galactocentric phase-space coordinates from Gaia parallax and proper motion measurements are available. We integrate the orbits of these satellites in the MW potential and note the impact velocity, $\vp$, angle of impact, $\thetap$, with respect to the normal, and the impact distance from the Solar neighborhood, $\xp$, during their penultimate, last and next disk crossings. We use these parameters to compute the direct response to the MW satellites and find that all of them excite bending modes and thus one-armed phase spirals in the Solar neighborhood, similar to that discovered in the Gaia data by \citet{Antoja.etal.18}. In the Solar vicinity, the largest direct response, by far, is due to the encounter with Sgr. The direct responses triggered by other satellites, most notably Hercules and the LMC, are at least $1-2$ orders of magnitude smaller. Hence, we conclude that, if the Gaia phase spiral was triggered by an encounter with a MW satellite, the strongest contender is Sgr. Although Sgr has been considered as the agent responsible for the Gaia phase spiral and other local asymmetries and corrugations, several studies have pointed out that it cannot be the sole cause of all these perturbations \citep[see e.g.,][]{Bennett.etal.21, Bennett.Bovy.21}. Our work argues, though, that the direct response in the Solar neighborhood from the other MW satellites, including the LMC, is not significant enough, at least in the range of actions covered by the Gaia snail. Of course, as discussed in section~\ref{sec::caveats}, the indirect response from the DM halo of the MW might play an important role especially for the more massive satellites such as Sgr and the LMC. Moreover the global response of a realistic disk will be different from that of the isothermal slab model considered here. We investigate the realistic disk response in Paper~II and leave a sophisticated analysis incorporating self-gravity and halo response for future work. It remains to be seen whether a combination of Sgr plus other (internal) perturbations due to for example spiral arms \citep[][]{Faure.etal.14} or the (buckling) bar \citep[e.g.,][]{Khoperskov.etal.19} can explain the fine-structure in the Solar neighborhood, or whether perhaps a solution requires modifying the detailed MW potential. It is imperative, though, to investigate the structure of phase spirals at other locations in the MW disk, in particular whether they are one-armed or two-armed. This would help to constrain both the time-scale and location of the perturbation responsible for the various out-of-equilibrium features uncovered in the disk of our MW. \section*{Acknowledgments} The authors are grateful to the anonymous referee for thoughtful comments and to Kathryn Johnston, Jason Hunt, Adrian Price-Whelan, Kaustav Mitra, Elena D'Onghia, Chris Hamilton and Dhruba Dutta-Chowdhury for insightful discussions and valuable suggestions. MW is supported by the National Science Foundation through Grant No. AST-1812689. FvdB is supported by the National Aeronautics and Space Administration through Grant No. 19-ATP19-0059 issued as part of the Astrophysics Theory Program. \bibliography{references_banik}{} \bibliographystyle{aasjournal} \appendix \section{Adiabatic limit of slab response} \label{App:ad_lim_resp} In the adiabatic/slow limit, the slab response can be computed by taking the $\omega_0\to 0$ limit and performing the $\tau$ integral in equation~(\ref{f1nk_isosol}) to obtain \begin{align} f_{1nk} = -i \pi\, \Phi_\rmN \calZ_n(I_z) \calX_k \left(\frac{n\Omega_z}{\sigma^2_z}+\frac{k v_x}{\sigma^2}\right) f_0(I_z,v_x,v_y)\, \delta(n\Omega_z+k v_x). \end{align} The Dirac delta function implies that only the resonant stars, i.e., those for which $n\Omega_z + k v_x=0$, contribute to the response in this slow limit. Substituting the expression for $f_0$ from equation~(\ref{f_iso}) in the above equation, integrating over $v_x$ and then summing over $n$, we obtain \begin{align} f_{1k} = -i\pi\,\Phi_\rmN \frac{\calX_k}{\left|k\right|} \sum_{n=-\infty}^{\infty} \calZ_n(I_z) \exp{\left[-\frac{n^2\Omega^2_z}{2 k^2\sigma^2}\right]} n\Omega_z \left(\frac{1}{\sigma^2_z}-\frac{1}{\sigma^2}\right) \exp{\left[i n w_z\right]}. \end{align} Substituting the Gaussian form for $\calX_k$ given in equation~(\ref{Phink_gaussian}) in the above expression, multiplying it by $\exp{\left[i k x\right]}$ and integrating over all $k$, we obtain the following final expression for the slab response in the slow limit: \begin{align} f_1(I_z,w_z,x) = -i \pi\,\Phi_\rmN \frac{\calX_k}{\left|k\right|} \sum_{n=-\infty}^{\infty} \calZ_n(I_z) \calJ_n(x)\, n\Omega_z \left(\frac{1}{\sigma^2_z}-\frac{1}{\sigma^2}\right) \exp{\left[i n w_z\right]}, \end{align} where \begin{align} \calJ_n(x) = \int_{-\infty}^{\infty} d k\, \frac{\exp{\left[i k x\right]}}{\left|k\right|} \exp{\left[-k^2\Delta^2_x/2\right]} \exp{\left[-\frac{n^2\Omega^2_z}{2 k^2\sigma^2}\right]}. \end{align} The above integral can be approximately evaluated in the small and large $x$ limits by the saddle point method to obtain the following asymptotic behaviour of $\calJ_n(x)$: \begin{align} \calJ_n(x) &\sim \begin{cases} \sqrt{\pi \sigma/2 \left|n\right| \Omega_z \Delta_x}\, \exp{\left[-\left|n\right|\Omega_z \Delta_x/\sigma\right]}\, \cos{\left(\sqrt{\frac{\left|n\right|\Omega_z}{\sigma \Delta_x}}x\right)}, & \text{small\;} x,\nonumber \\ \sqrt{2\pi}\,\frac{\Delta_x}{x} \exp{\left[-x^2/2\Delta^2_x\right]}, & \text{large\;} x. \end{cases} \end{align} \\ \section{Slab response to satellite encounters} \label{App:sat_disk_resp} The perturbing potential, $\Phi_\rmP$, at $(x,z)$ due to a satellite galaxy impacting the disk along a straight orbit with uniform velocity $\vp$ at an angle $\thetap$ with respect to the normal is given by equation~(\ref{Phip_sat}). Computing the Fourier transform, $\Phi_{nk}$, of $\Phi_\rmP$, and substituting this in equation~(\ref{f1nk_isosol}) yields \begin{align} f_{1nk}(I_z,v_x,v_y,t) &=i\frac{G \Mp}{\vp} f_0(v_x,v_y,E_z) \left(\frac{n\Omega_z}{\sigma^2_z}+\frac{k v_x}{\sigma^2}\right) \exp{\left[-i\left(n\Omega_z+k v_x\right) t\right]}\, \calF_{nk}(t), \label{f1nk_sat_1} \end{align} where \begin{align} \calF_{nk}(t) &= \frac{1}{{\left(2\pi\right)}^2} \int_0^{2\pi}d w'_z \exp{\left[-i n w'_z\right]} \int_{-\infty}^{\infty} d x' \exp{\left[-i k x'\right]} \int_{-\infty}^{t} d \tau\, \frac{\exp{\left[i\left(n\Omega_z+k v_x\right) \tau\right]}}{\sqrt{{\left(\tau-\frac{z'\cos{\thetap}+x'\sin{\thetap}}{\vp}\right)}^2+\frac{{\left(x'\cos{\thetap}-z'\sin{\thetap}\right)}^2}{\vp^2}}}. \end{align} The $\tau$ integral can be computed in the large $t$ limit to yield \begin{align} \calF_{nk}(t\to \infty) &= \frac{1}{2\pi^2} \int_0^{2\pi}d w'_z \exp{\left[-i n w'_z\right]} \int_{-\infty}^{\infty} d x' \exp{\left[-i k x'\right]} \nonumber \\ & \times \exp{\left[i \frac{\left(n\Omega_z+k v_x\right)\cos{\thetap} z'}{\vp} \right]} \exp{\left[i \frac{\left(n\Omega_z+k v_x\right)\sin{\thetap} x'}{\vp} \right]} K_0\left[\left(n\Omega_z+k v_x\right)\frac{\left(x'\cos{\thetap}-z'\sin{\thetap}\right)}{\vp}\right], \end{align} where $K_0$ denotes the zero-th order modified Bessel function of the second kind. Recalling that the unperturbed DF is isothermal, given by equation~(\ref{f_iso}), we integrate equation~(\ref{f1nk_sat_1}) over $v_x$ and $v_y$ to obtain \begin{align} &\int_{-\infty}^{\infty}d v_y\int_{-\infty}^{\infty}d v_x\, f_{1nk}(I_z,v_x,v_y,t) \approx \frac{\rho_c}{\sqrt{2\pi}\sigma_z} \exp{\left[-E_z/\sigma^2_z\right]} \frac{G\Mp}{\vp} \nonumber \\ &\times \frac{1}{2\pi^2} \int_0^{2\pi}d w'_z \exp{\left[-i n w'_z\right]} \exp{\left[i \frac{n\Omega_z\cos{\thetap} z'}{\vp} \right]} \int_{-\infty}^{\infty} d x' \exp{\left[-i k x'\right]} \exp{\left[i \frac{n\Omega_z\sin{\thetap} x'}{\vp} \right]} \nonumber \\ &\times \exp{\left[-\frac{1}{2}k^2\sigma^2 {\left(t-\frac{\calS}{\vp}\right)}^2\right]} \left[k^2 \left(t-\frac{\calS}{\vp}\right)+i\frac{n\Omega_z}{\sigma^2_z}\right] K_0\left[\left(n\Omega_z- i k^2 \sigma^2 \left(t-\calS/\vp\right)\right)\frac{\left(x'\cos{\thetap}-z'\sin{\thetap}\right)}{\vp}\right], \label{f1nk_sat_2} \end{align} where we have defined \begin{align} \calS=z'\cos{\thetap}+x'\sin{\thetap}. \end{align} Multiplying equation~(\ref{f1nk_sat_2}) by $\exp{\left[ikx\right]}$ and integrating over $k$ yields \begin{align} &\int_{-\infty}^{\infty} d k\,\exp{\left[i k x\right]}\int_{-\infty}^{\infty}d v_y\int_{-\infty}^{\infty}d v_x\, f_{1nk}(I_z,v_x,v_y,t) \approx \frac{\rho_c}{\sqrt{2\pi}\sigma_z} \exp{\left[-E_z/\sigma^2_z\right]} \frac{G\Mp}{\vp} \nonumber \\ &\times \frac{1}{2\pi^2} \int_0^{2\pi}d w'_z \exp{\left[-i n w'_z\right]} \exp{\left[i \frac{n\Omega_z\cos{\thetap} z'}{\vp} \right]} \times \sqrt{2\pi} \int_{-\infty}^{\infty} d \Delta x\, \frac{1}{\sigma t'} \exp{\left[-\frac{1}{2}\frac{{(\Delta x)}^2}{\sigma^2 t'^2}\right]} \left[\frac{1}{\sigma^2 t'}\left(1+\frac{{(\Delta x)}^2}{\sigma^2 t'^2}\right)+i\frac{n\Omega_z}{\sigma^2_z}\right] \nonumber \\ &\times \exp{\left[i \frac{n\Omega_z\sin{\thetap} x'}{\vp} \right]} K_0\left[\left(n\Omega_z+ i \frac{{(\Delta x)}^2}{\sigma^2 t'^3}\right)\frac{\left(x'\cos{\thetap}-z'\sin{\thetap}\right)}{\vp}\right], \end{align} where $\Delta x = x-x'$, and $t'=t-\calS/\vp$. In the large time limit, using the identity that $\lim_{t'\to \infty} \exp{\left[-{(\Delta x)}^2/2\sigma^2 t'^2\right]}\Big/\sigma t'=\sqrt{2\pi}\delta (\Delta x)$, the integration over $\Delta x$ is simplified. Upon performing this integral, multiplying the result by $\exp{\left[i n w_z\right]}$ and summing over all $n$, we obtain the following response: \begin{align} f_1(I_z,w_z,x,t) &\approx \frac{\rho_c}{\sqrt{2\pi}\sigma_z} \exp{\left[-E_z/\sigma^2_z\right]}\times \frac{2G\Mp}{\vp} \nonumber \\ &\times \sum_{n=-\infty}^{\infty} \left[\frac{1}{\sigma^2 t}+i\frac{n\Omega_z}{\sigma^2_z}\right]\, \Psi_n(x,I_z)\, \exp{\left[i\,\frac{n\Omega_z \sin{\thetap}}{\vp}x\right]} \exp{\left[i n\left(w_z-\Omega_z t\right)\right]}, \label{f1_sat_app} \end{align} where \begin{align} \Psi_n(x,I_z)&= \frac{1}{2\pi} \int_0^{2\pi} d w_z\, \exp{\left[-i n \left(w_z - \frac{\Omega_z \cos{\thetap} z}{\vp}\right)\right]} K_0\left[\,\left|\frac{n\Omega_z \left(x\cos{\thetap}-z\sin{\thetap}\right)}{\vp}\right|\,\right]. \label{Psi_n_app} \end{align} The above expression for $\Psi_n$ can be simplified by evaluating the $w_z$ integral under the epicyclic approximation (small $I_z$ limit), to yield the following approximate form, \begin{align} \Psi_n(x,I_z) &\approx K_0\left(\frac{\left|n\Omega_z \cos{\thetap}\right|}{\vp}x\right) \Phi_n^{(0)}(I_z) - i\frac{n\Omega_z \sin{\thetap}}{\vp} K'_0\left(\frac{\left|n\Omega_z \cos{\thetap}\right|}{\vp}x\right) \Phi_n^{(1)}(I_z)\nonumber \\ &- \frac{1}{2} {\left(\frac{n\Omega_z \sin{\thetap}}{\vp}\right)}^2 K''_0\left(\frac{\left|n\Omega_z \cos{\thetap}\right|}{\vp}x\right) \Phi_n^{(2)}(I_z)+...\,. \label{Psi_n_app_epi} \end{align} Here each prime denotes a derivative with respect to the argument of the function. $\Phi_n^{(j)}(I_z)$, for $j=0,1,2,...$, is given by \begin{align} \Phi_n^{(j)}(I_z) &= \frac{1}{2\pi} \int_0^{2\pi}d w_z\, z^j\, \exp{\left[-i n \left(w_z - \frac{\Omega_z \cos{\thetap} z}{\vp}\right)\right]} \nonumber \\ &\approx {\left(\frac{2I_z}{\nu}\right)}^{j/2} J_{n,j}\left(\frac{n\Omega_z \cos{\thetap}}{\vp}\sqrt{\frac{2I_z}{\nu}}\right). \label{Phin_sat_app} \end{align} Here the implicit relation between $z$, $w_z$ and $I_z$ given in equation~(\ref{z_wz_Iz}), which yields $z=\sqrt{2 I_z/\nu}\, \sin{w_z}$ for small $I_z$, has been used. $J_{n,j}$ denotes the $j^{\rm th}$ derivative of the $n^{\rm th}$ order Bessel function of the first kind, and $\nu=\sqrt{2}\,\sigma_z/h_z$ is the vertical epicyclic frequency. In equation~(\ref{f1_sat_app}), well after the encounter (large $t$), the term, $1/\sigma^2 t$, can be neglected relative to $i n\Omega_z/\sigma^2_z$ for $n\neq 0$, thus yielding the expression for the disk response to satellite encounters given in equation~(\ref{f1_sat}). \section{Detectability criterion for the phase spiral} \label{App:detect_crit} The demarcation between strong and weak amplitudes of a phase spiral is dictated by the minimum detectable relative response, $\delta$, which can be determined in the following way. Let there be a phase spiral that we want to detect with a total number, $N_*$, of stars by binning the phase-space distribution in the $\sqrt{I_z}\cos{w_z}-\sqrt{I_z}\sin{w_z}$ plane. Let us define the unperturbed DF, $f_0$, and the normalized unperturbed DF, $\bar{f}_0$, such that \begin{align} N_* = \iint f_0\, d I_z\, d w_z,\;\;\;\; \bar{f}_0 = \frac{f_0}{N_*}. \end{align} The perturber introduces a perturbation in the (normalized) DF, $\bar{f}_1$, which manifests as a spiral feature in the phase-space due to phase-mixing. To recover $\bar{f}_1$ we bin the data in $I_z$ and $w_z$, such that the perturbation in the number of stars in each bin ($\Delta I_z,\Delta w_z$) is given by \begin{align} N(\Delta I_z,\Delta w_z) = N_* \bar{f}_1 \Delta I_z \Delta w_z. \end{align} The optimum binning strategy can be determined as follows. The phase spiral is a periodic feature in both $I_z$ and $w_z$. Therefore, to pull out the periodicity in $I_z$, we need to sample with a frequency exceeding the Nyquist frequency, i.e., the bin size, $\Delta I_z$, should be less than $I_{z,\rm max}/N_{\rm wind}$, where $I_{z,\rm max}$ is the maximum $I_z$ in the sample and $N_{\rm wind}$ is the number of winds of the spiral. Moreover, $\Delta I_z$ is required to exceed the Gaia measurement error so that the error is dominated by Poisson noise, i.e., we require $\Delta I_z/I_z > \Delta_{\rm Gaia} \sim 10^{-2}$ \citep[see][for parallax and radial velocity errors, the two dominant sources of measurement errors in Gaia]{Luri.etal.18,Katz.etal.19}. Within each $I_z$ bin, the data is further divided into $N_a$ azimuthal bins, each of size $\Delta w_z=2\pi/N_a$. For optimum sampling in $w_z$, $N_a$ should be greater than $2n$ (for spiral mode $n$) and less than $2\pi/\Delta_{\rm Gaia}$. After binning the data as discussed above, a reliable detection of the phase spiral can be made with a given signal to noise ratio, $S/N$, when the perturbation in the number of stars in each bin, \begin{align} N(\Delta I_z,\Delta w_z) = N_* \times \frac{\bar{f}_1}{\bar{f}_0} \times \frac{2\pi \bar{f}_0(I_z) \Delta I_z}{N_a} \geq {\left(S/N\right)}^2. \end{align} Here we have assumed that the error in recovering the spiral feature is dominated by Poisson noise. This yields the following estimate for the minimum detectable relative response for an isothermal slab, \begin{align} \frac{\bar{f}_1}{\bar{f}_0} \geq \delta = 3.6\times 10^{-4} \times {\left(\frac{S/N}{3}\right)}^2 \left(\frac{10^6}{N_*}\right) \left(\frac{N_a}{10}\right) \left(\frac{0.1}{\Delta I_z/I_z}\right) \frac{h_z\sigma_z}{I_z}\, \exp{\left[\frac{E_z(I_z)}{\sigma^2_z}\right]}. \end{align} Provided that there are about a million stars in the Gaia data of the Solar neighborhood \citep[][]{Antoja.etal.18}, we consider $\delta=10^{-4}$ to be a rough estimate for the minimum detectable relative response. \label{lastpage}

Title: Optical spectroscopy of the extremely metal-deficient star-forming galaxy HSC J1631+4426: a test of the strong-line method

Abstract: Recently, Kojima and co-authors have reported a record low oxygen abundance, 12+logO/H=6.90+/-0.03 in the low-mass star-forming galaxy HSC J1631+4426. This exceptionally low oxygen abundance was obtained by the direct method, using the [OIII]4363 emission line. However, using the strong-line method by Izotov et al. (2019b), these authors have derived a significantly higher metallicity 12+logO/H=7.175+/-0.005. To clarify the situation, we have obtained new observations of HSC J1631+4426 with the Large Binocular Telescope (LBT)/Multi-Object Dual Spectrograph (MODS). We have derived a higher oxygen abundance, 12+logO/H=7.14+/-0.03, using the direct method, a value similar to the oxygen abundance obtained by the strong-line method. Thus, HSC J1631+4426 has a metallicity close to that of the well known blue compact dwarf galaxy IZw18.

https://export.arxiv.org/pdf/2208.08766

\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2022} \label{firstpage} \begin{keywords} galaxies: dwarf -- galaxies: starburst -- galaxies: ISM -- galaxies: abundances. \end{keywords} \section{Introduction}\label{sec:INT} Extremely metal-deficient (XMD) galaxies with active star formation constitute a rare and intriguing class of objects in the local Universe. We shall define here XMD galaxies as those having oxygen abundances 12~+~logO/H~$\leq$~7.3, or 4\% solar, taking the solar value to be 8.7 \citep{asp09}. Despite the fact that these low-$z$ XMD galaxies have a gaseous content that is not quite as pristine as that of primordial galaxies, they do represent their best local counterparts and thus, can be used to compare with the high-$z$ primeval objects to be observed in the near-future by the {\sl James Webb Space Telescope} ({\sl JWST}) and the 30 m-class ground-based telescopes. In the last few years, a number of nearby galaxies have been discovered with extremely low oxygen abundances 12~+~logO/H $\sim$ 7.0 (or 2\% solar). Thus, \citet{H16} have found 12~+~logO/H = 7.02\,$\pm$\,0.03 for the galaxy AGC~198691, and the value of 7.13\,$\pm$\,0.08 has been derived by \citet{H17} for the dwarf galaxy Little Cub. \citet{I18a}, \citet{Iz19a} and \citet{IzTGus21} have reported 12~+~logO/H = 6.98\,$\pm$\,0.02, 7.035\,$\pm$\,0.026 and 7.085\,$\pm$\,0.031 for the galaxies J0811$+$4730, J1234$+$3901 and J2229$+$2725, respectively. Recently, \citet{Kojima2020} have derived a record low oxygen abundance of 6.90\,$\pm$\,0.03 in the XMD galaxy HSC J1631+4426. Oxygen abundances in all these compact galaxies have been derived for the entire galaxy. On the other hand, there have also been cases where extremely low oxygen abundances have been derived for individual H~{\sc ii} regions in the same galaxy. Thus, oxygen abundances of 7.01\,$\pm$\,0.07, 6.98\,$\pm$\,0.06, 6.86\,$\pm$\,0.14 have been determined in three H~{\sc ii} regions of the XMD blue compact dwarf SBS 0335--052W \citep{I09}, while 6.96\,$\pm$\,0.09 has been found for one of the H~{\sc ii} regions in the dwarf irregular galaxy DDO~68 \citep{An19}. The most reliable method for abundance determination is the so-called ``direct method''. It requires the detection, with good accuracy, of the [O~{\sc iii}]$\lambda$4363 emission line which plays the role of an electron temperature indicator. For star-forming galaxies with a weak or undetected [O~{\sc iii}]$\lambda$4363 emission line, ``strong-line methods'' are used. As proposed originally by \citet{Pagel1979}, these methods are based on combinations of strong emission-line intensities of various elements. The oxygen abundance indicator R$_{23}$ = ([O~{\sc iii}]$\lambda$4959+$\lambda$5007 + [O~{\sc ii}]$\lambda$3727)/H$\beta$ suggested by \citet{Pagel1979} has met with widespread acceptance and use. For the strong-line method, grids of photoionization models have been used to calibrate the relation between the line intensities of strong oxygen lines and the oxygen abundance. However, different models by different authors give divergent calibrations, so it is best to base the calibration on oxygen abundances derived from observations through the direct method. The earliest calibrations were one-dimensional \citep[e.g., ][]{EP84,mccall85,D86}, i.e., the oxygen abundance depends on an unique parameter, R$_{23}$. Further development in reducing the scatter and improving the accuracy of the strong-line method is to introduce a correction for the ionization state of H~{\sc ii} regions. Indeed, as pointed out by \citet{M91}, the intensities of the oxygen emission lines depend not only on the metallicity of the H~{\sc ii} region, but also on its ionization state, which can be parameterized by the ratio O$_{32}$\,=\,[O~{\sc iii}]$\lambda$5007/[O~{\sc ii}]$\lambda$3727. This ratio is the observational proxy of the ionization parameter $U$. Many two-dimensional calibrations (the oxygen abundance depends on R$_{23}$ and O$_{32}$ or a parameter similar to O$_{32}$) were derived for application to galaxies in a wide range of metallicities, typically at oxygen abundances 12~+~log\,O/H~$\geq$\,7.4 and O$_{32}$~$\la$~5, and often at the expense of the accuracy in the abundance determination at the lowest metallicities and high O$_{32}$ \citep[e.g., ][]{PTh05,Nagao2006}. Recently, \citet{IzTGus21} have proposed an improved modification of the empirical strong-line calibration by \citet{Iz19b}, focussed on the oxygen abundance determination in very metal-poor galaxies, those in the XMD class with 12 + log O/H $\leq$ 7.3. It is found that all the XMD galaxies discussed above follow closely the statistical relation between oxygen abundance and strong line ratios, as derived by \citet{IzTGus21}. There is nevertheless a striking exception, concerning the star-forming galaxy HSC J1631$+$4426 \citep{Kojima2020}. With its oxygen abundance of 6.90\,$\pm$\,0.03 derived by the direct method, it strongly deviates from the strong-line relation. On the other hand, \citet{Kojima2020,Kojima2021} using the strong-line method by \citet{Iz19b}, derived a much higher oxygen abundance of 7.175\,$\pm$\,0.005. As the XMD galaxy HSC J1631$+$4426 has the lowest oxygen abundance ever reported for a star-forming object, it is crucial to resolve any possible inconsistency between the oxygen abundances derived by the direct and strong-line methods, especially when these are applied in the most extreme metallicity regime. To this end, we have obtained new spectroscopic observations of that galaxy with the LBT to detect the [O~{\sc iii}]$\lambda$4363 emission line with a high signal-to-noise ratio, use the direct method to derive an accurate oxygen abundance, and compare its value with the one derived by the strong-line method. \section{Observations and data reduction}\label{sec:observations} We have obtained LBT long-slit spectrophotometric observations of HSC J1631$+$4426 on 1 May, 2022 in the twin binocular mode, using the MODS1 and MODS2 spectrographs. Spectra were obtained in the wavelength range 3200~--~10000\AA\ with a 1.2 arcsec wide slit, resulting in a resolving power $R$ $\sim$ 2000. The seeing during the observations was 0.6 arcsec. Eight subexposures of approximately 900 s were obtained in both the blue and red ranges separately with MODS1 and MODS2, resulting in a total exposure time of 14048~s in the blue range and of 12252~s in the red range, counting separate exposures with both spectrographs. The airmass during observations was small, equal to 1.05. Thus, the effect of atmospheric refraction is small for all subexposures \citep[see ][]{Filippenko1982}. The spectrum of the spectrophotometric standard star BD+33~2642 was obtained with a 5 arcsec wide slit during the same night, for flux calibration and correction for telluric absorption in the red part. Bias subtraction, flat field correction, wavelength and flux calibration were done with the MODS Basic CCD Reduction package {\sc modsccdred} \citep{Pogge2019} and {\sc iraf}. After these reduction steps, MODS1 and MODS2 subexposures were co-added and one-dimensional spectra of HSC J1631$+$4426 in the blue and red ranges were extracted in a 1.2 arcsec aperture along the spatial axis. These spectra exhibit intense emission lines, including a strong [O~{\sc iii}]$\lambda$4363 emission line (see Fig. \ref{fig1} and insets therein). \section{Heavy element abundances}\label{sec:abundances} The observed emission-line fluxes and their errors were measured using the {\sc iraf} {\it splot} routine. They were corrected for extinction and underlying hydrogen stellar absorption, derived iteratively from the observed decrement of the hydrogen Balmer emission lines, following \citet*{ITL94}. In our iterative procedure, the equivalent widths of the underlying stellar Balmer absorption lines are assumed to be the same for all transitions. The extinction-corrected fluxes together with the extinction coefficient $C$(H$\beta$), the observed H$\beta$ emission-line flux $F$(H$\beta$), its rest-frame equivalent width EW(H$\beta$), and the equivalent width of the Balmer absorption lines are shown in Table~\ref{tab1}. In the Table, we have also given similar data obtained by \citet{Kojima2021} for their spectrum of HSC J1631$+$4426. Comparison between the two sets of fluxes shows that there is general good agreement for the strong lines. For example, the flux differences in the [O~{\sc iii}]$\lambda$4959, 5007 lines are $\sim$5--8 per cent. The fluxes of all other lines are in agreement within the 1$\sigma$ errors. Exceptions are the H$\alpha$/H$\beta$ flux ratio of \citet{Kojima2021} which is lower than the recombination value, and their higher [O~{\sc ii}]$\lambda$3727 (by about 20 per cent), probably due to their adopted higher extinction. Most relevant to this work, our flux of the [O~{\sc iii}]$\lambda$4363 emission line is $\sim$30 per cent smaller than the one obtained by \citet{Kojima2021}. This smaller flux value will have important consequences on the derived direct-method oxygen abundance. \input{tab1.tex} \input{tab2.tex} We follow the prescriptions of \citet{I06} to derive the electron temperature and density and heavy element abundances from extinction-corrected fluxes of emission lines in HSC J1631$+$4426. The electron temperature $T_{\rm e}$(O~{\sc iii}) is calculated from the [O~{\sc iii}]$\lambda$4363/($\lambda$4959 + $\lambda$5007) emission-line flux ratio. It is used to derive the abundances of O$^{2+}$, O$^{3+}$ and Ne$^{2+}$. The abundances of O$^{+}$, N$^{+}$, S$^{+}$ and Fe$^{2+}$ are derived with the electron temperature $T_{\rm e}$(O~{\sc ii}), using the relations of \citet{I06} between $T_{\rm e}$(O~{\sc ii}) and $T_{\rm e}$(O~{\sc iii}). To derive abundances of S$^{2+}$ and Ar$^{2+}$ we adopt the relation between $T_{\rm e}$(S~{\sc iii}) and $T_{\rm e}$(O~{\sc iii}) by \citet{I06}. The electron number density was derived from the [S~{\sc ii}]$\lambda$6717/$\lambda$6731 flux ratio. The electron temperatures $T_{\rm e}$(O~{\sc iii}), $T_{\rm e}$(O~{\sc ii}) and $T_{\rm e}$(S~{\sc iii}), and the electron number density $N_{\rm e}$(S~{\sc ii}) are shown in Table~\ref{tab2}. We obtain an electron temperature $T_{\rm e}$(O~{\sc iii}) = 20300\,$\pm$\,800K, compared to the exceptionally high electron temperature $T_{\rm e}$(O~{\sc iii}) = 25570\,$\pm$\,1100K obtained by \citet{Kojima2020}. % The ionic abundances, ionisation correction factors ($ICF$s) and total O, N, Ne, S, Ar and Fe abundances are obtained using relations by \citet{I06}. Using the direct method, we derive 12\,+\,logO/H\,=\,7.139\,$\pm$\,0.032 for HSC J1631$+$4426 (Table~\ref{tab2}), significantly higher than the oxygen abundance 12\,+\,logO/H\,=\,6.90\,$\pm$\,0.03 obtained by \citet{Kojima2020,Kojima2021}. The N/O, Ne/O, S/O, Ar/O and Fe/O abundance ratios are similar to those in other star-forming dwarf galaxies. To derive element abundances, we have used the prescriptions of \citet{I06} as they are based on fairly recent atomic data and photoionization models. We have compared our results with those obtained by using relations based on other models. There is nearly complete agreement between our results and those obtained with the \citet{S90} relations. For HSC J1631$+$4426, the difference in the electron temperature $T_{\rm e}$(O~{\sc ii}) would be only $\sim$25K, thus giving the same oxygen abundance. \citet{Kojima2020} used the empirical relation $T_{\rm e}$(O~{\sc ii})\,=\,0.7\,$\times$\,$T_{\rm e}$(O~{\sc iii})\,+\,3000 of \citet{CTM86}. With our derived $T_{\rm e}$(O~{\sc iii})\,=\,20300K, the latter relation would give $T_{\rm e}$(O~{\sc ii})\,=\,17210K. Then 12\,+\,logO/H\,=\,7.11, compared to 7.14 derived with the \citet{I06} relations. In summary, the difference between the oxygen abundances of HSC J1631$+$4426 derived by \citet{Kojima2020} and our group comes almost solely from the flux difference measured for the [O~{\sc iii}]$\lambda$4363 emission line, and not from the particular relation used to derive $T_{\rm e}$(O~{\sc ii}). Because our measured [O~{\sc iii}]$\lambda$4363 flux is $\sim$30 per cent smaller than that of their group, our derived temperature is lower, and our oxygen abundance is higher. Since there is a discrepancy between the [O~{\sc iii}]$\lambda$4363 fluxes measured by our two groups, we need to estimate the oxygen abundances in HSC J1631$+$4426 in another way, distinct from the direct method. We next discuss oxygen abundances derived by the strong-line method in XMD star-forming galaxies. \section{Strong-line method for oxygen abundance determination in XMD galaxies} \label{SLM} The direct $T_{\rm e}$ method is by far the most accurate method to determine oxygen abundances. However, in the case of galaxies with an undetected [O~{\sc iii}]$\lambda$4363 line, to determine oxygen abundances, one has to appeal to strong-line (or indirect) methods based on the fluxes of some of the brightest lines in the spectra of actively star-forming galaxies. The parameter R$_{23}$ = ([O~{\sc iii}]$\lambda$4959+$\lambda$5007 + [O~{\sc ii}]$\lambda$3727)/H$\beta$ is often used. Nonetheless, even using only data with [O~{\sc iii}]$\lambda$4363\AA\ measured with an accuracy better than 25\% to calibrate the relation, there remains a rather large scatter in the relation 12~+~logO/H vs. R$_{23}$. The next significant step in improving the accuracy of the indirect method for oxygen abundance determination is the introduction of a correction for the ionization state of H~{\sc ii} regions, as discussed by \citet{M91}. That correction can be quantified by the line flux ratio O$_{32}$\,=\,[O~{\sc iii}]$\lambda$5007/[O~{\sc ii}]$\lambda$3727. The use of an ionization correction does indeed reduce the scatter in the relation by a factor of $\ga$2 \citep{Iz19b}. Calibrations of the strong-line method have been performed by many authors over the years \citep[e.g., ][]{Pilyugin2000, PTh05, KD02, Nagao2006,Curti17}. One of the most recent calibrations is that of \citet{IzTGus21} who derive a new empirical relation for the strong-line method, specifically tailored for XMD galaxies. It has the form 12\,+\,logO/H\,=\,0.950\,log(R$_{23}$\,--\,$a$$_1$\,O$_{32}$)\,+\,6.805, where $a$$_1$\,=\,0.080\,--\,0.00078\,O$_{32}$. The scatter of the data points about this relation, which gives an idea of the abundance uncertainties, is $\sim$ 0.07 dex (Fig.~\ref{fig2}). We now discuss the oxygen abundance of HSC J1631$+$4426, for which we present new observations here. If the oxygen abundance 12~+~logO/H = 6.90\,$\pm$\,0.03 of \citet{Kojima2020}, derived by the direct method, is used, then the galaxy is strongly deviant from the mean 12~+~logO/H vs. log(R$_{23}$ -- $a$$_1$ O$_{32}$) relation (open red circle in Fig.~\ref{fig2}). However the agreement is considerably better if the value derived by the direct method from our LBT observations, 12~+~logO/H = 7.14 \,$\pm$\,0.03, is adopted (filled red circle in Fig.~\ref{fig2}). We note that the latter oxygen abundance is nearly the same as the values of 7.18 and 7.14 obtained respectively by using the strong-line calibration for XMD galaxies of \citet{IzTGus21}, and the strong-line $P$ method of \citet{Pilyugin2000} ($P$ is a quantity related to O$_{32}$). The good agreement between the oxygen abundances derived by the direct and indirect methods gives us confidence both in the accuracy of our direct method of oxygen abundance determination and in the reliability of the strong-line method. If our measurement 12~+~logO/H\,=\,7.14\,$\pm$\,0.03 is correct, then the metallicity of HSC J1631$+$4426 is close to that of the well-known blue compact dwarf galaxy I~Zw~18. The fact that intensive many-decades long searches for XMD star-forming galaxies have uncovered no object with a metallicity below 12~+~logO/H $\sim$ 7.0, either in the ionized gas or neutral gas component \citep*[e.g. ][]{T05} suggests a previous enrichment of the primordial gas to that oxygen abundance level, perhaps by Population III stars. This suggestion is supported by observations of Ly$\alpha$ absorbers which show similar oxygen abundances, 12~+~logO/H $\sim$ 7.0. Fig.~\ref{fig3} displays the position of HSC J1631$+$4426 in the 12~+~logO/H vs. $M_\star$ plane for XMD galaxIes. The Figure shows that HSC J1631$+$4426 has properties similar to that of other XMD galaxies. \section{Summary}\label{sec:conclusions} We have carried out spectroscopic observations of the low-mass star-forming galaxy HSC J1631$+$4426 with the Large Binocular Telescope (LBT)/Multi-Object Dual Spectrograph (MODS). \citet{Kojima2020} have reported a record lowest oxygen abundance 12~+~logO/H = 6.90\,$\pm$\,0.03 for this galaxy, using the direct method. However, this value is considerably smaller than the one derived by the indirect strong-line method. From our new observations, we have obtained a higher oxygen abundance, 12~+~logO/H = 7.14\,$\pm$\,0.03 for HSC J1631$+$4426, using the direct method. This value is also the same as the one obtained by the strong-line method, suggesting it is likely to be the correct one. \section*{Acknowledgements} N.G.G. and Y.I.I. acknowledge support from the National Academy of Sciences of Ukraine by its priority project No. 0122U002259. ``Fundamental properties of the matter and its manifestation in micro world, astrophysics and cosmology''. The Large Binocular Telescope (LBT) is an international collaboration among institutions in the United States, Italy and Germany. This paper used data obtained with the LBT/MODS spectrographs built with funding from National Science Foundation (NSF) grant AST-9987045 and the NSF Telescope System Instrumentation Program (TSIP), with additional funds from the Ohio Board of Regents and the Ohio State University Office of Research. {\sc iraf} is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. \section*{Data availability} The data underlying this article will be shared on reasonable request to the corresponding author. \input{ref.tex} \bsp \label{lastpage}

Title: BACCHUS Analysis of Weak Lines in APOGEE Spectra (BAWLAS)

Abstract: Elements with weak and blended spectral features in stellar spectra are challenging to measure and require specialized analysis methods to precisely measure their chemical abundances. In this work, we have created a catalog of approximately 120,000 giants with high signal-to-noise APOGEE DR17 spectra, for which we explore weak and blended species to measure Na, P, S, V, Cu, Ce, and Nd abundances and $^{12}$C/$^{13}$C isotopic ratios. We employ an updated version of the BACCHUS (Brussels Automatic Code for Characterizing High accUracy Spectra) code to derive these abundances using the stellar parameters measured by APOGEE's DR17 ASPCAP pipeline, quality flagging to identify suspect spectral lines, and a prescription for upper limits. Combined these allow us to provide our BACCHUS Analysis of Weak Lines in APOGEE Spectra (BAWLAS) catalog of precise chemical abundances for these weak and blended species that agrees well with literature and improves upon APOGEE abundances for these elements, some of which are unable to be measured with APOGEE's current, grid-based approach without computationally expensive expansions. This new catalog can be used alongside APOGEE and provide measurements for many scientific applications ranging from nuclear physics to Galactic chemical evolution and Milky Way population studies. To illustrate this we show some examples of uses for this catalog, such as, showing that we observe stars with enhanced s-process abundances or that we can use the our $^{12}$C/$^{13}$C ratios to explore extra mixing along the red giant branch.

https://export.arxiv.org/pdf/2208.00071

\title{BACCHUS Analysis of Weak Lines in APOGEE Spectra (BAWLAS)} \author[0000-0003-2969-2445]{Christian R. Hayes} \affiliation{NRC Herzberg Astronomy and Astrophysics Research Centre, 5071 West Saanich Road, Victoria, B.C., Canada, V9E 2E7} \author[0000-0002-6939-0831]{Thomas Masseron} \affiliation{Instituto de Astrof\'{i}sica de Canarias, 38205 La Laguna, Tenerife, Spain} \affiliation{Departamento de Astrof{\'i}sica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain} \author[0000-0002-4989-0353]{Jennifer Sobeck} \affiliation{Department of Astronomy, University of Washington, Box 351580, Seattle, WA 98195, USA} \author[0000-0002-1693-2721]{D. A. Garc\'{i}a-Hern\'{a}ndez} \affiliation{Instituto de Astrof\'{i}sica de Canarias, 38205 La Laguna, Tenerife, Spain} \affiliation{Departamento de Astrof{\'i}sica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain} \author{Carlos Allende Prieto} \affiliation{Instituto de Astrof\'{i}sica de Canarias, 38205 La Laguna, Tenerife, Spain} \affiliation{Departamento de Astrof{\'i}sica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain} \author[0000-0002-1691-8217]{Rachael L. Beaton} \altaffiliation{Carnegie-Princeton Fellow} \affiliation{Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ~08544} \affiliation{The Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA~91101} \author{Katia Cunha} \affiliation{Observat\'{o}rio Nacional, 77 Rua General Jos\'{e} Cristino, Rio de Janeiro, 20921-400, Brazil} \affiliation{Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA} \author{Sten Hasselquist} \affiliation{Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA} \author[0000-0002-9771-9622]{Jon A. Holtzman} \affiliation{New Mexico State University, Las Cruces, NM 88003, USA} \author[0000-0002-4912-8609]{Henrik J\"{o}nsson} \affiliation{Materials Science and Applied Mathematics, Malm\"{o} University, SE-205 06 Malm\"{o}, Sweden} \author{Steven~R.~Majewski} \affiliation{Department of Astronomy, University of Virginia, Charlottesville, VA, 22903, USA} \author[0000-0003-0509-2656]{Matthew Shetrone} \affiliation{University of California Observatories, University of California Santa Cruz, Santa Cruz, CA, 95064} \author[0000-0002-0134-2024]{Verne V. Smith} \affiliation{NSF's National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719, USA} \author{Andr\'es~Almeida} \affiliation{Department of Astronomy, University of Virginia, Charlottesville, VA, 22903, USA} \email{Christian.Hayes@nrc-cnrc.gc.ca} \email{chrishayesastro@gmail.com} \section{Introduction} Currently, there are several spectroscopic surveys that have undertaken the difficult task of measuring stellar parameters and abundances for large samples of stars spanning a wide range in stellar parameters, such as the Apache Point Observatory Galactic Evolution Experiment \citep[APOGEE;][]{apogee}, Galactic Archaeology with HERMES \citep[GALAH;][]{galah}, or Large sky Area Multi-Object fiber Spectroscopic Telescope survey \citep[LAMOST;][]{lamost}. Despite the difficulty of doing so, these surveys have had great success in producing large high-quality samples of stellar parameters and chemical abundances with which to study the large-scale properties of the Milky Way (MW), its stellar populations, and even the stellar populations of MW satellites. One limitation of such surveys, however, is that analyzing such large samples is time consuming and resource intensive. To provide high quality data for large samples, some simplifications or compromises must be made. This could be in the form of limiting the dimensionality of synthetic grids, synthesizing only one or a few elements at a time, among others. While these simplifications are often necessary, they typically make it difficult to measure those elements with weak and/or significantly blended lines , where detailed syntheses and a careful treatment of blends is critical to measuring abundances. Here we perform a careful analysis of APOGEE spectra to complement its exploration of chemical space by measuring weak and blended elements with a simultaneously flexible and detailed methodology. APOGEE is a dual-hemisphere, high-resolution (R$\sim$22500), H-band (1.5$-$1.7 $\mu$m) spectroscopic survey of the Milky Way and its nearby dwarf galaxies \citep{apogee} that began in as APOGEE-1 SDSS-III \citep{sdss3} and continued into SDSS-IV \citep{sdss4} with APOGEE-2. In the 17th data release of SDSS \citep[DR17;][]{dr17, Holtzman2021}, APOGEE has produced its final data release, which contains spectra, stellar parameters, and chemical abundances for up to 20 species. In addition to these 20 species, there are some chemical species that have features in APOGEE spectra, but were not satisfactorily measured by the APOGEE pipeline, including P, Cu, Nd, and the \cisotope{} ratio. In this work, we provide a complementary analysis of APOGEE spectra for high S/N red giants in the APOGEE sample that expands the set of elemental abundances for these stars. We do so by analyzing the elements and isotopic ratios that are difficult to assess with the APOGEE Stellar Parameters and Chemical Abundances Pipeline \citep[ASPCAP;][]{aspcap} methodology and are therefore not provided by APOGEE. We also reanalyze a few additional elements with weak and blended lines. For our analysis, we use the Brussels Automatic Code for Characterizing High accUracy Spectra \citep[BACCHUS][]{masseron2016} that measures line-by-line elemental abundances from on-the-fly spectral synthesis and provides additional quality flags for each line it analyzes (e.g., identifying when a line is severly blended, too weak to be measured, etc.). We follow up this BACCHUS analysis with a post-processing of the line-by-line abundances to remove erroneous or suspect measurements, and combine the high quality line-by-line measurements to provide sample of chemical abundances for weak and blended elements in APOGEE spectra. The final data set -- the BACCHUS Analysis of Weak Lines in APOGEE (BAWLAS) catalog -- can be found on the SDSS DR17 website\footnote{The table will be available at \url{https://www.sdss.org/dr17/data_access/value-added-catalogs/}}. This paper is outlined as follows. Section \ref{sec_data} describes the APOGEE data we use in this study. Section \ref{sec:bacchus} provides a brief overview of the BACCHUS code and our specific implementation of it in this work. Section \ref{sec:line_comb} explains our post-processing analysis, flagging, line combination, and error estimation. Section \ref{sec:results} explores our derived chemical abundances, their uncertainties, trends, and presents any known caveats. In Section \ref{sec:comparison} we compare our derived abundances with both APOGEE, for overlapping elements, and high-resolution literature measurements. Finally in Section \ref{sec:conclusion} we give a short summary of this work. We also provide some detailed information about our post-processing choices, flagging, and upper limit relations for each element in the Appendices. \section{Data} \label{sec_data} In this work, we use the spectra and stellar parameters from the final data release of the APOGEE survey \citep{apogee} in SDSS-IV's DR17 \citep{sdss4,dr17,Holtzman2021}. This data release includes all APOGEE spectra taken from both the Northern and Southern hemispheres using the APOGEE spectrographs \citep{wilson2019} on the SDSS 2.5-m \citep{gunn06} and the NMSU 1-m \citep{Holtzman2010} telescopes in the north, and the 2.5-m du Pont \citep{bv73} telescope in the south. Targeting for APOGEE survey is described in \citet{zas13,zas17}, \citet{Beaton2021}, and \citet{Santana2021}. Details of the data reduction pipeline for APOGEE can be found in \citet{dln15} and \citet{Holtzman2015}, for the main survey and 1-meter data respectively, which have been updated in DR17 to use the {\texttt{Doppler}}\footnote{\url{https://github.com/dnidever/doppler}} code for radial velocity determination \citep{Holtzman2021}. APOGEE's stellar parameters and chemical abundances are derived from ASPCAP \citep{aspcap}, based on the {\texttt{FERRE}}\footnote{\url{https://github.com/callendeprieto/ferre}} code \citep{AllendePrieto2006}. As in prior APOGEE data releases, ASPCAP uses a grid of MARCS stellar atmospheres \citep{marcs,Jonsson2020}, and an H-band line list from \citet{Smith2021} that updates the earlier APOGEE line list presented in \citet{shetrone2015}, which includes the Ce and Nd line identifications from \citet{Cunha2017} and \citet{Hasselquist2016}, respectively. These atmospheres and the line list are then used to generate a grid of synthetic spectra \citep{zamora2015} using the Synspec code \citep{Hubeny2011} and nLTE calculations for Na, Mg, Ca, and K from \citet{Osorio2020} that are fit to the observed spectra to determine stellar parameters and chemical abundances. From the full APOGEE survey, we use a sample of 126,362 spectra that we analyze using BACCHUS as described in Section \ref{sec:bacchus}. To build this sample, we mainly select high signal-to-noise APOGEE spectra, S/N $ > 150$ per pixel of giants with calibrated ASPCAP stellar parameters between $3500 \ {\rm K} < T_{\rm eff} < 5000$ K and $\log g < 3.5$. We also filter these stars so that we select the ones with the high quality spectra according to their APOGEE {\texttt{STARFLAG}} and {\texttt{ANDFLAG}} flags\footnote{A description of these flags can be found in the online SDSS DR17 bitmask documentation (\url{http://www.sdss.org/dr17/algorithms/bitmasks/})} by selecting stars with {\texttt{STARFLAG}} = 0 and {\texttt{ANDFLAG}} = 0. We specifically select these stars using the \texttt{allStar-dr17-synspec.fits}\footnote{details of the different \texttt{allStar} versions can be found in the following links: \url{https://www.sdss.org/dr17/irspec/apogee-libraries/\#Spectralgrids} and \url{https://www.sdss.org/dr17/irspec/caveats/\#wronglsf}} version of APOGEE data. While our sample is mostly driven by these selections, there are two caveats to the final sample regarding stars that have duplicate or repeat entries in APOGEE because they were observed in multiple distinct fields and therefore have multiple unique spectra \citep[i.e., stars that have multiple spectra that were processed separately through ASPCAP; for a further description of these stars see Sections 2 and 5.4 of][]{Jonsson2020}. 1) We have intentionally included a sample of duplicate/repeat spectra (of the same stars) in order to investigate some of the systematic uncertainties in our analysis, as described in Section \ref{sec:final_abund}. This is a set of 4,876 unique stars with a total of 11,752 observed spectra and corresponding entries in APOGEE's catalogs. 2) In some cases for stars with duplicate entries, an entry/spectrum that does not conform to our initial selections was included in our analysis (e.g., ASPCAP derives parameters outside of the above range, or the spectra do not conform to our S/N selection), this affects $\sim 4\%$ of the final sample (4,668 spectra). Figure \ref{fig:hrd} shows a spectroscopic Hertzsprung-Russell (HR) diagram of our full sample, which illustrates that in cases where duplicates have parameters outside of our initial selection range they typically fall close to our selection limits. For each star in our sample, we obtain the 1D combined and resampled APOGEE apStar and asStar spectra (for northern and southern targets respectively), including any duplicate spectra, and the ASPCAP derived, calibrated stellar parameters and chemical abundances from each of these spectra (using the APOGEE summary file version \texttt{allStar-dr17-synspec.fits}). The spectra are converted to air wavelengths according to the transformations given in \citet{shetrone2015}. Then, these spectra and calibrated stellar parameters are used in BACCHUS (with APOGEE's chemical abundances as starting guesses) to derive chemical abundances and isotopic ratios of weaker features in APOGEE spectra, as described in the following section. \section{BACCHUS Processing} \label{sec:bacchus} Throughout this analysis, we used the abundance determination module of the latest version of the BACCHUS code ($\sim$ v67). More specifically, this BACCHUS module consists of a shell script code that computes on the fly synthetic spectra for a range of abundances and compares these syntheses to observational data on a line-by-line basis, deriving abundances from different methods (e.g. using equivalent width, line depth, or $\chi^2$). The synthetic spectra are calculated using the 1D-LTE Turbospectrum radiative transfer code \citep{AlvarezPlez1998,Plez2012}, with spherically symmetric radiative transfer for giants, and the MARCS model atmosphere grids \citep{marcs}. From these synthetic spectra the BACCHUS code then identifies both continuum (or pseudo-continuum) points for normalizing the observed spectra, and the relevant pixels (i.e., a ``mask'' or ``window'') to use for the abundance determination of each line in each spectrum. The normalization of observed spectra is performed by selecting continuum wavelengths in the synthetic spectra, and fitting a linear relation across these continuum points within 30 \AA{} around the line of interest in the observed spectra. The spectral window for each line is determined by comparing where changing the elemental abundance causes significant changes in the synthetic spectra with the second derivative of the observed flux (i.e., to identify the local maxima on either side of the line of interest). The last, crucial and unique feature of this code is that it uses four methods to compare the observed and synthetic spectra within the selected window and provides an abundance measurement for each method. The four methods are: \begin{enumerate} \item The ``\chitwo{}'' method, which determines an abundance by minimizing the squared differences between synthetic and observed spectra. \item The ``\syn{}'' method, which looks for the abundance that makes the difference between the synthetic and the observed points zero. \item The ``\eqw{}'' method, which determines the abundance needed to match the equivalent widths of the synthetic spectra to the observations. \item The ``\intmethod{}'' method, which measures abundances by matching the line core in the synthetic and observe spectra. \end{enumerate} Each of these methods has benefits and limitations. For example, the \eqw{} method is not sensitive to uncertainties in spectral broadening (such as, instrumental or macroscopic velocity broadening), but it is more affected by blends in the wings of lines, whereas the inverse is true for the \intmethod{} method. Each of these methods also have diagnostic flags that indicate the quality of their fits which are listed in Table~\ref{tab:methodflagdesc} and are described in more detail in Section \ref{sec:method_flags}. \subsection{New features} Given the difficulties imposed by analyzing weak and strongly blended lines, we have also implemented some new features. One improvement is the incorporation of the latest version of Turbospectrum (v19.1.3), which includes Stark broadening (important for a Mg blend with one Nd line; see Appendix \ref{app:Nd}). Another improvement is the addition of the extended MARCS model atmosphere grid that has been specifically built for APOGEE that notably includes C and $\alpha$ abundances dimensions \citep[see][]{Jonsson2020}. While BACCHUS does interpolate atmospheric structure in T$_{\rm eff}$, $\log g$ and metallicity, for C and $\alpha$ it uses the atmospheric model with the closest value to the input C and $\alpha$ abundance. This interpolation first attempts to select the closest grid points in T$_{\rm eff}$, $\log g$ and metallicity, and when it encounters a hole in the grid it will try an alternative search in T$_{\rm eff}$ first, then $\log g$, and finally metallicity to find a set of grid points that are fully populated. If all of these attempts fails, i.e., there are too many missing grid points in the atmosphere grid around the stellar parameters of a given star, the star will be rejected and not processed by BACCHUS. We have also implemented a new, fifth method for deriving abundances in BACCHUS, ``\wln{}'' method. This method interpolates the synthetic spectra as a function of elemental abundance at the exact wavelength of the input line. This method has been particularly useful for the abundance determination of one of the Cu lines, because it bypasses the automatic mask determination, which is heavily biased in the strong blends affecting this line. However this method has the disadvantage of relying on one single pixel, and, therefore, is more sensitive to random or systematic errors in that pixel. \subsection{Specific features for the BACCHUS-APOGEE run} For the APOGEE spectra, we use the APOGEE DR17 Turbospectrum linelist, version \emph{180901t20}, with one modification (to reduce the strength of one C$_2$ line, the 15710.665 \AA{} line, changing its loggf from -1.502 to -2.502). To select a model atmosphere for each star, we interpolate the MARCS model atmosphere grid at the calibrated APOGEE T$_{\rm eff}$, $\log g$, and [M/H] assuming a microturbulence relation of \citep{Masseron2019}: \begin{displaymath} \rm \mu_t (km/s) = 2.488 -0.8665*\log g + 0.1567*\log g^2 \end{displaymath} To determine the convolution parameter, which is used to encompasses the instrumental, rotational and macroturbulence broadening with a Gaussian profile, we use Si lines rather than more typical Fe lines because the former are generally cleaner and stronger in the APOGEE spectra, even at low metallicities. Because they are responsible for blending and accurate continuum adjustment, we first determined the abundances for C, O, N and Mg in each star, in this specific order to establish a proper chemical equilibrium. We then measure the carbon isotopic ratio, V, S, Na, Ce, Nd, P, Yb, Zn and Cu, while locally adjusting possible blending features for each line. The code iterates whenever the measured abundance is out of the initial synthesis range, so when possible we use the ASPCAP calibrated abundances as a starting guess for each elemental abundance to help accelerate the convergence of the code. When ASPCAP calibrated abundances are not available for a given element \citep[typically because the star is in a temperature range where the abundances of that element are suspect abundances, see][for more details]{Holtzman2021}, we use an $\alpha$-scaled starting guess for $\alpha$-elements (O, Mg, Si, S, etc.) or a solar-scaled abundance for all other elements. While the code is expected to automatically iterate over an element until convergence, we force two iterations of the overall procedure to ensure self-consistency between the elements and also to avoid artificial stratification in the abundance diagrams when values approach the edge of BACCHUS search range that can occur for elements with multiple lines and enhancements over solar-abundances, such as Ce and Nd. \section{Line Combination and Flagging} \label{sec:line_comb} The BACCHUS processing produces line-by-line measurements of the \logeps{X} $\equiv \log_{10}({\rm N_X/N_H})$ abundance of each element, X, or the \cisotope{} ratio, and quality flags of that measurement from the five different methods mentioned in Section \ref{sec:bacchus}. This allows us flexibility in how to flag and remove individual line measurements when combining our line-by-line measurements for the highest quality results. Our process for combining the variety of line-by-line measurements and flags is as follows: \begin{enumerate} \item Identify which measurement method to use for each line (Section \ref{sec:meas_method}). \item Evaluate the quality of each line-by-line measurement using BACCHUS measurement flags (Section \ref{sec:method_flags}), flags designed to indicate the quality of the spectra around each line (Section \ref{sec:spectra_flags}), and upper limit relations to indicate measurements below our upper limit threshold (Section \ref{sec:upper_limits_relation}). \item Flag stars whose stellar parameters may be of suspect quality for deriving chemical abundances (Section \ref{sec:param_flags}). \item Derive and apply line-by-line zero-point abundance calibrations (Section \ref{sec:zero_points}). \item Finally, we average the zero-point calibrated abundances for the lines that pass all of our quality cuts to arrive at our final abundances and derive two separate estimates of the uncertainties on these values (Section \ref{sec:final_abund}). \end{enumerate} In each of the sections below, we describe our general methodology for the above steps, and we give specific details for what our flagging and combination choices are for each element in Appendix \ref{app:comb_settings}. \subsection{Measurement Method Choices} \label{sec:meas_method} BACCHUS uses several methods of measuring the abundance of each line in each star, from which we then select one method to use across all stars for a given line. Because BACCHUS returns abundance measurements from several different methods, we are able to select the optimal method for each line in our post-processing. This flexibility is beneficial, since some methods are more or less sensitive to different issues that may arise, such as, how badly blended a line is, if there are problems with the continuum around that line, etc. For all of the lines that we measure in this work, we either use the \chitwo{} method or the \wln{} method. In general, because it uses multiple pixels, the \chitwo{} method gives higher precision measurements and is less sensitive various effects that can impact the spectrum at the level of individual pixels. But, the \chitwo{} method can also provide poor measurements when a line has a poorly fit blend, especially on the wings. The \wln{} method however, because it is determined using a single pixel is more susceptible to noise and may be somewhat less precise, but performs better in cases where there are poorly fit blends on the wings of the line, or when lines are weak and only the central pixels on a line provide measurable signal. While we typically use the \chitwo{} method to take advantage of multiple pixels across a line, for particularly weak or blended lines we use the \wln{} method as discussed in more detail in Appendix \ref{app:comb_settings}. \subsection{BACCHUS Method Flags} \label{sec:method_flags} Each of the five abundance measurement methods employed in our BACCHUS analysis have integer flags that indicate the quality of the spectral line fit from that method. Every line of every star that is measured will be given one of the flags in Table \ref{tab:methodflagdesc} for each method (note that the \syn{} and \eqw{} method flags have the same general description). \begin{deluxetable}{r p{6cm}} \tabletypesize{\scriptsize} \tablewidth{0pt} \tablecolumns{2} \tablecaption{BACCHUS Method Flag Description \label{tab:methodflagdesc}} \tablehead{\colhead{Flag} & \colhead{Description}} \startdata \cutinhead{\syn{}/\eqw{}} 0 & Indicates when BACCHUS is only measuring an upper limit with this method\\ 1 & The measurement from this method is okay\\ 2 & The measurement is an extrapolation outside of the synthesized range of abundances\\ 3 & Large (wavelength) offset between the observed flux minimum (nominally the line core) and the synthetic flux minimum \\ \cutinhead{\intmethod{}} 0 & Indicates when BACCHUS is only measuring an upper limit with this method\\ 1 & The measurement from this line is okay\\ 2 & Large (wavelength) offset between the observed and synthesized line cores or strong line (line intensity is below 0.4 of the continuum)\\ \cutinhead{\chitwo{}} 0 & The measurement is beyond the synthesized range of abundances\\ 1 & The measurement from this line is okay\\ 2 & The \chitwo{} method crashed\\ \cutinhead{\wln{}} 0 & The syntheses are too close in flux to provide a reliable interpolation or extrapolation\\ 1 & The measurement from this line is okay\\ 2 & The measurement is an extrapolation outside of the synthesized range of abundances\\ \enddata \end{deluxetable} We use these flags to determine the quality of the BACCHUS fits to each line in a given star and we can improve this assessment by combining the information from the flags of multiple methods. For each line that we measure we have a list of method flags that we consider reflective of a ``good'' BACCHUS fit. In general, our default is to require that all methods have flag = 1 for a line's measurement in a star to be considered ``good''. So with this as an example if all method flags = 1 for a specific line in a star we will use the measurement of that line in our line combination (assuming said measurement passes our other quality criteria), and if any of the method flags $\neq$ 1, we will not use that line for the abundance measurement in that star (and consider it ``flagged,'' removing it from consideration). However, for some lines or elements we consider other combinations of these method flags on a case-by-case basis. For example, some lines lie on the wings of deeper blends (e.g., Cu 16006 \AA{} or Nd 16053 \AA{}). This can cause BACCHUS to give a \syn{} or \eqw{} flag = 3 because of the large wavelength difference between the input line center and the local minimum in the spectrum (i.e., the deeper line center of the blending line), despite providing a good fit to the line of interest. So, we would want to consider measurements with \syn{} and \eqw{} flag = 1 or 3 for these lines. In cases such as this example, we have used a modified selection of quality flags to appropriately treat the line in question. Appendix \ref{app:comb_settings} gives a full summary of our flag choices for each element and their lines as well as individual illustrations of the line profiles and blends. \subsection{Additional Spectra Flags} \label{sec:spectra_flags} In addition to the native flags produced by BACCHUS we supplement these with an additional set of quality flags defined to indicate the condition of the spectrum around the lines of interest that we are measuring. These flags are roughly categorized as those that indicate the state of any blends that may affect our line of interest (blend) and those that describe the quality of the continuum around each line of interest (cont). Both flags are defined such that flag = 1 means the line in a specific star is okay, whereas flag = 0 warrants further investigation (and typically we remove lines in stars with blend/cont flag = 0) The blend flags indicate poorly fit blends with other spectral lines in the star or a blend with a sky line or telluric feature that may have an imperfect subtraction, either of which can lead to incorrect abundance measurements. The continuum flags vary in meaning from line-to-line. For some lines this flag is set to 0 when the spectrum is consistent with being only continuum, i.e., the line of interest is too weak to be reasonably measured. In other lines this flag is defined to be 0 when the pseudo-continuum around a line is a poor match to observations, typically because the wings of a nearby blend are poorly fit. A detailed description of the spectra flags that we have implemented in our analysis can be found in Appendix \ref{app:spectra_flags}. \subsection{Upper Limit Relations} \label{sec:upper_limits_relation} To assess BACCHUS's fits in a programmatic way, we have also included a process for distinguishing detections from upper limit measurements. BACCHUS inherently provides an evaluation of upper limits based on a continuum threshold computed from the variance or the signal-to-noise of the observed spectrum. However, this procedure is not valid for blended lines, such as the lines used in this study, because the blends prevent the line growth from reaching the continuum threshold at very low abundances. Instead, we developed a procedure based on synthetic spectra computed with Turbospectrum. We synthesize individual elemental lines without any blending features in 9 steps of \logeps{X} from \logeps{X} = $-$2.0 to $+$2.0 dex around solar at five temperatures from 3500 K to 5500 K in steps of 500 K along the giant branch (with $\log$ g adjusted to have the syntheses appropriately lie on the RGB). At each temperature step we interpolate between these syntheses to determine the abundances when line depths are 1, 2, 3, 4, or 5$\%$ of the continuum. Thus, these are the upper limit of abundances we could measure with a t = X\% threshold. For each threshold, we then derive an upper limit relation for each line as a linear function of temperature following: \logeps{X}$_{\rm lim} = A_{\rm t} \cdot T_{\rm eff} + B_{\rm t}$, where t is the \% threshold level used. Figure \ref{fig:Na_upperlimit} illustrates this process for the Na 16374 \AA{} line, and Table \ref{tab:upper_limits} in Appendix \ref{app:upper_limits} gives the derived constants $A$ and $B$ for each of the values of t mentioned above for each element. Therefore, with a chosen threshold we can predict the abundance upper limit that is measurable for a given temperature for each line. If a line is measured with an abundance less than that limit for a star, we don't use that line in our combination of line-by-line abundance measurements. Instead, when considering the final, combined abundances in a star, if all of the lines for a given element in a star that pass the flagging criteria (i.e., the BACCHUS fit is considered okay) are upper limits, we record the minimum of the line-by-line upper limits as the abundance upper limit on said star. Note, that we did not compute limits in this way for \cisotope{}, because it is a more complex function of temperature and carbon abundance. Instead, we use a simple line-by-line lower limit for \cisotope{} based on the synthesis range we have used (see Appendix \ref{app:comb_settings}), but also include line-by-line detection threshold flag based on the observed normalised flux, and a more complicated lower limit relation set for the final combined \cisotope{} measurements (see Appendix \ref{app:spectra_flags}). As a default we have chosen to use a threshold of 1\%, requiring that lines should have abundances that produce at least a 1\% line depth. With a S/N ratio of $>$ 150, as in our sample, one may expect to use a threshold below 1\% as the detection limit, however because these relations do not account for the effect of blends, uncertainties in stellar parameters, etc., we have used this slightly more conservative upper limit threshold. Furthermore, for individual lines that are especially blended or sensitive to continuum placement, we have adjusted the threshold to even higher levels based on visual inspection of abundance measurements near the upper limit boundary to ensure that we have bona fide detections (see Section \ref{app:comb_settings} for more case-by-case details). \subsection{Star Flags} \label{sec:param_flags} In addition to line-by-line quality flags, we also flag some stars based on how they have been processed by BACCHUS, and therefore we do not report abundances for these stars. Specifically we flag stars that may have suspect fits to critical molecular blends and stars that require particularly high convolution value to explain their spectral features. \subsubsection{Missing C, N, and O Updates} Because the C, N, and O abundances of a star have a significant impact on the H-band spectrum and are critical for fitting blends with our weaker lines, we have used the ASPCAP C, N, and O abundances as an initial guess. But, with BACCHUS we rederive these abundances, given that their corresponding molecular features are highly temperature sensitive. This allows us to find the best solution for the C, N, and O abundances using the calibrated stellar parameters from ASPCAP (whereas the ASPCAP C, N, and O are derived using the uncalibrated stellar parameters). However, the C, N, and O abundances are not always updated with new values from BACCHUS and are fixed to their initial APOGEE DR17 values when fitting other elements (i.e., when C, N, and O features may be blended with the lines of an element of interest), which can occur for a variety of reasons, e.g., their lines are too weak for BACCHUS to provide a good measurement. To track this, we record which stars have had their C, N, and O abundances updated by BACCHUS measurements in integer flags {\texttt{UPDATE\_C}}, {\texttt{UPDATE\_N}}, and {\texttt{UPDATE\_O}} respectively, where flag = 1 indicates stars that were not updated and flag = 2 indicates stars that have been updated with BACCHUS measured values for that element. Figure \ref{fig:cno_update} shows the effective temperatures and metallicities of the BAWLAS sample and the stars that do not have their C, N, or O updated. These elements are updated in almost all of the stars in our sample. Those stars that do not have updated C, N, or O are preferentially at low metallicities or warm temperatures, where these features are weaker and less important to fit. While we have not removed any stars that have not had their C, N, or O abundances updated by BACCHUS, we have provided these flags so that stars whose molecular blends may be more poorly fit can be tracked. \subsubsection{Stars with Large Line Broadening} One of the parameters that we fit for each star with BACCHUS is a Gaussian convolution broadening parameter, {\texttt{CONVOL}}, that consolidates observational and stellar effects from multiple sources, including: instrument resolution, macroturbulent velocity, rotation, etc. In giants, most of this broadening (in our spectra) comes from the instrument resolution and macroturbulent velocities. Combined, these effects produce a median convolution broadening of 14.9 \kms{} across all of the stars in our sample with stars typically having a convolution broadening ranging between 14.1 \kms{} and 15.8 \kms{}, the 16$^{\rm th}$ and 84$^{\rm th}$ percentiles respectively. However, the broadening has a tail to even higher values, which could indicate that there are additional broadening effects, such as rotation, larger macroturblent velocities at stellar surfaces, or that APOGEE's stellar parameters do not adequately describe the observed spectra (e.g., red supergiants observed in the Magellanic Clouds, see Section \ref{sec:lmc_rsg}). Giants are not treated with rotation or other broadening parameters (other than macroturbulence) by ASPCAP, so stars that exhibit additional broadening may have erroneous stellar parameters, which can lead to incorrect abundances from BACCHUS as well. In order to flag these suspect stars we have removed final combined abundances for stars with {\texttt{CONVOL}} $> 18$ \kms{} (around 2\% of the BAWLAS sample), but we have retained their line-by-line abundances for the possibility of more detailed investigation. \subsection{Zero Point Calibration} \label{sec:zero_points} Similar to APOGEE DR17, we have chosen to calibrate our chemical abundances to a solar zero point derived from solar neighborhood samples (except for C, N and \cisotope{}), instead of using literature solar zero points or abundances derived from solar spectra by BACCHUS. Errors in loggf values or other linelist parameters, non-Local Thermodynamic Equilibrium (nLTE) effects, or systematic errors in stellar parameters or blend fitting can all lead to different line-by-line zero points, which would not be captured by using solar zero points from literature. Indeed, Figure \ref{fig:ce_zeropoint} illustrates that the abundances we derive can have even significant offsets from line to line, even when we narrow to a solar neighborhood sample (defined below) that should nominally have solar abundance ratios on average. Therefore, by using a single zero point between all lines we may still end up with systematic offsets in our abundances. In addition, because these sources of line-by-line zero point offsets may conceivably have trends with stellar parameters (such as nLTE effects, systematic errors in stellar parameters, etc.), analyzing a solar spectrum with BACCHUS and using that as a zero point for a red giant sample may not be appropriate. Instead, we have chosen to empirically calibrate our line-by-line abundance measurements such that the solar neighborhood stars in our sample have a solar ([X/M] $= 0$) abundances in each line. To establish our solar neighborhood sample, we use APOGEE stellar parameters and radial velocities combined with {\it Gaia} early data release 3 \citep[EDR3][]{gaiaedr3} parallaxes and proper motions. We calculate Galactocentric positions using inverted parallax distances from {\it Gaia} EDR3 parallaxes and assuming the sun is at $R_{\rm GC,\odot} = 8.122$ kpc \citep{gravity}. We also use these inverted parallax distances to convert APOGEE radial velocities and {\it Gaia} EDR3 proper motions to Galactocentric velocities using the prescription from \citet{JohnsonSoderblom1987} and assuming a solar motion of $(V_{r}, \ V_{\phi} \ V_{z})_{\odot} = (14, 253, 7)$ km s$^{-1}$ in right-handed notation \citep{Schonrich2010,Schonrich2012,Hayes2018c}. Our solar neighborhood sample is then defined, as those stars in our BAWLAS sample with: \begin{itemize} \item {\it Gaia} EDR3 relative parallax uncertainty below 10\% ($ 0 < \sigma_{\varpi}/\varpi < 0.1$) \item Metallicities, $-0.05 <$ [M/H] $< 0.05$ \item Galactocentric (cylindrical) radii, $8 < R_{\rm GC} < 9$ kpc \item Height above the MW midplane, $|z| < 0.5$ kpc \item Total space velocity within 40 km/s of the local standard of rest, assuming $(V_{R}, V_{z}, V_{\phi})_{\rm LSR} = (0, 0, 229)$ km/s using the value of $V_{\phi,\ {\rm LSR}} = 229$ km/s from \citet{Hayes2018c} \end{itemize} For each line of each element, we apply the quality flag and upper limit cuts described in the previous sections and calculate a zero-point from this solar neighborhood sample by taking the median abundance in that line. We use the median to limit the effects of outliers, either in the case of random scatter or in the case of elements like Ce and Nd to reduce the effect of stars with enhanced abundance ratios from processes other than Galactic chemical evolution (for more information about these stars see Section \ref{sec:cend_results}). \begin{deluxetable}{l c c c} \tablewidth{0pt} \tablecolumns{12} \tablecaption{Line-by-Line Abundance Zero Points \label{tab:zero_points}} \tablehead{\colhead{Element} & \colhead{Line (\AA{})} & \colhead{\logeps{X}$_{\rm zpt}$} & \colhead{Grev07 offset\tablenotemark{{\scriptsize a}}}} \startdata O & 15373.5 & 8.800 & 0.140 \\ O & 15391.0 & 8.902 & 0.242 \\ O & 15569.0 & 8.880 & 0.220 \\ O & 15719.7 & 8.928 & 0.268 \\ O & 15778.5 & 8.964 & 0.304 \\ O & 16052.9 & 8.814 & 0.154 \\ O & 16055.5 & 8.854 & 0.194 \\ O & 16650.0 & 8.798 & 0.138 \\ O & 16704.8 & 8.874 & 0.214 \\ O & 16714.5 & 8.888 & 0.228 \\ O & 16872.0 & 8.863 & 0.203 \\ O & 16909.4 & 8.938 & 0.278 \\ Na & 16373.9 & 6.367 & 0.197 \\ Na & 16388.8 & 6.399 & 0.229 \\ P & 15711.6 & 5.700 & 0.340 \\ P & 16482.9 & 5.474 & 0.114 \\ S & 15403.5 & 7.159 & 0.019 \\ S & 15422.3 & 7.151 & 0.011 \\ S & 15469.8 & 7.157 & 0.017 \\ S & 15475.6 & 7.242 & 0.102 \\ S & 15478.5 & 7.133 & -0.007 \\ V & 15924.8 & 4.022 & 0.022 \\ V & 16137.3 & 4.347 & 0.347 \\ V & 16200.1 & 4.209 & 0.209\\ Cu & 16005.5 & 4.228 & 0.018 \\ Cu & 16006.0 & 4.147 & -0.063 \\ Ce & 15784.8 & 1.615 & -0.085 \\ Ce & 16376.5 & 1.932 & 0.232 \\ Ce & 16595.2 & 1.717 & 0.017 \\ Ce & 16722.5 & 1.784 & 0.084 \\ Nd & 15368.1 & 1.769 & 0.319 \\ Nd & 16053.6 & 1.531 & 0.081 \\ Nd & 16262.0 & 1.860 & 0.410 \\ \enddata \tablenotetext{a}{Zeropoint offset from the \citet{Grevesse2007} solar abundances, reported as (\logeps{X}$_{\rm zpt}$ $-$ \logeps{X}$_{\rm \odot, Grev07}$)} \end{deluxetable} The zero-points that we calculate for each line are listed in Table \ref{tab:zero_points}, which we then use to calculate our ``bracket notation'' abundances e.g., \xh{X}$_{*}$ $=$ \logeps{X}$_{*}$ - \logeps{X}$_{\rm zpt}$ for each line. The exception to this is the C and N abundances and the \cisotope{} ratios that we derive. Because these three values are expected to change along the giant branch due to dredge-up, we do not necessarily expect that the solar neighborhood should have solar abundance or isotopic ratios. Instead, for C and N we have used the \citet{Grevesse2007} solar abundances for our zero point, and for \cisotope{} we use only the raw ratios that we calculate without applying any calibration. \subsection{Combined Abundances and Uncertainties} \label{sec:final_abund} Our final combined abundances are derived by averaging all of the ``good,'' unflagged, zero-point calibrated line measurements for each star (following the detailed, element-by-element and line-by-line selections given in Appendix \ref{app:comb_settings}). While we consider most of the lines that were attempted by BACCHUS, a few lines were rejected for all stars because they showed strong trends with temperature or their abundance patterns disagree strongly with the remaining lines (see Appendix \ref{app:comb_settings} for more details). For stars whose lines of a given element are all identified as upper limits, instead of combined abundances we provide an \xfe{X} upper limit defined as the minimum line-by-line upper limit values as mentioned in Section \ref{sec:upper_limits_relation} and adjusted using the appropriate zero point from Section \ref{sec:zero_points}. In order to estimate uncertainties on our combined abundance measurements, we use two different methods. (1) We use the line-by-line scatter (and the scatter between two of BACCHUS's measurement methods, \chitwo{} and \wln{}) as an estimate of the measurement uncertainty, and (2) we can estimate the uncertainty empirically from the scatter in repeat observations (and separate reductions/analyses) of the stars with duplicate entries in APOGEE. \subsubsection{Measurement Uncertainties} One way we estimate our abundance uncertainties is by measuring the line-by-line and method-to-method scatter in BACCHUS's abundance measurements. The ``measurement'' uncertainty that we report for each star is the standard error of the abundances as measured by the \chitwo{} and \wln{} methods of ``good,'' unflagged lines of a given element. Numerically this is calculated by taking the standard deviation of the abundance measurements from the \chitwo{} and \wln{} methods in each of the good lines and dividing this by $\sqrt{\rm N_{\rm lines}}$. This estimate of the uncertainty only includes the uncertainty that comes from line choice and measurement methodology, and does not account for more systematic errors, such as the uncertainties or errors in the input stellar parameters, linelist, choice of code, etc. For cases where only one or two lines of an element were measured in a star, we do note that these uncertainties may be underestimated, so on a star-by-star basis these uncertainties may need to be considered carefully. However, we do provide an additional estimate of the abundance uncertainties below, that is less sensitive to star-by-star cases. \subsubsection{Empirical Uncertainties} Motivated by what is done by APOGEE to estimate its abundance uncertainties, we also provide empirical uncertainty estimates determined from stars that have been observed multiple times in APOGEE but have their multiple spectra processed and analyzed separately. These stars may have ``random'' observational effects in their spectra, different stellar parameters from APOGEE, etc., so by processing them separately we can understand how these differences impact the derived abundances. This allows us to understand some of the impact of varying the input stellar parameters and observational noise on the abundances we measure. We take our sample of $\sim 4,900$ stars with $\sim 12,000$ different spectra and ASPCAP results, process them separately through BACCHUS and our post-processing and then compare the resulting abundances that we measure. Each pair of absolute differences in \xh{X} provides an estimate of the standard deviation in the abundance measurements for an individual star when multiplied by $\sqrt{\pi}/2$ for an unbiased estimator. Each star then provides an estimate, $\sigma_{\rm X} = \sqrt{\pi}/2 \ |\, \Delta {\rm [X/H]} \, |$, of the typical random abundance errors for that stars parameters. Therefore we can fit the ensemble distribution of these standard deviation estimators to derive an empirical relation for the typical uncertainties on our measurements as a function of various parameters. For this work we use a simple relationship to fit the uncertainty distributions, using the following equation (and substituting \cisotope{} for \xh{X} when calculating \cisotope{} uncertainties): \begin{displaymath} \ln{\sigma_{\rm [X/H]}} = A + B \cdot T^{'}_{\rm eff} + C \cdot (T^{'}_{\rm eff})^2 + D \cdot {\rm [X/H]} \end{displaymath} where $T^{'}_{\rm eff} = T_{\rm eff} - 4500$ K. We use a slightly different form than used by APOGEE (in ASPCAP) for their empirical uncertainties. Here we drop the S/N dependence since our sample is high S/N, and we use the [X/H] abundance instead of [M/H] in our formulation, because the abundance uncertainty should depend primarily on the amount of that element rather than the total metallicity\footnote{Although one can imagine that this could be refined by including metallicity and elements with dominant molecular features whose blends might also affect the errors on our derived abundances. Including more terms in this relation may be a promising way to improve this kind of uncertainty estimation in the future.}. As an example of our derivation of these relations, Figure \ref{fig:ce_emp_errs} shows the distribution of differences in Ce abundances for our repeat sample and the best fit empirical error relation for this element. \begin{deluxetable}{l c c c c} \tablewidth{0pt} \tablecolumns{5} \tablecaption{Empirical Error Relation Parameters \label{tab:uncertainties}} \tablehead{\colhead{Element} & \colhead{A} & \colhead{B} & \colhead{C} & \colhead{D} \\ \colhead{} & \colhead{} & \colhead{(10$^{3}$ K)$^{-1}$} & \colhead{(10$^{3}$ K)$^{-2}$} & \colhead{}} \startdata C & -3.528 & 0.894 & 0.214 & -0.110 \\ N & -2.864 & 0.942 & 0.639 & 0.516 \\ O & -3.841 & 1.405 & 2.666 & 0.356 \\ Na & -3.357 & 0.371 & -0.550 & -0.670 \\ P & -2.738 & -0.361 & -0.798 & 0.747 \\ S & -3.265 & -0.721 & -0.568 & 0.196 \\ V & -3.441 & 1.640 & 0.613 & -0.834 \\ Cu & -3.233 & 0.360 & 0.831 & -0.597 \\ Ce & -3.344 & 1.113 & 1.126 & 0.018 \\ Nd & -2.973 & 0.175 & 0.307 & 0.893 \\ \cisotope{} & -1.224 & 1.287 & 0.707 & 0.137 \\ \enddata \end{deluxetable} The coefficients of our best fit relations for each element can be found in Table \ref{tab:uncertainties}, which we then use to calculate the empirical uncertainties for our full sample. Specifically we report $\sigma_{\rm [X/Fe]}$ by summing the $\sigma_{\rm [X/H]}$ uncertainties we calculate in quadrature with the ASPCAP reported \xh{Fe} uncertainties.\\ We report both the measured and empirical uncertainties that we calculate so they can be applied as desired. The measured uncertainties have the benefit of capturing the variation in measurement from line to line, but do not account for possible variation in stellar parameters (or other systematics like the analysis pipeline, or observations with different instruments/wavelengths of course). On the other hand, the empirical uncertainties may not account for the conditions or line-to-line variations in an individual star, but do allow us to see what the general effect of different spectra and input stellar parameters has on the derived abundance variability. \section{BAWLAS Chemical Abundance Patterns and Trends} \label{sec:results} Our BAWLAS catalog of input parameters, calculated abundances, upper or lower (for \cisotope{}) limits, and errors (as well as the line-by-line abundances and flags) can be found on the SDSS DR17 Value Added Catalog (VAC) page\footnote{\url{https://www.sdss.org/dr17/data_access/value-added-catalogs/}}. Here we present some of the overall results. \subsection{Individual Abundance Patterns} In Figures \ref{fig:cno_abund} and \ref{fig:xfe_abund} we show the combined abundances from BAWLAS. Figure \ref{fig:cno_abund} shows the C, N, and O abundances that were calculated primarily for the purposes of fitting blends, and Figure \ref{fig:xfe_abund} shows the goal elements that we measure: \cisotope{}, Na, P, S, V, Cu, Ce, and Nd. Before delving into each element (discussed below), we point out some of the key features seen in these abundance distributions. Many of the elements do not cover the full metallicity range probed by our sample, which extends down to APOGEE's lower metallicity limit of \feh{} $= -2.5$. Instead, for many elements the number of stars for which we measure that element begins to drop off quite rapidly around a metallicity of \xh{Fe} $\sim -1$ to $-1.5$. This is in part because the underlying density of stars does begin to decrease at these metallicities. But, this also occurs because most of the elements we examine have relatively weak (and few) lines that become increasingly difficult to measure at low metallicities. Essentially only S, Ce, and \cisotope{} are measured below \xh{Fe} $\sim -1.5$, but even these measurements may be biased towards detecting stars with stronger spectral features, e.g, higher abundances for S and Ce, or lower values of \cisotope{}. Indeed the imprint of upper limit flagging for S appears as the diagonal line at low metallicities below which we do not populate any S abundances. P provides another example of a clear hard upper limit flagging, however upper limit flagging also affects the distribution of V, Cu, Ce, and Nd abundances in a less obvious manner. Figure \ref{fig:sce_upperlimits} shows an example of these upper limits for S and Ce. The temperature insensitivity of S is apparent as the very narrow spread of upper limits, which come in three tiers (of nearly constant \xh{S}), corresponding to the three S lines that are used, with the two weaker lines providing upper limits when the quality of the spectrum around the strongest line is too poor to measure or place upper limits on. Ce instead has more temperature sensitive upper limits, which means that \xh{Ce} upper limit as a function of temperature turns into a less localized spread of upper limits as a function of \feh{}. \subsubsection{Carbon and Nitrogen (C and N)} While C, N, and O were measured to fit blends, we can also examine the BAWLAS abundance patterns in these elements. In the C abundances of Figure \ref{fig:cno_abund} we see some internal structure at intermediate metallicities, \feh{} $\sim -0.5$, that appears to the superposition of the thin and thick disk populations. Whereas the N abundances do not show clear substructure in its chemical abundance pattern, instead showing a large spread, in part due to the larger uncertainties in warm stars. At low metallicities, C begins to decrease before showing a large scatter, whereas N shows a slightly rising trend with decreasing metallicity. In addition to the general trends at low metallicities some stars are found with particularly enhanced \xfe{C} or \xfe{N} ($\gtrsim 0.5-1.0$). In the case of N, many of these stars belong to globular clusters which are known to have C-N anti-correlations with high N abundances \citep[e.g.,][]{Smith1996, Gratton2001, Briley2004, Meszaros2015, Masseron2019, Meszaros2020}. As for the stars with high C abundances there are a few ways that stars may get C-enhancements. For example, carbon stars will often show higher C abundances, and can be formed in a few ways. Carbon stars can either be intrinsically enhanced in C, e.g., through the dredge-up of carbon rich material fused in a star's interior, or extrinsically enhanced by C-rich material accreted from an evolved companion, altering its surface chemistry \citep[for a more in depth discussion see][]{Wallerstein1998, LloydEvans2010}. At low metallicities there are also known to be carbon enhanced metal-poor (CEMP) stars (notable for their enhanced \xfe{C} ratios), which come in a number of astrophysical varieties \citep[discussed more in][]{Beers2005, Masseron2010, Frebel2015}. \subsubsection{Oxygen and Sulfur (O and S)} Examining O in Figure \ref{fig:cno_abund} and S in Figure \ref{fig:xfe_abund} we see abundance distributions that are fairly typical for $\alpha$ elements, with a nearly flat plateau at low metallicities that decreases at higher metallicities with a knee occurring around \feh{} $\sim -0.7$ to $-0.5$. In O we see a bifurcation in \xfe{O} at metallicities $> -0.7$ into high- and low-$\alpha$ sequences that are commonly attributed to the thick and thin disks, respectively. The \xfe{O} plateau appears to be quite flat, although we note that it seems to widen around metallicities between $-1.5$ and $-1$, which, as previously observed, suggests the presence of stars that have been accreted from dwarf galaxies \citep{Nissen2010,Hawkins2015,Hayes2018a}. In some ways S appears qualitatively similar to O. While there is not quite a bimodality in the \xfe{S} distribution, at metallicities around $-0.5$ (where the thin and thick disk are most chemically distinct in other $\alpha$-elements), the spread of the \xfe{S} distribution exceeds what would be expected from the reported errors alone. Therefore, this is likely a true astrophysical spread in the S abundances at these metallicities. Similar to the \xfe{O} bifurcation, this spread would seem to be tied to differing $\alpha$-abundances in the thin and thick disk. At low metallicities we see a slightly sloped plateau in \xfe{S}, but the upper limit flagging also begins to impact the completeness of our S abundance measurements below \feh{} $\sim -1.5$, so this sloped appearance may be somewhat artificial. \subsubsection{\cisotope{} Isotopic Ratio} \label{sec:c12c13_results} Looking at Figure \ref{fig:xfe_abund} we see that at high metallicities the \cisotope{} ratios measured in BAWLAS range from a few up to $\sim 20$. This range then narrows and the average \cisotope{} ratio decreases with decreasing metallicity. These \cisotope{} ratios are also expected to change as a function of $\log g$ due to mixing along the giant branch. Examining this distribution with $\log g$ in Figure \ref{fig:c12c13_logg}, we see two features, the red clump at $\log g \sim 2.5$ and a distribution of red giant stars at $\log g < 2.5$. We measure \cisotope{} ratios for very few $\log g \gtrsim 2.5$ stars, because the $^{13}C$ features become too weak at the warm temperatures of these stars, and therefore at low $\log g$ we predominantly report upper limits. For the stars where we do measure \cisotope{} ratios, the red clump spans a range of \cisotope{} ratios, but sits at slightly lower \cisotope{} ratios, than the red giants at slightly lower $\log g$. Along the giant branch, the \cisotope{} ratios appear to decrease with decreasing $\log g$. This is interesting because these trends generally do not match predictions from standard mixing as shown in Figure \ref{fig:c12c13_logg_models}. Instead, as has been noted in the past \citep[e.g.,][]{Szigeti2017}, we see that \cisotope{} changes along the red giant branch. This feature indicates that extra-mixing occurs along the red giant branch following the red giant branch ``bump,'' and, while the dominant source of this extra-mixing has yet to be settled, many different theoretical models have been proposed \citep[such as thermohaline mixing, rotational mixing, gravity waves, or magnetic fields][and references therein]{Charbonnel1998, Denissenkov2000, Charbonnel2007, Busso2007, Karakas2010, Lattanzio2015}. Here we compare with one of these extra-mixing models, thermohaline mixing, which (at least qualitatively) matches with the \cisotope ratios that we measure. We have selected two mass-metallicity combinations of stellar evolution models from \citet{Lagarde2012} to illustrate what stellar evolution trends we might expect for stars of different mass and metallicity. We show model tracks for mass-metallicity combinations of ($M$, \feh{}) $=$ (1.0 $M_{\odot}$,$-0.56$) and (1.25 $M_{\odot}$,$0.0$). These values are chosen to be reasonable for typical high-$\alpha$ thick disk stars, and low-$\alpha$ thin disk stars, given the distribution of thin and thick disk RGB star masses found by \citet{Pinsonneault2018}\footnote{\citet{Pinsonneault2018} found that thick disk RGB stars in the {\it Kepler} field are typically around a solar mass, while their thin disk sample spans a larger range in masses, but is more massive than one solar mass on average.}. While standard mixing models imply relatively constant \cisotope{} along the upper giant branch (with only significant changes in \cisotope{} as a star begins to ascend the giant branch at $\log g \sim 3.8$ and again slightly as it passes the tip of the giant branch), thermohaline mixing models predict that the surface \cisotope{} ratio should decline as a star evolves up the upper RGB post RGB-bump (occuring around or above $\log g \sim 2$ depending on mass and metallicity). Our \cisotope{} ratios also show a correlation with metallicity in red clump stars (at $\log g \sim 2.5$), which as shown in the right panel of Figure \ref{fig:c12c13_logg_models}, causing the large spread in \cisotope{} at these surface gravities, and again qualitatively agreeing with thermohaline mixing predictions. This is generally what we see in our \cisotope{} measurements, but one must take care when interpreting the \cisotope{} ratios exclusively as a function of $\log g$ or metallicity because of the biases in our sample and our detection limits. For example our sample is biased towards including metal-poor stars at lower $\log g$ and excluding those at high $\log g$. This comes both from the observational biases to observe more luminous, low $\log g$ metal-poor stars, and from our temperature cuts that remove warm, higher $\log g$ metal-poor stars. In addition, our detection limits, bias our sample to the stars that have lower \cisotope{} ratios (because that implies $^{13}C$ is more abundant and has stronger features), higher C abundances, or in cool stars, whose molecular features are stronger. So the trends we observe may be a combination of physical trends and observational and measurement selection biases. To investigate the significance of biases that might be present in our sample, Figure \ref{fig:c12c13_completeness} shows the average \cisotope{} ratio in bins of $\log g$ and metallicity in the left panel, and the fraction of stars which have measured \cisotope{} ratios within each of those bins in the right panel. This clearly shows that there are some biases in our sample. For example, at low metallicities, $\lesssim -1$, we only measure \cisotope{} in some of the lowest $\log g$ stars. It also shows that in the red clump ($\log g \sim 2.5$) at metallicities, \feh{} $\sim -0.5$, we measure very low \cisotope{} ratios ($\lesssim 5$) on average, but we are also measuring \cisotope{} ratios in less than 50\% of the stars with similar stellar parameters. In regions such as this we are only able to measure \cisotope{} ratios in the stars with the lowest ratios, and all other stars are upper limit measurements. Figure \ref{fig:c12c13_completeness}, however, shows that we are nearly complete in measuring \cisotope{} in our sample at metallicities between $-1$ and solar, and $\log g$ between 1.5 and 0 (with a small drop at low $\log g$ around \feh{} $ \sim -0.5$, because of luminous LMC stars that have been flagged with high convolution values that may have erroneous APOGEE stellar parameters). Further, this suggests that the gradients seen in the \cisotope{} ratios in this parameter range in the left panel are not driven by selection effects and are real trends in our data. Therefore, the decreasing \cisotope{} ratios with decreasing $\log g$ and metallicity in this range are qualitatively consistent with the expectations of the thermohaline mixing models shown above from \citet{Lagarde2012}. Future works should examine these \cisotope{} ratios more carefully to compare with theoretical predictions, ideally examining stars whose mass, metallicity, and stage of evolution are known such as stars in this sample that have astroseismology from {\it Kepler} or cluster stars, whose masses and ages can be relatively easily measured. This could provide a more detailed comparison with stellar evolution models. \subsubsection{Sodium (Na)} For Na we see a slightly decreasing trend in \xfe{Na} as a function of increasing metallicity, that begins to rise at super-solar metallicities. The other noticeable feature in the Na distribution are a cloud of stars with high \xfe{Na} ($\gtrsim 0.5$). { While high Na abundances are seen in some literature studies of field stars \citep[e.g.,][]{Duong2019} and globular clusters \citep{Gratton2001, Briley2004, Meszaros2015, Masseron2019, Meszaros2020} and some of the high Na measurements may be real, others may be suspect, because some of these stars are affected by an inaccurately subtracted sky line. In many cases we have attempted to flag stars where this sky line subtraction affects the Na abundance measurement (see Appendix \ref{sec:na16388blendflag} for more information), but in some of the more marginal cases, these stars have passed our flagging criteria and remain in our sample with erroneously high Na abundances. } In general these high \xfe{Na} stars should be treated cautiously. Many of these measurements are of only one line, the Na 16388 \AA{} line, which is affected by this sky subtraction issue particularly in stars with radial velocities between $\sim -110$ and $-60$ \kms{}. The stars that are truly enhanced in Na should have strong enough Na lines that one may expect Na to be measurable in both of the lines we use. So those interested in exploring stars enhanced in Na may want to examine stars that have both Na lines measured and preferably have radial velocities that are not in the range of $\sim -110$ and $-60$ \kms{}. \subsubsection{Phosphorus (P)} Most prominent in the P abundances we measure is the considerable upper limit on the \xh{P} values we are able to measure. Given the extent of the upper limit flagging on P, we are likely incomplete in our \xfe{P} coverage at nearly all metallicities. Nonetheless this does provide us the largest sample of stars with P measured present in the literature, and there is some preliminary evidence for a rising \xfe{P} with decreasing metallicity judging from the upper envelope of our observed sample. We can also see that there is a possible hint of an increase in the \xfe{P} ratios at super-solar metallicities. While our P distribution is limited in it's metallicity coverage, some of these features may still be able to provide further constraints to chemical evolution models and nucleosynthetic yields for P. Most of the theoretical P yields have previous been shown to poorly match observed P abundances in the MW \citep[e.g.,][]{Maas2019, Kobayashi2020}. These predictions typically find sub-solar \xfe{P} ratios at or above solar metallicities, which also seem to be ruled out by our measurements (which is unsurprising given that our P abundance pattern seems to agree reasonably well with past observations, see Section \ref{sec:lit_comp2}). This disagreement suggests that most theoretical yields do not produce enough P and more modeling is needed to reconcile these differences. However our data is qualitatively consistent with some of the features produced by predictions, such as the decreasing \xfe{P} ratios with increasing metallicity at sub-solar metallicities and the slight increase in \xfe{P} at super solar metallicities seen by \citet{Kobayashi2020}. \subsubsection{Vanadium and Copper (V and Cu)} V and Cu both show a similarly shaped abundance distribution pattern at sub-solar metallicities, with slightly elevated \xfe{X} at low metallicities and decreasing with metallicity. While V flattens out around solar metallicity, Cu instead begins to increase. In addition, the \xfe{V} abundances show a tighter distribution than Cu, however Cu also has larger uncertainties because the Cu lines are significantly weaker than the V lines that we use, likely inflating the Cu distribution. While the production of Cu is somewhat more complicated, it is close to the Fe-peak and produced similar to bona fide Fe-peak elements such as V. Both of these elements show chemical abundance patterns like that of other Fe-peak elements measured by APOGEE, such as Co and Ni, although here we don't see a rise in \xfe{V} at super solar metallicities, as seen in APOGEE's Co and Ni abundances and our BAWLAS Cu abundances. Some studies have suggested that several Fe-peak elements may have metallicity dependent nucleosynthetic production in Type Ia SNe at high metallicities \citep[including V][]{Weinberg2021}, {whereas other studies that have seen increases in Cu abundances at super-solar metallicities, like the increases seen here, have suggested that it is due to metallicity dependent yields of Cu in massive stars through the weak s-process \citep{Johnson2014, McWilliam2016, Xu2019}. In either case,} the stronger rise in Cu abundances at high metallicities may suggest that Cu has a stronger metallicity dependence in its nucleosynthetic production than V, but such modeling is beyond the scope of this work. \subsubsection{Cerium and Neodymium (Ce and Nd)} \label{sec:cend_results} The two neutron-capture elements in APOGEE spectra with the strongest lines, Ce and Nd, are both produced by a mix of the r- and s-process in the sun. Ce is a predominantly s-process element in the sun with percentages of 19\%/81\% r-/s-process, whereas Nd has a more even mix, with a slight s-process lean at percentages of 42\%/58\% r-/s-process \citep{Prantzos2020}. The Ce abundances measured here show one of the more complex abundance patterns seen in this study (partly because it is one of the most precise, non-CNO elements that we measure). At low metallicities, between $-2$ and $-1$ we see a slightly rising \xfe{Ce}. Then at higher metallicities, the bulk of the stars have an arched pattern with slightly lower \xfe{Ce} at \feh{} $\sim -0.7$ rising to peak at \feh $\sim -0.25$ before decreasing with increasing metallicity. This is similar to the \xfe{Ba} abundance pattern \citep[another predominantly s-process element in the sun,][]{Prantzos2020} seen by GALAH in their GALAH$+$ DR3 \citep{Buder2021}. On the other hand, while Nd is not measured to as low of metallicities, it does show a different chemical abundance pattern, likely because of its greater production in the r-process. The bulk of the stars with measured Nd form a relatively simple pattern of decreasing \xfe{Nd} abundance ratio with increasing metallicity. Qualitatively this is indeed between the abundance patterns of Ce and the predominantly r-process element, Eu (97\% r-process in the sun) seen with GALAH \citep[and, reassuringly, similar to their Nd abundance pattern][]{Buder2021}. We also find a population of stars with enhancements in Ce and Nd, extending in some cases to \xfe{Ce} or \xfe{Nd} of $+2.0$. Figure \ref{fig:cefendfe} shows that these enhancements are generally correlated, so that Ce-enhanced stars typically show Nd-enhancements and vice-versa. While Ce and Nd are measured from singly ionized transition lines and could be similarly sensitive to systematic parameter errors, \citet{Hasselquist2016} and \citet{Cunha2017} showed that the expected errors in Ce and Nd abundances should be $< 0.1$ dex for Temperature errors of $\sim 100$ K and gravity errors of $\sim 0.2$ dex (and the Ce and Nd errors are anti-correlated for $\log g$ errors). So systematic errors in $\rm T_{\rm eff}$ or $\log g$ don't seem to be able to account for the enhancements we see (even if there were quite large systematic errors). Instead, many of these stars with Ce- and Nd-enhancements are likely s-process enhanced stars which can occur because they have dredged up s-process rich material during their evolution \citep[such as N-type carbon stars;][]{LloydEvans2010} or have accreted s-process rich material from a companion, e.g., Ba stars, CH stars, or CEMP-s stars \citep[][and references therein]{Mcclure1984, Masseron2010, LloydEvans2010}. But it is also possible that stars with r-process enhancements may be present in this sample too (although there are fewer methods of producing r-process enhancements at these metallicities). It has long been seen that roughly $\sim 1\%$ of RGB stars are Ba/CH stars \citep{BoehmVitense1984}, and indeed with a very rough calculation we find a similar percentage of our sample is Ce-enhanced (we consider Ce over Nd, since Ce is measured in a larger fraction of our sample). The density of the bulk sample of stars with Ce measurements begins to drop off rapidly above \xfe{Ce} $> 0.5$, so if we take this as an indication of where we begin to see these various classes of s-process enhanced stars, we find about 1,500 of the 106,000 stars with measured Ce abundances would be considered s-process rich, about 1.4$\%$ of our red giant sample (all of this excluding LMC red supergiants for the reasons mentioned below). While this calculation can and should be done more carefully (though doing so is beyond the scope of this work), this simple order of magnitude estimate suggests that the number of Ce-enhanced stars (and Nd) agrees well with past predictions of the number of s-process enhanced red giants. There is also a notable overdensity of Ce-enhanced stars (and Nd-enhanced stars) at a metallicity of -0.5, which appears to be predominantly due to red supergiants in the LMC. While LMC stars at higher metallicities may in general be enhanced in s-process elements \citep[e.g.,][]{Hasselquist2021}, we suspect the LMC red supergiants have suspect stellar parameters, leading to erroneous chemical abundances not only in Ce and Nd but other elements too, as discussed below in Section \ref{sec:lmc_rsg}. \subsection{Effective Temperature Trends} \label{sec:teff_trends} As with all spectroscopic studies, systematic trends with stellar parameters can be difficult to avoid, particularly when analyzing weak and blended lines, or may be entirely unavoidable in cases where lines are affected by non-LTE effects for example. To investigate what kinds of trends may be present in this analysis, we show our derived abundances as a function of stellar effective temperature in Figure \ref{fig:xteff_abund}. Because temperature trends can appear due to selection biases, we have also highlighted our solar neighborhood sample (using the same selection criteria listed in Section \ref{sec:zero_points}, but without the S/N restriction to increase the sample size), which represents a relatively mono-abundance population that should have a solar \xfe{X} abundance ratio at all temperatures. In the solar neighborhood sample, we do not see particularly significant trends in Na, P, S or Ce, however, the upper giant branch is not well populated because such cool stars are not common in the solar neighborhood. Looking at the full sample for these elements there may be some hints of temperature trends in the coolest stars, e.g., a gradual rise in P abundances below $\sim 3900$ K, a downward trend in Ce below $\sim 3800$ K, etc. Because cooler red giants are more luminous and can therefore be seen to larger distances than warmer, fainter red giants (for the same S/N limit), these trends could be coming from selection biases, i.e., picking out cool stars in regions that are not well populated by the rest of the sample such as the MW bulge. Alternatively, these could be systematic errors because molecular features (the dominant source of blends) become especially strong in cool stars, so any errors in blend treatment could lead to erroneous abundance measurements. For the remaining elements, we do see some structure or trends in their abundances as a function of temperature. The impact of our detection threshold and limit flagging can be seen in \cisotope{}, V, and Nd, particularly at warm temperatures $\lesssim 4600$ where only low \cisotope{} ratios ($\lesssim$ 10) and high \xfe{V} or \xfe{Nd} abundances ($\gtrsim$ solar) can be measured. This compounds with the increasing uncertainties and scatter in V and Nd at low metallicities to cause a decreasing trend in \xfe{X} as a function of temperature below $\sim 4500$ K. At lower temperatures, there may be very slight \cisotope{}, V, and Nd trends with temperature, but it is difficult to assess, given the low number of cool stars with measurements in the solar neighborhood. Looking at the full sample, similar to Ce, we do see a decrease in \xfe{V} at temperatures below $\sim 3800$ K. We also see a change in the average \cisotope{} ratio of the full sample as a function of temperature, however, as mentioned in Section \ref{sec:c12c13_results}), the likely reflects astrophysical \cisotope{} trends as stars evolve up the giant branch due to internal mixing. Finally, Cu shows the strongest temperature trend of the elements we measure. The \xfe{Cu} rises from near solar abundances at 5000 K up to around 0.2 at 3500 K. This is perhaps unsurprising given that the Cu lines used here are quite blended. This trend is likely reflecting the difficulty of measuring these lines when they are strongly blended in cool stars. Therefore, any investigation of the Cu abundances measured here should consider these trends and what impact they may have. It may be advisable to use caution even for those elements that do not show obvious temperature trends, since precision and uncertainties also change with stellar parameters. Comparing stars with similar temperature ranges may provide one way of avoiding these biases. \subsection{Abundance Uncertainties} In Figures \ref{fig:cno_abund}, \ref{fig:xfe_abund}, and \ref{fig:xteff_abund}, we also show the typical measured and empirical abundance uncertainties in our sample as a function of metallicity and temperature (and as a function of \cisotope{} for its uncertainties). In general the typical measured and empirical uncertainties are of a similar magnitude, however, the range of measured uncertainties may be larger since they are determined on a star-by-star basis rather than via an empirical relationship. For some elements, we can see that the measured and empirical uncertainties differ more significantly. For example, in N the measured uncertainties are, on average, much smaller than the empirical uncertainties at high metallicities. This is because there are many N lines with which to measure these abundances, so we can very precisely measure the average N abundance in a star. However, since N is measured from molecular features, it is expected that the measured N abundances will be quite sensitive to the input stellar parameters, so when varying the input stellar parameters in our derivation of empirical abundance uncertainties we find a larger range of resulting N abundances. In Cu, we see a different case, where the measured uncertainties are significantly larger than the empirical uncertainties in cool stars (see Figure \ref{fig:xteff_abund}). This occurs because the Cu I 16006 \AA{} is contaminated by a strong nearby blend, and BACCHUS \chitwo{} method, which is used to calculate our measured uncertainties, is biased by this blend. It derives significantly different Cu abundances than either the other Cu line, or the BACCHUS \wln{} method that was also used to determine uncertainties. The large difference between line and method measurements for Cu in cool stars leads to these large measured uncertainties, despite the fact that the average abundances are not affected by this issue, since we use the \wln{} method for determining the abundances of the Cu I 16006 \AA{} line, and only included this \chitwo{} method for the purpose of determining uncertainties. The Cu measurement uncertainty in these cool stars may, therefore, be artificially inflated, but this can also indicate how uncertain the Cu values could be if blends are not properly treated. \subsection{LMC Supergiant Feature} \label{sec:lmc_rsg} One of the noticeable features in the abundance patterns of a few of the elements is the overdensity of stars at \feh{} $\sim -0.5$ that cover a wide range of abundance ratios. This can be seen most clearly in Ce and Nd, and somewhat less obviously in \cisotope{}, but it also occurs in the other elements to a lesser extent, such as S, covering a smaller spread of \xfe{X}. This feature is primarily from red supergiants (RSGs) in the Large Magellanic Cloud (LMC). These stars are relatively metal-rich stars for the LMC, and because they are all young, recently formed stars, they only cover a narrow range of metallicity. Many LMC RSGs have been observed by APOGEE \citep[particularly in the southern TESS continuous viewing zone contributed programs;][]{Santana2021} and analyzed here. Figure \ref{fig:lmc_cmd} shows a 2MASS CMD of LMC stars in our sample that have been selected according to the spatial and kinematic (proper motion and radial velocity) selections given in \citet{Hasselquist2021}. The patchwork distribution of stars in this CMD reflects the variety of programs and subprograms that have targeted LMC stars in APOGEE \citep{Nidever2020,Santana2021}, but for reference the tip of the red giant branch (TRGB) lies around 12 in K$_{\rm s}$ \citep[noting though that the TRGB is actually sloped; see e.g.,][]{Boyer2011,Hoyt2018}. The stars that are more luminous than the TRGB split into two branches with a redward branch around J-K$_{\rm s}$ $\sim 1.2$ and a blueward branch that is brighter and centered on J-K$_{\rm s}$ $\sim 1.0$. The redder stars seen here belong to various AGB populations, whereas the bluer stars that have been outlined in Figure \ref{fig:lmc_cmd} are red supergiants that have been selected photometrically following the criteria of \citet{Neugent2020}. In Figure \ref{fig:lmc_rsg_chemistry}, we show the S and Ce chemistry of these photometrically selected RSGs compared to the rest of the BAWLAS sample. Indeed the RSGs produce the overdensity of high \xfe{Ce} stars seen in Figure \ref{fig:xfe_abund}. The source of the Ce enhancement and the large spreads seen in Ce, S and other elements is unclear. It is possible that the \citet{Neugent2020} photometric selection may have some contamination by massive AGB stars that could have unusual abundances from internal mixing \citep[e.g., ][]{Plez1993}. However, this photometric selection is expected to have a relatively low contamination from AGB stars \citep[and was designed to be so,][]{Neugent2020}, and because most of this selection has unusual abundances (rather than just a few outliers in this selection), it suggests that this odd feature is related to the LMC RSGs specifically. To some extent it may be expected that young metal-rich stars in the LMC may have enhancements in s-process elements from many generations of AGB stars. For instance, \citet{Hasselquist2021} have showed that metal-rich LMC RGB stars (which are thought to be relatively young, but still slightly older than the massive RSGs) do show an enhancement in \xfe{Ce} over MW stars of the same metallicity. However, these enhancements are at \xfe{Ce} $\sim 0.2 - 0.3$ rather than the \xfe{Ce} $\sim 0.5 - 2.0$ seen in the RSGs here. Furthermore, this wouldn't explain the large spread seen in other elements, e.g., S, which extends to high \xfe{S} unexpectedly given that the metal-rich RGB stars in the LMC show relatively a small $\alpha$-element abundance spread at slightly super-solar \xfe{X}. Therefore, this large spread in multiple elements (beyond just S and Ce shown here) for such a narrow range in metallicity seems unlikely to be a physical abundance pattern and is more likely to be evidence that there are systematic errors for the RSGs. This could be in the form of incorrect stellar parameters, improper fitting of blends, heightened nLTE or 3D effects in these stars, etc. For example, because these stars are very luminous and low $\log g$, spherical radiation transfer might be needed (in addition to the spherical atmosphere models that APOGEE already uses) to properly derive stellar parameters for these stars, rather than the plane-parallel radiation transfer used to derive the DR17 ASPCAP stellar parameters \citep{Holtzman2021}. This could introduce systematic trends in stellar parameters of these stars that we see propagate into the abundances we derive here. We note that these stars also exhibit a spread in some APOGEE derived abundances, such as APOGEE's Al or S measurements, which indicates that this feature does not originate exclusively in our BACCHUS analysis. Fully investigating these stars, however, is beyond the scope of this work, but for completeness we warn that the abundances (and potentially even the stellar parameters of these stars) may be suspect and unphysical, and should be used with caution. \section{Literature Comparison} \label{sec:comparison} \subsection{APOGEE Comparison} \label{sec:apogee_comp} While our abundances are derived from the exact same spectra that are used to measure abundances in APOGEE, there are several key differences in our analyses that lead to differences in derived abundances. In addition to using a different abundance pipeline and spectral synthesis code for our analysis, we also use different input stellar parameters, employ line-by-line and star-by-star flagging that allows us to provide a more cleaned sample of abundances, and, in some cases, use a slightly different selection of lines. \begin{deluxetable*}{l c c c c c c l} \tablewidth{0pt} \tablecolumns{8} \tablecaption{BAWLAS Abundance Comparison to APOGEE DR17 and High-resolution Literature Measurements\label{tab:lit_comp}} \tablehead{\colhead{Element} & \multicolumn{3}{c}{APOGEE DR17} & \multicolumn{4}{c}{High-res Lit} \\ & Mean\tablenotemark{{\scriptsize a}} & St Dev & N$_{\rm stars}$ & Mean\tablenotemark{{\scriptsize a}} & St Dev & N$_{\rm stars}$ & References} \startdata $\Delta$\xh{C} & -0.09 & 0.10 & 122,425 & -0.16 & 0.22 & 45 & 1, 2 \\ $\Delta$\xh{N} & -0.20 & 0.16 & 122,071 & -0.21 & 0.14 & 41 & 1, 2 \\ $\Delta$\xh{O} & -0.01 & 0.11 & 114,591 & 0.03 & 0.13 & 116 & 1, 2, 3\\ $\Delta$\xh{Na} & -0.02 & 0.11 & 93,893 & 0.11 & 0.15 & 30 & 1, 2\\ $\Delta$\xh{V} & -0.15 & 0.10 & 67,115 & -0.12 & 0.13 & 69 & 1, 2, 4\\ $\Delta$\xh{S} & 0.03 & 0.09 & 104,319 & - & - & - & - \\ $\Delta$\xh{Cu} & - & - & - & 0.03 & 0.18 & 19 & 1\\ $\Delta$\xh{Ce} & -0.10 & 0.14 & 90,191 & 0.04 & 0.18 & 96 & 5\\ $\Delta$\cisotope{} & - & - & - & -0.1 & 4.7 & 23 & 6-14\\ \enddata \tablenotetext{a}{Calculated as (\xh{X}$_{\rm Ref}$ $-$ \xh{X}$_{\rm BAWLAS}$)} \tablecomments{References for high-resolution literature measurements: 1 -- \citet{daSilva2015}; 2 -- \citet{Brewer2016}; 3 -- \citet{Jonsson2017}; 4 -- \citet{Lomaeva2019}; 5 -- \citet{Forsberg2019}; 6 -- \citet{Briley1994}; 7 -- \citet{Briley1997}; 8 -- \citet{Smith2002}; 9 -- \citet{Pavlenko2003}; 10 -- \citet{Smith2007}; 11 -- \citet{Mikolaitis2012}; 12 -- \citet{Smith2013}; 13 -- \citet{Drazdauskas2016}; 14 -- \citet{Szigeti2017}} \end{deluxetable*} These analysis choices can all contribute to differences between the abundances measured by APOGEE and our work here for elements that are analyzed by both. In Table \ref{tab:lit_comp}, we tabulate some basic statistics on the differences between the APOGEE DR17 and BAWLAS abundances, showing the mean abundance differences and 1$\sigma$ standard deviation of these differences. A few elements see a moderate shift, such as C, N, V, and Ce, which can be attributed to dependence on input stellar parameters for C and Ce, and the removal of temperature trends in APOGEE data for V, as discussed below. For N the offset isn't obviously tied to one single source, and is likely a result of changing stellar parameters as well as differences in C and O which have an important in determining N abundances. The scatter in the differences between BAWLAS and APOGEE are typically larger than the combined uncertainties for most of the elements by $\sim 25-50\%$ (and slightly larger still for the more precisely measured molecular elements C, N and O). However, the reported uncertainties from BAWLAS and APOGEE are simply estimates of the true uncertainty, and there are some systematic differences between the two analyses, in terms of input stellar parameters, methodology, etc., either of which may account for the difference between random uncertainties and the scatter between the two sets of measurements. Figure \ref{fig:apogee_diffs} shows a few examples of the ways in which the abundances we derive differ from APOGEE's measurements. In particular, this shows APOGEE abundance measurements for the sample of stars we analyze, the measured BACCHUS abundances in this sample, and for the stars that are measured in both, we also show the differences between those measurements. \subsubsection{Systematic Effects of Stellar Parameter Choices} BAWLAS also differs from APOGEE because we use APOGEE's calibrated stellar parameters to derive abundances, while APOGEE has used its uncalibrated stellar parameters. The differences between the calibrated and uncalibrated stellar parameters are primarily in the effective temperatures and surface gravities, which are, respectively, calibrated to photometric temperature relations, and astroseismic surface gravities in ASPCAP's post-processing \citep[for giants;][]{Holtzman2021}. We also re-derive the C, N, and O abundances for fitting blends, since we are using different input stellar parameters, whereas C, N and $\alpha$ abundances (which adjust the O abundance by assuming [O/$\alpha$] = 0) are fit simultaneously by APOGEE in its stellar parameter fits. Not surprisingly, it appears that the change in stellar parameters, particularly the use of calibrated temperatures and gravities, affects some of the elements that we measure here. One clear example of this is Ce. While, in a broad sense, the chemical abundance patterns of Ce from APOGEE DR17 and BAWLAS are relatively similar (as seen in Figure \ref{fig:apogee_diffs}), we find that there is a trend in the differences between the two Ce measurements as a function of metallicity, with BAWLAS finding systematically higher \xfe{Ce} at lower metallicities. This metallicity-correlated offset seems to primarily be a result of using different $T_{\rm eff}$ and $\log g$. The Ce II lines in APOGEE's wavelength range are quite sensitive to surface temperatures and gravities, such that using a hotter $T_{\rm eff}$ or larger $\log g$ results in deriving higher Ce values \citep{Cunha2017}. This compounds with the APOGEE DR17 $T_{\rm eff}$ and $\log g$ calibrations, both of which are largest at low metallicities, to produce Ce differences that grow with decreasing metallicity. Figure \ref{fig:apogee_cediff} shows a clear correlation of the difference between APOGEE and BAWLAS \xfe{Ce} as a function of the difference between the uncalibrated and calibrated temperatures used in the respective analyses. Note that the scatter at low $\Delta T_{\rm eff}$ coming primarily from warm stars where the Ce lines are weaker and the abundance measurements are more uncertain, and there are a lack of cool stars, since APOGEE does not populate its \xfe{Ce} abundances for cool stars where the Ce is deemed unreliable \citep{Holtzman2021}. Similar trends also appear as a function of the $\log g$ calibration differences. We also find that C and O abundances appear to be affected by the differences in assumed stellar parameters. We also show the differences between the BAWLAS and APOGEE abundances for C and O as a function of the APOGEE temperature calibration in Figure \ref{fig:apogee_cediff}. While there are some differences between the trends for C, O, and Ce, each of them show a correlation with the temperature calibration. The variety in these correlations reflects the complexities in deriving these abundances and each of their individual dependencies on the input stellar parameters, line choice, the abundance of elements with molecular species (for molecular equilibrium or blending), etc. For example the larger spread as $\Delta {T_{\rm eff}}$ approaches zero is from warm stars, which have weaker features, increasing uncertainty and scatter in their resulting abundances. Another example is the increase in the spread of the O abundance differences around $\Delta {T_{\rm eff}} \sim 175$ K, which appears to be due to correlations between the O differences and the $\log g$ calibration (which is temperature dependent, hence the temperature trend for a fixed $\Delta {T_{\rm eff}}$), which has less of an effect on the C and Ce measurements. These systematic differences can be important to consider because they do have some impact on the chemical abundance patterns of these elements. Figure \ref{fig:apogee_co_comp} compares the C and O abundances patterns of APOGEE and BAWLAS for the present sample. While in both cases the BACCHUS measurements have more scatter, because fewer lines have been used than in APOGEE's analysis, we can see they also result in higher \xfe{X} abundances at low metallicities. For C this does not have a particularly significant effect on the appearance of the abundance pattern. However, this systematic difference at lower metallicities is important to consider, because different metallicity trends can affect the interpretation of C for stellar evolution \citep[e.g.,][]{Shetrone2019}. Interestingly, these changes for C bring the BAWLAS abundances into better agreement with APOGEE's atomic C measurements rather than the molecular ones that are compared to here. Some of this may be because BACCHUS measures a mix of atomic and molecular C features, but it may be worth investigating if calibrated temperatures and surface gravities bring atomic and molecular C abundances into better agreement than APOGEE's uncalibrated parameters. Similar to C, in Figure \ref{fig:apogee_co_comp} we see differences between the O abundances in BAWLAS and those in APOGEE, which also seem to be related to the choice of stellar parameters. In BAWLAS we see a higher \xfe{O} plateau of $\sim 0.5$ at low metallicities, as opposed to the $\lesssim 0.4$ plateau seen in APOGEE. At higher metallicities we also see a difference in the \xfe{O} abundance patterns of the thick disk/high-$\alpha$ and thin disk/low-$\alpha$ sequences. Both of these sequences appear flatter and plateau at \feh{} $\sim -0.7$ in APOGEE, whearas BAWLAS finds both of these sequences monotonically increasing towards lower metallicities. Interestingly, the BAWLAS \xfe{O} abundance patterns are more similar than APOGEE to what has been seen with high-resolution optical studies. Such studies, typically find steeper \xfe{O} trends in the high- and low-$\alpha$ sequences and a higher plateau at low metallicities than what APOGEE finds \citep[e.g., averaging a plateau value of \xfe{O} $\sim 0.6$][]{Bensby2014}, and instead more similar to what we see in BAWLAS. The difference in the \xfe{O} plateau is likely a result of using calibrated stellar parameters, but it is not clear what causes the differences between the abundance patterns of the high- and low-$\alpha$ sequences. These may be in part affected by using different parameters or may be a result of separating O into an individual dimension instead of changing it along with other $\alpha$-elements as is done in APOGEE. It is also possible that the lower precision of BAWLAS O abundances blur out the fine details that are apparent in APOGEE's O abundances. While we have used the calibrated APOGEE DR17 parameters, it is not entirely clear which set of stellar parameters might be best to use for abundance analysis. The calibrated stellar parameters used here may more accurately reflect the true stellar parameters of the stars we observe and help remove systematic trends with uncalibrated stellar parameters. On the other hand the spectroscopic stellar parameters are derived simultaneously and (theoretically) provide the best fit to the observed spectra. So the use of spectroscopic stellar parameters may reduce the impact of any systematic errors or unaccounted for physics in the model atmospheres or synthetic spectra that might lead to poorly fit blends, line shapes, etc. Therefore we simply note that there may be differences that arise from using different input stellar parameters, and that by comparing the differences in derived abundances, we may be able to explore some of these effects. \subsubsection{Cleaning Suspect Abundance Measurements} One significant difference between our analysis here and APOGEE's is our more detailed flagging that helps identify lines or stars whose abundances may be erroneous or highly uncertain. In APOGEE, there is no explicit flagging for lines that may be significantly affected by blends or for measurements that should be below upper limits. In cases such as these, the abundances APOGEE reports may be coming from poor fits or measuring noise. An example of this can be seen in Na. While the Na measurements agree reasonably well (within uncertainties) for stars measured in both APOGEE and BAWLAS, the effect of flagging and upper limits can clearly be seen in Figure \ref{fig:apogee_diffs}. At low metallicities, $\lesssim -0.5$ APOGEE reports abundances for many stars, however these stars show a large scatter, whereas with BAWLAS they are flagged as upper limits or as having poor quality fits (typically because the lines are weak and dominated by spectral noise). While we perform similar to APOGEE when stars are measured in BAWLAS, the benefit of our analysis is that we are able to flag stars with suspect measurements even at the level of indivdual lines on a star-by-star basis. Some systematic differences are also seen between APOGEE and BAWLAS as a function of metallicity (BAWLAS reports slightly higher \xfe{Na} at lower metallicities), which is similar to other elements as discussed in the previous section. The flagging and upper limits can then improve or simplify the interpretation of the chemical abundances such as Na. S, and to some extent N (aside from its systematic offset), are similar to Na, in that the chemical abundance patterns do not significantly change, but by removing suspect abundances we present a clearer picture of these patterns. \subsubsection{Holistic Treatment of Weak and Blended Features} The last element that overlaps with APOGEE in DR17 is V, which is also shown in Figure \ref{fig:apogee_diffs}. The V abundance pattern is significantly different between APOGEE and BAWLAS, and appears to be improved in the BAWLAS analysis here. In APOGEE the V abundances show a large spread (which is even larger at low metallicities where there is no clear pattern), whereas in BAWLAS we see a tight distribution at higher metallicities and that much of the low metallicity scatter seen in APOGEE has been cleaned by our flagging and upper limits at low metallicities. Comparing the V abundances as a function of temperature for stars that were measured by both APOGEE and BAWLAS in Figure \ref{fig:apogee_vdiff}, we see that there are clear systematic trends in APOGEE's V abundances as a function of temperature. This produces the large range of \xfe{V} and complex abundance pattern seen at higher metallicities in APOGEE. In BAWLAS, while we do see slight temperature trends (see Section \ref{sec:teff_trends}), they are significantly reduced, leading to a much tighter \xfe{V} pattern. This improvement is not clearly tied to any one difference between the APOGEE and BAWLAS analyses and is likely a result of a variety of improvements in localized blend treatment, line choice and flagging, upper limits, etc. This allows us to more precisely measure elements like V that are heavily blended, and in some cases only present themselves in weak lines, such as P and Nd as well as the \cisotope{} ratio, illustrating the strength the BAWLAS methodology here. \subsection{High-resolution External Literature Comparison} In addition to comparing with APOGEE, we have compiled high-resolution literature measurements for our elements of interest to compare with our derived abundances. \subsubsection{Star-by-Star Comparison} As with \citet{Jonsson2020,Holtzman2021} we compare star-by-star measurements of BAWLAS to those of \citet{daSilva2015} and \citet{Brewer2016} for C, N, O, Na, V and Cu (with Cu coming from \citealt{daSilva2015} only), and to the sample from \citet{Jonsson2017,Lomaeva2019,Forsberg2019} for O, V, and Ce respectively. For \cisotope{} we compare with values measured from several literature studies \citep{Briley1994, Briley1997, Smith2002, Pavlenko2003, Smith2007, Mikolaitis2012, Smith2013, Drazdauskas2016, Szigeti2017}; this sample may be relatively small and inhomogenous, but it can be used to indicate the quality of our \cisotope{} measurements. Our overlap with the literature for P, S, and Nd is too small (a couple stars if any) to draw any meaningful conclusions from. Table \ref{tab:lit_comp} gives the mean difference between our measurements and these high-resolution studies, and the $1\sigma$ standard deviation scatter around that mean. The mean differences are relatively small for most elements, and the scatter is similar to or smaller than APOGEE comparisons with literature values for the same elements \citep[as of DR16,][]{Jonsson2020}. The one exception is C, which has slightly larger scatter and is likely a reflection of the lower precision that we are able to achieve using only a handful of C lines. Because the high \cisotope{} ratios are more uncertain, this can skew the scatter in our literature comparison to follow the differences in stars with high \cisotope{} values. So, to compare our \cisotope{} ratios in another way, in Figure \ref{fig:c12c13_litcomp} we show a normalized histogram of the difference between BAWLAS and literature measurements of \cisotope{}, divided by the uncertainty on that difference (combining the reported literature uncertainty, where available, with the measurement uncertainty on our \cisotope{} ratios). This effectively shows how many $\sigma$ away from agreement these differences are. We also indicate the distribution we would expect to see if the differences were normally distributed according to the combined uncertainties. We find that the differences are close to within what we would expect from random errors alone, although there are a few outliers, which may be true outliers or have underestimated uncertainties (either in our study or in the literature studies that they were drawn from). \subsubsection{Chemical Abundance Pattern Comparison} \label{sec:lit_comp2} The above comparisons are for relatively small samples, so to expand our comparison with literature, we have also compiled a larger sample of high-resolution studies to compare their general chemical abundance patterns with those we measure here. This sample is made of measurements from the following studies: \begin{itemize} \item Na: \citet{Chen2000, Reddy2006, Adibekyan2012, Bensby2014, Roederer2014, daSilva2015, Brewer2016, Duong2019} \item P: \citet{Caffau2011, Roederer2014p, Maas2017, Caffau2019, Maas2019} \item S: \citet{Chen2002, Caffau2005, Nissen2007, CostaSilva2020} \item V: \citet{Chen2000, Reddy2006, Battistini2015, Roederer2014, daSilva2015, Brewer2016, Duong2019, Lomaeva2019} (we exclude the V measurements of \citealt{Adibekyan2012} since their V show strong systematic temperature trends) \item Cu: \citet{Reddy2006, Roederer2014, daSilva2015, Duong2019} \item Ce: \citet{Reddy2006, Mishenina2013, Roederer2014, Battistini2016, DelgadoMena2017, Forsberg2019} \item Nd: \citet{Mashonkina2004, Reddy2006, Mishenina2013, Roederer2014, Battistini2016, DelgadoMena2017, Duong2019} \end{itemize} The chemical abundances from this literature compilation are shown over our derived abundance distributions in Figures \ref{fig:xfe_litcomp} and \ref{fig:pfe_litcomp}. In general, we see that there is a fairly good agreement between the chemical abundance patterns measured in the literature and by our analysis here. {We note that, while we have good agreement with the Na abundances from the literature referenced here, there is some disagreement between our results for Na and those of bulge RGB stars from \citet{Johnson2014}, particularly in the metal-poor regime, where they report sub-solar \xfe{Na}. However, other optical literature abundances do not show sub-solar Na abundances even when considering bulge stars \citep[e.g.,][]{Duong2019}. Therefore given the good agreement between BAWLAS and the optical literature here, it suggests that there may be some systematics in the \citet{Johnson2014} Na abundance measurements particularly at low metallicities, as suggested by \citet{Duong2019}.} For elements where we see stars that have significantly elevated abundances like Ce and Nd, we also see high-resolution literature measurements of similarly enhanced stars (albeit fewer of them). For P we note that there are very few literature measurements, so it is difficult even with the literature to clearly uncover the underlying chemical abundance pattern. However the literature distribution does seem to reflect a similar pattern to what we see at the metallicities where we are able to measure P. The one element where we see some deviation from the literature is Cu. Literature measurements show a decreasing \xfe{Cu} abundance with decreasing metallicity at metallicities, \feh{} $\lesssim -0.8$, which we do not find in our sample. One possible explanation for these differences is that in the H-band, at APOGEE resolution, we are limited in how metal-poor we can measure Cu because of it's weak, blended lines. These limits begin to affect our measured \xfe{Cu} abundance pattern as metal-rich as \feh{} $\sim -0.7$. This biases our final Cu sample to stars with relatively higher \xfe{Cu} at low metallicities. Stars with low Cu, such as those in the decreasing literature trend, would only be measureable as upper limits in our analysis. This combination of effects might cause some of the differences between the Cu patterns we see in our analysis and those in literature. Another possible factor is that studies have found that nLTE effects should be more significant in optical Cu measurements at lower metallicities, while they are less significant in the H-band \citep{Yan2015,Andrievsky2018}. The necessary nLTE corrections would push the optically measured, literature, Cu to higher values, bringing the literature into better agreement with the trend we see with BAWLAS. However, these nLTE corrections are not quite large enough to completely account for the differences. {Cu is also one of the elements that we measure using the ``wln'' method in BACCHUS for one of its lines. This method relies on only a single pixel, and therefore could be more subject to systematics or random uncertainties, and could skew our results away from what is found in the literature. However, the differences in the Cu trend between BAWLAS and the literature are present in the Cu 16005.5 \AA{} measurements which are calculated with the ``chi2'' method, and persist even if we use the ``chi2'' method for the Cu 16006.0 \AA{} line (though the scatter is larger). Instead, systematic blending, the influence of upper limit flagging and optical nLTE effects are a more likely the source of the discrepancy between the optical literature Cu measurements and BAWLAS. P is also heavily reliant on the ``wln'' method, but we remark for completeness, that as with Cu, the abundance patterns of P do not change significantly if we use the ``chi2'' method, which is more robust to individual pixel noise.} We may also be seeing an affect of upper limits in Nd, insofar as we are limited to upper limits at the low metallicity end of our distribution. This in turn may make our distribution look relatively higher in Nd than the literature at low metallicities, due to the missing stars with lower Nd. Otherwise, the Nd abundance pattern we measure appears fairly similar to that seen in the literature. \section{Conclusion} \label{sec:conclusion} In this work we use the BACCHUS code to analyze $\sim$ 126,000 high S/N, $> 150$, APOGEE spectra of giant stars. Using the APOGEE calibrated stellar parameters of these stars we measure their \cisotope{} ratios and chemical abundances in C, N, O, Na, P, S, V, Cu, Ce, Nd. We provide both the line-by-line abundances measured by BACCHUS and combinations of these line-by-line measurements, where suspicious measurements have been identified and removed using native BACCHUS line flagging, additional spectral flags, and upper limit relations that we calculate for our elements of interest. Alongside these combined abundances, we also report upper limit measurements and two measures of the uncertainties on our abundance measurements. (1) We measure uncertainties from line-by-line scatter of the abundances in a given star. And (2) we report empirical uncertainties derived from the abundance scatter in stars with repeat observations. Because of the flexibility of our analysis and its careful treatment of spectroscopic details, such as blends and upper limits, we are able to provide our BAWLAS catalog that is complementary to the APOGEE DR17 catalog. More automated and rigid pipelines such as ASPCAP can have difficulty measuring elements that are weak or heavily blended, whereas our more time-intensive but focused analysis allows us to expand upon what is measurable from APOGEE spectra. This provides improvements for some of the elements that APOGEE already provides like Na, S and Ce, by cleaning suspect measurements, but also significantly improves the measurement of blended V lines, and can even measure elements that APOGEE is unable to like P, Cu, Nd, and \cisotope{}, providing the largest samples of P and \cisotope{} measured to date. In addition to providing improved abundances for elements that were measured by APOGEE, we can now explore other dimensions of chemical space previously unavailable with APOGEE. This includes examining r- {\it and} s-process abundances for APOGEE stars, studying dredge-up and stellar evolution along the giant branch with \cisotope{}, and investigating the relatively poorly studied P abundance distribution. There are, of course multiple uses for this data particularly, because this relatively large sample spans a range of stellar parameters, positions in the galaxy, and even extends to some of the more massive satellites of the Milky Way (such as Sagittarius and the LMC). Several large spectroscopic surveys are beginning soon or are planned, e.g., Milky Way Mapper (MWM) as part of SDSS-V \citep{sdss5}, the William Herschel Telescope Enhanced Area Velocity Explorer \citep[WEAVE;][]{weave}, the 4-metre Multi-Object Spectroscopic Telescope \citep[4MOST;][]{4most}, and Maunakea Spectroscopic Explorer \citep[MSE;][]{mse}. These surveys will necessarily require more and more efficient pipelines to handle the volume of data that they will observe. The current epoch of surveys, like APOGEE, GALAH, LAMOST, etc., have demonstrated that a great wealth of information is available, and can be extracted, and many new, quick methods are being developed to help facilitate this kind of information extraction in the next generation of surveys \citep[e.g., The Cannon, StarNet, astroNN, The Payne;][]{cannon, starnet, astronn, payne}. However with the present analysis we show that there is more information that can be extracted from spectroscopic surveys, when analyzed with more careful and meticulous methods. While methods such as ours here are time consuming and often require high quality data (e.g., high S/N spectra, precise stellar parameters, etc.), future surveys may benefit from having smaller scale, complementary analyses like what we present in this work to capitalize on their highest quality data. \section*{} DAGH and TM acknowledge support from the State Research Agency (AEI) of the Spanish Ministry of Science, Innovation and Universities (MCIU), and the European Regional Development Fund (FEDER) under grant AYA2017-88254-P. TM acknowledges financial support from the Spanish Ministry of Science and Innovation (MICINN) through the Spanish State Research Agency, under the Severo Ochoa Program 2020-2023 (CEX2019-000920-S) as well as support from the ACIISI, ConsejerГ­a de EconomГ­a, Conocimiento y Empleo del Gobierno de Canarias and the European Regional Development Fund (ERDF) under grant with reference PROID2021010128. This research made use of {\texttt{topcat}} \citep{topcat}, Astropy, a community-developed core Python package for Astronomy \citep{astropy}, NASA's Astrophysics Data System, and the SIMBAD database, operated at CDS, Strasbourg, France. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High Performance Computing at the University of Utah. The SDSS website is \url{www.sdss.org}. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard \& Smithsonian, the Chilean Participation Group, the French Participation Group, Instituto de Astrof\'isica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut f\"ur Astrophysik Potsdam (AIP), Max-Planck-Institut f\"ur Astronomie (MPIA Heidelberg), Max-Planck-Institut f\"ur Astrophysik (MPA Garching), Max-Planck-Institut f\"ur Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observat\'ario Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Aut\'onoma de M\'exico, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement. \facilities{Sloan (APOGEE-N spectrograph), Du Pont (APOGEE-S spectrograph)} \software{BACCHUS \citep{masseron2016}, TOPCAT \citep{topcat}, Astropy \citep{astropy}, Matplotlib (\url{http://dx.doi.org/10.1109/MCSE.2007.55}), Numpy (\url{http://scitation.aip.org/content/aip/journal/cise/13/2/10.1109/MCSE.2011.37})} \begin{appendix} \section{BACCHUS Flag and Combination Settings} \label{app:comb_settings} As mentioned in Section \ref{sec:line_comb}, here we describe the choices we have made to flag individual line measurements and combine them to derive the final elemental abundances that are presented in this paper. In many cases we have used a set of default settings that work well for most lines across the elements we measure. The line-by-line settings that we have to choose from fall into the following categories: (1) which lines did we use in our combination, (2) which BACCHUS measurement method was used for each line, (3) what BACCHUS method flags were use to identify satisfactory measurements, (4) what spectra flags were used for each line, and finally (5) what upper limit threshold was used for each line. Our default choices for how to flag and combine lines for each of these categories are: \begin{enumerate} \item Line selection: use all input lines of a given element \item BACCHUS measurement method: use the \chitwo{} method \item BACCHUS method flags: require that all reported BACCHUS method flags = 1 for a given measured line \item Spectra flags: require that all spectra flags = 1 (if such a flag is present for the star and line in question). \item Upper limit threshold: use the t = 1\% empirical threshold relation to define upper limits \end{enumerate} However, in some cases we have deviated from these default settings in order to better treat specific lines. Table \ref{tab:settings} lists all of the flag choices and settings that we have used in our line combination, and below we go through each element and describe the choices that differ from the defaults listed above. \startlongtable \begin{deluxetable*}{l c c c c c c c c c c} \tabletypesize{\scriptsize} \tablewidth{0pt} \tablecolumns{11} \tablecaption{Line Flag and Combination Settings \label{tab:settings}} \tablehead{\colhead{Line (\AA)} & \colhead{Use Line?} & \colhead{Measurement Method} & \multicolumn{5}{c}{Method Flags} & \multicolumn{2}{c}{Spectra Flags} & \colhead{Upper Limit Threshold} \\ \cline{4-8} \cline{9-10} \colhead{} & \colhead{} & \colhead{} & \colhead{\syn{}} & \colhead{\eqw{}} & \colhead{\intmethod{}} & \colhead{\chitwo{}} & \colhead{\wln{}} & \colhead{blend} & \colhead{cont} & \colhead{}} \startdata \cutinhead{C} 15578.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15775.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15783.9 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15978.7 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16004.9 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16021.7 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16185.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16397.2 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16481.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16614.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16836.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16890.4 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 17063.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 17448.6 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 17456.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ \cutinhead{N} 15119.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15210.2 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15222.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15228.8 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15242.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15251.8 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15309.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15317.6 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15363.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15410.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15447.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15462.4 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15466.2 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15495.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15514.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15581.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15636.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15659.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15706.9 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15708.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15825.7 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ \cutinhead{O} 15373.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15391.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15569.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15719.7 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 15778.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16052.9 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16055.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16650.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16704.8 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16714.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16872.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ 16909.4 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & BACCHUS \intmethod{} limit\\ \cutinhead{Na} 16373.9 & Yes & \wln{} & 1 & 1 & 1 & 1 & 1 & - & 1 & 1\% \\ 16388.8 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\% \\ \cutinhead{P} 15711.6 & Yes & \wln{} & 1 & 1 & 1 & 1 & 1 & 1 & - & 3\% \\ 16482.9 & Yes & \wln{} & 1 & 1 & 1 & 1 & 1 & - & 1 & 3\% \\ \cutinhead{S} 15403.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & 1 & - & 1\% \\ 15422.3 & No & \chitwo{} & 1 & 1 & 1 & 1 & 1 & 1 & - & 1\% \\ 15469.8 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & 1 & - & 1\% \\ 15475.6 & No & \chitwo{} & 1 & 1 & 1 & 1 & 1 & 1 & - & 1\% \\ 15478.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & 1 & - & 1\% \\ \cutinhead{V} 15924.8 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & 1 & 3\% \\ 16137.3 & No & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & 1\% \\ 16200.2 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & 1\% \\ \cutinhead{Cu} 16005.5 & Yes & \chitwo{} & 1,3 & all & all & 1 & 1 & 1 & - & 3\% \\ 16006.0 & Yes & \wln{} & all & all & all & 1 & 1 & 0,1 & - & 3\% \\ \cutinhead{Ce} 15784.8 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & 1 & - & 1\% \\ 16376.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & 1\% \\ 16595.2 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & 1\% \\ 16722.5 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & 2\% \\ \cutinhead{Nd} 15368.1 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & 0,1 & 0,1 & 2\% \\ 16053.6 & Yes & \wln{} & 1,3 & 1,3 & all & 1 & 1 & - & 1 & 1\% \\ 16262.0 & Yes & \chitwo{} & 1 & 1 & 1 & 1 & 1 & - & - & 3\% \\ \cutinhead{\cisotope{}} 15641.7 & Yes & \wln{} & 1,2,3 & 1,2,3 & all & 0,1 & 1,2 & - & 1 & \multirow{8}{2.25cm}{\centering Line-by-line lower limit of \cisotope{} $= 49$, require that line 16530 \AA{} is measured, and final lower limits based on a relation for 16530 \AA{}.} \\ 16121.4 & Yes & \chitwo{} & 1,2,3 & 1,2,3 & all & 0,1 & 1,2 & - & 1 & \\ 16323.4 & Yes & \wln{} & 1,2,3 & 1,2,3 & all & 0,1 & 1,2 & - & 1 & \\ 16326.0 & Yes & \wln{} & 1,2,3 & 1,2,3 & all & 0,1 & 1,2 & - & 1 & \\ 16327.3 & Yes & \wln{} & 1,2,3 & 1,2,3 & all & 0,1 & 1,2 & - & 1 & \\ 16530.0 & Yes & \chitwo{} & 1,2,3 & 1,2,3 & all & 0,1 & 1,2 & - & 1 & \\ 16741.2 & Yes & \wln{} & 1,2,3 & 1,2,3 & all & 0,1 & 1,2 & - & 1 & \\ 16744.7 & Yes & \wln{} & 1,2,3 & 1,2,3 & all & 0,1 & 1,2 & - & 1 & \\ \enddata \tablecomments{When multiple flags (comma separated) are given for a specific method, any lines flagged with those values were considered in line combination.} \tablecomments{Lines marked with a ``-'' in their blend or cont spectra flag column indicate that no such flag was defined for that line.} \end{deluxetable*} \subsection{Carbon, Nitrogen, and Oxygen (C, N, and O)} For C, N, and O, since these were used primarily for fitting blends, we have used a fairly simple selection of settings to combine these elements. For C, N, and O we use our default settings for the line selection (all lines), measurement method (\chitwo{}), and the method flag choices (flag = 1 for all methods). No spectra flags were defined or used for C, N, and O. For the upper limits of C, N, and O, because these are dominantly measured in molecular features, they are therefore sensitive to each other's relative abundance. For instance typical stars have abundance ratios with C $<$ O, but the molecular features in a star strongly differ when they have abundance ratio C $>$ O. However, our empirical upper limit relations are derived assuming solar abundances of all elements except the element in question, therefore driving a changing abundance ratio relative to other elements. Therefore for C, N, and O the empirical relations we derive are not appropriate because they are derived from spectra with strongly changing [C/N], [C/O], etc. and do not reflect the ratios seen in typical stars, leading to unrealistic limits. Instead we have used the native BACCHUS derived \intmethod{} method upper limit for each line in each star. While these upper limits can occasionally extend to very high values, but because C, N, and O have quite strong features, it is rare that a line is simultaneously flagged as well measured by BACCHUS and below these upper limits. \subsection{Sodium (Na)} An example of the two Na lines is shown in Figure ~\ref{fig:Nalines}. For Na we use the \wln{} method instead of \chitwo{} for the Na 16373 \AA{} line because it is a weak line with both blue-ward and red-ward blends (with the red-ward CO blend nearly coincident with the Na 16373 \AA{} line) and the \wln{} method allows us to measure Na from a point where the Na line is maximal while minimizing contamination from any poorly fit blends. \subsection{Phosphorus (P)} An example of the two P lines is shown in Figure ~\ref{fig:Plines}. Both of the P lines that we measure are weak and blended, so we have chosen to use the \wln{} method to measure them where the lines are strongest and minimize noise and contamination from incompletely fit blends (because even small mismatches in the underlying blend can have a large impact on calculated the abundances from weak P lines). Similarly because of the strong blends with the weak P lines we have raised the upper limit threshold to the 3\% level which has been verified with by-eye investigation of borderline measurements. \subsection{Sulfur (S)} An example of the five possible S lines analyzed here is shown in Figure ~\ref{fig:Slines}. While we use the default settings for our S lines, after inspection of the output line-by-line abundance ratios we have chosen to not include the measurements of lines 15422 \AA{} and 15475 \AA{}. These lines both show strong temperature trends even once removing stars that have been flagged and have chemical abundance patterns that differ from each other and the other three lines. \subsection{Vanadium (V)} An example of the three V lines we analyze is shown in Figure ~\ref{fig:Vlines}. Similar to S, for V we have not included the V 16137 \AA{} line in our combination, because it is a slightly weaker and more blended line, and shows strong temperature trends and abundance patterns that deviate from the other two (likely due to misfit blends). We also impose an upper limit threshold of t = 3\% to remove stars that end up with anomalously low V abundances that should be below our detection limit. This was implemented in addition to the continuum flag designed for the V 15924 \AA{} line, to only consider measurements of lines that are sufficiently strong to dominate over errors from the spectral normalization around this line (see Section \ref{sec:v15924contflag} for why these errors may arise). \subsection{Copper (Cu)} An example of the single Cu feature (a blend of two Cu lines) is shown in Figure ~\ref{fig:Culine}. The Cu 16005 \AA{} and 16006 \AA{} lines both are significantly blended both blue-ward by a combination of \ion{C}{1}, \ion{Ti}{2} and \ion{Fe}{1} lines, and red-ward by \ion{C}{1} and \ion{Fe}{1} lines. The blue-ward blend is weaker, so for the Cu 16005 \AA{} line, we can still use the \chitwo{} method with minimal contamination from the blue-ward blends, but the red-ward blend is typically stronger and can contaminate wings of the redder Cu 16006 \AA{} line. So for the Cu 16006 \AA{} line we can minimize this contamination by measuring the the Cu abundance at a point with minimal contribution from blends by using the \wln{} method. The blends also cause many method flags to be thrown by BACCHUS, so we have modified our flag selection for the Cu lines. For the \eqw{} and \intmethod{} flags, we accept all flags (i.e., we ignore these flags or do not exclude a measurement for any specific flag from these methods) for both of the Cu lines. Of the \syn{} flags, we allow any flags for the Cu 16006 \AA{} line, but because the blends are weaker for the Cu 16005 \AA{} line we only use \syn{} flag = 1 or 3 (many stars have \syn{} flag = 3 because the Cu line falls in between two blends and does not produce a local minimum in flux, and are therefore flagged despite decent fits by BACCHUS). To further reduce the impact of poorly fit blends that can be interpreted by BACCHUS as enhancements in Cu, we have also used a higher upper limit threshold of t = 3\%. Finally, while we defined a blend flag for the Cu 16006 \AA{} line we did not cut any stars based on this flag since there were no clear offsets in the Cu measurements of these (few) stars from the rest of the population. However all stars with blend flag = 0 were removed through other quality and flag cuts when calculating combined Cu abundances. \subsection{Cerium (Ce)} An example of the four Ce lines used in this study is shown in Figure ~\ref{fig:Celines}. For Ce we use the default settings for line flagging and combination, except for raising the upper limit threshold slightly to t = 2\% for the Ce 16722 \AA{} line. This helps account for stars that visually appear to be upper limits but are considered detections in warm stars at the t = 1\% threshold due to mismatches in the nearby \ion{Si}{1} blend. \subsection{Neodymium (Nd)} \label{app:Nd} An example of the three Nd lines used in this study is shown in Figure ~\ref{fig:Ndlines}. The Nd 16053 \AA{} line lies in the wings of an OH line, which is frequently included in a larger BACCHUS window around the Nd 16053 \AA{}. To avoid adding more noise to our Nd measurements and avoid contamination in cases of poorer fits to this OH feature, we have chosen to use the \wln{} method for this line, which appears to provide more consistent measurements and is fitting the Nd 16053 \AA{} at a wavelength that is minimally blended with any other lines. In addition to using the \wln{} method for Nd 16053 \AA{}, BACCHUS tends to focus on the OH line core as the nearest local minimum within it's fitting window. Therefore it will report \syn{} and \eqw{} flag = 3 for many stars when the fits to the Nd line are otherwise good, so we include \syn{} and \eqw{} flag = 1 or 3 when assessing Nd measurements of this line. Again, because BACCHUS focuses on the core of the OH blend for its \intmethod{} method, the \intmethod{} flags are inappropriate for the overall quality of the BACCHUS fit to the Nd line, so we allow any \intmethod{} flag for the Nd 16053 \AA{} line. As with the weak lines of some other elements, we've increased the upper limit threshold for the weaker Nd 15368 \AA{} and 16262 \AA{} lines because of coincident blends and continuum placement that can lead BACCHUS to derive good measurements where visual inspection reveals that the fits are likely upper limits. This particularly occurs for hotter stars where our linear empirical upper limit threshold relations may not be complex enough to encompass the change in line depths from synthetic spectra as a function of temperature. Particularly we've raised the thresholds to 2\% and 3\% for Nd 15368 \AA{} and 16262 \AA{} respectively, to flag hotter stars as much as possible (where the upper limits are not quite high enough when inspecting individual stars), while minimizing the number of cooler stars that are flagged with Nd abundances that do appear to be measurable. Finally, for Nd, while we created a blend and continuum flag for the Nd 15368 \AA{} line, we decided not to use these flags as they primarily flagged stars with parameters where the Nd measurements should be more reliable (the blend flag flags nearly the whole sample and the continuum flag flagged only cool stars leaving things below 4800 K where visual inspection suggests that these measurements are predominantly upper limits). This does, however, suggest that this line (or its blends) may be more poorly fit on average, possibly the source of the large zero-point offset relative to \citet{Grevesse2007} of 0.372. However, this line does show similar abundance trends to the other Nd lines so it has been retained. \subsection{\cisotope{} Isotopic Ratio} As mentioned in Section \ref{sec:method_flags}, because BACCHUS's flags are tuned typical abundances measurements of individual lines, the flags are not quite appropriate for measuring $^{13}$C features. Typically the flags are designed for narrow features, and for the broader features of \cisotope{}, especially those with a variety of blends, BACCHUS will often throw \syn{} flag = 3 and \eqw{} = 3. These flags are not necessarily a description of the quality of the fit for \cisotope{}, so we also allow these flags when combining \cisotope{} measurements. Furthermore, since the \intmethod{} method attempts to measure abundances from line core intensities, it is not particularly suited to the, typically broader, $^{13}$C features, so we allow any \intmethod{} flags in our combination. In general for \cisotope{}, because they have a relatively broad wavelength coverage and are therefore blended with multiple lines, we have chosen to use the \wln{} method for our measurements, measuring the most intense parts of these features. While the \chitwo{} method could provide more robust measurements by averaging multiple pixels, it is also more susceptible to blends which are more frequent when taking a larger wavelength range. The only two lines that we use the \chitwo{} method for are the $^{13}$C$^{16}$O 16121 \AA{} and 16530 \AA{} features, which are the strongest and least blended features in our list. Since $^{13}$C features have increasing depth with decreasing \cisotope{} values, measurement limits are actually lower limits, and our empirical relations are not appropriate for the \cisotope{} values we derive. Instead because our input range maxes out at \cisotope{} $= 50$ we put a default line-by-line lower limit of 49 (which can cut off stars that sit near the upper synthesis range) on the measurements of all lines. To add another safe guard that helps ensure the \cisotope{} measurements are reasonable, for all of the stars that we report a final, combined \cisotope{} value, we require that they have good measurements from the $^{13}$C$^{16}$O 16530 \AA{} feature which is the strongest, least blended line. So, if this line has not been measured, we posit that the other lines too shouldn't be measurable. Finally to provide useful lower limits, we have derived a lower limit relation based on the strongest $^{13}$C feature we analyze, the $^{13}$C$^{16}$O 16530 \AA{} feature. The form of our \cisotope{} ratio lower limit relation is $\log_{10}($\cisotope$) = A/T_{\rm eff} + {\rm [C/H]} + B$, where \xh{C} is that calculated by BACCHUS. By adding a factor of \xh{C}, this limit essentially is a limit on \xh{$^{13}$C}. In order to determine the value of the coefficient $A$, we synthesize the $^{13}$C$^{16}$O 16530 \AA{} region as described in Section \ref{sec:upper_limits_relation} for \cisotope{} ratios of: 1, 3, 5, 7, 10, 15, 20, 30, 50, and 70. Based on the temperature dependence of these syntheses we derive a value of $A = 37200$ K. We then set the coefficient $B = -7.0$ from a data driven determination, based on the cont flag for $^{13}$C, which provides a limit threshold at the flux level. Nominally the cont flag indicates when a $^{13}$C line should provide a measurement (cont flag = 1) or if it can only provide a limit (cont flag = 0). In theory we would expect that for a given temperature and \xh{C}, at some threshold, stars with higher \cisotope{} should be lower limits, and below that threshold we have measurements. However we find that there is some overlap, such that BACCHUS measures similar \cisotope{} ratios for stars of the same temperature and \xh{C} but with a mix of cont = 0 and 1 set. So we have chosen to set the lower limit relationship coefficient $B$ such that there are approximately an equal number of stars with cont = 1 but \cisotope{} ratios that are above the lower limit relation as there are stars with cont = 0 but BACCHUS \cisotope{} ratios for the $^{13}$C$^{16}$O 16530 \AA{} feature that lie below the lower limit relation. For the cont = 1 stars, these are stars where, based on the line flux, we should be able to measure \cisotope{} ratios, but according to our abundance based relation should be lower limits. On the other hand, for the cont = 0 stars, these are stars that the flux limit indicates should be lower limits, but BACCHUS's measurements suggest that they would have low enough \cisotope{} ratios that we should be able to measure them according to our lower limit relation. By setting the coefficient $B$ such that the number of stars in each of these cases is approximately the same, we have an equal number of stars where we believe these lower limits should be over- and underestimated. We then use this relation to set lower limits for our sample. Any stars with combined \cisotope{} ratios below this relation are replaced with these limits, and all stars that have cont = 0 for the 16530 \AA{} line, but no other bad method flags, are given lower limits according to this relation. Finally, the \cisotope{} measurements are not populated for any stars without C measurements, because it is critical to have measured C in order to provide a reliable \cisotope{} ratio. \subsection{Zn, Ge, Rb and Yb} There are three Zn lines at 15730.4, 16483.4 and 16505.5 $\rm \AA$ in the APOGEE spectra that are theoretically strong enough to allow abundance measurement. However, as shown in Fig.~\ref{fig:Znlines}, all those lines suffer from major caveats which do not allow any measurement. The 15730.4 $\rm \AA$ line is the strongest but is severely blended by an unknown feature, which may have lead to an overestimation of this line's astrophysical $\log{gf}$. We note that this feature appears at all temperature, including for T$\rm _{eff} > 5000$ K which strongly suggests that it is an atomic line. The 16483.4 $\rm \AA$ line is cleaner but appear to have a largely overestimated strength in the models, such that it is not detectable in the spectra. Finally, the 16505.5 $\rm \AA$ line is the weakest and blended by two Fe lines. Concerning Ge, Rb and Yb, the strongest known lines in APOGEE spectra -- and more generally in the H-band -- are located at respectively 16759.8, 15289.5 and 16498.4 $\rm \AA$. Unfortunately, those lines are very weak and following our upper limits procedure such as described in Sec.~\ref{sec:upper_limits_relation}, very few stars pass the upper limits flag even at solar metallicity with the lowest intensity threshold. So we do not consider these elements further. We still stress that those elements can become clearly measurable in neutron-capture enhanced stars (particularly Yb), but as they represent a minority of stars in this general purpose catalog, and cannot be calibrated or treated in a similar way to the other elements, we have decided not to provide their abundances. \section{Spectra Flag Definitions} \label{app:spectra_flags} \subsection{Sodium (Na)} \subsubsection{Na Cont Flag} \label{sec:na16388contflag} As illustrated in Fig.~\ref{fig:Nalines}, both Na lines are affected by a CO blend. In order to control whether the modelling of the CO lines is satisfactory, we build an identical flag for both Na lines such that the difference between the modelling and the observation of the nearest CO clean line (at 16375 \AA{}) is less than 20\% of the depth of the 16388 \AA{} Na line. This flag implies that the weaker the Na lines are, the better the modelling of the CO lines needs to be, for us to measure accurate Na abundances. \subsubsection{Na 16388 \AA{} Blend Flag} \label{sec:na16388blendflag} At certain radial velocities ($\sim$ $-110$ -- $-60$ \kms{}) a relatively strong sky line aligns with the Na 16388 \AA{} line. In many stars at these velocities this sky line is either subtracted so that the Na 16388 \AA{} line can be recovered or sky line is strong enough that these pixels have been masked in the APOGEE pipeline. However, in some cases the sky line has been over-subtracted and not masked. When this occurs, the flux uncertainty in the affected pixels are sometimes increased to reflect the higher uncertainty from the sky line subtraction, but BACCHUS does not account for the flux uncertainty in each pixel, so it will find an enhanced Na abundance from these sky contaminated lines. To flag stars where this occurs, we have defined a blend flag for this line that attempts to identify cases where the uncertainties at the Na 16388 \AA{} line are larger than typical, or when stars have improperly subtracted sky lines at other pixels. To flag cases where the flux uncertainties per pixel have been raised, we identify stars whose flux uncertainty is greater than 5\%. Because not all affected stars have elevated flux uncertainties, we also use two other points in the spectra that have near continuum flux and are at wavelengths where strong sky sit in stars at the same radial velocities where Na 16388 \AA{} is affected. At these two wavelengths, 15431 \AA{} and 16128 \AA{}, we compare the observed flux to that of a BACCHUS model. If the sky line at Na 16388 \AA{} is erroneously subtracted, we may also see significant differences between observations and the model at these wavelengths from other poorly subtracted sky lines. To evaluate this, we calculate a mean and standard deviation of the differences between model and observed spectra at these wavelengths for stars with velocities between 0$-$100 \kms{} (which won't be affected by sky contamination), and flag all stars (at all radial velocities) whose differences at either of these wavelengths deviate from the calculated mean differences by more than 5$\sigma$. Combining these two methods we create a final blend flag for Na 16388 \AA{} that records whether the uncertainty at the line center is large or if other spectral pixels that may be affected by sky contamination deviate significantly from their BACCHUS model. Stars that may be negatively impacted by sky lines in Na 16388 \AA{} have blend = 0 and those that pass the flagging have blend = 1. \subsection{Phosphorus (P)} Phosphorus is among the most challenging elements to be measured in the APOGEE spectra because the lines are particularly weak and blended as illustrated in Fig.~\ref{fig:Plines}. \subsubsection{P 15711 \AA{} Blend Flag} The 15711 \AA{} line is blended by a CN line at 15706.9 \AA{}. To improve modelling of that CN line, we adjust the N abundance to this specific line before letting the code determine the P abundance of that line. The blend flag of the 15711 P line is raised as bad (= 0) only when the difference between the N abundance from the 15706.9 \AA{} CN line and the N abundance from an average of several good N lines is larger than 0.5 dex. In other words, this flag indicates when there has been a local adjustment of the blend. \subsubsection{P 16482 \AA{} Blend Flag} The 16482 \AA{} P line is blended by a CO line. The procedure to control the potential impact of a poorly modelled CO is quite similar to the 15711 \AA{} line. The carbon abundance is locally adjusted to match the CO feature at 16481 \AA{} before the P abundance is determined. We then create a blend flag for the 16482 \AA{} P line in a similar fashion as for the 15711 \AA{} blend flag, i.e. it is raised as bad (= 0) when the difference between carbon abundance of the 16481 \AA{} CO line and the mean carbon abundance derived from other good lines is larger than 0.2 dex. However, it turned out that this flag has never been raised all over the sample (implying that the 16481 \AA{} CO line is a good representative of the C abundance) and we did not find useful to provide it. \subsubsection{P 16482 \AA{} Cont Flag} Depending on the radial velocity of the star, the 16482 \AA{} line can near the edge of the CCD, where the quality of the spectra can be poor. The cont flag of this line is set to 0 (i.e., bad) when the line is within 15 \AA{} of the chip edge. \subsection{Sulfur (S)} \subsubsection{S Blend Flag} As illustrated in Fig.~\ref{fig:Slines}, two of three lines used in this study are blended by an underlying OH line. While OH lines are negligible if present at all in the hottest stars ($\approx$ 5000~K), the strength of OH increases rapidly with decreasing temperature such that at 4000~K it dominates the absorption, preventing any S abundance measurement in nearly all such cool stars. In addition, this effect is also increased for low metallicity stars, where the O abundance is relatively higher than at solar metallicities. In order to measure the S abundance only in cases where the contribution from OH lines is small, we have created a blend flag for the 15403 \AA{} and 15470 \AA{} lines such that the the flag is set as bad (= 0) if the observed flux at 15409 \AA{} (which corresponds to the wavelength of a clean OH line) is more than 5\% of the continuum. Regarding the 15478 \AA{}, it is blended by Fe lines that we assumed to be always be well modelled. Consequently, we did not implement any special flagging for this line (hence always =1). \subsection{Vanadium (V)} \subsubsection{V 15924 \AA{} Cont Flag} \label{sec:v15924contflag} As illustrated in Fig.~\ref{fig:Vlines}, the 15924 \AA{} line is the best line (in contrast to the 16137 \AA{} severely blended and not used in our final combination, and the 16200 \AA{} line quite strong, but blended by CO lines). Nevertheless, in hotter stars or very metal-poor stars, the V 15924 \AA{} line is relatively weak, so a slight mismatch in the BACCHUS normalization can introduce errors in the continuum placement that lead to erroneous V 15924 \AA{} measurements. We have therefore defined a continuum flag to identify when the local observed continuum deviates significantly from the BACCHUS model, such that the poor normalization may lead to errors in the derivation of V abundances for this line. This flag is set as bad (= 0) if the average of the difference between the observed and model flux at 15924.9 $\pm$ 0.4 is more than 15\% of the depth of the observed flux in the core the the V 15924 \AA{} line. \subsubsection{Cu Blend Flag} Despite the fact that the Cu feature is blended on both its red and blue side (see Fig.~\ref{fig:Culine}) by a C and Fe line respectively, the derivation of Cu abundances is possible in many of the stars as long as the blends are appropriately modeled. To assure this, we have implemented a procedure as follows: the C atomic blend is locally adjusted by setting the carbon abundance to the ``\intmethod{}'' method value of the C 16004.9 \AA{} line. If this local carbon abundance is more than 0.4 dex different from the mean carbon abundance derived from the average of the many lines we use, then we consider this local carbon measurement unreliable. And therefore we consider the the model fitting for Cu suspect, and set the Cu blend flag to 0. \subsection{Cerium (Ce)} Four Ce lines have been used in this study (Fig.~\ref{fig:Celines}). All of the lines are blended, but from our line-by-line comparison tests, it appears that for three of them the blends are sufficiently weak and/or well-modelled so that the Ce measurements are not severely affected. However, the fourth line, Ce 15784 \AA{}, is affected in some cases, for which we have built a specific procedure and flag. \subsubsection{Ce 15784 \AA{} Blend Flag} As shown in the left panel of Fig.~\ref{fig:Celines}, the Ce 15784 \AA{} line is blended on its blue side by a CO line. To minimize the impact of an inaccurate modelling of this line, we adjust the carbon abundance to match the core of the 15783.9 \AA{} CO line with the ``\wln{}'' method. The use of the ``\wln{}'' method is particularly relevant for this CO line as its core is clearly distinct from the Ce line 15784 \AA{} and thus offer the best leverage to accurately constrain the fit of the CO line. Nevertheless, whenever the carbon abundance differs from the mean carbon abundance by more than 0.3 dex, then the Ce 15784 \AA{} blend flag is set to 0. \subsection{Neodymium (Nd)} Along with P, Nd is a very challenging element to measure precisely because the lines are weak and blended. Fortunately, there are several lines present in the APOGEE spectra, which has allowed some robustness in the Nd measurement in our sample (Fig.~\ref{fig:Ndlines}). While we developed a similar procedure for locally fitting the blending feature such as for P, Cu or Ce, the blends do not appear to significantly impact the results, implying that the blends were reasonably well modeled, so we have not implemented any blend flags for these Nd lines like these other elements. Nevertheless, we have still developed a procedure to control for the quality of the continuum placement of the 16054 \AA{} line. \subsubsection{Nd 16053 \AA{} Cont Flag} Similar to what has been done for the Na and V cont flags: if the difference in the synthesis and the observed spectrum in the pseudo-continuum region 16054.2-16054.7 \AA{} is more than 30\% of the average flux at the wavelength of the 16053 \AA{} Nd line, then the cont flag is set to 0 (i.e., the fit is likely to be poor). \subsection{\cisotope{} Isotopic Ratio} \subsubsection{\cisotope{} Cont Flags} \begin{deluxetable}{l c c c c} \tablewidth{0pt} \tablecolumns{5} \tablecaption{$\rm ^{13}C$ lines flux threshold} \label{tab:13Clinesthreshold} \tablehead{\colhead{Wavelength ($\rm \AA$)} & \colhead{Threshold} } \startdata 15641.7 & 0.85 \\ 16121.3 & 0.97 \\ 16323.4 & 0.7 \\ 16326.0 & 0.98 \\ 16327.3 & 0.985 \\ 16530.0 & 0.982 \\ 16741.2 & 0.97 \\ 16744.7 & 0.98 \\ \enddata \end{deluxetable} In addition to an upper limit relation, we incorporate a flux limit flagging into the continuum flags for $\rm ^{13}C$. For each line we use a flux threshold value for each of the $\rm ^{13}C$ features when considering whether to flag each line. If the observed, normalised flux around the central wavelength (averaged within a window of $\pm$~0.5~$\AA)$) is higher than the threshold, the corresponding $^{13}C$ continuum flag is raised (flag = 0), and if the flux is lower than the threshold, the continuum flag for that feature is set to flag = 1. As indicated in Appendix \ref{app:comb_settings} we require that cont flag = 1 for all $\rm ^{13}C$, providing limit flagging at the flux level rather than at the abundances. This, however, means that for stars where all $\rm ^{13}C$ features are flagged as not passing the flux limits, no \cisotope{} limits are provided. \section{Upper Limit Relations} \label{app:upper_limits} \startlongtable \begin{deluxetable*}{l c c c c c c c c c c c} \tabletypesize{\scriptsize} \tablewidth{0pt} \tablecolumns{12} \tablecaption{Upper Limit Relation Constants \label{tab:upper_limits}} \tablehead{\colhead{Element} & \colhead{Line} & \colhead{$A_1$} & \colhead{$B_1$} & \colhead{$A_2$} & \colhead{$B_2$} & \colhead{$A_3$} & \colhead{$B_3$} & \colhead{$A_4$} & \colhead{$B_4$} & \colhead{$A_5$} & \colhead{$B_5$} \\ \colhead{} & \colhead{\AA{}} & \colhead{dex/10$^{4}$ K} & \colhead{} & \colhead{dex/10$^{4}$ K} & \colhead{} & \colhead{dex/10$^{4}$ K} & \colhead{} & \colhead{dex/10$^{4}$ K} & \colhead{} & \colhead{dex/10$^{4}$ K} & \colhead{}} \startdata C & 15578.0 & 14.912 & 0.825 & 15.490 & 0.818 & 28.277 & -4.128 & 42.768 & -9.773 & 57.914 & -15.676 \\ C & 15775.5 & 11.826 & 1.902 & 14.417 & 1.055 & 15.009 & 0.947 & 14.754 & 1.153 & 19.638 & -0.712 \\ C & 15783.9 & 5.979 & 4.358 & 6.444 & 4.438 & 7.192 & 4.335 & 7.543 & 4.331 & 7.534 & 4.441 \\ C & 15978.7 & 11.197 & 2.097 & 13.251 & 1.468 & 14.614 & 1.051 & 14.729 & 1.109 & 14.122 & 1.439 \\ C & 16004.9 & 0.523 & 6.252 & 1.028 & 6.381 & 1.232 & 6.535 & 1.146 & 6.738 & 0.976 & 6.943 \\ C & 16021.7 & 1.517 & 6.020 & 2.350 & 6.018 & 2.364 & 6.189 & 2.217 & 6.392 & 2.240 & 6.513 \\ C & 16185.5 & 10.737 & 2.310 & 12.488 & 1.798 & 13.844 & 1.381 & 14.314 & 1.293 & 14.332 & 1.367 \\ C & 16397.2 & 10.992 & 2.212 & 12.616 & 1.773 & 13.793 & 1.450 & 14.065 & 1.457 & 13.657 & 1.717 \\ C & 16481.5 & 14.272 & 1.447 & 24.747 & -2.390 & 38.484 & -7.691 & 58.048 & -15.131 & 80.308 & -23.545 \\ C & 16614.0 & 11.002 & 2.279 & 12.364 & 1.976 & 13.000 & 1.895 & 13.004 & 2.025 & 13.032 & 2.122 \\ C & 16836.0 & 11.091 & 2.350 & 12.615 & 2.017 & 12.755 & 2.147 & 12.611 & 2.343 & 13.145 & 2.245 \\ C & 16890.4 & 0.246 & 6.017 & -0.121 & 6.372 & -0.485 & 6.725 & -0.807 & 7.053 & -1.037 & 7.325 \\ C & 17063.0 & 11.096 & 2.448 & 12.509 & 2.190 & 12.096 & 2.545 & 11.531 & 2.915 & 17.408 & 0.686 \\ C & 17448.6 & 12.033 & 2.718 & 42.485 & -9.100 & 77.479 & -22.768 & 124.107 & -40.439 & 170.862 & -58.125 \\ C & 17456.0 & 41.070 & -8.492 & 123.426 & -39.866 & 213.908 & -72.611 & 304.937 & -104.691 & 394.770 & -136.108 \\ N & 15119.0 & 8.484 & 2.636 & 8.359 & 3.000 & 7.671 & 3.490 & 7.495 & 3.715 & 7.627 & 3.775 \\ N & 15210.2 & 8.863 & 2.565 & 8.143 & 3.181 & 7.678 & 3.582 & 7.772 & 3.689 & 8.070 & 3.675 \\ N & 15222.0 & 8.495 & 2.633 & 8.394 & 2.984 & 7.702 & 3.473 & 7.512 & 3.702 & 7.634 & 3.765 \\ N & 15228.8 & 8.848 & 2.564 & 8.201 & 3.149 & 7.733 & 3.552 & 7.811 & 3.667 & 8.091 & 3.664 \\ N & 15242.5 & 8.576 & 2.648 & 8.153 & 3.142 & 7.551 & 3.598 & 7.512 & 3.764 & 7.734 & 3.785 \\ N & 15251.8 & 8.777 & 2.557 & 8.335 & 3.051 & 7.755 & 3.496 & 7.711 & 3.664 & 7.938 & 3.682 \\ N & 15309.0 & 8.679 & 2.678 & 7.862 & 3.345 & 7.485 & 3.711 & 7.669 & 3.781 & 7.975 & 3.768 \\ N & 15317.6 & 8.774 & 2.516 & 8.618 & 2.879 & 7.968 & 3.346 & 7.795 & 3.568 & 7.927 & 3.626 \\ N & 15363.5 & 8.561 & 2.574 & 8.667 & 2.826 & 7.996 & 3.301 & 7.745 & 3.557 & 7.801 & 3.649 \\ N & 15410.5 & 8.748 & 2.796 & 7.844 & 3.513 & 8.119 & 3.604 & 8.506 & 3.584 & 8.658 & 3.631 \\ N & 15447.0 & 8.676 & 2.742 & 7.683 & 3.487 & 7.517 & 3.762 & 7.835 & 3.774 & 8.141 & 3.758 \\ N & 15462.4 & 8.687 & 2.731 & 7.720 & 3.464 & 7.535 & 3.749 & 7.836 & 3.768 & 8.140 & 3.753 \\ N & 15466.2 & 8.687 & 2.731 & 7.720 & 3.464 & 7.538 & 3.747 & 7.841 & 3.765 & 8.143 & 3.752 \\ N & 15495.0 & 8.654 & 2.746 & 7.685 & 3.479 & 7.476 & 3.771 & 7.768 & 3.791 & 8.092 & 3.766 \\ N & 15514.0 & 8.604 & 2.669 & 8.056 & 3.214 & 7.507 & 3.646 & 7.525 & 3.783 & 7.781 & 3.787 \\ N & 15581.0 & 8.980 & 2.501 & 8.377 & 3.061 & 7.944 & 3.448 & 8.022 & 3.565 & 8.293 & 3.566 \\ N & 15636.5 & 8.690 & 2.787 & 7.655 & 3.548 & 7.668 & 3.744 & 8.063 & 3.719 & 8.365 & 3.698 \\ N & 15659.0 & 8.699 & 2.799 & 7.683 & 3.555 & 7.787 & 3.713 & 8.203 & 3.679 & 8.474 & 3.671 \\ N & 15706.9 & 8.101 & 3.275 & 8.209 & 3.577 & 8.742 & 3.544 & 8.826 & 3.646 & 8.542 & 3.883 \\ N & 15708.5 & 8.101 & 3.275 & 8.209 & 3.577 & 8.742 & 3.544 & 8.826 & 3.646 & 8.541 & 3.884 \\ N & 15825.7 & 8.546 & 2.630 & 8.441 & 2.980 & 7.819 & 3.438 & 7.677 & 3.646 & 7.820 & 3.701 \\ O & 15373.5 & 5.668 & 6.015 & 7.566 & 5.373 & 8.231 & 5.161 & 8.779 & 4.994 & 9.328 & 4.828 \\ O & 15391.0 & 5.416 & 6.079 & 7.112 & 5.486 & 8.125 & 5.152 & 8.665 & 4.974 & 9.205 & 4.795 \\ O & 15569.0 & 5.827 & 5.819 & 6.470 & 5.702 & 7.785 & 5.243 & 8.291 & 5.092 & 8.761 & 4.936 \\ O & 15719.7 & 5.832 & 5.829 & 6.624 & 5.650 & 7.898 & 5.211 & 8.384 & 5.063 & 8.871 & 4.902 \\ O & 15778.5 & 5.665 & 5.927 & 6.763 & 5.610 & 7.952 & 5.210 & 8.454 & 5.050 & 8.956 & 4.887 \\ O & 16052.9 & 5.807 & 5.813 & 6.303 & 5.772 & 7.594 & 5.326 & 8.149 & 5.154 & 8.586 & 5.014 \\ O & 16055.5 & 5.807 & 5.813 & 6.303 & 5.772 & 7.594 & 5.326 & 8.149 & 5.154 & 8.586 & 5.014 \\ O & 16650.0 & 5.781 & 5.918 & 7.393 & 5.392 & 8.259 & 5.112 & 8.845 & 4.921 & 9.430 & 4.731 \\ O & 16704.8 & 5.673 & 5.872 & 6.225 & 5.808 & 7.476 & 5.381 & 8.070 & 5.191 & 8.479 & 5.062 \\ O & 16714.5 & 5.686 & 5.861 & 6.210 & 5.812 & 7.456 & 5.387 & 8.066 & 5.192 & 8.487 & 5.060 \\ O & 16872.0 & 5.710 & 5.820 & 6.056 & 5.859 & 7.218 & 5.459 & 7.978 & 5.208 & 8.349 & 5.089 \\ O & 16909.4 & 5.666 & 5.887 & 6.302 & 5.784 & 7.594 & 5.345 & 8.117 & 5.180 & 8.547 & 5.046 \\ Na & 16373.9 & 5.295 & 3.106 & 5.243 & 3.418 & 4.766 & 3.806 & 4.694 & 3.985 & 4.844 & 4.046 \\ Na & 16388.8 & 4.650 & 3.060 & 5.310 & 3.112 & 5.397 & 3.252 & 5.185 & 3.468 & 4.962 & 3.672 \\ P & 15711.6 & 1.227 & 4.558 & 0.254 & 5.250 & 0.023 & 5.522 & -0.116 & 5.796 & -0.215 & 6.036 \\ P & 16482.9 & 2.198 & 3.866 & 1.280 & 4.673 & 0.651 & 5.088 & 0.648 & 5.229 & 0.664 & 5.384 \\ S & 15403.5 & -0.946 & 6.361 & -1.481 & 6.912 & -2.438 & 7.583 & -2.881 & 7.998 & -3.139 & 8.299 \\ S & 15422.3 & -1.311 & 6.127 & -1.426 & 6.604 & -1.915 & 7.054 & -2.736 & 7.606 & -3.344 & 8.052 \\ S & 15469.8 & -0.158 & 6.154 & -1.072 & 6.845 & -1.574 & 7.350 & -1.721 & 7.634 & -1.975 & 7.914 \\ S & 15475.6 & -0.374 & 5.970 & -0.626 & 6.423 & -1.479 & 7.012 & -2.053 & 7.472 & -2.360 & 7.792 \\ S & 15478.5 & -0.374 & 5.970 & -0.626 & 6.423 & -1.479 & 7.012 & -2.053 & 7.472 & -2.360 & 7.792 \\ V & 15924.8 & 11.592 & -2.154 & 12.350 & -2.180 & 12.732 & -2.148 & 12.675 & -1.988 & 12.375 & -1.754 \\ V & 16137.3 & 11.045 & -1.471 & 11.321 & -1.255 & 10.790 & -0.859 & 10.661 & -0.686 & 10.733 & -0.615 \\ V & 16200.1 & 11.768 & -2.062 & 12.548 & -2.073 & 12.489 & -1.859 & 12.040 & -1.537 & 11.768 & -1.320 \\ Cu & 16005.5 & 3.512 & 1.393 & 2.944 & 1.950 & 3.655 & 1.819 & 4.139 & 1.750 & 4.144 & 1.870 \\ Cu & 16006.0 & 3.512 & 1.393 & 2.944 & 1.950 & 3.655 & 1.819 & 4.139 & 1.750 & 4.144 & 1.870 \\ Ce & 15784.8 & 9.574 & -3.934 & 10.135 & -3.901 & 9.918 & -3.624 & 9.893 & -3.477 & 10.021 & -3.422 \\ Ce & 16376.5 & 10.590 & -4.227 & 10.393 & -3.855 & 10.529 & -3.730 & 10.880 & -3.742 & 11.066 & -3.707 \\ Ce & 16595.2 & 10.405 & -3.947 & 10.526 & -3.695 & 11.003 & -3.706 & 11.116 & -3.611 & 11.005 & -3.450 \\ Ce & 16722.5 & 9.727 & -3.563 & 9.638 & -3.219 & 9.990 & -3.169 & 10.155 & -3.096 & 10.165 & -2.988 \\ Nd & 15368.1 & 8.264 & -2.352 & 8.376 & -2.055 & 7.537 & -1.499 & 7.176 & -1.213 & 7.313 & -1.166 \\ Nd & 16053.6 & 9.867 & -3.079 & 9.474 & -2.616 & 8.941 & -2.203 & 8.918 & -2.056 & 9.238 & -2.072 \\ Nd & 16262.0 & 9.355 & -2.801 & 8.989 & -2.331 & 8.424 & -1.899 & 8.373 & -1.736 & 8.691 & -1.754 \\ \enddata \end{deluxetable*} In Table \ref{tab:upper_limits} we present the constants used in our upper limit relations of the form \logeps{X}$_{\rm lim} = A_{\rm t} \cdot T_{\rm eff} + B_{\rm t}$ where t is the chosen \% threshold of the continuum. We have calculated these relations for continuum threshold levels of 1, 2, 3, 4, and 5\%, and have reported the $A_{\rm t}$ in dex per 10$^4$ K in Table \ref{tab:upper_limits}. \end{appendix}

Title: Lunar Samples are Time Capsules of the Sun

Abstract: The Heliophysics Decadal survey should embrace the coming opportunity of sustained lunar surface exploration and facilitate cross-disciplinary efforts to unlock the secrets of the Sun that are held by the lunar surface. With planned Artemis efforts that include prioritization of samples of high interest and protocols for sample handling and analysis, input into the relevant solar signatures that would be most diagnostic and how best to obtain/retain them is incredibly important. Finally, leveraging the theoretical expertise of the two communities in ways that bring them together, such as through dedicated conferences and workshops, will let the two communities help each other learn more than they could alone.

https://export.arxiv.org/pdf/2208.13307

\pagestyle{plain} \pagenumbering{arabic} \raggedright \large Heliophysics Decadal Survey 2024 White Paper \linebreak \linebreak \huge Lunar Samples are Time Capsules of the Sun \linebreak \normalsize \textbf{Authors:} \textit{Prabal Saxena* (NASA Goddard), Natalie Curran (Catholic University/CRESST II/NASA Goddard), and Heather Graham (NASA Goddard)} \linebreak \textbf{Co-Signers:} \textit{Vladimir Airapetian (American University/NASA Goddard), D.L. Blank (University of Southern Queensland), Ian Crawford (Birbeck, University of London), Ben Davidson (SpaceWeatherNews, The Mobile Observatory Project), Brett W. Denevi (Johns Hopkins University Applied Physics Laboratory), Katherine Joy (University of Manchester), James Tuttle Keane (NASA JPL), Rosemary M. Killen (NASA Goddard), Jason L McLain (NASA Goddard), Liam Morrissey (Memorial University), Noah Petro (NASA Goddard), Carle M. Pieters (Brown University), Song Tan [\begin{CJK}{UTF8}{gbsn}и°­е®‹\end{CJK}] (Yunnan Observatories, Chinese Academy of Sciences)} \linebreak \textbf{Synopsis/Recommendation:} The history of the Sun is buried in the surface of the Moon, which presents a particularly exciting opportunity given the upcoming planned efforts for the return of humans to the Moon. These future efforts involve long term, sustainable human exploration of the Moon and promise a return of a large mass of diverse and new types of lunar samples. The Heliophysics Decadal survey should actively embrace this coming opportunity and facilitate cross-disciplinary efforts to unlock the secrets of the Sun held by the lunar surface. With planned Artemis efforts that include prioritization of samples of high interest and protocols for sample handling and analysis, input into relevant solar signatures that would be most diagnostic and how best to obtain/retain them is incredibly important. Finally, leveraging the theoretical expertise of the two communities in ways that bring them together, such as through dedicated conferences and workshops, will let the two communities help each other learn more than they could alone. \linebreak \vfill \textbf{*Contact Email: prabal.saxena@nasa.gov} \newpage \par \hspace{10pt} The history of the Sun is buried in the surface of the Moon. This is in part due to the profound effect the Sun has on bodies in the solar system \citep{1996ofm..book.....S, 2010IAUS..264..475M} - one recognized as a key goal of the previous heliophysics decadal survey ("Determine the interaction of the Sun with the solar system...") \citep{NAP13060}. While the Sun sculpts the atmospheres, surfaces and habitability of the larger atmosphere possessing bodies in our solar system, its interactions with the Moon's surface are more direct as they have been potentially mediated by an atmosphere or a global magnetic field for only brief periods of time \citep{2017E&PSL.474..198S, NEEDHAM2017175, Weiss1246753, 2021SciA....7.7647T, 2022NatAs...6..325E}. Consequently, interactions of both the energetic and more corpuscular (magnetized solar wind and energetic particles) outputs of the Sun are able to leave seemingly indelible signatures in the lunar surface. \par \hspace{10pt} A key advantage of signatures of the Sun left on the lunar surface is that there are very few processes (e.g. plate tectonics driven resurfacing, erosion, etc.) which have operated globally on the Moon and which could have wiped out such signatures. Processes such as volcanism and impacts have modified parts of the surface, but have also done so in a way which has been able to leave many key chronologically diagnostic signatures intact. Indeed in both cases, these process may actually serve to help highlight and constrain the chronological influence of the Sun in specific periods of time. For example, in the case of volcanism, lava flows from episodic volcanism may have essentially sealed off layers of the regolith from further interaction with the Sun \citep{2010Icar..207..595F} that leaves these overlaid portions of `paleoregolith' as invaluable snapshots into time periods preceding the event in which they were trapped (see figure \ref{fig:paleoreg} for an example from \citep{2021RSPTA.37990562C}). Similarly, while impacts of high enough energy may in some cases modify or eliminate signatures of interest in specific regions, the impact process also drives the overturning and stratification of the regolith \citep{1974LPSC....5.2365G, COSTELLO2018327} - a process which creates a chronological record of surface interactions with the Sun with depth. \par \hspace{10pt} The existence of potential chronologically diagnostic signatures of solar interaction with the lunar surface, particularly paleo-space weather, is especially exciting given the demonstrated ability to repeatedly actively obtain samples from the Moon. Samples from the lunar surface have been returned to the Earth by multiple space agencies \citep{1970GeCAS...1....1S, 1971LPSC....2....1V}, beginning roughly a half century ago, and continue to this day \citep{2021Sci...374..887C}. This is true for no other body outside of Earth in the solar system, and as discussed above, lunar samples have the advantage of preserving a relatively intact record of interaction with solar and solar system processes in the past. The ability of lunar samples to explore the nature of the Sun has been recognized by scientists since the Apollo program. In 1979, a conference on `The Ancient Sun: Fossil Record in the Earth, Moon, and Meteorites' \citep{1979LPICo.390.....R} included seminal initial presentations on topics including the variation in past solar flare activity, properties of the past solar wind, changes in the solar dynamo over time and changes in GCR flux as modulated by the Sun - all deduced from examination of lunar samples \citep{1979LPICo.390...17C, 1979LPICo.390...76R, 1979LPICo.390...36F, 1979LPICo.390...12C, 1979LPICo.390...82R}. Additional studies including some recent ones built on this work by noting the relative constancy of the solar wind in the recent past (up to 10's of millions of years) \citep{1972JGR....77..537R, 1977RSPTA.285..587C, 2018A&A...618A..96P}, but a significant number of studies, have noted evidence that the solar wind flux and consequently the Sun has not been constant farther in the past \citep{1979LPICo.390...76R, 1980E&PSL..47...34T, 1989GeCoA..53.1135B, 2006mess.book..829E}. \begin{table}[] \caption{Selected Lunar Sample Proxies and Relevant Solar Inputs} \label{tab:solarsignatures} \resizebox{\textwidth}{!}{% \begin{tabular}{|l|c|c|c|} \hline \multicolumn{1}{|c|}{\textbf{Proxy}} & \textbf{Meaning} & \textbf{Solar Assumptions} & \textbf{Solar Input} \\ \hline \begin{tabular}[c]{@{}l@{}}Cosmic Ray\\ Exposure Age\end{tabular} & \begin{tabular}[c]{@{}c@{}}Indicates exposure to cosmic rays \\ (down to several meters)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Assumes a constant \\ flux of GCRs and \\ SEPs over time\end{tabular} & \begin{tabular}[c]{@{}c@{}}SEP/GCR Flux/Spectrum over time,\\ Heliosphere properties that \\ modulate GCR flux/spectra\end{tabular} \\ \hline \begin{tabular}[c]{@{}l@{}}Particle/Fission\\ Tracks\end{tabular} & \begin{tabular}[c]{@{}c@{}}Indicates irradiation of a \\ particular sample over an\\ integrated time by\\ energetic particles\end{tabular} & \begin{tabular}[c]{@{}c@{}}Varying assumptions about \\ the flux and plausible \\ spectra of input GCRs/SEPs\end{tabular} & \begin{tabular}[c]{@{}c@{}}SEP/GCR Flux/Spectrum over time,\\ Heliosphere properties that \\ modulate GCR flux/spectra\end{tabular} \\ \hline Antiquity Age & \begin{tabular}[c]{@{}c@{}}Semi-quantifies formation age\\ of a breccia - the time at which a\\ soil is lithified into a breccia\end{tabular} & \begin{tabular}[c]{@{}c@{}}Dependent on isotope ratios \\ normalized to a constant\\ solar wind flux \end{tabular} & \begin{tabular}[c]{@{}c@{}}Overall SEP flux rate from Solar \\ Wind (particularly Argon), \\ Solar UV flux input for ionization\end{tabular} \\ \hline Maturity Index & \begin{tabular}[c]{@{}c@{}}Indicates surface exposure\\ of particular sample\\ (in top mm of the surface)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Dependent on assumption of\\ constant solar wind particle flux\end{tabular} & \begin{tabular}[c]{@{}c@{}}Solar wind flux rate - isotopes\\ of helium-3, neon-20 and argon-36\end{tabular} \\ \hline \end{tabular}% } \end{table} \par \hspace{10pt} Recent changes in understanding on how the Sun may have evolved over time have supported this past variable history and have complicated the ability to use long standing proxies used to interpret the history of and nature of lunar samples. Many of these proxies (a selection of which are listed in table \ref{tab:solarsignatures}) are dependent upon assumptions that recent research suggests are untrue, such as the assumption of constancy of energetic particle flux from the Sun over time or the constancy of the morphology of the heliosphere over time. These processes are dependent on fundamental processes and properties of the Sun, such as rotation rate, and a number of recent studies have suggested that relaxing such assumptions leads to dramatic impacts on the atmospheres and surfaces of planets in our solar system \citep{Airapetian2016, Lammer2018} and would obviously also change some interpretation of certain lunar samples using the proxies in table \ref{tab:solarsignatures}. \par \hspace{10pt} Equally intriguing is that the dependence of these proxies used to examine lunar samples on these Sun related properties \textbf{may enable the use of lunar samples to put constraints on the properties of the Sun over time}. For example, examination of solar analogue stars has suggested that the Sun may have had a range of evolutionary paths with respect to how its rotation changed over time (see figure \ref{fig:solarevocartoon} - with significant consequences for the planets in our solar system \citep{2013A&A...556A..36G, 2016A&A...587A.105A}. The dependence of the activity of the Sun and nature of the heliosphere on these different rotation evolution paths could lead to significantly different signatures in lunar samples, particularly if there is chronological granularity provided by samples. Indeed, initial work has attempted to place constraints on the likelihood the Sun had particular rotation states at much younger ages \citep{2019ApJ...876L..16S}. \par \hspace{10pt} The potential for lunar samples to hold diagnostic information regarding the nature of the Sun is especially promising giving planned return of humans to the Moon. Many of these plans stress the importance of understanding the Sun over time using lunar samples \citep{2021LPI....52.1261W}. These future efforts involve long term, sustainable human exploration of the Moon and promise a return of a large mass of diverse and new types of lunar samples. This windfall of information promises to offer insight into a number of different solar system processes, and particularly relevant to studies regarding the evolution of the Sun, may enable population studies that put constraints on these processes over time. The interpretation of such signatures will require the joint expertise and efforts of planetary scientists and heliophysicists. \par \hspace{10pt} \fbox{% \parbox{\textwidth}{ \textbf{The Heliophysics Decadal survey should actively embrace this coming opportunity and facilitate cross-disciplinary efforts to unlock the secrets of the Sun held by the lunar surface. With planned Artemis efforts that include prioritization of samples of high interest and protocols for sample handling and analysis, input into relevant solar signatures that would be most diagnostic and how best to obtain/retain them is incredibly important. Finally, leveraging the theoretical expertise of the two communities in ways that bring them together, such as through dedicated conferences and workshops, will let the two communities help each other learn more than they could alone. } } } \clearpage \bibliography{bibliography}{} \bibliographystyle{aasjournal}

Title: High Contrast and High Angular Imaging at Subaru Telescope

Abstract: Adaptive Optics projects at Subaru Telescope span a wide field of capabilities ranging from ground-layer adaptive optics (GLAO) providing partial correction over a 20 arcmin FOV to extreme adaptive optics (ExAO) for exoplanet imaging. We describe in this paper current and upcoming narrow field-of-view capabilities provided by the Subaru Extreme Adaptive Optics Adaptive Optics (SCExAO) system and its instrument modules, as well as the upcoming 3000-actuator upgrade of the Nasmyth AO system.

https://export.arxiv.org/pdf/2208.01809

\keywords{Adaptive Optics, High Contrast Imaging, Coronagraphy, Exoplanets} \section{Adaptive Optics at Subaru Telescope: Overview} \label{sec:AOoverview} % Adaptive optics at the Subaru Telescope \cite{2020SPIE11448E..0KO} started with the 36-element curvature system at the telescope's Cassegrain focus \cite{2004PASJ...56..225T}. A more capable 188-element system with both LGS and NGS modes was deployed in the 2000s at the telescope's infrared Nasmyth focus \cite{2008SPIE.7015E..10H,2010SPIE.7736E..0NH}, and is still in operation, feeding the Infrared Camera and Spectrograph (IRCS) \cite{1998SPIE.3354..512T, 2004SPIE.5492.1542T, 2000SPIE.4008.1056K} and Subaru Coronagraphic Extreme AO (SCExAO)\cite{2015PASP..127..890J,2020SPIE11448E..0NL} instruments. New adaptive optics capabilities are currently in development, focusing on three major directions : \begin{itemize} \item{{\bf Ground-Layer Adaptive Optics (GLAO)}, providing $\approx$ 0.2-arcsec image quality over a 20-arcmin field of view at the telescope's Cassegrain focus using laser guide stars (LGSs) and an adaptive secondary mirror. GLAO development is the core part of the ULTIMATE\cite{2020SPIE11203E..0GM,2020SPIE11450E..0OM} project currently in development phase, with first light anticipated in 2028.} \item{{\bf Laser Tomography Adaptive Optics (LTAO)}, delivering diffraction-limited imaging over $\approx$ 20-arcsec field of view at the telescope's IR Nasmyth focus using LGSs for full sky coverage. LTAO is part of the ULTIMATE project, and will be deployed in year 2023 at the IR Nasmyth platform with the ULTIMATE-START\cite{2020SPIE11448E..1OA} project.} \item{{\bf Extreme Adaptive Optics (ExAO)} providing high image quality for high contrast imaging and visible-light diffraction limited imaging over small field of views. ExAO is implemented in the Subaru Coronagraphic Extreme AO (SCExAO) platform and its instrument modules.} \end{itemize} We describe in this paper the current and near-future capabilities of narrow-field adaptive optics modes, with a focus on ExAO and including LTAO, and their implementation on the telescope's Nasmyth IR (NasIR) platform. We describe in \S \ref{sec:NasIR} the NasIR instrument configuration, including the upcoming deployment of the NasIR beam switcher, which allows for multiple instrument to be AO-fed. Observing capabilities are discussed in \S \ref{sec:obsmodes}. The main development activities and future perspectives are discussed in \S \ref{sec:dev}. \section{Nasmyth IR Platform Optical Configuration} \label{sec:NasIR} \subsection{Current Configuration} Narrow-field AO instrumentation is hosted at the telescope's NasIR platform, as shown in Figure \ref{fig:NasIR}. The current 188-element facility AO system provides NGS and LGS correction for the IRCS and SCExAO instruments. IRCS provides diffraction-limited imaging and spectroscopy in near-IR while SCExAO includes a second-stage Extreme AO correction with a 2000-element MEMS deformable mirror. The IRCS and SCExAO observing modes are mutually exclusive and cannot be scheduled within the same night. Switching between both instruments requires physically swapping the instruments between the storage and in-focus locations. The picture in Figure \ref{fig:NasIR} shows the SCExAO configuration, while the IRCS configuration would have IRCS at the AO188 focus. Switching between the two modes is done in daytime with a overhead crane (partially visible in Figure \ref{fig:NasIR} above AO188). This limitation will be addressed by deploying the Nasmyth beam switcher (NBS) behind the facility AO system. \subsection{Beam Switcher Configurations} The Nasmyth bean switcher \cite{Zheng2022} (NBS) will be deployed by 2024 to allow for on-the-fly switching between SCExAO and IRCS without requiring instrument craning, and will also support expanding the suite of AO-fed instruments. The NBS also allows for simultaneous operation of instrument modules by using dichroic beam splitters. Figure \ref{fig:NasIR-top} shows the NasIR instrument platform layout, and Figure \ref{fig:AOblockdiag} shows the light path through adaptive optics key elements and instruments. \section{Observing Modes and Capabilities} \label{sec:obsmodes} \subsection{Adaptive Optics Correction} All adaptive optics instrumentation at NasIR is first corrected by the AO3k system \cite{Lozi2022}. AO3k is an upgrade of the current 188-element system, where the 188-actuator bimorph DM is replaced by a 64x64 magnetic DM (model ALPAO DM3228), and high order visible and NIR wavefront sensors are added. AO3k supports the following modes: \begin{itemize} \item Visible light NGS single conjugated AO (NGS-VIS-SCAO) \item Near-infrared light NGS single conjugated AO (NGS-NIR-SCAO) \item LGS Single conjugated AO (LGS-SCAO) \item Laser tomographic AO (LTAO) \end{itemize} The high order visible light high-order WFS will be a non-linear curvature WFS \cite{2010PASP..122...49G}, located within the AO3k enclosure. The NIR pyramid WFS and LTAO LGS Shack-Hartman WFSs are located at the output of AO3k, before the NBS. Second-stage correction is performed by SCExAO with a 2000-actuator MEMS type mirror (model: Boston Micromachines 2k, to be upgraded in 2024 to a segmented 3k device), which provides high-speed high precision wavefront control for high contrast imaging and visible-light AO. \subsection{Near-IR instrumentation} The AO3k NIR corrected beam can be fed to the IRCS camera and spectrograph or the SCExAO NIR modules. IRCS provides wide field imaging (20mas or 50mas FOV) and spectroscopy over the full NIR wavelength range from J-band to M-band. The instrument supports both low-resolution GRISM spectroscopy and high resolution (R$\approx$ 20k) echelle spectroscopy. IRCS's relatively wide field of view is a good match to the upcoming LTAO correction mode. Narrower field of view capabilities are provided by SCExAO instrument modules which benefit from ExAO correction. Four high contrast imaging instrument modules are currently available for open use, all compatible with SCExAO's coronagraphy mode : \begin{itemize} \item The {\bf CHARIS}\cite{2015SPIE.9605E..1CG} integral field spectrograph, providing R=20 to 70 spectro-imaging over a 2 arcsecond field of view. \item The {\bf Fast-PDI}\cite{2020SPIE11448E..7CL} H-band high-speed polarimetric imaging camera \item The {\bf MKIDS exoplanet camera (MEC)}\cite{2020PASP..132l5005W}, a high speed photon-counting energy resolving camera \item The {\bf REACH high resolution spectroscopy}\cite{2020SPIE11448E..78K} mode proving R $=$ 100k NIR spectroscopy of exoplanets and companions \end{itemize} Together, these four modes provide exoplanet and disk detection and characterization by imaging, spectroscpy and polarimetry in NIR \cite{Currie2022,Kuzuhara2022,Steiger2021,Lawson2020,Lawson2021,Currie2020,Chaushev2022}. Example science observations are shown in Figure \ref{fig:NIRimaging}. The GLINT \cite{2020MNRAS.491.4180N, 2021NatCo..12.2465M} instrument, currently in development, is a NIR nulling interferometer providing access to exoplanet and faint sources in the 0.5 to 2 $\lambda / D$ separation range. GLINT will extend high contrast imaging capabilities to smaller angular separations than possible with conventional coronagraphy. A photonic lantern is under development for exoplanet imaging\cite{Lin2022} and high precision spectro-astrometry\cite{Kim2022}. \subsection{Visible Light instrumentation} Visible-light instrumentation includes the SCExAO/VAMPIRES dual-band imager, the SCExAO/FIRST spectro-interferometer and the Kyoto3DII integral field spectrograph. Kyoto3DII\cite{2010PASP..122..103S} was previously used behind AO188 and will be re-deployed on the top port of the NBS. It will benefit from AO3k correction in NGS, LGS and LTAO modes, which will provide near-diffraction-limited imaging performance in visible light with full sky coverage. SCExAO's extreme-AO correction delivers high quality diffraction-limited imaging at visible wavelengths for the VAMPIRES and FIRST instruments. VAMPIRES\cite{2015MNRAS.447.2894N} is available for open use observations, and supports polarimetric differential imaging \cite{2020SPIE11203E..0SN, Safonov2022}, H-$\alpha$ differential imaging \cite{2020JATIS...6d5004U} and coronagraphic imaging \cite{Lucas2022}. VAMPIRES routinely operates at the diffraction limit, providing sub-20mas angular resolution, as illustrated in Figure \ref{fig:Capella} (left). The FIRST\cite{2020SPIE11446E..29V} fiber-fed spectro-interferometer, under development, provides high precision imaging below the telescope's diffraction limit, as illustrated on Figure \ref{fig:Capella} (right). FIRST is being upgraded with a stable compact integrated optics device feeding a R=4000 spectrograph designed for H-$\alpha$ spetro-imaging \cite{2021sf2a.conf..135L, Barjot2022, Martin2022, Martin2022b, Lallement2022}. \section{Conclusions and Perspectives} \label{sec:dev} The AO188 system at Subaru Telescope feeds light to a wide range of visible and NIR instruments for imaging, spectroscopy and polarimetry. The upcoming upgrade of the 188-element system into a 3000-element system supporting NIR wavefront sensing, LGS and LTAO modes will provide significantly improved image quality over a wide range of targets. The Nasmyth beam switcher will allow efficient and simultaneous operation of multiple instruments modules, and support additional observing modes. Adaptive Optics developments at Subaru Telescope are both expanding scientific capabilities and prototyping new instrument concepts and techniques for future efforts. The ULTIMATE-Start LTAO effort is validating wide-field AO sensing and control solutions for the more ambitious ULTIMATE Subaru project. In high contrast imaging, SCExAO is prototyping techniques for future exoplanet-imaging systems to be installed on 30-m class telescopes \cite{2020SPIE11448E..0NL}, including the Thirty Meter Telescope (TMT). The SCExAO system is supporting development of promising WFS/C techniques for high contrast imaging, including focal plane speckle control\cite{Ahn2022SPIE} and PSF reconstruction\cite{Guyon2022} as illustrated in Figure \ref{fig:HCIdev}. High frame rate science cameras in the system are also used for wavefront sensing \cite{Vievard2022LAPD, Deo2022}. Thanks to the AO188 upgrade to AO3k, these advanced modes of operation will be deployed on-sky to provide improved high contrast detection capabilities. SCExAO is also validating astrophotonics technologies for high angular resolution and high contrast imaging \cite{2021SPIE11823E..0CV}. This includes the GLINT and FIRST instruments that use coherent waveguides and photonic chips for high precision spectro-interferometric imaging, and new wavefront sensing concepts using dispersed interferometry \cite{Vievard2022, Norris2022GLINT} and Photonic Lantern \cite{Lin2022coupling,Norris2022PL}. \acknowledgments % This work is based on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. The authors also wish to acknowledge the critical importance of the current and recent Subaru Observatory daycrew, technicians, telescope operators, computer support, and office staff employees. Their expertise, ingenuity, and dedication is indispensable to the continued successful operation of these observatories. The development of SCExAO was supported by the Japan Society for the Promotion of Science (Grant-in-Aid for Research \#23340051, \#26220704, \#23103002, \#19H00703 \& \#19H00695), the Astrobiology Center of the National Institutes of Natural Sciences, Japan, the Mt Cuba Foundation and the director's contingency fund at Subaru Telescope. KA acknowledges support from the Heising-Simons foundation. \bibliography{report} % \bibliographystyle{spiebib} %

Title: Balancing the efficiency and stochasticity of star formation with dust extinction in z > 10 galaxies observed by JWST

Abstract: Early observations with the James Webb Space Telescope (JWST) indicate an over-abundance of bright galaxies at redshifts z > 10 relative to Hubble-calibrated model predictions. More puzzling still is the apparent lack of evolution in the abundance of such objects between z ~ 9 and the highest redshifts yet probed, z ~ 13-17. In this study, we first show that, despite a poor match with JWST LFs, semi-empirical models calibrated to UVLFs and colours at 4 < z < 8 are largely consistent with constraints on the properties of individual JWST galaxies, including their stellar masses, ages, and rest-ultraviolet spectral slopes. We then show that order-of-magnitude scatter in the star formation rate of galaxies (at fixed halo mass) can indeed boost the abundance of bright galaxies, provided that star formation is more efficient than expected in low-mass halos. However, this solution to the abundance problem introduces tension elsewhere: because it relies on the up-scattering of low-mass halos into bright magnitude bins, one expects typical ages, masses, and spectral slopes to be much lower than constraints from galaxies observed thus far. This tension can be alleviated by non-negligible reddening, suggesting that - if the first batch of photometrically-selected candidates are confirmed - star formation and dust production could be more efficient than expected in galaxies at z > 10.

https://export.arxiv.org/pdf/2208.12826

\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2022} \begin{keywords} galaxies: high-redshift -- galaxies: luminosity function, mass function -- diffuse radiation \end{keywords} \section{Introduction} \label{sec:intro} JWST has long promised a revolution in our understanding of early galaxy formation, and with the public release of the first deep surveys in July 2022, that revolution is finally beginning. Several early analyses have extended studies of high-redshift galaxies to $z \ga 10$ -- and perhaps as far as $z \sim 17$ -- using early data primarily taken from the GLASS-JWST and CEERS surveys. For the most part, these early searches have focused on photometric Lyman-break selection, often checking candidates with SED-fitting codes to attempt to validate the redshift selection. These studies have extended estimates of the UV luminosity function to much earlier times than heretofore possible \citep{Donnan2022, Harikane2022}. Because of the limited depth of the early data, the most compelling candidates in these searches are relatively luminous (and hence presumably massive), and the biggest surprise has been the apparent abundance of such massive galaxies at $z \ga 10$, if the candidates are indeed primeval sources. Although the number counts are still small -- and so heavily affected by Poisson noise and cosmic variance -- they are consistent with the number density of bright galaxies remaining roughly constant at $z \ga 8$ \citep{Naidu2022, Donnan2022, Castellano2022, Atek2022, Harikane2022, Labbe2022}. This is very surprising in comparison to theoretical models, as the abundance of massive dark matter haloes should evolve very rapidly during this epoch \citep{Mason2022, BoylanKolchin2022,Lovell2022}. These sources must be interpreted with substantial caution, however. Many of the faint sources are sensitive to the selection criteria (e.g., see the discussion in \citealt{Harikane2022}), while the recent release of in-flight photometric calibration corrections will drive $\sim$ tens of per-cent level changes in previously published photometry. Even bright sources can be confused by ``low''-redshift interlopers. For example, \citet{Zavala2022} and \citet{Naidu2022b} show that dropout galaxies selected to lay at $z \sim 17$ may be confused with dusty, star-forming galaxies at $z \sim 5$. Indeed, \citet{Glazebrook2022} and \citet{Rodighiero2022} find an array of galaxies with surprising properties, e.g., extreme obscuration ($A_V \sim 5$ at $z \sim 5$) and very strong Balmer breaks and emission lines at $2 \lesssim z \lesssim 6$. Given the extreme rarity of massive, high-$z$ galaxies, reliable candidate selection requires comprehensive libraries of template spectra against which to compare \citep{Furlanetto2022} -- a process made all the more difficult as JWST reveals the diversity of galaxy properties in the early Universe (e.g., \citealt{Barrufet2022, Whitler2022}). Despite these caveats, it is intriguing to consider the implications of an ``overabundance'' of massive galaxies at $z \ga 10$ -- a phenomenon first suggested by the spectroscopically confirmed GN-z11 at $z \sim 11$ discovered in \emph{Hubble Space Telescope} (HST) data \citep{Oesch2016}. Theoretical models robustly predict a sharp decline in the abundance of such objects, simply because the halo mass function is evolving rapidly at these times. Indeed, \citet{BoylanKolchin2022} and \citet{Lovell2022} point out that the most extreme detections (from \citealt{Labbe2022}) appear to have much larger stellar masses than allowed by the standard cosmological paradigm, even if they have perfectly efficient star formation! \citet{Mason2022} show that the problem is less extreme for most sources but demonstrate that there is still a clear overabundance relative to the simplest models. Previous work has shown that, through the HST era, the $z \gtrsim 6$ luminosity function could be understood using the same basic mechanisms that explain the abundance of star-forming galaxies at lower redshifts, with accretion onto haloes and stellar feedback controlling the star formation rates (e.g., \citealt{Trenti2010, Mason2015, Furlanetto2017, Tacchella2018}). Such ``minimalist'' models offered simple but powerful ways to extrapolate our understanding of galaxy formation to early times and roughly matched the predictions from more complex models (e.g., \citealt{Behroozi2015, Yung2019}). But now it appears these kinds of models may significantly underpredict the abundance of massive sources, indicating that new physics is required to understand the first stages of galaxy formation -- an exciting conclusion indeed! So far, three mechanisms have been proposed to explain the overabundance. \citet{Mason2022} argue that it may be a result of scatter in the mapping of halo mass to UV luminosity -- or in other words through strong fluctuations in the star formation rate of galaxies at a fixed halo mass. Because there are so many small haloes, galaxies with upward (temporary) fluctuations in the star formation rate will appear as luminous sources. The steepness of the underlying mass function means there are many more intrinsically small sources that will ``up-scatter'' to high luminosities than massive galaxies that will ``down-scatter'' to fainter luminosities, flattening the luminosity function and amplifying the bright end. One would also find an overabundance of bright objects if galaxies in low-mass halos produce stars or UV photons more efficiently than expected \citep{Inayoshi2022}. Another potential explanation is dust. Measurements of dust obscuration in $z \sim 7$ galaxies by the REBELS survey show that a substantial fraction of star formation is obscured even at that time \citep{Inami2022, Algera2022}. \citet{Ferrara2022} argued that the overabundance of bright sources at $z \sim 10$ could be explained by evolution in the dust content: if very little obscuration occurred at early times (because the dust was not yet in place), the evolving dust content could roughly cancel the evolution in the halo mass function, causing the abundance of luminous galaxies to evolve only slowly. In this work, we take a deeper look at the implications of an overabundance of luminous, high-$z$ galaxies for our understanding of galaxy formation and evolution. In particular, we focus not just on the number counts themselves but on the inferred properties of these sources, which offer tantalizing clues about the mechanisms that may power these sources. In particular, we consider stellar mass estimates, inferred ages, and colours. We show that this additional information substantially complicates models that attempt to explain the overabundance of bright sources, requiring a linked set of non-trivial changes to the physics driving galaxy formation at early times. In section~\ref{sec:methods}, we describe the flexible framework through which we model the galaxy population and introduce the data sets to which we focus our comparison. We then show that models with typical assumptions cannot reasonably explain the new observations (\S\ref{sec:status}) and that small number statistics arguments do not easily solve the abundance problem (\S\ref{sec:stats}). We then study several extensions to the model in \S\ref{sec:extensions}, and we offer predictions for how each scenario's assumptions manifest in high-$z$ UVLFs and galaxy properties. We show that the wide range of measurements available with JWST will allow us to test these explanations in the near future. Finally, we conclude in section~\ref{sec:disc}. We adopt AB magnitudes throughout \citep{Oke1983}, and the following cosmology: $\Omega_m = 0.3156$, $\Omega_b = 0.0491$, $h = 0.6726$, and $\sigma_8=0.8159$, similar to the recent \citet{Planck2018} constraints. \section{Methods} \label{sec:methods} \subsection{Galaxy models} We employ a semi-empirical model calibrated to UVLFs at $4 \la z \la 8$ and colours described in \citet{Mirocha2017,Mirocha2020dust}, and implemented in the \textsc{ares} code\footnote{\url{https://github.com/mirochaj/ares}}. The basic approach is similar to other models in the recent literature, in that the star formation rate (SFR) of galaxies is assumed to be proportional to the mass accretion rate (MAR) of dark matter halos, modulo an efficiency factor that varies with redshift and/or halo mass (e.g., \citealt{Trenti2010, Mason2015, Tacchella2018}). We first initialize a set of halo assembly histories following the \citet{Furlanetto2017} approach, which assumes halos evolve at fixed number density and has been shown to agree well with the mean growth histories of halos in $N$-body simulations \citep{Trac2015,Mirocha2021}. We assume that galaxies occupy halos in a 1:1 fashion and that the halo mass function is that given by \citet{Tinker2010}, which we compute using the \textsc{hmf}\footnote{\url{https://hmf.readthedocs.io/en/latest/}} code \citep{Murray2013}. We assume that the baryonic mass accretion rate is the total MAR times the cosmic baryon fraction, i.e., $\dot{M}_b= f_b \dot{M}_h$, where $f_b \approx 0.16$ is the cosmic baryon fraction. For our fiducial models, we assume 0.3 dex log-normal scatter in the $\rm{MAR}$ at fixed $M_h$, which matches measurements in cosmological simulations \citep{Ren2019, Mirocha2021}, but do not allow any other scatter in the star formation process. We will explore scenarios with additional scatter in \S\ref{sec:extensions}. With halo growth histories in hand, the SFR is then taken to be $\dot{M}_{\ast} = f_{\ast} \dot{M}_b$, where the star formation efficiency $f_{\ast}$ is assumed to be a double-power law in halo mass, \begin{equation} \fstar(M_h) = \frac{f_{\ast,10} \ \mathcal{C}_{10}} {\left(\frac{M_h}{M_{\mathrm{p}}} \right)^{-\alphalo} + \left(\frac{M_h}{M_{\mathrm{p}}} \right)^{-\alphahi}} \label{eq:sfe_dpl} \end{equation} Here, $f_{\ast,10}$ is the SFE at $10^{10} M_{\odot}$ (which may be redshift-dependent), $M_p$ is the mass at which $\fstar$ peaks, and $\alphahi$ and $\alphalo$ describe the power-law index at masses above and below the peak, respectively. The constant $\mathcal{C}_{10} \equiv (10^{10} / M_p)^{-\alphalo} + (10^{10} / M_p)^{-\alphahi}$ re-normalizes the standard double-power law formula to $10^{10} M_{\odot}$ instead of the peak mass. We also allow star formation to be stochastic, with high-$z$ galaxies going through periods of star formation (with probability $f_{\rm duty}$ at any given time step) and dormancy. We model the build-up of dust in galaxies assuming that the metal production rate, $\dot{M}_Z$, is proportional to the SFR, and that the dust yield is a fraction $f_{\rm{dtmr}}$ of the metal yield. For an effective dust scale length, $R_d$, the dust optical depth can be written \begin{equation} \tau_{\nu} = \kappa_{\lambda} N_d = \kappa_{\lambda} \frac{3 \fdtmr M_Z}{4 \pi R_d^2} \label{eq:tau_d} \end{equation} where we take the dust opacity to be $\kappa \propto \lambda^{-1}$, similar to SMC-like attenuation curves in the rest-UV \citep{Weingartner2001}. We parameterize $R_d$ as a double power-law in halo mass, whose parameters (which may vary with redshift) are calibrated empirically jointly with the parameters of $f_{\ast}$ via fits to UVLFs at $4 \la z \la 8$ \citep[from][]{Bouwens2015} and colours \citep[from][]{Bouwens2014}. The full rest-ultraviolet spectrum of each halo in the model is synthesized over its past star formation history, including nebular continuum emission \citep[following standard procedures][]{Ferland1980}, as implemented within \textsc{ares} \citep{Mirocha2020dust,Sun2021}. This intrinsic spectrum is then reddened by the optical depth given in Eq.~ (\ref{eq:tau_d}). We quantify their colours in terms of the spectral index $\beta$, where the rest-UV specific flux density is approximated as $f_\lambda \propto \lambda^{\beta}$. We will explore two previously-established versions of this model to begin, both of which match UVLFs measured by HST, followed in section~\ref{sec:extensions} by several extensions that aim to understand the implications of early JWST constraints. We take as our first case a modified version of the \citet{Mirocha2020dust} model, which used UVLFs at $z \sim 4, 6$, and $8$, and colours at $z \sim 4$ and 6 \citep[from][]{Bouwens2015,Bouwens2014} to calibrate the free parameters of $f_{\ast}$ (assumed independent of redshift here) and $R_d$. Here, for simplicity we neglect dust, but still synthesize the full SED of all model galaxies to establish a baseline for their rest-UV colours. Second, we employ a more physically-motivated model, in which $f_{\ast}$ is given by the scalings appropriate for energy-regulated stellar feedback, $f_{\ast} \propto M_h^{2/3} (1+z)$ \citep[as in][]{Furlanetto2017}, and the dust scale length evolves as the halo virial radius, $R_d \propto M_h^{1/3} (1+z)^{-1}$. In this case, reconciling the model with UVLFs and colours requires evolution in both the star formation duty cycle ($\fduty$) and dust-to-metal-ratio ($\fdtmr$) \citep{Mirocha2020prospects}, because the shallow mass dependence of $\Rvir$ results in a dramatic over-reddening of the bright galaxy population \citep[see also, e.g.,][]{Somerville2012,Yung2019}. As we will see shortly, neither of these models describe all of the new JWST observations as-is -- both under-predict the $z \gtrsim 10$ galaxy abundance -- which is unsurprising given that our approach is similar to others in the literature already known to be in tension with the new observations \citep[as summarized in, e.g., Fig. 5 of][]{Naidu2022}. As a result, we explore several variants of the model in what follows, in an attempt to identify viable scenarios to account for the new observations. While our approach is admittedly phenomenological, it has the virtue of making testable predictions for other galaxy observables. \subsection{Comparison data} The early release of JWST data has been accompanied by a bevy of observational analyses. We focus our comparison in this work on a subset of these, largely because they present either UVLFs \citep[e.g.,]{Donnan2022, Harikane2022, Naidu2022} or a key set of galaxy observables inferred from SED fitting, especially stellar masses, UV colours, and ages of individual objects \citep[as reported in][]{Atek2022,Whitler2022,Naidu2022,Donnan2022,Harikane2022,Morishita2022,Cullen2022,Finkelstein2022}. These studies use three independent survey volumes. The GLASS-JWST survey uses a single JWST pointing, which is $9.7$~arcmin$^2$ \citep{Treu2022}. CEERS is a somewhat larger survey, which so far has four pointings covering $\sim 33$~arcmin$^2$ \citep{Finkelstein2022CEERS}. Finally, the SMACS J0723 field has a small volume (a single pointing of $9.7$~arcmin$^2$) but uses a foreground galaxy cluster to magnify distant sources \citep{Ebeling2010,Coe2019,Ferreira2022}. Several other groups have made similar measurements of galaxies in this era: for example, \citet{Castellano2022} found several candidate galaxies at $9 \la z \la 15$ in the GLASS field, and \citet{Furtak2022} focused on strongly-lensed galaxies at $z \sim 10$. \citet{Yan2022} used color selection to develop a large list of candidates but did not attempt to measure physical quantities from them, while \citet{Carnall2022}, \citet{Adams2022}, and \citet{Trussler2022} focused on the detailed properties of a small number of objects. However, because all these studies use a subset of the same fields, they cannot be treated as fully independent measurements of the galaxy abundance. We have therefore focused on the aforementioned subset for convenience, recognizing that the detailed list of candidate galaxies depends upon the analysis method (see the helpful comparison in \citealt{Harikane2022}). However, the apparent overabundance of luminous galaxies at $z \ga 8$ appears to be generic to all these analyses. We also note that several studies have focused on more detailed properties of galaxies at $z \la 8$ (e.g., \citealt{Dressler2022}). While eventually it will be crucial to match all of the properties of $z \ga 10$ galaxies onto their descendants (and to use faint analogs at $z \la 7$ to inform expectations for their higher-$z$ cousins through such studies as \citealt{Nanayakkara2022}), here we focus on the properties of these early galaxies inferred directly from their detections. In this work, we will also take the properties of these galaxies inferred through SED-fitting -- especially the stellar mass and age -- at face value. This may be problematic, as both quantities depend upon the assumed parameterization of the star formation (see especially \citealt{Whitler2022}) and upon assumptions about the star formation process (e.g., \citealt{Steinhardt2022}), so difficulties in matching the observations may ultimately point toward refinements needed in such inferences. In particular, halos hosting galaxies grow exponentially with redshift during this era in essentially every theoretical model because halo growth rates are roughly proportional to halo mass \citep[e.g.,][]{McBride2009,Goerdt2015}, so we expect the star formation history to at least roughly mirror this behavior \citep[see,e.g.,][]{Dekel2014,Furlanetto2021}. A fair comparison to galaxy models requires star formation histories that permit this form in the inference procedure, though that is not generally included. We note that these different star formation histories make the age comparison particularly fraught. Finally, we emphasize that even with the subset of measurements we have chosen to focus on, they all still ultimately draw from the same survey fields. Candidate galaxies are either detected by more than one study (in which case there may be conflicting measurements of the same source) or are detected by one study but not another. To highlight the latter, we use \citet{Harikane2022}, which appears to have the most conservative selection criteria, to highlight sources that are more marginal. \section{Status of Current Models} \label{sec:status} The most obvious challenge posed by early JWST observations, as pointed out previously \citep[e.g.,][]{Naidu2022,Mason2022,Ferrara2022}, is to explain the apparent over-abundance of bright $z \gtrsim 10$ galaxies relative to model predictions anchored to $z \lesssim 8$ datasets. In this section, we start by showing the predictions of our default semi-empirical models from $z \sim 4$ to $z \sim 13$ for the UVLF (\S\ref{sec:uvlfs} and Fig. \ref{fig:uvlfs}), and then focus on the same models' predictions for the properties of individual galaxies (\S\ref{sec:scaling_relations} and Fig. \ref{fig:scaling_relations}). \subsection{The UV luminosity function at $z \ga 10$} \label{sec:uvlfs} In Fig. \ref{fig:uvlfs}, we show UVLF predictions from our models at $z \sim 4-13$. UVLFs at $4 \lesssim z \lesssim 8$ agree with the plotted data \citep[from][]{Bouwens2015} by construction (top-left corner), as do the and $\MUV$-$\beta$ relations during that period (see below), but the higher-redshift data were not used for the calibration. The five panels at $z \ga 9$ thus demonstrate how HST-era models fare against the new observations. Again, we emphasize that while data points from JWST in each of these figures are drawn from different studies, the same galaxies will appear in multiple datasets, potentially with different inferred properties. We have made no attempt to eliminate repeated sources in this work, or to give preference to one study over any other. However, given that \citet{Harikane2022} impose the most conservative colour cuts, we use open plot symbols to mark galaxies that were identified in the indicated paper but \textit{not} by \citet{Harikane2022}. We will discuss uncertainties arising from these small survey areas in \S\ref{sec:stats}. The top row of Fig.~\ref{fig:uvlfs} shows that both of our fiducial models fare reasonably well at $z \lesssim 10$, despite some mild tension at $z \sim 10$ for the energy-regulated model (top right, solid line). However, neither capture the abundance of $z \gtrsim 11$ galaxies inferred in early JWST analyses. The JWST UVLFs are consistent with no evolution between $z \sim 10$ and $z \gtrsim 13$, at least in the brightest $\MUV$ bins. In contrast, both of our models predict an order-of-magnitude decrease in the abundance of bright galaxies from $z \sim 10$--13. This is most clear upon comparing to the dust-free universal SFE model prediction for $z = 10$, which is repeated in each panel of the bottom row to guide the eye (and which fits the $z \sim 10$ data well). This is a simple consequence of the rapid evolution in the halo mass function over this period: galaxies this luminous live inside massive haloes, which are undergoing rapid assembly throughout the Cosmic Dawn. This is the same tension highlighted by \citet{Naidu2022} and addressed previously by \citet{Mason2022} and \citet{Ferrara2022}. \subsection{Galaxy scaling relations} \label{sec:scaling_relations} We now step beyond the abundance of these sources and ask another set of questions about how well the models describe the \emph{properties} of these galaxies -- in particular, the relationships between UV magnitude $\MUV$, stellar mass, stellar age, and ultraviolet colour\footnote{The colour is quantified by a power-law fit to the rest-UV spectrum, i.e., $f_{\lambda} \propto \lambda^{\beta}$, which we estimate from a fit to mock photometry in the JWST medium filters.}, all shown in Figure~\ref{fig:scaling_relations}. We note that the $\MUV$-$\beta$ relation at $z \sim 4$ and $6$ was used to calibrate the model parameters so agrees with the plotted data from \citet{Bouwens2014} by construction. However, the stellar masses and ages are predictions at all redshifts, as these quantities have not been used at any stage of the model calibration. The observed $\MUV$-$\beta$ relation at $z \ga 8$ is also a prediction of the model. We do not show the $\Mstell$-$\beta$ relation explicitly, but the \citet{Mirocha2020dust} model agrees well with the measurements of \citet{Finkelstein2012} at $4 \lesssim z \lesssim 8$. In Fig. \ref{fig:scaling_relations}, we explore scaling relations between $\MUV$ and $\Mstell$ (top row), UV colour $\beta$ (middle row), and age (bottom row), where in the models we define the age as the timescale over which galaxies formed half their stellar mass. The half-mass time is slightly longer than mass-weighted ages (at least for exponential SFHs like ours), but of course shorter than the time since the first star formed, providing a middle-ground when comparing to the many observations in this study, in which each of these three definitions can be found. From left-to-right, we show our model predictions relative to observational constraints from $z \sim 4$ to $\gtrsim 13$. For clarity, we only include the level of scatter (as shaded regions) for the energy-regulated stellar feedback model, showing the median relations for the dust-free universal $f_{\ast}$ model as dashed curves only. We compare to constraints from a variety of recent studies \citep{Whitler2022,Morishita2022,Atek2022,Naidu2022,Finkelstein2022,Leethochawalit2022,Cullen2022}, as indicated in the legends. Several SED-fitting codes are represented among these studies, which may infer different galaxy properties given the same input photometry. The inferred masses and ages, for example, can depend strongly on the assumed parameterization of the star formation history, whereas the colours are a more empirical diagnostic of galaxy measurements. \citet{Whitler2022} provide constraints on stellar population properties for a variety of assumptions -- here, we adopt their results assuming a constant star formation rate with the \textsc{Beagle} code \citep{Chevallard2016}. Non-parametric histories, e.g., from \textsc{Prospector} \citep{Johnson2021}, generally result in smaller ages \citep[see also][]{Tacchella2022}, which may help alleviate some of the tension in the $\MUV$-age plane. We do not attempt to add these systematic uncertainties to Figure~\ref{fig:scaling_relations}, but we recommend the reader bear them in mind while comparing. Both of our models predict similar $\MUV$-$\Mstell$ and $\MUV$-age relationships (top and bottom rows, respectively), which are in good agreement with the latest JWST constraints, with the exception of a few very massive and old candidates at $z \sim 10$ (second column from left). The most noticeable and consistent tension in Fig.~\ref{fig:scaling_relations} is that the colours of the dust-free model are far too blue, with $\beta \simeq -3$, in comparison to the measurements. This aspect of the model comes as no surprise, given the exponentially rising (and continuous) star formation histories of galaxies in our theoretical models: without dust, the UV luminosity is dominated by very young, massive stars, which have an intrinsic colour $\beta \sim -3$. While a few observed galaxies have extreme colours consistent with $\beta \sim -3$, most are closer to $\beta \sim -2$ or $2.5$ \citep[see also, e.g.,][]{Topping2022}. The colours of galaxies in our energy-regulated model are in much better agreement with these observations, indicating that the dependencies of $\fduty$ and $\fdtmr$ on halo mass and redshift inferred from observations at $z \lesssim 8$ work reasonably well at $z \gtrsim 10$. There are some clear outliers, particularly at $z \sim 8$ and $z \sim 10$, but again, the agreement is reasonably good overall. In this model, dust reddens the starlight, even at high redshifts. This highlights how the observed colours present an immediate challenge to one potential solution to the overabundance problem -- in which a rapid increase in dust reddening from $z \sim 10$ to $z \sim 7$ roughly cancels the underlying halo mass function evolution \citep{Ferrara2022}. Such a solution would require a different way to redden the galaxies, likely some form of quenching to ensure that they lack young, blue stars. Given that the properties of individual galaxies are largely consistent with our model when dust is included, in the next section we explore the possibility that Poisson and cosmic variance could be responsible for the apparent over-abundance of $z \gtrsim 10$ galaxies. \section{Poisson Sampling and Cosmic Variance} \label{sec:stats} The first wave of JWST results has been drawn from a few relatively small fields, so one might reasonably worry about the impact of small number statistics and cosmic variance. In Fig.~\ref{fig:ngtx}, we explore these possibilities, with an emphasis on the probability of obtaining a $\sim$ constant number of galaxies over a wide redshift range. In this section, we continue to use our fiducial models calibrated to the $4 \la z \la 8$ measurements, as in section~\ref{sec:status}. However, because the expected redshift evolution of these models is generic to many models based on the rapidly evolving halo mass function, we take a step back and consider how some basic properties of the halo population evolve over this period. Starting in the top row, we show the predicted number of objects one expects to detect in a $10 \ \rm{arcmin}^2$ survey, of radial depth $\Delta z =1$, comparable to the first GLASS parallel field \citep[which uncovered one $z \sim 12-13$ candidate; ][]{Castellano2022,Naidu2022,Harikane2022} as a function of redshift. We generate 300 realizations of the halo population in each such volume, subject only to Poisson noise from the limited volume. We consider separately the effect of cosmic variance momentarily. We first focus solely on two halo properties: the halo mass $M_h$ (left panel) and mass accretion rate (center panel) (which also has an intrinsic variation in the model). In both panels, we have chosen the lower mass and accretion rate thresholds to yield (on average) $\sim 1$ galaxy at $z \sim 12$, as found in the observations. The larger threshold in each panel was chosen to highlight the behavior at even higher masses. Shaded regions indicate the variation in source counts at each redshift above a specified mass or MAR cut (gray and blue shaded regions) with 68\% (dark) and 95\% (light) probability. In the left panel, one can see clearly the rapid decline in the abundance of halos of a fixed mass, suggesting that a simple model in which $\MUV$ traces $M_h$ alone is not a viable solution to the JWST abundance problem: if a survey is sensitive enough to detect $\sim 1$ halo at $z \sim 12$, and if galaxy luminosity traces halo mass, it should find $\sim 10$ haloes at $z \sim 10$. In the middle column of Fig. \ref{fig:ngtx}, we shift focus to the mass accretion rate: in our model, the star formation rate (and hence luminosity) is directly tied to accretion rather than halo mass, so this is a better proxy for luminosity. The abundance of haloes above a fixed MAR threshold evolves more slowly than the abundance above a fixed mass threshold. This is because the MAR is a strongly increasing function of redshift at fixed $M_h$ (crudely, $\dot{M}_h/M_h \propto (1+z)^{5/2}$; \citealt{Neistein2008,Dekel2014}), which helps to partially cancel the rapid evolution of the mass function itself. Here, we can see that a halo MAR cut of $\dot{M}_h \geq 10^{2.7} \ M_{\odot} \ \rm{yr}^{-1}$ yields $\sim 1-2$ halos 95\% of the time between $\sim 9$ and $\sim 11.5$. However, one more step is needed to convert to UV luminosity. In most models, the efficiency of star formation is a strong function of halo mass (rather than MAR), which means that a fixed MAR threshold does \textit{not} yield a fixed $\MUV$ independent of redshift (because the MAR-$M_h$ relation evolves with redshift). We use our dust-free model to convert to UV luminosity in the final column of Figure~\ref{fig:ngtx}, which shows the expected number of galaxies in two $\MUV$ bins: the solid curve shows the mean expectation (matching the luminosity function shown in Fig.~\ref{fig:uvlfs}), while the shaded regions show the Poisson noise. Given the small expected number of sources in each redshift bin, it is not surprising that Poisson sampling imposes a wide uncertainty on the source counts. In blue, we see that one expects $0-1$ galaxies with $\MUV = -21$ over $z \sim 9-10.5$. But even including the Poisson variations, the likelihood to find a source that bright at $z \gtrsim 11$ is extremely small. Thus Poisson noise alone is not enough to rescue our default models. In the bottom row of Fig. \ref{fig:ngtx}, we perform the same exercise for a wider field of view, $50 \ \rm{arcmin}^2$, which is comparable to the total area of the first four CEERS fields \citep{Finkelstein2022CEERS}. The number of targets increases, as expected, but the trends remain largely the same -- the fluctuations in the counts of luminous sources are large, but not nearly large enough to explain the observations. In the final column of Fig. \ref{fig:ngtx}, we also show examples of the effect of cosmic variance, accounting for the possibility that the survey fields happen to be overdense regions with enhanced halo populations. The mean expected number of galaxies in each magnitude bin is indicated by a solid line, while the dashed line indicates the boost in counts expected in a region of the Universe that is a $2\sigma$ overdensity. These boosts are computed with \texttt{galcv} \citep{Trapp2020}, an analytic model that assumes galaxy counts in different regions reflect differences in the underlying dark matter halo population alone, i.e., no additional variations in galaxy properties are introduced to reflect environmental effects on galaxy formation\footnote{Here we assume the galaxy counts follow a gamma distribution, which more accurately reflects cosmic variance in the rare-source, high-bias limit and has a longer tail for clustered sources \citep{Steinhardt2021, Trapp2022}.}. This overdensity provides a modest boost to the source counts, but it is far too small to explain the $\sim$ constant abundance of bright $\MUV \sim -21$ objects revealed with JWST, especially for wider area samples assembled in, e.g., \citet{Harikane2022}. We must also note another problem for cosmic variance and Poisson sampling explanations for the overabundance of $z \ga 10$ sources: these excess sources appear across several redshift bins and multiple fields. While a statistical fluke could perhaps explain an overabundance in a single field and at a single redshift as an unlikely but plausible event, finding similar overabundances in many different volumes is extremely unlikely. \section{Model Extensions} \label{sec:extensions} From \S\ref{sec:status}-\ref{sec:stats}, it is clear that typical semi-empirical models calibrated to $z \lesssim 8$ datasets do not naturally explain the abundance of high-$z$ galaxies discovered with JWST. In this section, we explore several extensions of the model in order to identify plausible explanations for the JWST data. We do not attempt to constrain quantitatively the physical elements of these solutions; instead, we illustrate the complexity of explaining the observations with a few specific examples that highlight the major challenges. We record the key parameter values of each model extension in Table \ref{tab:params} for reference. \begin{table}% \begin{tabular}{| l | c | c | c | c | c | } \hline model & $f_{\ast}$ & $\sigma_{\rm{MAR}} $ & dust yield & relevant figure \\ \hline constant $f_{\ast}$ & 0.110 & 0.3 & 0.0 & \ref{fig:extension_nodust} \\ constant $f_{\ast}$ & 0.0066 & 1.0 & 0.0 & \ref{fig:extension_nodust} \\ constant $f_{\ast}$ & 0.150 & 0.3 & 0.1 & \ref{fig:extension_dust} \\ constant $f_{\ast}$ & 0.072 & 1.0 & 0.125 & \ref{fig:extension_dust} \\ \hline DPL $f_{\ast}$ & 0.055 & 0.3 & 0.4 & n/a \\ \hline \end{tabular} \caption{{\bf Summary of model extensions shown graphically in Fig. \ref{fig:extension_nodust} and \ref{fig:extension_dust}.} Note that these values are not determined via detailed fits. The key point is that (i) increased stochasticity (through $\sigma_{\rm{MAR}}$) requires a reduction in the star formation efficiency to avoid over-producing galaxies, and (ii) the introduction of dust requires a slight increase in $f_{\ast}$ to get roughly the same luminosity per SFR. We include for reference in the last row parameter values for the universal double power-law $f_{\ast}$ model from \citet{Mirocha2020dust} that fits $4 \lesssim z \lesssim 8$ UVLFs and colours measured from \textit{Hubble} data \citep{Bouwens2014,Bouwens2015}. In this case, the quoted $f_{\ast}$ refers to the value at $M_h = 10^{10} \ M_{\odot}$.} \label{tab:params} \end{table} \subsection{The star formation efficiency} We saw in Fig.~\ref{fig:ngtx} that the abundance of dark matter halos above a given MAR threshold can evolve quite slowly and plausibly produce $\sim 1-2$ objects over a broad range of redshifts $z \sim 9-13$, depending on the survey area. This same behaviour is not mirrored in the number counts of galaxies within a given UV magnitude bin in our models due to the strong halo mass dependence of the star formation efficiency (right column of Fig.~\ref{fig:ngtx}). We first therefore explore scenarios in which $f_{\ast}$ is a constant, independent of both redshift and halo mass, which sets up a scenario in which galaxy star formation rates are directly proportional to halo mass accretion rates and may thus evolve relatively slowly with redshift to better match JWST constraints. While we are motivated to fix $f_\ast$ by this phenomenological result, we must emphasize that a constant star formation efficiency is a substantial departure from the results of galaxy models at lower redshifts. For example, the empirical models of \citet{Behroozi2019} demonstrate that the star formation efficiency of low-mass haloes at $z \la 6$ increases with halo mass, while that of massive haloes decreases. The former effect is normally interpreted as ``feedback regulation,'' in which stellar radiation and/or supernovae prevents star formation and/or gas accretion in a way that depends upon the potential of the halo. If $f_\ast$ is independent of halo properties, star formation occurs without knowledge of this larger environment and must depend only on local physics. One example of a process that can help ``decouple'' star formation from the halo environment is burstiness, which can break the quasi-equilibrium relation between feedback and accretion \citep{Orr2019, Furlanetto2022a}. We note that \citet{Inayoshi2022} also considered a constant $f_\ast$ scenario to explain the JWST observations, with an emphasis on the interplay between $f_\ast$ and the efficiency of UV photon production. We will compare to their results momentarily. In comparison to our fiducial models, a constant $f_\ast$ boosts the efficiency of star formation in low-mass halos well above their nominal values and so inflates the pool of objects that could up-scatter into bright $\MUV$ bins. (Here, we assume that scattering is a result of variations in the MAR, which follows a log-normal distribution with a scatter of $\sigma=0.3$, as in the fiducial models.) The result of this exercise is shown in Fig. \ref{fig:extension_nodust}. The four panels show our model extension's predictions for UVLFs and relationships between $\MUV$ and $\Mstell$, stellar age, and UV colour, going clockwise from top-left to bottom-left. For simplicity, we first focus on scenarios without dust. The dashed curves show our revised model with a constant $f_\ast=0.11$, similar to the values found to roughly match the $z \sim 10$ JWST UVLFs in \citet{Inayoshi2022}. We find that in this model, tuned to roughly match the $z \sim 11$ UVLFs (magenta curves), the UVLF at $z \sim 13$ drops by more than an order of magnitude and cannot explain the \citet{Naidu2022} or \citet{Donnan2022} sources (dashed yellow curves). \citet{Inayoshi2022} showed that $f_{\ast} \sim 0.3$ may be required to explain the $z \sim 13-17$ UVLFs, at least for normal stellar populations. Next, we will consider an alternative possibility: that $f_{\ast}$ does not evolve over this interval but that the scatter in SFR at fixed halo mass is large. \subsection{Scatter in the halo--SFR relationship} We next explore the effect of increasing the scatter in the halo--galaxy mapping. To do so, we simply change the amount of log-normal scatter in the SFR of galaxies at fixed halo mass. Specifically, we increase the scatter from $\sigma = 0.3$, to $\sigma = 1$. This level of scatter has not been observed in models or simulations of dark matter halos at high redshift \citep{Ren2019,Mirocha2021}, but because ${\rm SFR} \propto f_{\ast} \times {\rm MAR}$ in our model, one can think of this scatter either as an increase in the MAR variations or as fluctuations intrinsic to the star formation process itself -- perhaps in the star formation efficiency $f_\ast$ or stochasticity in that process. To accommodate the increased stochasticity, we reduce the overall normalization of $f_{\ast}$ from 0.11 to 0.006 to roughly preserve the LF at fainter magnitudes $\MUV \sim -19$. The large decrease in the \emph{average} star formation efficiency is a manifestation of the importance of scatter, but we will see below it presents problems compared to the observations. The solid curves in Figure~\ref{fig:extension_nodust} show the effects of this increased scatter. In this case, the abundance of bright objects is boosted significantly and the evolution of the UVLF with redshift slows accordingly, as suggested by \citet{Mason2022}. The overall effect is to flatten the luminosity function at the bright end, because many more intrinsically small sources exist that can be up-scattered than massive sources that can be down-scattered. However, the resulting stellar masses and stellar ages are generally too small in comparison to the measured values. The latter is because up-scattered objects are undergoing a temporary increase in their star formation rate so have very young stellar populations -- in contrast to the measurements, where they are typically found to be $\sim 100$~Myr old. Similarly, the models underestimate the stellar masses because so much of the luminosity comes from the young stars -- whereas older populations require more mass to make up the difference, and because the overall star formation efficiency is much smaller. The most obvious problem, however, is with the UV colours: we have ignored dust, so the young stars driving luminous objects in the models have $\beta \simeq -3$, in severe tension with the JWST constraints (lower right panel). \subsection{The role of dust} Finally, in order to match the observed colors we re-introduce dust. As an example of how this affects the model results, we will explore the effects of changing the dust yield at $z \ga 10$, and keep $R_d$ fixed to a power-law with $R_d \propto M_h^{0.45}$ as in the fiducial model. We first tune $f_{\ast}$ to achieve agreement in the $\MUV$-$\Mstell$ relation (top right), then adjust the dust yield $\fdtmr$ until the typical colour is $\beta \simeq -2$. For the $\sigma=0.3$ case, we obtain $f_{\ast} = 0.15$ and $\fdtmr=0.1$. With $\sigma=1$, we obtain $f_{\ast} = 0.07$ and $\fdtmr=0.125$. (Note that introducing dust increases the overall star formation efficiency by an order of magnitude in the high-scatter case, easing the tension with the stellar mass measurements.) The results are shown in Figure~\ref{fig:extension_dust}, for both moderate and elevated scatter. Just as in the dust-free models, increasing the scatter slows the evolution in the UVLFs somewhat. However, the inclusion of dust partially counters the effects of scatter, making it more difficult to simultaneously match UVLFs and colours. The dust extinction increases with halo mass, acting more strongly on the most massive systems. This steepens the mass function, making it more difficult to match both bright and fainter sources at high redshifts. We note that the combination of increased (and constant) $f_\ast$, increased scatter, and dust presented here works reasonably well to match the observations but is by no means unique. For example, though we found $\fdtmr \simeq 0.1$ -- about four times smaller than the default $\fdtmr=0.4$ \citep[motivated by, e.g.,][]{Dwek2007} -- this parameter is degenerate with the normalization of the dust scale length (see eq. \ref{eq:tau_d}), a more difficult parameter to interpret or constrain. Additional degeneracies between the dust scale length and star formation parameters are also strong, and so other solutions may provide equally reasonable agreement. Their mutual dependencies should be explored more thoroughly in the future. Nevertheless, it is reassuring that the dust correction is relatively modest in comparison to models at later times, as dust typically forms over long timescales. \section{Discussion \& Conclusions} \label{sec:disc} Motivated by the apparent over-abundance of bright sources at $z \gtrsim 10$ implied by the first wave of results from JWST, we use a semi-empirical modeling framework to identify galaxy evolution scenarios that can reproduce this excess. Importantly, we consider both the UVLFs and observed constraints on the properties of individual galaxies, including the stellar ages, stellar masses, and rest-ultraviolet colours. We find that maintaining a constant number density of bright galaxies at $z \ga 10$ requires several adjustments to ``standard'' \emph{HST}-calibrated models. First, one must counteract the rapid decline of the abundance of massive haloes over this redshift range, e.g., by appealing to $M_h$-independent star formation efficiencies (see \S\ref{sec:stats} and Fig. \ref{fig:ngtx}). This moderates, but does not eliminate, the expected decline in halo counts. Second, substantial scatter in galaxy SFRs at fixed $M_h$ (over and above that found by variations in the mass accretion rate in cosmological simulations) improves the match to the shape of UVLFs and further slows the UVLF evolution. However, leveraging the abundance of low-mass halos to boost counts in bright $\MUV$ bins results in very young ages, low stellar masses, and blue colours that are inconsistent with JWST constraints on most individual galaxies \citep{Cullen2022,Topping2022,Nanayakkara2022}. As a result, we must also make a \emph{third} change: non-trivial scatter must be accompanied by non-negligible dust, which helps not only to alleviate the tension with JWST colours, but also with stellar masses and ages as well, because dust extinction requires an increase in $f_{\ast}$ to roughly preserve galaxy counts. We must be cautious in interpreting these observations, of course: it has already been argued that some high-$z$ candidates are interlopers from lower redshifts \citep{Zavala2022, Naidu2022b}, and there are good reasons to expect that contamination becomes a serious problem at $z \ga 12$ \citep{Furlanetto2022}. Moreover, we have taken inferences about stellar ages and masses from SED-fitting codes, which require an accurate set of template spectra (potentially a problem at high redshifts; \citealt{Steinhardt2022}) and accurate parameterization of the star formation histories (e.g., imposing a constant star formation history will overestimate $M_\ast$ compared to bursty models; \citealt{Whitler2022}). Spectra of these candidates will be crucial in rejecting interlopers and more robustly measuring the stellar populations. Only then will we pin down the UVLFs at $z \ga 10$; for now we can only make tentative conclusions. But, provided that the candidates are confirmed, we emphasize that, while any one of our three solutions can be invoked to explain the apparent (nearly redshift-independent) overabundance of bright galaxies at $z \ga 10$ \citep{Mason2022, Ferrara2022, Inayoshi2022}, it is much more difficult to create a scenario consistent with both the UVLFs and the properties of individual galaxies, even at this early stage of JWST observations. This is important because all three of our required changes are, in a sense, surprising: \emph{(i)} First, the steep dependence of $f_{\ast}$ on $M_h$, which is essential to fitting observed UVLFs at $z \la 8$, is generally attributed to stellar feedback. One interpretation of our results is that the JWST UVLFs support a picture in which stellar feedback is ineffective at $z \gtrsim 10$, so that $f_\ast$ is independent of halo mass. This is not wholly unexpected; halo dynamical times are comparable to the lifetimes of massive stars at high redshifts \citep[e.g.,][]{FaucherGiguere2018,Orr2019,Furlanetto2022}, which can break the equilibrium established at later times between supernova-driven outflows and cosmological inflows. The qualitative expectation is for star formation to occur in bursts as a result of this disconnect. This is one example of a mechanism that breaks the connection between the star formation rate and the halo environment and so can cause a constant $f_\ast$ -- a fundamental change in the nature of galactic-scale star formation. We also require significantly more efficient star formation in these haloes than feedback arguments suggest; interestingly, the $f_{\ast}$ values used here are comparable to those required to explain the EDGES detection of the global 21-cm signal \citep{Bowman2018, Mirocha2019}. \emph{(ii)} We also require significantly more scatter in the ${\rm SFR}(M_h)$ relation than previously expected (of about an order-of-magnitude, compared to 0.3~dex variation in the mass accretion rate of haloes measured by simulations). The excess scatter likely results from the star formation process itself, and it could either be attributed to short-term temporal variations in the SFR or stochasticity in the star formation process. This too is potentially consistent with a picture in which star formation occurs in bursts in this population, although of course other scenarios are possible. \emph{(iii)} Finally, an (almost) inevitable result of an increased scatter is to make luminous galaxies more blue (because their high luminosities are caused by very recent star formation episodes). In order to make the colours consistent with JWST measurements, we require non-neglible dust. This may itself be surprising at $z \gtrsim 10$, given the short timescales involved: it likely requires dust production in supernova explosions, because the systems are not old enough for dust to be produced in AGB stars. However, these systems are likely so compact that the total amount of dust does not need to be extreme, if it also remains compact. Indeed, our example scenario only requires dust production to be $\sim 25\%$ as efficient as in ``normal'' galaxies. Our solution is not unique: degeneracies exist between these three factors, which should be explored in more detail in the future. For example, the colours also depend upon the ages of the stellar populations. Very young stellar populations with strong nebular continuum emission could produce $\beta \sim -2.5$ colours even without dust. Similarly, stochastic star formation histories can result in colours redder than expected in galaxies that are between star formation episodes \citep[see,e.g.,][]{Kelson2022}. The fact that our model -- which includes nebular continuum emission and stochasticity -- still requires dust to jointly fit number counts and colours could be a byproduct of our assumed parameterizations. For example, we implement stochasticity as a random `flickering' in the SFR rather than as coherent bursts, which makes it very unlikely that all galaxies in a given $\MUV$ bin are uniformly young and dust free. As a result, it may be misleading to compare the $\MUV$-$\beta$ relation predictions for an entire galaxy population with the $\MUV$ and $\beta$ values for just a few individual galaxies. Finally, we note that by focusing on the UVLF, we have not incorporated all the early JWST analyses. We have ignored candidates selected for their large stellar masses, such as those of \citet{Labbe2022}, which pose substantial challenges to the standard galaxy evolution paradigm \citep[as pointed out by, e.g.,][]{BoylanKolchin2022,Lovell2022}. However, strong nebular line emission could also bias inferred stellar masses significantly \citep[see][]{Endsley2022}. Spectroscopic follow-up of such sources thus remains paramount. At the least, this analysis has demonstrated the complexity of interpreting the early JWST measurements. We expect our understanding to evolve rapidly as larger and more refined samples become available in the coming months. \section*{Data Availability} The data underlying this article is available upon request. \section*{Acknowledgments} We thank Adam Trapp for his help with \texttt{galcv}. SRF was supported by the National Science Foundation through award AST-1812458. In addition, this work was directly supported by the NASA Solar System Exploration Research Virtual Institute cooperative agreement number 80ARC017M0006. This work has made extensive use of NASA’s Astrophysics Data System (http://ui.adsabs.harvard.edu/) and the arXiv e-Print service (http://arxiv.org). \textit{Software:} numpy \citep{numpy}, scipy \citep{scipy}, matplotlib \citep{matplotlib}, hmf \citep{Murray2013}. \bibliography{references} \bibliographystyle{mn2e_short}

Title: Propagation of transverse waves in the solar chromosphere probed at different heights with ALMA sub-bands

Abstract: The Atacama Large Millimeter/sub-millimeter Array (ALMA) has provided us with an excellent diagnostic tool for studies of the dynamics of the Solar chromosphere, albeit through a single receiver band at one time presently. Each ALMA band consists of four sub-bands that are comprised of several spectral channels. To date, however, the spectral domain has been neglected in favour of ensuring optimal imaging, so that time-series observations have been mostly limited to full-band data products, thereby limiting studies to a single chromospheric layer. Here, we report the first observations of a dynamical event (i.e. wave propagation) for which the ALMA Band 3 data (centred at 3\,mm; 100\,GHz) is split into a lower and an upper sideband. In principle, this approach is aimed at mapping slightly different layers in the Solar atmosphere. The side-band data were reduced together with the Solar ALMA Pipeline (SoAP), resulting in time series of brightness-temperature maps for each side-band. Through a phase analysis of a magnetically quiet region, where purely acoustic waves are expected to dominate, the average height difference between the two side-bands is estimated as $73\pm16$~km. Furthermore, we examined the propagation of transverse waves in small-scale bright structures by means of wavelet phase analysis between oscillations at the two atmospheric heights. We find 6\% of the waves to be standing, while 54\% and 46\% of the remaining waves are propagating upwards and downwards, respectively, with absolute propagating speeds on the order of $\approx96$~km/s, resulting in a mean energy flux of $3800$\,W/m$^2$.

https://export.arxiv.org/pdf/2208.12070

\title{Propagation of transverse waves in the solar chromosphere probed at different heights with ALMA sub-bands} \author{Juan Camilo Guevara G\'omez \inst{1,2} \and Shahin Jafarzadeh \inst{3,1} \and Sven Wedemeyer \inst{1,2} \and Mikolaj Szydlarski \inst{1,2}} \authorrunning{Guevara G\'omez {et~al.}} \titlerunning{Transverse waves and height difference in ALMA sub-bands} \institute{Rosseland Centre for Solar Physics, University of Oslo, Postboks 1029 Blindern, 0315 Oslo, Norway\\ \email{j.c.g.gomez@astro.uio.no} \and Institute of Theoretical Astrophysics, University of Oslo, Postboks 1029 Blindern, 0315 Oslo, Norway \and Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 G\"{o}ttingen, Germany\\ } \date{Received: 30 June 2022 / Accepted: 20 August 2022} \abstract{The Atacama Large Millimeter/sub-millimeter Array (ALMA) has provided us with an excellent diagnostic tool for studies of the dynamics of the Solar chromosphere, albeit through a single receiver band at one time presently. Each ALMA band consists of four sub-bands that are comprised of several spectral channels. To date, however, the spectral domain has been neglected in favour of ensuring optimal imaging, so that time-series observations have been mostly limited to full-band data products, thereby limiting studies to a single chromospheric layer. Here, we report the first observations of a dynamical event (i.e. wave propagation) for which the ALMA Band 3 data (centred at 3\,mm; 100\,GHz) is split into a lower and an upper sideband. In principle, this approach is aimed at mapping slightly different layers in the Solar atmosphere. The side-band data were reduced together with the Solar ALMA Pipeline (SoAP), resulting in time series of brightness-temperature maps for each side-band. Through a phase analysis of a magnetically quiet region, where purely acoustic waves are expected to dominate, the average height difference between the two side-bands is estimated as $73\pm16$~km. Furthermore, we examined the propagation of transverse waves in small-scale bright structures by means of wavelet phase analysis between oscillations at the two atmospheric heights. We find 6\% of the waves to be standing, while 54\% and 46\% of the remaining waves are propagating upwards and downwards, respectively, with absolute propagating speeds on the order of $\approx96$~km/s, resulting in a mean energy flux of $3800$\,W/m$^2$.} \keywords{Sun: chromosphere -- Sun: radio radiation -- Sun: oscillations -- Magnetohydrodynamics (MHD) -- techniques: interferometrics } \section{Introduction} The Solar chromosphere is a highly dynamic environment where interactions between the magnetic fields and plasma occur across a broad range of spatial and temporal scales \citep{2009ApJ...706..148W, 2019ARA&A..57..189C}. In particular, waves and oscillations play an important role in transferring energy and momentum throughout the atmosphere, thus maintaining the energy balance of the chromosphere and beyond \citep{1993ApJ...413..811C, 2008ApJ...680.1542H}. While oscillatory phenomena and their propagation through the Solar chromosphere have readily been studied for more than half a century \citep{2015SSRv..190..103J, 2015LRSP...12....6K}, direct observations of their energy deposition, particularly on small scales, have been challenging \citep{2017ApJS..229....7G, 2017ApJS..229....9J}. This is partly due to the commonly used chromospheric diagnostics being subject to non-local thermodynamic equilibrium (non-LTE) effects, which have made it difficult to reliably infer the physical parameters \citep{2017SSRv..210..109D}. Alternatively, observations at millimetre wavelengths (which are formed under LTE conditions and are optically thick) would provide direct observations of brightness temperatures, serving as a close proxy for the local electron temperature \citep{2016SSRv..200....1W, 2017SoPh..292...88W,2019ApJ...881...99M,2020arXiv200512717C,2021A&A...652A..92N}. Since 2016, ALMA has provided high-quality, high-resolution observations of the Solar chromosphere at millimetre wavelengths, including the study of oscillations in Solar ALMA observations \citep{2020A&A...634A..86P,2020A&A...635A..71W, 2020A&A...644A.152E, 2021RSPTA.37900174J}. Thus far, however, Solar observations with ALMA have been limited to one receiver band at the time, only providing information about dynamical phenomena across the (relatively small) height range from the radiation that emerges at the observed wavelengths. It should be noted that the absolute heights of formation vary from location and location as well as over time, as a result of the chromosphere's intermittent and dynamic nature \citep{2021A&A...656A..68E}. For Solar observations, each ALMA receiver band is organised into four sub-bands, with each individual sub-band consisting of several spectral channels spanning a certain range of frequencies. Observations at each channel take place in the same time, meaning that there is no time delay between measurements at different frequencies. Due to the particular challenges of observing the Sun with ALMA as compared to other targets, for instance, the antenna beam being filled with complex emission that varies on very short time scales, until very recently, there was no standard reduction pipeline that could produce science ready time series of images. The standard approach so far is to use all the data to reconstruct one continuum map for each time step, thus neglecting the spectral domain in favour of higher image quality \citep{2022A&A...659A..31H}. In their studies, \citet{2019A&A...622A.150J} and \citet{2019ApJ...875..163R} split the data into the four sub-bands and analysed the resulting four sub-bands individually \citep[see also][]{2018A&A...617L...6R}. Since the first two sub-bands (i.e. SB-1 and SB-2) are directly adjacent in frequency, the corresponding SB-1 and SB-2 maps were found to be very similar. The same is true for the last two sub-bands (i.e. SB-3 and SB-4). However, close comparisons between individual sub-bands showed relatively low signal-to-noise ratios (S/N). Therefore, the Solar ALMA Pipeline (SoAP; Szydlarski et al., in prep.) was extended with an additional mode that reconstructs image time series by using all data for SB-1 and SB-2 combined (together forming the lower sideband, LSB) and respectively for SB-3 and SB-4 combined (together forming the upper sideband, USB), which are simultaneous observations of the same target. The resulting two image time series for LSB (SB-12) and USB (SB-34) are found to have a higher S/N. In this letter, we exploit the new imaging mode and estimate the average formation height difference between the LSB/USB maps by assuming that magnetically quiet regions are mostly dominated by purely acoustic waves. We further compare (in brief) the resulting height difference with that obtained from corresponding synthetic millimetre maps from magnetohydrodynamic simulations. Furthermore, we study transverse oscillations and we estimate the propagating speeds of the transverse wave and, ultimately, their average energy flux. \section{ALMA sub-band observations} The ALMA Band 3 (2.8-3.3\,mm) observation used in this study was carried out on 22 April 2017 between 17:20 and 17:55 UTC as part of program 2016.1.00050.S. This observation consists of a time series split into three scans (blocks of observation) with durations of about 10~min each and a cadence of 2~s. The spectral setup consists of four sub-bands with a bandwidth of 2\,GHz centred at different frequencies within the full-band range (SB1 is centred at 93\,GHz, SB2 at 95\,GHz, SB3 at 105\,GHz, and SB4 at 107\,GHz). For the purposes of this study, we combined the lower and upper pairs of sub-bands (within the reduction pipeline) to reconstruct the time series. Effectively, the first pair, hereafter referred to as LSB (SB-12), is a 4\,GHz band wide and centred at 94\,GHz; the second pair, hereafter referred to as USB (SB-34), is also 4\,GHz wide and centred at 106\,GHz. This approach allows to improve the S/N values compared to individual sub-bands. Moreover, it allows us to have co-temporal observations of the same region at two frequencies that are separated by a gap of 12\,GHz. The pixel size during the reconstruction of the time series was chosen to be 0.34\,arcsec for both LSB and USB. The spatial resolution is of about 2.1\,arcsec for the LSB and 1.9\,arcsec for the USB. The time series is reconstructed in such a way that the individual frames have the same size in LSB and USB, enabling a pixel-to-pixel comparison between them. Furthermore, as the interferometric observation only provides relative differences in brightness temperature, the absolute temperature values were obtained by shifting the zero point by 7418\,K in the case of LSB and 7277\,K in the case of USB, according to the average temperature values reported by \citet{2022A&A...661L...4A}. The resulting brightness temperature ranges are [1974-13648]\,K, with a mean of 7418\,K and a standard deviation of 1417\,K for the LSB, and [1362-13946]\,K, with a mean of 7277\,K and a standard deviation of 1381\,K for the USB, respectively. Figure~\ref{fig:ALMA_snaps} shows the same time frame for the LSB on the left and the USB on the middle, whereas the right panel shows the absolute temperature difference between the two sidebands. The latter clearly demonstrated that the LSB-USB differences provide valuable information regarding the thermal structure of the chromosphere. The ALMA maps are spatially-coaligned with observations from the Solar Dynamic Observatory (SDO) \cite{2012SoPh..275....3P}. The Solar coordinates of the centres of the field of views (FOV) are $(x,y)\,=(-246,267)$\,(arcsec). The observation samples mainly a plage region on the east side of NOAA AR12651 but also a small, magnetically quiet region (marked with the white squares in the figure). A full description of the same observation although in the form of continuum-only (full-band) time series can be found in \citep{2021RSPTA.37900184G,2021RSPTA.37900174J,MHD_Guevara_Gomez}. \section{Results} \subsection{Height differences} The radiation at millimetre wavelengths is mainly formed via the interaction of electrons with the Coulomb field of charged ions, plus a relatively small contribution by neutral hydrogen affected by the Coulomb field of electrons passing by. These processes occur basically under local thermodynamic (LTE) conditions, so that the observed temperature at a certain frequency samples the predominant height over the Solar surface where the radiation at that frequency originates \citep{2016SSRv..200....1W,2020FrASS...7...57N}. In general, for a monotonic increase of temperature with height in the chromosphere, the formation height of the LSB is expected to be above the formation height of the USB as the centre frequency of the latter is 12\,GHz higher than the centre frequency of the former. Despite the fact that it is not possible to truly define what are the absolute formation heights at ALMA wavelengths based on these observations alone, \citet{2020A&A...640A..57A} estimated that the height difference between SDO/AIA 1600\,$\AA$ and ALMA Band 3 observations (centred at $100$\,GHz) is about 1200\,km, which would put the absolute formation height of ALMA Band 3 in the upper chromosphere. Furthermore, the study of oscillations in temperature at the two different sidebands LSB and USB can be used to estimate the relative height differences between them. For this purpose, we have chosen a $(17\times17)$\,(arcsec) or $(50\times50)$\,(pixels) box of a magnetically quiet area (see the white squares in Fig.~\ref{fig:ALMA_snaps}) within which acoustic waves with a speed of $c_s = 8.0\pm 1.0$\,km/s \citep[e.g.][]{2002ApJ...564..508R,2019AnGeo..37..891S} are expected to be dominant. Thus, we performed a Fourier analysis on the entire time series, individually for each pixel inside the white box, and we computed the phase-angles between the two sidebands from their cross spectra. A zero phase-angle between two signals means that the signals are in phase, while a non-zero phase-angle can imply that one signal is either leading or lagging behind the other one. Each phase-angle $\phi\,\text{[deg]}$ has a corresponding frequency $f\,\text{[Hz]}$ in the Fourier space, such that the two quantities are related with a time delay, $\tau\,\text{[s],}$ between the signals via the following equation: \begin{equation} \phi\ = 360^{\circ} \,\tau\,f \label{eq:phase_freq} .\end{equation} The top panel of Fig.~\ref{fig:phase_slope} shows the detrended brightness temperatures for the LSB as a solid line and for the USB as a dotted line for the pixel marked with a crossed blue circle in Fig.~\ref{fig:ALMA_snaps}. This plot illustrates the typical behaviour of temperature within the white box corresponding to the magnetically quiet area. In the bottom panel of Fig.~\ref{fig:phase_slope}, the 2D phase spectrum of the corresponding temperature light curves between the LSB and the USB (for the quiet region) is shown. The black dots correspond to the centres of Gaussian curves fitted to vertical cuts in the spectrum and the error bars to their respective standard deviations. The solid black line corresponds to the linear regression fit to the black dots and shows the relation between phase-angle and frequency according to Eq.~\eqref{eq:phase_freq}. The slope of the line is given by $\Delta \phi / \Delta f = 360^{\circ} \,\tau$. Hence, the time delay between the ALMA LSB and USB is estimated to be $\tau = 9.1 \pm 1.6$\,s. The time delay between LSB and USB is an indication of the travel time of a propagating wave observed in brightness-temperature oscillations. Under the assumption that the analysed magnetically quiet region is dominated by acoustic wave, it is then possible to obtain the (average) height difference $\Delta H$ between the two ALMA sidebands as $\Delta H = c_s\tau$. Using the values derived above, we obtain $\Delta H \approx 73 \pm 16$\,km. This value falls within the range of formation height differences between synthetic continuum maps calculated for the public enhanced network Bifrost simulation \citep{2016A&A...585A...4C} for the same ALMA sidebands. Specifically, the height differences in the simulation for a relatively quiet region are predominantly distributed between 20\,km and 120\,km with the peak of the distribution at $\approx$60\,km. We note that based on the theory presented by \cite{2006ApJ...640.1153C}, \citet{2017ApJS..229...10J} used an identical method to estimate the height difference between the 300\,nm and the Ca\,{\sc{ii}}\,H 396.8\,nm passbands of the filter imager on board the {\sc{Sunrise}} balloon-borne solar observatory \citep{2010ApJ...723L.127S}. As such, we refer to \citet{2017ApJS..229...10J} for further details on this approach. \subsection{Propagation of transverse waves} A statistical study of the same ALMA observations (prepared, instead, as full-band maps) showed the possible presence of MHD transverse (kink) oscillations in small-scale bright features. Specifically, \citet{MHD_Guevara_Gomez} analysed $\approx$ 200 bright features in ALMA Band 3 (full-band), which exhibited transverse oscillations in the horizontal velocities. The amplitude of the oscillations spanned a range between 0.2-27.1\,km/s with an average oscillation period of 66\,s. These properties suggested that the transverse oscillations may be associated with kink MHD modes \citep[see e.g.][and references therein]{2015SSRv..190..103J}. In this letter, we have selected five magnetic bright features to study their transverse-oscillatory properties in the two sidebands. Their median locations are marked with blue crosses in Fig.~\ref{fig:ALMA_snaps}. Each of the features is visible and traceable in time in the two sidebands. The border of the features is defined by the contour at half of the maximum temperature of the features at each frame. The location of the features is computed as the centre of gravity (of intensity) using the temperatures within the feature borders. For each feature, the total horizontal velocity is calculated as $\mathrm{v_t} = \sqrt{\mathrm{v_x}^2 + \mathrm{v_y}^2,}$ where the velocities in $x$ and $y$ directions correspond to the displacement of the centre of gravity from frame to frame in each direction. The total horizontal velocities show a similar behaviour as those analysed in \cite{2021RSPTA.37900184G,MHD_Guevara_Gomez} suggesting the presence of MHD kink modes. For each individual feature, we have computed the phase-angles between velocity oscillations observed in the two sidebands by the means of a cross-wavelet transform analysis. Each phase-angle is associated to a dominant period of oscillation within the 95\% confidence level used in the wavelet (i.e. regions on the wavelet spectra where the power exceeds a 95\% confidence level and is outside the cone of influence). By putting together all the phase-period values identified in the five features, it is possible to draw a phase diagram of the horizontal displacements. We note that there are several phase angles associated to each bright feature. Figure~\ref{fig:feat_phase} shows in the top a typical transverse oscillation of a feature as a plot of the horizontal velocity versus time for the feature marked with a blue cross enclosed by a square in Fig.~\ref{fig:ALMA_snaps}. In the middle of the figure, we show the phase diagram in the form of a 2D histogram. The brightest part close to a zero phase and a period of 40\,s indicates the maximum occurrence of standing waves, namely, where there is no propagation. Then, the slightly less strong occurrence above and below the white dotted line would be due to upwardly and downward propagating waves, respectively. Although the absolute formation heights vary from location to location and they are shown to be higher in magnetic elements compared to quiet regions \citep{2015A&A...575A..15L, MHD_Guevara_Gomez}, their average height difference should not change considerably (see e.g. Figure~2 of \citet{2017ApJS..229...10J}, where the average height difference for quiet-Sun and plage models are similar). Therefore, it is practical to use the value of $\Delta H \approx 73 \pm 16$\,km that was previously found to estimate the velocity propagation (phase velocity $\mathrm{v_{ph}}$) of the transverse waves through the chromospheric layers probed with ALMA sidebands. To this end, we use the following equation for the phase velocity \begin{equation} \mathrm{v_{ph}}\,=\,\dfrac{360^{\circ}\Delta H}{T\varphi} \label{eq:vel} ,\end{equation} where $T$ is the period in seconds and $\varphi$ is the phase-angle in degrees. The distribution of the resulting phase velocity values is presented in the bottom panel of Fig.~\ref{fig:feat_phase}, where negative values correspond to downward propagation and positive values to upward propagation. The time lags corresponding to phases of 0$^{\circ}$ are interpreted as standing waves between the two layers and they are therefore excluded of the distribution. However, they represent about $6\%$ of the total number of occurrences. This procedure is similar to the method used by \citet{2022ApJ...930..129B} to estimate the phase velocities of transverse oscillations in spicules. The distribution of velocities shows that more than $72\%$ of the computed values are between -100 and 100\,km/s. The occurrence rate of non-standing waves were 54\% and 46\% for those corresponding to upwardly and downwardly propagation, respectively. The mean and median upward phase velocities are 94\,km/s and 36\,km/s. For the downward phase velocities the mean and the median are $-99$\,km/s and $-53$\,km/s, whereas the absolute mean velocity is 96\,km/s. Under the assumption that the horizontal oscillations are related to MHD kink waves, it is possible to estimate the energy flux $F$ that the identified kink waves carry between the two heights according to \citet{2015A&A...578A..60M} with the following equation: \begin{equation} F\,\approx \dfrac{1}{2}\,f\,(1+\ln{1/f})\,\rho\,\mathrm{v_{amp}^2}\,\mathrm{v_{ph}} \label{eq:Energy} ,\end{equation} where $\rho = 2.33\times 10^{-8}$\,kg/m$^3$ is the density just outside the waveguide, $f=0.045$ is a filling factor, and $\mathrm{v_{amp}}=4.3$\,km/s is the amplitude of the velocity oscillations. These quantities were taken from the statistical values reported by \citet{MHD_Guevara_Gomez}. We take $\mathrm{v_{ph}}=96$\,km/s as the absolute mean velocity and replace the values in Eq.~\eqref{eq:Energy}, obtaining an energy flux of about $3800$\,W/m$^2$. \section{Conclusions} We first studied the simultaneously observed temperature oscillations in a magnetically quiet region observed at two different frequencies with ALMA. To achieve this, we made used of a special procedure to reconstruct time series of two sidebands within the ALMA Band 3 receiver, that is, LSB (94\,GHz) and USB (106\,GH). Under the assumption that the temperature fluctuations in the relatively quiet region of the LSB and USB represent propagating acoustic waves with a speed of $8 \pm 1$\,km/s, we computed a height difference of $73 \pm 16$\,km between the two chromospheric layers from the phase differences between the temperatures oscillations in LSB and USB maps. This study demonstrates the potential diagnostic use of ALMA LSB and USB observations to probe the Solar atmosphere. Furthermore, we traced and studied the oscillatory properties of five small-scale bright magnetic features present in both sidebands. In particular, we compared the transverse oscillation of the features through wavelet analysis, resulting in the detection of positive and negative phase lags between them, namely, upwardly and downwardly propagating waves, with mean velocities of $94$\,km/s and $-99$\,km/s, respectively. These phase velocities are comparable to those of fast kink waves observed in spicules and fibrils (in the Solar chromosphere) with velocities on the order of 50-150\,km/s \citep{2009A&A...497..525H,2011ApJ...736L..24O,2012NatCo...3.1315M,2015SSRv..190..103J,2017ApJS..229....9J}. Taking into account that these velocities correspond to waves propagating upwardly and downwardly in the chromosphere, using a vertical or near-vertical magnetic field as wave guides (see the magnetic topology of the same data set in \citealt{2021RSPTA.37900174J}), we speculate that the observed features may be spicules seen from the top. The presence of standing waves with a strong occurrence over periods close to 40\,s was identified as well. The standing waves may be due to a superposition of upward and downward propagating waves, the latter being the product of reflections of the former somewhere near to the transition region, above the heights mapped with ALMA. Finally, we also estimated the energy flux carried by the propagating kink waves to be on average about $3.8\times 10^3$\,W/m$^2$, which is close to the value of $4\times 10^3$\,W/m$^2$ needed to compensate for radiative losses in the chromosphere according to \citet{1977ARA&A..15..363W}. However, this does not imply that the energy carried by these waves is completely dissipated in the chromosphere and therefore able to account for the radiative losses alone; it indicates, rather, that their contribution to sustain a hot chromosphere may be substantial. An in-depth analysis of these waves, as well as other MHD modes, from both ALMA sidebands observations and numerical simulations, is essential for identifying how they can contribute to Solar atmospheric heating and this will be the subject of a future work. \section*{Acknowledgments} This work is supported by the SolarALMA project, which received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 682462), and by the Research Council of Norway through its Centres of Excellence scheme, project number 262622. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2016.1.00050.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC(Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in co-operation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. We are grateful to the many colleagues who contributed to developing the Solar observing modes for ALMA and for support from the ALMA regional centres. \bibliographystyle{aa} \bibliography{ref.bib}

Title: The Distribution of UV Radiation Field in the Molecular Clouds of Gould Belt

Abstract: The distribution of ultraviolet (UV) radiation field provides critical constraints on the physical environments of molecular clouds. Within 1 kpc of our solar system and fostering protostars of different masses, the giant molecular clouds in the Gould Belt present an excellent opportunity to resolve the UV field structure in star forming regions. We performed spectral energy distribution (SED) fitting of the archival data from the Herschel Gould Belt Survey (HGBS). Dust radiative transfer analysis with the DUSTY code were applied to 23 regions in 14 molecular complexes of the Gould Belt, resulting in the spatial distribution of radiation field in these regions. For 10 of 15 regions with independent measurements of star formation rate, their star formation rate and UV radiation intensity largely conform to a linear correlation found in previous studies.

https://export.arxiv.org/pdf/2208.05250

\title{The Distribution of UV Radiation Field in the Molecular Clouds of Gould Belt} \correspondingauthor{Di Li, Ningyu Tang, Qijun Zhi} \email{dili@nao.cas.cn,nytang@ahnu.edu.cn,qjzhi@gznu.edu.cn} \author[0000-0003-3726-570X]{Jifeng Xia} \affiliation{School of Physics and Electronic Science, Guizhou Normal University, Guiyang, 550025, China} \affiliation{National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China} \affiliation{University of Chinese Academy of Sciences, Beijing 100049, China} \author[0000-0002-2169-0472]{Ningyu Tang} \affiliation{Department of Physics, Anhui Normal University, Wuhu, Anhui 241002, China} \author{Qijun Zhi} \affiliation{School of Physics and Electronic Science, Guizhou Normal University, Guiyang, 550025, China} \affiliation{Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, Guizhou Normal University, Guiyang 550001, China} \author[0000-0002-9151-1388]{Sihan Jiao} \affiliation{National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China} \affiliation{University of Chinese Academy of Sciences, Beijing 100049, China} \author[0000-0002-2738-146X]{Jinjin Xie} \affiliation{Shanghai Astronomical Observatory, CAS, Shanghai,200030, China} \author{Gary~A. Fuller} \affiliation{Jodrell Bank Centre for Astrophysics, Department of Physics \& Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom} \affiliation{I. Physikalisches Institut, University of Cologne, Z\"ulpicher Str. 77, 50937 K\"oln, Germany} \author{Paul F. Goldsmith} \affiliation{Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA} \author[0000-0003-3010-7661]{Di Li} \affiliation{National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China} \affiliation{University of Chinese Academy of Sciences, Beijing 100049, China} \keywords{ISM:clouds---ISM:UV intensity---Dust---Radiative transfer} \section{Introduction} % \label{sect:intro} Interstellar medium (ISM) is the cradle of star formation. Its evolution is strongly affected by the ultra-violet (UV) radiation. By ejecting electrons from dust grains and directly exciting atoms and molecules, UV photons ionize atoms, dissociate molecules and heat gases \citep[e.g.,][]{1985ApJ...291..722T, 2001RvMP...73.1031F, 2011piim.book.....D}. \citet{1998ARA&A..36..189K} found a tight correlation between UV radiation and the star formation rate in galaxies. In Galactic studies, most previous studies focus on the distribution of UV radiation field of individual nearby regions \citep[e.g. ][]{1999A&A...344..342L,2010A&A...521L..19P}. Within 1\,kpc of our Solar system, the Gould Belt containing a lot of molecular complexes provides an excellent opportunity to investigate the relationship between UV intensity and star formation rate under diverse environments \citep{2007PASP..119..855W}. Most molecular complexes of the Gould Belt harbor bright young OB stellar clusters or star-forming regions while the rest show little sign of star formation. For instance, Orion complex is a widely studied giant molecular region with abundant clustering OB stars and turbulent massive star formation \citep[e.g.,][]{1991ApJ...368..432L,1998ApJS..118..517T}. Similar to Orion, Serpens/Aquila Rift is a rich complex with well-known massive star-forming regions, e.g., W40 HII region that contains embedded young high-mass stars \citep[e.g.,][]{1999PASJ...51..851K}. Although the Cepheus region contains an OB star (HD 200775 in Cep 1172), it has been generally considered as a low to intermediate mass star-forming region. Lupus dark-cloud complex was found to be surrounded by about 70 T Tauri stars, with no indication of massive OB star inside \citep[e.g.,][]{1999PASJ...51..895H}. The Chamaeleon-Musca dark-cloud complex including Cha I, II, III, and the Musca dark lane, is a region with low-mass star formation \citep[e.g.,][]{1999A&A...345..965C, 2001PASJ...53.1071M}. IC 5146 is a filamentary dark cloud with scattered low star formation activity \citep[e.g.,][]{1994ApJS...95..419D}. Pipe Nebula has been a primary example of little signs of disturbance from star formation \citep[e.g.,][]{1999PASJ...51..871O}. No star formation was found in the Polaris flare, a high-latitude translucent cloud \citep[e.g.,][]{1990ApJ...353L..49H}. Utilizing the Spectral and Photometric Imaging Receiver (SPIRE) and Photodetector Array Camera and Spectrometer (PACS) instruments on board the {\it Herschel} \footnote{{\it Herschel} is an European Space Agency (ESA) space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.} Space Observatory, the {\it Herschel} Gould Belt Survey (HGBS) covered a substantial fraction of the Gould Belt. Specifically, images were taken at 250, 350, and 500 $\mu$m for regions with $A\rm_V > $ 3 mag with SPIRE and at 70 and 160 $\mu$m for those with $A\rm_V > $ 6 mag with PACS. This survey covers the following 14 molecular complexes: Aquila, Cepheus, Chamaleon, Corona Australis, IC 5146, Lupus, Musca, $\rho$ Oph, Orion, Perseus, Pipe nebula, Polaris, Taurus, and Serpens. The dust radiative transfer model, DUSTY \citep{1997MNRAS.287..799I, 2000ASPC..196...77N} accommodate different kinds of geometry and radiation parameters. \cite{2003ApJ...587..262L} demonstrated the utilities of the DUSTY code to derive the UV intensity of the Orion clouds based on fitting the dust temperature data. We further develop the recipe and apply it to the full set of the HGBS data. % This paper is organized as follows. In Section \ref{sec:data}, we introduce the data information of column density, dust temperature and star formation rate toward the HGBS molecular complexes. The DUSTY radiative transfer code and method for calculating the UV radiation intensity map are described in Section \ref{sec:method}. Results and further discussions are presented in Section \ref{sec:results} and \ref{sec:discussion}, respectively. The summary is in Section \ref{sec:summary}. \section{Data} \label{sec:data} \subsection{{\it Herschel} Dust Continuum Emission}\label{subsection:dust} Dust emission is almost always optically thin at (sub)millimete wavelengths and can thus act as a surrogate tracer of the total (gas + dust) mass along the line of sight (LOS) \citep[][]{2014A&A...562A.138R}. The HGBS took a census in the nearby (0.5\,kpc) molecular cloud complexes for an extensive imaging survey of the densest portions of the Gould Belt, down to a 5$\sigma$ column sensitivity $N_{H_{2}}$ $\sim$ 10$^{21}$\,cm$^{-2}$ or $A_{V}$ $\sim$ 1 mag \citep[][]{2010A&A...518L.102A}. We use the HGBS data to generate the column density map of 23 molecular clouds that belongs to 14 molecular complexes of the Gould Belt . The molecular clouds studied in this paper are Aquaila M2 \citep{2010A&A...518L.102A} , Cep 1151, Cep 1172, Cep 1228, Cep 1241, Cep 1251 \citep{2020ApJ...904..172D}, Cham I, Cham II, Cham III \citep{2012A&A...545A.145W}, \citep{2014A&A...568A..98A}, CraNS \citep{2018A&A...615A.125B}, IC 5146 \citep{2019A&A...621A..42A}, Lup I, Lup III, Lup IV \citep{2013A&A...549L...1R}, Musca \citep{2016A&A...590A.110C}, $\rho$ Oph \cite{2014A&A...562A.138R}, Orion B \citep{2020A&A...635A..34K}, Orion A(Jiao et al. accepted by SCPMA), Perseus \citep{2012A&A...547A..54P}, Pipe \citep{2012A&A...541A..63P}, Polaris \citep{2010A&A...518L.102A}, Taurus \citep{2019A&A...621A..42A}, and Serpens\citep{2021MNRAS.500.4257F}. \subsection{Deriving Dust Temperature and Column Density Based on SED Fitting} \label{subsection:sed} The dust temperature and column density map toward 22 of 23 regions were downloaded from HGBS Archive\footnote{The website of HGBS Archive: http://www.herschel.fr/cea/gouldbelt/en/index.php}. We obtained dust distribution and improved the image quality of the Orion A region, based on a novel image combination technique (Jiao et al.\ 2022, accepted). \footnote{The column density and dust temperature of Orion A was not published on the website of HGBS archive by 2022.} The procedure to derive the dust temperature and dust/gas column density images of Orion A is similar to that in \cite{2014A&A...562A.138R}. Before performing any SED fitting, all images at multiple bands were convolved into beam size of 36$''$.3 at 500 $\mu$m. We weighted the data points by the measured noise level in the least-squares fits. We adopted the dust opacity per unit mass at 300 $\mu$m of 0.1 cm$^{2}$ g$^{-2}$ \citep{1983QJRAS..24..267H}, and assumed a gas-to-dust mass ratio of 100. As modified black-body assumption, the flux density $S_{\nu}$ at a certain observing frequency $\nu$ is given by \begin{equation} S_{\nu} = \Omega_{m}B_{\nu}(T_{d})(1-e^{-\tau_{\nu}}),\label{eq:Snu} \end{equation} where $B_{\nu}(T_{d})$ is the Planck function at temperature $T_{d}$, $\Omega_{m}$ is the beam size. The total column density $N$ of gas and dust can be approximated by \begin{equation} N=\frac{\tau_{\nu}}{\kappa_{\nu}\mu m_{H}},\label{eq:N} \end{equation} where the dust opacity $\kappa_{\lambda}=\kappa_{\mbox{\tiny{300$\mu$m}}}(\lambda/300\,\mu m)^{\beta}$ ($\beta$ was fixed to a constant value of 1.8), $\mu$ = 2.8 is the mean molecular weight, $m_{H}$ is the mass of a hydrogen atom. The grey-body dust temperature ($T_d$) thus calculated has ignored the dependence of dust temperature on grain size \citep{1999ApJ...522..897L}, but has been shown to be within a couple of kelvins of the gas temperatures in well coupled regions \citep{2001ApJ...557..736G,2013ApJ...768L...5L,2020MNRAS.499.4432W, 2021SCPMA..6479511X}. The effect of scattering opacity \citep[e.g.,][]{2019ApJ...877L..22L} can be safely ignored in the case given that focusing on $>$0.05\,pc scale structures. \subsection{Star Formation Rates}\label{subsection:sfr} The SFRs were determined through $\mbox{SFR}$ = $N(\mbox{\scriptsize YSO}) \langle M \rangle \tau^{-1}$, in which $N(\mbox{\scriptsize YSO})$ is the YSOs number in the molecular region, $\langle M \rangle$ is the mean mass of stars, and $\tau$ is the relevant evolution timescale. The identification of YSOs requires careful discrimination against background stars and galaxies. Star-forming galaxies were the most problematic source of contaminants. A combination of color - color and color - magnitude diagrams of both Spitzer and 2MASS data were adopted to reject contaminants. The detailed descriptions of this process are shown in \citet{2009ApJS..181..321E}. \cite{2013AJ....145...94D} provides a YSO catalog for Gould Belt clouds. To convert number of YSOs to mass of forming stars, \cite{2014ApJ...782..114E} adopted mean mass of stars M$_\star$ = 0.5 M{$_ \odot $}, based on a fully sampled initial mass function \citep[][]{2002Sci...295...82K}. The relevant timescales were derived by classifying YSOs into standard SED classes based on 2 to 24 $\mu$m data. Table~\ref{tab:SF} lists the star formation rates \citep{2010ApJ...724..687L, 2014ApJ...782..114E} we adopted in this paper. \begin{table}[h!] \centering \begin{minipage}[]{100mm} \caption[]{Star formation rate of 15 molecular regions.\label{tab:SF}}\end{minipage} \begin{tabular}{lccccr} \hline\noalign{\smallskip} Cloud Name & RA & DEC & Distance$^a$ &Star Formation Rate$^a$ & Category$^b$ \\ & (deg)& (deg) & (pc) & (10$^{-6}$\Ms yr$^{-1}$) & \\ \hline Orion A & 84.52 & -7.03 & 423 & 715 & I \\ Orion B & 86.90 & 0.10 & 423 & 159 & I \\ Serpens & 278.78& 4.6E-06& 429& 56 & I \\ Perseus & 53.92 & 31.53 & 250 & 150 & I \\ $\rho$ Oph & 246.87 & -24.21 & 125 & 79 & I \\ Musca & 186.81 & -71.54 & 200 & 3 & I \\ Lupus III & 242.51 & -39.08 &200 & 17 & II \\ Lupus IV & 241.15 & -42.07 &150 & 3 & II \\ Aquila & 277.43 & -2.78& 260 & 322 & II \\ IC 5146 & 327.19 & 47.49 & 460 & 24 & II \\ Cham I & 165.46 & -77.40 & 150 & 20.5 & II \\ Cham II & 195.23 & -77.41 & 178 & 6 & II \\ Cham III & 190.47 & -79.82 & 150 & 1 & II \\ Pipe & 260.78 & -26.40 & 145 & 5 & II \\ Taurus & 65.08 & 27.72 & 153 & 84 & II \\ \hline\noalign{\smallskip} \noalign{\smallskip}\hline \end{tabular} \tablecomments{The coordinates are determined by the data of HGBS. \\ $^{a}$The values of distances and star formation rates are obtained from \citet{2010ApJ...724..687L} and \citet{2014ApJ...782..114E}. \\ $^{b}$The region with both OB stars and YSOs are defined as Category I; the region with YSOs only are defined as Category II.} \end{table} \section{Method} \label{sec:method} \subsection{The DUSTY code} \label{sec:dusty} The public DUSTY code \citep{1997MNRAS.287..799I} solves the radiative transfer problems through a fully scale-free method. By adopting this scaling method, the DUSTY code solves the spherically symmetric (1-D) problem with a single central radiation source and surrounding spherically symmetric dusty envelope, in which the radial dust density profile is arbitrary. Besides, a dusty plane-parallel slab with illumination from one or both sides at an arbitrary angle is available too. This code utilizes the scaling properties to minimize the number of input parameters. Parameters describing the external radiation, dust and gas properties, cloud geometry are needed for inputs. For the case of spherical geometry, they include number, spectral shape and flux of the external source, chemical composition of dust, the lower and upper limit of dust optical depth, and density distribution of dust. For the case of slab geometry, an extra selection of source side is needed. Once the input parameters are chosen, the DUSTY code outputs the value of dust temperature as a function of gas optical depth. \subsection{Parameter Setting in $\it$ DUSTY Code} \label{subsec:dusty-para} We adopted the slab geometry for our calculations. The incident angle between UV radiation and the slab, $\theta$ is treated as 0 degree, $\theta$= 0$^\circ$ , indicating perpendicular UV radiation toward the slab. The selected radiation wavelength of 0.365\,$\mu$m locates in the central wavelength of UV band. The dust sublimation temperature, which is the highest temperature the dust grains can exist, is chosen to be the common value of 1500 K. The value range of gas optical depth and dust temperature were chosen as [0,30] and [0, 50] K, separately. With the above selections, the UV radiation intensity, G$_0$ is the only parameter that determines the relationship between dust temperature and gas optical depth. As for the optional UV radiation flux received by one side of the slab, six sets of UV intensity (G$_0$= 1, 10, 31.6, 100, 316, and 1000) compared to standard Habing field \citep[e.g.,][]{2017A&A...602A..49H} of 1.6 $\times$ 10$^{-6}$\,W/m$^2$ were introduced for calculation. \subsection{Distribution of UV Intensity Field } \label{sec:contour} In order to produce the spatial distribution of UV radiation field of the Gould Belt, we combined the DUSTY model calculations and the HGBS data. The derived H$_2$ column density and dust temperature map of HGBS molecular complexes with pixel size of 3$^{\prime \prime}$ were convolved and re-sampled with the beam size of 36.3$^{\prime\prime}$ at 500 $\mu$m. We are focusing on dense molecular regions, the contribution from atomic hydrogen can be ignored. Thus the H$_2$ column density was converted into gas optical depth at V band through the following equation \citep{1978ApJ...224..132B, 1968nim..book..221G, 2009ApJS..180..125R} \begin{equation} \tau_V = \frac{A_V}{1.086}=\frac{1.07\times 10^{-21}N_{H_2}}{1.086}. \label{eq:tauv} \end{equation} As an example, the derived relationship between T$_{dust}$ and gas optical depth ($\tau_V$) of all pixels for $\rho$ Oph cloud can be found in Fig. \ref{fig:tau-dust}. We obtained the UV intensity of each pixel by interpolating the results from observations with that from DUSTY model calculations. The spatial distribution of specific UV radiation field of each HGBS molecular complex can be derived. \section{Results} \label{sec:results} After applying the procedures described in section \ref{sec:method}, we obtained the UV intensity distribution of 23 regions in 14 molecular complexes of the Gould Belt. \subsection{Spatial Distribution of UV Intensity} As an example, We presented spatial distribution of selected UV intensity (G$_0$= 1, 10, 31.6, 100, 316, and 1000) for three molecular complexes: Orion (Category I), Aquaila (Category II) and Polaris (Category III). Descriptions of the rest complexes are presented in Appendix. \subsubsection{Orion} Orion molecular complex is the most active star forming region within 500 pc \citep{2012AJ....144..192M}. Orion A and Orion B molecular clouds are covered by HGBS. Locating in Orion A, Orion Nebular cluster is a significant laboratory for understanding the initial mass function. The Orion B molecular cloud is one of the clouds scattered along the region named Orion-Eridanus superbubble, which was created by supernova explosions \citep{2020A&A...635A..34K}. The Orion B cloud is $\sim$ 423 pc away and covers an area of $\sim$ 6.8 $\times$ 8.6deg{$^2$} \citep{2014ApJ...782..114E}. The total mass of these two molecular clouds exceed 2 $\times$ 10{$^5$} M{$_{\odot}$} \citep{2012AJ....144..192M}. As shown in Fig. \ref{fig:orion}, the UV intensity of these two regions show tight correlation with the OB stars. \subsubsection{Aquila} The Aquila field is a very active star-forming region at a distance of about 260 pc \citep{2014ApJ...782..114E}. With size of $\sim$ 7.56 pc, the mass of this molecualr complex is about 24446 M$_{\odot}$, of which two-thirds are composed of dense cores \citep{2014ApJ...782..114E}. The existence of dense cloud leads to highest level of background cloud emission in HGBS. Cluster of YSOs but no OB stars were detected in this region. The number of YSOs in this region exceeds 1000. As shown in Figure \ref{fig:aquila}, the intensity of the radiation field distribution varies from G$_0$=1 to G$_0$= 1000. We found a certain correlation between the spatial distribution of YSOs and UV radiation field. The possible reason responsible for this is discussed in section \ref{sec:discussion}. \subsubsection{Polaris} The Polaris Flare has a distance of about 352 pc and total mass of about 5500 M$_\odot$ \citep{2019ApJ...879..125Z}. It is a high-latitude translucent cloud with little or no obvious star formation activity. It is expected to have the lowest level of background cloud emission \citep{1990ApJ...353L..49H}. No OB stars nor YSOs were detected in this region. As shown in Fig.\ref{fig:polaris}, the UV intensity G$_0$ of almost all region is smaller than 10. \subsection{UV Intensity $vs$ N(H$_2$) } \label{subsec:uv_NH2} We presented a statistical result between UV intensity $G_0$ and peak H$_2$ column density N(H$_2$) of each molecular cloud in Fig. \ref{fig:uv_Nh2}. The correlation between $G_0$ and N(H$_2$) can be fitted with a linear function of \begin{equation} \rm log(G_0)=(0.62\pm0.12)log(N(H_2))-(11.56\pm2.87) \>\>,\label{eq:logm1} \end{equation} for all complexes. It becomes \begin{equation} \rm log(G_0)=(0.94\pm0.17)log(N(H_2))-(18.76\pm3.91) \>\>,\label{eq:logm2} \end{equation} when Orion A was not included. \section{Discussion} \label{sec:discussion} In this paper, we obtained the distribution of UV radiation field toward 23 regions of 14 molecular complexes in Gould Belt through dust radiative analysis. \subsection{Uncertainty Analysis} The main assumption of such analysis is that the heating of dust was mainly due to UV radiation from massive stars. This has been demonstrated to be valid in the presence of massive stars, particularly, OB clusters \citep[e.g.,][]{1999ApJ...522..897L, 2003ApJ...587..262L}. In more quiescent regions, however, dynamic feedback through outflows and bubbles from low-mass stars have proven to be capable of sustaining the turbulence in, e.g.~Taurus \citep{2015ApJS..219...20L}, thus presumably may result in some dust heating. However, even suppose the gas and dust are closely coupled, there is no sign of elevated temperatures close to low-mass YSOs in Taurus \citep{2008ApJ...680..428G}. We thus consider the heating of dust from low-mass YSOs to be minor. In testing the model, we find that the incident angle affect more the absolute value of the derived UV field, rather than its distribution. To obtain robust results for the Gould belt sample in a systematic way, we set the incident angle to zero. Further investigation of the individual radiation geometry is warranted. \subsection {Compare with Star Formation Rate} We obtain the total UV fluxes in Aquila, Cham I, Cham II, Cham III, IC5146, Lupus III, Lupus IV, Serpens, Musca, $\rho$ Oph, Orion A, Orion B, Pipe, Perseus, and Taurus by adding up the DUSTY outputs for each region and compare them with the star formation rate in \citet{2010ApJ...724..687L,2014ApJ...782..114E}. UV emission is a direct tracer of the recent SFR since it traces the photospheric emission of young stars. The investigation of SFR of extragalaxies has been revolutionized with the observations of GALEX telescope \citep{2005ApJ...619L...1M}. By combining the UV data and Initial Mass Function (IMF), the relationship between SFR and UV luminosity \citep{2011ASPC..446...63H, 2012ARA&A..50..531K} can be described as \begin{equation} \rm log(\dot{M}_{\odot})=log(L_{\nu})-log(C) \>\>,\label{eq:logm} \end{equation} where $\dot{M_{\odot}}$ is SFR in unit of 10$^{-6}M_{\odot}$ year$^{-1}$, L$_{\nu}$(FUV) is FUV luminosity in unit of ergs s$^{-1}$ , log(C)= 43.35 is conversion constant. The relationship between UV intensity and SFR for 15 molecular regions of Gould Belt is presented in Fig. \ref{fig:sr-intensity}. The blue line denotes the correlation between UV flux by OB stars and star formation rate. The correlation conforms to the general expectation. Regions with prominent OB clusters tend to be more consistent with the expectation. The scatter is bigger where there is no OB star. Different regions in the same molecular cloud tend to have similar UV fields, but different star formation rate. The less massive YSOs seem to have little effect on the UV distribution. \section{Summary} \label{sec:summary} By interpreting dust continuum data through radiative transfer analysis, we obtained UV intensity distribution toward 23 molecular regions in the Gould Belt. The main results of this study are summarized as follows, \begin{enumerate} \item The UV intensity G$_0$ of molecular clouds ranges from 1 to over 1000, relative to the Habing interstellar field. \item The UV distribution in the majority of the molecular regions shows a tight correlation with that of OB stars and/or YSOs. \item The UV intensity of 10 molecular regions conforms to an expected linear correlation with the SFR. \end{enumerate} \normalem \begin{acknowledgements} This work is supported by the National Natural Science Foundation of China (NSFC) grant No. 11988101, No. 11725313, No. 11721303, No. U1731238, the International Partnership Program of Chinese Academy of Sciences grant No.114A11KYSB20160008, the National Key R\&D Program of China No. 2016YFA0400702, the Guizhou Provincial Science and Technology Foundation (Nos. [2016]4008, [2017]5726-37,), the Foundation of Guizhou Provincial Education Department (No. KY (2020) 003). This research was carried out in part at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research made use of {\sc APLpy}, an open-source plotting package for Python\citep{aplpy2012,aplpy2019}. \end{acknowledgements} \appendix \section{UV Distribution of the Gould Belt Complexes} \subsection{Cepheus} Located at a high declination, the Cepheus molecular complex includes many regions that have loose association with compact dark clouds. HGBS contains 5 regions in Cepheus molecular complex, Cep 1157 (also known as L1157, the same below), Cep 1172, Cep 1228, Cep 1241 and Cep 1251. The distance of the five regions are considered to be 200-300 pc. The masses of L1157, L1172, L1228, L1241, L1251 are 1400 M$_{\odot}$, 1900 M$_{\odot}$, 1600 M$_{\odot}$, 3200 M$_{\odot}$ and 1800 M$_{\odot} $ \citep{2020ApJ...904..172D}. L1172 is the host of the bright NGC 7023 reflection nebula, which contains the bright B star, HD200775 in these five regions. L1241 and L1251 lie within the Cepheus Flare Shell. L1241 is the only one region without YSOs nor OB star. Some YSOs are found in L1251. The L1228 locates at the edge of the Cepheus Flare Shell, while L1157 and L1172 locate outside to the Cepheus Flare Shell. The radiation field distribution of these five regions are quite different depending on the existence of massive stars and YSOs. As shown in Fig.\ref{fig:l1172}, the UV intensity of L1172 around HD 200775 can exceed 1000 G$_0$. For the L1157 ( Fig. \ref{fig:l1157}), L1228 (Fig.\ref{fig:l1172}) and L1251 (Fig.\ref{fig:l1251}) regions, obvious higher UV intensity can be found around the YSOs. The UV intensity in L1241 is smaller than 31.6 G$_0$ while there are no massive stars nor YSOs in this region. \subsection{Chamaleon} Chamaleon is a nearby low-mass star-forming region containing Cham I, Cham II and Cham III. The distance from the regions in Chamaleon is about 150 pc \citep{2014ApJ...782..114E}. The entire Cham I region covers an area about ~5 deg{$^2$}, containing about 200 known low-mass YSOs, making it one of the closest and richest star-forming regions. The total mass of this cloud is about 482 M$_{\odot}$, one third of which are dense gas \citep{2014ApJ...782..114E}. Though there is abundant low-mass YSOs in Cham I, there is only one B star: HD 97300 in the northern part of the cloud \citep{2012A&A...545A.145W}. Cham II region contains a smaller number($\sim$60) of YSOs compared to the Cha I region. The size of Cham II is about 1.78\,pc. The total mass of Cham II is about 637 M$_{\odot}$ while one tens is dense gas. As the largest cloud of the three regions with a total mass of 746 M$_{\odot}$, Cham III contain little dense gas and only a few YSOs\citep{2014A&A...568A..98A}. Due to the differences in containing YSOs and dense gas, the UV intensity distribution of Cham I, Cham II and Cham III are expected to vary significantly. As shown in Fig. \ref{fig:cham1}, \ref{fig:cham2}, and \ref{fig:cham3}, the maximum UV intensity value decrease from Cham I to Cham III, which is proportional with the existence of OB stars. \subsection{CraNS} With distance of around 130\,pc and being out of the Galactic plane, the CraNS (Corona Australias) molecular cloud is a low-mass star-forming region. The total mass of this region is about 279 M$_{\odot}$, half of which is dense gas \citep{2014ApJ...782..114E}. There is an OB star(V$^*$ R CrA) and a cluster of YSOs in this region \citep{2018A&A...615A.125B}. As shown in Fig. \ref{fig:crans}, the derived UV intensity G$_0$ around the OB star can exceed 1000. \subsection{IC 5146} With a distance of $\sim$ 460 pc, IC 5146 region covers an area of $\sim$ 3.1 $\times$ 2.5 deg{$^2$}. The total mass of this cloud is about 3.7 $\times$ 10 $^3$ M{$_ \odot $} \citep{2019A&A...621A..42A}. There is little dense gas nor OB star in this region \citep{2014ApJ...782..114E}. Abundant YSOs exist in this region. We present UV intensity distribution for IC 5146 in Fig. \ref{fig:ic5146}. The UV intensity G$_0$ can reach 31.6 around the YSOs. \subsection{Lupus} The distance of Lupus molecular complex is about 189 pc \citep{2019ApJ...879..125Z}. The HGBS surveyed three clouds in this complex: Lupus I, Lupus III and Lupus IV cloud. Among the three clouds, the Lupus I cloud is the youngest one. The mass of Lupus I is about 512 M$_{\odot}$. Lup III is the most evolved cloud with mass of $\sim$ 912 M$_{\odot}$. Lupus IV cloud has a middle property between Lupus I and III cloud. The mass of dense gas in Lupus IV cloud is 50 M$_{\odot}$, accounting for about one quarter of the cloud mass \citep{2014ApJ...782..114E}. These three clouds contain YSOs but there are no massive stars inside. As shown in Fig \ref{fig:lupus1}, \ref{fig:lupus3} and \ref{fig:lupus4}, the UV intensity distribution correlate with the existences of YSOs. \subsection{Musca} With distance of $\sim$ 200 pc, the Musca cloud is a 10.5 pc long filament with low-mass star formation \citep{2016A&A...590A.110C}. The mass of Musca molecular cloud is about 335 M$_{\odot}$ \citep{2014ApJ...782..114E}. There are no massive stars in this region. It is clear from Fig. \ref{fig:musca}, UV intensity increases in some dense regions. \subsection{$\rho$ Oph} With distance of 125\,pc \citep{2014ApJ...782..114E}, the $\rho$ Oph molecular cloud is one of the most conspicuous nearby regions where low and intermediate-mass star formation is taking place \citep{1992lmsf.book..159W}. The total mass of the $\rho$ Oph cloud is about 3128 M$_{\odot}$, one third of which is dense gas. The $\rho$ Oph cloud consists of two massive, centrally condensed cores, L1688 and L1689 \citep{1989ApJ...338..902L}. Being different from L1689 with little star formation activity, L1688 harbors a rich cluster of YSOs at various evolutionary stages and is distinguished by high star-formation efficiency \citep{1983ApJ...274..698W}. Two OB stars(HD 147889 and $\rho$ Oph A) are found in this region. The UV radiation distribution of $\rho$ Oph cloud is shown in Fig. \ref{fig:rho Oph}. A strong correlation between UV intensity and star distribution was found. The UV intensity G$_0$ can exceed 1000 in dense gas region and regions around OB stars. \subsection{Perseus} The Perseus molecular clouds is $\sim$ 250 pc away with sky coverage of $\sim$ 10 deg{$^2$}. It is a low and intermediate-mass star-forming region. The total mass of the Perseus molecular cloud is about 6586 M$_{\odot}$, one third of which is dense gas \citep{2014ApJ...782..114E}. As shown in Fig. \ref{fig:perseus}, UV radiation field correlates with the locations of OB stars and YSOs in this region. \subsection{Pipe} The Pipe Nebula has a distance of $\sim$ 145 pc \citep{2007A&A...470..597A}. Composed of an elongated dark cloud with length of 18 pc, the Pipe Nebula is one of the closest star-forming regions. The Pipe Nebula is an ideal target for investigating core formation. The mass of the Pipe nebula is about 1.7 $\times$ 10$^4$ M$_{\odot}$ \citep{ 2006A&A...454..781L}. A few identified YSOs were found in this region\citep{2012A&A...541A..63P}. As shown in Fig.\ref{fig:pipe}, the UV intensity is very low ($G_0<$ 31.6) toward most region of this molecular complex. \subsection{Serpens} The Serpens star-forming region located at $\sim$ 429 pc was covered in $\sim$ 15 deg{$^2$} by HGBS \citep{2014ApJ...782..114E}. Its total mass is $\sim$ 6583 M$_{\odot}$, two thirds of which is dense gas \citep{2014ApJ...782..114E}. About 81 percent of the prestellar cores are found in the filamentary structure of Serpens \citep{2021MNRAS.500.4257F}. Serpens is confirmed to be a low-mass and active star-forming region at a young age. As shown in Fig. \ref{fig:serpens}, lots of YSOs and an OB star were found in this region. \subsection{Taurus} The distance of Taurus cloud to our solar system is about 140\,pc \citep{2013A&A...550A..38P}. The total mass of Taurus molecular cloud is about 2-4 $\times$ 10$^4$ M$_{\odot}$, ten percent of which is dense gas \citep{2014ApJ...782..114E}. HGBS covered about 52 deg$^2$ of this region \citep{2013MNRAS.432.1424K}. No OB stars was found in the Taurus molecular cloud. As shown in Fig. \ref{fig:taurus}, the UV radiation intensity correlates with the distribution of dense gas.

Title: Comparing the Performance of a Solar Wind model from the Sun to 1 AU using Real and Synthetic Magnetograms

Abstract: The input of the Solar wind models plays a significant role in accurate solar wind predictions at 1 AU. This work introduces a synthetic magnetogram produced from a dynamo model as an input for Magnetohydrodynamics (MHD) simulations. We perform a quantitative study that compares the Space Weather Modeling Framework (SWMF) results for the observed and the synthetic solar magnetogram input. For each case, we compare the results for Extreme Ultra-Violet (EUV) images and extract the simulation data along the earth trajectory to compare with in-situ observations. We initialize SWMF using the real and synthetic magnetogram for a set of Carrington Rotations (CR)s within the solar cycle 23 and 24. Our results help quantify the ability of dynamo models to be used as input to solar wind models and thus, provide predictions for the solar wind at 1 AU.

https://export.arxiv.org/pdf/2208.13668

\thispagestyle{plain} \newcommand{\btx}{\textsc{Bib}\TeX} \newcommand{\thestyle}{\texttt{\filename}} \begin{center}{\bfseries\Large Reference sheet for \thestyle\ usage}\\ \large(Describing version \fileversion\ from \filedate) \end{center} \begin{quote}\slshape For a more detailed description of the \thestyle\ package, \LaTeX\ the source file \thestyle\texttt{.dtx}. \end{quote} \head{Overview} The \thestyle\ package is a reimplementation of the \LaTeX\ |\cite| command, to work with both author--year and numerical citations. It is compatible with the standard bibliographic style files, such as \texttt{plain.bst}, as well as with those for \texttt{harvard}, \texttt{apalike}, \texttt{chicago}, \texttt{astron}, \texttt{authordate}, and of course \thestyle. \head{Loading} Load with |\usepackage[|\emph{options}|]{|\thestyle|}|. See list of \emph{options} at the end. \head{Replacement bibliography styles} I provide three new \texttt{.bst} files to replace the standard \LaTeX\ numerical ones: \begin{quote}\ttfamily plainnat.bst \qquad abbrvnat.bst \qquad unsrtnat.bst \end{quote} \head{Basic commands} The \thestyle\ package has two basic citation commands, |\citet| and |\citep| for \emph{textual} and \emph{parenthetical} citations, respectively. There also exist the starred versions |\citet*| and |\citep*| that print the full author list, and not just the abbreviated one. All of these may take one or two optional arguments to add some text before and after the citation. \begin{quote} \begin{tabular}{l@{\quad$\Rightarrow$\quad}l} |\citet{jon90}| & Jones et al. (1990)\\ |\citet[chap.~2]{jon90}| & Jones et al. (1990, chap.~2)\\[0.5ex] |\citep{jon90}| & (Jones et al., 1990)\\ |\citep[chap.~2]{jon90}| & (Jones et al., 1990, chap.~2)\\ |\citep[see][]{jon90}| & (see Jones et al., 1990)\\ |\citep[see][chap.~2]{jon90}| & (see Jones et al., 1990, chap.~2)\\[0.5ex] |\citet*{jon90}| & Jones, Baker, and Williams (1990)\\ |\citep*{jon90}| & (Jones, Baker, and Williams, 1990) \end{tabular} \end{quote} \head{Multiple citations} Multiple citations may be made by including more than one citation key in the |\cite| command argument. \begin{quote} \begin{tabular}{l@{\quad$\Rightarrow$\quad}l} |\citet{jon90,jam91}| & Jones et al. (1990); James et al. (1991)\\ |\citep{jon90,jam91}| & (Jones et al., 1990; James et al. 1991)\\ |\citep{jon90,jon91}| & (Jones et al., 1990, 1991)\\ |\citep{jon90a,jon90b}| & (Jones et al., 1990a,b) \end{tabular} \end{quote} \head{Numerical mode} These examples are for author--year citation mode. In numerical mode, the results are different. \begin{quote} \begin{tabular}{l@{\quad$\Rightarrow$\quad}l} |\citet{jon90}| & Jones et al. [21]\\ |\citet[chap.~2]{jon90}| & Jones et al. [21, chap.~2]\\[0.5ex] |\citep{jon90}| & [21]\\ |\citep[chap.~2]{jon90}| & [21, chap.~2]\\ |\citep[see][]{jon90}| & [see 21]\\ |\citep[see][chap.~2]{jon90}| & [see 21, chap.~2]\\[0.5ex] |\citep{jon90a,jon90b}| & [21, 32] \end{tabular} \end{quote} \head{Suppressed parentheses} As an alternative form of citation, |\citealt| is the same as |\citet| but \emph{without parentheses}. Similarly, |\citealp| is |\citep| without parentheses. Multiple references, notes, and the starred variants also exist. \begin{quote} \begin{tabular}{l@{\quad$\Rightarrow$\quad}l} |\citealt{jon90}| & Jones et al.\ 1990\\ |\citealt*{jon90}| & Jones, Baker, and Williams 1990\\ |\citealp{jon90}| & Jones et al., 1990\\ |\citealp*{jon90}| & Jones, Baker, and Williams, 1990\\ |\citealp{jon90,jam91}| & Jones et al., 1990; James et al., 1991\\ |\citealp[pg.~32]{jon90}| & Jones et al., 1990, pg.~32\\ |\citetext{priv.\ comm.}| & (priv.\ comm.) \end{tabular} \end{quote} The |\citetext| command allows arbitrary text to be placed in the current citation parentheses. This may be used in combination with |\citealp|. \head{Partial citations} In author--year schemes, it is sometimes desirable to be able to refer to the authors without the year, or vice versa. This is provided with the extra commands \begin{quote} \begin{tabular}{l@{\quad$\Rightarrow$\quad}l} |\citeauthor{jon90}| & Jones et al.\\ |\citeauthor*{jon90}| & Jones, Baker, and Williams\\ |\citeyear{jon90}| & 1990\\ |\citeyearpar{jon90}| & (1990) \end{tabular} \end{quote} \head{Forcing upper cased names} If the first author's name contains a \textsl{von} part, such as ``della Robbia'', then |\citet{dRob98}| produces ``della Robbia (1998)'', even at the beginning of a sentence. One can force the first letter to be in upper case with the command |\Citet| instead. Other upper case commands also exist. \begin{quote} \begin{tabular}{rl@{\quad$\Rightarrow$\quad}l} when & |\citet{dRob98}| & della Robbia (1998) \\ then & |\Citet{dRob98}| & Della Robbia (1998) \\ & |\Citep{dRob98}| & (Della Robbia, 1998) \\ & |\Citealt{dRob98}| & Della Robbia 1998 \\ & |\Citealp{dRob98}| & Della Robbia, 1998 \\ & |\Citeauthor{dRob98}| & Della Robbia \end{tabular} \end{quote} These commands also exist in starred versions for full author names. \head{Citation aliasing} Sometimes one wants to refer to a reference with a special designation, rather than by the authors, i.e. as Paper~I, Paper~II. Such aliases can be defined and used, textual and/or parenthetical with: \begin{quote} \begin{tabular}{lcl} |\defcitealias{jon90}{Paper~I}|\\ |\citetalias{jon90}| & $\Rightarrow$ & Paper~I\\ |\citepalias{jon90}| & $\Rightarrow$ & (Paper~I) \end{tabular} \end{quote} These citation commands function much like |\citet| and |\citep|: they may take multiple keys in the argument, may contain notes, and are marked as hyperlinks. \head{Selecting citation style and punctuation} Use the command |\bibpunct| with one optional and 6 mandatory arguments: \begin{enumerate} \item the opening bracket symbol, default = ( \item the closing bracket symbol, default = ) \item the punctuation between multiple citations, default = ; \item the letter `n' for numerical style, or `s' for numerical superscript style, any other letter for author--year, default = author--year; \item the punctuation that comes between the author names and the year \item the punctuation that comes between years or numbers when common author lists are suppressed (default = ,); \end{enumerate} The optional argument is the character preceding a post-note, default is a comma plus space. In redefining this character, one must include a space if one is wanted. Example~1, |\bibpunct{[}{]}{,}{a}{}{;}| changes the output of \begin{quote} |\citep{jon90,jon91,jam92}| \end{quote} into [Jones et al. 1990; 1991, James et al. 1992]. Example~2, |\bibpunct[; ]{(}{)}{,}{a}{}{;}| changes the output of \begin{quote} |\citep[and references therein]{jon90}| \end{quote} into (Jones et al. 1990; and references therein). \head{Other formatting options} Redefine |\bibsection| to the desired sectioning command for introducing the list of references. This is normally |\section*| or |\chapter*|. Define |\bibpreamble| to be any text that is to be printed after the heading but before the actual list of references. Define |\bibfont| to be a font declaration, e.g.\ |\small| to apply to the list of references. Define |\citenumfont| to be a font declaration or command like |\itshape| or |\textit|. Redefine |\bibnumfmt| as a command with an argument to format the numbers in the list of references. The default definition is |[#1]|. The indentation after the first line of each reference is given by |\bibhang|; change this with the |\setlength| command. The vertical spacing between references is set by |\bibsep|; change this with the |\setlength| command. \head{Automatic indexing of citations} If one wishes to have the citations entered in the \texttt{.idx} indexing file, it is only necessary to issue |\citeindextrue| at any point in the document. All following |\cite| commands, of all variations, then insert the corresponding entry to that file. With |\citeindexfalse|, these entries will no longer be made. \head{Use with \texttt{chapterbib} package} The \thestyle\ package is compatible with the \texttt{chapterbib} package which makes it possible to have several bibliographies in one document. The package makes use of the |\include| command, and each |\include|d file has its own bibliography. The order in which the \texttt{chapterbib} and \thestyle\ packages are loaded is unimportant. The \texttt{chapterbib} package provides an option \texttt{sectionbib} that puts the bibliography in a |\section*| instead of |\chapter*|, something that makes sense if there is a bibliography in each chapter. This option will not work when \thestyle\ is also loaded; instead, add the option to \thestyle. Every |\include|d file must contain its own |\bibliography| command where the bibliography is to appear. The database files listed as arguments to this command can be different in each file, of course. However, what is not so obvious, is that each file must also contain a |\bibliographystyle| command, \emph{preferably with the same style argument}. \head{Sorting and compressing citations} Do not use the \texttt{cite} package with \thestyle; rather use one of the options \texttt{sort} or \texttt{sort\&compress}. These also work with author--year citations, making multiple citations appear in their order in the reference list. \head{Long author list on first citation} Use option \texttt{longnamesfirst} to have first citation automatically give the full list of authors. Suppress this for certain citations with |\shortcites{|\emph{key-list}|}|, given before the first citation. \head{Local configuration} Any local recoding or definitions can be put in \thestyle\texttt{.cfg} which is read in after the main package file. \head{Options that can be added to \texttt{\char`\\ usepackage}} \begin{description} \item[\ttfamily round] (default) for round parentheses; \item[\ttfamily square] for square brackets; \item[\ttfamily curly] for curly braces; \item[\ttfamily angle] for angle brackets; \item[\ttfamily colon] (default) to separate multiple citations with colons; \item[\ttfamily comma] to use commas as separaters; \item[\ttfamily authoryear] (default) for author--year citations; \item[\ttfamily numbers] for numerical citations; \item[\ttfamily super] for superscripted numerical citations, as in \textsl{Nature}; \item[\ttfamily sort] orders multiple citations into the sequence in which they appear in the list of references; \item[\ttfamily sort\&compress] as \texttt{sort} but in addition multiple numerical citations are compressed if possible (as 3--6, 15); \item[\ttfamily longnamesfirst] makes the first citation of any reference the equivalent of the starred variant (full author list) and subsequent citations normal (abbreviated list); \item[\ttfamily sectionbib] redefines |\thebibliography| to issue |\section*| instead of |\chapter*|; valid only for classes with a |\chapter| command; to be used with the \texttt{chapterbib} package; \item[\ttfamily nonamebreak] keeps all the authors' names in a citation on one line; causes overfull hboxes but helps with some \texttt{hyperref} problems. \end{description}

Title: Bremsstrahlung from the Cosmic Neutrino Background

Abstract: In this paper we discuss a detection method for the Cosmic Neutrino Background using bremsstrahlung from a neutrino scattering process which has no kinematic threshold, does not rely on a resonance and would in principle allow to measure the velocity distribution of the relic neutrinos. As a concrete example we calculate the rate for solar neutrinos scattering from a relic neutrino emitting a photon. We also provide the energy and angular distributions of the emitted photons.

https://export.arxiv.org/pdf/2208.01207

\title{Bremsstrahlung from the Cosmic Neutrino Background} \author{Konstantin Asteriadis}% \email{kasteriad@bnl.gov} \affiliation{% High Energy Theory Group, Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA }% \author{Alejandro Quiroga Trivi\~no}% \email{alejandro.quiroga@gapp.nthu.edu.tw} \affiliation{% Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan }% \author{Martin Spinrath}% \email{spinrath@phys.nthu.edu.tw} \affiliation{% Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan }% \affiliation{% Center for Theory and Computation, National Tsing Hua University, Hsinchu, 30013, Taiwan }% \section{Introduction} The direct observation of the Cosmic Neutrino Background (CNB) in a laboratory remains one of the great challenges of experimental particle cosmology. Given the tiny cross sections and energies of CNB neutrinos it was even called an ``apparently impossible experiment''~\cite{Melissinos:1999ew}. Some recent proposals include detecting the tiny force induced by the CNB ``wind'' using current gravitational wave detector technology~\cite{Domcke:2017aqj,Shergold:2021evs}, resonant scattering against ultra-high energetic cosmic neutrinos~\cite{Brdar:2022kpu}, cosmic birefringence induced by the CNB \cite{Mohammadi:2021xoh} and the absorption of CNB neutrinos on tritium~\cite{PTOLEMY:2018jst, Betts:2013uya}. The last is probably the most promising proposal at this time, see, e.g.~Refs~\cite{Gelmini:2004hg,Ringwald:2009bg,Vogel:2015vfa,Bauer:2022lri} for more comprehensive reviews. Currently proposed methods to detect the CNB often only work above kinematic thresholds, at specific resonances or rely on non-standard cosmologies. Here we study the feasibility of a method that has no such restrictions. We consider scattering between the two largest natural neutrino fluxes on earth: solar neutrinos and CNB neutrinos. Since the scattered neutrinos would still be hard to detect, we consider an additional bremsstrahlung photon in the final state that is comparatively easy to detect and that can be produced at any energy. In the Standard Model of particle physics (SM) including Dirac neutrino masses neutrinos couple to photons via loop induced magnetic dipole moments. The neutrino magnetic moment is tiny and so is the cross section for the considered process. For Majorana neutrinos, however, the magnetic moment is exactly zero. In this case, the scattering cross section is still non-zero if one considers transition magnetic moments, see, e.g.~Ref.~\cite{Czakon:1998rf}, and we expect the cross sections to be similar to the (simpler) Dirac case. Measuring this process would be more conceivable if the neutrino magnetic moment would be substantially enhanced by some new physics. We, therefore, consider an enhanced neutrino magnetic moment, slightly below the current upper bound, but still orders of magnitude larger than in the SM. \section{Bremsstrahlung from Neutrino-neutrino scattering} We study the processes \begin{equation} \label{eq:processes} \nu_\odot \; + \stackrel{\brabar}{\nu}_{\text{CNB}} \to \nu \; +\stackrel{\brabar}{\nu} + \; \gamma \end{equation} at leading order, neglecting the exchanged momentum with respect to the $Z$-boson mass in the propagators. Here $\nu_\odot$ is a solar neutrino and $\nu_{\text{CNB}}$ and $\bar{\nu}_{\text{CNB}}$ are relic neutrinos and anti-neutrinos, respectively. In the SM and standard cosmology the CNB is expected to consist of neutrinos and anti-neutrinos to equal parts and we assume them to be left-helical today, cf.~Ref.~\cite{Long:2014zva}. The solar neutrinos are assumed to be purely left-chiral and for the sake of simplicity we will neglect any flavor effects. That means we treat solar and CNB neutrinos to consist of only one flavor. We consider massive Dirac neutrinos with a mass set to $0.05$~eV that is compatible with current limits. Hence, the CNB neutrinos are non-relativistic and we can neglect their velocity in our calculation. Interestingly though, non-vanishing CNB velocities would lead to corrections to our result which would, in principle, allow to measure their velocity distribution. In our setup, the photons couple to the neutrinos via an effective magnetic dipole moment with the effective Lagrangian~\cite{Giunti:2014ixa} \begin{align} \mathcal{L}_{\text{eff}} = - \ci M_\nu \, \bar{\nu} \, \sigma_{\alpha \beta} \, q^\alpha \nu \, A^\beta \;, \end{align} where $\sigma_{\alpha \beta}$ is the anti-symmetric combination of $\gamma$-matrices, $q^\alpha$ is the momentum carried away by the photon field $A^\beta$, and $M_\nu$ is the magnetic moment of the neutrino. The coupling then reads \begin{align} \label{eq:vertex} \vcenter{\hbox{\includegraphics[scale=.41,trim=25 0 0 0,clip]{nna}}} \widehat{=} \hspace{7pt} -\frac{\ci}{2} ( \gamma_\beta \slashed{q} - \slashed{q} \gamma_\beta) M_\nu \;. \end{align} In the SM the coupling~Eq.~\eqref{eq:vertex} occurs for Dirac neutrinos via loops with an effective coupling constant of $M_\nu^\textrm{SM} \lesssim 3.8 \times 10^{-19} \mu_B$~\cite{Giunti:2014ixa}. Here we assume it to be additionally enhanced by some new physics. The parameter $M_\nu$ is experimentally constrained to be $M_\nu < 0.28 \times 10^{-10} \, \mu_B$ at 90\%~CL~\cite{ParticleDataGroup:2020ssz}. Accordingly, we write $M_\nu = f_M \times 10^{-11} \, \mu_B$ for some $f_M \lesssim 1$. Note that, for Majorana neutrinos $f_M = 0$, but considering multiple flavours and transition magnetic moments instead would also imply the existence of the considered process with expected results similar to the case of Dirac neutrinos. Neutrinos can also have other electromagnetic moments. For instance, they could have a tiny electric charge. However, the current upper bound is so low that these contributions should be orders of magnitude smaller (even considering possible enhancements close to infrared singularities). For this reason, and for simplicity, we will neglect such complications here. For this set of assumptions, the relation between the cross sections for CNB neutrinos $\sigma(\nu\nu)$ and CNB anti-neutrinos $\sigma(\nu\bar{\nu})$ and the photon production rate $R$ reads \begin{align} R &= 3 \, n_{\text{CNB}} \int \frac{\diff \Phi_{\odot} }{\diff E_\nu} (\sigma(\nu \nu) + \sigma (\nu \bar{\nu})) \diff E_\nu \;, \end{align} where the local CNB density $n_{\text{CNB}} = f_n \times 56$/cm$^3$. The factor $f_n$ parametrizes potential overdensities which are nevertheless not expected to be very large, see, e.g.\ Ref.~\cite{Ringwald:2004np}. The factor three is from summing over the three flavors of the CNB. Finally, $\diff \Phi_{\odot}/\diff E_\nu$ are the solar neutrino fluxes. In our calculations we consider an observed volume of 1~km$^3$, at an earth-like distance from the sun, isolated from the surroundings (it will still contain CNB photons) and want to see how many photons are produced in this volume within a year and how their energies and angles are distributed. Numerical tables for the solar neutrino fluxes are taken from Ref.~\cite{Vitagliano:2019yzm}. The final state phase space integration is performed numerically and getting the desired distributions is straightforward. \section{Results} In Fig.~\ref{fig:dRdEg} we show the energy distribution of the emitted bremsstrahlung photons, separating the different components of the solar neutrino flux. It follows from this figure that they can be clearly distinguished from each other in many cases. This is quite a unique feature which could potentially be used to separate the bremsstrahlung photons of this process from other potential backgrounds. We also see that the higher energies of the $^8$B and the hep neutrinos, implying larger cross sections, cannot compensate for the much larger flux of the pp neutrinos. We checked numerically that in the relevant energy range the cross section grows quadratically with the incoming neutrino energy to a good approximation. This increase is much weaker than the flux decrease for the high energy solar neutrinos. For that reason we also do not consider other naturally occurring neutrino fluxes such as atmospheric neutrinos which are much smaller than the solar flux~\cite{Vitagliano:2019yzm}. In Fig.~\ref{fig:dRdthg} we show the angular distribution with respect to the direction towards the center of the sun, assuming the incoming neutrino momenta to be parallel. As expected the distribution is very much peaked at $\theta_\gamma = 0$ which is due to the ``fixed target'' nature of our setup and the very large energies of the incoming neutrinos compared to the target neutrino mass. This is another feature which could, in principle, be used to distinguish a signal from potential background photons. The width of the angular distribution is similar to the deviations that would be induced by the neutrino production zones in the sun which have a similar angular diameter in the sky~\cite{Lin:2022jyv}. Therefore, a fully realistic treatment would widen the distribution somewhat but there would still be a very strong directional dependence. Nevertheless, what becomes apparent from both figures is that the expected rates are extremely small. In fact, the total expected rate is about $1.9 \times 10^{-45}$ bremsstrahlung photons per year and km$^3$ of target volume assuming no neutrino overdensities and a neutrino magnetic dipole moment slightly below the current bound, to be precise $f_M = f_n = 1$. That makes a discovery of the CNB in this way rather unlikely as one would need a neutrino beam with significantly higher energies, flux and/or observed volume to get a reasonable rate. The dependence of the rate on the model parameter $f_M$ and $f_n$ is only quadratic and linear, respectively. Increasing the rate by orders of magnitude would also require increasing these parameters by orders of magnitudes which is not expected neither from experiments nor simulations. What might be more promising to improve the rate is to increase the neutrino flux. Given that the biggest neutrino flux on earth are the CNB neutrinos themselves, self-scattering of CNB neutrinos may be an option. Such a process with similar kinematics to the process studied above would be a massless neutrino flavour scattering from a massive one. We can provide a rough estimate for this case. The flux would be roughly a factor 25 larger than the solar neutrino flux. On the other hand the cross section would drop by a factor $m_\nu^2/E_\odot^2 \approx 2.5 \times 10^{-15}$ where we used for the neutrino mass $m_\nu = 0.05$~eV and for the solar neutrino energy $E_\odot \approx 10^6$~eV. This estimate is only an upper bound because the neutrino mass is used as the energy scale of the CNB self-scattering process instead of the much smaller kinetic energy. We conclude from the above numbers that this process would be even more rare compared to the one involving solar neutrinos. This said, this little thought experiment shows how the rates for other sources can be estimated as long as the center of mass energy is below the $Z$-boson mass and no other new physics scenarios are considered. \section{Conclusions} In this work we presented a way to search for neutrinos from the very early universe, the Cosmic Neutrino Background. From our results it is once again clear why this is such a big challenge. We computed the cross section for bremsstrahlung photons produced in the scattering of solar neutrinos on the CNB and, although we chose a magnetic dipole moment which is strongly enhanced compared to the SM and just slightly below the current experimental bound, the obtained cross section is still tiny and somewhat discouraging. Nevertheless, we want to emphasize the principle advantages of our proposal: Our method has no kinematic thresholds and is not related to any resonance. Both computed neutrino energy and angular distributions show clear features from the considered process that could be used to distinguish bremsstrahlung photons from potential background. Furthermore, the velocity distribution of the relic neutrinos would lead to corrections to the photon spectra, which therefore could be measured, at least in theory. These are important benefits compared to other proposals. We also showed how the bremsstrahlung photon rate for other energetic neutrino sources can be easily estimated assuming a similar set of assumptions. Other neutrino beams or sources, non-standard cosmology, additional new physics contributions, a larger target volume and a combination thereof could lead to a rate closer to being measurable. We leave the study of these cases for further investigations. \begin{acknowledgments} The research of KA is supported by the United States Department of Energy under Grant Contract DE-SC0012704. AQT and MS are supported by the Ministry of Science and Technology (MOST) of Taiwan under grant number MOST 110-2112-M-007-018 and MOST 111-2112-M-007-036. \end{acknowledgments} \bibliography{refs.bib}

Title: Evolution of Urca Pairs in the Crusts of Highly Magnetized Neutron Stars

Abstract: We report on the effects of strong magnetic fields on neutrino emission in the modified Urca process. We show that the effect of Landau levels on the various Urca pairs affects the neutrino emission spectrum and leads to an angular asymmetry in the neutrino emission. For low magnetic fields the Landau levels have almost no effect on the cooling. However, as the field strength increases, the electron chemical potential increases resulting in a lower density at which Urca pairs can exist. For intermediate field strength there is an interesting interference between the Landau level distribution and the Fermi distribution. For high enough field strength, the entire electron energy spectrum is eventually confined to single Landau level producing dramatic spikes in the emission spectrum.

https://export.arxiv.org/pdf/2208.09573

command. \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \shorttitle{Urca Pairs in Magnetic Fields} \shortauthors{Famiano et al.} \graphicspath{{./}{figures/}} \begin{document} \title{Evolution of Urca Pairs in the Crusts of Highly Magnetized Neutron Stars} \author[0000-0003-2305-9091]{Michael A. Famiano} \email{michael.famiano@wmich.edu} \affiliation{Western Michigan University, Kalamazoo, MI 49008-5252 USA} \affiliation{National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588 Japan} \affiliation{Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements, USA} \author[0000-0002-3164-9131]{Grant Mathews} \email{gmathews@nd.edu} \affiliation{Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements, USA} \affiliation{Center for Astrophysics, Department of Physics and Astronomy, University of Notre Dame, Notre Dame, IN 46556, USA} \author[0000-0002-2999-0111]{A. Baha Balantekin} \email{baha@physics.wisc.edu} \affiliation{National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588 Japan} \affiliation{Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706 USA} \author[0000-0002-8619-359X]{Toshitaka Kajino} \email{kajino@buaa.edu.cn} \affiliation{School of Physics, Beihang University, 37 Xueyuan Road, Haidian-qu, Beijing 100083, China} \affiliation{National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588 Japan} \affiliation{Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033 Japan} \author[0000-0003-3083-6565]{Motohiko Kusakabe} \email{kusakabe@buaa.edu.cn} \affiliation{School of Physics, Beihang University, 37 Xueyuan Road, Haidian-qu, Beijing 100083, China} \author[0000-0003-2595-1657]{Kanji Mori} \email{kanji.mori@fukuoka-u.ac.jp} \affiliation{Research Institute of Stellar Explosive Phenomena, Fukuoka University,\\ 8-19-1 Nanakuma, Jonan-ku, Fukuoka-shi, Fukuoka 814-0180, Japan} \keywords{PPISN -- PISN -- black holes -- massive stars -- nuclear physics} \section{Introduction} It is by now widely accepted that soft gamma-ray repeaters (SGRs) and anomalous X-ray pulsars (AXPs) correspond to a class of neutron stars known as magnetars. These objects are warm, isolated, and slowly rotating neutron stars of age $\sim 10^5$ yr with unusually strong surface magnetic fields. Indeed, both pulsars and magnetars have strong magnetic fields at their surface that can be as large as $10^{12}$ to $10^{16}$ G \citep{Kouveliotou98, Turolla15}. Moreover, the interior field can be as large as the equipartition limit of order $10^{18}$ G \citep{Lai91,Chanmugam92}. Among possible explanations for the cooling of magnetars, % decay processes leading to neutrino or anti-neutrino emission such as the direct Urca (DU) process ($n \rightarrow p + e^{-} + {\bar \nu}_e$, $p + e^{-} \rightarrow n + \nu_e$), the modified Urca (MU) process ($n + N \rightarrow p + e^{-} + N^{\prime} + {\bar \nu}_e$, $p + e^{-} + N \rightarrow n + N^{\prime} + {\nu}_e$ ) \citep{haensel94,yakovlev95,yakovlev01}, or the neutrino-pair emission process ($N_1+ N_2 \rightarrow N_1^{\prime} + N_2^{\prime} + \nu + {\bar \nu}$) are possible. Early in the cooling process the DU process is a viable candidate to explain the rapid cooling of NSs \citep{Boguta81,LPPH91,Maruyama}. However, in this paper we are more concerned with the MU process. The DU process takes place at high temperature and density and for certain ratios of constituents. However, under less extreme conditions when the DU process is diminished, the MU process can continue to occur. In particular, the Urca cycles of electron capture and $\beta$ decay on pair nuclei at a depth of about 150 m in neutron stars were shown to occur in \cite{schatz14} although this mechanism had been previously discussed in the context of white dwarfs~\citep{tsuruta70}, Type Ia supernovae~\citep{paczynski72,woosley86} and electron-degenerate supernovae~\citep{jones2013}. The Urca cycle operates to cool the outer neutron star crust by emitting neutrinos while also thermally decoupling the surface layers from the deeper crust. This cooling eliminates the possibility that interior heating produces the unexpectedly short recurrence times of energetic thermonuclear bursts on neutron stars. The Urca cycles also indicate that the ignition scenario of superbursts by $^{12}$C+$^{12}$C fusion reaction, whose reaction rate is severely limited in recent theoretical studies~\citep{mori2019}, would require another heat source because of higher neutrino emissivity for cooling. In previous studies, however, the effects of possible strong magnetic fields on the modified Urca process have not been considered. Here we show that the appearance of Landau levels for electrons experiencing strong magnetic fields significantly alters the operation of the various Urca pairs proposed by \cite{schatz14} and affects the neutrino emission spectrum leading to an angular asymmetry in the neutrino emissivity. For low magnetic fields the Landau levels have almost no effect on the cooling. However, as the field strength increases, the electron chemical potential increases resulting in a lower density at which Urca pairs can exist. For intermediate field strength there is an interesting interference between the Landau level distribution and the Fermi distribution. For high enough field strength, eventually the entire electron energy spectrum is confined to single Landau level producing dramatic spikes in the emission spectrum. This paper is organized as follows: The ingredients of the model for the MU are summarized in Section 2. The results are presented in Section 3. Our discussion and conclusions are in Section 4. \section{The Model} \subsection{Weak Interaction Rates in External Fields} In a homogeneous plasma at a given temperature and density, the electron chemical potential will change in the presence of a strong magnetic field. This will ultimately change the effectiveness of a particular Urca pair for a given environment. If a dipole field is assumed, then the electron chemical potential, which depends on the magnetic field, will in turn depend on the angular position with respect to the magnetic pole for a constant density and temperature within the crust. The electron number density with the electron transverse momentum components constrained to Landau levels is \citep{famiano20,grasso01,kawasaki12}: \begin{eqnarray} \label{derv} n_e = n_- - n_+ &= \frac{1}{4\pi^2}\int_{0}^{\infty}d^3p {\left( \left[{\exp{\left(\frac{E-\mu}{T}\right)}+1}\right]^{-1} - \left[{\exp{\left(\frac{E+\mu}{T}\right)}+1}\right]^{-1} \right)}\\\nonumber &= \int_{0}^{\infty}dp_xdp_ydp_z {\left( \left[{\exp{\left(\frac{E-\mu}{T}\right)}+1}\right]^{-1} - \left[{\exp{\left(\frac{E+\mu}{T}\right)}+1}\right]^{-1} \right)}\\\nonumber &=\frac{eB}{2\pi^2}\sum\limits_{n=0}^{\infty}g_n\int_{0}^{\infty}dp_z \left( \left[{\exp{\left(\frac{\sqrt{p_z^2 + m_e^2 + 2n eB}-\mu}{T}\right)}+1}\right]^{-1} \right. \\\nonumber &- \left.\left[{\exp{\left(\frac{\sqrt{p_z^2 + m_e^2 + 2n eB}+\mu}{T}\right)}+1}\right]^{-1} \right) \end{eqnarray} where $g_n$ is the degeneracy of individual Landau levels $n$: \begin{equation} g_n = \left\lbrace \begin{array}{cl} 1, & n=0\\ 2, & n>0 \end{array} \right. . \end{equation} Natural units are used in this manuscript ($k=\hbar=c=1)$. Weak interaction rates are computed as in \cite{famiano20} and \cite{arcones10}. In the presence of an external field, the electron energy density is reconfigured by the presence of the field such that the components of the electron momentum perpendicular to the external magnetic field, $p_\perp^2=p_x^2+p_y^2$, are placed into individual levels, $p_{\perp,n}^2 = neB$. This quantization results in a shift in the electron chemical potential, $\mu_e$, and ultimately the weak interaction rates \citep{luo20,grasso01,fassio69}. At low fields, $B\lesssim B_c$, where $B_c\sim {m_e^2}/{e}=4.4\times 10^{13}$ G, the Fermi distribution of the electrons is very similar to the distribution in which B=0. In this region, the spacing between individual Landau levels is small, and the phase space distributions in Equation (\ref{derv}) between the magnetized plasma and the non-magnetized plasma are similar \citep{luo20,grasso01,fassio69}: \begin{equation} dn \propto \frac{d^3p}{(2\pi)^3} = \sum\limits_{n=0}^{\infty}\left(2 - \delta_{n0}\right)\frac{eB}{2\pi^2}dp_z. \end{equation} From this, the Fermi-Dirac distribution for the $n^{th}$ Landau level is rewritten: \begin{equation} f_{FD}(E,\mu_e) = \frac{1}{\exp{\left[\frac{\sqrt{E^2 + 2neB}-\mu_e}{T}\right]}+1}. \end{equation} Following the prescription of \cite{arcones10} and \cite{famiano20}, the weak interaction rates are approximated as: \begin{eqnarray} \label{mag_beta_minus} \Gamma_{\beta^-} & = &\kappa \frac{eB}{2} \sum\limits_{n=0}^{N_{max}}\left(2-\delta_{n0}\right)\int\limits_{\omega_\beta}^{Q}\frac{E(Q-E)^2}{\sqrt{E^2 - m_e^2-2neB}}\left(1-f_{FD}(E,\mu_e)\right)\left(1-f_{FD}(Q-E,-\mu_\nu)\right)dE, \\ \label{mag_beta_plus} \Gamma_{\beta^+} & = &\kappa \frac{eB}{2} \sum\limits_{n=0}^{N_{max}}\left(2-\delta_{n0}\right)\int\limits_{\omega_\beta}^{-Q}\frac{E(-Q-E)^2}{\sqrt{E^2 - m_e^2-2neB}}\left(1-f_{FD}(E,-\mu_e)\right)\left(1-f_{FD}(-Q-E,-\mu_\nu)\right)dE, \\ \label{mag_beta_EC} \Gamma_{EC} & = &\kappa \frac{eB}{2} \sum\limits_{n=0}^{N_{max}}\left(2-\delta_{n0}\right)\int\limits_{\omega_{EC}}^\infty\frac{E(E-Q)^2}{\sqrt{E^2 - m_e^2-2neB}}f_{FD}(E,\mu_e)\left(1-f_{FD}(E-Q,\mu_\nu)\right)dE, \\ \label{mag_beta_PC} \Gamma_{PC} & = &\kappa \frac{eB}{2} \sum\limits_{n=0}^{N_{max}}\left(2-\delta_{n0}\right)\int\limits_{\omega_{PC}}^\infty\frac{E(E+Q)^2}{\sqrt{E^2 - m_e^2-2neB}}f_{FD}(E,-\mu_e)\left(1-f_{FD}(E+Q,-\mu_\nu)\right)dE, \end{eqnarray} in which the following are defined: \begin{eqnarray} \omega_{EC/PC} &\equiv & \mbox{max}\left[\pm Q,m_e\right], \\\nonumber \omega_{\beta} &\equiv & \sqrt{m_e^2+2neB}, \\\nonumber N_{max}&\le &\frac{Q^2 - m_e^2}{2eB}, \\\nonumber \kappa & \equiv & \frac{B\ln 2}{K m_e^5}, \\\nonumber B & \equiv & 1+3g_A^2 = \left\lbrace \begin{array}{ll} 5.76, & \text{nucleons}, \\ 4.6, & \text{nuclei}, \end{array} \right. \\\nonumber K &\equiv & \frac{2\pi^3\hbar^7\ln 2}{G_V^2 m_e^5} = 6144 \mbox{ s} \end{eqnarray} and $Q$ is the nuclear mass difference between the parent and daughter nucleus. In this evaluation, we recognize that the above rates are semi-classical approximations in which nuclear structure and sums over excited states are ignored. We take this approach to examine the overall gross effects of the external field. The rates evaluated above can be adapted to individual sums over individual transitions from parent to daughter states if desired. In this work, we simplify the above to examine ratios of transition rates in an external field to those without a field. In a charge-neutral plasma at temperature $T$ and electron charge density $\rho Y_e/N_A$, the electron chemical potential is calculated from Eq.~(\ref{derv}). The chemical potential is then used in Eqs.~(\ref{mag_beta_minus}) -- (\ref{mag_beta_PC}) to evaluate the modified rates. \subsection{Urca Pairs} For an Urca pair to exist \citep{schatz14}, a pair of nuclei, $^AZ$ and $^A(Z+1)$ must be linked by weak interactions \begin{eqnarray} ^AZ\rightarrow ^A(Z+1)+e^-+\bar{\nu}_e\\\nonumber ^A(Z+1)+e^-\rightarrow ^AZ+\nu_e ~~. \end{eqnarray} The electron energy distribution in the plasma must also be non-degenerate at energies less than the $\beta^-$ decay Q value and non-zero at energies greater than the electron-capture Q value. In other words, the electron phase space must be simultaneously available for both $\beta^-$ decays and electron capture. For a plasma in which this condition is met $\mu_e\approx Q_{\beta^-}=-Q_{EC}$. The finite plasma temperature results in the availability of electron states at energies below Q and electrons occupying states at energies above Q. Under these conditions, the Urca pair will undergo captures and decays, resulting in enhanced cooling via neutrino emission. Captures may occur to low-lying states of the daughter nucleus~\citep{schatz14}. In the absence of a magnetic field, the electron chemical potential is constrained by the temperature, $T$, and the charge density $\rho Y_e$ of the plasma. That is, for a specific temperature, there is only a one-to-one relationship between electron chemical potential and charge density. In fact, there exists a range of charge densities that define an Urca pair because of the finite plasma temperature. The Urca pair is constrained to the region where $Q-kT\lesssim\mu\lesssim Q+kT$. Thus, for a specific temperature, $T$, and electron chemical potential, $\mu$, a range of densities for which an Urca pair exists can be defined, $\rho Y_e(T,\mu-kT)\lesssim \rho Y_e\lesssim \rho Y_e(T,\mu+kT)$. It has been found ~\citep{schatz14,deibel16} that this density range results in a thin layer in which a particular pair can exist. However, this constraining relationship is broken by the introduction of an external magnetic field, $\mu = \mu(T,\rho Y_e, B)$. This may have multiple effects because: \begin{itemize} \item The electron chemical potential depends on the external field, the density at which an Urca pair forms for a particular temperature changes, resulting in a changed emissivity. \item Weak rates may change at various field strengths, changing the neutrino emissivity. \item For a specific T, $\rho Y_e$, and B, new Urca pairs may result from changes in the chemical potential. In addition, existing Urca pairs may by prohibited as the field changes. \item EC rates on exited state nuclei will change with the magnetic field strength. Concurrent changes in the $\beta$-decay rates of the daughter nucleus may result in similar rates between exited states and the existance of Urca pairs which involve formerly inaccessible excited states in nuclei. \item For a non-uniform magnetic field on the surface of a highly-magnetized neutron star, the location of Urca pairs in the ocean and crust of a neutron star may depend on the location on the surface of the star. \end{itemize} For a specific Urca pair, the emissivity and luminosity can be calculated following the method of \cite{deibel16}: Our model for the neutron star and the emissivity in the modified Urca process are directly dependent on the weak interaction rates~\citep{tsuruta70,deibel16}: \begin{equation} \epsilon_{\pm}\approx m_e \Gamma_\pm . \end{equation} In this work, we define the quantity $\epsilon_{22}$: \begin{equation} \epsilon_{22}\equiv \frac{\left(\epsilon_-+\epsilon_+\right)}{10^{22}} \end{equation} The geometric thickness of an Urca layer can be shown to be quite thin. In this case, the luminosity (in units of 10$^{32}$ erg s$^{-1}$ as a function of polar angle, assuming a relativistic correction is: \begin{equation} L_{32}(\theta) \equiv \varepsilon\times2\pi R^2\sin(\theta)\Delta\theta\Delta R/(10^{32} \mbox{erg s}^{-1}) \end{equation} The radial thickness of each zone is calculated using the formulation of \cite{schatz14} and is found to be $\sim$ 1 m. \section{Results} The $\beta^-$ decay rate ratio $\lambda(B)/\lambda(B=0)$ is shown as a function of the magnetic field in Figure \ref{beta_ratios}(a) for $T_9$=0.51, $Y_e$=0.5, and $\rho$= 4$\times 10^{10}$ g cm$^{-3}$. The oscillations in the decay rate occurs with a change in the field as fewer Landau levels contribute to the electron energy spectrum and as Landau levels shift across the spectrum. The decay rate ratio as a function of density is shown in Figure \ref{beta_ratios}(b) for $T_9$=0.51 and B=10$^{14}$ G. While the oscillatory behavior of Figure \ref{beta_ratios}(a) can be explained by Landau levels shifting into or out of the electron energy spectrum as the field changes, in Figure \ref{beta_ratios}(b), the oscillatory behavior can be explained by the shift in electron chemical potential as the density increases. The change in chemical potential results in the electron energy spectrum shifting with respect to the existing Landau levels. The same behavior is shown for a field of 10$^{15}$ G in Figure \ref{beta_ratios}(c). Figure \ref{latitude} shows the ratio of electron capture rates $\lambda(B)/\lambda(B=0)$ as a function of polar angle about the axis of the neutron star for two different values of field (as indicated at the pole). The field is assumed to be a dipole field. Electron capture rates are computed at the stellar surface. For each case, $Q_{EC}$ = 5 MeV. For a weaker field, more Landau levels are present in the electron phase space, and as the field changes across the surface of the star a larger number of Landau levels are shifted out of the electron energy spectrum resulting in the larger number of fluctuations in the EC rate. However, for a larger field, fewer Landau levels are included in the electron energy spectrum, and as the field shifts across the surface of the star, fewer Landau levels shift into or out of the electron energy spectrum. Thus, there are fewer oscillations in the EC rate across the surface of the star. In order for efficient Urca pair cooling to occur, EC and $\beta^-$ decay rates must be similar. We thus examine these rates as a function of magnetic field for several Urca pairs. In Figure \ref{mg33_fig}, rates are shown as a function of magnetic field for two presumed Urca pairs~\cite{schatz14}. In each figure, the black line corresponds to $\beta^-$ decay rates while the red line corresponds to EC rates. Optimal Urca pair cooling occurs if the rates are similar. However, as the field increases, there can be oscillations in the rates, reducing the efficiency of the associated pair. For example, consider the $^{29}$Mg$\leftrightarrow^{29}$Na pair in Figure \ref{mg33_fig} at the temperature, density, and electron fraction indicated. At a field of B$\approx 10^{15.8}$, the EC rate exceeds the $\beta^-$ rate by about five orders of magnitude. However, at a field strength slightly lower than this, the rates intersect on the graph. At very high fields, the rates diverge significantly, as only one Landau level contributes to the electron energy spectrum. A similar comparison can be made for the $^{33}$Al$\leftrightarrow ^{33}$Mg pair in the same figure. The introduction of magnetic fields can also create Urca pairs where they would not have otherwise existed. This is shown in Figure \ref{mg29_fig}, for two pairs at densities and temperatures not conducive to the formation of Urca pairs for the nuclei shown at low fields. However, as the field increases, the rates can match and possibly form Urca pairs. The additional degree of freedom from the magnetic field extends the range of possible conditions at which Urca pairs can form within the NS crust. Finally, in addition to beta decay, Urca pairs can form for EC daughter excited states. For excited states in the EC daughter nucleus, the EC rate drops. The daughter nucleus de-excites almost immediately and decays via $\beta^-$ decay back to the original nucleus. However, because the $\beta^-$ rate exceeds the EC rate for excited states, the cooling efficiency is lower. The rate oscillations, however, may open Urca pair transitions to excited states in the EC daughter nucleus. This is shown in Figure \ref{Mg31_fig}. EC and $\beta^-$ rates to the ground state of $^{31}$Mg are shown as a function of magnetic field, as well as to the first two excited states of $^{31}$Mg. While Urca pairs including excited states may not be possible at low field strength, higher fields may allow for Urca pairs between excited states. This is particularly interesting because EC to excited states may open up transition rates that are more favored. As an example, Figure \ref{mu_theta_fig} shows the electron chemical potential as a function of polar angle for a layer of constant density and temperature within the crust. For all panels in this figure, a typical temperature of $T_9=0.51$ is adopted. For the top row of panels, $\rho Y_e=4.70\times 10^{10}$ g cm$^{-3}$, and for the bottom row $\rho Y_e=3.73\times 10^{10}$ g cm$^{-3}$. The shaded regions in these figures indicate those for which the chemical potential is within $kT$ of the EC Q-values of the indicated reactions, $Q_{EC}-kT\le\mu_e\le Q_{EC}+kT$. With this evaluation, an Urca pair is not consistent across the entire surface of the NS at a given temperature and density. At high enough magnetic fields, an Urca pair may dominate a particular angular band. This may also have the effect of making additional Urca pairs possible in various angular regions. Because of the shift in electron chemical potential with magnetic field, the location or Urca pairs within the neutron star crust varies with field and latitude (polar angle on the stellar surface). As an example, consider Figure \ref{mu_angle_dens}, which shows the electron chemical potential as a function of polar angle for various surface magnetic fields (at the poles), latitudes, and densities within a NS crust. Because the chemical potential changes with both field and density, the physical location of Urca pairs may change. This will also result in a differentiation in the neutrino luminosity of various stars. For example, for a very large field of 10$^{16.5}$ G as indicated in the figure, the density for which an Urca pair occurs is highest at the poles of the star, resulting in a larger polar neutrino luminosity, whereas, the stellar equator will form Urca pairs at a lower density. However, for the $^{63}$Cr$\leftrightarrow ^{63}$V Urca pair, it can be seen that the minimum luminosity occurs at $\theta\approx$ 1.2 rad. The magnetic field can result in a shift in the locations of Urca pairs as a function of latitude as seen in this figure. It can also be seen that magnetic fields will enable the existence of new pairs. For a constant density and temperature, changes in magnetic field can result in a change in the electron chemical potential (Equation \ref{derv}). This changes the overall phase space for electrons in $\beta^-$ decay and electron capture. If an Urca pair is defined as a $\beta^-$-EC pair for which $\mu_e\sim Q$, then the density at which an Urca pair can exist also depends on the environmental magnetic field. For example, the crusts and oceans of neutron stars have been modeled for various dipole fields defined by the field at the NS polar region. The density at which various Urca pairs are viable for various polar fields is shown in Figure \ref{rho_em_lum} as a function of polar angle on the NS surface. For an assumed dipole field, the field is dependent on the NS polar angle at the surface. A 12 km diameter star with M=1.4 M$_\odot$ was considered. A constant temperature and $Y_e$ of $T_9=0.51$ and $Y_e$=0.41 are assumed in each case. For these fixed parameters and a fixed dipole field, the density at which $\mu_e\sim Q$ is found. This is plotted in the left column of Figure \ref{rho_em_lum} for fields of $B(\theta=0)$ of 10$^{14}$ G, 10$^{15}$ G, and 10$^{16}$ G. Because the field is not constant across the NS surface, the electron chemical potential is not constant for constant density. Using the densities computed in Figure \ref{rho_em_lum} (left panels), the neutrino emissivity as a function of polar angle in the NS ocean is plotted in the same figure (center). Here, the emissivity is normalized to the mass fraction of the relevant nuclei in these panels, $\varepsilon/X$. The emissivity is computed using the approximation of \cite{deibel16}. Because we are interested in bulk behaviors of the neutrino emissivity and luminosity, we adopt the approximate weak rates of \cite{arcones10} using the phase space extracted from Equation \ref{derv}~\citep{famiano20}. However, given the extraction of the $ft$ values from the rates presumed in \cite{deibel16}, the rate ratios given by the phase space differences with and without the magnetic fields, $\lambda(B\ne 0)/\lambda(B=0)$ are expected to be independent of any nuclear structure effects in this evaluation. The luminosity, $L_{32}$ is shown on the right side of Figure \ref{rho_em_lum} assuming an axisymmetric field for polar angle increments of $\Delta\theta=\pi/200$ radians. The radial thickness of each zone is calculated using the formulation of \cite{schatz14}. For this calculation, a temperature of T$_9$=0.51, an electron fraction of $Y_e=0.41$, a crust radius of 12 km, and a local gravity of $g=1.85\times 10^{14}$ cm s$^{-2}$ are assumed. The smaller overall solid angle near the poles results in an overall reduction of the total luminosity, though the emissivity may be larger in this region. The value of the neutrino emissivity, $\varepsilon$, is proportional to the weak interaction rates and is computed following the prescription of \citep{deibel16}. There are multiple effects contributing to the behavior of the neutrino luminosity as a function of field. At the densities necessary for an Urca pair to exist, only the high(low)-energy tail of the electron energy spectrum is relevant for $\beta^-$(EC) decay. Also, a change in the local magnetic field will shift the overall electron chemical potential as shown in Figure \ref{mu_theta_fig}. In this case, shifting the electron chemical potential in one direction or the other will result in a shift in the electron phase space, resulting in $\beta$ decays or electron captures being impossible. As the magnetic field increases, the overlap between the Landau level distribution, the Fermi distribution, and the electron momentum space can interfere constructively or destructively, resulting in dramatic shifts in the optimum density at which an Urca pair can exist and changing the overall emissivities. This will be discussed below. In this figure one can see that, for very high fields ($\sqrt{eB}\gtrsim$ Q), the entire electron energy spectrum is confined to a single Landau level. As the field increases near the poles, the electron chemical potential decreases, resulting in a lower density at which an Urca pair can exist. This can result in a reduced emissivity. In addition, as the Landau level spacing increases, and the tail of the lowest Landau level ($E=m_e $) is more prominent in the electron energy spectrum. As the field decreases, and the Landau levels move closer together (but with a level spacing still greater than the Q value), the tail of the lowest Landau level becomes less prominent in the electron spectrum, resulting in a decrease in the overall rates. For a very low field, $\sqrt{eB}\ll Q$, the effects of the Landau level spacing on the overall electron spectrum are minimal. Here, any change in field along the surface of the NS has little or no effect on the electron chemical potential, and the optimal density for an Urca pair does not change significantly. The ``intermediate field'' regime ($\sqrt{eB}\sim Q$) is particularly interesting as the interference between the Landau level distribution and the Fermi distribution becomes prominent. In this regime, a small shift in the magnetic field can result in a shift in the optimum density of an Urca pair through a shift in the electron chemical potential. This contributes to a shift in the emissivity. Because the location of Landau levels strongly affects the availability of electrons to decay or capture, the effect can be magnified. This can be seen in Figure \ref{integrand_B}, which shows the electron phase space distribution for $^{23}$Ne decay in the $^{23}$Na$\leftrightarrow^{23}$Ne Urca pair. In each panel in this figure, the optimum density for this Urca pair to exist is computed (i.e., $\mu_e=Q_\beta$), and the resulting electron phase space spectrum (the integrand in the classical decay rate integral) is shown. For a low field of 10$^{14}$ G, the Landau level spacing is slightly less than the energy range of available electrons. As the field changes, the Landau level spacing changes, as do the positions of individual Landau levels. One sees that as the field increases above 10$^{14}$ G, Landau levels can change location with respect to the edge of the Fermi distribution. At 10$^{14.5}$ G, a Landau level exists at the right edge of the Fermi distribution with a less-prominent contribution from a level near 4.4 MeV. Most of the energy spectrum is dominated by the tail of the lower-energy Landau level. At a field of 10$^{14.7}$ G, the Landau level spacing is larger than the available phase space created by the Fermi distribution and the electron momentum distribution. For this reason, much of the available electron energy spectrum is dominated by the tail of a low-energy Landau level. As the field increases towards 10$^{15}$ G, the degeneracy of available Landau levels increases as electrons are placed in only the lowest few Landau levels, and the occupancy of the tails of the distributions becomes more important in the electron energy spectrum. When the field is high enough such that only the lowest Landau level is occupied, the tail of the distribution becomes quite prominent, and the electron energy spectrum is dominated by the single Landau level. \section{Conclusions} We explored the dependence and evolution of Urca pairs in crusts and oceans of magnetized neutron stars. Because the electron chemical potential depends on the environmental magnetic field, the presence or absence of Urca pairs must also depend on the external field. Urca pairs which may exist at a certain temperature and density, $\rho Y_e$, at zero magnetic field may not exist for a non-zero field. Conversely, Urca pairs which do not exist at specific temperatures and densities may appear in the presence of an external field. The presence of a magnetic field may also make excited states available in EC-$\beta^-$ pairs. In this case, two nuclei which may not transition to low-lying states, may undergo transitions to excited states in the presence of an external field. We also explored the evolution of Urca pairs along the surface of a magnetized neutron star. While we have assumed a dipole field for simplicity, this is sufficient to convey the concept that the variations in the field on the surface of a NS may change Urca pair locations within the crust/ocean of the star. Here, we have calculated the density at which an Urca pair may exist in a NS crust/ocean for a constant crust temperature of $T_9 = 0.51$. Of course, we note that the temperature of the crust varies with depth, so we acknowledge the simplification in this evaluation. This variation in density will result in a change in the neutrino emissivity and subsequent luminosity as a function of polar angle. We find that the emissivity (normalized to the mass fraction of the Urca pair studied) subsequently depends on the NS latitude, resulting in changes in the luminosity as a function of latitude. However, because the density at which an Urca pair may exist in a NS is not constant, this ultimately affects the actual presence of an Urca pair at a location in a NS. This is because the density and the mass fraction changes with depth in the NS crust. At one location on the NS surface, an Urca pair may exist at one density while it exists at a different density at another location in the NS. However, the mass fractions of the Urca pair also vary at different densities/radii. For this reason, the emissivity is presented as normalized to the mass fraction, and a true computation of the emissivity must be multiplied by the mass fraction for the radius at which a specific density exists. Figure \ref{rho_em_lum} can be used to scale Urca pair emissivity as a function of latitude for a known mass fraction, X. Thus, the emissivity can be expressed as a functional, $\epsilon = \epsilon(T,\rho Y_e, B, X) = \epsilon(T, \rho(r,\theta) Y_e(r), B(\theta))= \epsilon(T,r,\theta, B_0)$, where $B_0$ is the magnetic field at the NS pole. It may very well be possible that the mass fraction in the region at which an Urca pair may be viable is zero, while it is non-zero in another viable region. Thus, an Urca pair may exist in one part of the NS, but not in another. We have shown that the addition of a magnetic field to NS luminosity calculations adds another layer of complexity to the total emissivity calculation, and ultimately may result in uneven NS cooling along the surface of the star. There may be several ramifications of this result which will be explored in subsequent papers. First, we find that the bulk luminosity of a NS is dependent on its magnetic field. This can ultimately change the cooling curve of the star. Perhaps variations in NS luminosities are a result of the surface field. Likewise, limits can be placed on NS surface fields from observations of their cooling. Second, uneven neutrino emissivity on the NS surface can result in uneven cooling and heating in the crust. This can have affect the thermal equation of state on the surface and may be a way of explaining NS crust quakes or possibly even NS kicks. Certainly, an exhaustive treatment of the overall complexity of this problem is beyond the scope of the present work. More precise work is needed. In particular, a thorough evaluation of the presence or absence of Urca pairs as a function of density, temperature, and magnetic field is needed. From this, luminosity maps can be produced for NSs with various field configurations. In order to produce these, realistic models of ocean/crust abundances can be developed. While these will be explored in future work, our current results introduce the possibility that magnetized neutron stars may have uneven cooling as a result of variations in the interior magnetic field, and the overall bulk luminosity changes with the field configuration and magnitude. \bibliography{sample631}{} \bibliographystyle{aasjournal} \begin{acknowledgements} T.K. is supported in part by Grants-in-Aid for Scientific Research of JSPS (17K05459, 20K03958). A.B.B. is supported in part by the U.S. National Science Foundation Grants No. PHY-2020275 and PHY-2108339. M.A.F. is supported by National Science Foundation Grant No. PHY-1712832 and by NASA Grant No. 80NSSC20K0498. K.M. is supported by Research Institute of Stellar Explosive Phenomena at Fukuoka University and JSPS KAKENHI Grant Number JP21K20369. M.A.F., G.J.M. and A.B.B. acknowledge support from the NAOJ Visiting Professor program. Work at the University of Notre Dame (G.J.M.) supported by DOE nuclear theory grant DE-FG02-95-ER40934. \end{acknowledgements}

Title: Can primordial parity violation explain the observed cosmic birefringence?

Abstract: Recently, the cross-correlation between $E$- and $B$-mode polarization of the cosmic microwave background (CMB), which is well explained by cosmic birefringence with rotation angle $\beta\approx 0.3$ deg, has been found in CMB polarization data. We carefully investigate the possibility of explaining the observed $EB$ correlation by the primordial chiral gravitational waves (CGWs), which can be generated in the parity-violating theories in the primordial Universe. We found that the CGWs scenario does not work due to the overproduction of the $BB$ auto-correlation which far exceeds the observed one by SPTPol and POLARBEAR.

https://export.arxiv.org/pdf/2208.08101

\preprint{APS/123-QED} \title{Can primordial parity violation explain the observed cosmic birefringence?} \author{Tomohiro Fujita} \affiliation{Waseda Institute for Advanced Study, Shinjuku, Tokyo 169-8050, Japan}% \affiliation{Research Center for the Early Universe, The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan}% \email{tomofuji@aoni.waseda.jp} \author{Yuto Minami}% \affiliation{% Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka, 567-0047, Japan }% \author{Maresuke Shiraishi}% \affiliation{% School of General and Management Studies, Suwa University of Science, 5000-1 Toyohira, Chino, Nagano 391-0292, Japan}% \author{Shuichiro Yokoyama} \affiliation{Kobayashi Maskawa Institute, Nagoya University, Chikusa, Aichi 464-8602, Japan} \affiliation{Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan} \section{\label{sec:intro}Introduction} Measurements of linear polarization patterns of the cosmic microwave background (CMB) have provided rich information of our Universe~\cite{Komatsu:2014ioa,Aghanim:2018eyx, POLARBEAR:2022dxa, Polarbear:2020lii, ACT:2020gnv, SPT:2019nip,Dutcher:2021vtw,BICEP:2021xfz,Ade:2021cyk}. The CMB photons gained the linear polarization at the last scattering surface (LSS) at redshift $z\approx1100$. In addition, the polarization pattern may have been affected by some interactions, which the CMB photons experienced during the travel to observers. The linear polarization patterns of CMB can be decomposed into two orthogonal components: parity even $E$-mode and parity odd $B$-mode. Having them, we can construct two parity even correlations, $EE$ and $BB$, and a parity odd correlation, $EB$. If the parity symmetry is conserved for the CMB photons, the $EB$ cross-correlation should vanish, and thus it is a good probe of parity-violating physics. Recently, a hint of parity violation was measured in the $EB$ correlation of the CMB photons~\cite{Minami:2020odp,Diego-Palazuelos:2022dsq,Eskilt:2022cff}, where a parity-violating physics, so-called ``cosmic birefringence'' was assumed. Cosmic birefringence is that the linear polarization plane of the CMB photons rotates by angle $\beta$ while they travel between the LSS and observers. This rotation produces a non-zero $EB$ cross-correlation from $EE$ and $BB$ spectra generated at the LSS as $ C_\ell^{EB,\rm{obs}} = \sin(4\beta) (C_\ell^{EE} - C_\ell^{BB})/2$~\cite{Minami:2019ruj}. The recent careful data analysis indicates $\beta\approx 0.3\,\deg$ with about 3$\sigma$ significance~\cite{Eskilt:2022cff}. No significant dependence of $\beta$ on the CMB photon frequency was found~\cite{Eskilt:2022wav}. To explain this cosmic birefringence, several models have been proposed~\cite{Fujita:2020ecn,Takahashi:2020tqv,Fung:2021wbz,Choi:2021aze, Obata:2021nql,Gasparotto:2022uqo}. However, the non-zero detection of the $EB$ correlation does not necessarily imply that this parity-violating signal was caused by cosmic birefringence. It is important to consider whether an alternative explanation is possible or not. Primordial chiral gravitational waves (CGWs) are known to produce $EB$ correlation before the LSS because their imbalance between right- and left-handed circular polarization mode breaks the parity symmetry~\cite{Lue:1998mq}. A number of models, which generate CGWs with various spectrum shapes in the primordial Universe, have been studied~\cite{Alexander:2004wk,Takahashi:2009wc,Anber:2012du,Adshead:2012kp,Namba:2015gja,Obata:2016tmo,Dimastrogiovanni:2016fuu,Adshead:2017hnc,Fujita:2018ndp,Machado:2018nqk,McDonough:2018xzh,Iarygina:2021bxq}. It is apparently possible to reproduce the observed $EB$ spectrum $C_{\ell}^{EB,{\rm obs}}$ by considering CGWs with a suitable spectrum shape. In this short paper, we investigate if CGWs can consistently explain the observed $EB$ spectrum. To test the $EB$ spectrum induced by CGWs with the observed $EB$, one needs to re-estimate the miscalibration angles of detectors, which are simultaneously determined in the measurements of $\beta$~\cite{Minami:2020odp,Diego-Palazuelos:2022dsq,Eskilt:2022cff}. Because CGWs produce not only the $EB$ spectrum but also the other spectra between $T,E, B$, an appropriate likelihood function that includes all spectra produced by CGWs is required, which is a very complicated task. However, we have noticed that the induced $BB$ spectrum becomes much larger than the observed value in the CGW scenario. Therefore, we adopted a strategy of focusing on the compatibility of the $EB$ and $BB$ spectra induced by CGWs which are tuned to mimic the cosmic birefringence signal of $\beta=0.3\,\deg$. We do not specify the generation mechanism of CGWs but introduce a spectrum template of CGWs with a sufficient number of parameters to obtain the desired $EB$ spectrum. We shall show that such CGWs lead to the overproduction of the $BB$ spectrum. This paper is organized as follows. In Sec.~\ref{sec:PCGW}, we introduce our template of CGWs. In Sec.~\ref{sec:result}, we tune the parameters of the CGW spectrum and obtain the $EB$ spectrum similar to the observed one. In Sec.~\ref{BB result}, we compute the $BB$ spectrum induced by the CGW spectrum and show that it exceeds the observed value. Sec.~\ref{sec:conclusion} is devoted to summary and discussion. \section{\label{sec:PCGW}Chiral Gravitational Waves} Chiral gravitational waves (CGWs), which violate the parity symmetry, produce $EB$ cross-correlation in the CMB polarization anisotropy if they are generated before the recombination era. In this section, we introduce our parameterization of the primordial spectrum of CGWs and illustrate how they induce CMB polarization correlations. CGWs can be generated in the early Universe in various models, which predict diverse CGW spectrum shapes~\cite{Alexander:2004wk,Takahashi:2009wc,Anber:2012du,Adshead:2012kp,Namba:2015gja,Obata:2016tmo,Dimastrogiovanni:2016fuu,Adshead:2017hnc,Fujita:2018ndp,Machado:2018nqk,McDonough:2018xzh,Iarygina:2021bxq}. In this paper, however, we adopt a model-independent approach. We consider fully chiral and log-normal spectra of primordial CGWs and superpose them with different heights and peak positions as \begin{align} \mathcal{P}_h^{L}(k) &\simeq 0, \notag\\ \mathcal{P}_h^{R}(k) &=\mathcal{P}_\zeta \sum_i r_{i} \exp \left[-\frac{1}{2\sigma^2}\ln^2\left(\frac{k}{k_i}\right)\right], \label{eq:Ph_template} \end{align} where $\mathcal{P}_h^{L/R}$ denotes the dimensionless power spectrum of the left/right-handed circular polarization modes of the primordial gravitational waves on super-horizon scales. $\mathcal{P}_\zeta=2.2\times 10^{-9}$ is the curvature power spectrum on the CMB scale. $r_i, k_i, \sigma$ parameterize the amplitude, peak scale, and width of $\mathcal{P}_h^{R}$, respectively. Our template \eqref{eq:Ph_template} can accommodate a sufficiently large parameter space to try mimicking the observed $EB$ spectrum, though we only introduce the single parameter $\sigma$ to control the width for simplicity. It should be stressed that we do not propose Eq.~\eqref{eq:Ph_template} as a natural spectrum shape of CGWs. Instead, we will show that even such a highly fine-tuned spectrum fails to explain the observation, and hence it is even harder for more realistic spectra. Note that we consider the parity violation with $\mathcal{P}_h^{R} \gg \mathcal{P}_h^{L}$ so that CGWs induce positive $EB$ cross-correlation, which is favored by the measurements~\cite{Minami:2020odp, Diego-Palazuelos:2022dsq,Eskilt:2022cff}. Primordial CGWs produce all of the auto- and cross-correlations between the CMB temperature $T$ and linear polarization $E$ and $B$. Among them, we focus on $EB$ and $BB$ angular power spectra in this paper. The contributions from the CGWs to them are written as e.g.,~\cite{Pritchard:2004qp,Namba:2015gja,Thorne:2017jft} \begin{align}\begin{split} C_{\ell}^{EB} &= 4\pi \int {\rm d} (\ln k) \left[\mathcal{P}_h^{L}(k)-\mathcal{P}_h^{R}(k)\right] \Delta^E_\ell(k) \Delta^B_\ell(k),\\ C_{\ell}^{BB} &= 4\pi \int {\rm d} (\ln k) \left[\mathcal{P}_h^{L}(k)+\mathcal{P}_h^{R}(k)\right] \Delta^B_\ell(k) \Delta^B_\ell(k), \label{eq:EBBB} \end{split}\end{align} where $\Delta^{E/B}_\ell(k)$ is the tensor transfer function of the CMB $E$/$B$-mode, and the contributions from the scalar and vector % modes are ignored. Since $C_{\ell}^{EB}$ and $C_{\ell}^{BB}$ are linear functions of $\mathcal{P}_h^{R}$, each term in our template \eqref{eq:Ph_template} independently contributes to them. \section{\label{sec:result} Reproducing $EB$ spectrum} In this section, we numerically compute the $EB$ spectrum, which is contributed by the CGWs parametrized in Eq.~\eqref{eq:Ph_template}, using a modified version of a publicly available Boltzmann code CAMB \cite{Lewis:1999bs,camb} where Eq.~\eqref{eq:EBBB} is implemented. We shall fix the parameters, $k_i, r_i$ and $\sigma$ in Eq.~\eqref{eq:Ph_template}, in order to maximally mimic the observed $EB$ spectra. In Fig.~\ref{fig:EBandBB}, we plot $D_\ell^{XY} = {\ell(\ell+1)} C_\ell^{XY}/{2\pi}~ (XY=EB, BB)$, and the left panel is for the $EB$ spectrum and the right panel shows the $BB$ spectrum. In the left panel, the black solid line represents the target $EB$ spectrum inferred by the birefrigence angle reported by Refs.~\cite{Minami:2020odp,Diego-Palazuelos:2022dsq,Eskilt:2022cff}. One observes that the target $EB$ has % five peaks up to $\ell \sim 1500$, which is the $\ell_\mathrm{max}$ used in the measurements of $\beta$~\cite{Minami:2020odp,Diego-Palazuelos:2022dsq,Eskilt:2022wav,Eskilt:2022cff}, that inherits from the intrinsic $EE$ spectrum, as cosmic birefringence induces, $ C_\ell^{EB,\rm{obs}} \simeq \sin(4\beta)\, C_\ell^{EE}/2$. The colored lines % denote the calculated $EB$ spectra produced by the CGWs. In mimicking the target $EB$ with CGWs, we introduce five terms in $\mathcal{P}_h^{R}$ to individually fit these peaks. To obtain narrow $EB$ peaks from the CGWs, we first set the width parameter to a small value $\sigma=0.2$, while we will vary it later. Then, we search for the value of the peak scale $k_i$ and the peak amplitude $r_i$, to ensure that the calculated $EB$ spectrum is adjusted to the position and the height of each peak in the target $EB$. In this manner, we determine five sets of parameters as $\{k_i\,\SI{}{Mpc},\, r_i\} = \{\SI{9.60e-3},\, \SI{5.68e+0}\}, \{\SI{2.82e-2},\, \SI{8.02e2}\},$ $\{\SI{5.00e-2},\,\SI{ 5.50e+3}\}, \{\SI{8.70e-2},\,\SI{4.80e+04}\},$ $\{\SI{1.10e-1},\,\SI{1.02e+5}\}$. One observes that these $EB$ spectra indueced by the CGWs show faster damped oscillations than the target one due to a distinctive feature of the tensor transfer function. Note that their peak position and height do not completely coincide with the target peaks, because they are very sensitive to the free parameters and we did not pursue such a fine-tuning, which would not affect our conclusion. One might wonder if a similar multiple peak structure in the $EB$ spectrum could be reproduced by the tensor transfer functions without superposing five different log-normal spectra of CGWs. However, a single wide CGW spectrum leads to a $EB$ spectrum that has only a couple of peaks for $\ell \lesssim 200$ but no large peaks $\ell \gtrsim 600$ due to the highly damping nature of the tensor transfer function (see e.g.\cite{Pritchard:2004qp,Namba:2015gja,Thorne:2017jft}). Hence, it is difficult to reproduce these $EB$ peaks without a tuning. Although it might be possible to find a suitable oscillating spectrum of CGWs, that would not impact on our conclusion drawn below. \section{\label{BB result}Overproduction of $BB$ spectrum} In this section, we consider $BB$ power spectrum which must be simultaneously produced through Eq.~\eqref{eq:EBBB}. Since CMB observations have measured the $BB$ power spectrum for the relevant $\ell$ range, one should seek an appropriate parameter set of the CGWs, which does not produces a too large $BB$, while reproducing $EB$. In the right panel of Fig.~\ref{fig:EBandBB}, the colored lines show the calculated $BB$ spectra induced by the CGWs for the same sets of the CGW parameters as the previous section. The observed $BB$ data taken by SPTPol~\cite{SPT:2019nip} (blue points) and POLARBEAR~\cite{Adachi:2019mjv} (red points) are also shown with error bars. From this panel, one can see that the induced $BB$ spectra from the CGWs far exceed the measured amplitude and thus the corresponding parameters should be excluded. We note that even the blue line, which is adjusted to the first peak and has the smallest height, also overproduces the $BB$ spectrum. This result implies that it is hard for the CGWs to explain the observed $EB$ spectrum without conflicting with the observed $BB$. We have fixed the width parameter $\sigma$ for the CGWs as $\sigma=0.2$ so far. Does a different value of $\sigma$ alleviate the incompatibility between $EB$ and $BB$? To test this possibility, we compute the ratio of the maximum value of $D_\ell^{BB}$ to that of $D_\ell^{EB}$ by increasing $\sigma$ for each contribution in Eq.~\eqref{eq:Ph_template} separately. As apparent from Fig.~\ref{fig:EBandBB}, since the peak height in the observed $D_\ell^{EB}$ is a few times $0.1 \, \mu \mathrm{K}^2_\mathrm{CMB}$ and the maximum value of the observed $D_\ell^{BB}$ is $0.24 \, \mu \mathrm{K}^2_\mathrm{CMB}$, this ratio $\mathrm{max}[D_\ell^{BB}]/\mathrm{max}[D_\ell^{EB}]$ should be less than $2.4$ at least. The result is shown in Fig.~\ref{fig:BBperEB}. Although the ratio changes depending on $k_i$, it never becomes smaller than $21.5$ for $\sigma\ge 0.2$. For larger $\sigma$, the ratios converges to the same value around $36.5$, because the CGW spectrum $\mathcal{P}_h^R$ becomes flatter and less dependent on its peak position $k_i$. Note that this ratio does not depend on $r_i$. Therefore, we find that larger $\sigma$ does not mitigate the problem of the overproduction of $BB$ spectrum. Even though we can make the $\mathrm{max}[D_\ell^{BB}]/\mathrm{max}[D_\ell^{EB}]$ ratio smaller with $\sigma$ smaller than $0.2$, the peaks of the $EB$ spectrum from the CGWs would be too sharp to explain the target $EB$ spectrum. \section{\label{sec:conclusion} Summary and discussion} Recently, the $EB$ power spectrum, which is well explained by cosmic birefringence with rotation angle $\beta\approx \SI{0.3}{deg}$, has been observed in CMB data. In this paper, we investigated the possibility that this observed $EB$ spectrum is produced by primordial chiral gravitational waves (CGWs) instead of cosmic birefringence. However, we found that if CGWs produced a similar $EB$ spectrum to the observed one, they would inevitably overproduce a $BB$ spectrum whose amplitude is much larger than the measured value by SPTPol and POLARBEAR. Therefore, it is difficult to attribute the observed $EB$ spectrum to CGWs. To parameterize the CGW spectrum, we superposed five log-normal spectra with different peak heights and positions in Eq.~\eqref{eq:Ph_template} and analyzed it. Nonetheless, we expect that our conclusion does not depend on the detailed shape of the CGW spectrum, because the $EB$ and $BB$ spectra are linear function of $\mathcal{P}_h^R(k)$. Moreover, this CGW template enabled us to illustrate that even CGWs reproducing only one peak of the $EB$ spectrum lead to overproduction of $BB$ and should be excluded. The reason for the difficulty of the CGW scenario can be understood as follows. The newly observed $EB$ spectrum is smaller than the standard $EE$ spectrum mainly contributed by the scalar perturbation by a factor of $C_\ell^{EB,\mathrm{obs}}/C_\ell^{EE,\mathrm{scalar}}=\sin(4\beta)/2 \approx 10^{-2}$ for $\beta \approx 0.3^\circ$. On the other hand, it has been known that scale-invariant primordial gravitational waves produce the $EE$ and $BB$ spectra of roughly equal size, $C_\ell^{EE,\mathrm{tens}}\simeq C_\ell^{BB,\mathrm{tens}}\simeq 10^{-4} r\, C_{\ell}^{EE,\mathrm{scalar}}$ where and hereafter we consider $\ell\approx 600$, and $r$ is the tensor-to-scalar ratio. Thus, we expect that CGWs produce a $EB$ spectrum, $C_\ell^{EB,\mathrm{tens}}\simeq 10^{-2} r\, C_{\ell}^{EB,\mathrm{obs}}$, and in order to reproduce the observed $EB$ spectrum, $r\simeq 10^2$ is necessary, which leads to $C_\ell^{BB,\mathrm{tens}}\simeq 10^{-2}\, C_{\ell}^{EE,\mathrm{scalar}}$. This is incompatible with an observed fact at high $\ell$ that $C_\ell^{BB,\mathrm{obs}}\simeq C_\ell^{BB,\mathrm{lens}}\simeq 10^{-3}C_{\ell}^{EE,\mathrm{scalar}}$ with $C_\ell^{BB,\mathrm{lens}}$ the lensing $B$-mode spectrum. Note that the above argument assumed scale-invariant CGWs and derived smaller amplitude parameter than $r_i\sim 10^{3}$ obtained in Sec.~\ref{sec:result}. Nonetheless, it illustrates the basic reason why CGWs cause the incompatibility between the $EB$ and $BB$ spectra. In this paper, we did not study the case with an extremely small width parameter, $\sigma < 0.2$. The trend in Fig.~\ref{fig:BBperEB} infers that smaller $\sigma$ would increase the ratio, $\mathrm{max}[D_\ell^{BB}]/\mathrm{max}[D_\ell^{EB}]$, for the fourth (red) and fifth (purple) peaks and thus worsen the conflict between $EB$ and $BB$. Even if the ratio decreases for much smaller $\sigma$, such a spike-like $D_\ell^{EB}$ would not be responsible for the entire observed $EB$ spectrum. While a dedicated analysis should be done to rigorously dismiss this possibility, we expect that smaller $\sigma$ would not give a viable solution. \begin{acknowledgments} This work was supported in part by Japan Society for the Promotion of Science (JSPS) KAKENHI, Grants Nos.~JP18K13537 (T.F.), JP20H05854 (T.F.), JP20H01932 (S.Y.), JP20K03968 (S.Y.), JP20K14497 (Y.M.), JP19K14718 (M.S.) and JP20H05859 (M.S.). The authors thank the Yukawa Institute for Theoretical Physics at Kyoto University. Discussions during the YITP workshop YITP-T-21-08 on ``Upcoming CMB observations and Cosmology'' were useful to complete this work. The authors are grateful to Yuji Chinone for suggestions on the plots of power spectra. M.S. acknowledges the Center for Computational Astrophysics, National Astronomical Observatory of Japan, for providing the computing resources of Cray XC50. \end{acknowledgments} \bibliography{main}%

Title: Magnetic field measurements of sharp-lined Ap stars

Abstract: Previous observations suggested that Ap and Bp stars exhibit a bimodal distribution of surface magnetic field strengths and that actually only few or no stars exist with magnetic dipole field strengths below 300 G down to a few Gauss. As the number of Ap and Bp stars currently known to possess weak magnetic fields is not large, it is necessary to carry out additional spectropolarimetric studies of Ap and Bp stars to prove whether the assumption of the existence of a critical value for the stability of magnetic fields is realistic. In this study, we present high-resolution HARPSpol magnetic field measurements for a sample of Ap stars with sharp spectral lines with a view to characterize the strengths of their magnetic fields. Out of the studied seven sharp-lined stars, two stars, HD 174779 and HD 203932, exhibit a rather weak longitudinal magnetic field with $\left< B_{\rm z}\right>=-45\pm3$ G and $\left< B_{\rm z}\right>=21\pm4$ G, respectively. Additionally, TESS observations were used to test previous conclusions on the differentiation of rotation periods of Ap and Bp stars. Apart from HD 189832 and HD 203932, all other studied sharp-lined stars have long rotation periods. Since an explanation for the slow rotation of Ap stars is currently missing, additional studies of slowly rotating Ap and Bp stars are necessary to improve our understanding of the formation and evolution of Ap and Bp stars.

https://export.arxiv.org/pdf/2208.14013

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} stars: chemically peculiar -- stars: individual: HD\,70702, HD\,89393, HD\,137949, HD\,138633, HD\,174779, HD\,176196, HD\,189832, HD\,203932, HD\,217522 -- stars: magnetic fields -- stars: rotation \end{keywords} \section{Introduction} \label{sec:intro} Large-scale organised magnetic fields with strengths ranging from several tens of Gauss to several kG are present in 10 to 15\% of all stars of spectral types O to early F \citep[e.g.][]{Grunhut2017, Schoeller2017, book2021}. Magnetic fields cover the whole stellar surface, and their geometry often resembles a single dipole whose axis is inclined with respect to the stellar rotation axis. Understanding how these stars acquired magnetic fields and why they are only present in a fraction of upper main-sequence stars, as well as how these fields affect stellar evolution, will have significant implications for a wide range of astrophysical areas, from galactic evolution to exoplanets. Studies of upper main-sequence Ap and Bp stars, constituting the most populous group of magnetic early-type stars, are also of great importance because such stars display the most extreme manifestations of the effects of magnetic fields on stellar atmospheres. Specifically, they exhibit strong abundance anomalies, including both the horizontal accumulation and vertical stratification of abundances of various chemical elements. Based on mean longitudinal magnetic field measurements (i.e.\ measurements of the line-intensity weighted average over the stellar disc of the component of the magnetic vector along the line of sight) of a small sample of 28 magnetic Ap and Bp stars with poorly constrained magnetic field strengths, \citet{Auriere} concluded that there exists a critical dipole field strength, $B_{\rm d}\approx300$\,G, which corresponds to the minimum field strength for a star to maintain the stability of its magnetic field. Only two stars in the sample of 28 Ap stars showed a dipole strength below 300\,G. The authors suggested that the magnetic dichotomy of intermediate-mass stars -- i.e., a dichotomy in the distribution of the observed magnetic field between the kG dipoles of Ap and Bp stars and the sub-Gauss magnetism of Vega and Sirius -- arise from the development of non-axisymmetric instabilities separating stable strong field configurations observed in Ap and Bp stars from unstable weaker field configurations, whose surface average field becomes very weak after the destabilization. However, in contrast to this study, a number of Ap and Bp stars have been reported to possess very weak longitudinal magnetic fields on the order of a few tens of Gauss \citep[e.g.,][]{Donati1990, Donati2006, Alecian2016}. Also, for higher mass Bp stars, rather weak magnetic fields were reported by \citet{Fossati2015}. As discussed by \citet{JerCan}, such weak fields can be consistent with dynamo fields generated in subsurface convection zones. Since the reported dichotomy may be due to observational incompleteness, there is certainly a need for more representative studies of Ap and Bp stars. Such studies are especially important for the theoretical understanding of the origin of magnetic Ap and Bp stars. The other important question for the understanding of the origin of magnetic Ap and Bp stars with sharp lines is whether the majority of stars with weak magnetic fields are slow rotators. If these stars are not observed close to the rotation pole, then we expect that a fraction of stars with very sharp spectral lines have longer rotation periods. The occurrence of very slow rotation in weakly magnetic Ap and Bp stars has not been investigated in detail yet, but is becoming an important subject of current studies of correlations between rotation rate and magnetic field strength, following the recent discovery of unbiased Transiting Exoplanet Survey Satellite \citep[TESS;][]{TESSSPIE} samples of slowly rotating Ap stars by \citet{MathysTESS, Mathys2022}. According to \citet{Hubrig2007}, stronger magnetic fields tend to be found in hotter, younger and more massive stars, as well as in stars with shorter rotation periods. \citet{Mathys2017} confirmed that Ap stars with very strong magnetic fields never achieve extremely slow rotation: Ap stars with $\langle B\rangle >7.5$\,kG have rotation periods shorter than 150\,d whereas Ap stars with $\langle B\rangle <7.5$\,kG have periods longer than 150\,d. Thus, the differentiation of rotation in Ap and Bp stars is a possible key to the understanding of the origin of their magnetic fields. In this work, we discuss new mean longitudinal magnetic field measurements for a sample of sharp-lined Ap stars using the High Accuracy Radial velocity Planet Searcher polarimeter, \citep[HARPS\-pol;][]{Harps} fed by the ESO 3.6-m telescope. In the spectra of several of the Ap and Bp stars obtained in the framework of a systematic search for Ap stars with resolved magnetically split lines, spectral lines appear very sharp. This indicates that magnetic line broadening, which is proportional to the absolute value of the magnetic field strength, should be very small. For a number of stars in our sample, photometric data have recently been provided by the TESS mission, enabling us to search for variability in their light curves. As the surfaces of Ap and Bp stars are covered by long-lived chemical spots that rotate in and out of view, the light curves show rotational modulation. Usually, the light curves exhibit different shapes and amplitudes, depending on the size of the spots and their location with respect to the line of sight. The availability of space-based photometric observations is especially valuable in studies of Ap and Bp stars, as their rotation periods are most frequently determined from light curves. Among the stars in our sample, magnetic field measurements are carried out for the first time for the sharp-lined stars HD\,89393, HD\,174779, and HD\,189832. In the spectra of HD\,138633, HD\,176196, HD\,203932, and HD\,217522, the spectral lines are also sharp and do not show any hint of resolution into their magnetic Zeeman components, but the possible presence of relatively weak magnetic fields was already discussed in previous studies. The very slowly rotating star HD\,137949, which possesses a rather strong magnetic field and exhibits spectral lines resolved into magnetically split components, was previously reported to have a rotation period of more than 14\,yr by \citet{Mathys2017}. However, their published magnetic field measurements for this star were taken in 1997. The strongly magnetic star HD\,70702, with a magnetic field modulus of about 15\,kG, was selected in our observations as a standard star to check the proper technical functionality of the analysing optics of HARPS\-pol during our observing run in 2019. Observations of this star at two different rotation phases indicate strong changes in the magnetic field strengths and variable spectral appearance. Strongly magnetic stars are of special interest because only in these stars the effect of the magnetic field on the stellar atmosphere can be studied in great detail. Since magnetism affects atomic diffusion \citep[e.g.][]{AlecianStift}, the obtained information on the magnetic field geometry is frequently used to study the horizontal accumulation and vertical stratification of abundances of various chemical elements \citep[e.g.][]{Hubrig2018,silva2020}. In the following sections we present the acquired HARPS\-pol observations, their reduction and the analysis, describe the measurement results for each individual target, and discuss their usefulness for a better understanding of stellar magnetism in Ap and Bp stars. All available TESS data are also used to identify any significant periodicities. \section{Observations} \label{sec:obs} \subsection{HARPS\-pol observations} \begin{table*} \caption{ The logbook of observations and the results of the magnetic field measurements for the investigated stars based on line lists that include all lines. The first column gives the name of the star followed by the spectral classification from \citet{Renson}. The third column presents the MJD values at the middle of the exposure, while the fourth column presents the signal to noise ratio measured in the spectral region shown in Fig.~\ref{fig:region}. The remaining columns show the total number of lines in each line mask, the average effective Land\'e factor $\overline{g}_{\rm eff}$ calculated for each line mask, the measured LSD mean longitudinal magnetic field strength, the false alarm probability (FAP) value for each measurement, and the detection flag, where DD means definite detection, MD marginal detection, and ND no detection. \label{tab:obsall} } \centering \begin{tabular}{lllr lr r@{$\pm$}l cc} \hline \multicolumn{1}{c}{Object} & \multicolumn{1}{c}{Spectral} & \multicolumn{1}{c}{MJD} & \multicolumn{1}{c}{$S/N$} & \multicolumn{1}{c}{N. of} & \multicolumn{1}{c}{$\overline{g}_{\rm eff}$} & \multicolumn{2}{c}{$\left< B_{\rm z}\right>$} & \multicolumn{1}{c}{FAP} & \multicolumn{1}{c}{Det.} \\ \multicolumn{1}{c}{} & \multicolumn{1}{c}{Type} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{lines} & \multicolumn{1}{c}{} & \multicolumn{2}{c}{(G)} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{flag} \\ \hline HD\,89393 & A0\,SrCrEu & 58\,646.094 & 434 & 86 & 1.20 & 314 & 7 & $<10^{-10}$ & DD \\ & & 58\,649.023 & 351 & & & 321 & 6 & $<10^{-10}$ & DD \\ HD\,137949 & F0\,SrEuCr & 56\,145.020 & 175 & 173 & 1.18 & 1543 & 19 & $<10^{-10}$ & DD \\ HD\,138633 & F0\,SrEuCr & 58\,650.051 & 239 & 116 & 1.21 & $-$205 & 9 & $<10^{-10}$ & DD \\ HD\,174779 & A0\,Si & 58\,647.316 & 324 & 80 & 1.21 & $-$45 & 3 & $<10^{-10}$ & DD \\ HD\,176196 & B9\,EuCr & 58\,648.375 & 309 & 96 & 1.21 & 120 & 18 & $<10^{-10}$ & DD \\ & & 58\,650.344 & 402 & & & 109 & 17 & $<10^{-10}$ & DD \\ HD\,189832 & A6\,SrCrEu & 58\,647.406 & 311 & 141 & 1.22 & 143 & 4 & $<10^{-10}$ & DD \\ HD\,203932 & A5\,SrEu & 58\,648.430 & 290 & 141 & 1.20 & 21 & 4 & $<10^{-10}$ & DD \\ HD\,217522 & A5\,SrEuCr & 56\,148.297 & 218 & 120 & 1.24 & $-$401 & 6 & $<10^{-10}$ & DD \\ & & 58\,647.438 & 363 & & & $-$323 & 6 & $<10^{-10}$ & DD \\ HD\,70702 & B9\,EuCrSr & 58\,648.008 & 188 & 72 & 1.19 & 4388 & 169 & $<10^{-10}$ & DD \\ & & 58\,650.973 & 319 & & & 1233 & 102 & $<10^{-10}$ & DD \\ \hline \end{tabular} \end{table*} All spectropolarimetric observations used in our work were acquired with HARPS\-pol on the ESO 3.6-m telescope on La Silla, which has a resolving power of $R=115\,000$ and a wavelength coverage from 3780 to 6910\,\AA{}, with a small gap between 5259 and 5337\,\AA. Observations were carried out in 2019 June (Prg.~ID 0103.C-0240). Additionally, we discuss archival observations for HD\,137949 and HD\,217522, which were obtained in 2012 August (Prg.~ID 089.D-0383). The data reduction was carried out on La Silla using the HARPS data reduction pipeline. The normalization of the spectra to the continuum level is described in detail by \citet{Hubrig2013}. More details on the observations are presented in Table~\ref{tab:obsall}. For each star in our sample, HARPS\-pol Stokes~$I$ and Stokes~$V$ spectra in the spectral region containing spectral lines characteristic for Ap stars are presented in Fig.~\ref{fig:region}. \subsection{UVES and CES} To check the short- and long-term line profile variability of HD\,138633, we extracted from the ESO archive high-resolution observations obtained with UVES (the Ultraviolet and Visual Echelle Spectrograph) mounted on Unit Telescope\,2 (UT2) of the Very Large Telescope (VLT) at Cerro Paranal, Chile (Prg.~ID 072.D-013, carried out on 2004 March 5), as well as two observations with the CES Very Long Camera, which was previously installed at the 1.4\,m CAT telescope (Prg.~ID 68.D-0445, carried out on 2002 January 25). The original UVES data consist of a large amount of very short, noisy exposures covering the wavelength range 4959--7071\,\AA{} and a spectral resolution $R=107\,000$. The individual UVES files were combined into one final spectrum with $S/N=456$ using the ESO Phase 3 UVES pipeline\footnote{http://www.eso.org/rm/api/v1/public/releaseDescriptions/163}. Wavelength calibrations for the CES spectra were executed by utilising the Th-Ar comparison spectra obtained immediately before and after recording the stellar spectrum. The first spectrum has an exposure time of 600\,s, whereas the second spectrum was obtained 14 minutes later with an exposure time of 780\,s. \subsection{TESS photometry} For three stars in our sample, HD\,176196, HD\,203932, and HD\,217522, an analysis of TESS photometry was presented in the past. No variability was detected by \citet{MathysTESS} in the available TESS data for HD\,176196 and HD\,217522. Both stars were classified by these authors as super slowly rotating Ap (ssrAp) stars. \citet{Cunhaetal} and \citet{Hold2021} analysed TESS data for HD\,203932 and reported a rotation period of $6.44\pm0.01$\,d, and pulsation periods of 2.6985 and 2.8048 mHz, confirming that this star is a rapidly oscillating Ap (roAp) star. Three stars -- HD\,70702, HD\,89393, and HD\,174779 -- were observed during Year~1 of the TESS mission, in sectors~8 and 9, sector~9, and sector~13, respectively\footnote{These observations occurred between 2018 July to 2019 July.}, but no analysis of the data for these stars has been reported in the past. In our study, the 2-minute cadence TESS observations in sectors 8--9 are used to search for periodicity in the light curve of HD\,70702. HD\,89393 has 2-min cadence data from sector 9 and 10-min full frame image (FFI) data from sector 36, while HD\,174779 has 2-minute cadence data from sector 13. No TESS data exist for HD\,137949, but it was observed as a part of the Kepler \citep{kepler} K2 \citep{k2} Campaign~15, from mid- to late-2017, in 1-minute cadence. In addition, we discuss the analysis of TESS data of HD\,138633, which was observed in sector 51 in 2022 May. We do not have TESS data for HD\,189832, which will be observed in sector 67, probably in 2023 July. The shorter-cadence data from both Kepler and TESS are available in both simple aperture photometry (SAP) and presearch data conditioning SAP (PDCSAP) forms. Data processing was done using the Science Processing Operations Center (SPOC) pipeline at the NASA Ames Research Center \citep{jenkinsSPOC2016}. With the PDCSAP light curves, we performed a Discrete Fourier Transform \citep[DFT; see, e.g.][]{kurtz:dft} to identify the dominant frequency components of the signal and their amplitudes. \section{Magnetic field measurements and detected periodicities} To increase the $S/N$ in our polarimetric measurements, we employed the least-squares deconvolution (LSD) technique. The details of this technique, as well as how Stokes~$I$ and Stokes~$V$ parameters are calculated, were presented by \citet{Donati1997}. The line masks were constructed using the Vienna Atomic Line Database \citep[VALD3;][]{kupka2011}. For each star, we carefully checked that the selected lines are indeed present in the Stokes~$I$ spectra and do not show severe blending. The presence of a magnetic field is evaluated following \citet{Donati1992}, who defined that a Zeeman profile with a false alarm probability (FAP) $\leq 10^{-5}$ is considered as a definite detection (DD), $10^{-5} <$ FAP $\leq 10^{-3}$ as a marginal detection (MD), and FAP $> 10^{-3}$ as a non-detection (ND). Previous studies of magnetic Ap and Bp stars showed that the lines of different elements with different abundance distributions across the stellar surface sample the magnetic field in different ways. Some elements, such as rare-earth elements (REEs), are frequently observed to concentrate close to the magnetic poles, whereas other elements cluster in regions closer to the magnetic equator. Combining lines of all elements together in the LSD line masks may lead to the dilution of the magnetic signal or even to its (partial) cancellation, if enhancements of different elements occur in regions of opposite magnetic polarities. Therefore, it is advisable to use in the measurements line masks constructed for individual elements. Among the REEs, Pr, Nd, and Eu are well known to be concentrated in surface spots, whereas Fe shows a rather uniform surface distribution. Using line masks constructed for \ion{Fe}{i}, \ion{Fe}{ii}, \ion{Pr}{iii}, \ion{Nd}{iii}, and \ion{Eu}{ii} for the magnetic field measurements, it is possible to get information about the differences in the individual surface distributions of these ions. We note that we do not actually know the magnetic field geometries of the stars in our sample, but we should still be able to identify the locations of surface element spots if different field strengths are derived by the application of the LSD technique to different line masks. \begin{table*} \caption{ The LSD mean longitudinal magnetic field measurements for all targets using five different line masks are presented along with the MJD values, number of lines in each mask, the average Land\'e factors, and the FAP values. DD means definite detection, MD marginal detection, and ND no detection. \label{tab:Bzelem} } \centering \begin{tabular}{llr r@{$\pm$}l lc llr r@{$\pm$}l lc} \hline \multicolumn{1}{c}{MJD} & \multicolumn{1}{c}{Mask} & \multicolumn{1}{c}{$\overline{g}_{\rm eff}$} & \multicolumn{2}{c}{$\left< B_{\rm z}\right>$} & \multicolumn{1}{c}{FAP} & \multicolumn{1}{c}{Det.} & \multicolumn{1}{c}{MJD} & \multicolumn{1}{c}{Mask} & \multicolumn{1}{c}{$\overline{g}_{\rm eff}$} & \multicolumn{2}{c}{$\left< B_{\rm z}\right>$} & \multicolumn{1}{c}{FAP} & \multicolumn{1}{c}{Det.}\\ \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{2}{c}{(G)} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{flag} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{2}{c}{(G)} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{flag} \\ \hline \multicolumn{7}{c}{HD\,89393} & \multicolumn{7}{c}{HD\,189832} \\ 58\,646.094 & \ion{Fe}{i} (49) & 1.16 & 343 & 9 & $<10^{-10}$ & DD & 58\,647.406 & \ion{Fe}{i} (97) & 1.22 & 151 & 4 & $<10^{-10}$ & DD \\ & \ion{Fe}{ii} (14) & 1.17 & 262 & 14 & $<10^{-10}$ & DD & & \ion{Fe}{ii} (17) & 1.11 & 160 & 11 & $<10^{-10}$ & DD \\ & \ion{Pr}{iii} (5) & 1.11 & 246 & 20 & $<10^{-10}$ & DD & & \ion{Pr}{iii} (10) & 0.96 & 184 & 21 & $<10^{-10}$ & DD \\ & \ion{Nd}{iii} (6) & 1.04 & 260 & 43 & 0.052 & ND & & \ion{Nd}{iii} (2) & 1.09 & 166 & 35 & 0.081 & ND \\ & \ion{Eu}{ii} (12) & 1.54 & 436 & 20 & $<10^{-10}$ & DD & & \ion{Eu}{ii} (15) & 1.50 & 84 & 20 & $1\times10^{-6}$ & DD \\ 58\,649.023 & \ion{Fe}{i} (49) & 1.16 & 349 & 10 & $<10^{-10}$ & DD & \multicolumn{7}{c}{HD\,203932} \\ & \ion{Fe}{ii} (14) & 1.17 & 240 & 16 & $<10^{-10}$ & DD & 58\,648.430 & \ion{Fe}{i} (98) & 1.21 & 3 & 5 & $<10^{-10}$ & DD \\\ & \ion{Pr}{iii} (5) & 1.11 & 292 & 18 & $<10^{-10}$ & DD & & \ion{Fe}{ii} (10) & 1.17 & 38 & 13 & $<10^{-10}$ & DD \\ & \ion{Nd}{iii} (6) & 1.04 & 249 & 39 & 0.026 & ND & & \ion{Pr}{iii} (8) & 0.92 & 99 & 42 & $<10^{-10}$ & DD \\ & \ion{Eu}{ii} (12) & 1.54 & 461 & 17 & $<10^{-10}$ & DD & & \ion{Nd}{iii} (24) & 1.19 & 37 & 15 & $<10^{-10}$ & DD \\ \multicolumn{7}{c}{HD\,137949} & & \ion{Eu}{ii} (5) & 1.65 & 150 & 19 & $<10^{-10}$ & DD \\ 56\,145.020 & \ion{Fe}{i} (98) & 1.20 & 1548 & 32 & $<10^{-10}$ & DD & \multicolumn{7}{c}{HD\,217522} \\ & \ion{Fe}{ii} (19) & 1.16 & 1291 & 55 & $<10^{-10}$ & DD & 56\,148.297 & \ion{Fe}{i} (80) & 1.23 & $-$358 & 7 & $<10^{-10}$ & DD \\ & \ion{Pr}{iii} (34) & 1.03 & 1941 & 48 & $<10^{-10}$ & DD & & \ion{Fe}{ii} (9) & 1.10 & $-$429 & 36 & $<10^{-10}$ & DD \\ & \ion{Nd}{iii} (14) & 1.24 & 1091 & 38 & $<10^{-10}$ & DD & & \ion{Pr}{iii} (6) & 1.07 & $-552$ & 59 & $<10^{-10}$ & DD \\ & \ion{Eu}{ii} (8) & 1.54 & 1368 & 30 & $<10^{-10}$ & DD & & \ion{Nd}{iii} (20) & 1.30 & $-$343 & 6 & $<10^{-10}$ & DD \\ \multicolumn{7}{c}{HD\,138633} & & \ion{Eu}{ii} (5) & 1.60 & $-$423 & 8 & $<10^{-10}$ & DD \\ 58\,650.051 & \ion{Fe}{i} (84) & 1.21 & $-$204 & 10 & $<10^{-10}$ & DD & 58\,647.438 & \ion{Fe}{i} (80) & 1.23 & $-$264 & 7 & $<10^{-10}$ & DD \\ & \ion{Fe}{ii} (11) & 1.07 & $-$160 & 27 & $<10^{-10}$ & DD & & \ion{Fe}{ii} (9) & 1.10 & $-$345 & 30 & $<10^{-10}$ & DD \\ & \ion{Pr}{iii} (11) & 1.07 & $-$257 & 30 & $<10^{-10}$ & DD & & \ion{Pr}{iii} (6) & 1.07 & $-$426 & 40 & $<10^{-10}$ & DD \\ & \ion{Nd}{iii} (4) & 1.06 & $-$174 & 26 & $7\times10^{-4}$& MD & & \ion{Nd}{iii} (20) & 1.30 & $-$309 & 6 & $<10^{-10}$ & DD \\ & \ion{Eu}{ii} (6) & 1.75 & $-$182 & 28 & $<10^{-10}$ & DD & & \ion{Eu}{ii} (5) & 1.60 & $-$413 & 8 & $<10^{-10}$ & DD \\ \multicolumn{7}{c}{HD\,174779} & \multicolumn{7}{c}{HD\,70702} \\ 58\,647.316 & \ion{Fe}{i} (21) & 1.25 & $-$75 & 12 & $<10^{-9}$ & DD & 58\,648.008 & \ion{Fe}{i} (20) & 1.26 & 4629 & 222 & $<10^{-10}$ & DD \\ & \ion{Fe}{ii} (14) & 1.20 & $-$11 & 3 & $<10^{-10}$ & DD & & \ion{Fe}{ii} (16) & 1.23 & 4144 & 153 & $<10^{-10}$ & DD \\ & \ion{Pr}{iii} (7) & 1.08 & $-$171 & 25 & $<10^{-10}$ & DD & & \ion{Pr}{iii} (22) & 1.02 & 4031 & 290 & $<10^{-10}$ & DD \\ & \ion{Nd}{iii} (35) & 1.17 & $-$69 & 5 & $<10^{-10}$ & DD & & \ion{Nd}{iii} (8) & 1.15 & 5788 & 467 & $<10^{-9}$ & DD \\ & \ion{Eu}{ii} (3) & 1.62 & 5 & 53 & 0.687 & ND & & \ion{Eu}{ii} (6) & 1.54 & 3223 & 183 & $<10^{-10}$ & DD \\ \multicolumn{7}{c}{HD\,176196} & 58\,650.973 & \ion{Fe}{i} (20) & 1.26 & 930 & 140 & $<10^{-10}$ & DD \\ 58\,648.375 & \ion{Fe}{i} (43) & 1.27 & 123 & 34 & $<10^{-10}$ & DD & & \ion{Fe}{ii} (16) & 1.23 & 833 & 74 & $<10^{-10}$ & DD \\ & \ion{Fe}{ii} (9) & 1.13 & 18 & 20 & $<10^{-10}$ & DD & & \ion{Pr}{iii} (22) & 1.02 & 1295 & 164 & $6\times10^{-7}$ & DD \\ & \ion{Pr}{iii} (11) & 1.08 & 177 & 47 & $<10^{-10}$ & DD & & \ion{Nd}{iii} (8) & 1.15 & 1000 & 185 & $2\times10^{-5}$ & MD \\ & \ion{Nd}{iii} (31) & 1.17 & 141 & 37 & $<10^{-10}$ & DD & & \ion{Eu}{ii} (6) & 1.54 & 2279 & 160 & $<10^{-9}$ & DD \\ & \ion{Eu}{ii} (2) & 1.61 & 129 & 68 & $8\times10^{-6}$& DD & & & & \multicolumn{2}{c}{} & & \\ 58\,650.344 & \ion{Fe}{i} (43) & 1.27 & 96 & 33 & $<10^{-10}$ & DD & & & & \multicolumn{2}{c}{} & & \\ & \ion{Fe}{ii} (9) & 1.13 & 60 & 20 & $<10^{-10}$ & DD & & & & \multicolumn{2}{c}{} & & \\ & \ion{Pr}{iii} (11) & 1.08 & 188 & 61 & $<10^{-10}$ & DD & & & & \multicolumn{2}{c}{} & & \\ & \ion{Nd}{iii} (31) & 1.17 & 117 & 35 & $<10^{-10}$ & DD & & & & \multicolumn{2}{c}{} & & \\ & \ion{Eu}{ii} (2) & 1.61 & 90 & 77 & $5\times10^{-4}$& MD & & & & \multicolumn{2}{c}{} & & \\ \hline \end{tabular} \end{table*} The LSD longitudinal magnetic field $\left< B_{\rm z}\right>$ measurements based on line masks that include the combined spectral lines belonging to different elements are listed in Table~\ref{tab:obsall}, and the corresponding LSD profiles are shown in Fig.~\ref{fig:IVNall}. The results of the LSD magnetic field measurements for all stars using individual line masks with the spectral lines of \ion{Fe}{i}, \ion{Fe}{ii}, \ion{Pr}{iii}, \ion{Nd}{iii}, and \ion{Eu}{ii} are presented in Table~\ref{tab:Bzelem}. For each star, the results of our magnetic field measurements and the available information on the periodicities are discussed in the following subsections. When studying the stars' periodicities using TESS observations, we identified a frequency as significant if it has a signal-to-noise ratio $S/N \geq 4.5$ and marginal if $3 \leq S/N < 4.5$. Any peaks with $S/N < 3$ were discarded as being consistent with noise and therefore merited no further investigation. \subsection{HD\,89393} Not much is known about this star. It is classified as A0\,SrCrEu in \citet{Renson}. Magnetic field measurements were not reported for this star in the past. In Fig.~\ref{fig:IVNHD89393}, we present the LSD Stokes~$I$, Stokes~$V$, and diagnostic null $N$ profiles for this star, calculated using five different masks. The mean longitudinal field strengths obtained at both epochs are almost identical (see also Tables~\ref{tab:obsall} and \ref{tab:Bzelem} and Fig.~\ref{fig:IVNall}), indicating the absence of any short-timescale variability. The amplitude of the Zeeman features and the measured field values clearly depend on the line masks constructed for individual elements, as the strongest mean longitudinal magnetic field is detected in the measurements using the \ion{Eu}{ii} line mask, suggesting a slightly different location (probably closer to the magnetic pole) of the Eu concentration on the stellar surface in comparison to other elements. The TESS light curve of HD\,89383 is shown in the left panel of Fig.~\ref{fig:PHD89383}. The \texttt{CROWDSAP} parameter suggests that there is significant contamination from a nearby star (TYC\,7712-2715-1). The \texttt{CROWDSAP} parameter refers to the fraction of flux that falls in the optimal aperture for the 2-minute data that is not directly attributable to the star in question (i.e.\ a ``crowding parameter'') and is calculated as part of the SPOC pipeline when data products are produced. The Gaia BP--RP colour index \citep{gaia2018} for this nearby contaminating star is greater than 1, suggesting that it is a cooler star. The periodogram shows just one marginal peak with a calculated $S/N$ of $\sim4$ at a period of $0.34116\pm0.00001$\,d. Since our metric is the relative amplitude of the peaks in the periodograms, we do not attach any physical interpretation to this marginal peak and are not able to conclude whether this frequency could represent a pulsation mode or arises from some other physical phenomenon, such as rotation of either HD\,89393 or the contaminating star. We assume that HD\,89393 likely has a long rotation period; this must at least be longer than the length of the TESS observations over 27\,d. Also the examination of the 10\,min cadence TESS data in sector 36 did not reveal any significant peak in our periodogram. \subsection{HD\,137949 = 33\,Lib} The star is classified as F0\,SrEuCr in \citet{Renson}. It is a roAp star with a pulsation period of 8.3\,min \citep{Kurtz82}. The study of \citet{Kervella} indicated the presence of a companion from an analysis of proper motions in the Gaia Data Release~2 \citep[GDR2;][]{gaia2018} and Hipparcos \citep{hipparcos} catalogues. A strong mean longitudinal magnetic field with a value of more than 1\,kG was initially discovered in this star by \citet{Babcock}. Given the presence of the strong magnetic field and the slow rotation, numerous spectral lines are resolved into their magnetically split components. The most complete review of the previous magnetic studies of 33\,Lib is presented in \citet{Mathys2017}. Their seven measurements of the mean magnetic field modulus $\left<B\right>$ using magnetically resolved lines in observations acquired between 1996 and 1998 appear to be constant in the range from 4.63 to 4.69\,kG, whereas the $\left<B_{\rm z}\right>$-values from the same observations range from 1.47 to 1.68\,kG. Taking into account all previous measurements of $\left<B_{\rm z}\right>$-values, \citet{Mathys2017} suggest that this star has a rotational period $P_{\rm rot}$ of 5195\,d (approximately 14.25\,yr). On the other hand, \citet{Giarrusso} recently reported that $\left<B\right>$ and $\left<B_{\rm z}\right>$-values have shown a slight increase over several years, indicating that the rotation period should be longer than 27\,yr. Our measurement of the mean field modulus using the HARPS\-pol observations obtained in 2012 August, $\left<B\right>=4.76\pm0.02$\,kG, is of the same order as reported by \citet{Giarrusso}. As an illustration, we show the Zeeman subcomponents of the resolved magnetically split \ion{Fe}{ii} line at 6149.258\,\AA{} in Fig.~\ref{fig:BmodHD137949}. No measurements of the mean longitudinal magnetic field after 2007 were presented by \citet{Giarrusso}. Our measurement of the mean longitudinal field strength using all lines, $\left<B_{\rm z}\right>=1.54\pm0.02$\,kG, does not confirm the slight increase, but the small differences in the field values can be caused by the choice of spectral lines used in the measurements. As we see in Fig.~\ref{fig:IVNHD137949}, which presents the LSD Stokes~$I$, Stokes~$V$, and diagnostic null $N$ profiles calculated using five different masks, and in Table~\ref{tab:Bzelem}, the amplitude of the Zeeman features and the measured field values for this star depend on the line masks constructed for individual elements: We measure $\left<B_{\rm z}\right>=1.94\pm0.05$\,kG using 34 \ion{Pr}{iii} lines, but only $\left<B_{\rm z}\right>=1.09\pm0.04$\,kG using 14 \ion{Nd}{iii} lines. In any case, the comparison of our measurements with previous measurements confirm a long rotation period for HD\,137949. A full analysis of short-period oscillations (typical for roAp stars) was done for HD\,137949 by \citet{K2-HD137949} based on 60-s cadence K2 observations. The authors found that the low-frequency peaks were aliases of the frequency splitting, and that the rotation period was likely extremely long. The amplitude spectra of 33\,Lib were presented in Figures~1 and 2 of \citet{K2-HD137949}. \subsection{HD\,138633} This star is classified as F0\,SrEuCr in \citet{Renson}. The presence of a weak magnetic field of the order of 0.7\,kG was announced by \citet{Titarenko}. They report that in contrast to ordinary Ap stars that exhibit strong REE lines in their spectra, these elements are represented only very poorly in the atmosphere of HD\,138633. A photometric study using the STEREO satellites suggested that this star is constant or probably constant \citep{Wraight2012}. On the other hand, \citet{Romanyuk2017} report on the change of the field polarity in the measurements of the mean longitudinal magnetic field in spectra acquired in 2010 on two different nights separated by 5 days, with $\bz=310\pm30$\,G and $\bz=-290\pm20$\,G. To check the short- and long-term line profile variability of HD\,138633, we compared the line profile shapes observed with HARPS\-pol in 2019 with those observed using UVES on 2004 March 5 and CES on 2002 January 25. In Fig.~\ref{fig:HD138633reg}, we display a few line profiles observed with these spectrographs in the same spectral region as shown in Fig.~\ref{fig:region}. Some tiny changes in the line shapes between the two CES spectra separated in time by 14 minutes seem to exist, but obviously higher $S/N$ data are needed to achieve any conclusion about any rapid spectral variability of this star. In Fig.~\ref{fig:IVNHD138633} we present the LSD Stokes~$I$, Stokes~$V$, and diagnostic null $N$ profiles calculated using five different masks. Using all lines for the measurements, we obtain a positive longitudinal magnetic field on the order of 200\,G, which is comparable to measurements carried out using different line masks (see Tables~\ref{tab:obsall}--\ref{tab:Bzelem}). The strongest field, $\left<B_{\rm z}\right>=257\pm30$\,G, is detected for the \ion{Pr}{iii} line mask. Based on the strengths of the LSD Stokes~$I$ profiles of the studied REEs, we cannot confirm the finding of \citet{Titarenko} that these elements are represented very poorly in the atmosphere of HD\,138633. Due to significant scattered light from both the Moon and the Earth entering Camera 1, the short-cadence TESS data of HD\,138633 suffers from a data gap of approximately 10 days. Using the available photometry, we performed a DFT (see, Fig.~\ref{fig:PHD138633}) and were unable to find any significant peaks in the periodogram that could correspond to either pulsations or rotation. This suggests that this star has a long rotational period that is at least longer than the length of the TESS observations. Due to the short length and poor quality of the TESS short-cadence data, we are unable to identify any evidence for the finding of \citet{Titarenko} that this star belongs to the group of rapidly oscillating Ap stars and possesses a pulsation period of $\sim17$\,min. \subsection{HD\,174779} The star is classified as A0\,Si in \citet{Renson}. \citet{CatRen} noted that it exhibits variability in its luminosity and/or colour, but do not report an associated period. The study of \citet{Kervella} indicated the presence of a companion from an analysis of proper motions in the GDR2 and Hipparcos catalogues. Magnetic field measurements have not been reported for this star in the past. Our measurements presented in Tables~\ref{tab:obsall} and \ref{tab:Bzelem} and in Figs.~\ref{fig:IVNall} and \ref{fig:IVNHD174779} show that the magnetic field is very weak, with $\left<B_{\rm z}\right>=-45\pm3$\,G measured using all lines. It is striking that for the \ion{Pr}{iii} line mask our measurements yield a much higher field strength, $\left<B_{\rm z}\right>=-171\pm25$\,G, while the measurements using other masks are all below $-75$\,G. These results show that great care should be taken in the selection of line lists for the measurements of weak magnetic fields. The analysis of the TESS observations of HD\,174779 reveals two marginal peaks in the periodogram (see Fig.~\ref{fig:PHD174779}). The first corresponds to a period of $3.7933\pm0.0003$\,d with S/N of 4.11. The second, higher-frequency peak with a similar S/N corresponds to the period of $1.11473\pm0.0001$\,d. We avoid any physical interpretation of these two peaks due to their marginal significance, and note that there is no clear, salient evidence of rotation from the light curve. \subsection{HD\,176196} This star is classified as B9\,EuCr in \citet{Renson}. The study of \citet{Kervella} indicated the presence of a companion from an analysis of proper motions in the GDR2 and Hipparcos catalogues. First detections of the mean longitudinal magnetic field, $\left<B_{\rm z}\right>=258\pm69$\,G and $\left<B_{\rm z}\right>=174\pm58$\,G, were reported by \citet{Hubrig2006} using low resolution spectropolarimetry with the FORS\,1 instrument installed at the ESO VLT. Our measurements, separated by two nights, are presented in Tables~\ref{tab:obsall} and \ref{tab:Bzelem} and in Figs.~\ref{fig:IVNall} and \ref{fig:IVNHD176196} and show almost identical, rather low, mean longitudinal field strengths on the order of 100--120\,G. For the measurements using the \ion{Pr}{iii} line mask, the field strength reaches 180--190\,G. Given the small change in the field strength since the first observations by \citet{Hubrig2006}, it is very likely that the rotation period of HD\,176196 is quite long. \subsection{HD\,189832} Not much is known for this star. It is classified as A6\,SrCrEu in \citet{Renson}. The study of \citet{Kervella} indicated the presence of a companion from an analysis of proper motions in the GDR2 and Hipparcos catalogues. A rotational period $P_{\rm rot}$ of 18.89\,d was mentioned in \citet{ManMat}, who used ground-based photometry to arrive at this value. Magnetic field measurements have not been reported for this star in the past. Similar to HD\,176196, our measurements for this star, displayed in Tables~\ref{tab:obsall} and \ref{tab:Bzelem} and in Figs.~\ref{fig:IVNall} and \ref{fig:IVNHD189832}, show the presence of a positive weak magnetic field, with a somewhat stronger field, $\left<B_{\rm z}\right>=184\pm21$\,G, measured using the \ion{Pr}{iii} line mask. \subsection{HD\,203932} This star is classified as A5\,SrEu in \citet{Renson}. It is a roAp star \citep{Kurtz84}, with a pulsation period of about 6.2\,min. \citet{Hold2021} reported a rotational period of 6.44\,d, based on TESS data. \citet{MatHub97} were unable to identify the presence of a magnetic field. Later on, \citet{Hubrig2004b} reported $\left<B_{\rm z}\right>=-267\pm72$\,G measured in low-resolution FORS\,1 polarimetric spectra acquired in 2002. Our analysis, shown in Tables~\ref{tab:obsall} and \ref{tab:Bzelem} and in Figs.~\ref{fig:IVNall} and \ref{fig:IVNHD203932}, yields the lowest mean longitudinal field strength when all lines are used in the measurement, but the field strength is as high as 150\,G in the measurements using only the \ion{Eu}{ii} line mask. Notably, the shape of the calculated Zeeman features in Figs.~\ref{fig:IVNall} and \ref{fig:IVNHD203932} corresponds to a typical crossover profile, which results from the correlation between the Zeeman effect and the rotation-induced Doppler effect across the stellar surface. \subsection{HD\,217522} This star is classified as A5\,SrEuCr in \citet{Renson}. It is a rapidly pulsating star with a pulsation period of about 13.7\,min \citep{Medupe}. \citet {Hubrig2002} showed that HD\,217522 is very similar to Przybylski’s star (HD\,101065), which exhibits the most complex spectra known, with numerous lines of lanthanides and also rotates extremely slowly, with a probable $P_{\rm rot}$ of about 188\,yr \citep{Hubrig2018}. The study of \citet{Kervella} indicated the presence of a companion from an analysis of proper motions in the GDR2 and Hipparcos catalogues, but no companion candidate was detected using diffraction-limited near-infrared imaging with NAOS-CONICA at the VLT by \citet{Schoeller2012}. The first detection of a magnetic field of about $-400$\,G was reported by \citet{MatHub97} using the ESO Cassegrain Echelle Spectrograph (CASPEC), fed by the ESO 3.6\,m telescope, for observations acquired in 1992. One additional observation was obtained in 1997 with $\left<B_{\rm z}\right>=-559\pm63$\,G \citep{Hubrig2002}. Later, \citet{Hubrig2004b} reported $\left<B_{\rm z}\right>=-725\pm88$\,G using low-resolution spectropolarimetry with FORS\,1. No variability of the field strength over the pulsation cycle was detected by \citet{Hubrig2004a} using a FORS\,1 spectropolarimetric time series. Our analysis presented in Tables~\ref{tab:obsall} and \ref{tab:Bzelem} and in Figs.~\ref{fig:IVNall} and \ref{fig:IVNHD217522} is based on two HARPS\-pol observations, one from 2012 August and the second one from 2019 June. A comparison of the measurements indicates that the field in the measurements using all lines has slightly decreased from $\left<B_{\rm z}\right>=-401\pm6$\,G to $\left<B_{\rm z}\right>=-323\pm6$\,G. Assuming that the maximum field strength was reached in the year 2004, the expected rotation period should be at least of the order of a few tens of years. The strongest field strength was measured using the \ion{Pr}{iii} line mask, reaching $\left<B_{\rm z}\right>=-552\pm59$\,G in the observations acquired in 2012. \subsection{HD\,70702} This star is classified as B9\,EuCrSr in \citet{Renson} and was used in our observations as a standard star. It possesses an extremely strong magnetic field with a mean field modulus of the order of 14--15\,kG. Because of the strength of this field, numerous lines in the spectra of HD\,70702 appear resolved into their magnetically split components \citep{Elkin}. Both of our HARPS spectra are rather noisy and, in addition, the precision of the measurements is limited by the distortion of the lines as a result of the combination of the Zeeman effect and the rotational Doppler effect. The measurements of the mean magnetic field modulus, $\left<B\right>=15.2\pm0.9$\,kG (from a spectrum obtained on 2019 July 13) and $\left<B\right>=14.0\pm0.5$\,kG (from a spectrum obtained on 2019 July 16), are presented in Fig.~\ref{fig:BmodHD70702}. They are in good agreement with the previously published value of 15\,kG in \citet{Elkin}. In Fig.~\ref{fig:IVNHD70702}, we present the LSD Stokes~$I$, Stokes~$V$, and diagnostic null $N$ profiles calculated for both observing epochs using five different masks. The strong variability of HD\,70702 is obvious and is clearly reflected in the measurement results presented in Tables~\ref{tab:obsall} and \ref{tab:Bzelem}, as well as in the distinct changes of the amplitude of the Stokes~$V$ profiles between both epochs. The strongest mean longitudinal magnetic field is detected using the \ion{Nd}{iii} line mask in the observations from the first epoch, suggesting that this element, in comparison to other elements, is concentrated in a region of the surface located closer to the magnetic pole. In the DFT for this star, we combined the data from Sectors 8 and 9 to find three significant periods. Contamination from other stars is likely minimal, since the \texttt{CROWDSAP} parameter suggests that only 1.5\% of the flux in the optimal aperture does not belong to HD\,70702. We observe a strong rotational modulation with $P_{\rm rot} = 3.7601 \pm 0.0007$\,d, as well as a short-period modulation -- perhaps due to pulsation -- at $P_{\rm puls} = 0.39019 \pm 0.00001$\,d. The other significant peak, at $P = 1.8782 \pm 0.0002$\,d, has a lower $S/N$ than the others ($\sim5$), but still meets our significance criterion. This peak can be identified as a harmonic of $P_{\rm rot}$, given that it is exactly twice the frequency (to within the uncertainty of the data) of the lower-frequency peak. Using the same argument, the small peak visible near $\nu = 5$\,d$^{-1}$ is almost certainly a harmonic of $P_{\rm puls}$. The light curve and Fourier transform for HD\,70702 are shown in Fig.~\ref{fig:PHD70702}. Since the TESS data show a clear rotation period, we can calculate the phases of our HARPS observations. Using as initial epoch $T_{0}$ the value TJD\,1564.13520327 (=JD\,2458564.13520327), which corresponds to the time of minimum light, we calculate $\varphi=0.306$ for the first observation and $\varphi=0.094$ for the second observation. Since this star has a rather short rotation period and possesses an extremely strong magnetic field, it is an excellent candidate for future spectropolarimetric monitoring to map its magnetic field and chemical spots using Zeeman Doppler Imaging. \section{Discussion and conclusions} \label{sec:dis} Among the three sharp-lined stars not previously studied for the presence of a magnetic field (HD\,89393, HD\,174779, and HD\,189832), the weakest mean longitudinal magnetic field, $\left< B_{\rm z}\right>=-45\pm3$\,G was detected in HD\,174779 in the LSD measurements using all lines, i.e.\ based on the line mask combining the five individual line masks. The magnetic field measurements of HD\,189832 yielded a field strength of 140\,G, whereas the field strength for HD\,89393 was 300\,G. Considering the field measurements using the five line masks separately, the strongest magnetic field was always detected for a line mask corresponding to one of the REEs: in the \ion{Eu}{ii} lines for HD\,89393 and in the \ion{Pr}{iii} lines for HD\,174779 and HD\,189832. TESS observations of HD\,89393 and HD\,174779 revealed only the presence of marginal peaks at low $S/N$, indicating that the rotation periods of these stars are likely long. HD\,189832 was reported to have a $P_{\rm rot}=18.89$\,d from ground-based photometry in \citet{ManMat}. The presence of weak magnetic fields in the sharp-lined stars HD\,138633, HD\,176196, HD\,203932, and HD\,217522 was already discussed in previous studies, but the detections were, in most cases, based on low-resolution spectropolatimetric observations. The weakest mean longitudinal magnetic field, $\left< B_{\rm z}\right>=21\pm4$\,G, was detected in HD\,203932 using all lines for the measurements. Field strengths of the order of 120\,G and 200\,G were measured for HD\,176196 and HD\,138633, respectively. The mean longitudinal magnetic field of HD\,217522 decreased from $-400$\,G to $-323$\,G over the seven years from 2012 to 2019. As for the field measurements using five line masks separately, similar to the three above discussed sharp-lined stars lacking previous magnetic field determinations, the strongest magnetic field was always detected in a line list corresponding to one of the REEs, in \ion{Pr}{iii} lines for three stars and in \ion{Eu}{ii} lines for HD\,203932. These results clearly show that the most promising way to detect weak magnetic fields in sharp-lined Ap stars is to apply the LSD technique to a line mask belonging to REEs. Our finding that stronger fields are detected using REE line masks is probably explained by the location of surface REE patches in the vicinity of the magnetic poles. TESS observations show a short $P_{\rm rot}$ of 6.44\,d for HD\,203932, which exhibits in the polarimetric spectrum a Zeeman feature with a typical crossover profile. Longer rotation periods are suggested for HD\,89393, HD\,138633, HD\,174779, HD\,176196, and HD\,217522, either from TESS observations or spectropolarimetric observations. A very long rotation period, on the order of tens of years, is suggested for the strongly magnetic roAp star HD\,137949. Summarising the results of our study, out of seven sharp-lined stars, two stars, HD\,174779 and HD\,203932, exhibit a rather weak longitudinal magnetic field. We should, however, keep in mind that the dipole strengths for these two stars are not known yet, due to the absence of their $\left< B_{\rm z}\right>$ phase curves. The fact that the majority of the studied stars are slow rotators -- apart from HD\,189832 and HD\,203932, all other studied sharp-lined stars have long rotation periods -- is in agreement with our expectations that the inclination angles of the rotation axes of our sample stars with respect to our line of sight are randomly distributed. This is also in agreement with the previous work of \citet{Hubrig2007}. On the other hand, the absence of rotational variability can also be expected if the magnetic obliquity $\beta$ with respect to the rotation axis is very small \citep[e.g.][]{MathysTESS}. Interestingly, the study by \citet{Hubrig2007} on the evolution of magnetic fields in stars across the upper main sequence using a sample of 90 Ap and Bp stars with accurate Hipparcos parallaxes and definitely determined longitudinal magnetic field phase curves revealed that the angle $\beta$ is smaller than 20\degr{} in slower rotating stars. The rotation periods of Ap stars span up to five or six orders of magnitude \citep[e.g.][]{MathysSVOS}, but no evidence was previously found for any loss of angular momentum during the main-sequence lifetime \citep[e.g.][]{Hubrig2007}. \citet{MathysPASE} denoted all Ap and Bp stars with rotation periods longer than 50\,d as super-slowly rotating Ap (ssrAp) stars and presented accurate periods for 33 such targets, with the 29\,yr period of HD\,50169 being the longest of them \citep{Mathys2019}. A rotation period of about 35462.5\,d ($\sim97$\,yr) was suggested by \citet{gammaEqu} for the Ap star $\gamma$\,Equ (=HD\,201601), and of about 188\,yr for Przybylski’s star (=HD\,101065) by \citet{Hubrig2018}. It was also suggested that weak-field Ap stars may potentially represent a significant fraction of the group of very slowly rotating stars, with rotation periods reaching several hundred years \citep{MathysSVOS}. An explanation for the extremely slow rotation of a fraction of Ap stars is currently missing, although a first theoretical attempt to understand this phenomenon was recently presented by \citet{Kitchatinov}, who suggested a possible scenario in terms of a longitudinal drift of the unstable disturbances of a kink-type (Tayler) instability of the stellar internal magnetic field. Our studied sample of sharp-lined Ap stars is rather small and does not allow us to decide whether a critical value for the stability of a large-scale magnetic field indeed exists, or if previous observations are incomplete, insofar as they are missing a sizeable population of chemically peculiar sharp-lined stars without any magnetic or Doppler line broadening. Therefore, additional systematic, multi-epoch $\bz$ determinations for the known Ap and Bp stars with sharp unresolved spectral lines are needed to characterise the distribution of the slowly rotating Ap stars, both in terms of their magnetic field strengths and rotation periods, in order to provide essential clues for the theoretical understanding of the formation and evolution of these stars. \section*{Acknowledgements} We thank the anonymous referee for their helpful comments. We also thank Gautier Mathys for the discussion on several stars in our sample. Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 68.D-0445, 072.D-0138, 089.D-0383, and 0103.C-0240. This paper includes data collected by the TESS mission. Funding for the TESS mission is provided by the NASA Science Mission Directorate. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center to produce the SPOC data products. This work has made use of the VALD, operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and the University of Vienna. \section*{Data Availability} The data obtained with ESO facilities are available in the ESO Archive at http://archive.eso.org/ and can be found with the instrument and object name. TESS light curves are publicly available through the Mikulski Archive for Space Telescopes (https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html). \label{lastpage}

Title: The Yarkovsky effect in REBOUNDx

Abstract: To more thoroughly study the effects of radiative forces on the orbits of small, astronomical bodies, we introduce the Yarkovsky effect into REBOUNDx, an extensional library for the N-body integrator REBOUND. Two different versions of the Yarkovsky effect (the "Full Version" and the "Simple Version") are available for use, depending on the needs of the user. We provide demonstrations for both versions of the effect and compare their computational efficiency with another previously implemented radiative force. In addition, we show how this effect can be used in tandem with other features in REBOUNDx by simulating the orbits of asteroids during the asymptotic giant branch phase of a 2 $M_{\odot}$ star. This effect is made freely available for use with the latest release of REBOUNDx.

https://export.arxiv.org/pdf/2208.09987

\title{The Yarkovsky effect in REBOUNDx} \author{Noah Ferich} \affiliation{Department of Astrophysical \& Planetary Sciences, University of Colorado Boulder, Boulder, CO 80309, USA \\} \author{Stanley A. Baronett} \affiliation{Department of Physics and Astronomy, University of Nevada, Las Vegas, Box 454002, 4505 S. Maryland Pkwy., Las Vegas, NV 89154-4002, USA\\} \affiliation{Nevada Center for Astrophysics, University of Nevada, Las Vegas, Box 454002, 4505 S. Maryland Pkwy., Las Vegas, NV 89154-4002, USA\\} \author{Daniel Tamayo} \affiliation{Department of Astrophysical Sciences, Princeton University, 4 Ivy Ln, Princeton, NJ 08544, USA} \author{Jason H. Steffen} \affiliation{Department of Physics and Astronomy, University of Nevada, Las Vegas, Box 454002, 4505 S. Maryland Pkwy., Las Vegas, NV 89154-4002, USA\\} \affiliation{Nevada Center for Astrophysics, University of Nevada, Las Vegas, Box 454002, 4505 S. Maryland Pkwy., Las Vegas, NV 89154-4002, USA\\} \correspondingauthor{Noah Ferich} \email{noah.ferich@colorado.edu} \keywords{Astronomy data modeling (1859) --- Astronomy software (1855) --- N-body simulations (1083) --- Radiative processes (2055) --- Asteroid dynamics (2210) --- Orbital evolution (1178)} \section{Introduction} \label{sec:intro} Radiation pressure \citep{Nichols1903}, Poynting-Robertson drag \citep{Poynting1904, Robertson1937}, the Yarkovsky effect \citep{Radzievskii1954, Peterson1976}, and the YORP (Yarkovsky-O’Keefe-Radzievskii-Paddack) effect \citep{Radzievskii1954, Paddack1969, OKeefe1976} are four radiative forces that can be important for the orbital evolution of small bodies. Both radiation pressure and Poynting-Robertson drag (PR drag) arise from the absorption and scattering of radiation by dust and other particles \citep{Burns1979}. Radiation pressure is generally a force that propels particles away from their host stars, while PR drag is a force that causes particles to lose momentum and spiral inwards \citep{Veras2015}. Unlike radiation pressure and PR drag, the Yarkovsky and YORP effects appear only when considering objects' angular momentum \citep{Vokrouhlicky2015}. Both phenomena occur after a radiation-induced temperature gradient forms on the surface of a body, causing an uneven emission of radiation between its hemispheres. As an object rotates relative to the substellar point, this imbalance in the emission creates a net force on the object. While both effects are based on this principle, the Yarkovsky effect changes the orbit of a body, while the YORP effect creates a torque that changes the body's spin. In addition, the YORP effect only operates on non-spherical, asymmetric objects while the Yarkovsky effect can operate on spherical ones. Even with the differences between them, all of these effects can have important consequences on the dynamical evolution of small bodies; this motivates developing tools to numerically model these phenomena. \reb\footnote{Documentation is available at \url{https://rebound.readthedocs.io}.} is an open-source, \Nbody\ integrator for gravitational dynamics \citep{Rein2012}. Flexible and efficient, \reb\ can be used in \C\ and is also available as a \python\ package. \rebx\footnote{Documentation is available at \url{https://reboundx.readthedocs.io}.} is an extensional library to \reb\ that contains additional effects and forces (general relativity, tidal dissipation, etc.) that users can add to simulations \citep{Tamayo2020}. This library includes radiative forces \citep{Tamayo2020}, but contained only radiation pressure and PR drag and was lacking both the Yarkovsky and YORP effects. This paper introduces the Yarkovsky effect as another radiative force available in \rebx\ and shows its capabilities and potential for future work. We present two separate versions of the Yarkovsky effect available for use in \rebx: the Full Version and the Simple Version. While both of these versions are based on the Yarkovsky effect model derived in \citet{Veras2015}, the Simple Version includes modifications done by \citet{Veras2019} that compromise long-term accuracy for a reduction in computational time and the number of necessary parameters. More detail will be given in later sections on the differences between these two versions and when to appropriately use them. Please note that rotational perturbations created by the YORP effect are not included within this \rebx\ module. YORP requires detailed models of the surface of a body \citep{Nesvorny2007, Scheeres2008} that are beyond the scope of this paper. For now, we assume that all bodies are spherical and have a uniform density. Adding the YORP effect into \rebx\ will be left for future work. The Yarkovsky effect has previously been added to other \Nbody\ integrators, including \mercury\ \citep{Chambers1999} and \swift\ \citep{Duncan1997}. While these integrators have their own advantages, \reb\ contains features that make it better suited for a wide variety of studies. \reb\ contains a larger array of different integrating schemes, providing nine different options for numerical integrators while \swift\ and \mercury\ only provide four and five, respectively. In addition, \reb\ comes standard with features such as collision tracking and the ability to create simulation archives that save particle and orbital parameters at certain times during an ongoing simulation. Features like these would have to be manually added to these other codes by users. \reb\ will also always yield results that are identical across all platforms and compilers. Finally, \rebx\ gives users greater flexibility when running simulations in \reb, allowing them to easily add multiple effects at once to simulations (see Section \ref{sec:combining_effects}). Section~\ref{sec:yark_in_rebx} describes the main equations that govern the Yarkovsky effect in \rebx. In Section~\ref{sec:full_version}, we introduce the Full Version of the effect and demonstrate one of its possible applications, and in Section~\ref{sec:simple_version}, we introduce the Simple Version and show how an analytic equation describing the evolution of an object's semi-major axis can be derived from it under specific circumstances. Finally, we show how the Yarkovsky effect can be combined with other previously implemented \rebx\ effects and test how they perform together in Section~\ref{sec:combining_effects}. \section{The Yarkovsky Effect in REBOUNDx} \label{sec:yark_in_rebx} Equation (27) from \citet{Veras2015} provides the model for calculating the acceleration on a body due to the Yarkovsky effect. It is implemented by both versions of the Yarkovsky effect in \rebx. \begin{equation} \left(\frac{d\textit{\textbf{v}}}{dt}\right)= \frac{3kL(1-\alpha)}{16\pi \rho Rcr^2}\mathbb{Y}\textit{\textbf{i}}, \label{eq:veras2015} \end{equation} where $L$ is the luminosity of the object's host star\footnote{$L$ can be constant or time-dependent. Section~\ref{sec:yark_effect_parameter_interpolation} gives an example of how the luminosity can be configured to change over time in a simulation.} and $c$ is the speed of light. In addition, $\alpha$, $R$, and $\rho$ are the bond albedo, radius, and density of the body, and $r$ is its distance from the star. The 3 $\times$ 3 matrix $\mathbb{Y}$ is the rotation matrix that contains the physics of the Yarkovsky effect, and $\textit{\textbf{i}}$ is the relativistically-corrected direction of the incoming radiation. The parameter $k$ is a constant between 0 and 0.25; it was created to avoid the need to model the complex spin behavior of most asteroid-like objects. Realistically, its value will change over time, but in these simulations, it remains constant to reduce complexity. If the target’s rotation speed approaches the critical rotation speed at which the target will begin to break up \citep{Walsh2008, Vokrouhlicky2015}, then $k \rightarrow 0$. If the target’s rotation period approaches its orbital period, then $k \rightarrow 0.25$. Equation~(\ref{eq:veras2015}) includes the physics for both the diurnal and seasonal components of the Yarkovsky effect. Both of these contributions are based on the same mechanism: the heated side of a body will release photons that are both more abundant and more energetic than those from the dark side. This leads to a net force on the body opposite the emission direction \citep{Bottke2002}. The changes in the orbital parameters of an object depend on how this heated side is oriented when this radiation is emitted. The diurnal contribution comes from the temperature differences created in the body due to its rotation with respect to its orbit. This component can cause an increase or decrease in the semi-major axis of a body, depending on the orientation of its spin axis. The seasonal contribution comes from temperature differences created due to the specific angular momentum of an object in its orbit. This component will always cause a decrease in the semi-major axis of the body and does not depend on the orientation or magnitude of the object's spin. Figure 1 from \citet{Bottke2002} provides a visual diagram of these two contributions, and Section~\ref{sec:full_version_demonstration} provides more information on how they compare and depend on different model parameters. Both of these components are required for a complete model of the Yarkovsky effect. We will now discuss the equations that comprise both $\mathbb{Y}$ and $\textit{\textbf{i}}$. The vector $\textit{\textbf{i}}$ is described by the following: \begin{equation} \textit{\textbf{i}} = \left(1-\frac{\textit{\textbf{v}} \cdot \textit{\textbf{r}}}{cr}\right)\frac{\textit{\textbf{r}}}{r}-\frac{\textit{\textbf{v}}}{c}, \end{equation} where $\textit{\textbf{r}}$ is the position vector of the body and $\textit{\textbf{v}}$ is its velocity vector. If $\lvert \lvert \textit{\textbf{v}} \rvert \rvert << c $ then $\textit{\textbf{i}} \approx \textit{\textbf{r}} / r$, which is the unit vector in the direction of the incoming radiation. The rotation matrix $\mathbb{Y}$ is the product of two matrices: \begin{equation} \mathbb{Y} = \mathbb{R}_{Y}(\textit{\textbf{s}}, \phi)\mathbb{R}_{Y}(\textit{\textbf{h}}, \xi), \end{equation} where $\mathbb{R}_{Y}(\textit{\textbf{s}}, \phi)$ is the contribution from the diurnal component of the Yarkovsky effect and $\mathbb{R}_{Y}(\textit{\textbf{h}}, \xi)$ is the contribution from the seasonal component of the effect. The vector $\textit{\textbf{s}}$ is the spin axis of the body, $\phi$ is the thermal lag angle along the equator of the body, $\textit{\textbf{h}}$ is the object's specific angular momentum axis, and $\xi$ is the thermal lag angle in the orbital plane of the body. The equations for $\phi$ and $\xi$ are based off the specific 1D conduction model found in \citet{Broz2006}: \begin{equation} \tan(\phi) = \left(1+\frac{1}{2}\left(\frac{\sigma \epsilon}{\pi ^ 5}\right)^\frac{1}{4}\left(\frac{\Sigma^\frac{1}{2}}{\Gamma}\right)\left(\frac{L(1-\alpha)}{r^2}\right)^\frac{3}{4}\right)^{-1}, \label{eq:phi} \end{equation} \begin{equation} \tan(\xi) = \left(1+\frac{1}{2}\left(\frac{\sigma \epsilon}{\pi ^ 5}\right)^\frac{1}{4}\left(\frac{\Pi^\frac{1}{2}}{\Gamma}\right)\left(\frac{L(1-\alpha)}{r^2}\right)^\frac{3}{4}\right)^{-1}, \label{eq:xi} \end{equation} where $\sigma$ is the Stefan-Boltzmann constant, $\epsilon$ is the object's emissivity, $\Sigma$ is its rotational period, $\Pi$ is its orbital period, and $\Gamma$ is its thermal inertia. Both $\phi$ and $\xi$ only have meaningful values between $0^\circ$ and $45^\circ$. The full equations and derivations for $\mathbb{R}_{Y}(\textit{\textbf{s}}, \phi)$ and $\mathbb{R}_{Y}(\textit{\textbf{h}}, \xi)$ can be found in \citet{Veras2015}. Section~\ref{sec:full_version} and Section~\ref{sec:simple_version} will describe how the two versions of the Yarkovsky effect are implemented in \rebx. \section{Full Version} \label{sec:full_version} The Full Version of the Yarkovsky effect in \rebx\ is directly based on the model for the Yarkovsky effect found in \citet{Veras2015}. Setting a \rebx\ parameter called "ye\_flag" to 0 on bodies in \reb\ simulations will apply this version of the effect to these chosen targets. Users must input all of the necessary parameters contained in the model for this effect to work. As is standard for all effects in \rebx, the parameters must be inputted with the same units as the \reb\ simulation where the effects are being added. The large number of parameters included in this version of the effect provides a high level of detail for the perturbations created by the Yarkovsky effect on a particular object. However, this means long simulations containing many bodies with the Full Version active will require a large amount of computational time (see Section~\ref{sec:time_performance}). \subsection{Demonstration} \label{sec:full_version_demonstration} To demonstrate the Full Version of the effect, we perform a simple parameter study that explores how the obliquity, radius, and thermal inertia of an asteroid affect the strength of the Yarkovsky effect. This is similar to a study \citet{Carruba2017} performed to create Figure 15 in their study of the Veritas asteroid family. We create a 0.01 Myr-long simulation that contains the Sun and an asteroid in a circular, non-inclined orbit at a distance of 3.165802 au, the semi-major axis of 7231 Veritas \citep{Carruba2017}. This value was chosen to place the asteroid at a realistic distance within the Veritas asteroid family. Table~\ref{tab:parameter_table} lists the unchanging physical properties of the asteroid. The density, rotation period, and albedo are the same values that \citet{Carruba2017} used in their creation of Figure 15. Because a value for the emissivity was not stated in \citet{Carruba2017}, we have chosen to use an estimate for the emissivity of 101955 Bennu from \citet{Emory2014}. Given that 101955 Bennu and the members of the Veritas family are all carbonaceous asteroids and generally similar in composition, the emissivity of 101955 Bennu should be a reasonable estimate for the emissivity of a Veritasian asteroid. Each time the simulation is run, the asteroid is given a new combination for its obliquity, radius, and thermal inertia. The object can have radii values of 10 m, 100 m, and 1000 m; obliquity values of $0^\circ$, $30^\circ$, $60^\circ$, $90^\circ$, $120^\circ$, $150^\circ$, and $180^\circ$; and a distribution of 50 thermal inertia values ranging from 0.1 to 4000. The change in semi-major axis of the asteroid is recorded at the end of each simulation. \begin{table} \centering \caption{Unchanging Physical Properties of the Asteroid in the Full Version Parameter Study} \begin{tabular}{lcc} \hline \hline Property & Value and Uncertainty & Reference\\ \hline Density & $1300$ \footnote{in units of kg m$^{-3}$.} & \citet{Carruba2017}\\ Rotation Period & \textbf{$6$} \footnote{in units of hours.} & \citet{Carruba2017}\\ Emissivity & $0.90 \pm 0.05\ $ & \citet{Emory2014}\\ Albedo & $0.07$ & \citet{Carruba2017}\\ \hline \end{tabular} \label{tab:parameter_table} \end{table} Figure~\ref{fig:fig1} shows how the change in semi-major axis of these simulated asteroids depends on these changing parameters. As expected from Equation~(\ref{eq:veras2015}), the change in semi-major axis increases as the radius of the body decreases. In addition, Equations~(\ref{eq:phi}) and~(\ref{eq:xi}) describe how bodies with small thermal inertias will radiate away too much heat before they rotate enough for the effect to activate, leading to a nearly negligible change in semi-major axis from the Yarkovsky effect. Figure~\ref{fig:fig1} also shows how the strength of the diurnal and seasonal components of the Yarkovsky effect depend on an object's physical properties. For example, the lines in Figure~\ref{fig:fig1} begin to noticeably trend downwards as thermal inertia reaches values above 1000\footnote{in SI units.}. Objects with extremely large thermal inertias will hold onto thermal energy for longer periods of time as they move throughout their orbits, leading to an increased contribution from the seasonal part of the Yarkovsky effect when compared to the diurnal portion. This, combined with the fact that the seasonal component always leads to a decrease in the body's semi-major axis, explains the increased negative change in semi-major axis for bodies with higher thermal inertias. In addition to thermal inertia, obliquity plays a large role in determining the relative strengths of the seasonal and diurnal components of the Yarkovsky effect. Bodies with obliquities at $0^\circ$ and $180^\circ$ will feel the strongest perturbations in their semi-major axis from the diurnal component. As the obliquity moves towards $90^\circ$, the diurnal portion's effect on the object's semi-major axis will begin to decrease and its effect on the object's inclination will start to increase. At an obliquity of $90^\circ$, only the seasonal component will affect the body's semi-major axis. This is why objects at $90^\circ$ in Figure~\ref{fig:fig1} remain relatively unperturbed until their thermal inertias reach a high enough value where the seasonal component becomes much more significant. In between these extreme values, the body will experience the combined effects of the diurnal and seasonal components. However, because the object's orbital mean motion is small compared to its rotational frequency, the diurnal portion will dominate for most obliquity and thermal inertia values and be the largest determining factor for the object's change in semi-major axis. This is why, in Figure~\ref{fig:fig1}, the object's overall change in semi-major axis is greater for obliquity values that are closer to $0^\circ$ or $180^\circ$ where the diurnal component is strongest in the plane of the object's orbit. Users that want to artificially decrease or eliminate the effects of the diurnal component to make the seasonal component more prominent can simply increase the body's rotation period to an arbitrarily high value in their simulations. From this model, observational information about changes in the orbital distance of an orbiting body can be used to constrain the physical properties of that body, as was done by \citet{Vokrouhlicky2008} to constrain several physical parameters of the asteroid 1992 BF and \citet{Tardioli2017} to constrain the obliquities of multiple near-Earth Asteroids. The Full Version of the effect can also be used to approximate the age of asteroid families by determining how the orbits of their constituent asteroids evolved since the family's creation. This information can then be used to determine the time when these asteroids began diverging from the collision that created them. \citet{Nesvorny2004} and \citet{Carruba2017} used this technique to decrease the uncertainty in the ages of the Karin and Veritas asteroid families, respectively. \section{Simple Version} \label{sec:simple_version} The Simple Version of the Yarkovsky effect in \rebx\ is based on modifications made to Equation~(\ref{eq:veras2015}) by \citet{Veras2019}. Recall from Equation~(\ref{eq:veras2015}) that $\mathbb{Y}$ is the Yarkovsky matrix that describes an object's rotation and thermal emission. Calculating this matrix between each time step of a \reb\ simulation is time-consuming, and using it requires many parameters. To prevent the need for such a complicated heat conduction model, \citet{Veras2019} placed constant entries into this matrix. While these modifications lead to a less detailed model of the Yarkovsky effect, they decrease the computational time needed for calculations. Two of these prescriptions (models A and B) are available for use in the Simple Version of the Yarkovsky effect. These matrices have been modified slightly to remove the effects of PR drag by changing the diagonal terms from 1 to 0. PR drag is already a feature in the previously implemented Radiation Forces effect in \rebx\ \citep{Tamayo2020}, so its inclusion in the Simple Version would be superfluous. In addition, while PR drag is important to consider for dust-sized particles, the Yarkovsky effect plays a larger role in the orbital evolution of larger bodies. \citet{Veras2015} showed that the Yarkovsky effect should only be active in bodies with diameters greater than 10 m that are spinning moderately fast. In addition, \citet{Veras2015} show that the Yarkovsky effect is proportional to $1/c$ while PR drag is proportional to $1/c^2$. Therefore, when the Yarkovsky effect is active on a body, it is reasonable to ignore the effects of PR drag. \begin{equation} \mathrm{Model\ A:\ } \mathbb{Y} = \Large \begin{bmatrix} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \\ \end{bmatrix}; \label{eq:model_A} \end{equation} \begin{equation} \mathrm{Model\ B:\ } \mathbb{Y} = \Large \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \\ \end{bmatrix}. \label{eq:model_B} \end{equation} Model A and Model B are shown in Equations~(\ref{eq:model_A}) and~(\ref{eq:model_B}), respectively. Model A is configured to push targets outward (corresponding to prograde rotation), while Model B is configured to drive targets inward (corresponding to retrograde rotation). Setting the \rebx\ parameter "ye\_flag" to 1 on an object in the simulation will apply model A to $\mathbb{Y}$, while setting "ye\_flag" to -1 will apply model B to $\mathbb{Y}$. In addition to the simplifications made to $\mathbb{Y}$, it's assumed that $k = 0.25$ for this version of the effect. The equation for the Simple Version of the effect is then \begin{equation} \left(\frac{d\textit{\textbf{v}}}{dt}\right)= \frac{3L(1-\alpha)}{64\pi \rho Rcr^2}\mathbb{Y}\textit{\textbf{i}}, \label{eq:simple_veras2015} \end{equation} where $\mathbb{Y}$ will either be Model A or Model B depending on what the user chooses. The following section will show how the change in semi-major axis of an asteroid under the influence of this version of the effect can be represented analytically while Section~\ref{sec:yark_effect_parameter_interpolation} gives a demonstration of using this version of the effect in combination with the parameter interpolation feature in \rebx. \subsection{Analytic Comparison} \label{sec:analytic_comparison} If an object is in a circular orbit with no inclination around a star whose mass and luminosity are constant, then an analytic equation describing the time-evolution of the object's semi-major axis can be derived from the equations of the Simple Version. This derivation is made possible due to the Simple Version's modifications to the Yarkovsky matrix, and it cannot be performed using the equations from the Full Version. While this analytic equation only applies to the specific situation described above, we can use it to show that the Simple Version of the effect properly applies Equation~(\ref{eq:simple_veras2015}) to targets in \reb\ simulations. Equation (A2) in \citet{Veras2015} shows how the average Yarkovsky-induced time rate of change of a body's semi-major axis depends on the entries in $\mathbb{Y}$: \begin{equation} \left(\frac{da}{dt}\right)= \frac{kAL(1-\alpha)}{4\pi mnca^2(1-e^2)} \begin{bmatrix}\mathbb{Y}_{21} - \mathbb{Y}_{12} \\\ \mathbb{Y}_{32} - \mathbb{Y}_{23} \\\ \mathbb{Y}_{31} - \mathbb{Y}_{13} \\ \end{bmatrix} \cdot \begin{bmatrix} \cos(i) \\\ \sin(i)\sin(\Omega) \\\ \sin(i)\cos(\Omega)\\ \end{bmatrix}, \label{eq:Veras_A4} \end{equation} where $a$ is the semi-major axis of the body, $n$ is the mean motion of the body, $i$ is the object's inclination, and $\Omega$\ is the target's longitude of ascending node. If we assume that the object's inclination and eccentricity are both 0 and that the luminosity of its host star is constant with time, then the equation simplifies to \begin{equation} \left(\frac{da}{dt}\right)= \frac{kAL(1-\alpha)}{4\pi mnca^2}(\mathbb{Y}_{21} - \mathbb{Y}_{12}). \label{eq:Veras_A4_with_n} \end{equation} To eliminate all time dependency in variables other than $a$, we must get $n$ in terms of $a$: \begin{equation} n = \frac{\sqrt{G(M+m)}}{a^{3/2}}, \label{eq:mean_motion} \end{equation} where $G$ is the gravitational constant and $M$ is the mass of the primary. If we assume that the mass of the body is small compared to the mass of the primary, the body orbits a 1 $M_{\sun}$ star, and the units we use are astronomical units, $M_{\sun}$, and years so that G has a value of $4\pi^2$, then Equation~(\ref{eq:mean_motion}) simplifies to \begin{equation} n = \frac{2\pi}{a^{3/2}}. \label{eq:mean_motion_simplified} \end{equation} We can now replace $n$ in Equation~(\ref{eq:Veras_A4_with_n}) with Equation~(\ref{eq:mean_motion_simplified}) to get the following separable differential equation: \begin{equation} \left(\frac{da}{dt}\right)= \frac{kAL(1-\alpha)}{8\pi^2 mca^{1/2}}(\mathbb{Y}_{21} - \mathbb{Y}_{12}). \label{eq:Veras_A4_difeq} \end{equation} Solving this differential equation, we get \begin{equation} \frac{2a^{3/2}}{3} = \frac{kAL(1-\alpha)t}{8\pi^2 mc}(\mathbb{Y}_{21} - \mathbb{Y}_{12}) + C, \label{eq:Veras_A4_difeq_solved} \end{equation} where $t$ is time and $C$ is a constant of integration. We can transform Equation~(\ref{eq:Veras_A4_difeq_solved}) into a form that more resembles Equation~(\ref{eq:simple_veras2015}) by assuming $A = \pi R^{2}$ and $k$ has a value of 0.25: \begin{equation} \frac{2a^{3/2}}{3} = \frac{R^2L(1-\alpha)t}{32\pi mc}(\mathbb{Y}_{21} - \mathbb{Y}_{12}) + C, \label{eq:Veras_A4_difeq_solved_simplified} \end{equation} where $R$ is the radius of the body. By isolating $a$ and solving for $C$ when $t = 0$, the equation becomes \begin{equation} a(t) = \left(\frac{3R^2L(1-\alpha)t}{64\pi mc}(\mathbb{Y}_{21} - \mathbb{Y}_{12}) + a_{0}^{3/2}\right)^{2/3}, \label{eq:yark_analytic} \end{equation} where $a_{0}$ is the starting semi-major axis of the body. This is the final form of the equation and can be used to replicate and validate certain results obtained from the Simple Version of the effect. To demonstrate this, we created a simple simulation containing an asteroid in a flat, circular orbit around a 1 $M_{\sun}$ star. The asteroid has a density of 3000 kg m$^{-3}$, a radius of 1 km, and a semi-major axis of 1 au. The star has a constant luminosity of 100,000 $L_{\sun}$ to make the effects of the Yarkovsky effect more noticeable on a shorter timescale. For this simulation, we used Equation~(\ref{eq:model_A}), which will push the body's orbit outward. The simulation lasts for 1 Myr and records how the semi-major axis of the body changes over time due to the Simple Version of the Yarkovsky effect. We then compared the results from the simulation to the results when using the exact same parameters in Equation~(\ref{eq:yark_analytic}). Figure~\ref{fig:fig2} shows both the results from the simulation and the curve created by the analytic equation. The two results are virtually identical for at least 1 Myr, showing that the Simple Version will properly apply Equation~(\ref{eq:simple_veras2015}) to targets in \reb\ simulations with substantial durations. \section{Combining Effects} \label{sec:combining_effects} \rebx\ provides users with the capability to easily use multiple forces, effects, and operators in a single simulation, allowing for detailed studies of situations that require a combination of different physics (giant branch studies, protoplanetary disks, etc.). Section~\ref{sec:yark_effect_parameter_interpolation} shows an example that combines the Yarkovsky effect with parameter interpolation while Section~\ref{sec:time_performance} shows how the different versions of the Yarkovsky effect perform when combined with the Radiation Forces effect in \rebx. \subsection{Yarkovsky effect and Parameter Interpolation} \label{sec:yark_effect_parameter_interpolation} \rebx\ effects can be used in conjunction with the parameter interpolator in \rebx, a feature that allows one to use imported time-series data for parameters that will change throughout the course of a simulation \citep{Baronett2022}. Based on the cubic spline algorithm from \citet{Press1992}, this interpolator takes user-inputted data for a particular parameter and creates a cubic spline that can be called upon at any arbitrary time during a simulation to obtain a value for that parameter. This interpolator is written in C, is machine-independent, and supports both forward and backwards integration. To demonstrate the combined capabilities of parameter interpolation and the Yarkovsky effect, we run a simple simulation based on studies done by \citet{Veras2019} of asteroids under the influence of the Yarkovsky effect during the late-stage phases of stellar evolution. The simulation contains a star with a mass of around 2 $M_{\sun}$\footnote{The actual starting mass is 1.951859 $M_{\sun}$.} that is about to ascend the asymptotic giant branch (AGB) and asteroids placed in circular orbits at 1 au, 3 au, and 10 au. These asteroids each have a density of 3000 kg m$^{-3}$ and can have radii of 10 m, 100 m, and 1000 m. The Simple Version of the Yarkovsky effect is applied to these nine asteroids for the entirety of the simulation, with the effect set to propel these bodies outward. We also add three control asteroids without the Yarkovsky effect active on them at these three different starting positions. To save computational time and keep the focus of the simulation on the effects of stellar mass loss and the Yarkovsky effect, all asteroids in this simulation are semi-active.\footnote{Setting particles to semi-active in \reb\ simulations eliminates their gravitational influence on other bodies while retaining the gravitational influence of active particles onto them.} The simulation lasts for 1.8 Myr and begins about 1.7 Myr before the star reaches the tip of the AGB. The time-series data for the mass and luminosity of the originally 2 $M_{\sun}$ star were obtained from the \sse\ stellar evolution code \citep{Hurley2000}. During the simulation, the star will lose approximately 67\% of its starting mass, which will cause all asteroid orbits to expand adiabatically by roughly a factor of three. The luminosity of the star will reach values thousands of times greater than the luminosity of the Sun, which will cause an increase in the strength of the Yarkovsky effect for the duration of the AGB. The top two panels of Figure~\ref{fig:fig3} show the mass and luminosity of the star during the course of the AGB phase. The bottom panel of Figure~\ref{fig:fig3} shows the semi-major axes of the asteroids as a function of simulation time and the age of the star. As expected, asteroids with smaller radii are propelled outward much faster by the Yarkovsky effect, and the combination of the Yarkovsky effect and stellar mass loss can push small asteroids to orbits that are over 100 times greater than their starting positions. The three asteroids not under the influence of the Yarkovsky effect remain in relatively unchanging orbits until near the end of the simulation, when the star quickly loses a large portion of its mass. This simulation and previous studies \citep{Zuckerman2010, Frewen2014, Veras2022} have shown that the Yarkovsky effect plays a role in determining the final arrangement of rocky material around white dwarfs and is an important mechanism for phenomena such as white dwarf pollution. The Yarkovsky effect in \rebx\ will be a useful tool to further explore the role this radiative force has on small bodies during the late stages in a star's life. \subsection{Time Performance} \label{sec:time_performance} \begin{table} \centering \caption{Time Performance of WHfast Simulations with Different Combinations of \rebx\ Effects} \begin{tabular}{lccc} \hline \hline Effects & Avg. Runtime & Std. Dev. & Ratio \\ & (s) & (s) & \\ \hline None & 0.116 & $\pm 0.010$ & ... \\ RF & 0.153 & $\pm 0.012$ & 1.322\\ SV & 0.154 & $\pm 0.010$ & 1.324\\ FV & 0.327 & $\pm 0.034$ & 2.820\\ RF \& SV & 0.212 & $\pm 0.020$ & 1.831\\ RF \& FV & 0.357 & $\pm 0.015$ & 3.078\\ \hline \end{tabular} \label{tab:whfast_time_table} \end{table} \begin{table*} \centering \caption{Average Simulation Durations for Different \rebx\ Effects and Integrators} \begin{tabular}{lccccc} \hline \hline Effects & IAS15 & JANUS & SABA & EOS & Leapfrog\\ \hline None & 3.863 $\pm\ 0.221$\footnote{all entries have units of seconds.} & 0.242 $\pm\ 0.013$ & 0.381 $\pm\ 0.023$ & 0.071 $\pm\ 0.005$ & 0.035 $\pm\ 0.003$\\ RF & 6.252 $\pm\ 0.070$ & 0.384 $\pm\ 0.020$ & 0.591 $\pm\ 0.022$ & 0.127 $\pm\ 0.020$ & 0.053 $\pm\ 0.005$\\ SV & 6.569 $\pm\ 0.165$ & 0.673 $\pm\ 0.045$ & 0.755 $\pm\ 0.027$ & 0.174 $\pm\ 0.010$ & 0.088 $\pm\ 0.008$\\ FV & 12.578 $\pm\ 0.093$ & 1.987 $\pm\ 0.063$ & 1.952 $\pm\ 0.051$ & 0.473 $\pm\ 0.042$ & 0.237 $\pm\ 0.011$\\ RF \& SV & 9.158 $\pm\ 0.212$ & 0.924 $\pm\ 0.050$ & 1.058 $\pm\ 0.030$ & 0.237 $\pm\ 0.019$ & 0.118 $\pm\ 0.008$ \\ RF \& FV & 16.466 $\pm\ 0.168$ & 2.286 $\pm\ 0.078$ & 2.255 $\pm\ 0.059$ & 0.524 $\pm\ 0.016$ & 0.287 $\pm\ 0.037$\\ \hline \end{tabular} \label{tab:time_table} \end{table*} As explained in Section~\ref{sec:simple_version}, the equations contained in the Simple Version of the effect are less detailed than the ones in the Full Version to save computational time during longer simulations. To demonstrate this difference in time performance, we apply each version of the effect to a 0.01 Myr-long simulation and record its duration. The simulation contains a test particle orbiting the Sun in a circular orbit at 1 au and uses the WHfast integrator \citep{RT2015, RTB2019} with a fixed time step of 0.05 yr. In addition to measuring the two versions of the Yarkovsky effect, we also measure the time performance of the Radiation Forces effect in \rebx, which applies both PR Drag and radiation pressure to chosen particles in a simulation. Measuring this effect will let us compare the performance of our new effect with previously implemented radiative forces. The following configurations of effects were applied to the simulation: (1) no effects; (2) Radiation Forces; (3) the Simple Version; (4) the Full Version; (5) Radiation Forces and the Simple Version; and (6) Radiation Forces and the Full Version. We performed 10 runs for each combination and calculated their average runtimes and standard deviations. Table~\ref{tab:whfast_time_table} shows the results for each configuration. As expected, simulations using the Simple Version require less computational time than ones using the Full Version. On average, we find that the Full Version increases a simulation's duration by a factor of about 2.820 while the Simple Version increases it by a factor of 1.324, which is comparable to the factor of 1.322 from Radiation Forces. We also see a difference in computational times when Radiation Forces is combined with the two different versions of the Yarkovsky effect. Radiation Forces combined with the Simple Version increases the length of a simulation by an average factor of 1.831, which is significantly less than the factor of 3.078 from Radiation Forces combined with the Full Version. These results show that the Simple Version of the effect is more useful for long-term simulations containing many bodies, while the Full Version is more useful for short, highly-detailed simulations with fewer bodies. For the convenience of users, we also provide time performance data for five other integrators available in \reb: IAS15 \citep{Rein2015}, JANUS \citep{Rein2018}, SABA\footnote{specifically SABA(10,6,4) (default option).} \citep{RTB2019}, EOS \citep{Rein2020}, and Leapfrog. Apart from the integrators, the setup for the simulations and methods for data collection remain the same. Table~\ref{tab:time_table} shows the average durations and standard deviations for simulations using these integrators with the same set of varied combinations of \rebx\ effects active. \section{Conclusion} Two different forms of the Yarkovsky effect have been added to the external library \rebx: the Full Version and the Simple Version. The Full Version will be useful for constraining the physical properties of asteroids \citep{Vokrouhlicky2008, Tardioli2017}, estimating the ages of asteroid families \citep{Nesvorny2004, Carruba2017}, and in general, short but detailed simulations containing fewer objects. The Simple Version will be useful for studying the evolution of smaller objects during the post-main sequence \citep{Veras2019} -- which could have important consequences for white dwarf pollution \citep{Zuckerman2010, Frewen2014, Veras2022} -- and in general, longer simulations with more particles that can tolerate lower levels of detail. We hope these new capabilities for \rebx\ will enable new studies into the role of radiative forces in planetary systems, and we encourage others to make contributions to \rebx\ as well. \hspace{1.5 pt} We thank Dimitri Veras for his insight and helpful discussions and an anonymous reviewer for constructive comments that improved this article. \textit{Software}: Simulations in this paper used the \sse, \reb, and \rebx\ codes, which are all freely available at \url{https://astronomy.swin.edu.au/~jhurley/bsedload.html}, \url{http://github.com/hannorein/rebound}, and \url{https://github.com/dtamayo/reboundx}. This paper also made use of the open-source projects \jupyter\ \citep{Jupyter}, \texttt{IPython} \citep{IPython}, and \texttt{matplotlib} \citep{matplotlib1, matplotlib2}. All of the data and \python\ scripts used to generate the figures in this article are available at \url{https://github.com/Nofe4108/REBOUNDx_Paper}. \bibliographystyle{aasjournal} \bibliography{references}{}

Title: Modeling Supermassive Primordial Stars with MESA

Abstract: Supermassive stars forming at $z \sim$ 15 - 20 are one of the leading contenders for the origin of the first quasars, over 200 of which have now been discovered at $z >$ 6. These stars likely form in pristine, atomically cooled haloes immersed in strong Lyman-Werner UV backgrounds or in highly supersonic baryon streaming flows. Atomic cooling triggers catastrophic baryon collapse capable of building up stars at rates of up to $\sim$1 M$_{\odot}$ yr$^{-1}$. Here we examine the evolution of supermassive stars with a much larger and finer grid of accretion rates than in previous studies with the MESA stellar evolution code. We find that their final masses range from 3.5 $\times$ 10$^3$ M$_{\odot}$ - 3.7 $\times$ 10$^5$ M$_{\odot}$ at accretion rates of 0.001 M$_{\odot}$ yr$^{-1}$ - 1 M$_{\odot}$ yr$^{-1}$, respectively. We also find that supermassive star evolution diverges at accretion rates of 0.01 M$_{\odot}$ yr$^{-1}$ - 0.02 M$_{\odot}$ yr$^{-1}$, above which they evolve as cool red hypergiants along the Hayashi track and collapse via the general relativistic instability during central hydrogen burning, and below which they evolve as hot blue supergiants and collapse at the end of their nuclear burning lifetimes after exiting the main sequence.

https://export.arxiv.org/pdf/2208.00008

\pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} quasars: general --- black hole physics --- early universe --- dark ages, reionization, first stars --- galaxies: formation --- galaxies: high-redshift \end{keywords} \section{Introduction} Supermassive stars (SMSs) are one of the leading candidates for the origin of the first quasars, more than 200 of which have been discovered at $z >$ 6 \citep{fan03,fan06}, including eight at $z >$ 7 \citep{mort11,wu15,ban18,smidt18,mats19,zhu20,wang21}. Until recently, they were thought to form at $z \sim$ 10 - 20 in primordial halos immersed in high Lyman-Werner UV backgrounds \citep{latif14,agarw15,anna15,anna17} or in highly supersonic baryon streaming motions \citep[BSMs;][]{lns14,hir17,srg17} that suppress normal star formation until they reach masses of 10$^7$ \Ms\ and virial temperatures of 10$^4$ K. These temperatures trigger atomic cooling that causes gas to collapse to the center of the halo at rates of up to $\sim$ 1 \Ms\ yr$^{-1}$ \citep{bl03,ln06,rh09b,latif15b}. Such flows create massive, hot accretion disks that build up either a single SMS or binaries and small multiples \citep{wise19, ret20,latif20a,pat21b,pat21a}. However, it has now been found that the rare haloes capable of forming quasars by $z \sim$ 6 can form SMSs without the need for UV backgrounds, BSMs, or even atomic cooling \citep{latif22b} SMSs have been the subject of analytical studies since the 1960s \citep[e.g.,][]{iben63,chandra64,fowler64,fowler66} and numerical simulations since the 1970s \citep[e.g.,][]{af72a,st79,fuller86,baum99,sun18,but18}, but it has only been recently that they have been studied in the extreme flows in which they form \citep{hos13,um16,tyr17,hle18b,hle18a,tyr22a,tyr21a}. These models indicate that rapidly accreting SMSs can reach masses of a few 10$^5$ \Ms\, and lifetimes of 1 - 2 Myr before collapsing to direct-collapse black holes (DCBHs). SMSs are the leading contenders for the seeds of the first quasars because it is difficult for ordinary Population III (Pop III) star BHs to grow rapidly after birth because they form in low densities and in some cases are ejected from their host halos \citep{wan04,ket04,awa09,wf12,srd18}. DCBHs are born with much larger masses and in much higher densities in halos that retain their fuel supply, even when it is heated by X-rays \citep{jet13}. Previous studies have mostly found that rapidly-accreting primordial stars evolve as cool, red hypergiants along the Hayashi limit, with surface temperatures of 5,000 - 10,000 K due to H$^-$ opacity in their atmospheres, at least until they reach $\sim$ 10$^5$ \Ms\ \citep{hos13}. \citet{hle18b} found that SMSs can remain cool even above these masses, reaching luminosities greater than 10$^{10}$ \Ls. In other studies, SMSs evolving from similar initial conditions soon settle onto hotter and bluer tracks with temperatures of 20,000 - 40,000 K \citep{tyr17}. \citet{hle18b} found that stars growing at low rates ($\lesssim$ 0.005 \Ms\ yr$^{-1}$) also evolve along blue tracks, as can stars with clumpy accretion due to fragmentation or turbulence in the disk (\citealt{sak15} -- but see \citealt{sak16b}). It is not clear if these differences arise from the opacities used in the models, code physics (such as the numerical treatment of convection), or resolution \citep[see][for further discussion of benchmarking efforts]{titans}. The relatively low surface temperatures of SMSs in most models to date imply that radiation from the star cannot overcome the enormous ram pressures of catastrophic infall and that they form, evolve and die in the accretion flows that create them. This has been corroborated at the initial stages of SMS formation by radiation hydrodynamical simulations by \citet{ard18} and \citet{luo18} \citep[see also][]{aaron17}. Any ionizing UV from the star would also only heat the gas to at most a few times the temperatures at which it already falls onto the star without forming the usual hot, high-pressure bubbles that drive all the baryons out of less massive Pop III star-forming halos \citep{suh09,susa13,hir13,hir15,sug20,latif21a,latif22a}. Pop III SMSs accreting at rates of 0.1 - 1 \Ms\ yr$^{-1}$ would have extremely large luminosities that could, in spite of their low surface temperatures, be detected in the near infrared (NIR) by the {\em James Webb Space Telescope} ({\em JWST}) and large ground-based telescopes in the coming decade \citep{jlj12a,hos13,sur18a,sur19a,wet20a}. Their BHs could also be found in the NIR at $z \sim 15 - 20$ by {\em JWST} \citep{pac15,nat17,bar18,wet20b} and at $z \sim$ 6 - 8 by {\em Euclid} and the {\em Roman Space Telescope}, although lensing by massive galaxies and galaxy clusters in their wide fields could extend these detections up to $z \sim$ 15 \citep{vik22a}. DCBHs will only be marginally visible to the Square Kilometre Array or next-generation Very Large Array at $z \gtrsim$ 6 - 8 \citep{wet20a,wet21a} but would become more luminous after growing to larger masses at later times \citep{wet22b}. They could also be detected out to $z \sim$ 10 by future X-ray missions such as the {\em Advanced Telescope for High-Energy Astrophysics} ({\em ATHENA}) and {\em Lynx} \citep{athena,lynx}. Previous studies of supermassive primordial star evolution considered small representative samples of 4 - 7 accretion rates in the range of 0.001 \Ms\ yr$^{-1}$ - 10 \Ms\ yr$^{-1}$. Here we revisit the evolution of supermassive primordial stars with the Modules for Experiments in Stellar Astrophysics (\texttt{MESA}) code by examining a much larger and finer grid of accretion rates than previously explored. We describe the physics in our runs and their setups in Section~2. The evolution of the stars is examined in detail in Section~3. We tally final masses for SMSs as a function of accretion rate in Section~4 and conclude in Section~5. \section{Numerical Method} \label{section:method} \texttt{MESA} is a one-dimensional (1D) Lagrangian stellar evolution code that solves equations of stellar structure that are implicitly coupled to convective mixing and nuclear burning \citep[version 12778;][]{paxt11,paxt13,paxt18}. \texttt{MESA} has implicit hydrodynamics that can solve for the velocity of each zone in the model at the onset of collapse to produce more accurate final masses for the star. We use the 21-isotope APPROX network, which includes the pp chain, triple-alpha burning, and the CNO cycle. Our models have an equation of state (EOS) that is a composite of several datasets such as the OPAL/SCVH tables \citep{rogers02,saumon1995}, which are used at lower densities and temperatures like those in the outer regions of the star and its atmosphere, and the \texttt{HELM} and \texttt{PC} EOSs \citep{ts00,potekhin10}, which are applied to high densities and temperatures like those in the core of the star \citep[see Figure 1 and Section 4.2 of][]{paxt11}. We use the Henyey method for convective mixing length theory, with the mixing length parameter $\alpha_{\mathrm{MLT}} =$ 2, and an overshooting parameter set to 0.1. We also use the Ledoux criterion for convection along with the \texttt{MESA} prescription for smoothing composition gradients in non-convective regions that trail retreating convection zones. This measure reduces unphysical steps in entropy that can develop in the interior of the star over time. We include the post-Newtonian Tolman-Oppenheimer-Volkoff (TOV) correction to the equation of hydrostatic equilibrium, \begin{equation} \frac{dP}{dr} = -\frac{Gm\rho}{r^2}, \label{eqn:HE} \end{equation} by replacing the gravitational constant, $G$, with \begin{equation} G_{\textrm{rel}} = G\left(1 + \frac{P}{\rho c^2} + \frac{4\pi Pr^3}{m_r c^2} \right)\left(1 - \frac{2Gm_r}{rc^2}\right)^{-1}. \label{eqn:grel} \end{equation} $G_{\textrm{rel}}$ is computed in every zone of the star every time step in the user-defined \textit{run\_star\_extras} subroutine in the \texttt{MESA}\_{\em star} subroutine and then used to update Equation~\ref{eqn:HE}. We activate the TOV correction to $G$ at the beginning of each run. \texttt{MESA} adaptively rezones the stars as they evolve. Mesh refinement is triggered when gradients in temperature, pressure, and $^4$He abundances between adjacent zones exceed preset values: $\delta \log P/P >$ 1/30, $\delta \log T/T >$ 1/80, and $\delta \log(\chi + 0.01) >$ 1/20 $\log \chi$, where $\chi$ is the $^4$He mass fraction \citep[see section 6.4 of][]{paxt11}. These criteria result in a larger number of zones usually being allocated near the center of the star to resolve nuclear burning and convective processes, and the stars are typically partitioned into about 1400 mass zones. Accretion flows onto the star have a primordial composition that does not change as the star evolves. Their entropy is matched to that of the surface of the star so it is assumed that the accretion luminosity is radiated away rather than deposited at the surface. We exclude mass loss due to stellar winds because they are thought to be negligible at primordial compositions \citep{vink01,bhw01}. We initialise all the stars as 20 \Ms\ fully-convective, $n =$ 1.5 polytropes with temperatures below 10$^6$ K to preclude nuclear burning \citep[Section 6.1 of][]{paxt11}. They have primordial compositions, with mass fractions $\chi_{\mathrm{H}} =$ 0.7516 and $\chi_{\mathrm{He}} =$ 0.2484. Our 16 models are uniformly partitioned in log accretion rate over 3 decades: $10^{-3}$ \Ms\ yr$^{-1}$, $10^{-2}$ \Ms\ yr$^{-1}$ and $10^{-1}$ \Ms\ yr$^{-1}$. Models with accretion rates at or above 0.02 \Ms\ yr$^{-1}$ are first evolved up to 10,000 \Ms\ without hydrodynamics and then stopped. Hydrodynamics is not activated until the star is in stable nuclear burning because it can lead to numerical instabilities if the core is still contracting or burning is about to begin. Each model is then branched into two parallel runs: one with implicit hydrodynamics to determine at what mass pulsational instabilities driven by the GRI cause the star to collapse and one without it to determine the mass of the star at which softening of the EOS in the core instead predicts the star would collapse. Stars with hydrodynamics are evolved up to the mass at which they are on the verge of collapse due to the GRI and are halted again. Stars without hydrodynamics are evolved up to the fatal softening of the EOS in the core. During this second stage of evolution, pulsational instabilities due to post-Newtonian corrections can lead to unphysical, premature collapse if the time steps are too small. Adjusting the time resolution in our high accretion rate models ensures that MESA only follows pulsations that become strong enough to build up internal shocks when the GRI would cause the star to collapse. Anomalous temperature spikes can also induce motions in the cores of some stars that artificially trigger collapse before the GRI would. Applying more resolution to the core mitigated these spikes. To follow the collapse of the star in the third and final phase, the time step must again be reduced or the implicit hydrodynamics can return large, unphysical infall velocities. We halt the simulations for good when infall velocities exceed 1000 km s$^{-1}$. GR is turned on in these runs at all times. Stars with accretion rates below 0.02 \Ms\ yr$^{-1}$ never reach masses at which they encounter post-Newtonian instabilities so GR is turned off in these models. They are evolved until their central hydrogen mass fractions fall to 1\%, and hydrodynamics is turned on from the beginning of the run in zones of the stars whose temperatures exceed 10$^7$ K. After the depletion of central hydrogen, our models experience numerical difficulties as they begin more advanced phases of burning. Because of these instabilities, and the fact that later phases of burning are so short-lived that the stars do not grow much more in mass by accretion, we take their final masses to be when they reach central hydrogen fractions of 1\% without following their collapse. We activated a control that adds more resolution to the models when log $T_c > 8.15$, by which point the central hydrogen fraction in most of the stars has fallen to below 30\%. Pressure must be imposed on the outer boundaries of these stars to prevent superluminal velocities from arising in their low-density outermost zones and ensure the stability of the star \citep{tyr17,tyr20a}. This pressure, $P$, is parametrised by $P_{\mathrm{extra\_factor}}$, \begin{equation} P = \frac{\tau g}{\kappa}\left(1 + P_{\textrm{extra}}\right), \end{equation} where \begin{equation} P_{\textrm{extra}} = P_{\textrm{extra\_factor}}\frac{\kappa}{\tau} \ \frac{M}{L} \left(6\pi c G\right)^{-1}, \end{equation} and $g = GM/R^2$, the surface gravity of the star, $\kappa$ and $\tau$ are the opacity and optical depth, and $M$ and $L$ are the mass interior to those layers and their luminosity, respectively. We set $P_{\mathrm{extra\_factor}} = 1.36$ for our low accretion rate models, 0.01 - 0.001 \Ms\ yr$^{-1}$. \section{SMS Evolution} We show Hertzsprung-Russel (HR) plots of the evolution of our stars at accretion rates of 0.001 \Ms\ yr$^{-1}$ - 1 \Ms\ yr$^{-1}$ in Figures~\ref{fig:HRmain} and \ref{fig:HRsubplot} and Kippenhahn diagrams of their structures over time in Figures~\ref{fig:kipp_highAR} - \ref{fig:kipp_lowAR}. How the stars evolve primarily depends on how accretion timescales, $t_{\textrm{acc}} = M/\dot{M}$, compare to Kelvin-Helmholz (KH) contraction times, $t_{\textrm{KH}} \approx GM^2/RL$, early in their lives where $\dot{M}$ and $L$ are the accretion rate and luminosity of the star, respectively. If the protostar gains enough mass during contraction it will have sufficient opacity in the outer layers to expand as a cool red SMS. If not, it will remain compact and hot and evolve along bluer tracks. At first, our protostars are fully convective, with central temperatures below those required for deuterium burning. They contract under accretion until their temperatures are high enough to ignite proton-proton (PP) burning. However, as shown by the early dips in all the HR tracks in Figure~\ref{fig:HRmain}, PP burning does not generate enough energy to halt contraction, which continues until the stars begin CNO burning, which is quickly followed by the triple alpha process. The onset of the CNO cycle thus marks the beginning of stable nuclear burning in the star. The PP chain raises central temperatures that drive changes in internal opacities that create a short-lived radiative core. The mass at which the star forms a radiative core increases with accretion rate because the core acquires more mass during initial contraction. However, CNO burning dramatically increases energy production so the core becomes convective to transport this energy outward more efficiently. The stars eventually develop large convective cores and high-entropy outer envelopes like those in \citet{tyr17} and \citet{hle18b}. As in \citet{hle18b}, each star converges to a monotonic mass-luminosity evolution after an early period of reconfiguration that, as discussed above, follows either a cool red track or a hot blue one. As shown in Figures~\ref{fig:HRmain} and \ref{fig:HRsubplot}, stars growing at rates above 0.01 \Ms\ yr$^{-1}$ - 0.02 \Ms\ yr$^{-1}$ branch off to evolve as red hypergiants while those below these rates become compact blue supergiants. The transition from one regime to the other occurs at $\sim$ 0.02 \Ms\ yr$^{-1}$, as this SMS oscillates between red and blue phases. As in \citet{hle18b}, effective temperatures above this rate remain at $\sim$ 10$^4$ K even as they reach luminosities of $10^{9} - 10^{10}$ \Ls\ as the stars evolve along the Hayashi limit. Below 0.01 \Ms\ yr$^{-1}$ the stars evolve onto the zero-age main sequence (ZAMS) and reach temperatures of up to $10^{5.5}$ K and luminosities of $10^8 - 10^{10}$ \Ls. The two tracks bifurcate during initial contraction, prior to any nuclear burning. This transition can also be seen in the interiors of the stars shown in Figure~\ref{fig:kipp_midAR}. Above 0.02 \Ms\ yr$^{-1}$ there are more radiation dominated structures in which less of the interior resides within the convective core. At lower accretion rates the stars are almost fully convective, and most of the star is a burning core. It is clear from Figure~\ref{fig:HRmain} that high accretion rate stars grow significantly in radius, and this is due to the fact that they gain mass more quickly than they can contract. Low accretion rate stars contract more quickly than they gain mass so their radii decrease as they settle onto the ZAMS. \subsection{$\dot{M} =$ 0.1 - 1 \Ms\ yr$^{-1}$} As shown in the top panel of Figure~\ref{fig:HRsubplot}, these cool red stars evolve very quickly over a narrow range of effective temperatures. As they grow in mass, they encounter opacity bumps that lead to excursions in the HR diagram until they converge to a direct track along the Hayashi limit. Energy production by the PP, CNO and triple alpha chains causes the stars to grow in radius by a factor of ten during main sequence burning. This expansion is driven by temperature sensitive H$^{-}$ opacity in their atmospheres, which leads to efficient transport of energy from the interior to the outer layers. In Figure~\ref{fig:kipp_highAR}, the internal structures of these models change with decreasing accretion rate. At lower accretion rates more of the stellar interior is subsumed by the convective core. The staircase-like growth of the central convective zone is due to the finite resolution of our models and the limitations of our 1D prescription for convective instability. \subsection{$\dot{M} =$ 0.02 - 0.1 \Ms\ yr$^{-1}$} As shown in the middle panel of Figure~\ref{fig:HRsubplot}, most of the stars in this accretion range evolve in the same manner as our highest accretion rate stars, with cool red tracks and radial expansion after converging to the Hayashi limit. The transition from red to blue SMSs occurs at the lower end of this range, as we discuss in greater detail in Section 3.6. The star accreting at 0.02 \Ms\ yr$^{-1}$ alternates between red and blue tracks because densities and temperatures in its atmosphere at times are like those in high accretion rate models, but the star does not gain sufficient mass over time to sustain them and periodically contracts back down to a bluer state. These transitions also cause \texttt{MESA} to readjust its timesteps upon excursion from one effective temperature to another. The large swings in the structure of the 0.02 \Ms\ yr$^{-1}$ star prematurely end its run but adjustments to the pressure at its surface allow it to be evolved to the end of main sequence burning, as discussed in Section 3.6. There we show that the star continues to cycle between red and blue states before finally settling onto a blue track and becoming hot. As shown in Figure~\ref{fig:kipp_midAR}, a significant fraction of the mass of the 0.02 \Ms\ yr$^{-1}$ star resides within its convective core. At lower accretion rates the percentage of the star that is convective rises. Nuclear energy generation in the convective core is mostly due to CNO, triple alpha and nitrogen burning. Outside the convective zone, weak energy generation proceeds by the PP chain but is dwarfed by core burning. It is clear from Figure~\ref{fig:kipp_midAR} that the 0.02 \Ms\ yr$^{-1}$ star is evolving towards the low accretion rate regime, as this model is comparatively much older once it reaches 10$^4$ \Ms. \subsection{$\dot{M} =$ 0.001 - 0.01 \Ms\ yr$^{-1}$} HR tracks for the SMSs in the lowest accretion range are shown in the bottom panel of Figure~\ref{fig:HRsubplot}. They evolve as hot blue stars that can be more than a hundred times smaller than red SMSs of comparable mass. At early stages in their evolution these stars have similar radii as they approach the ZAMS. Just before reaching the post-main sequence, they begin to expand as they deplete their central hydrogen supply, with those with the highest rates having the largest radii and luminosities. We show the internal structures of these stars in Figure~\ref{fig:kipp_lowAR}. Most of the mass of these compact blue stars resides in their convective cores. As at high accretion rates, the staircase-like growth of the central convective zone is again due to the finite resolution of our models and the limitations of our 1D prescription for convective instability. \subsection{Collapse} As discussed in greater detail in the next section, SMSs growing at 0.02 - 1 \Ms\ yr$^{-1}$ become unstable during hydrogen burning because of changes in central temperatures and densities induced by pulsations due to the post-Newtonian instability and collapse before the end of the main sequence. Once collapse proceeds, infall rates in the core rapidly reach 1000 km s$^{-1}$ as shown in Figure~\ref{fig:coll}. Stars growing at 0.02 - 0.1 \Ms\ yr$^{-1}$ evolve further along the main sequence before collapse, with the 0.02 \Ms\ yr$^{-1}$ SMS reaching final central hydrogen fractions of 0.12. The 0.01 \Ms\ yr$^{-1}$ SMS marks the transition to low accretion rate evolution in which the star reaches the central hydrogen fraction cutoff of 0.01 and develops a helium core as it advances to later stages of burning. Stars that grow at 0.001 - 0.01 \Ms\ yr$^{-1}$ reach the post-main sequence, evolving in a similar manner as massive Pop III stars. They never encounter the post-Newtonian instability and, although we do not model it here, are expected to collapse during He, C or O burning. Accretion rates of 0.01 \Ms\ yr$^{-1}$ - 0.02 \Ms\ yr$^{-1}$ thus mark a key divide in SMS evolution: whether they evolve as cool red hypergiants or compact blue supergiants and if they collapse via the GRI or because of core fuel depletion. \subsection{Hot CNO cycle} If core temperatures in SMSs reach $2 \times 10^8$ K while still at high central hydrogen fractions, energy generation due rapid proton (rp) captures can rival that of the CNO cycle and become several hundred times greater at temperatures above $5 \times 10^8$ K because CNO reaction rates are limited by the half-life of one of its beta decays \citep[$\beta$-limited CNO;][]{fuller86}. However, as shown in Figure~\ref{fig:Tc} core temperatures calculated with the 21-isotope APPROX network never exceed $2 \times 10^8$ K in our stars except for brief episodes late in their lives when central hydrogen fractions are low. In principle, the inclusion of more isotopes and rp captures (the hot CNO, or hCNO, cycle) could have produced larger core temperatures than those in our models. We ran the 0.001 \Ms\ yr$^{-1}$ and 1.0 \Ms\ yr$^{-1}$ stars with the 44-isotope HBURN network with one hCNO cycle and the reduced HCNO network with just the p-p chain, triple alpha chain, CNO and hCNO cycles to compare core temperatures and energy production rates, which are plotted in Figure~\ref{fig:TCNO} (we also ran the 1.0 \Ms\ yr$^{-1}$ star with the 8-isotope BASIC network, which has the p-p chain, triple alpha chain, and CNO cycle, for comparison). Unlike non-accreting SMS models, which exhibit core temperatures at which the hCNO cycle becomes important \citep[e.g.,][]{fuller86,nag22}, we find that the inclusion of the hCNO cycle in our models does not produce central temperatures at which it becomes important. Indeed, there is little difference in core temperatures with the three networks. The inclusion of more isotopes stabilizes energy production in the core at earlier times but results in the same evolution and final mass for the star. \subsection{0.02 \Ms\ / yr SMS Atmosphere Effects} As shown in Figure~\ref{fig:HRsubplot}, the 0.02 \Ms\ yr$^{-1}$ star alternates between red and blue tracks, suggesting that this is the accretion rate at which SMSs transition from blue to red. Its evolution is sensitive to the choice of atmosphere and surface pressure. We evolved the star with two versions of the standard \texttt{MESA} T($\tau$) atmosphere, one in which the opacity is constant throughout the atmosphere ('fixed', which was used for all the models in our study) and one in which it varies in a manner that is consistent with the local pressure and temperature ('varying'). For the fixed case we tested two extra pressures at the surface of the star that are set by $P_{\textrm{extra\_factor}} =$ 1.2 and 1.3, as discussed at the end of Section 2. We also consider an alternate formulation for the energy equation, De/Dt, that better conserves energy \citep{Paxton19}. Figure \ref{fig:002atmosphere} shows that changing only the pressure boundary for this star can lead to either a hot blue or cool red SMS after the onset of main sequence burning. The varying atmosphere option stabilises the thermal oscillations and leads to a track that is intermediate to the cool red and hot blue regimes. Testing these options in our other models produced much smaller deviations in evolution in the 0.01 \Ms\ yr$^{-1}$ and 0.03 \Ms\ yr$^{-1}$ stars and few if any in the others, confirming that 0.02 \Ms\ yr$^{-1}$ is the transitional accretion rate for rapidly accreting SMSs. During the protostellar phase the 0.02 \Ms\ yr$^{-1}$ SMS initially contracts on a shorter timescale than it can accrete and evolves as a hot blue star. After central nuclear burning begins, the choice of atmosphere leads to deviations in evolution by sending the outer layers of the star into regions of $T$ - $\rho$ space that favor or suppress H$^{-}$ formation, whose opacity intercepts energy from the core of the star and can cause the outer layers to expand along the Hayashi limit. In the original model, the surface layers of the star were at temperatures and densities that were marginally favorable to H$^{-}$ formation. As the star expanded its outer layers migrated into regions of $T$ - $\rho$ space in which H$^{-}$ tended to be destroyed (likely because falling densities decreased the two-body H$^{-}$ formation rate) and the star again began to contract to a hotter, bluer phase. Increasing the pressure on the outer boundary moves these layers into regions in $T$ - $\rho$ space that are less favorable to H$^{-}$ formation and cause the star to settle onto a blue track at earlier times. Excursions between red and blue states are dampened in the varying case and the star settles onto an intermediate track at early times because the temperature structure of the atmosphere can respond quickly to changes in opacity due to those in density. The run with De/Dt settles onto the hot blue track at early times because the energy equation produces consistently higher surface temperatures that prevent H$^{-}$ from forming and expanding the star. The varying case suggests that the oscillations of this star in the HR diagram are probably not in reality as large as the other test runs suggest but, as noted earlier, this and the De/Dt option had little effect of the evolution of the other stars. \section{Discussion} High accretion rate stars encounter pulsations due to the GRI at masses above 10,000 \Ms. These pulsations are initially mild, and can drive shocks into the core with speeds of a few tens of km s$^{-1}$. The shocks only induce small changes in central densities and temperatures from which the star can easily recover, unless code time steps that are shorter than nuclear burning times but too long for the hydrodynamics to return accurate velocities lead to unphysically large infall speeds, as discussed earlier. However, as the star grows in mass the pulsations become more violent and drive shocks into the core with velocities of hundreds of km s$^{-1}$. These shocks elevate central temperatures and densities that in turn exacerbate the GRI and lead to even stronger pulsations. The stars reach their final masses when a pulsation finally triggers collapse. In tests with no hydrodynamics, there are no shocks to induce changes in central densities or temperatures so the model ends when the GRI causes enough softening of the EOS in the core, usually at significantly larger masses as discussed below. We note that both high and low accretion rate stars could be prone to other types of pulsations that did not appear in our models because they had characteristic timescales that were shorter than the time steps taken by our simulations. Further study is required to determine if they occur and what impact they have on the evolution of the star, such as if they can lead to collapse at lower masses than those due to the GRI. \label{section:finalmass} \label{section:hydro} We show final masses for the stars in Figure~\ref{fig:finalmass} with those from previous studies at the same accretion rates. As in earlier work, our final masses rise monotonically with accretion rate, and above 0.01 \Ms\ yr$^{-1}$ they fall in between those of \citet{hle18b} and \citet{tyr17}. We modeled stars growing at above 0.01 \Ms\ yr$^{-1}$ with and without hydrodynamics to test its effects on their evolution and final masses. From 0.01 \Ms\ yr$^{-1}$ - 0.1 Ms\ yr$^{-1}$ they converge to essentially the same mass. Stars growing at $\ge$ 0.1 \Ms\ yr$^{-1}$ without hydrodynamics have consistently larger final masses that are closer to those of \citet{um16} because they do not manifest the pulsations that would have collapsed the star at earlier times and because the code can take larger time steps that can mitigate other effects of the GRI. As noted earlier, at accretion rates of 0.01 \Ms\ yr$^{-1}$ stars begin to collapse because of hydrogen depletion while at rates of 0.02 \Ms\ yr$^{-1}$ they collapse via the GRI. What are the relative roles of the two processes in collapse over this range? Either one can lead to excessive changes in central densities and temperatures that drive catastrophic reductions in time step that signify the death of the star. When we ran the 0.02 \Ms\ yr$^{-1}$ star without the TOV approximation it evolved further into hydrogen depletion and encountered numerical difficulties as it entered the post main sequence, but it did not develop large infall velocitiies we would associate with collapse. Likewise, when we ran the 0.01 \Ms\ yr$^{-1}$ star without post-Newtonian corrections, the star struggled through hydrogen depletion and then collapsed. These two tests indicate that the GRI was primarily to blame for the collapse of the first star and depletion of core hydrogen caused the collapse of the second. Both likely contribute to collapse at 0.01 \Ms\ yr$^{-1}$ - 0.02 \Ms\ yr$^{-1}$. As mentioned in the Introduction, SMS are not expected to lose mass to line-driven winds because of the low opacity of primordial gas. However, the stars soon become highly convective and heavier elements produced by the core could be dredged up and drive winds later in the life of the star. But it is unlikely that these winds would lead to significant mass loss because they would be modest and probably be overcome by the ram pressure of the infalling gas. Consequently, the final mass of the star will just be the mass it accrued over its lifetime. SMSs in a narrow range of masses around 55,000 \Ms\ in which accretion has subsided and the star has thermally relaxed \citep[e.g.,][]{tyr20a} have been found to explode in previous studies \citep{montero12,wet12a,jet13a,wet13b,wet13d,chen14b}. However, the stars in our study collapse without exploding because their cores are too massive and compact to overcome their own binding energy and the ram pressure of infall, even if collapse drives explosive nuclear burning. Nevertheless, it would be premature to conclude that rapidly accreting SMSs cannot explode because models to date exclude rotation and magnetic fields that could improve chances for an explosion by expanding the star or forming jets during collapse. While none of our stars made SNe, there could be other observational signatures of their collapse. For example, if the star rotates and core collapse drives a jet that pierces its outer layers, it could imprint distinctive nucleosynthetic patterns on surrounding gas that could later be found in the atmospheres of low mass stars forming in it \citep[e.g.,][]{jet09b}. Rapid burning during collapse could also produce a strong neutrino signal far brighter than that unleashed by conventional core-collapse SNe, but it would still only be detectable at Mpc distances \citep{sfh98,mun21,nag21}. Although we do not follow the collapse of our stars to late times, \citet{nag21} found that event horizon formation during the collapse of a 10$^4$ \Ms\ SMS intially encloses 40 - 50 \Ms\ of the core of the star, evoking the possibility of the birth of a quasi-star \citep[e.g.,][]{begel08,vb10}. In this picture, X-rays from the BH support the rest of the star from prompt collapse and form a stable envelope that would appear to be a cool, red giant star to an external observer. Such stars can grow to $\sim$ 10$^6$ \Ms\ before the BH becomes so massive that a hydrostatic envelope is no longer possible. However, even if the BH initially had similar masses in our stars it is unlikely that X-rays could halt the collapse of the star because so much of its mass is falling inward at velocities above several hundred km s$^{-1}$, as shown in Figure~\ref{fig:coll}. Consequently, DCBHs are born with the mass at which their progenitors die. \section{Conclusion} We have carried out a systematic study of the evolution of rapidly accreting SMS with the publicly-available stellar evolution code \texttt{MESA}. Our grid of models spans 3 decades in accretion rate that bracket the range expected for primordial environments conducive to SMS formation. We find that SMSs evolve along two different pathways, as cool red hypergiants or compact blue supergiants, at accretion rates above and below 0.01 $\leq \dot{M} \leq$ 0.02 \Ms\ yr$^{-1}$, respectively. This range also marks the transition between stars collapsing because of the depletion of core fuel after the end of the main sequence at low rates and collapse via the GRI during the main sequence at higher rates. We also find that hydrodynamics is crucial to capturing how the GRI causes the death of the star at higher accretion rates: triggering pulsations that eventually lead to its collapse. Without hydrodynamics, the GRI still leads to the collapse of the star but at significantly higher masses by softening the EOS in the core and triggering ingestion events that raise central densities and temperatures and destabilise the core. High accretion rate models with hydrodynamics encounter fatal unstabilities at lower final masses before ingestion events occur. When our SMS models reach collapse, large infall velocities enclose most of the mass of the star so it will go into the BH soon after birth, preventing the formation of a quasistar that could rapidly drive up the mass of the DCBH by a factor of ten. Our results are broadly consistent with previous, more sparsely-sampled accretion rates \citep{tyr17,hle18b} and, critically, provide a framework for future SMS modelling efforts with open-source tools. \section*{Acknowledgements} We thank Bill Paxton and the \texttt{MESA} community for valuable discussions that made this work possible. N.P.H. acknowledges funding from the European Research Council for the Horizon 2020 ERC Consolidator Grant project ICYBOB, grant number 818940. D.J.W. was supported by the Ida Pfeiffer Professorship at the Institute of Astrophysics at the University of Vienna and by STFC New Applicant Grant ST/P000509/1. T.E.W. acknowledges support from the NRC-Canada Plaskett Fellowship. \textit{Software:} \texttt{MESA} \citep{Paxton15,Paxton19}, MESASDK 20.12.1 \citep{mesasdk} and py\_mesa\_reader \citep{pymesa}. \bibliographystyle{mnras} \bibliography{refs} \bsp % \label{lastpage}

Title: Multi-wavelength Campaign on the Super-Eddington NLS1 RX J0134.2-4258 -- II. an Archetypal Weak-Line Seyfert Galaxy

Abstract: RX J0134.2-4258 is one of the most super-Eddington narrow-line Seyfert 1 (NLS1) galaxies, on which we conducted a monitoring campaign from radio to X-rays. In this paper, we present a detailed analysis of its optical/UV spectra and broadband spectral energy distribution (SED). Our study shows that the preferred black hole mass of RX J0134.2-4258 is $\sim 2~\times~10^{7}~M_{\odot}$, giving a mass accretion rate through the outer disc of $\dot{m}_{\rm out} \sim 20$ (assuming zero spin) compared to the observed luminosity ratio $L_{\rm bol}/L_{\rm Edd} \sim 6$. This reduction in radiative efficiency is expected for super-Eddington flows, as power can be lost via advection and/or winds. We find that the optical/UV lines of RX J0134.2-4258 resemble those from weak-like quasars (WLQs), as it has notably weak C IV and N V emission lines. It also has dramatic X-ray variability, again similar to that recently observed in some other WLQs. However, WLQs have systematically higher masses ($\gtrsim 10^8~M_{\odot}$), and lower Eddington ratios ($\dot{m}_{\rm out} \sim 1-3$) than RX J0134.2-4258. We compare instead to the most extreme NLS1s, with similarly large $\dot{m}_{\rm out}$ but smaller masses. These show similarly large reductions in radiative efficiency but their UV lines are not similarly wind-dominated. We suggest a new category of weak-line Seyfert (WLS) galaxies to describe sources like RX J0134.2-4258, and interpret its (so far unique) properties in a model where the lower disc temperature in the higher-mass black holes leads to UV line driving enhancing the super-Eddington radiation pressure driven wind.

https://export.arxiv.org/pdf/2208.06581

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} accretion, accretion discs - galaxies: active - galaxies: nuclei. \end{keywords} \section{Introduction} \label{sec-intro} \subsection{Narrow-line Seyfert 1 Galaxies} Active galactic nuclei (AGN) are powered by accretion onto a super-massive black hole (SMBH). This converts some fraction of the gravitational potential energy into radiation, powering the observed activity. The multi-wavelength properties of AGN are mainly determined by three parameters, namely the black hole mass, black hole spin and mass accretion rate. The inclination angle also plays a significant role in observations (e.g. \citealt{Luo.2015, Jin.2017b}). Narrow-line Seyfert 1 (NLS1) galaxies are a subtype of AGN characterized by relatively narrow broad lines such as H$\beta$ and relatively weak narrow lines such as [O {\sc iii}]$\lambda$5007 (\citealt{Osterbrock.1985}; \citealt{Boroson.2002}). Comparing with the entire AGN population, NLS1s tend to have small black hole masses of $10^{6-7}~M_{\odot}$ and high mass accretion rates (e.g. \citealt{Pounds.1995}; \citealt{Mathur.2001}; \citealt{Boroson.2002}; \citealt{Jin.2012a}). In the X-ray band, it is common to observe a strong soft X-ray excess in NLS1s (e.g. \citealt{Boller.1996, Brandt.1997}), which can often be modelled with an ionized disc reflection component (e.g. \citealt{Miniutti.2004, Ross.2005, Crummy.2006, Fabian.2013}), and/or a separate warm Comptonisation component (e.g. \citealt{Laor.1997}; \citealt{Magdziarz.1998, Done.2012, Jin.2013, Jin.2016, Jin.2017a, Jin.2021}). Complex absorption (partially ionised material partially covering the source) can also shape the soft X-ray emission in some AGN (e.g. \citealt{Miller.2007, Turner.2007, Tatum.2012}). NLS1s themselves form two subtypes, including the X-ray {\it simple} NLS1s and X-ray {\it complex} NLS1s (\citealt{Gallo.2006}). The X-ray {\it simple} NLS1s have smooth and steep X-ray spectra, while the X-ray {\it complex} NLS1s show more complicated absorption and emission features. Meanwhile, NLS1s with high mass accretion rates, especially super-Eddington, are likely to have a geometrically-thick (i.e. puffed-up) inner disc structure and disc wind (\citealt{Ohsuga.2011, Takeuchi.2014, Jiang.2016}), which can obscure the intrinsic X-ray emission and introduce additional spectral complexities and variability (e.g. \citealt{Hagino.2016, Done.2016, Jin.2017b, Parker.2021}). Therefore, the difference between X-ray {\it simple} and {\it complex} NLS1s can be explained by their different inclination angles, which lead to different line-of-sight to the X-ray corona (\citealt{Done.2016, Jin.2017b}). Supporting evidence for these subtypes being intrinsically the same is that their optical/UV emission is the same, suggesting that their intrinsic disc properties should indeed be similar (\citealt{Done.2016}). \subsection{Weak-line Quasars} A similar physical scenario of a puffed-up inner disc with significant winds is proposed to explain the properties of weak-line quasars (WLQs, e.g. \citealt{Fan.1999, Plotkin.2010, Wu.2011, Wu.2012, Luo.2015, Ni.2018}). WLQs are characterized by their weak UV high-ionization emission lines, e.g. rest-frame equivalent width (REW) of C {\sc iv} $\lesssim 10$ \AA, and/or REW of Ly $\alpha~+$ N {\sc v} is $\lesssim 15$ \AA\ (e.g. \citealt{Ni.2018}). Winds are clearly indicated as the peak of the weak C {\sc iv} lines is often highly blue-shifted (\citealt{Richards.2011, Rankine.2020}). WLQs have high black hole masses of $10^{8-9}M_\odot$, and also fairly high but not extreme mass accretion rates of $L_{\rm bol}/L_{\rm Edd}\sim 1$ (e.g. \citealt{Luo.2015}). The empirical $\alpha_{\rm ox}-L_{\rm 2500\text{\AA}}$ relation (e.g. \citealt{Lusso.2016}) implies that these are somewhat X-ray weak compared to less luminous quasars, but $\sim$35\% of the WLQ population show X-ray emission which is at least a factor 6 below this expectation (\citealt{Pu.2020}). This fraction of X-ray weakness is significantly higher than in the non-WLQ AGN population. A plausible explanation is that the high Eddington ratio causes the inner accretion disc of a WLQ to puff up, which partially shields the X-ray emission from near the black hole. Then the observed X-ray emission will depend on the viewing angle, in which case an X-ray weak WLQ will have a higher inclination angle, so that the line-of-sight to the X-ray corona is obscured by the geometrically thick inner disc (e.g. \citealt{Wu.2011, Luo.2015, Ni.2018}). Therefore, WLQs and super-Eddington NLS1s share some similar properties (\citealt{Leighly.2007b, Jin.2017b}), yet they do also differ significantly in black hole masses and, more importantly, in Eddington ratios, so the disc structure and geometry need not be the same. A more detailed comparison between these two AGN populations would allow us to better understand the evolution of super-Eddington accretion flows with black hole mass and mass accretion rate. \subsection{The Multi-wavelength Campaign on \rxj0134} We conduct a new multi-wavelength campaign from radio to hard X-rays to observe one of the most extreme super-Eddington NLS1s, namely \rxj0134 in order to deepen our understanding about super-Eddington accretion. This campaign involves new observations with \xmm, \nustar, \swift, {\it ATCA} and the 2.3-m telescope in the Sliding Spring Observatory (SSO), as well as a large set of archival multi-wavelength data (see Section~\ref{sec-obs} and \citealt{Jin.2022}, here after: Paper-I). \rxj0134 was discovered by \citet{Voges.1999} in the {\it ROSAT} all sky survey. Its key properties are summarized below, while a more detailed introduction can be found in Paper-I. This NLS1 lies at the redshift of 0.237, and it appears as an unresolved source in optical. It has a black hole mass of $M_{\rm BH}\simeq1.5\times10^{7}M_{\odot}$ and an extremely high Eddington ratio of $L_{\rm bol}/L_{\rm Edd}\simeq10.0$ (\citealt{Grupe.2010}). It has a steep hard X-ray slope ($\Gamma \simeq 2.2$, Paper-I), typical of NLS1, but has only an extremely weak soft X-ray excess, which is both peculiar and puzzling. In addition, it also exhibits drastic X-ray variability in terms of both spectral shape and flux (Paper-I). Its optical/UV properties such as the extremely weak [O {\sc iii}] $\lambda$5007 and blue-shifted C {\sc iv} were shown to be similar to the WLQ PHL 1811 by \citet{Leighly.2007b}. The latest simultaneous \xmm\ and \nustar\ observations in our campaign caught \rxj0134\ in its one of the lowest X-ray flux states in history, thus we conducted a detailed X-ray spectral-timing analysis. As shown in Paper-I, we found that the time-average X-ray spectra in the low-flux state has excess flux above 4 keV, which is lagged by $\sim$4 ks behind the soft X-rays. The spectral-timing properties in both low and high-flux states can be well modelled under the warm Comptonisation plus a distant neutral reflection scenario, or by a partial covering absorption scenario. Both scenarios require a clumpy disc wind in this super-Eddington accretion system. Here we perform a detailed multi-wavelength study from infrared, through optical/UV and then to X-rays to provide independent constraints on the global properties of the accretion flow. For example, the optical/UV continuum can be used to measure the mass accretion rate through the outer disc (e.g. \citealt{Davis.2011, Done.2016}). The optical/UV emission/absorption lines can provide information about the broad-line region and outflows (e.g. \citealt{Bottorff.1997, Pancoast.2011, Pancoast.2014, Grier.2017, Li.2018}), and can also be used to measure virial black hole mass (e.g. \citealt{Peterson.2004, Vestergaard.2006, Peterson.2014, Du.2019}). The infrared emission can be used to constrain the properties of the dusty torus (e.g. \citealt{Fuller.2016, Collinson.2017, Martinez.2017, Landt.2019}). The broadband spectral energy distribution (SED) can be used to estimate the black hole mass and Eddington ratio (e.g. \citealt{Jin.2012a, Jin.2016, Jin.2017b}), which can then be used to measure the global radiative efficiency ($\mu$, e.g. \citealt{Davis.2011}). In this work, we collate a large multi-wavelength dataset to study \rxj0134. \subsection{The Scope of This Paper} This paper presents a detailed study on the optical/UV and broadband SED properties of \rxj0134, as well as a detailed comparison with some representative super-Eddington NLS1s and WLQs. We will suggest a new category of weak-line Seyfert (WLS) galaxies, and demonstrate that \rxj0134\ is an archetypal WLS. The structure of this paper is as follows. Firstly, we describe the multi-wavelength datasets used in this work, and then briefly describe the data reduction procedures. Then we present a detailed estimate of the black hole mass of \rxj0134\ because it is a key parameter. Section 4 presents a detailed multi-component broadband SED modelling, in order to derive key parameters such as the bolometric luminosity, mass accretion rate and Eddington ratio. In Section 5, we first compare \rxj0134\ with WLQs, and propose it as an archetypal WLS, i.e. a new category of AGN. Then we use a small sample to conduct a more general comparison between the super-Eddington NLS1 population and the more typically Eddington WLQ population. In Section 6, we propose a picture for super-Eddington accretion flows with different parameters. We show how the disc properties, such as the disc structure, wind and global radiative efficiency, may depend on the black hole mass and mass accretion rate. Section 7 summarizes the main results of this paper. A detailed optical/UV spectral analysis is presented in the appendix. We adopt a flat universe model throughout this work, with the Hubble constant H$_{0} = 72$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\Lambda} = 0.73$ and $\Omega_{\rm M} = 0.27$. \begin{table} \centering \caption{The multi-wavelength dataset of \rxj0134\ used in this work. $T_{\rm obs}$ is the total observing time. For \nustar\ the Earth occultations and south atlantic anomaly passages have been excluded. A complete list of all the observations used by this research project can be found in Paper-I.} \begin{tabular}{@{}lccr@{}} \hline Instrument & Obs-Date & $T_{\rm obs}$ & Waveband \\ & & (ks) &\\ \hline \multicolumn{4}{c}{New Observations} \\ {\it NuSTAR} FPMA/FPMB& 2019-12-19 & 98.3 & Hard X-ray \\ \xmm\ EPIC/OM & 2019-12-19 & 134.3 & X-ray/UV \\ {\it Swift} XRT/UVOT & 2019-12-19 & 1.6 & X-ray/UV/Optical \\ {\it SSO 2.3-m Telescope} & 2019-12-19 & 1.8 & Optical \\ \hline \multicolumn{4}{c}{Archival Observations} \\ \xmm\ EPIC/OM & 2008-12-11 & 32.1 & X-ray/UV/Optical \\ {\it HST} FOS & 1996-09-21 & 1.7 & UV (G130H) \\ {\it HST} FOS & 1996-09-21 & 2.1 & UV (G130H) \\ {\it HST} FOS & 1996-09-21 & 0.2 & UV (G160L) \\ {\it HST} FOS & 1996-09-21 & 1.5 & UV (G190H) \\ {\it HST} FOS & 1996-09-21 & 1.2 & UV (G270H) \\ {\it HST} FOS & 1996-09-21 & 1.0 & Optical (G400H) \\ {\it HST} FOS & 1996-09-21 & 0.6 & Optical (G570H) \\ {\it WISE} & 2010-06-20 & -- & Infrared (Band 1-4) \\ {\it 2MASS} & 1999-08-27 & -- & Infrared ({\it J}, {\it H}, {\it K})\\ \hline \end{tabular} \label{tab-obs} \end{table} \section{Observations and Data Reduction} \label{sec-obs} We use a large number of observations, from both our new campaign and previous observations. These datasets are listed in Table~\ref{tab-obs}. The \xmm\ and \nustar\ data have been used in Paper-I for detailed X-ray spectral-timing analysis, where their data reduction are described in more detail. \subsection{X-ray Observations} There are two \xmm\ (\citealt{Jansen.2001}) observations for \rxj0134, whose observation dates differ by 11 years. The first observation in 2008 is referred to as Obs-1, and the second in 2019 is Obs-2. These observations also have simultaneous optical/UV data from various filters with the optical monitor (OM). We summarize the data reduction procedures below. The data are downloaded from the \xmm\ Science Archive (XSA), and reprocessed with the {\tt epproc} and {\tt empproc} tasks in the \xmm\ Science Analysis System (SAS v18.0.0). The source extraction region was chosen to be a circle of 35 arcsec radius, and no pile-up effect was detected during the two observations. The source and background spectra were extracted with the {\tt evselect} task, and the response and auxiliary files were produced by the {\tt rmfgen} and {\tt arfgen} tasks. The data obtained by the optical monitor (OM) were reprocessed with the {\tt omichain} task. The \nustar\ (\citealt{Harrison.2013}) observation of \rxj0134\ was conducted simultaneously with the \xmm\ observation in 2019. The {\tt nupipline} task inside the HEASoft package (v6.27.2, \citealt{Blackburn.1995}) was used to reprocess the data. The source extraction region was chosen to be a circle with 1 arcmin radius, and the background was extracted from a nearby circular source-free region with the same radius. The {\tt nuproducts} task was used to extract the spectra. There are 51 \swift\ (\citealt{Gehrels.2004}) observations on \rxj0134\ from 2019-12-19 to 2021-08-22. In this work we only use the observation conducted on 2019-12-31, because it is simultaneous with the \xmm\ and \nustar\ observations. A complete analysis of all the \swift\ observations will be present in a following paper (Panessa et al. in preparation, hereafter: Paper-III). Six filters were used in the \swift\ Ultra-violet Optical Telescope (UVOT) during this \swift\ observation (i.e. UVW2, UVM2, UVW1, U, B and V). The HEASsoft (v6.27.2) package was used to reduce the data. The \swift\ X-ray Telescope (XRT) data were reprocessed with the {\tt xrtpipeline}. The source spectrum was extracted from a circular region of 30 arcsec radius. For the UVOT photometric data, a circular aperture of 5 arcsec radius was adopted. Background was chosen from nearby source-free regions with larger areas. We also ran the standard sensitivity check for UVOT, in order to ensure that the data are not affected by the regions on the detector where the throughputs are degraded due to the contamination of dust/debris (\citealt{Edelson.2015}). \subsection{Optical/UV/Infrared Observations} \label{sec-optuv-spec} {\it Hubble} Space Telescope ({\it HST}) observed \rxj0134\ in 1996 with the Faint Object Spectrograph (FOS), which covered the spectral range of 970 -- 5500 \AA\ in the AGN rest-frame. The calibrated data were downloaded from the Mikulski Archive for Space Telescopes (MAST), from which the spectra were extracted with the IRAF/STSDAS tasks following the standard procedure\footnote{https://www.stsci.edu/instruments/wfpc2/Wfpc2\_dhb/intro\_ch36.html}. We obtained a new optical spectrum of \rxj0134\ with the SSO 2.3-m telescope on 2019-12-19. Infrared photometry from {\it WISE} (band: 1 -- 4) and {\it 2MASS} (band: {\it J}, {\it H}, {\it K}) were downloaded from the NASA/IPAC Infrared Science Archive (IRSA). Then we analyze the SSO optical and {\it HST} UV spectra, fitting for the lines and continuum components. Details are given in Appendix~\ref{sec-optuvfit}, with spectra shown in Figure~\ref{fig-optspec1}a, de-reddened with $E(B-V)=0.0144$ for the Fitzpatrick \& Massa (2007) reddening curve for $R_{\rm V}$ = 3.1, and de-redshifted for $z=0.237$. These data are separated by 23 years but the overall difference of normalization is only 7\%. Below 4000\AA, the two spectra match almost perfectly after removing this 7\% difference, while above 4000\AA\ the {\it HST} spectrum is weaker by another 5\%. Therefore, we can connect the {\it HST} spectrum (scaled up by 1.07) and the SSO spectrum at 4000\AA\ to derive a broad optical/UV spectrum, which is shown in Figure~\ref{fig-optspec1}b. The optical spectrum of \rxj0134\ resembles a typical NLS1 galaxy. According to the empirical eigenvector 1 of AGN (e.g. \citealt{Boroson.2002, Jin.2012c}), the strong Fe {\sc ii} and weak [O {\sc iii}]$\lambda$5007 lines imply that \rxj0134\ should have a very high mass accretion rate. Another key property of \rxj0134\ is that its UV spectrum has very weak and blue-shifted C {\sc iv} and Ly$\alpha$+N{\sc v} emission lines, like WLQs. We performed a detailed multiple Gaussian+Lorentzian profile decomposition for different emission lines. The methods and results are described in Appendix~\ref{sec-optuvfit}. The optical/UV line decomposition and best-fit parameters can be found in Figures~\ref{tab-optfit}, \ref{tab-uvfit} and Tables~\ref{tab-optfit}, \ref{tab-uvfit}. \section{The Black Hole Mass} \label{sec-bhmass} \citet{Grupe.2010} reported a virial mass of $M_{\rm BH}~=~1.47\times10^{7}M_{\odot}$ for \rxj0134, which is based on the single-epoch H$\beta$ full width at half-maximum (FWHM) of 1160 km s$^{-1}$, and the radius-luminosity (R-L) relation reported by \citet{Kaspi.2000}. We use the latest SSO optical spectrum and measure the H$\beta$ FWHM to be 1140 $\pm$ 20 km s$^{-1}$ for the Lorentzian decomposition, and 1410 $\pm$ 70 km s$^{-1}$ for the Gaussian decomposition. The monochromatic luminosity at the rest-frame 5100 \AA\ is measured to be $(8.86\pm0.92)\times10^{44}$ erg s$^{-1}$. For the two H$\beta$ FWHM measurements and using a later R-L relation reported by \citet{Vestergaard.2006}, we obtain a black hole mass of $M_{\rm BH}~=~(3.1-4.8)\times10^{7}M_{\odot}$. However, recent reverberation mapping studies have shown that for super-Eddington AGN, the observed radius of the broad line region (BLR) is smaller than expected from the classic R-L relation (\citealt{Du.2018}). This is likely due to changes in the disc structure and radiation as the flow becomes super-Eddington. Firstly the accretion flow has intrinsically lower radiative efficiency than a standard disc due to advection and/or winds, and secondly the inner disc may become geometrically thick, which, together with any wind, can provide a shielding mechanism for the ionization of BLR (e.g. \citealt{Abramowicz.1988, Wang.2003, Jiang.2014, Done.2016, Jin.2016, Jin.2017b}). As a result, previous versions of R-L relation can lead to an over-estimate of the black hole mass in the super-Eddington regime. Recently, \citet{Du.2019} reported a new R-L relation which includes the Fe {\sc ii} to H$\beta$ (broad component) flux ratio ($R_{\rm FeII}$) as an additional parameter. This new relation provides lower mass estimates than traditional relations for super-Eddington AGN. For \rxj0134, $R_{\rm FeII}$ is found to be 1.74 $\pm$ 0.15 (see Section~\ref{sec-opt-spec}), which is larger than most of the sources in \citet{Du.2019}. Then it is necessary to choose a value for the virial factor $f_{\rm BLR}$, which depends on the morphology of the host galaxy. Since the host galaxy of \rxj0134\ cannot be resolved, different $f_{\rm BLR}$ values need to be tried to understand its impact on the mass estimate (see Table~\ref{tab-mass-mdot}). \citet{Ho.2014} reported $f_{\rm BLR}=0.7\pm0.2$ for AGN in pseudo-bulges, which leads to a mass of $(0.79\pm0.25)\times10^{7}M_{\odot}$. Then for AGN in classic bulges or ellipticals with $f_{\rm BLR}=1.5\pm0.4$, the mass is $(1.70\pm0.54)\times10^{7}M_{\odot}$. If we choose $f_{\rm BLR}=1.12$ for the sample of 93 NLS1s reported by \citet{Woo.2015}, then the mass is found to be $(1.27\pm0.22)\times10^{7}M_{\odot}$. Therefore, it is clear that the uncertainty of $f_{\rm BLR}$ affects the single-epoch virial mass significantly. The intrinsic scatter of 0.2 dex of this new R-L relation introduces further uncertainty, and so there is no significant difference between these virial masses. The rapid X-ray variability provides an independent method to estimate the black hole mass. Various studies have shown that the mass scales with the X-ray rms (\citealt{Lu.2001, Zhou.2010, Ponti.2012, Jin.2016}). We calculate the X-ray rms for different variability timescales, leading to a mass range of $(0.8-2.5)\times10^{7}M_{\odot}$ and a mean value of $1.7\times10^{7}M_{\odot}$ (see Paper-I). This mass estimate is subject to an intrinsic scatter of 0.7 dex. Overall, a typical mass estimate of $M_{\rm BH}\sim 2\times10^{7}M_{\odot}$ for \rxj0134\ should be statistically consistent with all the mass estimates presented above. \begin{table} \centering \caption{Different mass estimates for \rxj0134, and the corresponding values for some other key parameters, including the mass accretion rates ($\dot{m}_{\rm out}$), Eddington ratio ($L_{\rm bol}/L_{\rm Edd}$) and radiative efficiency ($\mu$). The best-fit SED for Obs-1 assumes $a_{*}=0$ and $\mu_0=0.057$, and has $L_{\rm bol}=1.63\times10^{46}$ erg s$^{-1}$. Typical uncertainties are provided for $M_{\rm BH}$ and propagated into $\dot{m}_{\rm out}$, but the intrinsic 0.2 dex scatter is not included. Systematic uncertainties should dominate $L_{\rm bol}$ and $\mu$, so their errors are not provided.} \begin{tabular}{@{}lccccc@{}} \hline Method & $M_{\rm BH}$ & $L_{\rm bol}/L_{\rm Edd}$ & $\dot{m}_{\rm out}$ & $\mu/\mu_{\rm 0}$ & $\mu$ \\ & ($10^{7}M_{\odot}$) & & \\ \hline Best-fit SED & 2.00 fixed & 6.3 & 20.6 $^{+0.3}_{-0.6}$ & 0.31 & 0.017 \\ X-ray Rms & 1.70 $\pm$ 0.80 & 7.4 & 28.5 $^{+0.4}_{-0.8}$ & 0.26 & 0.015 \\ \multicolumn{3}{@{}l}{R-L Relation from \citet{Du.2019}} \\ ($f_{\rm BLR}=0.70$) & 0.79 $\pm$ 0.25 & 15.9 & 132.0 $^{+1.9}_{-3.8}$ & 0.12 & 0.007 \\ ($f_{\rm BLR}=1.12$) & 1.27 $\pm$ 0.22 & 9.9 & 51.1 $^{+0.7}_{-1.5}$ & 0.19 & 0.011 \\ ($f_{\rm BLR}=1.50$) & 1.70 $\pm$ 0.54 & 7.4 & 28.5 $^{+0.4}_{-0.8}$ & 0.26 & 0.015 \\ \hline \end{tabular} \label{tab-mass-mdot} \end{table} \section{Multi-wavelength Properties} \label{sec-multiwavelength} \subsection{Broadband Spectral Energy Distribution} \label{sec-sed} \subsubsection{Preparation of the Multi-wavelength Data} The study of spectral energy distribution (SED) can provide crucial information about the accretion system, such as the black hole mass and spin, mass accretion rate, the energy budget in different wavebands and spectral components (e.g. \citealt{Jin.2012a, Done.2012, Done.2013}). The abundant multi-wavelength data collected from our new observations and public data archives allow us to reconstruct the broadband SED of \rxj0134. The dataset used to construct the SED is listed in Table~\ref{tab-obs}, which includes \xmm\ EPIC-pn and five OM filters (UVW2, UVM2, UVW1, U and B) from Obs-1. We neglect Obs-2, with its factor of $\sim$ 4 lower EPIC-pn count rate, as it has a more complex X-ray shape which is most likely due to absorption variability rather than intrinsic spectral change (see Paper-I). We note that the corresponding simultaneous UV fluxes from the OM UVW1 filter are more similar, with Obs-2 being 16 per cent brighter than in Obs-1. The remaining datasets used are \swift\ XRT and UVOT, {\it ROSAT} PSPCB spectrum, {\it HST} FOS spectra, {\it 2MASS} and {\it WISE} photometric points. \rxj0134\ shows all kinds of emission lines in its optical/UV spectrum, which need to be removed so that the continuum can be used for the SED fitting. These lines will also contribute to the optical/UV photometry, thus we need to perform corrections for all the optical/UV photometric fluxes. We visually inspect the optical/UV spectrum, and choose a series of data points to define the underlying continuum, as shown by the yellow points and blue dash line in Figure~\ref{fig-optspec1}b. The cyan region between this continuum and the observed spectrum is considered to come from emission lines. The line-free continuum is converted into {\sc xspec}-readable spectral file for the SED fitting. We also calculate the fraction of continuum flux in every optical/UV band, and then correct the photometric data of the \xmm/OM filters to remove the emission line flux. We calculate the line flux contribution to each filter by convolving the spectra with the full response files of each optical/UV filters, read from the response files stored in the latest calibration database at \xmm\footnote{https://heasarc.gsfc.nasa.gov/FTP/xmm/data/responses/om}. We calculated the correction factor for every OM and UVOT filter, and then apply it to the corresponding photometric data. The typical correction factor is 5 -- 10 per cent, increasing from UV to optical. Finally these data are used as inputs for the SED fitting. \subsubsection{The Accretion Disc Model} \rxj0134\ is an extreme super-Eddington NLS1 (\citealt{Grupe.2010}). In the inner region of such a super-Eddington accretion flow, energy advection and/or disc winds can take away a significant amount of the accretion energy (e.g. \citealt{Poutanen.2007, Hagino.2016, Done.2016, Jin.2017b}). There are several AGN SED models in {\sc xspec} (\citealt{Arnaud.1996}), such as {\tt optxagnf} (\citealt{Done.2012}), {\tt agnsed} (\citealt{Kubota.2018}) and {\tt agnslim} (\citealt{Kubota.2019}). Only the latter adopts a slim disc emissivity, where the surface luminosity is kept at the local Eddington limit within a critical radius (\citealt{Abramowicz.1988, Watarai.2000, Wang.2003, Sadowski.2011, Kubota.2019}). {\tt agnslim} uniquely combines this maximum emissivity with the ability to change the local emission to change between blackbody, soft Comptonisation and hard Comptonisation, in order to model the disc, soft excess and hard X-ray corona emission, respectively. Another major difference between a standard disc and a slim disc is that the inner radius of the disc ($R_{\rm in}$) is determined by the gas pressure more than by the black hole spin for highly super-Eddington discs (e.g \citealt{Watarai.2000}). This removes the most obvious signature of black hole spin, so we conservatively fix spin at zero here, with the consequence of minimizing the inferred Eddington ratio of the flow. By default, in the {\tt agnslim} model the seed photon temperature of the hard X-ray Comptonisation is set to be the temperature of inner disc photons. However, recent studies of X-ray {\it simple} super-Eddington NLS1s show that their X-ray spectral-timing properties are better modelled if the warm corona, rather than the inner disc, provides seed photons for the hot corona (\citealt{Jin.2013, Jin.2016, Jin.2017a, Jin.2021}). Thus we made a small modification to {\tt agnslim} to link the seed-photon temperature of the hard X-ray Comptonisation to the electron temperature of the warm corona. We refer to this modified {\tt agnslim} model as {\tt agnslimhot}, and use it in our subsequent SED analysis. {\tt agnslimhot} inherits the full set of parameters of {\tt agnslim} (see Table~\ref{tab-sedfit}). The black hole mass $M_{\rm BH}$ is fixed at $2\times 10^7 M_{\odot}$. The comoving distance is fixed at 937.1 Mpc for redshift $z=0.237$. The inclination angle $\theta_{\rm inc}$ is fixed at 60\degr, which is larger than 30\degr\ as often assumed for normal X-ray {\it simple} super-Eddington NLS1s. This is because the enigmatic X-ray variability of \rxj0134\ implies complex and variable absorption, which is more likely to happen at a larger inclination angle. The electron temperature of the hot corona $kT_{\rm e, hot}$ is fixed at 200 keV, and the overall normalization is fixed at 1. The remaining free parameters include the electron temperature $kT_{\rm e, warm}$, photon index $\Gamma_{\rm warm}$ and radius $R_{\rm warm}$ of the warm corona; the photon index $\Gamma_{\rm hot}$ and radius $R_{\rm hot}$ of the hot corona; the mass accretion rate through the outer disc $\dot{m}_{\rm out}$ and the outer radius of the disc $R_{\rm out}$. \subsubsection{Additional Components in the Broadband SED} \rxj0134\ was originally classified as a radio-loud (RL) AGN with a radio loudness of $R=71$ (\citealt{Grupe.2000}), thus there is possibility that the X-ray emission might also include some contribution from the jet, such as the synchrotron self-Compton (SSC) and external Compton (EC) emission (e.g. \citealt{Kynoch.2018}). However, we did not find any evidence of jet emission from X-ray spectral-timing analysis (see Paper-I), and our ongoing radio/optical monitoring campaign shows that \rxj0134\ has returned to a radio-quiet state (see Paper-III). Therefore, the spectral components in the {\tt agnslimhot} model should be enough to fit the nuclear emission. To model the hot dust emission in the near infrared, we take the hot dust template from \citet{Silva.2004}, and import it into {\sc xspec} as a local {\tt zagndust} model. The host galaxy is not resolved in optical images, but it is still possible to identify its flux contribution in the spectrum. We assume it is an Sb galaxy similar to the famous NLS1 \rej1034, and adopt a corresponding galaxy spectral template from \citet{Polletta.2007}, which is loaded into {\sc xspec} as the local {\tt hostgal} model. Since not all the datasets are simultaneous or observed by the same instrument, there can be normalization discrepancies caused by e.g. long-term variability, different aperture size and flux calibration. Thus we use a free constant to account for the normalization differences between the {\it ROSAT}, \xmm\ and {\it HST} data. The data points from {\it 2MASS}, {\it WISE} and {\it HST} join smoothly with each other, and so we use the same constant for these three datasets. This combination of accretion flow, host galaxy and hot dust, describe the intrinsic continuum, but these spectra are further modified by absorption and reddening along the line of sight. We use {\tt tbabs}/{\tt ztbabs} (\citealt{Wilms.2000}) to model the gas absorption from the Milky Way and host galaxy, respectively, with $N_{\rm H,gal}$ fixed at 1.77$\times10^{20}$ cm$^{-2}$ for our line of sight (\citealt{Willingale.2013}), and $N_{\rm H,host}$ left free. The absorption cross-sections were set to the values of \citet{Verner.1996}. While this model is a good approximation to the X-ray absorption, it is less good at modelling the impact of this same gas in the UV due to its assumption that the material is completely neutral (but the interstellar medium is multiphase, e.g. \citealt{McKee.1995, Wolfire.2003}) and that the UV absorption is dominated by bound-free edges rather than lines. Nonetheless, both the Galactic column and the host-galaxy column here are rather small (see Section~\ref{sec-sedfit1} below), so this mis-modelling of its UV absorption is not very important. We also use {\tt redden}/{\tt zredden} to model the dust reddening associated with the gas in the Milky Way and the host galaxy, respectively. The Galactic reddening $E(B-V)_{\rm gal}$ is fixed at 0.0144 (\citealt{Schlegel.1998}), while $E(B-V)_{\rm host}$ is left as a free parameter. We note that {\tt zredden} may not be appropriate to describe the effect of dust in the host galaxy if this is associated with the nuclear region rather than in the interstellar medium (see e.g. \citealt{Collinson.2015}), but this has little impact here as the UV is clearly a very blue spectrum, so the reddening is most probably limited. \begin{table} \centering \caption{The best-fit SED parameters of \rxj0134\ in Obs-1. The errors indicate 90 per cent confidence limits. `fixed' indicates that the parameter is fixed at the given value. $C_{\rm ROSAT}$ and $C_{\rm HST}$ are the scaling factors for the {\it ROSAT} and {\it HST} data.} \begin{tabular}{llcc} \hline Component & Parameter & Value & Unit \\ \hline {\tt tbabs} & $N_{\rm H, gal}$ & 1.77 fixed & $10^{20}$ cm$^{-2}$ \\ {\tt redden} & $E(B-V)_{\rm gal}$ & 1.44 fixed & $10^{-2}$ \\ {\tt ztbabs} & $N_{\rm H, host}$ & 0.56 $^{+0.43}_{-0.27}$ & $10^{20}$ cm$^{-2}$ \\ {\tt zredden} & $E(B-V)_{\rm host}$ & 0.40 $^{+0.27}_{-0.24}$ & $10^{-2}$ \\ {\tt zagndust} & norm & 1.90 $^{+0.09}_{-0.08}$ & $10^{-5}$ \\ {\tt hostgal} & norm & 1.05 $^{+0.30}_{-0.28}$ & $10^{-2}$ \\ {\tt agnslimhot} & $M_{\rm BH}$ & 2.0 fixed & $10^{7}M_{\odot}$ \\ {\tt agnslimhot} & log($\dot{m}_{\rm out}$) & 1.31 $^{+0.01}_{-0.01}$ & \\ {\tt agnslimhot} & $a_{*}$ & 0.0 fixed & \\ {\tt agnslimhot} & cos $\theta_{\rm inc}$ & 0.5 fixed \\ {\tt agnslimhot} & $kT_{\rm e, warm}$ & 0.22 $^{+0.37}_{-0.08}$ & keV \\ {\tt agnslimhot} & $kT_{\rm e, hot}$ & 200 fixed & keV \\ {\tt agnslimhot} & $\Gamma_{\rm hot}$ & 2.22 $^{+0.05}_{-0.04}$ & \\ {\tt agnslimhot} & $\Gamma_{\rm warm}$ & 2.84 $^{+1.46}_{-1.62}$ & \\ {\tt agnslimhot} & $R_{\rm hot}$ & 4.65 $^{+0.13}_{-0.16}$ & $R_{\rm g}$ \\ {\tt agnslimhot} & $R_{\rm warm}$ & 5.44 $^{+0.87}_{-0.28}$ & $R_{\rm g}$ \\ {\tt agnslimhot} & log($R_{\rm out}$) & 4.97 $^{+0.24}_{-0.22}$ & $R_{\rm g}$ \\ {\tt $C_{\rm ROSAT}$} & & 0.75 $^{+0.04}_{-0.05}$ & \\ {\tt $C_{\rm HST}$} & & 1.16 $^{+0.05}_{-0.05}$ & \\ \multicolumn{2}{l}{$\chi^{2}_{\nu}$} & 607.3/475 & \\ \hline \end{tabular} \label{tab-sedfit} \end{table} \subsection{Results of the SED Modelling} \label{sec-sedfit1} Based on the above datasets and model configurations, we obtain the best-fit broadband SED for \rxj0134. Figure~\ref{fig-sedfit}a shows this SED model, where both model and data are corrected for the Galactic and intrinsic extinction/absorption, and shown in the AGN rest-frame. The best-fit parameters are listed in Table~\ref{tab-sedfit}. It is clear that the near-IR emission is dominated by the hot dust components, while the UV continuum is well-fitted by the accretion disc component. There is a small ($\sim 10$ per cent) contribution from host galaxy star light between these two in the optical/near-IR band. At higher energies, the soft X-ray emission observed by {\it ROSAT} below 0.3 keV is dominated by the emission from the inner disc. The hard X-rays above 2 keV are dominated by a hot corona with photon index of 2.22$^{+0.05}_{-0.04}$. There is some evidence for a warm Comptonisation component, with electron temperature of 0.22$^{+0.37}_{-0.08}$ keV and photon index 2.84$^{+1.46}_{-1.62}$. All these Comptonisation parameters are typical for X-ray {\it simple} super-Eddington NLS1s (e.g. \citealt{Jin.2013, Jin.2016, Jin.2017a}), apart from the soft X-ray excess being much weaker relative to the disc and hot corona as discussed in Paper-I (see also the explicit comparison to \rx04\ in Section \ref{sec-wls}). The mass accretion rate through the outer disc ($\dot{m}_{\rm out}$) is completely determined by the observed optical/UV emission for the fixed black hole mass and spin, giving $\dot{m}_{\rm out}=20.6^{+0.3}_{-0.6}$, confirming that the accretion flow is highly super-Eddington. A higher value of black hole spin will only increase this. The only way to significantly reduce $\dot{m}_{\rm out}$ is to go to higher black hole mass, as the monochromatic luminosity on the Rayleigh-Jeans part of the standard (multi-temperature blackbody) disc continuum has $L_{\rm\nu} \propto (M_{\rm BH}~\dot{M})^{2/3} \propto (M_{\rm BH}^2~\dot{m}_{\rm out})^{2/3}$, where $\dot{m}_{\rm out}=\dot{M}/\dot{M}_{\rm Edd}$ (e.g. \citealt{Shakura.1973, Davis.2011, Kubota.2019}). Increasing the mass by a factor of 2 (i.e. $4\times 10^7 M_\odot$) then reduces $\dot{m}_{\rm out}$ by a factor of 4, but then $\dot{m}_{\rm out}$ is still $\sim 5$, so the disc is still super-Eddington even a factor of 2 away from our preferred black hole mass. The observed bolometric luminosity is derived by integrating the best fit model, and gives $L_{\rm bol}=1.63\times 10^{46}$~ergs s$^{-1}$. Thus $L_{\rm bol}/L_{\rm Edd}=6.3$, substantially below the $\dot{m}_{\rm out}=20.6$ derived for the accretion flow itself. This is clear evidence for a loss of power through advection and/or winds, as expected for a strongly super-Eddington flow. \begin{table*} \centering \caption{Comparison of optical/UV emission line properties among super-Eddington NLS1s, WLQs and the composite QSO spectra from \citet{Francis.1991} (F19). The line fitting method is described in Appendix~\ref{sec-optuvfit}. The reported equivalent widths are for the emission lines of Ly$\alpha$ plus the N {\sc v} $\lambda$1238/1243 doublet, C {\sc iv} $\lambda$1548/1551 doublet, Si {\sc iv} $\lambda$1393/1402 doublet, Mg {\sc ii} $\lambda$2797/2803 doublet, Fe {\sc ii} (4434 -- 4684 \AA) and [O {\sc iii}] $\lambda$5007. $v_{\rm blue}$ is the velocity of the blue-shifted Gaussian component in that emission line. $R_{\rm FeII}$ is the flux ratio between the Fe {\sc ii} (4434 -- 4684 \AA) and H$\beta$ broad Gaussian component.} \begin{tabular*}{\textwidth}{@{}lcccccccccc@{}} \hline Source & Ly$\alpha$ + N {\sc v} REW & Si {\sc iv} REW & C {\sc iv} REW & C {\sc iv} $v_{\rm blue}$ & Mg {\sc ii} REW & Fe {\sc ii} REW & H$\beta$ REW & H$\beta$ FWHM & [O {\sc iii}] REW & $R_{\rm FeII}$ \\ & (\AA) & (\AA) & (\AA) & (km s$^{-1}$) & (\AA) & (\AA) & (\AA) & (km s$^{-1}$) & (\AA) & \\ \hline RE10 & 85.6 $\pm$ 8.6 & 22.6 $\pm$ 3.2 & 62.1 $\pm$ 5.3 & -620 $\pm$ 550 & 22.6 $\pm$ 1.6 & 24.4 $\pm$ 1.8 & 31.7 $\pm$ 1.4 & 620 $\pm$ 20 & 33.1 $\pm$ 2.3 & 0.79 $\pm$ 0.07 \\ \ph1092 & 23.5 $\pm$ 2.4 & 6.6 $\pm$ 2.1 & 12.1 $\pm$ 2.5 & -9900 $\pm$ 2400 & -- & 53.2 $\pm$ 4.2 & 26.1 $\pm$ 5.4 & 1700 $\pm$ 300 & 4.5 $\pm$ 1.3 & 2.33 $\pm$ 0.45 \\ \phl1811 & 28.6 $\pm$ 2.9 & 7.9 $\pm$ 2.0 & 7.9 $\pm$ 2.5 & -1900 $\pm$ 1200 & 13.8 $\pm$ 2.0 & 33.6 $\pm$ 3.0 & 36.1 $\pm$ 3.9 & 1900 $\pm$ 200 & 3.3 $\pm$ 2.8 & 1.11 $\pm$ 0.15 \\ RX04 & 37.3 $\pm$ 3.7 & 7.5 $\pm$ 2.0 & -- & -- & -- & 27.6 $\pm$ 2.0 & 24.3 $\pm$ 2.5 & 4000 $\pm$ 800 & 6.3 $\pm$ 0.7 & 1.34 $\pm$ 0.16 \\ 1H07 & 39.1 $\pm$ 3.9 & 6.9 $\pm$ 4.5 & 21.6 $\pm$ 5.9 & -1200 $\pm$ 1100 & 9.8 $\pm$ 1.3 & 44.1 $\pm$ 3.3 & 25.4 $\pm$ 7.0 & 680 $\pm$ 60 & 4.7 $\pm$ 3.0 & 2.01 $\pm$ 0.47 \\ RX01 & 24.4 $\pm$ 2.4 & 8.4 $\pm$ 1.2 & 7.6 $\pm$ 1.5 & -7600 $\pm$ 5700 & 12.1 $\pm$ 2.1 & 41.5 $\pm$ 3.4 & 23.8 $\pm$ 1.2 & 1410 $\pm$ 70 & 2.7 $\pm$ 0.8 & 1.74 $\pm$ 0.16 \\ (F19) & 49.7 $\pm$ 5.0 & 9.0 $\pm$ 4.2 & 32.6 $\pm$ 7.0 & -830 $\pm$ 510 & 26.7 $\pm$ 2.7 & 31.2 $\pm$ 9.3 & 67.5 $\pm$ 4.1 & 3210 $\pm$ 40 & 17.9 $\pm$ 8.2 & 0.53 $\pm$ 0.15 \\ \hline \end{tabular*} \label{tab-comp-line} \end{table*} \begin{table*} \centering \caption{Comparison of the broadband SED properties among super-Eddington NLS1s, WLQs and mean quasar properties. For individual sources, typical black hole masses are adopted from the literatures listed below. Other parameters are measured from the SEDs shown in Figure~\ref{fig-sed-compare}. $L_{\rm bol}$ and $\dot{m}_{\rm out}$ are the bolometric luminosity and mass accretion rate through the outer disc. We do not provide errors for $L_{\rm bol}$, $\dot{m}$ or related parameters as they should be dominated by systematic SED model uncertainties which are difficult to estimate. $\mu$ is the observed radiative efficiency. $\mu_{0}=0.057$ is for the standard \citet{Shakura.1973} disc with zero spin. $k_{\rm 2-10keV}$, $k_{\rm 0.5-1keV}$ and $k_{5100\textup{\AA}}$ are the bolometric corrections for 2-10 keV, 0.5-1 keV and 5100 \AA. $f_{\rm IR}$ is the fraction of infrared dust luminosity in 1-50 $\mu$m relative to $L_{\rm bol}$. $\alpha_{\rm ox}$ and $\alpha_{\rm optir}$ are the optical-to-X-ray and optical-to-infrared spectral indices. Their statistical 1-$\sigma$ errors are provided. The relative uncertainty of infrared luminosity is assumed to be 10 per cent. Zero spin is assumed for all the parameter values listed in this table.} \begin{tabular}{ccccccccccccc} \hline & $M_{\rm BH}$ & $L_{\rm bol}$ & $L_{\rm bol}/L_{\rm Edd}$ & $\dot{m}_{\rm out}$ & $\mu$ & $\mu/\mu_{\rm 0}$ & $k_{\rm 2-10keV}$ & $k_{\rm 0.5-1keV}$ & $k_{\rm 5100\textup{\AA}}$ & $\alpha_{\rm ox}$ & $\alpha_{\rm optir}$ & $f_{\rm dust}$ \\ & ($10^{7}M_{\odot}$) & ($10^{45}$ erg s$^{-1}$) & & & & & & & & & & (\%) \\ \hline \multicolumn{13}{@{}l}{RE J1034+396: (\rej1034, $z=0.042$, moderate-$\dot{m}$ X-ray {\it simple} NLS1)} \\ Obs-1 & 0.2 & 0.5 & 1.7 & 1.7 & 0.057 & 1.00 & 226 & 64 & 60 & 1.33 $\pm$ 0.07 & 1.23 $\pm$ 0.15 & 16 \\ \hline \multicolumn{13}{@{}l}{PHL 1092 (\ph1092, $z=0.396$, WLQ)} \\ Obs-1 & 10.0 & 29.8 & 2.3 & 2.3 & 0.057 & 1.00 & 399 & 61 & 23 & 1.69 $\pm$ 0.07 & 0.94 $\pm$ 0.08 & 26 \\ Obs-2 & 10.0 & 29.1 & 2.2 & 2.2 & 0.057 & 1.00 & 56900 & 12700 & 23 & 2.49 $\pm$ 0.25 & 0.87 $\pm$ 0.08 & 22 \\ \hline \multicolumn{13}{@{}l}{PHL 1811 (\phl1811, $z=0.192$, WLQ)} \\ Obs-1 & 15.0 & 89.2 & 4.6 & 7.4 & 0.035 & 0.62 & 16500 & 37900 & 17 & 2.46 $\pm$ 0.11 & 0.67 $\pm$ 0.08 & 17 \\ \hline \multicolumn{13}{@{}l}{RX J0439.6-5311 (\rx04, $z=0.242$, high-$\dot{m}$ NLS1: X-ray {\it simple})} \\ Obs-1 & 0.7 & 5.2 & 5.7 & 11.3 & 0.029 & 0.50 & 90 & 10 & 24 & 1.22 $\pm$ 0.04 & 0.91 $\pm$ 0.11 & 11 \\ \hline \multicolumn{13}{@{}l}{1H 0707-495 (\1h07, $z=0.041$, high-$\dot{m}$ NLS1: X-ray {\it complex})} \\ Obs-1 & 0.2 & 2.1 & 8.0 & 22.7 & 0.020 & 0.35 & 320 & 69 & 44 & 1.46 $\pm$ 0.04 & 0.57 $\pm$ 0.09 & 4 \\ Obs-2 & 0.2 & 2.6 & 9.9 & 31.2 & 0.018 & 0.32 & 1270 & 340 & 44 & 1.85 $\pm$ 0.06 & 0.45 $\pm$ 0.09 & 3 \\ Obs-3 & 0.2 & 1.7 & 6.5 & 26.4 & 0.014 & 0.25 & 2100 & 1460 & 33 & 2.02 $\pm$ 0.08 & 0.55 $\pm$ 0.09 & 5 \\ \hline \multicolumn{13}{@{}l}{RX J0134.2-4258 (\rxj0134, $z=0.237$, high-$\dot{m}$ NLS1 \& WLS)}\\ Obs-1 & 2.0 & 16.3 & 6.3 & 20.6 & 0.017 & 0.31 & 75 & 149 & 22 & 1.40 $\pm$ 0.02 & 0.66 $\pm$ 0.08 & 12 \\ Obs-2 & 2.0 & 17.3 & 6.7 & 26.2 & 0.015 & 0.26 & 319 & 600 & 20 & 1.70 $\pm$ 0.02 & 0.66 $\pm$ 0.08 & 13 \\ \hline \multicolumn{13}{@{}l}{Quasar Mean SED (\citealt{Elvis.1994})} \\ -- & -- & -- & -- & -- & -- & -- & 27 & 48 & 6 & 1.38 & 0.80 & 34 \\ \hline \multicolumn{13}{@{}l}{Quasar Mean SED (\citealt{Richards.2006})} \\ -- & -- & -- & -- & -- & -- & -- & 103 & 109 & 4 & 1.53 & 0.85 & 37 \\ \hline \end{tabular} \\References for the redshifts and black hole mass estimates: \rej1034: \citealt{Jin.2021}; \ph1092: \citealt{Miniutti.2012, Marinello.2020}; \phl1811: \citealt{Leighly.2007a, Leighly.2007b}; \rx04: \citealt{Jin.2017a}, \citealt{Jin.2017b}; \1h07: \citealt{Done.2016}; \rxj0134: this work; Quasar mean SEDs: \citealt{Elvis.1994}, \citealt{Richards.2006}. \label{tab-comp-sed} \end{table*} \section{Discussion} \subsection{Weak UV Lines and the SEDs of Super-Eddington Flows} \label{sec-wls} The weak UV lines of \src1\ resemble those from WLQs, defined as Ly$\alpha$ + N {\sc v} REW $\le$ 15 \AA, and C {\sc iv} REW $\le$ 10 \AA\ (\citealt{Fan.1999, Diamond-Stanic.2009, Plotkin.2010, Wu.2011, Wu.2012, Luo.2015, Ni.2018}). In this section, we compare \src1\ with some typical WLQs and NLS1s (see Table~\ref{tab-comp-sed})\footnote{Note that the estimates of the mass, mass accretion rate and bolometric luminosity are all subject to various systematic uncertainties, so we only show their typical values and do not provide their uncertainties in Table \ref{tab-comp-sed}.}, in order to obtain a full understanding of their similarities and differences. \subsubsection{PHL 1811 (hereafter: \phl1811): typical WLQ} So far, most of the known WLQs lie at relatively high redshifts with $z>2.2$. This redshift distribution is partly a selection effect as the C {\sc iv} line is redshifted into the more easily accessible optical band where wide area surveys such as Sloan Digital Sky Survey (SDSS, \citealt{York.2000}, 3000 -- 9200 \AA) detect a large number of sources. However, it is also possible to find WLQs at lower redshifts from pointed UV spectroscopic observations (e.g. \citealt{McDowell.1995, Londish.2004}), such as the classic WLQ \phl1811\ at $z=0.192$, whose black hole mass is $\sim1.5\times10^{8}M_{\odot}$ and mass accretion rate $\sim {7}$ (\citealt{Leighly.2007a, Leighly.2007b}; \citealt{Wu.2012, Luo.2015}). \citet{Leighly.2007a} first noticed the similarity of weak UV lines between \phl1811\ and \src1\ in the wavelength range of 1000-1600\AA. Here we compare the entire optical/UV spectra of \src1\ (black) and \phl1811\ (orange), as shown in Figure~\ref{fig-optuv-compare} panels a1 and b1. It is clear that these two AGN have remarkably similar optical/UV lines, including the strong optical Fe {\sc II} lines, extremely weak oxygen forbidden lines, and very weak and blue-shifted UV C {\sc iv}, C {\sc iii}] and N {\sc v} lines. Their optical/UV underlying continua also have similar shapes. Their optical continua, after correcting for the Galactic reddening, are consistent with a standard thin disc model (dotted grey curve: \citealt{Shakura.1973}), while their UV continua appear flatter at $\lesssim$ 2300 \AA. We apply the same line fitting method as described in Section~\ref{sec-uv-spec} to measure the REWs of the UV lines of \phl1811. We find Ly$\alpha$ + N {\sc v} REW = 28.6 $\pm$ 2.9 \AA\ and C {\sc iv} REW = 7.9 $\pm$ 2.0 \AA\ (for C {\sc iv} $\lambda$1548 + C {\sc iv} $\lambda$1551) for \phl1811, which are remarkably similar to \src1 (see Table~\ref{tab-comp-line}). However, \phl1811\ has a black hole mass which is around an order of magnitude higher than \src1. We fit the broad band SED of \phl1811\ with the same models as used for \src1, fixing black hole spin at $a_{*}=0$, so the model parameters are all directly comparable. This gives $\dot{m}_{\rm out}=7.4$ (see Table \ref{tab-comp-sed}), a factor of 3 lower than that of \src1. We compare their broad band SEDs as the solid black and solid orange lines in Figure \ref{fig-sed-compare}a. The lower black hole mass and higher $\dot{m}_{\rm out}$ of \src1\ predict a higher inner disc temperature, but it is clear that the WLQ \phl1811\ has much less X-ray emission relative to the disc peak, and that the X-rays have complex shape. More quantitatively, we adopt the optical-to-X-ray index ($\alpha_{\rm ox}$, e.g. \citealt{Lusso.2010}) as, \begin{equation} \alpha_{\rm ox} = - \frac{\rm{log}(L_{\rm 2 keV}/L_{\rm 2500\textup{\AA}})}{\rm{log}(\nu_{\rm 2 keV}/\nu_{\rm 2500\textup{\AA}})} \end{equation} where $L_{\rm 2 keV}$ and $L_{\rm 2500\textup{\AA}}$ are the luminosities at 2 keV and 2500 \AA. We find $\alpha_{\rm ox}$ is 2.46 $\pm$ 0.11 for \phl1811, but is only 1.40 $\pm$ 0.02 for Obs-1 of \src1. Paper-I shows that \src1\ also has a complex X-ray spectral shape when it is X-ray weaker. We include this Obs-2 spectrum as the dashed black line in Figure~\ref{fig-sed-compare}a. Clearly it is not so extremely X-ray weak as \phl1811, as confirmed by its $\alpha_{\rm ox}$ of 1.70 $\pm$ 0.02, but the same mechanisms may well be at work, most likely absorption and scattering in a clumpy wind (\citealt{Done.2016, Jin.2017b}). Such winds are clearly expected from super-Eddington sources. Then we compare the broadband SED and optical/UV spectra of \src1\ to some other well-known super-Eddington AGN. \subsubsection{PHL 1092 (hereafter: \ph1092): typical WLQ} \ph1092, lying at $z=0.396$, is often discussed as a WLQ, though its UV line REWs are slightly larger than the formal definition. Its mass is $M_{\rm BH}=1.0\times10^{8}M_{\odot}$, similar to \phl1811, but is one order of magnitude larger than \src1. Its mass accretion rate is only mildly super-Eddington at $\dot{m}_{\rm out}=2.2$ (\citealt{Miniutti.2012, Marinello.2020}, see Table \ref{tab-comp-sed}). Nonetheless, Figure~\ref{fig-optuv-compare} panels a2 and b2 shows its UV/optical spectra (green line) are very similar in both continuum shape and line emission to \src1\ (black line). Its broadband SED shows dramatic soft and hard X-ray variability. One SED observation (Obs-1 of \ph1092, Figure~\ref{fig-sed-compare}a, green solid line) shows a strong soft X-ray excess, together with harder X-ray emission which has $\alpha_{\rm ox}=1.69$, similar ratio to the disc power as in \src1. The other SED observation (Obs-2 of \ph1092, green dashed line) shows a much reduced soft X-ray flux, together with extremely weak and complex harder X-ray emission, whose $\alpha_{\rm ox}$ increases to 2.49 $\pm$ 0.25, similar to the X-ray weak WLQ \phl1811. \subsubsection{1H 0707-495 (hereafter: \1h07): high-$\dot{m}$ NLS1} A dramatic change in the soft and hard X-ray fluxes is also seen in the famous NLS1 \1h07. This source shows a strong soft X-ray excess and stochastic dips in the 0.3-10 keV light curve as observed by e.g. \xmm\ and {\it eROSITA} (\citealt{Wilkins.2014, Done.2016, Boller.2021}). The drastic X-ray variability may be originated from complex absorption in the wind (\citealt{Hagino.2016, Parker.2021}). Besides, it has a similar super-Eddington accretion flow as \src1, with $\dot{m}\sim20-30$, by an order of magnitude lower mass, $M_{\rm BH}=2\times10^{6}~M_{\odot}$ (\citealt{Done.2016}, see table \ref{tab-comp-sed}). We show its UV/optical spectra in magenta in Figure~\ref{fig-optuv-compare} panels a3 and b3. Plainly its optical continuum and lines are very similar to \src1\ and WLQs. However, the UV lines are now not so much alike, in that \1h07\ shows a substantial REW core which is not strongly blue-shifted (see table \ref{tab-comp-line}). Assuming that the blue-shifted wing of the C {\sc iv} is showing the wind strength, then \1h07\ has a less strong wind despite the accretion flow being similarly super-Eddington to \src1, and much more super-Eddington than \phl1811\ and \ph1092. Figure~\ref{fig-sed-compare}b shows three distinct SEDs of \1h07\ (magenta lines and data, similar to those reported by \citealt{Done.2016}) compared to both observations of \src1\ (black solid and dashed lines). This clearly shows that not only are the hard X-rays affected by this complex absorption/scattering in the wind, but the shape and strength of the soft X-rays can also change. As a result, the optical-to-X-ray index $\alpha_{\rm ox}$ varies between 1.4 and 2.0. Thus it may also be that the strangely weak soft X-ray excess in \src1\ is due to some obscurers in our line-of-sight rather than being intrinsic. \subsubsection{RX J0439.6-5311 (hereafter: \rx04): high-$\dot{m}$ NLS1} We explore this further using another super-Eddington source, \rx04\ ($\dot{m}_{\rm out}=5.9$ for $M_{\rm BH}=7\times10^{6}~M_{\odot}$: \citealt{Jin.2017b}, see Table \ref{tab-comp-sed}). This is an archetypal X-ray {\it simple} NLS1 so has very little complex X-ray variability, making it likely that we have a clean line-of-sight to this object. Figure~\ref{fig-optuv-compare} panels a4 and b4 shows the UV and optical spectra of \rx04\ (red spectra) compared to \src1. Again there is a continuum bend in the UV, indicating that this is an intrinsic feature rather than being due to reddening in the host galaxy. Unfortunately, the {\it HST} spectra do not cover the C {\sc iv} line, but the Ly$\alpha$+N {\sc v} REW is larger than the formal definition of a WLQ despite the similarity in Si {\sc iv} line shape and REW (see Table~\ref{tab-comp-line}). Figure~\ref{fig-sed-compare}c shows the broadband SED of \rx04\ (red line) compared to the Obs-1 SED of \src1\ (black line). The soft X-ray excess of \rx04\ is much stronger than \src1, which is quantitatively confirmed by the smaller optical-to-X-ray index $\alpha_{\rm ox}$ of \rx04\ (see Table~\ref{tab-comp-sed}). A possible explanation is that the soft excess of \src1\ is intrinsically present, but is severely suppressed due to complex and variable obscuration/scattering in the wind. Indeed, this is also seen in the variability of \ph1092\ and \1h07\ that the extent of the soft X-ray emission can be dramatically reduced. Especially, the soft X-ray bright state of \ph1092\ (its Obs-1, green solid line in Figure~\ref{fig-sed-compare}c) looks like \rx04, while the soft X-ray weak state of \ph1092\ is more like \src1. However, while wind obscuration/scattering seems to be a good explanation for the extremely weak soft excess of \src1, it remains difficult to understand how its hard X-rays are apparently seen directly, i.e. not suppressed by the wind as much as the soft excess does. \subsubsection{RE J1034+396 (hereafter: \rej1034): moderate-$\dot{m}$ NLS1} Alternatively, we explore the idea that the very steep soft X-ray emission of \src1\ is intrinsic. One of the steepest soft X-ray AGN is \rej1034, a source which uniquely shows a persistent X-ray quasi-periodic oscillation in AGN (QPO: \citealt{Gierlinski.2008}; \citealt{Middleton.2011}; \citealt{Alston.2014}; \citealt{Jin.2020}). This is a very low mass black hole, with $\dot{m}_{\rm out}=1.7$ for $M_{\rm BH}=2\times10^{6}~M_{\odot}$ (\citealt{Jin.2021}). Figure~\ref{fig-optuv-compare} panels a5 and b5 shows its UV and optical spectra (blue spectra) compared to \src1. Plainly the optical spectrum is very different, probably due to strong host galaxy contamination (e.g. \citealt{Czerny.2016, Jin.2021}). The UV continuum shape is similar to \src1 (black line) but the UV lines are much stronger, making it unlike a WLQ (see Table~\ref{tab-comp-line}). Figure~\ref{fig-sed-compare}c also includes the broadband SED of \rej1034\ (blue line), with a similar hard X-ray shape as in \src1, but a much stronger steep soft X-ray excess. The optical-to-X-ray index $\alpha_{\rm ox}$ of \rej1034\ is 1.33 $\pm$ 0.07. The very steep soft X-ray emission here is most likely the inner edge of the disc, with perhaps a small contribution from a warm Comptonisation region (see e.g. \citealt{Done.2012}). This very steep soft X-ray spectrum is a feature of the newly discovered Quasi-Periodic Eruptions (QPE), seen in a very rare class of AGN. Intriguingly, the characteristic eruptions are clearly marked by a dramatic increase in soft X-ray excess emission (\citealt{Miniutti.2019, Arcodia.2021}). \subsubsection{AGN Composite Spectra} Finally, we compare the composite QSO spectrum from \citet{Francis.1991} with \src1, as shown in the cyan line in Figure~\ref{fig-optuv-compare} panels a6 and b6. In the optical band, the composite spectrum has stronger oxygen lines and weaker Fe {\sc ii} lines, indicating that the average Eddington ratio of the QSO sample is smaller. In the UV band, the composite spectrum has a continual shape similar to \src1\ and \phl1811, making it quite unlikely that the bend away from the standard disc shape in the UV is due to host galaxy reddening as this would require the \src1, \phl1811\ and the composite spectrum to have very similar E(B-V). Instead, the bend looks like an intrinsic spectral feature. The composite spectrum also has stronger and less blue-shifted emission lines than WLQs. For example, we measure Ly$\alpha$ + N {\sc v} REW = 49.7 \AA\ and C {\sc iv} REW = 32.6 \AA\ for the composite spectrum, which are even larger than \1h07\ and \rx04\ (Table~\ref{tab-comp-line}). The SED shape is very different to all the super-Eddington SEDs, with no strong evidence for a larger extreme-UV (EUV) component (Figure~\ref{fig-sed-compare}d). \subsubsection{Weak-Line Seyfert Galaxy} To summarize this section: \src1 has weak and blue-shifted UV emission lines sufficient to class it as a WLQ, but it has much lower black hole mass and higher mass accretion rate than most WLQs. While at the moment this is a unique object, we propose that such sources be called Weak Line Seyferts (WLS). In comparison, objects with similarly super-Eddington mass accretion rates as \src1, but lower masses have UV emission lines which are not so weak and blue-shifted, so would not be defined as WLS, but rather classed as super-Eddington NLS1. Most super-Eddington AGN with both low and high black hole masses show strong and complex soft and hard X-ray variability, plausibly due to an absorption/scattering in a clumpy wind. Our viewing angle with respect to this wind as well as the wind properties will determine the impact of this extrinsic variability (separate to the intrinsic variability of the corona/soft X-ray emission region) on the observed X-ray spectrum. This impacts on the derived optical-to-X-ray index $\alpha_{\rm ox}$, by reducing the observed X-ray flux when the wind material is in the line of sight, so that these events can be identified by the source being X-ray weaker than expected at a given 2500 \AA\ luminosity. Interestingly, the dusty torus offers an independent viewing angle which is nearly edge-on (\citealt{Antonucci.1993}). It reprocesses the emission from the inner accretion flow and re-emits in the near infrared, thus the intensity of infrared torus emission may also provide clues about the properties of accretion flow. Indeed, Figure~\ref{fig-sed-compare}a shows that the relative infrared luminosities of \src1\ and \phl1811\ are very similar. More quantitatively, we adopt the optical-to-infrared index ($\alpha_{\rm optir}$, \citealt{Castello-Mor.2017}) as, \begin{equation} \alpha_{\rm optir} = - \frac{\rm{log}(L_{\rm 2500\textup{\AA}}/L_{\rm 5\mu m})}{\rm{log}(\nu_{\rm 2500\textup{\AA}}/\nu_{\rm 5\mu m})} \end{equation} where $L_{\rm 2500\textup{\AA}}$ and $L_{\rm 5\mu m}$ are the luminosities at 2500 \AA\ and 5 $\mu$m. Assuming an uncertainty of 10 per cent for the infrared luminosity, we find $\alpha_{\rm optir}$ is 0.67 $\pm$ 0.08 for \phl1811\ and 0.66 $\pm$ 0.08 for both SEDs of \src1\ (see Table~\ref{tab-comp-sed}). Hence it is likely that their torii also see similar SEDs from their separate accretion flows, which increases the global similarity between WLS and WLQs. \subsection{Efficiency of Super-Eddington Accretion Flows in AGN} \label{sec-disc} The global radiative efficiency ($\mu$) is a key parameter of the accretion flow around SMBHs, which indicates the fraction of accreted energy that is converted into radiation. $\mu$ can be estimated by measuring the difference between the observed Eddington ratio and mass accretion rate (\citealt{Davis.2011}). In Table~\ref{tab-comp-sed} we have estimated the radiative efficiency for \src1\ from its best-fit SEDs, which is found to be $\sim$ 30 per cent of the theoretical efficiency $\mu_{0}=0.057$ of a standard thin disc for $M_{\rm BH}=2\times10^{7}~M_{\odot}$ and zero spin. Likewise, we measure $\mu$ for the other AGN mentioned in this work based on their best-fit SEDs, which are shown in Table~\ref{tab-comp-sed}. We find that as $\dot{m}_{\rm out}$ increases, $\mu$ decreases significantly. This is qualitatively consistent with theoretical expectation that as the accretion flow becomes more super-Eddington, its properties deviate from the standard thin disc more significantly, the advection and wind may carry away a larger fraction of accretion energy, thus the global radiative efficiency becomes smaller. For instance, \citet{Poutanen.2007} derived a simple equation, $L_{\rm bol}/L_{\rm Edd}=1+x\cdot$ln$(\dot{m}_{\rm out})$, to describe the relation between the Eddington ratio and mass accretion rate for super-Eddington accretion flows. The $x$ factor is directly related to $\mu$. For a super-Eddington disc with advection but without outflow, the $x$ factor is 1.0; while for a disc with outflow but without advection, the $x$ factor is 0.6. In Figure~\ref{fig-mdot-eddr} we plot our small AGN sample on the parameter space of $\dot{m}_{\rm out}$ and $L_{\rm bol}/L_{\rm Edd}$, and compare them with various theoretical relations. We find that for a fixed $\dot{m}_{\rm out}$, the observed Eddington ratio is a factor of few higher than the disc+advection and disc+outflow model, and the deviation increases as AGN becomes more and more super-Eddington. This suggests that while the actual radiative efficiency of a super-Eddington accretion flow is significantly lower than that of a standard thin disc (\citealt{Shakura.1973}), it remains higher than those predicted by previous super-Eddington disc models. Based on the six AGN in Figure~\ref{fig-mdot-eddr}, we perform second-order polynomial model fit to derive an empirical relation between $\dot{m}_{\rm out}$ and $L_{\rm bol}/L_{\rm Edd}$, \begin{equation} \label{eq-1} \textup{log}(L_{\rm bol}/L_{\rm Edd})=a_0~[\textup{log}(\dot{m}_{\rm out})]^2+a_1~\textup{log}(\dot{m}_{\rm out})+a_2 \end{equation} where $a_0 = -0.234^{+0.237}_{-0.136}$, $a_1 = 0.919^{+0.170}_{-0.189}$, and $a_2 = 0.044^{+0.009}_{-0.104}$ are derived from the parameter values and inferred uncertainties. This equation can be used to infer the radiative efficiency for an AGN with $1.7\le \dot{m}_{\rm out} \lesssim 50$, but it is limited by the small sample size, and so it should be refined by future large sample studies. Figure~\ref{fig-mdot-eddr} also shows the inferred uncertainty region for every source. The uncertainty of $\mu$ is mainly caused by the measurement accuracies of the black hole mass $M_{\rm BH}$, spin $a_{*}$ (we assumed spin 0 in this work as a conservative limit) and bolometric luminosity $L_{\rm bol}$. The typical measurement uncertainty of $L_{\rm bol}$ is a few tens of per cent for a well defined SED of an unobscured super-Eddington AGN with high-quality optical/UV and soft/hard X-ray data (e.g. \citealt{Jin.2012c, Jin.2013, Jin.2017b, Jin.2021}). The black hole spin is largely unknown, but \citet{Davis.2011} showed that spin can introduce a few tens of per cent uncertainty on $\dot{m}_{\rm out}$. However, this can be easily overwhelmed by the uncertainty of black hole mass estimate, because this uncertainty can be a factor of few, and we have $\dot{m}_{\rm out} \propto M_{\rm BH}^{-2}$ for an observed optical/UV luminosity, so the uncertainty propagated from $M_{\rm BH}$ to $\dot{m}_{\rm out}$ is square-amplified. The uncertainty regions in Figure~\ref{fig-mdot-eddr} are based on these black hole mass ranges and $\pm$ 50 per cent uncertainty of $L_{\rm bol}$, and we have also assumed that $L_{\rm bol}/L_{\rm Edd}$ cannot exceed $\dot{m}_{\rm out}$. We find that except \rej1034\ and \ph1092\ whose $\dot{m}_{\rm out}$ values are only 1.7 and 2.3, all the other uncertainty regions lie between the standard thin disc model and the disc+advection/outflow models. Therefore, considering various measurement uncertainties, it appears robust that the observed radiative efficiencies of our super-Eddington AGN sample are indeed lower than the prediction of standard thin disc model, but significantly higher than predictions of theoretical super-Eddington disc models. It is also useful to compare our results with numerical simulations of super-Eddington accretion flows. For example, \citet{Jiang.2014} performed three-dimensional (3D) radiation magneto-hydrodynamical (MHD) simulations for an AGN with $M_{\rm BH}=4.2\times10^{6}~M_{\odot}$ and $\dot{m}_{\rm out}=12.5$. These parameters are comparable with those of \rx04. \citet{Jiang.2014} reported $L_{\rm bol}/L_{\rm Edd} \sim 10$, which then leads to $\mu = 0.045$, as shown by the star symbol in Figure~\ref{fig-mdot-eddr}. This radiative efficiency is significantly higher than those predicted by most theoretical models. It is caused by the inclusion of magnetic buoyancy in the simulation, which enhances the vertical advection of radiation, thereby increasing the global radiative efficiency. The efficiency we found for \rx04\ is 0.029, which is also much higher than previous theoretical models, but is $\sim$ 55 per cent lower than the 3D MHD simulation. The observed correlation in Figure~\ref{fig-mdot-eddr} for the extreme super-Eddington regime is broadly consistent with the moderate correlation reported by \citet{Davis.2011} using a much larger sample of both sub and super-Eddington Palomar-Green quasars. \citet{Davis.2011} also reported a significant correlation between the radiative efficiency and the black hole mass, which can be explained by the mass-spin correlation predicted by the cosmic evolution of SMBH (\citealt{Fanidakis.2011}). We do not find such a relation in our small sample, which is probably because these six AGN only cover a narrow mass range, and so the effect of mass-spin correlation is negligible. \subsection{A Proposed Picture for Super-Eddington Accretion Flows Depending on $M_{BH}$ and $\dot{m}_{\rm out}$} \label{sec-disc2} Super-Eddington NLS1s and WLQs are both characterized by their high mass accretion rates, but their black hole masses differ by more than one order of magnitude, thus comparing these two types of AGN can bring new insights on the super-Eddington accretion flow of SMBH. Based on their different multi-wavelength properties, a puffed-up inner disc scenario was proposed for both types of SMBH accretion systems (e.g. \citealt{Luo.2015} for WLQs and \citealt{Jin.2017b} for NLS1s). This was discussed even in the original paper of \citet{Shakura.1973}, where they show that the flow forms a quasi-spherical funnel when the disc reaches the local Eddington limit, within the spherization radius $r_{\rm sp}~=~R_{\rm sp}/R_{\rm in}~\sim~\dot{m}_{\rm out}$ where $R_{\rm in}$ is the inner radius (see also \citealt{Poutanen.2007}, Kubota \& Done 2019). We can derive a rough estimate of this radius for each object. Such an inner disc structure will produce a geometric collimation of the inner disc/soft X-ray/hard X-ray emission. This inclination dependence will be enhanced by any wind from the super-Eddington flow, and strong evidence for a wind is seen in the rapid and complex X-ray variability. However, both the funnel and the super-Eddington wind might be expected to depend only on $\dot{m}_{\rm out}$ rather than on mass, yet we showed examples in the previous section where sources at similar high Eddington fractions have weaker/more blue-shifted UV lines at higher masses, so there should be another intrinsic factor at work as well. One possibility is that there is an additional wind from the disc due to UV line driving. This is already implicated in the WLQ by the fact that the UV lines (C {\sc iv} and Ly$\alpha$+N {\sc v}) are blue-shifted. UV line driving is sensitively dependent on the disc temperature which depends on $M_{\rm BH}$ as well as $\dot{m}_{\rm out}$. A continuum SED which peaks in the UV (as in the mildly super-Eddington high mass AGN) is much more efficient in UV line driving than one that peaks in the EUV/soft X-rays (as in the strongly super-Eddington low-mass NLS1s). Hence this gives a component which depends on mass as well as $\dot{m}_{\rm out}$. Based on the above ideas, we compare the inferred structure of super-Eddington accretion discs for different black hole masses and mass accretion rates, which is shown in Figure~\ref{fig-discwind}. The first row is the puffed-up disc scenario proposed for WLQs such as \phl1811\ and \ph1092\ (e.g. \citealt{Wu.2012, Luo.2015, Ni.2018}). Since their mass accretion rates are close to or slightly above the Eddington limit, the puffed up region is quite small (a few tens of $R_{\rm g}$), and the Eddington wind is not very powerful. However, these structures shield the rest of the disc from the hottest emission (inner funnel and X-ray corona). The maximum outer disc temperature is given at the radius where the puffed up region starts. To infer the SED shape outside the spherization radius, we adopt the {\tt optxagnf} model (\citealt{Done.2012}) and take the best-fit $M_{\rm BH}$ and $\dot{m}_{\rm out}$ listed in Table~\ref{tab-comp-sed} as inputs\footnote{As a rough comparison, we take the average $\dot{m}_{\rm out}$ of 25 for \1h07, and 20.6 for \src1.}, and then truncate the SED at $R_{\rm sp}$. As shown in Figure~\ref{fig-outer-disc}, for WLQs with masses of $\sim 10^8$, this outer disc emits in the UV, and is just below Eddington. This should power an extremely strong UV line-driven disc wind. This shields the standard BLR from the outer disc from the inner UV/X-ray emission, so the standard BLR which makes the core of the lines is weak, so the UV lines are dominated by the wind emission, giving the characteristic blueshift of the BLR line profile. In Figure~\ref{fig-discwind}, we also show the failed dust-driven wind from the model of \citet{Czerny.2011}, where it is suggested that the BLR is itself a (failed) wind, but driven by dust rather than by the UV\footnote{This disc scenario may also be applicable for another famous AGN called PDS 456. It is a low-redshift AGN located at $z=0.184$ with $M_{\rm BH}\sim\times10^{9}~M_{\odot}$ and $L_{\rm bol}/L_{\rm Edd}\sim 1.0$ (\citealt{Simpson.1999, Reeves.2000}). It is famous for showing extreme X-ray variability and X-ray absorption features indicating ultra-fast outflows (\citealt{Reeves.2020} and references therein). Besides, it is known to show weak [O {\sc iii}] $\lambda 5007$ line with REW $<$ 2 \AA\ (\citealt{Simpson.1999}), as well as weak UV high-ionization lines such as C {\sc iv} REW $=14.7$ \AA\ and broad UV absorption lines (\citealt{Hamann.2018}). These multi-wavelength properties are similar to \src1\ and \phl1811, although its C {\sc iv} is still stronger than the definition of WLQ. PDS 456 also shows a low level of radio emission, although it can be classified as a radio-quiet AGN (\citealt{Vignali.2000, Yang.2021}).}. The second row of Figure~\ref{fig-discwind} shows the disc scenario for the archetypal super-Eddington NLS1 \rx04\ (\citealt{Jin.2017b}). Comparing to \phl1811, its black hole mass is one order of magnitude lower, and its mass accretion rate is one order of magnitude higher, thus we expect the disc to be much hotter ($T_{\rm eff}^{4} \propto M_{\rm BH}^{-1}~\dot{m}$) and the puffed-up disc region to be larger ($\sim$ 50 $R_{\rm g}$), and so the super-Eddington disc wind may be more powerful. This again shields the outer disc from the hottest parts of the disc, but the disc temperature just outside of the funnel region is somewhat hotter than seen in WLQs (see Figure~\ref{fig-outer-disc}). This stronger emission below 200\AA\ could be enough to over-ionise the disk wind, so the UV line driving is not efficient. The UV wind then does not shield the core of BLR as efficiently, so the core of the UV lines (e.g. C {\sc iv}) are stronger, as well as the blue-wing of the lines being weaker (due to the weaker UV wind), so these line profiles do not meet the criteria for a WLQ. The above picture is further supported by the multi-wavelength properties of the WLS \src1, whose black hole mass and mass accretion rates are both higher than \rx04, as shown in the third row of Figure~\ref{fig-discwind}. In this case, higher mass accretion rate means that the disc can begin to puff up at an even larger radius of $\sim$ 100 $R_{\rm g}$, and so an even stronger super-Eddington wind is expected. This clumpy wind absorbs the soft X-rays of \src1\ and causes its drastic X-ray variability, similar to those observed in \ph1092\ and PDS 456. But the emission from the disk outside of the funnel is now similar to that of the WLQ (see Figure~\ref{fig-outer-disc}), so this can have a similarly strong UV line driven wind, and so weak core lines from the BLR, and UV lines dominated by the UV line driven disc wind. Then the UV line-driving mechanism still works efficiently at outer/cooler disc region, making \src1\ the so far unique WLS. This also explains why \src1\ and \phl1811\ show similar infrared luminosity from hot torus relative to their optical luminosity. A jet is also drawn in this picture, because the radio-loudness of \rej1034\ was reported to be $\sim~17$, and so it was a marginally radio-loud NLS1 more than one decade ago (\citealt{Gelbord.2009}). But our latest observation campaign shows that now it becomes a radio-quiet source, and so its jet emission may be episodic (see more details in Paper-III). The last row of Figure~\ref{fig-discwind} shows the disc scenario for \1h07, whose $\dot{m}_{\rm out}$ is similar to \src1, but with one order-of-magnitude lower black hole mass. In this case, the accretion disc is the hottest among all the AGN mentioned above, and so the super-Eddington wind is also the strongest, which leads to drastic X-ray variability (e.g. \citealt{Hagino.2016, Done.2016, Boller.2021, Parker.2021}). The disc emission outside of the funnel region is now similar to \rx04\ (see Figure~\ref{fig-outer-disc}), just a little too high for the UV line driving to be efficient, so the blue-shifted line from the wind is not strong and the core BLR is not shielded. This explains why \1h07\ does not turn out to be a WLS. Finally, we emphasize that our study of the similarities and differences between super-Eddington NLS1s and WLQs is just at the beginning. More similarities/differences may be found by future studies, including detailed photoionization analyses of the BLRs of super-Eddington NLS1s and WLQs assuming different illuminating SEDs. If the above scenarios are generally correct, we speculate that further studies may find more WLS, as well as more WLQs showing drastic X-ray variability. \section{Summary and Conclusions} \label{sec-conclusion} We carried out a multi-wavelength campaign on the enigmatic super-Eddington NLS1 \rxj0134\ from radio to optical/UV to X-rays, using both space and ground-based telescopes. In this work, we present a detailed optical/UV spectroscopic analysis as well as broadband SED analysis from infrared to X-rays, and compare these multi-wavelength properties with other super-Eddington NLS1s and WLQs, thereby yielding deeper understanding about the super-Eddington accretion flows around SMBHs. The main results of this paper are summarized below. \begin{itemize} \item the optical/UV spectra of \rxj0134\ show extremely weak UV high-ionization lines such as C {\sc iv}, Si {\sc iv} and N {\sc v}, which are consistent with the definition of a WLQ. Together with other similarities such as the drastic X-ray variability and optical-to-infrared flux ratio, we propose \rxj0134\ as a new category of AGN, namely the weak-line Seyfert (WLS), which can be considered as the low mass and higher mass accretion rate counterpart of WLQs. \item we build the broadband SED of \rxj0134, which shows that the soft excess of \rxj0134\ is more than one order of magnitude weaker than in X-ray {\it simple} super-Eddington NLS1s. For a preferred black hole mass of $2 \times10^{7}M_{\sun}$ and $a_{*}=0$, the mass accretion rate $\dot{m}_{\rm out}$ is found to be $\sim$ 20, $L_{\rm bol}/L_{\rm Edd}$ is $\sim$ 6, so the radiative efficiency is only $\sim$ 30 per cent of that of a standard thin disc. \item by performing a systematic comparison within a small but representative super-Eddington AGN sample, we find that the most extreme NLS1s with similarly large $\dot{m}_{\rm out}$ but smaller masses than \rxj0134\ do not show similarly weak and wind-dominated UV high-ionization lines as WLQs and WLS do. Thus the properties of accretion flow should depend on both black hole mass and mass accretion rate. \item in the super-Eddington regime, the observed global radiative efficiency of the accretion flow decreases significantly as the mass accretion rate increases. The measured efficiency is higher than expected from the standard thin disc model and previous super-Eddington disc models, but lower than previous 3D MHD simulations of super-Eddington AGN discs. \item we propose a picture to show the dependence of super-Eddington accretion flows on the black hole mass, mass accretion rate and inclination angle, which can be used to qualitatively understand the multi-wavelength spectral differences between different subtypes of super-Eddington AGN, including super-Eddington NLS1s, WLQs and WLS. \end{itemize} The multi-wavelength long-term variability of \rxj0134\ from radio to X-rays as revealed by our {\it ATCA} observations and ongoing \swift\ observations will be presented in our next paper (Paper-III). \section*{Acknowledgements} CJ thanks Niel Brandt, Jianfeng Wu, Jianmin Wang, Yanrong Li and Erlin Qiao for valuable discussions, and thanks Peng Jiang, Hongyan Zhou and Weimin Yuan for helping with the coordination of optical observations. We thank Murilo Marinello for sharing with us the high-quality optical/UV/IR spectra of \ph1092. We thank the teams of the {\it Swift} satellite and the Siding Spring Observatory for approving and conducting the target-of-opportunity observations, as well as helping with the data reduction. We also thank the \xmm\ team for helping to investigate instrumental issues related to the OM data. CJ acknowledges the National Natural Science Foundation of China through grant 11873054, and the support by the Strategic Pioneer Program on Space Science, Chinese Academy of Sciences through grant XDA15052100. CD acknowledges the Science and Technology Facilities Council (STFC) through grant ST/T000244/1 for support. HL acknowledges the support by Chinese Postdoctoral Science Foundation (2021M693203), and the National Natural Science Foundation of China through grant 12103061. This work is based on observations conducted by \xmm, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA). This work also makes use of data from the \nustar\ mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by the National Aeronautics and Space Administration. This research has made use of the XRT Data Analysis Software (XRTDAS) developed under the responsibility of the ASI Science Data Center (ASDC), Italy. \section*{Data Availability} The data underlying this article are publicly available from the High Energy Astrophysics Science Archive Research Center (HEASARC) at https://heasarc.gsfc.nasa.gov, the \xmm\ Science Archive (XSA) at https://www.cosmos.esa.int/web/xmm-newton/xsa, the Barbara A. Mikulski Archive for Space Telescopes (MAST) at https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html, the Sloan Digital Sky Survey (SDSS) at http://skyserver.sdss.org/dr12/en/home.aspx. The SSO optical spectrum in this article will be shared on reasonable request to the corresponding author. \appendix \section{Optical/UV Spectral Analysis} \label{sec-optuvfit} \subsection{Optical Spectral Analysis} \label{sec-opt-spec} The optical spectrum of \rxj0134\ resembles a typical NLS1 galaxy. In order to determine the parameters of various emission lines and continuum, we perform a detailed fitting to the optical spectrum. Firstly, the spectrum is corrected for the Galactic dust reddening with $E(B-V) = 0.0144$ (\citealt{Schlafly.2011}) and the extinction model in \citet{Fitzpatrick.2007}. Then the wavelength and flux are de-redshifted to AGN's rest frame with $z=0.237$. Then we adopt the prescription described in \citet{Dong.2008} to fit the spectrum. The model of \citet{Dong.2008} comprises a power law with a slope of -2.5 for the underlying continuum, and an Fe {\sc ii} template from \citet{Veron.2004}. Since no stellar absorption features can be identified in the spectrum, star lights from host galaxy should not affect our emission line fitting, and so we do not include a host galaxy component. This is also confirmed by the extremely weak host galaxy component in the best-fit SED model in Figure~\ref{fig-sedfit}. All the narrow emission lines are fitted with Gaussian profiles of the same shape. [O {\sc iii}]$\lambda$5007 is fitted with two Gaussian profiles, one for the central component, and the other for the blue-shifted component. The intensity ratios of emission line doublets such as [O {\sc iii}]$\lambda$4959/5007 and [N {\sc ii}]$\lambda$6550/6585 are fixed at their theoretical values. For the broad Balmer lines, we first try Lorentzian profiles to fit the broad component (\citealt{Zhou.2006, Goad.2012, Rakshit.2017}). The velocity shifts and line widths of H$\alpha$ and H$\beta$ are kept the same, but their fluxes are left as free parameters. This model configuration assumes that different line components can have different Balmer decrements, which is possible because different emission line regions may have different electron densities and ionization states (e.g. \citealt{Kwan.1979, Mathews.1980, Jin.2012a, Jin.2012b}). Figure~\ref{fig-optfit} shows the fitting results for the entire optical spectrum. All the line parameters are listed in Table~\ref{tab-optfit}. We can see that Balmer lines (e.g. H$\alpha$, H$\beta$) are well fitted by this Lorentzian decomposition. The FWHM of the broad component is 1140 $\pm$ 20 km s$^{-1}$, which is consistent with the value reported by \citet{Grupe.2010}. The rest-frame equivalent width (REW) of the narrow component (NC) of H$\beta$ is only 0.1 $\pm$ 0.5 \AA, and its Balmer decrement is 28$^{+18}_{-3}$. In comparison, the Balmer decrement of the broad component (BC) is 2.76 $\pm$ 0.16, which is a typical value of AGN's broad line region (BLR, e.g. \citealt{Shen.2011, Jin.2012b, Shen.2014}). A possible explanation is that at least part of the NC flux is from a region in the host galaxy where the dust extinction is severe, thus it contributes a lot more NC flux in H$\alpha$ than H$\beta$. One way to test this is to check other hydrogen lines such as Pa$\alpha$ in the near infrared, but no spectral data is available in this band. However, it is also possible that the large Balmer decrement of NC is a consequence of line decomposition, which is often ambiguous. As a further test, we follow the procedure of \citet{Liu.2019} to fit the Balmer lines by replacing the BC Lorentzian profile with two broad Gaussian profiles, i.e. an intermediate component (IC) and a BC (e.g. \citealt{Zhu.2009, Jin.2012a}). As shown in Figure~\ref{fig-gaussfit}, this model can also provide good fits to the Balmer lines. The FWHMs of NC, IC and BC are found to be 300 $\pm$ 20, 1140 $\pm$ 70 and 3520 $\pm$ 170 km s$^{-1}$, respectively. The Balmer decrement is 5.7 $\pm$ 1.8, 2.9 $\pm$ 0.4 and 2.1 $\pm$ 0.3 for the three components. Thus NC also appears heavily reddened. In this case, the ratio of H$\beta$ NC/[O{\sc iii}]$\lambda$5007 is 0.6 $\pm$ 0.4, which is much larger than 0.06 $\pm$ 0.11 as found in the Lorentzian fit and in most AGN (e.g. \citealt{Shen.2014}). Hence it is possible that the NC flux is overestimated in this line decomposition, and so we slightly prefer the Lorentzian method. The FWHM of H$\beta$ after subtracting the NC is 1410 $\pm$ 70 km s$^{-1}$, which is slightly broader than in the Lorentzian decomposition. Therefore, there is indeed some ambiguity in the width of Balmer lines caused by different line decomposition methods. Our spectral fits also confirm the existence of strong Fe {\sc ii} emission, the REW of which is $41.5\pm3.4$ \AA\ within 4434 -- 4684 \AA. The flux ratio between the Fe {\sc ii} line in 4434 -- 4684 \AA\ and the broad H$\beta$ line is $R_{\rm FeII}=1.74\pm0.16$. After removing Fe{\sc ii} lines, [O{\sc iii}]$\lambda$5007 is found to be extremely weak, with the total REW being only 2.7 $\pm$ 0.8 \AA. \subsection{UV Spectral Analysis} \label{sec-uv-spec} A special property of \rxj0134\ is its weak and strongly blue-shifted UV emission lines. We perform local spectral fitting to individual UV lines, such as the Mg {\sc ii} $\lambda$2797/2803, C {\sc iv} $\lambda$1548/1551, N {\sc v} $\lambda$1238/1243 doublets. The fits are plotted in Figure~\ref{fig-uvfit}, and the line parameters are listed in Table~\ref{tab-uvfit}. The Mg {\sc ii} $\lambda$2797/2803 doublet can be well fitted with two Lorentzian components of the same shape and flux, as shown in Figure~\ref{fig-uvfit}a. The velocity shift is found to be -380 $\pm$ 90 km s$^{-1}$, indicating that the line is slightly blue-shifted. The FWHM is found to be 1170 $\pm$ 240 km s$^{-1}$, which is similar to the width of H$\beta$. The C {\sc iv} line is much more blue-shifted and extended. It contains two adjacent lines at 1548 and 1551 \AA. Each of the two lines are fitted with two Gaussian components, including a narrow core component and a broad and blue-shifted wing component. As shown in Figure~\ref{fig-uvfit}b, this model can fit the line very well. The core component has a FWHM of 3250 $\pm$ 320 km s$^{-1}$, much broader than the BC of H$\beta$, and its velocity shift is -1780 $\pm$ 70 km s$^{-1}$. The wing component has a velocity shift of -5160 $\pm$ 660 km s$^{-1}$ and a much larger FWHM of 7800 $\pm$ 800 km s$^{-1}$. These two components have a total REW of 3.8 $\pm$ 1.5 \AA. The total REW of the C {\sc iv} doublet is much smaller than that measured from the major AGN population (e.g. 30 \AA\ in the quasars' composite spectrum: \citealt{Luo.2015, Coatman.2016}). The weak and blue-shifted C {\sc iv} doublet resembles WLQs. The Si {\sc iv} $\lambda$1393/1402 doublet has a different shape from C {\sc iv}. The velocity shift of the core component is -710 $\pm$ 230 km s$^{-1}$, and its FWHM is 860 $\pm$ 320 km s$^{-1}$. The broad component is shifted by -1890 $\pm$ 20 km s$^{-1}$, with a FWHM of 3660 $\pm$ 40 km s$^{-1}$. Therefore, the line width and velocity shift of Si {\sc iv} are less extreme than C {\sc iv}. This is consistent with the fact that the ionization energy of Si {\sc iv} is lower, and so its emission region may have a larger radius, where the radiation pressure is weaker and the outflow speed is smaller. The N {\sc v} $\lambda$1238/1243 doublet contains two broad lines. These two lines are often heavily blended with the broad Ly$\alpha$ line at 1216 \AA, making them difficult to decompose. We assume that N {\sc v} has the same line profile as C {\sc iv}, and then fit only the red side of the line hump within 1230 -- 1250 \AA. As shown in Figure~\ref{fig-uvfit}d, the result indicates that the line blend within 1150 -- 1250 \AA\ mostly come from the strong N {\sc v} $\lambda$1238/1243 doublet. The flux ratio of N {\sc v}/C {\sc iv} is 2.2 $\pm$ 0.4 , which is larger than the typical AGN value of less than unity (e.g. \citealt{Shemmer.2002}). This suggests that either the outflow in \rxj0134\ is more metal rich, or the flux of N {\sc v} in the line blend is over-estimated. Unfortunately, the spectral quality of this waveband is not good enough for a more accurate decomposition of Ly$\alpha$ and N {\sc v}. \begin{table} \centering \caption{Best-fit parameters of some optical emission lines shown in Figures~\ref{fig-optspec1}. {\it tied} means the value is tied to the corresponding component of H$\alpha$ during the spectral fitting. Errors represent 1$\sigma$ confidence limits. } \begin{tabular}{@{}lccc@{}} \hline Component & Parameter & Value & Unit \\ \hline \multicolumn{4}{c}{H$\alpha$} \\ Narrow Gauss & $v_{\rm line}$ & -260 $\pm$ 20 & km s$^{-1}$ \\ & FWHM & 280 $\pm$ 50 & km s$^{-1}$ \\ & Flux & 560 $\pm$ 160 & $10^{-17}$erg cm$^{-2}$ s$^{-1}$ \\ & REW & 8.6 $\pm$ 2.5 & \AA \\ Broad Lorentz & $v_{\rm line}$ & -250 $\pm$ 10 & km s$^{-1}$ \\ & FWHM & 1140 $\pm$ 20 & km s$^{-1}$ \\ & Flux & 8540 $\pm$ 240 & $10^{-17}$erg cm$^{-2}$ s$^{-1}$ \\ & REW & 131.3 $\pm$ 3.6 & \AA \\ \multicolumn{4}{c}{H$\beta$} \\ Narrow Gauss & $v_{\rm line}$ & -260 {\it tied} & km s$^{-1}$ \\ & FWHM & 280 {\it tied} & km s$^{-1}$ \\ & Flux & 20 $\pm$ 70 & $10^{-17}$erg cm$^{-2}$ s$^{-1}$ \\ & REW & 0.1 $\pm$ 0.5 & \AA \\ Broad Lorentz & $v_{\rm line}$ & -250 {\it tied} & km s$^{-1}$ \\ & FWHM & 1140 {\it tied} & km s$^{-1}$ \\ & Flux & 3090 $\pm$ 160 & $10^{-17}$erg cm$^{-2}$ s$^{-1}$ \\ & REW & 24.7 $\pm$ 1.2 & \AA \\ \multicolumn{4}{c}{$[$O {\sc iii}$]~\lambda$5007} \\ Core Gauss & $v_{\rm line}$ & -260 {\it tied} & km s$^{-1}$ \\ & FWHM & 280 {\it tied} & km s$^{-1}$ \\ & Flux & 40 $\pm$ 40 & $10^{-17}$erg cm$^{-2}$ s$^{-1}$ \\ & REW & 0.3 $\pm$ 0.3 & \AA \\ Wing Gauss & $v_{\rm line}$ & -310 $\pm$ 290 & km s$^{-1}$ \\ & FWHM & 770 $\pm$ 270 & km s$^{-1}$ \\ & Flux & 290 $\pm$ 90 & $10^{-17}$erg cm$^{-2}$ s$^{-1}$ \\ & REW & 2.4 $\pm$ 0.7 & \AA \\ \hline \end{tabular} \label{tab-optfit} \end{table} \begin{table} \centering \caption{Best-fit parameters of some UV emission lines shown in Figures~\ref{fig-optspec1}. For the Mg {\sc ii} $\lambda$2797/2803, C {\sc iv} $\lambda$1548/1551 and Si {\sc iv} $\lambda$1393/1402 doublets, a line ratio of 1:1 is adopted, and the reported flux and REW are only for one line. The N {\sc v} $\lambda$1238/1243 doublet parameters are not listed here because the line is assumed to have the same shape as C {\sc iv} $\lambda$1548/1551, except that its flux is higher by a factor of 2.2 $\pm$ 0.4. Errors are 1$\sigma$ confidence limits.} \begin{tabular}{@{}lccc@{}} \hline Component & Parameter & Value & Unit \\ \hline \multicolumn{4}{c}{Mg {\sc ii} $\lambda$2797} \\ Single Lorentz & $v_{\rm line}$ & -380 $\pm$ 90 & km s$^{-1}$ \\ & FWHM & 1170 $\pm$ 240 & km s$^{-1}$ \\ & Flux & 2480 $\pm$ 610 & $10^{-17}$erg cm$^{-2}$ s$^{-1}$ \\ & REW & 6.1 $\pm$ 1.5 & \AA \\ \multicolumn{4}{c}{C {\sc iv} $\lambda$1548} \\ Core Gauss & $v_{\rm line}$ & -1780 $\pm$ 70 & km s$^{-1}$ \\ & FWHM & 3250 $\pm$ 320 & km s$^{-1}$ \\ & Flux & 1800 $\pm$ 350 & $10^{-17}$erg cm$^{-2}$ s$^{-1}$ \\ & REW & 1.7 $\pm$ 0.9 & \AA \\ Wing Gauss & $v_{\rm line}$ & -5160 $\pm$ 660 & km s$^{-1}$ \\ & FWHM & 7800 $\pm$ 800 & km s$^{-1}$ \\ & Flux & 2280 $\pm$ 400 & $10^{-17}$erg cm$^{-2}$ s$^{-1}$ \\ & REW & 2.1 $\pm$ 1.2 & \AA \\ \multicolumn{4}{c}{Si {\sc iv} $\lambda$1393} \\ Core Gauss & $v_{\rm line}$ & -710 $\pm$ 230 & km s$^{-1}$ \\ & FWHM & 860 $\pm$ 320 & km s$^{-1}$ \\ & Flux & 530 $\pm$ 300 & $10^{-17}$erg cm$^{-2}$ s$^{-1}$ \\ & REW & 0.5 $\pm$ 0.2 & \AA \\ Wing Gauss & $v_{\rm line}$ & -1890 $\pm$ 20 & km s$^{-1}$ \\ & FWHM & 3660 $\pm$ 40 & km s$^{-1}$ \\ & Flux & 4360 $\pm$ 530 & $10^{-17}$erg cm$^{-2}$ s$^{-1}$ \\ & REW & 3.7 $\pm$ 1.2 & \AA \\ \hline \end{tabular} \label{tab-uvfit} \end{table} \bsp % \label{lastpage}

Title: Universal Scaling Laws for Solar and Stellar Atmospheric Heating: Catalog of Power-law Index between Solar Activity Proxies and Various Spectral Irradiances

Abstract: The formation of extremely hot outer atmospheres is one of the most prominent manifestations of magnetic activity common to the late-type dwarf stars, including the Sun. It is widely believed that these atmospheric layers, the corona, transition region, and chromosphere, are heated by the dissipation of energy transported upwards from the stellar surface by the magnetic field. This is signified by the spectral line fluxes at various wavelengths, scaled with power-law relationships against the surface magnetic flux over a wide range of formation temperatures, which are universal to the Sun and Sun-like stars of different ages and activity levels. This study describes a catalog of power-law indices between solar activity proxies and various spectral line fluxes. Compared to previous studies, we expanded the number of proxies, which now includes the total magnetic flux, total sunspot number, total sunspot area, and the F10.7 cm radio flux, and further enhances the number of spectral lines by a factor of two. This provides the data to study in detail the flux-flux scaling laws from the regions specified by the temperatures of the corona (log(T/K)=6-7) to those of the chromosphere (log(T/K)~4), as well as the reconstruction of various spectral line fluxes of the Sun in the past, F-, G-, and K-type dwarfs, and the modeled stars.

https://export.arxiv.org/pdf/2208.10511

command. \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \submitjournal{ApJS} \shorttitle{Scaling Laws for Solar-stellar Atmospheric Heating} \shortauthors{Toriumi et al.} \graphicspath{{./}{figures/}} \begin{document} \title{Universal Scaling Laws for Solar and Stellar Atmospheric Heating: Catalog of Power-law Index between Solar Activity Proxies and Various Spectral Irradiances} \correspondingauthor{Shin Toriumi} \email{toriumi.shin@jaxa.jp} \author[0000-0002-1276-2403]{Shin Toriumi} \affiliation{Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan} \author[0000-0003-4452-0588]{Vladimir S. Airapetian} \affiliation{Sellers Exoplanetary Environments Collaboration, NASA Goddard Space Flight Center, Greenbelt, MD, USA} \affiliation{Department of Physics, American University, Washington, DC, USA} \author[0000-0002-1297-9485]{Kosuke Namekata} \affiliation{National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan} \author[0000-0002-0412-0849]{Yuta Notsu} \affiliation{Laboratory for Atmospheric and Space Physics, University of Colorado Boulder, 3665 Discovery Drive, Boulder, CO 80303, USA} \affiliation{National Solar Observatory, 3665 Discovery Drive, Boulder, CO 80303, USA} \affiliation{Department of Earth and Planetary Sciences, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan} \keywords{G dwarf stars (556); Solar analogs (1941); Solar magnetic fields (1503); Solar chromosphere (1479); Solar transition region (1532); Solar corona (1483); Solar spectral irradiance (1501); Stellar chromospheres (230); Stellar coronae (305)} \section{Introduction}\label{sec:intro} Late-type dwarf stars, including the Sun, commonly exhibit magnetic activity in a variety of forms. In their turbulent outermost envelope, the convection zone, magnetic flux is generated and enhanced by the dynamo mechanism \citep{2017LRSP...14....4B,2020LRSP...17....4C,2021LRSP...18....5F}. The produced magnetic flux emerges to the photosphere and builds up active regions, including sunspots/starspots \citep{2003A&ARv..11..153S,2005LRSP....2....8B,2009A&ARv..17..251S,2014LRSP...11....3C,2014PASJ...66S...6T}. Active regions contain highly concentrated magnetic flux that drives eruptive processes such as flares and coronal mass ejections via magnetic reconnection \citep{2010ARA&A..48..241B,2011SSRv..159...19F,2011LRSP....8....6S,2012Natur.485..478M,2016ApJ...829...23D,2017LRSP...14....2B,2017ApJ...834...56T,2019LRSP...16....3T}, and coronal mass ejections accompanying flares expand into interplanetary space \citep{1989MNRAS.238..657C,2011LRSP....8....1C,2016SoPh..291.1761H,2021NatAs...5..697V,2022NatAs...6..241N}. It is widely believed that the magnetic flux covering the entire stellar surface transports the energy from the surface upwards and heats the outer stellar atmospheres, known as the chromosphere, transition region, and corona \citep{2004A&ARv..12...71G}. However, the exact mechanism of atmospheric heating is still unclear \citep{2006SoPh..234...41K}. The comparison between the full-disk magnetograms and the associated X-ray and extreme ultraviolet (EUV) images of the Sun clearly shows that the rate of atmospheric heating strongly depends on the surface magnetic flux. Empirical relationships between the surface magnetic flux and quasi-steady X-ray emission flux of the Sun and Sun-like stars have been well characterized as a function of the rotation period and average magnetic field strength. For example, by measuring the total unsigned magnetic flux ($\Phi$) and X-ray flux ($F_{\rm X}$) of various structures such as the quiet Sun, X-ray bright points, active regions, entire solar disk, G, K, M dwarfs, and T Tauri stars, \citet{2003ApJ...598.1387P} found that the two parameters showed a power-law scaling with a power-law index in excess of unity, $F_{\rm X} \propto \Phi^{\alpha}$, where $\alpha=1.15$. Similar values ($\alpha\gtrsim 1$) were obtained by other studies \citep{1998ApJ...508..885F,2014MNRAS.441.2361V,2022A&A...662A..41R}. It was found that the X-ray flux of late-type dwarf stars decreases with the rotation period or the Rossby number $Ro$ (defined as the rotation period divided by the convective turnover time: \citealt{1984ApJ...287..769N}) in the regime of $Ro\gtrsim 0.1$, while it is saturated for $Ro\lesssim 0.1$ \citep{2003A&A...397..147P,2011ApJ...743...48W,2014MNRAS.441.2361V,2014ApJ...794..144R,2020ApJ...901...70T}. Recently, studies also investigated the dependence of magnetic field strength on $Ro$ \citep{2022A&A...662A..41R}. These relationships can be attributed to the stellar evolution \citep{1972ApJ...171..565S}. The rotation speed, which is fastest immediately after starbirth, determines the efficiency of stellar dynamo, and hence, the average magnetic field strength and X-ray luminosity driven by the magnetic heating of the corona. As the stellar evolution progresses, the stellar wind driven by the magnetic field carries away the angular momentum, which decreases the rotation speed. As a result, the dynamo action, average field strength, and X-ray luminosity weaken. For detailed discussions on the stellar evolution and activity, see reviews by \citet{2007LRSP....4....3G}, \citet{2015RSPTA.37340259T}, \citet{2017LRSP...14....4B}, and \citet{2021LRSP...18....3V}. High-energy radiation can create detrimental conditions for habitability of exoplanets around active host stars. Specifically, the X-ray and EUV radiations, collectively referred to as XUV, emitted from active regions and stellar flares can evaporate the planetary atmospheres by the photoionization-driven heating that expands the exosphere, thereby igniting ionospheric and hydrodynamic escape. Therefore, investigating the dependence of spectral line irradiances on the stellar magnetic activity is important for elucidating the stellar atmospheric heating as well as understanding their effects on exoplanets \citep{2019LNP...955.....L,2020IJAsB..19..136A}. Despite its importance for the exoplanetary atmospheric evolution and habitability, it is difficult to observe stellar EUV flux, especially of wavelengths longer than 360 {\AA} owing to the strong absorption by the interstellar medium (see, e.g., \citealt{1974ApJ...187..497C} for absorption cross-section). Therefore, EUV spectrum is estimated and reproduced by using the scaling laws between EUV and other observable wavelengths, such as X-ray and Ca II K, or by obtaining the differential emission measure distributions \citep{2011A&A...532A...6S,2014ApJ...780...61L,2015Icar..250..357C,2016ApJ...824..101Y,2017ApJ...843...31Y,2020ApJS..250...16A,2021ApJ...908..205A,2021A&A...649A..96J}. However, considering the stellar atmospheric heating is moderated by the surface magnetic field, physical correspondence can be obtained directly by measuring the scaling relations between irradiances and the surface magnetic flux. Accordingly, \citet[][hereafter TA22]{2022ApJ...927..179T} derived the power-law correlations between the total unsigned magnetic flux of the Sun over 10 years and the irradiances of emission lines of various wavelengths, i.e., various temperature domains. As a result, it was found that the acquired correlations strikingly replicated in Sun-like G-type stars at five spectral lines: X-rays, \ion{Fe}{15} 284 {\AA}, \ion{C}{2} 1335 {\AA}, Ly$\alpha$, and \ion{Mg}{2} h \& k. This indicated that the extremely hot outer atmospheres of the Sun and Sun-like stars are heated by a common mechanism, which is independent of the stellar age or activity level. The obtained power-law index for the soft X-ray band in \citetalias{2022ApJ...927..179T}, $\alpha=1.16\pm 0.03$, is highly consistent with the preceding studies by, e.g., \citet{2003ApJ...598.1387P}, wherein the exponent was $\alpha=1.15$. Furthermore, we found that the other coronal line fluxes can be consistently scaled with the above-unity exponents. Such values have been explained using theoretical models based on RTV scaling laws \citep{1998ApJ...508..885F,2020A&A...640A.119Z,2020ApJ...901...70T} and numerical simulations, wherein Alfv\'{e}n waves propagating in the corona loop heat the atmosphere via turbulent dissipation \citep{2021A&A...656A.111S}. For chromospheric lines, the $\alpha$ values lie below unity in \citetalias{2022ApJ...927..179T}, which is in agreement with previous studies \citep{1975ApJ...200..747S,1989ApJ...337..964S,1999ApJ...515..812H,2007A&A...466.1131R,2018A&A...619A...5B}. In \citetalias{2022ApJ...927..179T}, we examined the correlations of multiple lines to the total unsigned magnetic flux of the Sun and compared them with the stellar observations. However, by expanding the activity proxy to the historical records of sunspot number, sunspot area, and the F10.7cm radio flux, and by further enhancing the number of lines to be investigated, we can provide the means to synthesize the spectral irradiances over a wide range of wavelength, based on the combination of the obtained power-law indices and proxies of the Sun in the past, other Sun-like stars, and numerical models. Therefore, in this study, we create a catalog of power-law scaling factors for various lines and activity proxies by analyzing solar synoptic observations. Considering the number of lines was particularly small for the transition region temperatures in \citetalias{2022ApJ...927..179T}, this study also leads to a better understanding of how $\alpha$ changes as the temperature changes from the chromosphere to the corona. In Section \ref{sec:data}, we provide detailed descriptions of the data that are analyzed, while Section \ref{sec:analysis} explains how we measure the power-law correlations. Section \ref{sec:catalog} provides the catalog of the power-law index. We show the temperature and wavelength dependence of the power-law index in Section \ref{sec:dependence} based on the obtained scalings and demonstrate how to synthesize the line and band irradiances in Section \ref{sec:application}. Finally, Section \ref{sec:summary} summarizes and discusses the implications of the obtained results. \section{Data}\label{sec:data} In this study, we investigated the thermal responses of upper atmospheres to the magnetic flux on the surface by comparing the light curves of spectral lines and bands of various wavelengths, or various formation temperatures, with the multiple proxy data representing the solar magnetic activity. As proxies, we adopted two kinds of the total unsigned magnetic flux, both of which were derived from the line-of-sight (LOS) magnetic field, total sunspot number, total sunspot area, and the F10.7cm radio flux between May 2010 and February 2020. Table \ref{tab:observables} summarizes the key information of the proxies and spectral lines/bands, such as the formation temperature, central wavelength and spectral window for calculating the irradiance, and the data source. All spectral irradiances were converted to values at 1 au. The F10.7cm flux was used both as a proxy of solar activity and as a light curve data representing the solar atmospheres. \begin{deluxetable*}{lccccccl} \tabletypesize{\footnotesize} \tablecaption{Summary of the Observables\label{tab:observables}} \tablewidth{0pt} \tablehead{ \colhead{Feature} & \colhead{$\log{(T/{\rm K})}$} & \colhead{Wavelength ({\AA})} & \colhead{Basal} & \colhead{Minimum} & \colhead{Maximum} & \colhead{Unit} & \colhead{Source} } \decimalcolnumbers \startdata Radial magnetic flux & 3.8 & 6173.3 & $1.18\times 10^{23}$ & $1.16\times 10^{23}$ & $3.35\times 10^{23}$ & Mx & SDO/HMI\\ LOS magnetic flux & 3.8 & 6173.3 & $7.02\times 10^{22}$ & $6.85\times 10^{22}$ & $2.52\times 10^{23}$ & Mx & SDO/HMI\\ Sunspot number & 3.8 & WL & 0 & 0 & $220$ & -- & WDC-SILSO (ver 2.0)\\ Sunspot area & 3.8 & WL & 0 & 0 & $3120$ & MSH & USAF/NOAA\\ F10.7cm radio & $\sim$6 & $10.7\times 10^{8}$ & $68.83$ & $63.67$ & $466.57$ & sfu & DRAO\\ Total solar irradiance & 3.8 & WL & -- & $1358.5$ & $1362.3$ & W m$^{-2}$ & SORCE/TIM\\ \hline X-rays 1--8 {\AA} & 6--7 & 1--8 & 0 & $1.00\times 10^{-9}$ & $4.81\times 10^{-5}$ & W m$^{-2}$ & GOES/XRS\\ X-rays 5.2--124 {\AA} & 6--7 & 5.2--124 & $2.11\times 10^{-4}$ & $1.85\times 10^{-4}$ & $1.01\times 10^{-3}$ & W m$^{-2}$ & SORCE/XPS\\ Fe XV 284 {\AA} & 6.4 & $284.15\pm 1.50$ & $9.36\times 10^{-6}$ & $5.68\times 10^{-6}$ & $1.27\times 10^{-4}$ & W m$^{-2}$ & SORCE/XPS\\ Fe XIV 211 {\AA} & 6.3 & $211.32\pm 1.50$ & $1.20\times 10^{-5}$ & $9.88\times 10^{-6}$ & $6.75\times 10^{-5}$ & W m$^{-2}$ & SORCE/XPS\\ X-rays (XRT) & $6.2\pm 0.1$ & 5--60 & $5.00\times 10^{-5}$ & $4.71\times 10^{-5}$ & $1.01\times 10^{-3}$ & W m$^{-2}$ & Hinode/XRT\\ Fe XII 193$+$195 {\AA} & 6.2 & $193.50\pm 2.50$ & $6.16\times 10^{-5}$ & $5.66\times 10^{-5}$ & $1.72\times 10^{-4}$ & W m$^{-2}$ & SORCE/XPS\\ Fe XII 1349 {\AA} & 6.2 & $1349.40\pm 1.00$ & $3.64\times 10^{-6}$ & $3.23\times 10^{-6}$ & $5.66\times 10^{-6}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ Fe X 174 {\AA} & 6.1 & $174.53\pm 1.50$ & $5.64\times 10^{-5}$ & $5.40\times 10^{-5}$ & $0.90\times 10^{-4}$ & W m$^{-2}$ & SORCE/XPS\\ Fe XI 180 {\AA} & 6.1 & $180.41\pm 1.50$ & $4.57\times 10^{-5}$ & $4.31\times 10^{-5}$ & $0.95\times 10^{-4}$ & W m$^{-2}$ & SORCE/XPS\\ F10.7cm radio & $\sim$6 & $10.7\times 10^{8}$ & $68.83$ & $63.67$ & $466.57$ & sfu & DRAO\\ Fe IX 171 {\AA} & 5.9 & $171.07\pm 1.50$ & $5.50\times 10^{-5}$ & $5.32\times 10^{-5}$ & $0.73\times 10^{-4}$ & W m$^{-2}$ & SORCE/XPS\\ N V 1238 {\AA} & 5.3 & $1238.90\pm 1.15$ & $1.62\times 10^{-5}$ & $1.55\times 10^{-5}$ & $2.39\times 10^{-5}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ N V 1242 {\AA} & 5.3 & $1242.95\pm 1.00$ & $1.04\times 10^{-5}$ & $9.89\times 10^{-6}$ & $1.54\times 10^{-5}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ C IV 1548 {\AA} & 5.1 & $1548.25\pm 1.20$ & $1.11\times 10^{-4}$ & $1.07\times 10^{-4}$ & $1.53\times 10^{-4}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ C IV 1551 {\AA} & 5.1 & $1550.73\pm 0.95$ & $6.58\times 10^{-5}$ & $6.38\times 10^{-5}$ & $9.02\times 10^{-5}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ C III 1175 {\AA} & 5.0 & $1175.70\pm 1.75$ & $5.52\times 10^{-5}$ & $5.35\times 10^{-5}$ & $8.24\times 10^{-5}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ He II 256 {\AA}$+$blends & 4.9 & $256.30\pm 3.00$ & $5.53\times 10^{-5}$ & $5.20\times 10^{-5}$ & $1.21\times 10^{-4}$ & W m$^{-2}$ & SORCE/XPS\\ He II 304 {\AA} & 4.9 & $304.00\pm 1.00$ & $4.25\times 10^{-4}$ & $4.09\times 10^{-4}$ & $6.19\times 10^{-4}$ & W m$^{-2}$ & SORCE/XPS\\ Si IV 1393 {\AA} & 4.9 & $1393.85\pm 1.30$ & $4.45\times 10^{-5}$ & $4.27\times 10^{-5}$ & $7.66\times 10^{-5}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ Si IV 1402 {\AA} & 4.9 & $1402.85\pm 0.85$ & $2.32\times 10^{-5}$ & $2.25\times 10^{-5}$ & $3.91\times 10^{-5}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ Si III 1206 {\AA} & 4.8 & $1206.60\pm 1.25$ & $8.59\times 10^{-5}$ & $8.32\times 10^{-5}$ & $1.66\times 10^{-4}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ He I 10830 {\AA} & 4.5 & $10830.40\pm 0.25$ & $0.0292$ & $0.0270$ & $0.0308$ & W m$^{-2}$ & SORCE/SIM \& SOLIS/ISS\\ C II 1335 {\AA} & 4.3 & $1335.25\pm 1.90$ & $1.57\times 10^{-4}$ & $1.52\times 10^{-4}$ & $2.46\times 10^{-4}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ H I 1216 {\AA} (Ly$\alpha$) & 4.3 & $1215.70\pm 2.00$ & $5.73\times 10^{-3}$ & $5.60\times 10^{-3}$ & $8.94\times 10^{-3}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ O I 1302 {\AA} & 4.2 & $1302.20\pm 0.85$ & $4.16\times 10^{-5}$ & $3.93\times 10^{-5}$ & $5.40\times 10^{-5}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ O I 1305 {\AA} & 4.2 & $1305.50\pm 1.75$ & $9.14\times 10^{-5}$ & $8.77\times 10^{-5}$ & $1.17\times 10^{-4}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ Mg II k 2796 {\AA} & (3.9) & $2796.38\pm 0.78$ & $0.0136$ & $0.0135$ & $0.0180$ & W m$^{-2}$ & SORCE/SOLSTICE\\ Mg II h 2803 {\AA} & (3.9) & $2803.48\pm 0.65$ & $0.0097$ & $0.0096$ & $0.0126$ & W m$^{-2}$ & SORCE/SOLSTICE\\ Cl I 1351 {\AA} & (3.8) & $1305.50\pm 1.75$ & $9.06\times 10^{-6}$ & $8.57\times 10^{-6}$ & $1.17\times 10^{-5}$ & W m$^{-2}$ & SORCE/SOLSTICE\\ Ca II K 3934 {\AA} & (3.8) & $3933.66\pm 0.50$ & $0.0114$ & $0.0111$ & $0.0130$ & W m$^{-2}$ & SORCE/SIM \& SOLIS/ISS\\ Ca II H 3968 {\AA} & (3.8) & $3968.47\pm 0.50$ & $0.0139$ & $0.0139$ & $0.0155$ & W m$^{-2}$ & SORCE/SIM \& SOLIS/ISS\\ H I 6563 {\AA} (H$\alpha$) & (3.8) & $6562.80\pm 0.50$ & $0.0369$ & $0.0360$ & $0.0448$ & W m$^{-2}$ & SORCE/SIM \& SOLIS/ISS\\ Ca II 8542 {\AA} & (3.8) & $8542.10\pm 0.50$ & $0.0347$ & $0.0346$ & $0.0392$ & W m$^{-2}$ & SORCE/SIM \& SOLIS/ISS\\ \enddata \tablecomments{Listed above the horizontal line are the solar activity proxies, while the rest are the spectral lines and bands whose irradiances are compared with the proxies. F10.7cm radio flux is registered as both proxy and spectral band. The temperatures of optically thick chromospheric lines are given in parentheses. All irradiances were converted to the values at the distance of 1 au from the Sun.} \end{deluxetable*} \subsection{SDO/HMI} To calculate the total unsigned magnetic flux in the visible hemisphere of the Sun, we used the full-disk magnetograms obtained by the Helioseismic and Magnetic Imager (HMI; \citealt{2012SoPh..275..207S,2012SoPh..275..229S}) aboard the Solar Dynamics Observatory (SDO), which was launched in February 2010 and began observations in May 2010. This determines the beginning of the target period in this study. HMI obtains full-disk continuum images, magnetograms, and Dopplergrams with cadences of 45 s and 720 s by acquiring the spectropolarimetric signals of the \ion{Fe}{1} 6173.3 {\AA} line. In this study, we analyzed four LOS magnetograms of 720 s cadence at 0, 6, 12, and 18 UT for each day, which were reduced from the original $4096\times 4096$ pixels to $1024\times 1024$ pixels by averaging over a $4\times 4$ pixel tile.\footnote{\url{http://jsoc.stanford.edu/data/hmi/fits}} By integrating the LOS field strength $B_{\rm LOS}$ over the entire solar disk, two types of total magnetic flux were obtained: One is the radial unsigned magnetic flux, wherein the radial field strength at each pixel, which is estimated by correcting the viewing angle from the disk center ($\theta$), $B_{\rm LOS}/\cos{\theta}$, is integrated over the disk, $\Phi_{\rm rad}=\int|B_{\rm LOS}/\cos{\theta}|\,dS$; The other is the LOS unsigned magnetic flux, where the LOS field strength is simply integrated over the disk, $\Phi_{\rm LOS}=\int|B_{\rm LOS}|\,dS$. In both cases, the noise levels were estimated by fitting a Gaussian function to the distribution of the field strength in each magnetogram, as in \citet{2001ApJ...555..448H}. The reductions of magnetic flux due to binning the magnetograms from the original $4096\times 4096$ pixels to $1024\times 1024$ pixels were 18.9\% and 23.9\% for the solar maximum (2014 October 23) and minimum (2019 March 1), respectively. Therefore, a typical reduction of 20\% is expected to occur. \subsection{WDC-SILSO}\label{subsec:wdc} In 2015, the daily sunspot number was recalibrated and released as a new dataset (version 2). We obtained this dataset from the WDC-SILSO webpage.\footnote{\url{https://www.sidc.be/silso/datafiles}} Refer to \cite{2014SSRv..186...35C} for a general account on the sunspot number and recalibrated record. \subsection{USAF/NOAA}\label{subsec:noaa} Since 1977, the areas of sunspot groups were measured and recorded by the US Air Force (USAF) and the National Oceanic and Atmospheric Administration (NOAA),\footnote{\url{http://solarcyclescience.com/activeregions.html}} following the record by the Royal Greenwich Observatory. Using this dataset, we calculated the daily total sunspot area on the visible hemisphere of the Sun. The sunspot areas are measured in units of millionths of the solar hemisphere (MSH), which is equivalent to $3\times 10^{6}\ {\rm km}^{2}$. \subsection{SORCE/TIM} The daily total solar irradiance (TSI) data (level 3, version 19) obtained by the Total Irradiance Monitor (TIM; \citealt{2005SoPh..230..129K}) on board the Solar Radiation and Climate Experiment (SORCE; \citealt{2005SoPh..230....7R}) were downloaded from the data archive.\footnote{\url{https://lasp.colorado.edu/home/sorce/data/}} SORCE operated from February 2003 to February 2020, which determines the end of the analysis period in this study. However, there are some gaps in observation owing to the degradation of the battery capacity (longest one from August 2013 to February 2014; \citealt{2021SoPh..296..127W}). Whereas the TSI increases as the solar activity increases, it is occasionally reduced owing to individual sunspot transit and does not correlate well with other proxies such as the total magnetic flux and total sunspot number. Therefore, the TSI was used for reference purposes only. \subsection{GOES/XRS} As one of the X-ray datasets, we analyzed the soft X-ray flux over 1--8 {\AA}, measured by the X-Ray Sensor (XRS) onboard the GOES satellite. In this study, we used the daily-averaged ``science quality'' level 2 data, acquired by the GOES-15 satellite from May 2010 to February 2020.\footnote{\url{https://www.ngdc.noaa.gov/stp/satellite/goes-r.html}} To determine the noise level, we referred to the value of $\lesssim 3\times 10^{-9}\ {\rm W\ m}^{-2}$ at $10^{-5}\ {\rm W\ m}^{-2}$ or less provided by \citet{2015SoPh..290.3625S}. \subsection{SORCE/XPS and SOLSTICE} The irradiances of emission lines and bands from X-rays to near UV were derived using the XUV Photometer System (XPS; \citealt{2005SoPh..230..375W}) and the Solar Stellar Irradiance Comparison Experiment (SOLSTICE; \citealt{2005SoPh..230..225M}) on board the SORCE satellite. The data were obtained from the SORCE data archive. From the XPS daily spectral data (level 4, version 12), which spans over 1 to 400 {\AA} with a spectral resolution of 1 {\AA}, we measured the irradiances of X-rays 5.2--124 {\AA} (ROSAT heritage band), \ion{Fe}{9} 171 {\AA}, \ion{Fe}{10} 174 {\AA}, \ion{Fe}{11} 180 {\AA}, \ion{Fe}{12} 193$+$195 {\AA} (combined), \ion{Fe}{14} 211 {\AA}, \ion{He}{2} 256 {\AA}$+$blends, \ion{Fe}{15} 284 {\AA}, and \ion{He}{2} 304 {\AA}. From the SOLSTICE daily spectral data (level 3, version 18), which covers from 1150 to 3100 {\AA} with a resolution of 1 {\AA}, we estimated the irradiances of \ion{C}{3} 1175 {\AA}, \ion{Si}{3} 1206 {\AA}, \ion{H}{1} 1216 {\AA} (Ly$\alpha$), \ion{N}{5} 1238 {\AA}, \ion{N}{5} 1242 {\AA}, \ion{O}{1} 1302 {\AA}, \ion{O}{1} 1305 {\AA}, \ion{C}{2} 1335 {\AA}, \ion{Fe}{12} 1349 {\AA}, \ion{Cl}{1} 1351 {\AA}, \ion{Si}{4} 1393 {\AA}, \ion{Si}{4} 1402 {\AA}, \ion{C}{4} 1548 {\AA}, \ion{C}{4} 1551 {\AA}, \ion{Mg}{2} k 2796 {\AA}, and \ion{Mg}{2} h 2803 {\AA}. In this dataset (i.e., SORCE/SOLSTICE daily spectral data: level 3, version 18), the geocoronal effects were removed from Ly$\alpha$. For each line, we referred to \citet{2021ApJ...908..205A} for the central wavelength and spectral window to calculate the irradiance and the CHIANTI database for the corresponding formation temperature. All irradiances have been corrected to their respective value at 1 au. The noise levels were estimated by referring to the irradiance uncertainty shown in the dataset. \subsection{Hinode/XRT} The X-Ray Telescope (XRT; \citealt{2007SoPh..243...63G}) aboard the Hinode satellite captures the full-disk synoptic soft X-ray images roughly twice a day at 6 UT and 18 UT except for the interruption periods owing to, e.g., CCD bakeout and other engineering operations \citep{2016SoPh..291..317T}. Montana State University provides the daily averaged electron temperature ($T_{\rm e}$), emission measure ($EM$), and soft X-ray irradiance (5--60 {\AA}) data,\footnote{\url{http://solar.physics.montana.edu/takeda/XRT_outgoing/irrad/}} which are derived with the filter ratio method based on the isothermal spectrum (5--60 {\AA}) under the assumption of coronal elemental abundance in the CHIANTI atomic database (version 10: \citealt{2021ApJ...909...38D}). The filter pairs used for this method are Ti\_poly/Al\_mesh from February 2008 to May 2015 and Al\_poly/Al\_mesh from June 2015 to June 2021. The correction factors for the stray light and filter contamination are selected to ensure that the $T_{\rm e}$ and $EM$ values in the Cycle 24/25 minimum (around 2019) are nearly the same as those in the Cycle 23/24 minimum (around 2008). Considering the filter-ratio method diagnoses the plasmas over a wide temperature range \citep{2011SoPh..269..169N}, we determined the XRT temperature to be $\log{(T/{\rm K})}=6.2 \pm 0.1$ by taking the average and standard deviation of the $T_{\rm e}$ values between May 2010 and February 2020. \subsection{F10.7cm Radio Flux}\label{subsec:f107} The 10.7 cm (2.8 GHz) band radio flux, F10.7cm, is an excellent proxy of solar activity and has been measured consistently in Canada since 1947 \citep{2013SpWea..11..394T}. The transparency of the Earth's atmosphere to this microwave signal makes it possible to monitor solar activity with a high duty cycle. In this study, we used the daily F10.7cm flux data obtained by the Dominion Radio Astrophysical Observatory (DRAO), specifically, the ``adjusted'' data, which are corrected to the values at 1 au.\footnote{\url{https://www.spaceweather.gc.ca/forecast-prevision/solar-solaire/solarflux/sx-en.php}} The data are shown in solar flux units (sfu), which corresponds to $10^{-22}\ {\rm W}\ {\rm m}^{-2}\ {\rm Hz}^{-1}$. When the Sun is quiet with no flaring activity, the formation of F10.7cm flux can be described as a combination of thermal radiation from the transition region to the upper chromosphere (temperatures of 20,000--30,000 K), gyroresonance radiation from active regions, and thermal radiation from the active region corona ($> 1$ MK) \citep{1994ApJ...420..903G}. The variation component, which is used in this study, is mostly due to the active region corona, and hence, we assumed the corresponding temperature to be $\log{(T/{\rm K})}\sim 6$. For the data uncertainties, we assumed that the average error was no more than 0.5\% by referring to \citet{1994SoPh..150..305T}. \subsection{SORCE/SIM and SOLIS/ISS} For the chromospheric lines from the visible to near infrared range, we analyzed the daily spectral data of \ion{Ca}{2} K 3934 {\AA}, \ion{Ca}{2} H 3968 {\AA}, \ion{H}{1} 6563 {\AA} (H$\alpha$), \ion{Ca}{2} 8542 {\AA}, {\ion{He}{1} 10830 {\AA} measured by the Integrated Sunlight Spectrometer (ISS; \citealt{2011SoPh..272..229B}) of the Synoptic Optical Long-term Investigations of the Sun (SOLIS). These spectra are provided as relative intensities with respect to the nearby continuum levels. Therefore, the daily spectral irradiance data of the SORCE's Spectral Irradiance Monitor (SIM; \citealt{2005SoPh..230..141H}) (level 3, version 27), spanning from 2400 to 24200 {\AA} with a resolution of 10--340 {\AA}, were incorporated to obtain the absolute intensities. Note that the SOLIS/ISS observation was terminated in October 2017. \section{Derivation of Power-law Index}\label{sec:analysis} \subsection{Light Curve}\label{subsec:lightcurves} All the daily data used in this study, i.e., the activity proxies and line/band light curves, are shown in Figure \ref{fig:lc}, whereas the minimum and maximum values of these observables are shown in Table \ref{tab:observables}. As shown in Figure \ref{fig:lc}, the irradiance of each line/band varies as the solar activity waxes and wanes. The spikes in the curves indicate that when active regions and other magnetic elements transit across the solar disk, the surface magnetic flux, spot number, and spot area increase (dimming in case of TSI), whereas the $EM$ and irradiances in the Sun's upper atmospheres are enhanced. In contrast, some lines present weak correlations with the solar activity. In particular, H$\alpha$ and \ion{Ca}{2} 8542 {\AA} increase brightness only during the declining phase of the cycle, and the long-term trend of \ion{He}{1} 10830 {\AA} shows an almost inverse correlation with the activity. This may be because these chromospheric lines usually appear in absorption on the disk \citep{1994IAUS..154...35A,1996SoPh..163...79B}. \subsection{Power-law Index}\label{subsec:powerlaw} To obtain the scaling relationships between the activity proxies ($P$) and irradiances ($F$), we first obtained the basal fluxes ($P_{0}$ and $F_{0}$) and daily variations (residuals: $\Delta P=P-P_{0}$ and $\Delta F=F-F_{0}$). Then, we created a scatter plot of the residuals for each pair of the proxies and irradiances ($\Delta F$ vs. $\Delta P$). The basal fluxes can be considered as the surface magnetic flux and the resultant magnetically-driven high-temperature emissions that are always present as background components. Therefore, they can be measured during the deepest solar minimum. The residuals indicate the appearance of magnetic fields, such as active regions and plages, and the associated heating of the upper atmospheres. Additionally, it is possible to set wide dynamic ranges for scatter plots by taking residuals. The basal flux was defined as, of the total of 3,592 days, from May 2010 to February 2020, the median of the values on the days that met the following conditions: \begin{itemize} \item The final one year, i.e., the deepest solar minimum from March 2019 to February 2020; \item When the total sunspot number is 0; and \item When the radial total unsigned magnetic flux is less than the 10th percentile for the entire period. \end{itemize} As a result, the number of unspotted days that satisfy these conditions was 86. However, depending on the observables, the actual number of unspotted days that was used for taking the medians may differ. As the observation of SOLIS/ISS was terminated in 2017, for the chromospheric lines observed by this telescope, we considered the median of the 268 days that met the following conditions: \begin{itemize} \item One year centered on December 2008; and \item When the total sunspot number is 0. \end{itemize} The basal fluxes for all observables are summarized in Table \ref{tab:observables} and denoted by horizontal dashed lines in Figure \ref{fig:lc}. We set the basal fluxes for the total sunspot number, total sunspot area, and the GOES soft x-ray flux (1--8 {\AA}) as 0, 0 MSH, and 0 W m$^{-2}$, respectively. As described above, the basal flux for each time series was calculated as the median of spotless day data. This is because it is not known whether the minimum value in a time series is truly the lowest value due to data gaps. Therefore, the minimum values in Table \ref{tab:observables} are smaller than the basal fluxes in most cases. Figures \ref{fig:cc_magc} to \ref{fig:cc_srf} show the scatter plots of irradiances (residual: $\Delta F$) vs. the solar activity proxies (residual: $\Delta P$) of radial total unsigned magnetic flux (Figure \ref{fig:cc_magc}), LOS total unsigned magnetic flux (Figure \ref{fig:cc_mag}), total sunspot number (Figure \ref{fig:cc_tsn}), total sunspot area (Figure \ref{fig:cc_spotarea}), and the F10.7cm flux (Figure \ref{fig:cc_srf}). Here, only the data points where both $\Delta P$ and $\Delta F$ were positive are plotted. The fractions of data points that were not used due to negative values of $\Delta P$ or $\Delta F$ are typically 13--17\% for the SORCE data and 51--60\% for the SOLIS/ISS data. Each figure shows the result of a linear fit to a double logarithmic plot: The linear fit was applied to the $(\log{\Delta P}, \log{\Delta F})$ data, where both $\Delta P$ and $\Delta F$ were positive, to obtain $\alpha$ and $\beta$ as in the following equation: \begin{eqnarray} \Delta P = 10^{\beta} \Delta F^{\alpha}, \end{eqnarray} or equivalently, \begin{eqnarray} \log{\Delta P} = \beta + \alpha \log{\Delta F}. \end{eqnarray} We assumed that both $\log{\Delta P}$ and $\log{\Delta F}$ have errors. Also, we applied a uniform weight for each observable because giving weights to smaller data points allows for wider dynamic ranges over which the linear fit is performed.\footnote{We also tested the differential weighting method, which puts more weight on larger data. However, the fitting results were not much different from the uniform weighting cases, especially for $\log{\Delta P}$ with broad dynamic ranges such as the total radial unsigned magnetic flux. Therefore, we adopted the uniform weighting method in favor of the effective dynamic ranges.} The degree of dispersion of the data points was also examined by measuring the linear Pearson correlation coefficient, ${\rm CC}(\log{\Delta P}, \log{\Delta F})$. It should be noted here that the observation data for which the power-law scalings were calculated are not evenly distributed between May 2010 and February 2020: There exist observational gaps for each observable as they appear as gaps in the light curves in Figure \ref{fig:lc}. \begin{deluxetable*}{lcccccc} \tablecaption{Power-law Indices and Correlations between Irradiance and Total Radial Magnetic Flux\label{tab:powerlaw_magc}} \tablewidth{0pt} \tablehead{ \colhead{Feature} & \colhead{$\log{(T/{\rm K})}$} & \colhead{Power-law Index $\alpha$} & \colhead{Offset $\beta$} & \colhead{Correlation Coefficient CC} & \colhead{Data Points $N$} & \colhead{LS Deviation} } \decimalcolnumbers \startdata X-rays 1--8 {\AA} & 6--7 & $1.42\pm 0.04$ & $-38.6\pm 0.8$ & 0.893 & 3243 & 0.431\\ X-rays 5.2--124 {\AA} & 6--7 & $1.16\pm 0.03$ & $-29.9\pm 0.7$ & 0.926 & 2994 & 0.247\\ Fe XV 284 {\AA} & 6.4 & $1.15\pm 0.03$ & $-30.6\pm 0.7$ & 0.919 & 3009 & 0.258\\ Fe XIV 211 {\AA} & 6.3 & $1.15\pm 0.03$ & $-30.9\pm 0.7$ & 0.924 & 2998 & 0.248\\ X-rays (XRT) & $6.2\pm 0.1$ & $0.95\pm 0.03$ & $-25.2\pm 0.6$ & 0.932 & 2966 & 0.222\\ Fe XII 193$+$195 {\AA} & 6.2 & $1.14\pm 0.03$ & $-30.5\pm 0.7$ & 0.925 & 2998 & 0.246\\ Fe XII 1349 {\AA} & 6.2 & $0.71\pm 0.02$ & $-22.3\pm 0.5$ & 0.836 & 2978 & 0.236\\ Fe X 174 {\AA} & 6.1 & $1.15\pm 0.03$ & $-31.1\pm 0.7$ & 0.924 & 2998 & 0.248\\ Fe XI 180 {\AA} & 6.1 & $1.15\pm 0.03$ & $-30.9\pm 0.7$ & 0.925 & 2998 & 0.247\\ F10.7cm radio & $\sim$6 & $1.24\pm 0.03$ & $-26.8\pm 0.7$ & 0.939 & 3200 & 0.225\\ Fe IX 171 {\AA} & 5.9 & $1.15\pm 0.03$ & $-31.4\pm 0.7$ & 0.925 & 2998 & 0.247\\ N V 1238 {\AA} & 5.3 & $0.82\pm 0.02$ & $-24.3\pm 0.5$ & 0.888 & 3014 & 0.233\\ N V 1242 {\AA} & 5.3 & $0.85\pm 0.02$ & $-25.1\pm 0.5$ & 0.882 & 2989 & 0.239\\ C IV 1548 {\AA} & 5.1 & $0.85\pm 0.02$ & $-24.2\pm 0.5$ & 0.898 & 3080 & 0.233\\ C IV 1551 {\AA} & 5.1 & $0.81\pm 0.02$ & $-23.6\pm 0.5$ & 0.875 & 3072 & 0.248\\ C III 1175 {\AA} & 5.0 & $0.82\pm 0.02$ & $-23.6\pm 0.5$ & 0.903 & 3055 & 0.218\\ He II 256 {\AA} & 4.9 & $1.14\pm 0.03$ & $-30.8\pm 0.7$ & 0.923 & 3001 & 0.249\\ He II 304 {\AA} & 4.9 & $1.15\pm 0.03$ & $-30.4\pm 0.7$ & 0.923 & 2998 & 0.250\\ Si IV 1393 {\AA} & 4.9 & $0.90\pm 0.02$ & $-25.3\pm 0.5$ & 0.923 & 3089 & 0.215\\ Si IV 1402 {\AA} & 4.9 & $0.83\pm 0.02$ & $-24.1\pm 0.5$ & 0.914 & 3096 & 0.214\\ Si III 1206 {\AA} & 4.8 & $0.89\pm 0.02$ & $-24.7\pm 0.5$ & 0.924 & 3126 & 0.214\\ He I 10830 {\AA} & 4.5 & $1.09\pm 0.06$ & $-28.2\pm 1.4$ & 0.453 & 1419 & 0.381\\ C II 1335 {\AA} & 4.3 & $0.79\pm 0.02$ & $-22.5\pm 0.5$ & 0.924 & 3102 & 0.193\\ H I 1216 {\AA} (Ly$\alpha$) & 4.3 & $0.89\pm 0.02$ & $-23.3\pm 0.5$ & 0.939 & 3105 & 0.193\\ O I 1302 {\AA} & 4.2 & $0.84\pm 0.02$ & $-24.6\pm 0.5$ & 0.822 & 2971 & 0.300\\ O I 1305 {\AA} & 4.2 & $0.83\pm 0.02$ & $-24.1\pm 0.5$ & 0.815 & 3011 & 0.307\\ Mg II k 2796 {\AA} & (3.9) & $0.95\pm 0.02$ & $-24.4\pm 0.5$ & 0.949 & 3120 & 0.187\\ Mg II h 2803 {\AA} & (3.9) & $0.97\pm 0.03$ & $-25.2\pm 0.6$ & 0.944 & 3097 & 0.200\\ Mg II k$+$h & (3.9) & $0.96\pm 0.02$ & $-24.5\pm 0.6$ & 0.951 & 3120 & 0.187\\ Cl I 1351 {\AA} & (3.8) & $0.83\pm 0.02$ & $-24.9\pm 0.5$ & 0.783 & 2928 & 0.312\\ Ca II K 3934 {\AA} & (3.8) & $0.87\pm 0.03$ & $-23.1\pm 0.8$ & 0.723 & 1755 & 0.214\\ Ca II H 3968 {\AA} & (3.8) & $0.86\pm 0.04$ & $-22.7\pm 0.9$ & 0.539 & 1624 & 0.273\\ H I 6563 {\AA} (H$\alpha$) & (3.8) & $-1.46\pm 0.14$ & $ 29.9\pm 3.1$ & $-0.152$ & 1487 & 0.643\\ Ca II 8542 {\AA} & (3.8) & $-1.52\pm 0.45$ & $31.3\pm 10.1$ & $-0.014$ & 1678 & 0.714\\ \enddata \tablecomments{The first and second columns show the spectral lines and their formation temperatures, respectively. Columns 3, 4, 5, and 6 provide the power-law index $\alpha$, offset $\beta$, correlation coefficient CC, and the number of data points $N$ of each double logarithmic scatter plot of irradiance versus total radial magnetic flux. Column 7 presents the least-square deviation of the linear fit to the double logarithmic plot.} \end{deluxetable*} \begin{deluxetable*}{lcccccc} \tablecaption{Power-law Indices and Correlations between Irradiance and Total LOS Magnetic Flux\label{tab:powerlaw_mag}} \tablewidth{0pt} \tablehead{ \colhead{Feature} & \colhead{$\log{(T/{\rm K})}$} & \colhead{Power-law Index $\alpha$} & \colhead{Offset $\beta$} & \colhead{Correlation Coefficient CC} & \colhead{Data Points $N$} & \colhead{LS Deviation} } \decimalcolnumbers \startdata X-rays 1--8 {\AA} & 6--7 & $1.43\pm 0.04$ & $-38.7\pm 0.9$ & 0.887 & 3230 & 0.443\\ X-rays 5.2--124 {\AA} & 6--7 & $1.16\pm 0.03$ & $-29.8\pm 0.7$ & 0.910 & 2982 & 0.271\\ Fe XV 284 {\AA} & 6.4 & $1.15\pm 0.03$ & $-30.5\pm 0.7$ & 0.905 & 2997 & 0.279\\ Fe XIV 211 {\AA} & 6.3 & $1.15\pm 0.03$ & $-30.8\pm 0.7$ & 0.911 & 2986 & 0.269\\ X-rays (XRT) & $6.2\pm 0.1$ & $0.96\pm 0.03$ & $-25.3\pm 0.6$ & 0.932 & 2953 & 0.221\\ Fe XII 193$+$195 {\AA} & 6.2 & $1.14\pm 0.03$ & $-30.4\pm 0.7$ & 0.911 & 2986 & 0.267\\ Fe XII 1349 {\AA} & 6.2 & $0.71\pm 0.02$ & $-22.3\pm 0.5$ & 0.829 & 2966 & 0.238\\ Fe X 174 {\AA} & 6.1 & $1.15\pm 0.03$ & $-31.0\pm 0.7$ & 0.911 & 2986 & 0.269\\ Fe XI 180 {\AA} & 6.1 & $1.15\pm 0.03$ & $-30.8\pm 0.7$ & 0.911 & 2986 & 0.268\\ F10.7cm radio & $\sim$6 & $1.23\pm 0.03$ & $-26.3\pm 0.7$ & 0.928 & 3193 & 0.243\\ Fe IX 171 {\AA} & 5.9 & $1.15\pm 0.03$ & $-31.2\pm 0.7$ & 0.911 & 2986 & 0.268\\ N V 1238 {\AA} & 5.3 & $0.83\pm 0.02$ & $-24.3\pm 0.5$ & 0.881 & 3001 & 0.238\\ N V 1242 {\AA} & 5.3 & $0.85\pm 0.02$ & $-24.9\pm 0.5$ & 0.871 & 2980 & 0.250\\ C IV 1548 {\AA} & 5.1 & $0.84\pm 0.02$ & $-23.9\pm 0.5$ & 0.896 & 3070 & 0.232\\ C IV 1551 {\AA} & 5.1 & $0.82\pm 0.02$ & $-23.6\pm 0.5$ & 0.866 & 3059 & 0.257\\ C III 1175 {\AA} & 5.0 & $0.82\pm 0.02$ & $-23.6\pm 0.5$ & 0.893 & 3041 & 0.228\\ He II 256 {\AA} & 4.9 & $1.15\pm 0.03$ & $-30.6\pm 0.7$ & 0.909 & 2989 & 0.271\\ He II 304 {\AA} & 4.9 & $1.15\pm 0.03$ & $-30.4\pm 0.7$ & 0.909 & 2986 & 0.272\\ Si IV 1393 {\AA} & 4.9 & $0.90\pm 0.02$ & $-25.2\pm 0.5$ & 0.912 & 3078 & 0.229\\ Si IV 1402 {\AA} & 4.9 & $0.84\pm 0.02$ & $-24.2\pm 0.5$ & 0.908 & 3083 & 0.220\\ Si III 1206 {\AA} & 4.8 & $0.89\pm 0.02$ & $-24.6\pm 0.5$ & 0.928 & 3111 & 0.207\\ He I 10830 {\AA} & 4.5 & $1.07\pm 0.06$ & $-27.6\pm 1.3$ & 0.472 & 1419 & 0.374\\ C II 1335 {\AA} & 4.3 & $0.80\pm 0.02$ & $-22.6\pm 0.5$ & 0.927 & 3092 & 0.189\\ H I 1216 {\AA} (Ly$\alpha$) & 4.3 & $0.90\pm 0.02$ & $-23.2\pm 0.5$ & 0.939 & 3095 & 0.191\\ O I 1302 {\AA} & 4.2 & $0.85\pm 0.02$ & $-24.6\pm 0.5$ & 0.837 & 2962 & 0.288\\ O I 1305 {\AA} & 4.2 & $0.83\pm 0.02$ & $-24.0\pm 0.5$ & 0.817 & 2998 & 0.303\\ Mg II k 2796 {\AA} & (3.9) & $0.95\pm 0.02$ & $-24.3\pm 0.6$ & 0.945 & 3110 & 0.194\\ Mg II h 2803 {\AA} & (3.9) & $0.98\pm 0.03$ & $-25.2\pm 0.6$ & 0.943 & 3087 & 0.200\\ Mg II k$+$h & (3.9) & $0.96\pm 0.02$ & $-24.4\pm 0.6$ & 0.947 & 3109 & 0.193\\ Cl I 1351 {\AA} & (3.8) & $0.83\pm 0.02$ & $-24.8\pm 0.5$ & 0.781 & 2919 & 0.312\\ Ca II K 3934 {\AA} & (3.8) & $0.85\pm 0.03$ & $-22.5\pm 0.8$ & 0.737 & 1755 & 0.209\\ Ca II H 3968 {\AA} & (3.8) & $0.84\pm 0.04$ & $-22.1\pm 0.9$ & 0.570 & 1624 & 0.264\\ H I 6563 {\AA} (H$\alpha$) & (3.8) & $-1.44\pm 0.15$ & $ 29.2\pm 3.3$ & $-0.126$ & 1487 & 0.653\\ Ca II 8542 {\AA} & (3.8) & $1.50\pm 0.44$ & $-36.9\pm 9.9$ & $0.014$ & 1678 & 0.714\\ \enddata \tablecomments{The first and second columns show the spectral lines and their formation temperatures, respectively. Columns 3, 4, 5, and 6 provide the power-law index $\alpha$, offset $\beta$, correlation coefficient CC, and the number of data points $N$ of each double logarithmic scatter plot of irradiance versus total LOS magnetic flux. Column 7 presents the least-square deviation of the linear fit to the double logarithmic plot.} \end{deluxetable*} \begin{deluxetable*}{lcccccc} \tablecaption{Power-law Indices and Correlations between Irradiance and Total Sunspot Number\label{tab:powerlaw_tsn}} \tablewidth{0pt} \tablehead{ \colhead{Feature} & \colhead{$\log{(T/{\rm K})}$} & \colhead{Power-law Index $\alpha$} & \colhead{Offset $\beta$} & \colhead{Correlation Coefficient CC} & \colhead{Data Points $N$} & \colhead{LS Deviation} } \decimalcolnumbers \startdata X-rays 1--8 {\AA} & 6--7 & $1.98\pm 0.06$ & $-9.8\pm 0.1$ & 0.819 & 2682 & 0.419\\ X-rays 5.2--124 {\AA} & 6--7 & $1.41\pm 0.04$ & $-6.1\pm 0.1$ & 0.815 & 2578 & 0.293\\ Fe XV 284 {\AA} & 6.4 & $1.44\pm 0.04$ & $-7.0\pm 0.1$ & 0.809 & 2585 & 0.306\\ Fe XIV 211 {\AA} & 6.3 & $1.42\pm 0.04$ & $-7.3\pm 0.1$ & 0.813 & 2581 & 0.298\\ X-rays (XRT) & $6.2\pm 0.1$ & $1.11\pm 0.03$ & $-5.5\pm 0.1$ & 0.833 & 2479 & 0.228\\ Fe XII 193$+$195 {\AA} & 6.2 & $1.42\pm 0.04$ & $-7.0\pm 0.1$ & 0.814 & 2581 & 0.297\\ Fe XII 1349 {\AA} & 6.2 & $0.94\pm 0.03$ & $-7.9\pm 0.1$ & 0.738 & 2538 & 0.236\\ Fe X 174 {\AA} & 6.1 & $1.42\pm 0.04$ & $-7.5\pm 0.1$ & 0.813 & 2581 & 0.298\\ Fe XI 180 {\AA} & 6.1 & $1.42\pm 0.04$ & $-7.4\pm 0.1$ & 0.813 & 2581 & 0.297\\ F10.7cm radio & $\sim$6 & $1.46\pm 0.04$ & $-1.1\pm 0.1$ & 0.867 & 2861 & 0.258\\ Fe IX 171 {\AA} & 5.9 & $1.42\pm 0.04$ & $-7.8\pm 0.1$ & 0.813 & 2581 & 0.297\\ N V 1238 {\AA} & 5.3 & $1.10\pm 0.03$ & $-7.5\pm 0.1$ & 0.816 & 2557 & 0.232\\ N V 1242 {\AA} & 5.3 & $1.11\pm 0.04$ & $-7.7\pm 0.1$ & 0.792 & 2547 & 0.249\\ C IV 1548 {\AA} & 5.1 & $1.14\pm 0.04$ & $-6.9\pm 0.1$ & 0.814 & 2578 & 0.243\\ C IV 1551 {\AA} & 5.1 & $1.08\pm 0.03$ & $-7.0\pm 0.1$ & 0.812 & 2575 & 0.232\\ C III 1175 {\AA} & 5.0 & $1.13\pm 0.03$ & $-7.0\pm 0.1$ & 0.801 & 2580 & 0.250\\ He II 256 {\AA} & 4.9 & $1.42\pm 0.04$ & $-7.2\pm 0.1$ & 0.814 & 2582 & 0.297\\ He II 304 {\AA} & 4.9 & $1.42\pm 0.04$ & $-6.8\pm 0.1$ & 0.812 & 2581 & 0.299\\ Si IV 1393 {\AA} & 4.9 & $1.15\pm 0.04$ & $-7.0\pm 0.1$ & 0.815 & 2583 & 0.247\\ Si IV 1402 {\AA} & 4.9 & $1.05\pm 0.03$ & $-7.1\pm 0.1$ & 0.826 & 2579 & 0.217\\ Si III 1206 {\AA} & 4.8 & $1.08\pm 0.03$ & $-6.5\pm 0.1$ & 0.831 & 2591 & 0.222\\ He I 10830 {\AA} & 4.5 & $1.17\pm 0.08$ & $-5.5\pm 0.1$ & 0.352 & 1344 & 0.399\\ C II 1335 {\AA} & 4.3 & $1.02\pm 0.03$ & $-6.3\pm 0.1$ & 0.837 & 2574 & 0.205\\ H I 1216 {\AA} (Ly$\alpha$) & 4.3 & $1.14\pm 0.04$ & $-4.9\pm 0.1$ & 0.805 & 2585 & 0.250\\ O I 1302 {\AA} & 4.2 & $1.17\pm 0.04$ & $-7.6\pm 0.1$ & 0.754 & 2525 & 0.287\\ O I 1305 {\AA} & 4.2 & $1.18\pm 0.04$ & $-7.2\pm 0.1$ & 0.722 & 2539 & 0.310\\ Mg II k 2796 {\AA} & (3.9) & $1.21\pm 0.04$ & $-5.0\pm 0.1$ & 0.828 & 2588 & 0.250\\ Mg II h 2803 {\AA} & (3.9) & $1.23\pm 0.04$ & $-5.2\pm 0.1$ & 0.822 & 2584 & 0.259\\ Mg II k$+$h & (3.9) & $1.23\pm 0.04$ & $-4.8\pm 0.1$ & 0.828 & 2590 & 0.254\\ Cl I 1351 {\AA} & (3.8) & $1.12\pm 0.04$ & $-8.1\pm 0.1$ & 0.738 & 2517 & 0.277\\ Ca II K 3934 {\AA} & (3.8) & $0.85\pm 0.04$ & $-4.7\pm 0.1$ & 0.656 & 1659 & 0.218\\ Ca II H 3968 {\AA} & (3.8) & $0.85\pm 0.04$ & $-4.7\pm 0.1$ & 0.531 & 1529 & 0.254\\ H I 6563 {\AA} (H$\alpha$) & (3.8) & $-1.60\pm 0.19$ & $-0.5\pm 0.3$ & $-0.097$ & 1399 & 0.667\\ Ca II 8542 {\AA} & (3.8) & $ 1.61\pm 0.38$ & $-6.0\pm 0.7$ & $ 0.023$ & 1586 & 0.696\\ \enddata \tablecomments{The first and second columns show the spectral lines and their formation temperatures, respectively. Columns 3, 4, 5, and 6 provide the power-law index $\alpha$, offset $\beta$, correlation coefficient CC, and the number of data points $N$ of each double logarithmic scatter plot of irradiance versus total sunspot number. Column 7 presents the least-square deviation of the linear fit to the double logarithmic plot.} \end{deluxetable*} \begin{deluxetable*}{lcccccc} \tablecaption{Power-law Indices and Correlations between Irradiance and Total Sunspot Area\label{tab:powerlaw_spotarea}} \tablewidth{0pt} \tablehead{ \colhead{Feature} & \colhead{$\log{(T/{\rm K})}$} & \colhead{Power-law Index $\alpha$} & \colhead{Offset $\beta$} & \colhead{Correlation Coefficient CC} & \colhead{Data Points $N$} & \colhead{LS Deviation} } \decimalcolnumbers \startdata X-rays 1--8 {\AA} & 6--7 & $1.18\pm 0.04$ & $-9.2\pm 0.1$ & 0.796 & 2621 & 0.423\\ X-rays 5.2--124 {\AA} & 6--7 & $0.83\pm 0.03$ & $-5.7\pm 0.1$ & 0.767 & 2532 & 0.317\\ Fe XV 284 {\AA} & 6.4 & $0.83\pm 0.03$ & $-6.5\pm 0.1$ & 0.760 & 2534 & 0.322\\ Fe XIV 211 {\AA} & 6.3 & $0.82\pm 0.03$ & $-6.8\pm 0.1$ & 0.770 & 2532 & 0.311\\ X-rays (XRT) & $6.2 \pm 0.1$ & $0.65\pm 0.02$ & $-5.1\pm 0.1$ & 0.770 & 2415 & 0.251\\ Fe XII 193$+$195 {\AA} & 6.2 & $0.82\pm 0.03$ & $-6.5\pm 0.1$ & 0.770 & 2532 & 0.311\\ Fe XII 1349 {\AA} & 6.2 & $0.56\pm 0.02$ & $-7.6\pm 0.0$ & 0.650 & 2484 & 0.263\\ Fe X 174 {\AA} & 6.1 & $0.82\pm 0.03$ & $-7.1\pm 0.1$ & 0.770 & 2532 & 0.311\\ Fe XI 180 {\AA} & 6.1 & $0.82\pm 0.03$ & $-6.9\pm 0.1$ & 0.770 & 2532 & 0.311\\ F10.7cm radio & $\sim$6 & $0.88\pm 0.03$ & $-0.7\pm 0.1$ & 0.807 & 2804 & 0.297\\ Fe IX 171 {\AA} & 5.9 & $0.82\pm 0.03$ & $-7.3\pm 0.1$ & 0.770 & 2532 & 0.311\\ N V 1238 {\AA} & 5.3 & $0.66\pm 0.02$ & $-7.2\pm 0.1$ & 0.725 & 2504 & 0.276\\ N V 1242 {\AA} & 5.3 & $0.66\pm 0.02$ & $-7.4\pm 0.1$ & 0.700 & 2492 & 0.285\\ C IV 1548 {\AA} & 5.1 & $0.68\pm 0.02$ & $-6.5\pm 0.1$ & 0.694 & 2515 & 0.302\\ C IV 1551 {\AA} & 5.1 & $0.63\pm 0.02$ & $-6.7\pm 0.1$ & 0.718 & 2514 & 0.270\\ C III 1175 {\AA} & 5.0 & $0.66\pm 0.02$ & $-6.6\pm 0.1$ & 0.712 & 2515 & 0.282\\ He II 256 {\AA} & 4.9 & $0.81\pm 0.03$ & $-6.8\pm 0.1$ & 0.771 & 2532 & 0.308\\ He II 304 {\AA} & 4.9 & $0.82\pm 0.03$ & $-6.3\pm 0.1$ & 0.769 & 2532 & 0.312\\ Si IV 1393 {\AA} & 4.9 & $0.66\pm 0.02$ & $-6.6\pm 0.1$ & 0.744 & 2524 & 0.270\\ Si IV 1402 {\AA} & 4.9 & $0.63\pm 0.02$ & $-6.8\pm 0.0$ & 0.737 & 2521 & 0.258\\ Si III 1206 {\AA} & 4.8 & $0.65\pm 0.02$ & $-6.2\pm 0.1$ & 0.728 & 2529 & 0.273\\ He I 10830 {\AA} & 4.5 & $0.74\pm 0.05$ & $-5.3\pm 0.1$ & 0.311 & 1332 & 0.415\\ C II 1335 {\AA} & 4.3 & $0.59\pm 0.02$ & $-6.0\pm 0.0$ & 0.733 & 2507 & 0.245\\ H I 1216 {\AA} (Ly$\alpha$) & 4.3 & $0.67\pm 0.02$ & $-4.6\pm 0.1$ & 0.708 & 2521 & 0.291\\ O I 1302 {\AA} & 4.2 & $0.70\pm 0.02$ & $-7.2\pm 0.1$ & 0.645 & 2458 & 0.328\\ O I 1305 {\AA} & 4.2 & $0.71\pm 0.03$ & $-6.9\pm 0.1$ & 0.627 & 2475 & 0.346\\ Mg II k 2796 {\AA} & (3.9) & $0.72\pm 0.02$ & $-4.6\pm 0.1$ & 0.745 & 2528 & 0.293\\ Mg II h 2803 {\AA} & (3.9) & $0.74\pm 0.02$ & $-4.9\pm 0.1$ & 0.744 & 2522 & 0.300\\ Mg II k$+$h & (3.9) & $0.73\pm 0.02$ & $-4.4\pm 0.1$ & 0.748 & 2528 & 0.294\\ Cl I 1351 {\AA} & (3.8) & $0.66\pm 0.02$ & $-7.8\pm 0.1$ & 0.666 & 2463 & 0.300\\ Ca II K 3934 {\AA} & (3.8) & $0.53\pm 0.03$ & $-4.5\pm 0.1$ & 0.492 & 1643 & 0.262\\ Ca II H 3968 {\AA} & (3.8) & $0.53\pm 0.03$ & $-4.5\pm 0.1$ & 0.410 & 1514 & 0.282\\ H I 6563 {\AA} (H$\alpha$) & (3.8) & $-1.00\pm 0.13$ & $-0.8\pm 0.3$ & $-0.093$ & 1388 & 0.667\\ Ca II 8542 {\AA} & (3.8) & $1.01\pm 0.35$ & $-5.7\pm 0.9$ & $0.011$ & 1567 & 0.698\\ \enddata \tablecomments{The first and second columns show the spectral lines and their formation temperatures, respectively. Columns 3, 4, 5, and 6 provide the power-law index $\alpha$, offset $\beta$, correlation coefficient CC, and the number of data points $N$ of each double logarithmic scatter plot of irradiance versus total sunspot area. Column 7 presents the least-square deviation of the linear fit to the double logarithmic plot.} \end{deluxetable*} \begin{deluxetable*}{lcccccc} \tablecaption{Power-law Indices and Correlations between Irradiance and F10.7cm Radio Flux\label{tab:powerlaw_srf}} \tablewidth{0pt} \tablehead{ \colhead{Feature} & \colhead{$\log{(T/{\rm K})}$} & \colhead{Power-law Index $\alpha$} & \colhead{Offset $\beta$} & \colhead{Correlation Coefficient CC} & \colhead{Data Points $N$} & \colhead{LS Deviation} } \decimalcolnumbers \startdata X-rays 1--8 {\AA} & 6--7 & $1.21\pm 0.03$ & $-8.1\pm 0.0$ & 0.918 & 2978 & 0.334\\ X-rays 5.2--124 {\AA} & 6--7 & $0.89\pm 0.02$ & $-5.0\pm 0.0$ & 0.932 & 2853 & 0.207\\ Fe XV 284 {\AA} & 6.4 & $0.86\pm 0.02$ & $-5.8\pm 0.0$ & 0.938 & 2856 & 0.192\\ Fe XIV 211 {\AA} & 6.3 & $0.87\pm 0.02$ & $-6.1\pm 0.0$ & 0.935 & 2852 & 0.199\\ X-rays (XRT) & $6.2\pm 0.1$ & $0.78\pm 0.02$ & $-4.7\pm 0.0$ & 0.912 & 2741 & 0.218\\ Fe XII 193$+$195 {\AA} & 6.2 & $0.87\pm 0.02$ & $-5.8\pm 0.0$ & 0.935 & 2852 & 0.198\\ Fe XII 1349 {\AA} & 6.2 & $0.59\pm 0.02$ & $-7.1\pm 0.0$ & 0.811 & 2771 & 0.227\\ Fe X 174 {\AA} & 6.1 & $0.87\pm 0.02$ & $-6.3\pm 0.0$ & 0.935 & 2852 & 0.199\\ Fe XI 180 {\AA} & 6.1 & $0.87\pm 0.02$ & $-6.2\pm 0.0$ & 0.935 & 2852 & 0.198\\ Fe IX 171 {\AA} & 5.9 & $0.87\pm 0.02$ & $-6.6\pm 0.0$ & 0.935 & 2852 & 0.198\\ N V 1238 {\AA} & 5.3 & $0.70\pm 0.02$ & $-6.6\pm 0.0$ & 0.881 & 2812 & 0.216\\ N V 1242 {\AA} & 5.3 & $0.72\pm 0.02$ & $-6.8\pm 0.0$ & 0.865 & 2797 & 0.232\\ C IV 1548 {\AA} & 5.1 & $0.72\pm 0.02$ & $-5.9\pm 0.0$ & 0.877 & 2852 & 0.233\\ C IV 1551 {\AA} & 5.1 & $0.67\pm 0.02$ & $-6.1\pm 0.0$ & 0.881 & 2845 & 0.212\\ C III 1175 {\AA} & 5.0 & $0.69\pm 0.02$ & $-6.0\pm 0.0$ & 0.886 & 2826 & 0.207\\ He II 256 {\AA} & 4.9 & $0.88\pm 0.02$ & $-6.0\pm 0.0$ & 0.932 & 2854 & 0.204\\ He II 304 {\AA} & 4.9 & $0.88\pm 0.02$ & $-5.6\pm 0.0$ & 0.934 & 2852 & 0.200\\ Si IV 1393 {\AA} & 4.9 & $0.73\pm 0.02$ & $-6.0\pm 0.0$ & 0.901 & 2847 & 0.209\\ Si IV 1402 {\AA} & 4.9 & $0.67\pm 0.02$ & $-6.2\pm 0.0$ & 0.904 & 2855 & 0.190\\ Si III 1206 {\AA} & 4.8 & $0.72\pm 0.02$ & $-5.6\pm 0.0$ & 0.896 & 2871 & 0.215\\ He I 10830 {\AA} & 4.5 & $0.81\pm 0.05$ & $-4.6\pm 0.1$ & 0.438 & 1416 & 0.386\\ C II 1335 {\AA} & 4.3 & $0.65\pm 0.02$ & $-5.5\pm 0.0$ & 0.907 & 2845 & 0.180\\ H I 1216 {\AA} (Ly$\alpha$) & 4.3 & $0.73\pm 0.02$ & $-4.0\pm 0.0$ & 0.892 & 2863 & 0.222\\ O I 1302 {\AA} & 4.2 & $0.70\pm 0.02$ & $-6.5\pm 0.0$ & 0.802 & 2773 & 0.283\\ O I 1305 {\AA} & 4.2 & $0.72\pm 0.02$ & $-6.2\pm 0.0$ & 0.760 & 2788 & 0.321\\ Mg II k 2796 {\AA} & (3.9) & $0.78\pm 0.02$ & $-4.0\pm 0.0$ & 0.910 & 2872 & 0.214\\ Mg II h 2803 {\AA} & (3.9) & $0.79\pm 0.02$ & $-4.2\pm 0.0$ & 0.908 & 2856 & 0.218\\ Mg II k$+$h & (3.9) & $0.78\pm 0.02$ & $-3.8\pm 0.0$ & 0.914 & 2868 & 0.209\\ Cl I 1351 {\AA} & (3.8) & $0.72\pm 0.02$ & $-7.2\pm 0.0$ & 0.773 & 2744 & 0.299\\ Ca II K 3934 {\AA} & (3.8) & $0.65\pm 0.03$ & $-4.2\pm 0.0$ & 0.707 & 1752 & 0.220\\ Ca II H 3968 {\AA} & (3.8) & $0.64\pm 0.03$ & $-4.1\pm 0.0$ & 0.549 & 1620 & 0.271\\ H I 6563 {\AA} (H$\alpha$) & (3.8) & $-1.09\pm 0.11$ & $-1.7\pm 0.2$ & $-0.133$ & 1483 & 0.651\\ Ca II 8542 {\AA} & (3.8) & \nodata & \nodata & $ 0.000$ & 1674 & \nodata \\ \enddata \tablecomments{The first and second columns show the spectral lines and their formation temperatures, respectively. Columns 3, 4, 5, and 6 provide the power-law index $\alpha$, offset $\beta$, correlation coefficient CC, and the number of data points $N$ of each double logarithmic scatter plot of irradiance versus radial F10.7cm radio flux. Column 7 presents the least-square deviation of the linear fit to the double logarithmic plot.} \end{deluxetable*} \section{Catalog of Power-law index}\label{sec:catalog} Tables \ref{tab:powerlaw_magc} to \ref{tab:powerlaw_srf} summarize the power-law index $\alpha$, offset $\beta$, correlation coefficient CC, number of data points $N$, and least-square deviation of the linear fit for all scatter plots in Figures \ref{fig:cc_magc} to \ref{fig:cc_srf}. The overall trend is that the higher temperature lines and bands show higher CCs. For each line, among different proxies, the total magnetic fluxes and F10.7cm flux tend to show higher CCs compared to the sunspot number and the area. Because \ion{He}{1} 10830 {\AA} often falls below its basal flux level (i.e., $\Delta F$ often becomes negative), we created scatter plots by taking the absolute value of $\Delta F$. For F10.7cm vs. \ion{Ca}{2} 8542 {\AA} (Figure \ref{fig:cc_srf}), the scaling factors $\alpha$ and $\beta$ are not provided in Table \ref{tab:powerlaw_srf} owing to the failure of the linear fit. These chromospheric lines and H$\alpha$, \ion{Ca}{2} K 3934 {\AA}, and \ion{Ca}{2} H 3968 {\AA} generally had poorer CCs and least-square deviations. \section{Dependence of Power-law Index}\label{sec:dependence} \subsection{Temperature Dependence}\label{subsec:temperature} Figure \ref{fig:pl} shows the exponent of irradiances with respect to the total radial unsigned magnetic flux, plotted as a function of temperature. Note that H$\alpha$ and \ion{Ca}{2} 8542 {\AA} are omitted because they exhibited negative proportionalities with the magnetic flux (i.e., $\alpha<0$). As \ion{He}{1} 10830 {\AA} showed an anti-phased variation with the activity proxies (Figure \ref{fig:lc}) and a weak anti-correlation (Table \ref{tab:powerlaw_magc}), we plotted $\alpha$ calculated by taking the absolute value of $\Delta F$.\footnote{In this study, we measured the irradiance at the line core of \ion{He}{1} 10830 {\AA} and found a weak anti-correlation, while \citet{2007ApJ...657.1137L} showed a strong correlation between its equivalent width and the solar activity.} Compared to the previous study (Figure 3 in \citetalias{2022ApJ...927..179T}), the increase in the number of observables, especially for the transition region temperatures, allows for scrutinizing the change of $\alpha$ from the chromosphere to the corona. For the coronal temperatures, $\alpha >1$ for most observables, which is in agreement with many previous studies (see Section \ref{sec:intro}). However, for Hinode/XRT, $\alpha$ was slightly below unity owing to several possible reasons. For example, the field of view of XRT was only about $2048\arcsec\times2048\arcsec$, and hence, if there is a bright coronal structure outside the limb, XRT may miss its contribution and underestimate irradiance, especially during the solar maximum. The exclusion of images that contain saturated pixels due to flares may also lead to the underestimation of irradiance. Furthermore, the combination of filters used to create the XRT light curve was changed, making it difficult to compare the long-term evolution. \ion{Fe}{12} 1349 {\AA} also had a coronal formation temperature at $\log{(T/{\rm K})}=6.2$, but $\alpha$ was well below unity, even smaller than the chromospheric line in the same wavelength range. This may be attributed to the fact that this line is much weaker than the other lines, owing to which the irradiance cannot be easily determined. The result that the $\alpha$ values for most chromospheric lines take less than unity also supports the previous analyses (see Section \ref{sec:intro}). However, it is newly found that most of the transition region lines also take $\alpha <1$ as in the chromosphere. Herein the formation temperature of \ion{He}{1} 10830 {\AA} was set to $\log{(T/{\rm K})}=4.2$, however, it should be noted that this line was formed by the combination of multiple mechanisms \citep[e.g.,][]{1997ApJ...489..375A}: (1) EUV photons in the corona invade the upper chromosphere and photoionize the neutral He atoms. When the generated He ions are recombined, they form a group of \ion{He}{1} lines; (2) When electrons with temperatures of 20,000 K or higher collide with the He atoms between the chromosphere and corona, collisional excitation occurs, and as the electrons return to the ground state, \ion{He}{1} lines are produced. Therefore, the fact that $\alpha$ of \ion{He}{1} 10830 {\AA} is close to the coronal values (i.e., $\alpha >1$) indicates that the mechanism (1) is more effective. This may also be related to that the other He lines (\ion{He}{2} 256 {\AA} and 304 {\AA}) show $\alpha$ values that are above unity. \subsection{Wavelength Dependence}\label{subsec:wavelength} Figure \ref{fig:pl2} shows the dependence of the power-law index $\alpha$ on the spectral line wavelength. As shown in Figure \ref{fig:pl}, \ion{He}{1} 10830 {\AA} was plotted despite its inverse proportionality against the solar activity proxies, while F10.7cm ($=10.7\times 10^{8}$ {\AA}) radio flux is shown in the infrared range for visualization purposes only. As seen in the figure, $\alpha$ displays a V-shaped profile with the apex located at the near UV range around 1000--2000 {\AA}. The value increases from below unity to above unity as the wavelength shifts from near UV both towards the EUV and X-rays and the infrared and radio waves. This is because the corresponding spectral lines and bands are sensitive to increasingly higher temperature plasmas. \section{Applications: Reconstruction of Solar XUV Irradiances}\label{sec:application} We determined the scaling laws between the solar activity proxies and irradiances of various lines and bands. That is, using the obtained $\alpha$ and $\beta$ values, it is possible to calculate the irradiance of these lines/bands from any of these proxies, expressed as: \begin{eqnarray} F=10^{\beta}(P-P_{0})^{\alpha}+F_{0}. \end{eqnarray} We can even estimate the irradiances from proxies for targets having no observation of the upper atmospheres. For example, irradiances can be estimated from surface magnetic field distributions calculated by the solar dynamo models or surface flux transport models, the surface magnetic field distribution acquired by the stellar Zeeman-Doppler Imaging, or the starspot sizes estimated from the visible light curve of the Sun-like stars. XUV irradiance estimates are often based on scaling relationships with other spectral lines or bands \citep[e.g.,][see also Section \ref{sec:intro}]{2007SpWea...5.7005C,2020SpWea..1802588C,2014ApJ...780...61L}. However, since the model in this work uses the daily solar activity proxies, although it cannot be used for short time scales like solar and stellar flares, longer-term variations such as rotational modulations and solar cycle variations can be estimated based on more physical relationships, i.e., atmospheric heating owing to surface magnetic field. To demonstrate this approach, Figure \ref{fig:bc} shows the ``backcasting'' of solar irradiances in the past centuries based on the long-term solar observations. In fact, the reconstruction of spectral radiations using the historical records has been one of the key scientific targets for understanding the atmospheric/chemical interactions of the Earth and planets \citep[see, e.g.,][]{2021SoPh..296...60K}. Here we used the total spot number (Section \ref{subsec:wdc}) since January 1749, total spot area (Section \ref{subsec:noaa}) since May 1874, and the F10.7cm radio flux (Section \ref{subsec:f107}) since February 1947. The irradiances that were reconstructed are those whose scaling laws were verified by a comparison with stellar data in \citetalias{2022ApJ...927..179T}, i.e., X-rays 5.2--124 {\AA}, \ion{Fe}{15} 284 {\AA}, Ly$\alpha$, and \ion{Mg}{2} k 2796 {\AA}. Although it is possible to reconstruct daily irradiances by using the daily proxy data, for a better visualization, we synthesized monthly light curves based on the monthly-averaged proxies. The relative difference between two of the synthesized irradiances is expressed as: \begin{eqnarray} d_{\rm TSN, TSA}=\frac{|F^{\rm TSN}-F^{\rm TSA}|}{(F^{\rm TSN}+F^{\rm TSA})/2}, \end{eqnarray} \begin{eqnarray} d_{\rm TSN, F10.7}=\frac{|F^{\rm TSN}-F^{\rm F10.7}|}{(F^{\rm TSN}+F^{\rm F10.7})/2}, \end{eqnarray} where $F^{\rm TSN}$, $F^{\rm TSA}$, and $F^{\rm F10.7}$ are the irradiances based on the total sunspot number, total sunspot area, and the F10.7cm radio flux, respectively. For the period during which the irradiances are derived from multiple proxies, the median values of the relative differences are $d_{\rm TSN, TSA}=14.5$\% and $d_{\rm TSN, F10.7}=52.9$\% for X-rays 5.2--124 {\AA}, $d_{\rm TSN, TSA}=22.2$\% and $d_{\rm TSN, F10.7}=63.7$\% for \ion{Fe}{15} 284 {\AA}, $d_{\rm TSN, TSA}=4.3$\% and $d_{\rm TSN, F10.7}=18.6$\% for Ly$\alpha$, and $d_{\rm TSN, TSA}=2.7$\% and $d_{\rm TSN, F10.7}=12.3$\% for \ion{Mg}{2} k 2796 {\AA}. These values, up to approximately 20\% for the transition-region and chromospheric lines and up to approximately 50\% for the coronal lines, can be referred to as typical errors when reconstructing irradiances using this method. Possible sources of errors for this irradiance reconstruction method include the errors in the proxy data \citep[see,][for errors in the sunspot number data]{2014SSRv..186...35C} and those in the power-law indices (i.e. $\alpha$ and $\beta$ in Tables \ref{tab:powerlaw_magc} to \ref{tab:powerlaw_srf}). Also, the fact that the power laws were derived only for the Cycle 24, which showed a very weak activity, may cause additional errors (see Section \ref{sec:summary} for further discussion). \section{Summary and Discussion}\label{sec:summary} In this study, we used the methodology described in \citetalias{2022ApJ...927..179T} to derive the scaling laws between the solar activity proxies (not only the radial magnetic flux but also the LOS flux, total sunspot number, total sunspot area, and the F10.7cm flux) and the irradiances of various spectral lines and bands. By further increasing the number of lines, especially of the transition region temperatures, we investigated the variation of power-law index $\alpha$ from the chromospheric to coronal temperatures, as shown in Figure \ref{fig:pl}. Our results provide the framework for estimating spectral irradiances from the proxy data. If one of the five proxies is given, one can estimate the line/band irradiances by using the power-law indices $\alpha$ and offsets $\beta$ provided in Tables \ref{tab:powerlaw_magc} to \ref{tab:powerlaw_srf}. For instance, we can estimate the irradiances from the total magnetic flux or total sunspot area of the Sun-like stars obtained from modeling and observations. To demonstrate the usefulness of this method, we reconstructed selected irradiances over the past centuries based on the historical records of solar observations (Figure \ref{fig:bc}). The relative differences between the synthesized irradiances was up to 20\% for the chromospheric and transition-region lines and up to 50\% for the coronal lines, which can be considered as the typical errors of the method. It is also necessary to specify the limitations of this method. The scaling laws were obtained from daily solar synoptic data over the last decade. Therefore, this method can only be applied for reconstructing irradiance variations of time scales longer than a day (i.e., quasi-stationary component) and not for synthesizing transient brightenings, such as solar and stellar flares (time scales of tens of minutes to hours). Additionally, because the last 10 years was one of the weakest solar activity cycles in the last few hundred years \citep[e.g.,][]{2020JSWSC..10...60P}, one has to extrapolate the scalings to obtain the irradiances of stronger cycles, as shown in Figure \ref{fig:bc}. In addition, irradiances can only be reproduced for stars with almost the same parameters as the current Sun. For example, the chemical abundance is fixed to that of the current Sun, and hence, reproducing irradiances of stars with significantly different abundances can be challenging. Nonetheless, it has been verified by \citetalias{2022ApJ...927..179T} that the scalings are universal among G-type stars, regardless of age or activity level. Therefore, the method discussed here can be used as far as the irradiance synthesis is conducted for the main-sequence G-dwarfs. Another limitation is that H$\alpha$ and \ion{Ca}{2} 8542 {\AA} cannot be reproduced as they brighten only in the declining phase of the solar cycle (Figure \ref{fig:lc}) and show weak CCs against activity proxies. Based on the Sun-as-a-star monitoring, \citet{2019A&A...627A.118M} reported that H$\alpha$ and other Balmer lines (H$\beta$ and H$\gamma$) are inversely correlated with the sunspot number and \ion{Ca}{2} K intensity. However, these authors only used data over three years. \citet{2009A&A...501.1103M} analyzed the data for several cycles and showed that although H$\alpha$ and \ion{Ca}{2} indices were positively correlated with the activity cycle in the long term, their CCs varied with the phase of the activity cycle. Therefore, the negative or no correlations for H$\alpha$ and \ion{Ca}{2} 8542 {\AA} found in this study may be attributed to the timescale or the activity phase of our sampling. It is also important to analyze spatially resolved data of the Sun to investigate how individual structures such as plages, filaments, and sunspots affect the chromospheric lines and spectra of the Sun as a whole \citep[e.g.,][]{2022A&A...661A.107D}. For the active G-type main-sequence stars that emit superflares, \citet{2019ApJ...876...58N} found a strong positive correlation between the brightness variation amplitude of visible light curves, which is an indicator of the starspot size, and the \ion{Ca}{2} 8542 {\AA} and H \& K intensities, as opposed to the expectation from this study. It is possible that the solar \ion{Ca}{2} 8542 {\AA} line fluxes are in the saturated regime in the atmospheres of solar-like stars, where they only show a weak dependence on the \ion{Ca}{2} K intensity \citep[see Figure 5 of][]{2012PASJ...64..130T}. \citet{2007A&A...469..309C}, who studied various stars ranging from F to M, showed that although H$\alpha$ and \ion{Ca}{2} H \& K were strongly correlated as a whole, this general trend was lost for individual stars. \citet{2022A&A...662A..41R} showed that H$\alpha$ in M-dwarfs had a positive correlation with the magnetic flux with an exponent of $\alpha = 1.43$. However, these authors noted that H$\alpha$ requires a minimum average magnetic field strength of several hundred G to ensure a detectable emission. Therefore, the chromospheric lines that appear in absorption on the Sun may have different formation mechanisms compared to the chromospheres of active stars. One possible explanation of this difference can be attributed to the frequency of occurrence of coronal flare events in active stars, which can heat the chromosphere via electron beams and excite hydrogen line emissions. In contrast, frequent solar microflares can mostly heat the transition region and do not contribute much to the chromospheric heating. This points to the importance of estimating spectral irradiances using the scaling laws as well as examining the relationships between starspots and the upper atmospheric variations by actually conducting long-term monitoring of stars at multiple wavelengths. To this end, \citet{2020ApJ...902...36T} proposed the methodology of estimating the size of stellar active regions by acquiring the light curves for many different rotational phases, not only in the visible band but also in the XUV band. Recently, it has become possible to track the growth of starspots based on the long-term changes of dips in stellar visible light curves and the starspot mapping technique \citep{2019ApJ...871..187N,2020ApJ...891..103N}; however, if there is contemporaneous XUV observation, we can also obtain clues to understand how active region atmospheres evolve. For instance, whether the rotational modulations of visible light and H$\alpha$ are correlated, uncorrelated, or anti-correlated is a key to probe the chromospheric activity of starspots \citep{2021PASJ...73...44M,2022ApJ...926L...5N,2022A&A...663A..68S}, which should be expanded to the XUV range. In this study, we derived the scaling laws between the solar activity proxies and the irradiances. However, the mutual relations between irradiances of different spectral lines may also be utilized to investigate the physical processes of the solar and stellar atmospheres \citep[e.g.,][]{2014ApJ...780...61L,2020ApJ...902....3L,2017ApJ...843...31Y,2018ApJS..239...16F}. In \citetalias{2022ApJ...927..179T}, the power-law indices were also obtained by dividing the total 10-year period into four phases according to the solar activity, and it was found that $\alpha$ was smallest during the cycle maximum and largest during the minimum. Although the $\alpha$ values were derived only for the entire 10-year period in this study, it is possible that $\alpha$ depends on activity phase, and this may cause differences between the Sun and other stars. Future studies on such mutual relations and cycle dependence are expected. Another possible direction is to reconstruct the XUV irradiances using radio fluxes. Currently, observations of the radio photosphere are performed for a limited sample of G-type stars \citep{2014ApJ...788..112V}. However, the next generation Very Large Array (ngVLA) can supposedly detect radio photospheres of many more main-sequence stars \citep{2019BAAS...51c.243C}. In this study, strong correlations were found between the F10.7cm (2.8 GHz) radio flux and the XUV irradiances, which indicates that the radio fluxes can be useful proxies for reconstructing stellar XUV line fluxes. Understanding of the basal fluxes requires further investigations. For the late-type stars, \citet{1987A&A...172..111S} found the power-law scalings of the X-ray emission with the \ion{Ca}{2} and \ion{Mg}{2} emissions by subtracting the basal fluxes for the chromospheric lines. \citet{1987A&A...172..111S} interpreted the basal fluxes as a component due to pure acoustic heating of unmagnetized atmosphere. In this study, however, we defined the basal fluxes as the medians of unspotted values in the minimum of the solar activity cycle and, by subtracting the basal fluxes from the light curves, we derived the power-law scalings. Magnetic fluxes are ubiquitously distributed even in the quiet Sun during the cycle minimum, causing the atmospheric heating above. Therefore, in this study, the basal fluxes can be represented by the minimum magnetic flux and associated heating \citep{2012A&A...540A.130S}. This view may be supported by the fact that non-thermal broadening is detected for the chromospheric and transition-region lines during the cycle minimum \citep[e.g.,][]{2021ApJ...916...36A} \citep[for further discussions, see][]{2015RSPTA.37340259T,2019LNP...955.....L}. In \citetalias{2022ApJ...927..179T} and this study, the scalings were examined only for selected lines and bands of the chromospheric to coronal temperatures. However, it is important to extend these relations for the continuum components by evaluating the scaling relationships between the entire XUV spectrum and the activity proxies, such as the total magnetic flux, for every single wavelength bin, including the continuum, rather than extracting the emission lines only. This would make it possible to reconstruct the whole XUV spectra for the F-, G-, and K-type stars. Radiative energy distributions over the wavelength for planet-hosting stars can provide critical information for assessing the efficiency of atmospheric escape from the (exo)planets orbiting them. Derivation of such scalings requires further analysis that we defer to forth-coming publications. \begin{acknowledgments} The authors acknowledge the comments suggested by the referee, which helped to improve the quality of the paper. The authors would like to thank Dr. Aki Takeda for useful comments on the XRT synoptic data. Data are courtesy of the science teams of SDO, WDC-SILSO, USAF/NOAA, SORCE, GOES, Hinode, DRAO, and SOLIS. HMI is an instrument on board SDO, a mission for NASA's Living With a Star program. Hinode is a Japanese mission developed and launched by ISAS/JAXA, collaborating with NAOJ as a domestic partner, NASA and STFC (UK) as international partners. Scientific operation of the Hinode mission is conducted by the Hinode science team organized at ISAS/JAXA. This team mainly consists of scientists from institutes in the partner countries. Support for the post-launch operation is provided by JAXA and NAOJ (Japan), STFC (U.K.), NASA, ESA, and NSC (Norway). ISS data were acquired by SOLIS instruments operated by NISP/NSO/AURA/NSF. This work was supported by JSPS KAKENHI Grant Nos. JP20KK0072 (PI: S. Toriumi), JP21H01124 (PI: T. Yokoyama), JP21H04492 (PI: K. Kusano), JP21J00316 (PI: K. Namekata), and JP21J00106 (PI: Y. Notsu). V.S.A. was supported by the GSFC Sellers Exoplanet Environments Collaboration (SEEC), which is funded by the NASA Planetary Science Division's Internal Scientist Funding Model (ISFM), NASA's TESS Cycle 1, HST Cycle 27 and NICER Cycle 2 project funds. \end{acknowledgments} \vspace{5mm} \bibliography{toriumi2022}{} \bibliographystyle{aasjournal}

Title: Fundamental Reference AGN Monitoring Experiment (FRAMEx) III: Radio Emission in the Immediate Vicinity of Radio Quiet AGNs

Abstract: We present follow-up results from the first Fundamental Reference AGN Monitoring Experiment (FRAMEx) X-ray/radio snapshot program of a volume-complete sample of local hard X-ray-selected active galactic nuclei (AGNs). Here, we added 9 new sources to our previous volume-complete snapshot campaign, two of which are detected in the 6 cm Very Long Baseline Array (VLBA) observations. We also obtained deeper VLBA observations for a sample of 9 AGNs not detected by our previous snapshot campaign. We recovered 3 sources with approximately twice the observing sensitivity. In contrast with lower angular resolution Very Large Array (VLA) studies, the majority of our sources continue to be undetected with the VLBA. The sub-parsec radio (6 cm) and X-ray (2-10 keV) emission show no significant correlation, with L_R/L_X ranging from 10^-8 to 10^-4, and the majority of our sample lies well below the fiducial 10^-5 relationship for coronal synchrotron emission. Additionally, our sources are not aligned with any of the proposed "fundamental" planes of black hole activity, which purport to unify black hole accretion in the M_BH-L_X-L_R parameter space. The new detections in our deeper observations suggest that the radio emission may be produced by the synchrotron radiation of particles accelerated in low luminosity outflows. Non-detections may be a result of synchrotron self-absorption at 6 cm in the radio core, similar to what has been observed in X-ray binaries (XRBs) transitioning from the radiatively inefficient state to a radiatively efficient state.

https://export.arxiv.org/pdf/2208.05848

command. \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \newcommand{\borus}{\texttt{Borus}} \newcommand{\mytorus}{\texttt{MYTorus}} \newcommand{\xspec}{\textsc{xspec}} \newcommand{\chandra}{\textit{Chandra}} \newcommand{\nustar}{\textit{NuSTAR}} \newcommand{\suzaku}{\textit{Suzaku}} \newcommand{\xmm}{\textit{XMM-Newton}} \newcommand{\mbh}{$M_\mathrm{BH}$} \newcommand{\nbmc}{$N_\mathrm{BMC}$} \usepackage{booktabs} \usepackage{multirow} \usepackage{array} \usepackage{hyperref} \newcolumntype{H}{>{\setbox0=\hbox\bgroup}c<{\egroup}@{}} \usepackage{efbox,graphicx} \usepackage{stackengine,xcolor} \efboxsetup{linecolor=black,linewidth=1pt} \usepackage[caption=false]{subfig} \definecolor{nsgreen}{rgb}{0.1,0.5,0.1} \newcommand{\njs}[1]{\textcolor{nsgreen}{#1}} \fboxsep=0.1mm% \fboxrule=0.6pt% \shorttitle{FRAMEx III: High-sensitivity VLBA observation of radio quiet AGNs} \shortauthors{Shuvo et al.} \graphicspath{{./}{figures/}} \begin{document} \title{Fundamental Reference AGN Monitoring Experiment (FRAMEx) III:\\[0.05cm] Radio Emission in the Immediate Vicinity of Radio Quiet AGNs} \correspondingauthor{Onic Islam Shuvo} \email{oshuvo@gmu.edu} \author[0000-0003-4727-2209]{Onic I. Shuvo} \affiliation{U.S. Naval Observatory, 3450 Massachusetts Ave NW, Washington, DC 20392-5420, USA} \affiliation{Department of Physics and Astronomy, George Mason University, MS3F3, 4400 University Drive, Fairfax, VA 22030, USA} \author[0000-0002-4146-1618]{Megan C. Johnson} \affiliation{U.S. Naval Observatory, 3450 Massachusetts Ave NW, Washington, DC 20392-5420, USA} \author[0000-0002-4902-8077]{Nathan J. Secrest} \affiliation{U.S. Naval Observatory, 3450 Massachusetts Ave NW, Washington, DC 20392-5420, USA} \author[0000-0002-8818-9009]{Mario Gliozzi} \affiliation{Department of Physics and Astronomy, George Mason University, MS3F3, 4400 University Drive, Fairfax, VA 22030, USA} \author[0000-0002-3365-8875]{Travis C. Fischer} \affiliation{AURA for ESA, Space Telescope Science Institute, Baltimore, MD, USA, 3700 San Martin Drive, Baltimore, MD 21218, USA} \author[0000-0002-8736-2463]{Phillip J. Cigan} \affiliation{Department of Physics and Astronomy, George Mason University, MS3F3, 4400 University Drive, Fairfax, VA 22030, USA} \affiliation{U.S. Naval Observatory, 3450 Massachusetts Ave NW, Washington, DC 20392-5420, USA} \author[0000-0002-0819-3033]{Luis C. Fernandez} \affiliation{U.S. Naval Observatory, 3450 Massachusetts Ave NW, Washington, DC 20392-5420, USA} \affiliation{Department of Physics and Astronomy, George Mason University, MS3F3, 4400 University Drive, Fairfax, VA 22030, USA} \author[0000-0002-5604-5254]{Bryan N. Dorland} \affiliation{U.S. Naval Observatory, 3450 Massachusetts Ave NW, Washington, DC 20392-5420, USA} \keywords{Radio astrometry (1337), Active galaxies (17), Radio active galactic nuclei (2134), X-ray active galactic nuclei (2035)} \section{Introduction} \label{sec:intro} The Fundamental Reference AGN Monitoring Experiment, or FRAMEx, is a research collaboration between the U.S. Naval Observatory (USNO) and other institutions that aims to monitor and characterize the physical properties of Active Galactic Nuclei (AGNs) powered by accretion onto supermassive black holes (SMBHs) at the centers of galaxies. To understand the physical nature and features of SMBHs, their surrounding media, and mutual interactions with the host galaxies that influence their luminosity and variability, FRAMEx used ground and space-based telescopes to observe AGNs in the X-ray and radio wavelengths at multiple time epochs~\citep{Dorland_2020jsrs.conf..165D}. Periods of AGN activity have a profound impact on the nuclear environment of galaxies, heating, ionizing, and blowing out gas and dust, regulating star-formation and gas accumulation. This feedback process in turn regulates the accretion rates of AGNs, leading to a positive correlation between the masses of SMBHs and their host galaxy stellar bulges over cosmic time \citep[for a review, see][]{Kormendy_2013ARA&A..51..511K}. Despite the substantial progress that has been made in recent years, the accretion mechanism of AGNs is still an active area of research, in part due to the wide range of morphologies and spectral energy distributions that AGNs exhibit, from compact, thermally-dominant quasars to large elliptical galaxies displaying jets and radio lobes extending to Mpc scales. Correlations between emission mechanisms at various wavelengths and properties of the black hole itself, such as the mass, have led to the exciting prospect that black holes exhibit self-similar accretion properties for all masses, from stellar-mass black holes in X-ray binaries (XRBs), to billion-plus \(\textup{M}_\odot\) SMBHs at the centers of the largest galaxies. A notable attempt to unify black hole accretion is the ``Fundamental Plane of Black Hole Activity'' \citep[e.g.,][hereafter the FP]{Merloni_2003MNRAS.345.1057M,Gultekin_2009ApJ...706..404G}, which purports to place all black holes in a single accretion parameter space, with one axis being X-ray emission, another being radio, and the third the black hole mass. The apparent FP relation from the previous lower angular resolution studies needs to be investigated further to understand the radiative processes and the physical environment very close to the black hole using observations at finer physical scales. To achieve this goal, in \citet[][hereafter, Paper~I]{Fischer_2021ApJ...906...88F}, we obtained simultaneous Swift X-ray Telescope (XRT) and Very Long Baseline Array (VLBA) radio observations for a snapshot (1-hour on-source) survey of 25 nearby AGNs ($<40$~Mpc) making up a volume-complete ($L_\mathrm{14-195~keV}>10^{42}$~erg~s$^{-1}$) sample at our declination range from $-30\arcdeg < \delta < +60\arcdeg$. One of the surprising results from Paper~I was that, despite being at the same radio frequency (C-band: 6~cm) and X-ray energies (2--10~keV) that the % fundamental plane of black hole activity was defined at, we found that the FP breaks down at the angular resolution of the VLBA ($\sim3$~mas), calling into question its validity. Archival VLA data show that these objects do align with the FP at lower angular resolution ($\sim500$~mas, or $\gtrsim30-100$~pc), suggesting that whatever is responsible for the apparent correlation between SMBH mass and X-ray/radio emission paradoxically occurs on larger physical scales, at least beyond the largest angular scale of the VLBA observations ($\sim50$~mas, or $\gtrsim3-10$~pc for the typical distances of the sample), far larger than the scales of the AGN corona where X-ray emission is generally considered to arise. On the other hand, for radio loud (RL) AGNs, X-ray emission could be a superposition of different components such as synchrotron radiation from a jet, synchrotron self-Compton, emission from the accretion flow and inverse Compton scattering of lower energy photons off a corona \citep{Plotkin_2012MNRAS.419..267P}, but for our sample AGNs, which are almost entirely radio-quiet (RQ) (the exception is NGC~1052), the most likely explanation for the X-ray emission is the accretion disk corona and so far no current or planned future missions can resolve the X-ray emitting corona region in AGNs \citep[although see][]{2021ExA....51.1081U}. Despite the proximity of AGNs in the volume-complete sample and their selection at hard X-rays \citep{Oh_2018ApJS..235....4O}, only 9 out of the 25 AGNs were detected in the VLBA observations with a sensitivity level of $\sim20$~$\mu$Jy bm$^{-1}$ from Paper~I, raising the prospect of true ``radio-silent'' AGNs. In this paper, we present results from a follow-up VLBA observing campaign at C-band to observe 9 out of 16 initially non-detected AGNs with a much deeper sensitivity of $\sim10$~$\mu$Jy bm$^{-1}$ (4 hours on-source per target), to probe whether or not we were sensitivity limited in our initial snapshot campaign. Moreover, we added 9 new snapshot observations similar to our Paper I campaign of 1-hour on-source time integrations. We discuss the correlation between radio and X-ray luminosities ($L_\mathrm{6 cm}/L_\mathrm{2-10keV}$) in our sub-parsec scale study of higher sensitivity radio observations. Additionally, we explore radio-loudness as a function of a source's accretion rate and the FP proposed for the highly accreting black holes. To conclude, we discuss a physically motivated model to describe the non-detections as a result of the self-synchrotron absorption of particles accelerated in shocks or outflows. % The sample selection, reliable measurements of X-ray data, higher angular resolution VLBA radio observations, and data calibration details are discussed in Section~\ref{sec:Methodology}, and we present our results and discussion in understanding the origin of the radio emission for this FRAMEx sample comprised of radio-quiet AGNs in Section~\ref{sec:res} and Section~\ref{sec:dis}, respectively. \section{Methodology} \label{sec:Methodology} \subsection{Sample Selection} \label{subsection: Sample Selection} \begin{deluxetable*}{lrrlcccHHH} \caption{Observation Source List} \tablehead{\colhead{Target} & \colhead{R.A.\ (ICRS)} & \colhead{Decl.\ (ICRS)} & \colhead{Type} & \colhead{Redshift} & \colhead{Distance} & \colhead{log($M_\mathrm{BH}$)} \\%& \colhead{log($L_\mathrm{H\alpha}$)}\\% & \colhead{$\mathrm{SFR_{max}}$} & \colhead{$\mathrm{SNR_{max}}$}\\ [-0.3cm] & \colhead{(deg)} & \colhead{(deg)} & & & \colhead{(Mpc)} & \colhead{[$M_{\sun}$]}}% \startdata \hline New Snapshots$^\star$ \\ \hline MCG-05-23-016 & 146.91720558 & $-$30.94884951 & Sy1.9 & 0.0085 & 36.6 & 7.65 & 39.71 & & \\ NGC 2273 & 102.53602642 & 60.84582513 & Sy2 & 0.0061 & 26.2 & 7.99 & $\ldots$ & & \\ NGC 3147 & 154.22355150 & 73.40075317 & Sy2 & 0.0093 & 40.1 & 8.81 & $\ldots$ & & \\ NGC 3516 & 166.69775929 & 72.56867643 & Sy1.2 & 0.0088 & 37.9 & 7.39 & 39.63 & & \\ NGC 4102 & 181.59597242 & 52.71100601 & Sy2 & 0.0028 & 12.0 & 7.84 & 39.75 & & \\ NGC 4138 & 182.37416229 & 43.68524117 & Sy2 & 0.0030 & 12.9 & 7.71 & 38.97 & & \\ NGC 5728 & 220.59945800 & $-$17.25306162 & Sy1.9 & 0.0093 & 40.1 & 8.25 & 39.68 & & \\ NGC 7172 & 330.50781558 & $-$31.86965392 & Sy2 & 0.0087 & 37.5 & 8.15 & 39.52 & & \\ UGC 6728 & 176.31634979 & 79.68154046 & Sy1.2 & 0.0065 & 28.0 & 5.79 & 39.84 & & \\ \hline Deep Integrations$^\dagger$ &&&&&&\\ \hline NGC 1320 & 51.2028681 & $-$3.04226840 & Sy2 & 0.0089 & 38.4 & 7.96 & 40.49 & 0.24 & $ 4.9\times10^{-3}$\\ NGC 2782 & 138.5212787 & $+$40.11369022 & Sy2 & 0.0085 & 36.6 & 6.07 & 41.51 & 2.62 & $ 5.2\times10^{-2}$\\ NGC 3081 & 149.8731005 & $-$22.82631476 & Sy2 & 0.0080 & 34.5 & 7.74 & 40.97 & 0.74 & $ 1.5\times10^{-2}$\\ NGC 3089 & 149.9028701 & $-$28.33129443 & Sy2? & 0.0090 & 38.8 & 6.55 & \nodata & \nodata & ~~\nodata \\ NGC 4388 & 186.4449188 & $+$12.66215153 & Sy2 & 0.0084 & 36.2 & 6.94 & 41.57 & 2.93 & 5.9$\times10^{-2}$\\ NGC 4593 & 189.9143400 & $-$5.34417010 & Sy1 & 0.0090 & 38.8 & 6.88 & 40.65 & 0.35 & 7.0$\times10^{-3}$\\ NGC 6814 & 295.6690092 & $-$10.32345792 & Sy1 & 0.0052 & 22.4 & 7.04 & 39.35 & 0.02 & 3.5$\times10^{-4}$\\ NGC 7314 & 338.9424567 & $-$26.05043820 & Sy1.9 & 0.0048 & 20.6 & 6.76 & 39.68 & 0.04 & 7.5$\times10^{-4}$\\ NGC 7465 & 345.5039963 & $+$15.96477472 & Sy2 & 0.0066 & 28.4 & 6.54 & 40.44 & 0.22 & 4.3$\times10^{-3}$\\ \enddata \tablecomments{$^\star$Observation information for the additional 9 targets observed after Paper I \citep{Fischer_2021ApJ...906...88F}.\\ $^\dagger$Subset of sample chosen from Paper I for the follow up VLBA 4 hour on source integration time observation.} \label{tab:sample} \end{deluxetable*} Since the publication of Paper~I, we have added 9 AGNs to our sample, each observed with the VLBA for 1 hour of on-source integration following the same observing setup as described in Paper~I. These objects are comprised of four sources above $+60\arcdeg$ (to $\sim +80\arcdeg$) declination, as the original $+60\arcdeg$ limit was based on an initial intent to monitor these objects with The United Kingdom Infra-Red Telescope (UKIRT), which was ultimately not used in the study, plus five sources that are now found to fall within the 40 Mpc distance limit owing to improved redshifts. These 9 sources are hereafter called ``additional snapshot" observations. We proposed for Target of Opportunity (ToO) time to obtain simultaneous Neil Gehrels Swift Observatory X-ray Telescope (\emph{Swift} XRT) observations for these 9 sources in the same way as we did in Paper~I. % Alongside the new observations of these additional sources, we re-observed 9 of the 16 non-detected AGNs from Paper~I using the VLBA, with deeper 4-hour on-source integration times % (see Table \ref{tab:sample}). In order to maximize the likelihood of recovering emission on the milli-arcsecond angular scales of the VLBA we used the archival hundred-parsec-scale C-band, A-array VLA radio structure maps \citep[see Fig.~3 in][]{Fischer_2021ApJ...906...88F} to select the most point like AGNs for this deeper observation campaign. We illustrate our selection method in Figure~\ref{fig:detected_sample}, and sample sources along with their global properties are listed in Table \ref{tab:sample}. % \subsection{X-ray Data} \begin{deluxetable*}{lcc|cc|ccH} \setlength{\tabcolsep}{4pt} \caption{X-ray Data} \tablehead{ \colhead{~} & \colhead{BAT} & \colhead{~} \vline & \colhead{XRT} & \colhead{~} \vline & \colhead{NuSTAR$^\dagger$}& & \colhead{~}\\ [-0.1cm] \hline \colhead{Target~~~~~~~~~} & \colhead{F$_{14-195~keV}$} & \colhead{L$_{14-195~keV}$} \vline & \colhead{F$_{2-10~keV}$} & \colhead{L$_{2-10~keV}$} \vline & \colhead{F$_{2-10~keV}$} & \colhead{L$_{2-10~keV}$} & \\%\colhead{M$_{BH}$}\\ [-0.1cm] \colhead{~} & \colhead{$\times~10^{-11}$} & \colhead{$\times~10^{42}$} \vline& \colhead{$\times~10^{-11}$} & \colhead{$\times~10^{42}$} \vline& \colhead{$\times~10^{-11}$} & \colhead{$\times~10^{42}$} & \\%\colhead{$\times~10^{6}$}\\ [-0.1cm] \colhead{~} & \colhead{(erg s$^{-1}$ cm$^{-2}$)} & \colhead{(erg s$^{-1}$)} \vline& \colhead{(erg s$^{-1}$ cm$^{-2}$)} & \colhead{(erg s$^{-1}$)} \vline& \colhead{(erg s$^{-1}$ cm$^{-2}$)} & \colhead{(erg s$^{-1}$)} & }% \startdata \hline New Snapshots & & & & & & & \\ \hline MCG-05-23-016 & $13.9^{+0.30}_{-0.30}$& $22.3^{+0.50}_{-0.50}$ & $7.70^{+0.50}_{-0.50}$~~~ & $12.34^{+0.80}_{-0.80}$ & $9.54\pm 0.02$ & $15.3\pm0.03$ & $13.5\pm0.5$ \\ NGC 2273 & $0.30^{+0.20}_{-0.20}$& $0.25^{+0.20}_{-0.20}$ & $0.020^{+0.007}_{-0.005}\,^{**}$ & $0.016^{+0.006}_{-0.004}$ & $3.60\pm1.30$ & $32.0\pm11.0$ & $19.5\pm5.8$ \\ NGC 3147 & $0.80^{+0.30}_{-0.30}$& $1.50^{+0.60}_{-0.60}$ & $0.14^{+0.07}_{-0.05}$~~~ & $0.27^{+0.13}_{-0.10}$ & $0.28\pm0.04$ & $0.55\pm0.01$ & \nodata~~ \\%$0.55\pm0.21$^{*} \\ NGC 3516 & $10.9^{+0.30}_{-0.30}$& $18.7^{+0.50}_{-0.50}$ & $3.60^{+0.30}_{-0.30}$~~~ & $6.19^{+0.52}_{-0.52}$ & $0.60\pm0.05$ & $1.00\pm0.08$ & \nodata~~ \\%$1.4\pm0.5$^{*} \\ NGC 4102 & $1.20^{+0.10}_{-0.10}$& $0.21^{+0.02}_{-0.02}$ & $1.70^{+0.80}_{-0.60}\,^{**}$ & $0.29^{+0.14}_{-0.10}$ & $0.92\pm0.03$ & $0.16\pm0.01$ & \nodata~~ \\%$0.29\pm0.15$ \\ NGC 4138 & $2.80^{+0.40}_{-0.40}$ &$0.56^{+0.08}_{-0.08}$ & \nodata~~ & \nodata & \nodata & \nodata &\nodata~~ \\ NGC 5728 & $7.00^{+3.00}_{-2.00}$& $14.0^{+6.00}_{-4.00}$ & $5.00^{+3.00}_{-3.00}\,^{**}$ & $9.62^{+5.77}_{-5.77}$ & $4.12\pm0.08$ & $7.95\pm0.16$ & $8.3\pm2.9$ \\ NGC 7172 & $21.4^{+0.50}_{-0.50}$& $36.0^{+0.80}_{-0.80}$ & $0.40^{+0.30}_{-0.20}$~~~ & $0.67^{+0.50}_{-0.34}$ & $8.38\pm0.08$ & $14.0\pm0.10$ & $10.3\pm3.5$ \\ UGC 6728 & $2.70^{+0.20}_{-0.20}$ & $2.50^{+0.20}_{-0.20}$ & $0.60^{+0.10}_{-0.10}$~~~ & $0.56^{+0.09}_{-0.09}$ & $1.44\pm0.02$ & $1.35\pm0.02$ & $1.5\pm0.5$ \\ \hline Deep Integrations$^{**}$ & & & & & & & \\ \hline NGC 1320 & $1.34^{+0.04}_{-0.07}$ & $2.36^{+0.07}_{-0.12}$ & \nodata~~ & \nodata & $0.54\pm0.05$ & $0.95\pm0.09$& $1.0\pm0.4$\\ NGC 2782 & $1.20^{+0.04}_{-0.07}$ & $1.92^{+0.06}_{-0.11}$ & $0.13^{+0.03}_{-0.02}$ & $0.21^{+0.05}_{-0.03}$ & \nodata & \nodata& \nodata~~\\ NGC 3081 & $7.50^{+0.04}_{-0.08}$ & $10.68^{+0.06}_{-0.011}$ & $3.16^{+0.14}_{-0.27}$ & $4.50^{+0.20}_{-0.38}$ & $6.57\pm0.66$ &$9.26\pm0.09$& $9.5\pm3.6$\\ NGC 3089 & $0.70^{+0.02}_{-0.08}$ & $1.26^{+0.04}_{-0.14}$ & $0.20^{+0.02}_{-0.01}$ & $0.36^{+0.04}_{-0.02}$ & \nodata & \nodata & \nodata~~\\ NGC 4388 & $26.98^{+0.06}_{-0.06}$& $42.31^{+0.09}_{-0.09}$ & $2.60^{+0.80}_{-0.60}$ & $4.10^{+1.00}_{-0.90}$ & $1.39\pm0.06$ &$2.17\pm0.09$& $4.3\pm1.6$\\ NGC 4593 & $7.73^{+0.11}_{-0.11}$ & $13.93^{+0.20}_{-0.20}$ & $3.55^{+0.05}_{-0.05}$ & $6.40^{+0.09}_{-0.09}$ & $2.28\pm0.02$ &$4.10\pm0.04$& $3.9\pm1.3$\\ NGC 6814 & $5.52^{+0.12}_{-0.03}$ & $3.31^{+0.07}_{-0.02}$ & $2.09^{+0.03}_{-0.03}$ & $1.26^{+0.02}_{-0.02}$ & $3.48\pm0.03$ &$2.07\pm0.02$& $1.8\pm0.6$\\ NGC 7314 & $4.62^{+0.09}_{-0.04}$ & $2.35^{+0.05}_{-0.02}$ & $2.04^{+0.03}_{-0.03}$ & $1.04^{+0.02}_{-0.02}$ & $3.80\pm0.04$ &$1.93\pm0.02$& $1.3\pm0.5$\\ NGC 7465 & $1.90^{+0.05}_{-0.07}$ & $1.83^{+0.05}_{-0.07}$ & $1.12^{+0.02}_{-0.02}$ & $1.08^{+0.02}_{-0.02}$ & $1.26\pm0.02$ &$1.22\pm0.02$& $1.2\pm0.4$\\ \enddata \tablecomments{$^{**}$Archive fluxes with inflated errors based on 105-month \emph{Swift} BAT variability (see Section \ref{subsection: XRT} for details).\\ $^\dagger$Nustar X-ray spectra results for individual sources is added in Appendix~\ref{appendix:a}. } \label{tab:xray_data} \end{deluxetable*} \subsubsection{\emph{Swift} BAT Data} We use archival BAT data from the \emph{Swift} BAT 105-month Hard X-ray Survey \citep{Oh_2018ApJS..235....4O}, to compare the long term luminosity for each object. When solely examining the BAT data, $N_{\rm H}$ is difficult to constrain except in the case for Compton thick sources at this energy range. Therefore, we fit the BAT data with archival \nustar\ data. This improves the overall fit for each object and provided a more accurate normalization. We fit the BAT spectrum using a similar method as described in Paper~I using a simple power law model. However, for some of the targets this did not suffice and they needed an additional reflective component that allowed the cutoff energy to vary. Therefore, we utilized the model \texttt{pexrav} to accomplish this. See Table \ref{tab:xray_data} for resulting fluxes and luminosities. \subsubsection{\emph{Swift} XRT Data}\label{subsection: XRT} We obtained \emph{Swift} XRT time using the ToO proposal mechanism for the additional VLBA snapshot sources but we did not propose for simultaneous \emph{Swift} XRT observations for the deeper integration time objects shown in Table \ref{tab:sample}, and not all of the additional snapshot targets were observed with ToO due to competing schedule priorities. For the unobserved targets, archival XRT data is used. Analysis followed the same procedure as described in Section 2.6 of Paper~I. Unlike in Paper~I, for some of the additional snapshot targets and for all of the deep integration objects, the archival XRT data are not contemporaneous with the radio observations, so we add an additional uncertainty term in quadrature to the formal flux uncertainties, corresponding to each source variability. To estimate this, we use the source X-ray light curves from the BAT 105-month catalog, and for each source determine the intrinsic scatter term such that the reduced $\chi^2$ of the source light curve is unity. This is typically about $\sim0.1-0.2$~dex. % Table \ref{tab:xray_data} lists the X-ray fluxes and luminosities achieved from the spectral fits. \subsubsection{NuSTAR Data}\label{subsubsec:NuSTAR} Determining the spectral properties of the X-ray emitting corona in AGNs is of crucial importance to shed light on their central engine and on the link between accretion and ejection phenomena. In recent years, an important role has been played by the Nuclear Spectroscopic Telescope Array (\nustar), a focusing hard X-ray telescope launched in 2012 with a large effective area and excellent sensitivity in the energy range of 3--80 keV, which makes it possible to tighly constrain the contributions of absorption and reflection and hence measure the intrinsic properties of the primary X-ray emission. The goal of our \nustar\ data analysis was % to use the column densities, which are generally less variable, and to determine if XRT is seeing the intrinsic X-ray continuum or heavily absorbed or scattered continuum. In addition, we compared the \nustar\ data with Swift XRT data to assess whether the X-ray luminosities might be a source of FP discrepancy. Eight out of nine of the additional VLBA snapshot targets and seven out of nine of the deeper VLBA observation targets possess archival \nustar\ data (3--79~keV) from observations carried out recently or reprocessed very recently and therefore did not need to be reprocessed with the data analysis pipeline \texttt{nupipeline}. From the calibrated event files we extracted light curves and spectra, along with the RMF and ARF files necessary for the spectral analysis, using the \texttt{nuproduct} script. All spectra were binned with a minimum of 20 counts per bin using the HEASoft task \texttt{grppha} 3.0.1 for the $\chi^2$ statistics to be valid. We performed the X-ray spectral analysis using the \xspec\ \texttt{v.12.9.0} software package \citep{arnaud_1996ASPC..101...17A}, and the errors quoted on the spectral parameters represent the 1$\sigma$ confidence level. Our baseline model, expressed in the \xspec\ syntax , is: \begin{verbatim} phabs * (atable(Borus) + MYTZ*BMC) \end{verbatim} where the first absorption model \texttt{phabs} accounts for our Galaxy contribution, the \borus\ table model parametrizes the continuum scattering and fluorescent emission line components associated with the torus \citep{Balokovi__2018}, and \texttt{MYTZ} models the absorption and Compton scattering acting on the transmitted primary emission \citep{Murphy_2009MNRAS.397.1549M}, which is described by the Comptonization model \texttt{BMC} \citep{titarchuk_1997ApJ...487..834T}. For objects not classified as Seyfert 2, the \texttt{MYTZ} component is substituted by a \texttt{zphabs} model left free to vary. Depending on the complexity of the X-ray spectra, additional components (such as individual lines, additional absorption and scattering components, or the fraction of primary emission directly scattered towards the observer by a putative optically thin ionized medium) may be included and described in the text for individual sources (See Appendix~\ref{appendix:a}). \subsection{VLBA Data} \label{subsection: VLBA} \subsubsection{Observations} \begin{deluxetable*}{llcrccHH} \label{tab:radio_obs} \tablecaption{VLBA Observations} \tablehead{\colhead{Target} & \colhead{Frequency} & \colhead{Restoring Beam} & \colhead{Beam angle} & \colhead{RMS} & \colhead{Calibrator}\\% & \colhead{R.A.} & \colhead{Decl.}\\ [-0.2cm] & \colhead{(GHz)} & \colhead{($\alpha \times \delta$; mas)} & \colhead{(deg)} & \colhead{($\mu$Jy bm$^{-1}$)} & \colhead{IERS Name}} \startdata \hline New Snapshots$^{*}$ \\ \hline MCG-05-23-016 & 5.86747 & 9.37$\times$3.76 & $-$3.76 & 32.6 & J0948$-$2901 & \\ NGC 2273 & 5.84801 & 3.66$\times$1.58 &$-$23.6 & 31.2 & J0638+5933 & 102.536026417 & 60.8458251306 \\ NGC 3147 & 5.83532 & 3.53$\times$1.61 & 21.2 & 46.0 & J1027+7428 & 154.223551500 & 73.4007531694 \\ NGC 3516 & 5.86788 & 4.80$\times$2.29 & $-$23.0 & 29.5 & J1048+7143 & \\ NGC 4102 & 5.86787 & 3.20$\times$1.96 & $-$24.3 & 16.9 & J1200+5300 & 181.595972417 & 52.7110060111 \\ NGC 4138 & 5.86740 &4.43$\times$2.19 & $-$8.3 & 24.9 & J1221+4411 & \\ NGC 5728 & 5.86568 & 7.29$\times$3.11 & $-$9.2 & 28.1 & J1445$-$1629 & 220.599458000 & -17.2530616194 \\ NGC 7172 & 5.86068 & 3.61$\times$2.91 & 11.1 & 35.3 & J2158$-$3013 & 330.507815583 & -31.8696539194 \\ UGC 6728 & 5.86786 & $3.98\times1.86$ & $-$35.3 & 22.6 & J1058+8114 & 176.316349792 & 79.6815404611 \\ \hline Deep Integrations &&&&&&\\ \hline NGC 1320 & 5.86337 & 7.47$\times$2.02 &$-$16.1~~ & 16.2~~ & J0321$-$0526 & 50.499459824(8.17) & $-$5.4367857(222.2)\\ NGC 2782 & 5.86358 & 4.20$\times$2.04 & $-$8.5~~ & 13.7~~ & J0916+3854 & 139.203769060(7.04) & 38.9078184(115.6) \\ NGC 3081 & 5.86592 & 6.94$\times$3.40 & $-$2.5~~ & 16.7~~ & J1006$-$2159 & 151.693390348(4.80) &$-$21.9890028(109.5)\\ NGC 3089 & 5.87245 & 8.27$\times$3.82 & 4.8~~ & 20.4~~ & J1011$-$2847 & 152.772989020(7.74) &$-$28.7945604(246.8)\\ NGC 4388 & 5.86727 & 5.37$\times$2.49 & $-$18.1~~ & 19.4~~ & J1225+1253 & 186.265597247(4.1) & 12.8869831(138)\\ NGC 4593 & 5.86771 & 6.66$\times$2.37 & $-$12.9~~ & 14.7~~ & J1248$-$0632 & 192.095731923(4.98) & $-$6.5360605(154.7)\\ NGC 6814 & 5.80882 & 7.55$\times$3.13 &$-$8.5~~ & 27.1~~ & J1939$-$1002 & 294.988569046(4.17) &$-$10.04486683(98.1)\\ NGC 7314 & 5.80288 & 9.02$\times$2.94 &$-$8.3~~ & 27.8~~ & J2243$-$2544 & 340.860036591(6.44) &$-$25.7418576(189.9)\\ NGC 7465 & 5.80877 & 2.93$\times$1.40 &$-$5.5~~ & 43.0~~ & J2300+1655 & 345.179129648(7.01) & 16.9206644(148.4)\\ \enddata \tablecomments{$^{*}$Observed in September 2020. } \end{deluxetable*} Using the EVN Calculator, four hours of on-source integration time with the VLBA at 6 cm (C-band) produces a theoretical thermal noise rms of $\sim$10 $\mu$Jy/beam, more sensitive than the observations in our initial snapshot program by a factor of two, allowing for potential 10-sigma or higher detections for the proposed 9 deep integration targets. These new observations have $uv$-coverage similar to our initial snapshot program which interleaved observations of multiple targets, but now by sampling the $uv$-plane more finely over fuller, 4-hour long on-source integrations per target. We used phase referencing with known nearby calibrator sources for accurate phase calibration, which required total telescope times of roughly 6$-$6.5 hours per schedule per source. This resulted in extending the sampling of parallactic angles and $uv$-coverage for high fidelity imaging. The estimated absolute flux density calibration is within 5-10\%, which is the nominal VLBA flux calibration uncertainty. \subsubsection{Calibration and Imaging} We used National Radio Astronomy Observatory (NRAO) Astronomical Image Processing System \citep[\textsc{aips};][]{Van_1996ASPC..101...37V} release 31DEC19 to calibrate our VLBA data. Each dataset was calibrated independently by target -- phase calibrator pair. Bad data were flagged and calibration was performed using the standard \textsc{aips} VLBA procedures, following the prescription outlined in Paper~I. % We used the \textsc{aips} task {\sc imagr} to make images of the calibrators and sources, cleaning the images until the rms approached the theoretical thermal limit. % We achieved a S/N ratio of $>$10 for all our detected sources. For more information on our calibration and imaging procedures, please see Sections 2.2.1 and 2.2.2, respectively, in Paper~I. \section{Results} \label{sec:res} \begin{deluxetable*}{l l c c c c l r H H H } \tabletypesize{\scriptsize} \tablecaption{Radio Image Properties of the Detected Sources} \label{tab:radio_xray_luminosity} \tablehead{\colhead{Name} & \colhead{F$_{peak}^a$} & \colhead{Log\,F$_{peak}$} & \colhead{Log\,L$_{peak}$}& \colhead{S$_{int}^b$} &\colhead{T$_{b}$}& \colhead{Source R.A.\ (ICRS)} & \colhead{Source Decl.\ (ICRS)}\\ % [-0.2cm] & \colhead{(mJy bm$^{-1}$)} &\colhead{($\times10^{-17}$\,erg\,s$^{-1}$\,cm$^{-2}$)} & \colhead{(erg s$^{-1})$} & \colhead{(mJy)} & (K) & \colhead{(HMS)} & \colhead{(DMS)}}% \startdata \hline New Snapshots &&&&&&&\\ \hline NGC 3147 & ~~~~$6.698\pm0.042$ &39.085$\pm$0.245 &37.86 & $6.228\pm0.077$ & 10$^{7.6}$& 10:16:53.650476$\pm$ 0.000001 & 73:24:02.69529$\pm$0.00001 & 3.43 & 1.54 & 23 \\ NGC 3516 & ~~~~$0.672\pm0.029$ &3.943$\pm$0.170 & 36.82& $1.274\pm0.083$ & 10$^{6.3}$ & 11:06:47.46346$\pm$0.00001 & 72:34:07.2783$\pm$0.0002& 6.75 & 3.08 & 170 \\ \hline Deep Integrations &&&&&&&\\ \hline NGC 2782 & ~~~~0.098$\pm$0.007 & 0.575$\pm$0.041 & 35.95 & 0.336$\pm$0.033 & 10$^{5.6}$ &09:14:05.10270$\pm$0.00001 & 40:06:49.3224$\pm$0.0005 & 10.93 & 3.25 & 4 \\ NGC 4388 & ~~~~0.214$\pm$0.018 &1.256$\pm$0.105 & 36.28 & 0.563$\pm$0.065 &10$^{5.8}$ & 12:25:46.78136$\pm$0.00001 & 12:39:43.7611$\pm$ 0.0004 & 9.40 & 3.45 & 9 \\ NGC 4593 & ~~~~0.424$\pm$0.023 &2.488$\pm$0.135 & 36.64& 0.498$\pm$0.049 & 10$^{6.0}$ & 12:39:39.443589$\pm$0.000004 & $-$05:20:39.0347$\pm$0.0002 & 6.58 & 2.63 & 168 \\ \enddata \tablecomments{$^{a}$Peak flux values are derived from CASA's 2-D Gaussian model fitting algorithm and\\ $^{b}$integrated flux densities are determined from 5$\sigma$ outermost contour region shown in Figure~\ref{fig:radio_det_sources}.} \end{deluxetable*} In Table \ref{tab:radio_obs}, we list the radio observation properties for our final calibrated images displayed in Figure \ref{fig:radio_det_sources} for the additional snapshot and deep integration samples. Table~\ref{tab:radio_xray_luminosity} lists the peak flux values of the detections derived from CASA’s 2-D Gaussian model fitting algorithm and the integrated flux densities measured using the 5$\sigma$ outermost contour shown in our images. For the additional snapshot observation sample, the nuclei for both the detections (NGC 3147 and NGC 3516) appear unresolved, but with follow up higher sensitivity deep observations, the core for NGC 2782 and NGC 4388 appear slightly elongated. The central component for NGC 4593 remains unresolved, similar to the large scale C-band VLA archival\footnote{\url{http://www.aoc.nrao.edu/~vlbacald/src.shtml}} image shown in Paper I. In the C-band VLA image of NGC 4388, a resolved additional component separated by $\sim$200 pc from central core is seen towards the south-west direction. Our VLBA image also shows a similar elongated feature towards the same direction, but closer to the central peak ($\sim$1 pc). For NGC 2782, a large kpc-scale wing-like structure primarily spread in the north-south direction in the VLA A-configuration\footnote{\url{https://public.nrao.edu/vla-configurations/}} C-band image also exhibited a similar elongation in our VLBA deep observation on very small scales (pc, as compared with kpc) around the central core of the AGN. However, NGC 4593, even with the higher sensitivity observation, remains an unresolved point-like source with a luminous radio core contained within. The region generating the radio emission in its nucleus is extremely small ($\leq$1 pc), and comparable in size to the accretion disk~\citep[][]{Hawkins_2007A&A...462..581H}{}. % Prior to our data analysis, we looked over our sample again to confirm that the sources are free from any biasing in determining the radio detections within the redshift range (0.028 $<$ z $<$ 0.093) of our sample. Figure \ref{fig:detected_sample_rms} shows the radio luminosity distribution of our total volume-complete sample of 34 AGNs as a function of the ratio between the physical and angular size ($\sim$redshift) out to our 40 Mpc limit. The colored circles enclosed with boxes are the detections and those with the arrows are the 5$\sigma$ upper limits for the non detections. The black solid line is the fit for the average 5$\sigma$ rms for the non-detected sources and demonstrates the limit for any detection in the resulting images. We explored the possibility that some of our distant objects may suffer from ``beam dilution'' due to the lack of exposure time. For a distant point-like source with a small solid angle ($\Omega_{s}$) compared to the telescope's beam solid angle ($\Omega_{s}$ $\propto$ beam size), the measured signal is ``diluted" by the ratio of the solid angles of the source and beam, respectively. If that were true in our case, the detections would be expected to be primarily from the nearby objects with a drop off towards more distant sources. However, Figure~\ref{fig:detected_sample_rms} shows the luminosities of our observations as a function of the physical scale to beam solid angle thus demonstrating that our observations likely did not suffer beam dilution. In addition, all the detections needed to be higher than the upper limit of 10$^{36.2}$ erg s$^{-1}$ (dotted line) for non-detections, across the whole distance range. Radio luminosities for the detections except for NGC 1068 and NGC 2782 all lie above this 5$\sigma$ rms line. NGC 1068 is one of the nearest sources from the initial snapshot sample (16.3 Mpc), so its proximity combined with the high sensitivity of the data and extended morphology enabled its detection despite its low radio luminosity. NGC 2782 is one of the more distant deep observations, and its radio luminosity is comparable to the maximum upper limit level. Both are considered to host AGN that are heavily obscured, with NGC 1068 especially so. Figure~\ref{fig:soft_xray_comparison} compares our XRT and \nustar~X-ray (2-10 keV) luminosities from Table \ref{tab:xray_data} for our additional snapshot (red circles) and deep integration (blue circles) observations. NGC 2273, NGC 7172, and to a lesser extent NGC 3516 appear as outliers in Figure \ref{fig:soft_xray_comparison}, but we note that NGC 2273 is known % as a Compton-thick AGN and hence its intrinsic X-ray luminosity can not be estimated at the softer X-ray energy range of XRT. % On the other hand, for NGC 7172, a short-time variability ($\sim$~30\%) in 2-10 keV flux was reported by \citet{Guainazzi_1998MNRAS.298..824G} and a recent study \citep{Mehdipur_2022ApJ...925...84M} explained the new (since 2017) low-flux state and X-ray spectrum variability of NGC 3516 in detail. With the exception of the three sources discussed above, the 2-10 keV luminosities obtained from the \nustar\ are consistent with the \emph{Swift} XRT measurements within their respective uncertainties (see section \ref{subsection: XRT}). Although the \nustar\ observations are not simultaneous with the radio ones, which were taken at the same time as the \emph{Swift} XRT pointings, the good correlation between \nustar\ and XRT values indicates that the X-ray variability does not play a significant role for our sample. We used \nustar\ flux data (reported from Table \ref{tab:xray_data}) to properly account for the effects of absorption and reflection, and all the individual \nustar\ spectral analysis for the sources are shown in Appendix~\ref{appendix:a}. \section{Discussion} \label{sec:dis} \subsection{Radio$-$X-ray Correlation} In Figure 7 (left panel) of Paper I, we showed the relationship between X-ray (2-10 keV) and radio continuum (6 cm) luminosities for the initial volume-limited sample selected for FRAMEx together with a sample of local Seyfert galaxies from ~\citet[][]{Panessa_Giroletti_2013MNRAS.432.1138P}{} and Galactic and extragalactic black holes from ~\citet[][]{Merloni_2003MNRAS.345.1057M}{}. The main conclusion drawn from this figure was that the majority of the VLBA observed AGNs at parsec scales that were underluminous in radio as compared to the lower-resolution studies, found well below the ~$L_\mathrm{R}$/$L_\mathrm{X}$ = 10$^{-5}$ scaling relationship for pure coronal emission~\citep[][]{Guedel_1993ApJ...405L..63G,Laor_Behar_2008MNRAS.390..847L,panessa_2019NatAs...3..387P}{}. In this work, we recreated a similar figure (Figure~\ref{fig:radio_vs_soft_xray_comparison}) to explore this relationship further focusing only on the extragalactic sources. The higher sensitivity of the deeper 4-hour on-source integrations allowed for the recovery of three additional sources (blue filled circles), which were not previously detected in the 1-hour snapshot program (Paper I). The radio luminosities of these deeper observations are found to be lower compared to those obtained in previous snapshot detections from Paper~I (red squares), and in additional snapshot detections from this paper (red filled circles). In Figure~\ref{fig:radio_vs_soft_xray_comparison}, we plot the coronal emission relationship of~$L_\mathrm{R}$/$L_\mathrm{X}$ = 10$^{-5}$~\citep[][]{Guedel_1993ApJ...405L..63G}{} (green dash-dotted line) and the redefined radio-loudness parameter of ~$\log_{10} R_\mathrm{X} = \log_{10}(L_\mathrm{6~cm} / L_\mathrm{2-10~keV}) = -2.755$ from~\citet{panessa_2007A&A...467..519P} (blue dotted line), which is a more rigid threshold as compared to the classical divison of~$\log_{10} R_\mathrm{X} = -4.5$ between RL and RQ AGNs by~\citet{Terashima_2003ApJ...583..145T}. The latter division was derived based on a fixed ratio between the monochromatic luminosities at radio and optical frequencies (R $\equiv$ $L_\mathrm{6~cm} / L_\mathrm{B}$ = 10), and ~\citet[][]{panessa_2007A&A...467..519P}{} claimed that one should avoid the use of optical bands where absorption occurs naturally and might overestimate the value of R. The redefined boundary of the radio-loudness parameter was derived from comparing a sample of local Seyfert galaxies with a sample of low luminosity radio galaxies (LLRGs). Both the samples showed a similar correlation slope of $L_\mathrm{X} \propto L_\mathrm{R}^{0.97}$, but Seyfert galaxies were three orders of magnitude less luminous in the radio band than LLRGs. Both the samples exhibited two different distributions in R$_\mathrm{X}$, where the maximum separation between them was located at~$\log_{10} R_\mathrm{X} = -2.755$. The majority of the LLRGs were found on the radio-loud side and a common non-thermal origin for radio and X-ray emission has been suggested (e.g., synchrotron radiation from a relativistic jet). On the other hand, the origin of radio-quieter Seyfert sample was attributed to the disk-corona system. With our improved measurements and additional sources, we find that radio luminosities range widely over 3 to 4 orders of magnitude. The black solid line represents the best fit to the data including the non-detections, which are treated as upper limits, obtained with the ASURV regression analysis package~\citep[]{Lavalley_1992BAAS...24..839L} and determined in the same way as Paper I. Our best-fit line yields log $L_\mathrm{6 cm}$ = (1.02) log $L_\mathrm{2-10 keV}$ $-$ 6.67 or $L_\mathrm{X} \propto L_\mathrm{R}^{0.98}$, which is similar to the correlation slope for radio quiet Seyferts found by~\citet[][]{panessa_2007A&A...467..519P}{}. However, our sample contains radio nuclei that are $\sim$2 orders of magnitude lower in radio luminosity. Indeed, the higher-resolution VLBA data lie well below the radio-loud jet domination line, and only a few sources are close to the coronal line with the majority of our sample lying below. Our findings are broadly consistent with the results from Paper I and suggest that no significant coronal synchrotron radio emission is produced in these AGNs, or that the emission is attenuated such that it is not detectable at our sensitivity limits. \subsection{The Dependence of the Fundamental Plane of Black Hole Activity and Radio-Loudness on the Accretion Rate} There is some evidence that supermassive black holes in AGNs behave similarly to stellar mass black holes in X-ray binaries (XRBs)~\citep[see, e.g.,][]{Done_2005MNRAS.364..208D,kording_2006MNRAS.372.1366K,McHardy_2006Natur.444..730M}. More specifically, different classes of AGNs show similarities in their X-ray and radio properties with XRBs in different spectral states, which are driven by the accretion rate \citep[see][for a review]{McClintock_2006csxs.book..157M}. For this reason, it is important to compare the different AGN classes with XRBs in the appropriate spectral state. % Since our volume complete sample is mostly made of sources optically classified as Seyfert, which are accreting at a moderately high rate, it is probably more appropriate to compare it to the FP introduced by \citet{Dong_2014} that is restricted to XRBs in their high-accreting state and bright AGNs. On the other hand, the \citet[][]{Merloni_2003MNRAS.345.1057M} FP used a more heterogeneous sample possibly dominated by low-accreting sources. However, both of these FP relations were derived from samples observed at relatively large physical scales, based on VLA and ATCA radio observations, which probe scales of hundreds to thousands of parsecs around the AGNs while VLBA/VLBI measurements probe much more localized sub-parsec scales. Several other FP studies on parsec to tens of parsecs scales provide some useful context. For example, \citet[][]{Saikai_2018A&A...616A.152S}{} observed a sample of 76 low-luminosity active galactic nuclei (LLAGN) with the VLA at 15 GHz, consisting of low-luminosity Seyfert galaxies, low ionization nuclear emission region (LINER) objects, and transition nuclei. This low-luminosity AGN (LLAGN) sample appears to follow a FP, however, differently from our study, the radio fluxes were extracted from larger regions compared to our milli-arcsecond study and the radio frequency used was 15 GHz (as opposed to our 5 GHz). In the recent work from \citet[][]{Gultekin_2019ApJ...871...80G}{}, the FP was refined using a sample restricted to objects with BH masses determined directly from dynamical methods. Differently from our work, \citet[][]{Gultekin_2019ApJ...871...80G}{} utilize lower radio resolution observations obtained with the VLA and different radio frequencies transformed into 5 GHz by assuming canonical single power-law spectra. However, very recently, \citet[][]{panessa_2022MNRAS.tmp.1693P}{} in a study based on radio frequencies ranging from 5 to 15 GHz, demonstrated that the radio spectra of radio quiet AGNs are described by different models including convex models and broken power laws, not only by single power laws (see Table 3 in~\citet[][]{panessa_2022MNRAS.tmp.1693P}{}). These findings suggest that the radio fluxes derived by \citet[][]{Gultekin_2019ApJ...871...80G}{} using a power-law canonical model should be taken with caution. Similarly to Figure 4 (left panel) of Paper I, we show our sample and the local Seyfert galaxies observed with VLBI from \citet[][]{Panessa_Giroletti_2013MNRAS.432.1138P} on the FP described by \citet{Merloni_2003MNRAS.345.1057M} in the left panel in Figure \ref{fig:FP_compare} and in the right panel of the same figure we plot the same targets with the \citet[][]{Dong_2014} FP. % Admittedly, a large scatter is present in both plots, but, with the exception of a few objects that are likely radio loud (above the classical division between RL and RQ AGNs \citep{Terashima_2003ApJ...583..145T}; red dashed line), our data appeared to be comparatively closer to the fundamental plane of black holes activity described by \citet[][]{Dong_2014}. In agreement with Paper~I, we conclude that any fundamental plane of black hole activity breaks down when the radio emission is resolved to parsec scales with no discernible correlation between black hole mass or X-ray luminosity, similar to the $L_R$/$L_X$ relation. Interestingly, the radio loudness parameter,~$R_\mathrm{X} = L_\mathrm{6~cm} / L_\mathrm{2-10 keV}$, shows a negative trend when plotted as a function of the Eddington ratio $L_{bol}/L_{Edd}$, % which parametrizes the source's accretion rate as shown in Figure~\ref{fig:Rx}. % The figure shows $R_\mathrm{X}$ versus the Eddington ratio, with the \citet{panessa_2007A&A...467..519P} demarcation line $R_{X}$ = $-$2.755, shown along with the $R_{X}$ = $-$4.5 line representing the traditional separation between RL and RQ AGNs (which corresponds to a value of 10 for the optically-based radio-loudness parameter R). This is the trend observed in XRBs in their transition from the ``low/hard'' state (a low-luminosity state with a hard, non-thermal X-ray spectrum), which is consistently associated with steady radio emission attributed to a jet, to the ``high/soft" state (a high luminosity state with a soft, thermal X-ray spectrum) where the radio emission is suppressed. This inverse correlation was previously seen by \citet[][and the references therein]{Sikora_2007ApJ...658..815S}, using the traditional optically-based radio loudness parameter instead of the X-ray based one used here. This anti-correlation was also described by \citet[][]{Wang_2004ApJ...615L...9W}, expressing the idea of an inter-connection between radio jet and the Eddington ratio of the accretion disks. Both of these properties were found to be inversely correlated in a sample of 35 Blazars with VLBI observations. From Figure \ref{fig:Rx}, we find that most of our non-detections are grouped together in the highest accretion and lowest radio loudness region, the bottom right corner of the plot. The high accretion regime in black hole systems appears to disfavor the formation of radio emitting jets. This conclusion is also in agreement with the wind-jet inverse relation in radio-loud AGNs reported by \citet[][]{Mehdipur_2019A&A...625A..25M}, and may be a possible explanation for our radio quiet AGN sample. % \subsection{Flux Variability and Absorption} In FRAMEx Paper II~\citep[][]{Fernandez_2022ApJ...927...18F}, a nearby radio quiet AGN, NGC 2992, was observed simultaneously in X-ray (2-10 keV) and radio (5 cm) over a period of six months with an angular resolution similar to ours and showed a large drop in radio luminosity (over a factor of $>$ 3) within a few months timescale. Interestingly, this radio dimming event was observed shortly ($\sim$ a month) after an increase in the X-ray 2$-$10 keV flux, indicating an increase of ejected materials near the source and higher opacity at radio frequencies by free-free absorption. Interestingly, the importance of self-absorbed synchrotron emission at~$\sim$GHz frequencies is predicted by the model proposed by~\citet[]{Ishibashi_2011A&A...525A.118I}, where shocks accelerate relativistic electrons within the accretion flow of radio quiet AGNs, the opacity decreases with the increase of the frequency and the transition from optically thick to optically thin regimes lies around a few tens of GHz. For our sources, which are observed at $\sim$5 GHz, we thus expect free-free or self-absorbed synchrotron radiation or a combination of both as one of the major reasons for such low radio luminosity values. Alternatively, we simply might have observed our radio quiet sources during a similar ``off'' state (drop in radio luminosity) seen in NGC 2992 during the 6 months monitoring campaign. To quantify the opacity of the absorber near our AGNs, we calculated new absorbed radio-X-ray correlation lines depending on the strengths of absorption or optical depth ($\tau$) values. We assumed that without any absorption at all, the AGNs would follow the jet dominated radio-X-ray correlation of $L_\mathrm{R}$/$L_\mathrm{X}$ = 10$^{-2.755}$, similarly to radio loud AGNs. In the left panel of Figure~\ref{fig:radio_vs_xray_comparison}, our calculated $\tau$ lines are shown on top of the $L_\mathrm{R}$/$L_\mathrm{X}$ plot for a range of optical depth values ($2 < \tau < 10$), where the higher the absorption, the larger the deviation is seen from the intrinsic radio luminosity line. Detected sources were found to be distributed over a broader optical depth range ($2 < \tau < 8$), whereas the non-detections are clustered around a higher optical depth range ($6 < \tau < 10$). NGC 1068 is the only detected source found beyond the $\tau$ = 8 line. In the right panel, we include optical depth lines to the FP plot described by ~\citet[][]{Dong_2014}{}, but for an average black hole mass of $10^{7}$~M$_{\odot}$. This is roughly the average of all black hole masses for our radio quiet AGN sample. The scatter seen in the plot with the the optical depth lines, demonstrates a similar pattern to the radio-to-X-ray correlation plot. % In a simulation of sub-Eddington accretion disks around a supermassive black hole of mass $5\times10^8 ~{\mathrm{M_\odot}}$, \citet[][]{Jiang_2019ApJ...885..144J}{} showed that for an increase in the accretion rate, the disk becomes thicker and the total optical depth (the sum of absorption and scattering opacity) increases. % Finally, using different random distributions of optical depths, we generated synthetic data points to be compared with the observed ones. In Figure~\ref{fig:model_obs}, we illustrate the comparison between our model set and real observation set for two different random distributions of optical depth. We found that a power law distribution with a mean optical depth $\sim$10, yields a better agreement with the observed detections ($\approx$ 71\%) and non-detections ($\approx$ 65\%) compared to a normal distribution. In particular, it appears that the power-law model shows a good agreement with the real data when $8 < \tau < 12$. This result suggests that the presence of high optical depth regions producing absorption in radio emission may play a prominent role in reducing the rate of detections and in producing weak radiation. Throughout the paper, we have discussed different possible origins of radio emission from our sub-parsec scale observations, such as relativistic particles accelerated in shocks and winds or low-luminosity outflows. To shed some light on the dominant emission mechanism, we calculated the brightness temperature from the measured fluxes of our detected sources (see Table~\ref{tab:radio_xray_luminosity}). We find brightness temperatures of $10^{7.6}$~K and $10^{6.3}$~K from our two VLBA snapshot detections (NGC 3147 and NGC 3516), indicating either compact jet-associated or corona-associated synchrotron radiation as their likely dominant radio emission mechanism. This is in agreement with % the correlation plot between the X-ray and sub-parsec scale radio emission in Figure~\ref{fig:radio_vs_soft_xray_comparison}. On the other hand, for the deep observation objects, the brightness temperature range of $10^{6}$~K$~ \geq T_b\geq10^{5.6}$~K suggests that thermal free-free emission from accretion disk winds/shocks may be the primary source of radio emission. Overall, our data are inconclusive in distinguishing between competing emission mechanisms.% \section{Conclusions} \label{sec:conclusions} In this work, we have expanded the investigation of the X-ray and radio properties of the original volume-limited sample of AGNs presented in Paper I by adding VLBA deep integrations (4 hours per source) of 9 objects and including VLBA snapshot observations of 9 additional AGNs with declination limits of $-30\arcdeg$ and $+80\arcdeg$ and D~$\leq$~40 Mpc. Our three different X-ray data sets, which come from \emph{Swift}~BAT, \emph{Swift}~XRT and NuSTAR, have made it possible to derive good-quality broad-band X-ray spectra and absorption-corrected 2-10 keV luminosities that were compared with the respective radio luminosities and then used in the FP of BH activity for radiatively efficient radio-quiet AGNs. Our main findings are listed below: {\begin{enumerate} \item With the help of improved, deeper VLBA radio observations ($\mathrm{RMS}\sim8~\mu$Jy), we recovered three sources that were not detected in our initial snapshot observations and we found two more detections from our additional snapshot observations with a similar observation depth to our previous campaign ($\mathrm{RMS}\sim20~\mu$Jy). Despite the increased sensitivity of our observations, the majority of our sample is still undetected. Unlike strong jet-dominated radio loud AGNs, our sources lie well below the stringent threshold proposed by \citet[][]{panessa_2007A&A...467..519P}, as well as the $R_\mathrm{X}$ = $-$4.5 line commonly used in the literature. \item We compared the radio luminosities as a function of X-ray emission from our radio-quiet AGN sample and found a correlation slope of $L_\mathrm{X} \propto L_\mathrm{R}^{0.98}$ for these high resolution radio observations. This is similar to the slope found by~\citet[][]{panessa_2007A&A...467..519P}{} for a sample of radio-quiet Seyfert galaxies but our sources were found to be $\sim$3 orders of magnitude less luminous in the radio. Consistent with our previous findings from Paper I, the~$L_\mathrm{R}/L_\mathrm{X}$ values are between $10^{-8}$ to $10^{-4}$ and the majority of our sample lies well below the fiducial $10^{-5}$ relationship for coronal synchrotron emission, showing no significant coronal synchrotron radio emission is produced in these AGNs. \item In agreement with the results from our previous snapshot campaign, our work confirms that when radio fluxes are measured on sub-parsec scales, these radio-quiet AGN fall out of the FP, regardless of the relation chosen. % \item When the X-ray based radio loudness ~$R_{X}\equiv L_{R}/L_{X}$ is compared to the accretion rate~$\sim$~$L_{bol}/L_{Edd}$, an anti-correlation is revealed, in agreement with the findings of \citet[][]{Sikora_2007ApJ...658..815S} , as well as with those from \citet[][]{Panessa_Giroletti_2013MNRAS.432.1138P} based on VLBI observations. Our sample sources are mostly clustered in the highest accretion and lowest radio loudness region. Similarly to the high accreting XRBs in their ``high/soft'' state where the radio emitting jets get suppressed, the radio-quietness of our sample sources can be interpreted as the absence of jets and presence of wind/shocks due to interactions between disk outflows and accretion flows. \item The ``turned off'' or ``silent'' state observed in a nearby radio-quiet AGN, NGC 2992, during a 6 month VLBA monitoring campaign by~\citet[][]{Fernandez_2022ApJ...927...18F}{} may explain the non-detections of our sources at such high angular resolutions. In addition to this, we expect free-free and self-absorption synchrotron radiation in $\sim$5 GHz frequency observations as one of the major reasons behind the low radio luminosities. We have shown that synthetic data obtained from a power-law distribution with optical depths of the order of 8$-$12 are consistent with our observed data plotted in the $L_{R}-L_{X}$ diagram shown in Figure~\ref{fig:radio_vs_soft_xray_comparison}. The high values inferred for $\tau$ support a scenario where the radio emission is produced in highly opaque outflows/winds, when the source accretes at a relatively high rate, whereas optically-thin radio jets are more likely produced by low-accreting systems. \end{enumerate}} Our deeper VLBA radio observations have allowed us to look into the sub-parsec regime of our sample of AGNs and has made it possible to study their core radio emission and some existing correlations which are still not well understood at these spatial scales. A future study of well-sampled radio spectral energy distributions can help us determine the true source of radio emission and disentangle contributions from potentially multiple components. Our next goal is to study high-resolution multi-wavelength observations at radio frequencies, as the radio spectral index, $\alpha$, will provide insights into a critical piece of the overall emission mechanisms of these radio quiet AGNs. \acknowledgments This work supports USNO's ongoing research into the celestial reference frame and geodesy.\\ The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The authors acknowledge use of the Very Long Baseline Array under the US Naval Observatory's time allocation. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. \vspace{5mm} \facilities{VLBA, Swift, VLA, EVLA} \software{\textsc{aips}, Astropy~\citep{Astropy_2013A&A...558A..33A}, \textsc{casa}, \textsc{xspec} } \appendix \section{Nustar Spectral Analysis} \label{appendix:a} \setcounter{table}{0} \renewcommand{\thetable}{\Alph{section}\arabic{table}} \renewcommand*{\theHtable}{\thetable} In the following, we summarize the specific details on the \nustar\ spectral analysis and \mbh\ determination of the individual sources. The main spectral parameters obtained by fitting this baseline model are reported in Table~\ref{tab:xray_spectra}. The X-ray scaling method was successfully applied to a sample of heavily obscured AGN, providing \mbh\ values consistent with those determined using disk maser measurements \citep{gliozzi_2021MNRAS.502.3329G}. The basic assumption of this method is that the Comptonization process producing the X-rays is the same in all black holes systems regardless of their mass and that the photon index $\Gamma$ is a faithful indicator of the accretion state of the source; specific details of this method are provided in \citet{gliozzi_2021MNRAS.502.3329G} and refences therein.\\ \noindent{\bf MCG-05-23-016} (obsid: 60001046006): The \nustar\ spectrum of this bright Seyfert galaxy is well fitted by our baseline model, where the \texttt{MYTZ} component is substituted by a \texttt{zphabs} model. The \mbh\ value derived from the X-ray scaling method $(1.4\pm0.5)\times10^7~{\mathrm{M_\odot}}$ is consistent within the respective uncertainties with the virial value of $2.6\times10^7~{\mathrm{M_\odot}}$ obtained by \citet{Garcia_Bernete_2019MNRAS.486.4917G}. \noindent{\bf NGC 2273} (obsid: 60001064002): The \nustar\ spectrum of this Seyfert 2 galaxy is adequately fitted by our baseline model describing a Compton thick scenario, and the \mbh\ value derived from the X-ray scaling method $(1.9\pm0.6)\times10^7~{\mathrm{M_\odot}}$ is broadly consistent with the $(7.5\pm0.4)\times10^6~{\mathrm{M_\odot}}$ value obtained from mega-maser measurements by \citet{Kuo_2020MNRAS.498.1609K}, as explicitly demonstrated in the formal comparison between these two methods by \citet{gliozzi_2021MNRAS.502.3329G}. \noindent{\bf NGC 3147} (obsid: 60101032002): This AGN has been classified as a true type 2 AGN, i.e., a source with an optical spectrum without any broad line and at the same time without any significant obscuration. The \nustar\ spectrum confirms that the X-ray primary emission is unabsorbed and hence the genuine lack of a BLR, which is thought to be a consequence of the very low accretion rate \citep{Bianchi_2019MNRAS.488L...1B}. Since the X-ray scaling method cannot be applied in this accretion regime, the small value derived $(5.5\pm2.1)\times10^5~{\mathrm{M_\odot}}$ should not be considered reliable. \noindent{\bf NGC 3516} (obsid: 60002042004): This source, optically classified as Seyfert 1.5 galaxy, is a changing-look AGN, which is characterized by extreme changes in flux and spectrum. Since the X-ray spectral variability does not follow the standard softer-when-brighter trend typical of Seyfert galaxies, the \mbh\ value derived from the X-ray scaling method $(1.4\pm0.5)\times10^6~{\mathrm{M_\odot}}$ should be not considered reliable, and indeed it is about one order of magnitude smaller than the reverberation mapping value obtained by \citet{Feng_2021ApJ...909...18F}. \noindent{\bf NGC 4102} (obsid: 60160472002): This source has been classified as a LINER but also as type 2 Seyfert galaxy. The \nustar\ spectrum, which is well parametrized by a heavily absorbed but non Compton thick primary component, yields a fairly low \mbh\ of $(2.9\pm1.5)\times10^5~{\mathrm{M_\odot}}$. Considering the possible LINER nature of the source and hence its intrinsically low accretion rate, in this case, we caution about the use of the \mbh\ derived from the X-ray scaling method, because this method can only be applied to sources that are in the moderate or high accreting regime, which are characterized by the softer-when-brighter spectral transition, as shown by \citet{Jang_2014MNRAS.443...72J}. \noindent{\bf NGC 5728} (obsid: 60662002002): The \nustar\ spectrum of this source confirms that it is Compton thick, as suggested by \citet{Comastri_2010ApJ...717..787C}. The \mbh\ value derived from the X-ray scaling method $(8.3\pm2.9)\times10^6~{\mathrm{M_\odot}}$ is consistent with the constraints obtained by \citet{Kuo_2020MNRAS.498.1609K} based on mega-maser measurements. \noindent{\bf NGC 7172} (obsid: 60061308002): The \nustar\ spectrum of this Seyfert 2 galaxy is well parametrized by a Compton thin scenario. The \mbh\ value derived from the X-ray scaling method $(1.03\pm0.35)\times10^7~{\mathrm{M_\odot}}$ is broadly consistent but lower than the value $5.5\times10^7~{\mathrm{M_\odot}}$ derived by \citet{Marinucci_2012ApJ...748..130M} utilizing the $M-\sigma_\star$ correlation. This seems to confirm the tendency of this correlation to overestimate the \mbh\ in type 2 AGN, as suggested by \citet{Ricci_2017MNRAS.471L..41R}. \begin{deluxetable}{lcrrrrrHc} \setlength{\tabcolsep}{4pt} \caption{NuSTAR X-ray Spectral Result } \label{tab:xray_spectra} \tablehead{\colhead{Source} & \colhead{Exposure} & \colhead{$\log(N_{{\textrm{H}},{\textrm{Bor}}})$} & \colhead{$N_{{\textrm{H$_{MYTZ}$}}}$} & \colhead{$\Gamma$} & \colhead{$N_\textrm{BMC}$} & \colhead{($\chi^2/$dof)} & % &\colhead{{log($M_\mathrm{BH}$)}}\\ [-0.1cm] \colhead{} & \colhead{(ks)} & \colhead{~} & \colhead{($10^{24}$ cm$^{-2}$)} & \colhead{~} & \colhead{~} & \colhead{~} &% & \colhead{($M_{{\textrm{$\odot$}}}$)}\\ [-0.1cm] \colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} & \colhead{(6)} & \colhead{(7)} & % & \colhead{(8)}} \startdata \hline New Snapshots & & & & & & & \\ \hline MCG-05-23-016 & $98$ & $23.91\pm0.01$ & $0.02\pm0.01$ & $1.95_{-0.01}^{+0.01}$ & $1.1_{-0.9}^{+0.7}\times10^{-3}$ & 1779.5/1707 & $95.4\pm0.2$ & 7.13\\%$13.5\pm0.5$\\ NGC 2273 & $23$ & $25.00\pm0.70$ & $6.80\pm0.40$ & $1.95_{-0.05}^{+0.05}$ & $3.1_{-0.2}^{+0.2}\times10^{-3}$ & 54/57& $36.0\pm13.0$ & 7.29\\%$19.5\pm5.8$\\ NGC 3147 & $49$ & \nodata~~~~ & $0.49\pm0.49$ & $1.75_{-0.03}^{+0.03}$ & $2.3_{-0.1}^{+0.2}\times10^{-5}$ & 217.2/270& $2.8\pm0.4$ & \nodata~\\ NGC 3516 & $21$ & $23.84\pm0.04$ & $0.02\pm0.01$ & $1.88_{-0.01}^{+0.01}$ & $6.4_{-0.2}^{+0.2}\times10^{-5}$ & 726.9/647& $6.0\pm0.5$ & \nodata~\\ NGC 4102 & $21$ & $24.13\pm0.06$ & $0.64\pm0.06$ & $1.53_{-0.07}^{+0.06}$ & $5.6_{-1.6}^{+2.7}\times10^{-5}$ & 167.1/176 & $9.2\pm0.3$ & \nodata~\\ NGC 5728 & $25$ & $24.30\pm0.04$ & $1.09\pm0.04$ & $1.88_{-0.06}^{+0.05}$ & $4.5_{-0.9}^{+1.5}\times10^{-4}$ & 328.4/386& $41.2\pm0.8$ & 6.91\\%$8.3\pm2.9$\\ NGC 7172 & $32$ & $24.14\pm0.03$ & $0.10\pm0.01$ & $1.90_{-0.01}^{+0.02}$ & $6.7_{-0.6}^{+0.9}\times10^{-4}$ & 1155/1150& $83.8\pm0.8$& 7.01\\%$10.3\pm3.5$\\ UGC 6728 & $58$ & $24.06\pm0.06$ & $0.010\pm0.002$ & $1.85_{-0.04}^{+0.03}$ & $1.6_{-0.4}^{+0.3}\times10^{-4}$ & 867.4/831& $14.4\pm0.2$ & 6.18\\%$1.5\pm0.5$\\ \hline Deep Integrations & & & & & & & \\ \hline NGC 1320 & $28$ & $24.50\pm0.09$ & $1.19\pm0.08$ & $1.70_{-0.07}^{+0.06}$ & $3.9_{-0.7}^{+0.7}\times10^{-5}$ & 73.6/61 & $5.4\pm0.5$ & 6.00\\%$1.0\pm0.4$\\ NGC 3081 & $55$ & $23.08\pm0.10$ & $0.79\pm0.01$ & $1.72_{-0.01}^{+0.01}$ & $4.9_{-0.2}^{+0.2}\times10^{-4}$ & 868.1/863& $65.7\pm6.6$ & 6.98\\% % NGC 4388 & $21$ & $23.59\pm0.05$ & $0.45\pm0.04$ & $1.66_{-0.04}^{+0.04}$ & $3.3_{-0.4}^{+0.5}\times10^{-4}$ & 435.2/420& $13.9\pm0.6$ & 6.63\\%$4.3\pm1.6$\\ NGC 4593 & $23$ & $25.50\pm0.60$ & $0.0002\pm0.0002$ & $1.88_{-0.01}^{+0.01}$ & $2.3_{-0.1}^{+0.1}\times10^{-4}$ & 671/679& $22.8\pm0.2$ & 6.59\\%$3.9\pm1.3$\\ NGC 6814 & $148$ & $24.26\pm0.04$ & $0.008\pm0.001$ & $1.88_{-0.01}^{+0.01}$ & $3.0_{-0.1}^{+0.1}\times10^{-4}$ & 1598/1520 & $34.8\pm0.3$ & 6.25 \\%$1.8\pm0.6$\\ NGC 7314 & $100$ & $24.23\pm0.04$ & $0.012\pm0.001$ & $2.03_{-0.01}^{+0.01}$ & $4.9_{-0.1}^{+0.1}\times10^{-4}$ & 1310.8/1276& $38.0\pm0.4$ & 6.11\\%$1.3\pm0.5$\\ NGC 7465 & $21$ & $23.83\pm0.06$ & $0.01\pm0.01$ & $1.87_{-0.02}^{+0.02}$ & $1.2_{-0.1}^{+0.1}\times10^{-4}$ & 445.9/466& $12.6\pm0.2$ & 6.08\\%$1.2\pm0.4$\\ \enddata \tablecomments{Columns: 1 = AGN name. 2 = \nustar\ FPMA exposure. 3 = column density calculated with the \borus\ model that parametrizes the continuum scattering and fluorescent emission line components. 4 = column density acting on the transmitted primary emission calculated with the \mytorus\ model; for type 1 AGN the \texttt{phabs} is used instead. 5 = photon index. 6 = normalization of the BMC model. 7 = $\chi^2$ divided by degrees of freedom. 8 = black hole mass.} \end{deluxetable} \noindent{\bf UGC6728} (obsid: 60376007002): In the best fit model of this Seyfert 1.2 galaxy the \texttt{MYTZ} component, which is more appropriate for heavily obscured AGN, was substituted by a \texttt{zphabs} model. The \mbh\ value derived from the \nustar\ spectral data, $(1.5\pm0.5)\times10^6~{\mathrm{M_\odot}}$ is consistent within the respective uncertainties with the reverberation mapping value $(7\pm4)\times10^5~{\mathrm{M_\odot}}$ \citep{Bentz_2016ApJ...831....2B}. \noindent{\bf NGC 1320} (obsid: 60061036004): The best fit of this heavily absorbed Seyfert 2 galaxy was obtained by adding to our baseline model a Gaussian line at 6.4 keV (in the source's rest frame) and a scattered component produced by a putative optically thin ionized medium with a fraction of $\sim 4 \%$. The black hole mass derived with the X-ray scaling method $(1.0\pm0.4)\times10^6~{\mathrm{M_\odot}}$ is broadly consistent with the dynamical value ($5.3\times10^6 ~{\mathrm{M_\odot}}$) obtained from maser measurements, which however cannot considered as an accurate estimate because of the complex morphology and dynamics of the maser spots \citet{Gao_2017ApJ...834...52G}. For completeness, we also compared our \mbh\ estimate to the value ($1.3\times10^5 ~{\mathrm{M_\odot}}$) obtained using the most recent version of the Fundamental Plane for Black Hole Activity, based exclusively on objects with \mbh\ dynamically constrained \citep{Gultekin_2019ApJ...871...80G}, which appears to underestimate \mbh\ by one order of magnitude. \noindent{\bf NGC 3081} (obsid: 60561044002): The \nustar\ spectrum of this Seyfert 2 galaxy required two \borus\ components to properly fit the complex X-ray data. The \mbh\ value derived from the X-ray scaling method $(9.5\pm3.6)\times10^6~{\mathrm{M_\odot}}$ is fully consistent with the value ($1.6\times10^7 ~{\mathrm{M_\odot}}$) obtained from gas dynamics \citep{Beifiori_2012MNRAS.419.2497B} and inconsistent with the Fundamental Plane for Black Hole Activity estimate ($2.8\times10^4 ~{\mathrm{M_\odot}}$), which underestimates \mbh\ by nearly three orders of magnitude. \noindent{\bf NGC 4388} (obsid: 60061228002): The \nustar\ spectrum of this Seyfert 2 galaxy, one of the few with a disk maser measurement, was fitted with a scattered component (with a fraction of $\sim 17 \%$) added to our baseline model. The \mbh\ value derived from the X-ray scaling method $(4.3\pm1.6)\times10^6~{\mathrm{M_\odot}}$ is in agreement with the dynamical value derived from maser measurements \citep{kuo_2011ApJ...727...20K}, as explicitly shown in \citet{gliozzi_2021MNRAS.502.3329G}. The estimate based on the Fundamental Plane for Black Hole Activity ($2.1\times10^5 ~{\mathrm{M_\odot}}$) one more time severely underestimates the \mbh. \noindent{\bf NGC 4593} (obsid: 60001149002): In the best fit model of this Seyfert 1 galaxy the \texttt{MYTZ} component, which is more appropriate for heavily obscured AGN, was substituted by a \texttt{zphabs} model. The \mbh\ value derived from the X-ray scaling method $(3.9\pm1.3)\times10^6~{\mathrm{M_\odot}}$ is broadly consistent with the reverberation mapping \mbh\ estimate ($(9.8\pm2.1)\times10^6 ~{\mathrm{M_\odot}}$) obtained by \citet{Denney_2006ApJ...653..152D}, as well as the model-independent value ($(5.8\pm2.1)\times10^6 ~{\mathrm{M_\odot}}$) derived from X-ray variability\citep{middei_2019MNRAS.483.4695M}, whereas it is inconsistent with with the Fundamental Plane for Black Hole Activity value ($3.5\times10^5 ~{\mathrm{M_\odot}}$). \noindent{\bf NGC 6814} (obsid: 60201028002): Similarly to the other Seyfert 1 galaxy of this sample, the \texttt{zphabs} model was used to parametrize the intrinsic absorption acting on the transmitted primary X-ray emission. Once more the \mbh\ value derived from the X-ray scaling method $(1.8\pm0.6)\times10^6~{\mathrm{M_\odot}}$ is fully consistent with the reverberation mapping value ($2.34\times10^6 ~{\mathrm{M_\odot}}$) obtained by \citet{pancoast_2014MNRAS.445.3073P}. Again the value derived from the Fundamental Plane for Black Hole Activity ($4.4\times10^4 ~{\mathrm{M_\odot}}$) substantially underestimates \mbh. \noindent{\bf NGC 7314} (obsid: 60201031002):The \nustar\ spectrum of this source which has been optically classified as Seyfert 1.9 but also as a Narrow Line Seyfert 1 galaxy because of its pronounced variability, is well fitted by our baseline model with the addition of a gaussian line at 6.38 keV (in the source's rest frame) and a \texttt{zphabs} model instead of \texttt{MYTZ} . Because of the relatively steep photon index, the 2005 decaying outburst of GRO J1655-40 cannot be used to determine \mbh. Instead, we estimated the black hole mass using all the other available patterns described in \citet{gliozzi_2011ApJ...735...16G} (the rising phase of the 2005 outburst of GRO J1655-40, the 2003 decaying phase and the 2004 rising phase of GX 339-4, and the rising phase of the 1998 outburst of XTE J1550-564) and then computed the average value. The \mbh\ value derived with the X-ray scaling method $(1.3\pm0.5)\times10^6~{\mathrm{M_\odot}}$ is consistent with the virial estimate ($1.74\times10^6 ~{\mathrm{M_\odot}}$) obtained by \citet{Onori_2017MNRAS.468L..97O} by measuring NIR lines in the BLR and assuming a constant virial factor of 4.31, whereas it is inconsistent with with the Fundamental Plane for Black Hole Activity value ($3.9\times10^4 ~{\mathrm{M_\odot}}$). \noindent{\bf NGC 7465} (obsid: 60160815002):The \nustar\ spectrum of this Seyfert 2 galaxy is reasonably well fitted by our baseline model without any additional component. The \mbh\ value derived from the X-ray scaling method $(1.2\pm0.4)\times10^6~{\mathrm{M_\odot}}$ is in general agreement with the the virial estimate ($3.47\times10^6 ~{\mathrm{M_\odot}}$) obtained by \citet{Onori_2017MNRAS.468L..97O}, whereas the value derived from the Fundamental Plane for Black Hole Activity value ($1.7\times10^5 ~{\mathrm{M_\odot}}$) again underestimates \mbh\ by more than one order of magnitude. \bibliography{radio_quiet_agn}{} \bibliographystyle{aasjournal}

Title: Simplified Method for the Identification of Low Mass Ratio Contact Binary Systems that are Potential Red Nova Progenitors

Abstract: The study presents a simplified method to identify potential bright red nova progenitors based on the amplitude of the light curve and infrared (J-H) colour of a contact binary system. We employ published criteria for contact binary orbital instability to show that the amplitude of the light curve for a given contact system with a low mass (< 1.4Msun) primary must be less than a specified value for it to be potentially unstable. Using this we search the photometric data of a large survey to identify about 50 potential bright red nova progenitors. We analyse the survey photometry of each to determine the mass ratio and from the estimated mass of the primary other physical parameters of the systems. We show that each system has physical characteristics indicating potential orbital instability. Using the absolute parameters from our sample we model the expected instability separation and period for low mass contact binary systems

https://export.arxiv.org/pdf/2208.00626

\sloppy \title{Simplified Method for the Identification of Low Mass Ratio Contact Binary Systems that are Potential Red Nova Progenitors} \author{Surjit S. Wadhwa\textsuperscript{1*}, Ain Y. De Horta\textsuperscript{1}, Miroslav D. Filipovi\'c\textsuperscript{1}, Nick F. H. Tothill\textsuperscript{1}, Bojan Arbutina\textsuperscript{2}, Jelena Petrovi\'c\textsuperscript{3} and Gojko Djura\v sevi\'c\textsuperscript{3}} \affilOne{\textsuperscript{1}School of Science, Western Sydney University, Locked Bag 1797, Penrith, NSW 2751, Australia.\\} \affilTwo{\textsuperscript{2}Department of Astronomy, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia.\\} \affilThree{\textsuperscript{3}Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia\\} \twocolumn[{ \corres{19899347@student.westernsydney.edu.au} \keywords{Contact Binary, Low Mass Ratio, light curve solution} }] \doinum{12.3456/s78910-011-012-3} \artcitid{\#\#\#\#} \volnum{000} \year{0000} \pgrange{1--} \setcounter{page}{1} \lp{1} \section{Introduction} The number of known contact binary systems has, and still is, growing at a phenomenal rate given the new discoveries resulting from sky surveys. As an example, All Sky Automated Survey (ASAS) \citep{2002AcA....52..397P} and the All Sky Automated Survey - Super Nova (ASAS-SN) \citep{2014ApJ...788...48S, 2020MNRAS.491...13J} have added more than 100000 new discoveries. Theoretical models such as \cite{2003ApJ...582L.105S, 2012JASS...29..145E, 2011A&A...531A..18S, 1977MNRAS.179..359R} predict that contact binary systems with extremely low mass ratios are likely to merge into a single rapidly rotating relatively cool giant star. The merger event is thought to result in a transient nova like event that evolves to remain bright in the red and infrared bands. The event is usually termed a red nova. Although Galactic merger events are predicted to occur commonly (once every 2-3 year), brighter events that are likely to be available for study are more limited at once a decade \citep{2014MNRAS.443.1319K}. There has been only one confirmed observation linking typical red nova like transient to a contact binary progenitor, that of V1309 Sco \citep{2011A&A...528A.114T}. Other examples such as V4432 Sgr \citep{1999AJ....118.1034M}, V838 Mon \citep{2002IAUC.7785....1B} and OGLE2002-BLG-360 \citep{2013A&A...555A..16T} are postulated to represent stellar mergers although their progenitors remain unidentified. In addition there exist possible historical and extra galactic examples \citep{2019A&A...630A..75P, 2014MNRAS.443.1319K}. Since the recognition of V1309 Sco there has been heightened interest in the theoretical basis of orbital instability and the identification of low mass ratio contact binary systems \citep{2021MNRAS.501..229W,2021MNRAS.502.2879G,2022MNRAS.tmp..527C}. In two papers \citet{1993PASP..105.1433R,2001AJ....122.1007R} showed that the shape of contact binary light curves is dependant on three main geometrical factors namely the mass ratio ($q$), the fill-out ($f$) (degree of contact - or the thickness of the contact neck region) and the inclination ($i$). In addition the other somewhat minor determinant is the temperature difference between the components. He deduced that the maximum amplitude for any given system was seen if the system was observed to have a complete eclipse. In addition, he showed that only the mass ratio, fill-out and to a lesser extent (in the presence of complete eclipses) inclination determined the amplitude of the light curve with other factors such as temperature of the components having little impact. We combine the three techniques noted above namely the theoretical instability parameters, survey photometric data and the photometric amplitude distribution of contact binary systems to derive a simplified method of identifying potential contact binary systems that show signs of orbital instability (potential red nova progenitors). For the purpose of this study we define a contact binary system with a mass ratio at or below the theoretical critical level as being potentially unstable. The paper is divided into five sections. In section two we model theoretical light curves to derive a relationship between the mass of the primary and the maximum amplitude of a potentially unstable system. In section 3 we employ the relationship to the ASAS-SN survey to identify potential bright red nova candidates among low mass ($0.6M_{\odot} < M_1 < 1.4M_{\odot}$) contact binary systems. % In section 4 we define an average potential red nova progenitor in addition to comparing and contrasting our sample of bright potential red nova progenitors with other comparable systems. In section 5 we briefly discuss the historical development of mass ratio as a determinant of orbital instability, limitations of the present study and further ongoing search of red nova progenitors along with a summary and conclusion of the current work. \section{Amplitude Distribution of Potential Red Nova Progenitors} Critical to understanding the orbital evolution of contact binary system is predicated on knowledge of parameters such as the mass ratio, masses of the components and the geometry of the orbit such as inclination, degree of contact and temperature variation between the components. Many, if not all, of these parameters can be derived from light curve analysis but only if the light curve demonstrates a complete eclipse \citep{2005Ap&SS.296..221T}. As such, we limit our modelling of systems that demonstrate a complete eclipse and would be applicable to observed light curves Current light curve analysis tools can incorporate many tens of different parameters, however, as noted above in the case of contact binary systems only 4 main parameters are critical. Therefore in modelling the amplitude distribution of unstable systems we have neglected complications associated with star spots and other stellar activity. We used the 2009 version of the Wilson-Devinney code as incorporated into the Windows front end utility WDwin56d \citep{2021NewA...8601565N} to model all light curves. The gravity darkening coefficients $g_1 = g_2 = 0.32$, the bolometric albedos $A_1 = A_2 = 0.5$ were fixed \citep{1967ZA.....65...89L} and simple reflection treatment applied \citep{1969PoAst..17..163R}. As per \citet{2015IBVS.6134....1N} logarithmic limb darkening coefficients interpolated from \citet{1993AJ....106.2096V} were used. To confirm the findings of \citet{1993PASP..105.1433R,2001AJ....122.1007R} with respect to the effects of fill-out and inclination on the amplitude of the light curve we modelled light curves of an idealised contact binary system with primary of one solar mass with mass ratio ($q$) 0.1 and equal temperature of the components ($T_1 = T_2 = 5770K$) to record the effects of inclination ($i$) and degree of contact ($f=0 - 1$). As we are only interested in systems that display a total eclipse we modelled the system with an inclinations of $90^{\circ}$ which would yield the maximum eclipse time and $72^{\circ}$ which would yield a small total eclipse between phase 0.49 to 0.51. The change in inclination results in a reduction in the duration of the secondary eclipse with the lower inclination reducing the total amplitude slightly. We also modelled fill-out $f=0$ and $f=1$ for each inclination to determine the variation in amplitude and again to confirm previous findings using the current accepted modelling code. The fill-out has a significant effect on the amplitude of the light curve with high fill-out yielding the highest amplitude. This is as expected, because higher the degree of contact the thicker the neck of the contact region. The neck bears some luminosity and thicker the neck the greater the luminosity that is eclipsed. We did not model stars of different mass (hence different $T_1$) because as noted by \citet{2001AJ....122.1007R} the combination chosen is a reasonable representation of the light curve of low mass contact binaries and confirms the previous findings that the maximum amplitude of a contact binary system occurs at high inclination and high degree of contact. The results are illustrated in Figure 1 and summarised in Table 1. Having established the condition of high inclination and high fill-out for high amplitude we next modelled the effects of the difference in temperature of the components. The presence of a common envelope usually results in good thermal contact between the components so there is usually little difference in the temperatures of the components. Recently \citet{2021ApJS..254...10L} compiled a catalogue of published light curve solutions of contact binaries. Using a sub-sample of the catalogue for primary stars between ($0.6M_{\odot} < M_1 < 1.4M_{\odot}$) we determined the median temperature difference between the components of approximately $200K$. Adopting the commonest temperature difference between the components we next modelled light curves for our idealised system with ($i=90^{\circ}$ and $f=1$) and temperatures of the secondary either 200K higher ($5970K$) or 200K lower ($5570K$). The results are illustrated in Figure 1 and summarised in Table 1. It is clear that the typical temperature difference has little effect on the maximum amplitude of the light curve with both the cooler or warmer secondaries having minimal impact relative to the result with the components being of same temperature. We can deduce from the above that for any given mass ratio the maximum amplitude will be achieved with high inclination, high fill-out and the secondary slightly warmer. \begin{table}[ht] \centering \begin{tabular}{|c|c|c|c|} \hline Inclination($^\circ$) &fill-out&$T_2$& Max Ampl (Mag) \\ \hline 90&1& Eq & 0.36\\ \hline 72&1& Eq & 0.31\\ \hline 90&0& Eq & 0.27\\ \hline 90&1&CS & 0.36\\ \hline 90&1& HS & 0.36\\ \hline \end{tabular} \caption{Effects of inclination, fillout and temperature of the secondary on the maximum amplitude of a contact binary system with mass ratio 0.1. Eq = Equal component temperatures, CS = Cold Secondary, HS = Hot secondary, $r_{1,2}$ = mean fractional radii of primary and secondary.} \end{table} Having establish the modelling criteria for the maximum amplitude at a given mass ratio we employ this to determine the maximum amplitude of a potential red nova progenitor. Recently \citet{2021MNRAS.501..229W} linked the instability mass ratio of contact binary systems with the mass of the primary component. They demonstrated that the mass ratio at which instability ($q_{inst}$) is likely can be determined by a simple quadratic relationship for high and low level of contact: \begin{equation} \label{eq:qinst-f1} q_{inst}=0.1269M_{1}^2-0.4496M_{1}+0.4403\ (f=1). \end{equation} \begin{equation} \label{eq:qinst-f0} q_{inst}=0.0772M_{1}^2-0.3003M_{1}+0.3237\ (f=0). \end{equation} Using this relationship say for a system with primary of one solar mass the instability mass ratio at high fill-out would be approximately 0.12. If we now model a light curve with the following parameters $T_1 = 5770K, T_2 = 5970K, i = 90^{\circ}, f = 1$ and $q = 0.12$ this will give us the maximum amplitude at which such a system is possibly unstable. As amplitude increases with increasing mass ratio (see below and \citet{1993PASP..105.1433R,2001AJ....122.1007R}), any system with amplitude higher will likely have a higher mass ratio and therefore would be stable. As inclination of a system drops eventually a complete eclipse is lost and light from both components is observed throughout the orbital cycle and the overall variation (amplitude) of the light curve drops \citep{2001AJ....122.1007R}. Therefore a system with amplitude significantly below the maximum amplitude is unlikely to have a complete eclipse and therefore not suitable for photometric light curve analysis. We extended our modelling of the maximum amplitude for systems with primary star masses between $0.6M_{\odot}$ and $1.4M_{\odot}$ and the calculated instability mass ratio at high fill-out. We adopted values of $T_1$ based on the main sequence calibration from \citep{2013ApJS..208....9P} + 200K. The results are summarised in Table 2 and the line of best fit is as shown in Eq 3 and graphically shown in Figure 2. It is clear that higher the mass ratio the greater the amplitude therefore any system with an amplitude higher than that predicated at the theoretical instability mass ratio will likely have a mass ratio above the instability value and therefore be likely stable. \begin{table}[ht] \centering \begin{tabular}{|c|c|c|} \hline Mass ($M_1) (M_{\odot}$) &$q_{inst} (f=1)$&Max Ampl\\ \hline 0.6&0.22&0.63\\ \hline 0.7&0.19&0.60\\ \hline 0.8&0.16&0.52\\ \hline 0.9&0.14&0.45\\ \hline 1.0&0.12&0.43\\ \hline 1.2&0.08&0.32\\ \hline 1.4&0.06&0.22\\ \hline \end{tabular} \caption{Summary of the maximum amplitude at the instability mass ratio ($q_{inst}$) for systems with low mass primary component.} \end{table} \begin{equation} \label{eq:maxAmpl} MaxAmpl\ (mag)=-0.5179M_1 + 0.945 \end{equation} \section{Search for Potential Red Nova Progenitors} As noted above we confined our search to bright systems with lower mass ($0.6M_{\odot} < M_1 < 1.4M_{\odot}$) primaries. The ASAS-SN variable database provides a friendly user interface to select variable stars of different types. We used this to select all contact binaries brighter than 13.5 magnitude. We limited our search to brighter examples as random review of the ASAS-SN light curves indicated that fainter examples had too much scatter and in most cases it was impossible to visually confirm the presence of complete eclipses and/or the amplitude of the light curve. A secondary benefit for favouring brighter examples is the potential ease in obtaining long term follow up monitoring. The selected systems are within the reach of modest instruments and could potentially be observed regularly by campus based telescopes or even by advanced amateurs. The systems were then ordered by amplitude and all systems with amplitude greater than 0.65 were excluded as the maximum amplitude for a 0.6$M_{\odot}$ primary with a mass ratio at the instability level ($q_{inst} = 0.22)$ is 0.63 magnitudes. It is well established that the primary component of a contact binary systems follows in general a main sequence profile \citep{2013MNRAS.430.2029Y}. The ASAS-SN database also provided the J and H magnitude for each system. We calculated the J-H magnitude for each system and using the calibration for low mass (F3-K9) main sequence stars of \citet{2013ApJS..208....9P} we interpolated the mass and effective temperature of the primaries. All systems with masses outside our inclusion criteria were excluded. The remaining systems were examined visually to determine the presence of a total eclipse. Only those systems where a clear total eclipse were included. We note that such a crude selection system is likely to include or exclude some systems in the final sample. A more robust system of checking the variability within a defined time interval of the eclipses proved unworkable due to the cadence and scatter in many systems. The final sample totaled 189 contact binary systems. Having established the mass of the primaries for our selected 189 samples we next determined the instability mass ratio for each using Eq 1 and the maximum amplitude using Eq 3. We next compared the amplitude of the survey light curve against the maximum amplitude for potential instability. If the observed amplitude was significantly ($>5\%$) higher than the maximum instability amplitude than such a system would be expected to have a mass ratio higher than the instability mass ratio and not be considered a potential merger candidate. All systems identified as such were excluded. The 5\% leeway is arbitrary to account for the scatter normally present in survey data. This step left a sample of 65 systems of potential red nova progenitors. Even though the ASAS-SN light curves were of reasonable quality we searched the VSX database as well as the TESS and Kepler variable databases for all available light curves for the 65 systems. All light curves found from other databases were compared with the available ASAS-SN light curves and if they offered better phase coverage and clearer eclipses they were chosen for the formal light curve analysis instead of the ASAS-SN curves. Although where TESS data was available it provided the cleanest curves we did in some cases use the ASAS-SN, Catalina survey \citep{2017MNRAS.469.3688D} and SWASP Survey \citep{2006PASP..118.1407P} photometry. All the selected light curves from the final sample were analysed using the Wilson-Devenney code as noted above. We used the standard mass ratio search grid method to find the probable mass ratio for each system. Temperature of the primary was fixed according to the main sequence calibration of \citet{2013ApJS..208....9P}. Logarithmic limb darkening coefficients were interpolated from \citet{1993AJ....106.2096V}. The TESS and Kepler photometric data is provided as a flux and this was converted to magnitudes using the calibrations from \citet{2021AJ....162..170H} and \citet{2015MNRAS.447.2880A} respectively. We used the MIT Quick Look pipeline for the optimal aperture for the TESS data \citep{2020RNAAS...4..204H} and K-2 data when using Kepler mission data. The TESS photometry was acquired over a broad red to infra-red window centered on the standard $I_c$ band (786.5nm) \citep{2015JATIS...1a4003R}. We used limb darkening coefficients for the $I_c$ band when analysing the TESS data. The Kepler and SWASP photometry was acquired with wide-band filters from blue to red \citep{2010AAS...21542002V, 2006PASP..118.1407P} and we used limb darkening coefficients for the central V band. Given the scatter and potential incomplete phase coverage some of the systems upon analysis had a mass ratio above the instability mass ratio suggesting that the amplitude is probably higher than that measured by survey photometry. As noted by \citet{2022MNRAS.tmp..527C} a small uncertainty in the mass of the primary can result in a modest uncertainty in the instability mass ratio. Accordingly, for the purpose of this study we consider any system with a mass ratio below or up to 10\% above the theoretical maximum instability mass ratio to be potentially a red nova candidate. From our initial sample of 65 there were 45 that met the instability criteria and the basic parameters of these are summarised in Table 3. Abbreviation and cross matching of individual systems are presented in Table 5. The list will of course in the future be refined as the mass of the primary of these systems is more accurately determined. It must be stressed that this study only covers the ASAS-SN variable database. Given the sheer number of bright contact binaries (magnitude $\leq 13.5$) listed on the VSX ($\approx 20000$) at this time we have not systematically reviewed other survey data (some with poor search interfaces) for more examples. We hope to do this over time and add to the list progressively. For completeness we do add some examples to the list from the existing literature as described below. \section{Absolute Parameters and the Average Potential Red Nova Progenitor} \subsection{Absolute Parameters} We determined the absolute parameters for each system from the light curve solution and period of the system. As noted above mass of the primary was estimated from the J-H colour of each system. The mass ratio provides the mass of the secondary. The relative radii of the components are dependant on the mass ratio and Roche geometry as by definition both components overflow their inner Roche lobes in a contact binary system. The light curve solution provides fractional radii of the components ($a_{1,2},b_{1,2},c_{1,2}$) in three orientations. The geometric mean of these was used to estimate $r_{1,2}= \sqrt[3]{a_{1,2}b_{1,2}c_{1,2}}$. The separation ($A$) between the components was determined using Kepler's third law and the absolute radii of the components were determined as per \citep{2005JKAS...38...43A} $R_{1,2}$ = $A\times r_{1,2}$. By way of comparison with other contact binary systems we looked at our list of possible red nova progenitors with low mass ($0.6M_{\odot}<M_1<1.4M_{\odot}$) primary contact binary systems listed by \citet{2021ApJS..254...10L}. We accepted as true the masses determined by the publishing authors regardless of the methodology employed. Allowing for a 10\% margin the final list of 300 systems includes 9 systems that would be classified as potential red novas progenitors based on the instability criteria outlined above. Of these 9, three were already included in our list while the others were either fainter than our cut-off limit, had poor phase coverage and in two cases were too bright for the ASAS-SN survey equipment. We have added these to our final list of potential red nova progenitors relying on the published absolute parameters. As the catalogue of \citet{2021ApJS..254...10L} covers literature to the early part of 2021 we performed a literature search from March 2021 to March 2022 for any new reported contact binary systems that may be regarded as red nova progenitors. This resulted in the addition three more potential systems. In total we catalogue 54 low mass ratio contact binary systems that maybe regarded as potential red nova progenitors (Table 3). \begin{table*} \centering \scriptsize \begin{center} \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|} \hline Name & Period & $q$ & $q_{inst}$ range & $T_1$ & $T_2$ & $M_1(M_{\odot})$ & $R_1 (R_{\odot})$ & $R_1/ZAMS$ & Survey & References\\ \hline A0006 & 0.38318 & 0.115 & 0.108 - 0.128 & 5700 & 5699 & 0.95 & 1.34 & 1.39 & TESS & ~\\ \hline LM Psc & 0.34013 & 0.096 & 0.082 - 0.094 & 6075 & 6241 & 1.13 & 1.33 & 1.19 & ASAS-SN & ~\\ \hline A0346 & 0.30717 & 0.148 & 0.139 - 0.171 & 5120 & 5043 & 0.77 & 1.05 & 1.29 & TESS & ~\\ \hline A0458 & 0.33348 & 0.086 & 0.082 - 0.093 & 6100 & 5625 & 1.14 & 1.3 & 1.16 & TESS & ~\\ \hline A0514 & 0.34572 & 0.127 & 0.116 - 0.138 & 5600 & 5626 & 0.9 & 1.22 & 1.32 & TESS & ~\\ \hline NSVS 470 & 0.35576 & 0.078 & 0.095 - 0.110 & 5900 & 6231 & 1.04 & 1.37 & 1.32 & TESS & ~\\ \hline V644 Pup & 0.33056 & 0.14 & 0.132 - 0.161 & 5300 & 5824 & 0.8 & 1.12 & 1.33 & TESS & ~\\ \hline A0842 & 0.33353 & 0.1 & 0.086 - 0.098 & 6040 & 6315 & 1.1 & 1.3 & 1.23 & TESS & ~\\ \hline A1037 & 0.34370 & 0.09 & 0.070 - 0.085 & 6200 & 6081 & 1.21 & 1.34 & 1.14 & CATALINA & ~\\ \hline A1214 & 0.39850 & 0.085 & 0.099 - 0.116 & 5850 & 5786 & 1.01 & 1.43 & 1.41 & KEPLER & ~\\ \hline A1249 & 0.37191 & 0.095 & 0.091 - 0.104 & 5950 & 5948 & 1.07 & 1.39 & 1.3 & TESS& ~\\ \hline A1251 & 1.05207 & 0.085 & 0.073 - 0.082 & 6200 & 5599 & 1.21 & 2.89 & 2.45 & TESS & ~\\ \hline SSS1315 & 0.38281 & 0.075 & 0.108 - 0.128 & 5700 & 5330 & 0.95 & 1.39 & 1.44 & CATALINA & ~\\ \hline A1407 & 0.36358 & 0.088 & 0.086 - 0.098 & 6040 & 6105 & 1.1 & 1.39 & 1.27 & TESS & ~\\ \hline A1446 & 0.35170 & 0.09 & 0.116 - 0.138 & 5600 & 5369 & 0.9 & 1.28 & 1.39 & TESS & ~\\ \hline A1517 & 0.32518 & 0.1 & 0.132 - 0.161 & 5300 & 5368 & 0.8 & 1.13 & 1.34 & ASAS-SN & ~\\ \hline A1531 & 0.83309 & 0.085 & 0.082 - 0.093 & 6100 & 5812 & 1.14 & 2.43 & 2.17 & KEPLER & ~\\ \hline V396 Lup & 0.36324 & 0.132 & 0.120 - 0.145 & 5500 & 5794 & 0.88 & 1.27 & 1.39 & TESS & ~\\ \hline A1629 & 0.31077 & 0.059 & 0.104 - 0.122 & 5800 & 6115 & 0.98 & 1.22 & 1.23 & ASAS & ~\\ \hline A1651 & 0.35321 & 0.152 & 0.129 - 0.157 & 5300 & 5158 & 0.82 & 1.19 & 1.38 & TESS & ~\\ \hline V565Dra & 0.39032 & 0.092 & 0.091 - 0.104 & 5970 & 6055 & 1.07 & 1.42 & 1.34 & TESS & ~\\ \hline A1751 & 0.93521 & 0.105 & 0.095 - 0.110 & 5900 & 5551 & 1.04 & 2.46 & 1.49 & ASAS & ~\\ \hline A1846 & 0.30284 & 0.162 & 0.130 - 0.160 & 5250 & 5148 & 0.8 & 1.06 & 1.25 & KEPLER & ~\\ \hline A1847 & 0.68961 & 0.077 & 0.077 - 0.087 & 6150 & 5917 & 1.18 & 2.18 & 1.89 & ASAS-SN & ~\\ \hline A1907 & 0.76308 & 0.06 & 0.093 - 0.106 & 5970 & 5619 & 1.07 & 2.32 & 2.18 & ASAS-SN & ~\\ \hline A1928 & 0.32002 & 0.08 & 0.082 - 0.093 & 6100 & 6120 & 1.14 & 1.26 & 1.13 & SWASP & ~\\ \hline A1946 & 0.38378 & 0.1 & 0.120 - 0.144 & 5520 & 5524 & 0.88 & 1.3 & 1.43 & ASAS-SN & ~\\ \hline A2001 & 0.35088 & 0.12 & 0.123 - 0.140 & 5450 & 5614 & 0.86 & 1.18 & 1.33 & ASAS-SN & ~\\ \hline A2003 & 0.45720 & 0.127 & 0.107 - 0.128 & 5720 & 5613 & 0.95 & 1.52 & 1.58 & ASAS & ~\\ \hline A2044 & 0.34289 & 0.103 & 0.099 - 0.116 & 5800 & 5751 & 1 & 1.29 & 1.28 & TESS & ~\\ \hline A2044 & 0.37054 & 0.07 & 0.082 - 0.093 & 6100 & 6076 & 1.14 & 1.45 & 1.29 & ASAS-SN & ~\\ \hline NSVS 114 & 0.35470 & 0.096 & 0.116 - 0.138 & 5580 & 5572 & 0.9 & 1.28 & 1.38 & ASAS-SN & ~\\ \hline A2048 & 0.80583 & 0.115 & 0.104 - 0.122 & 5800 & 5870 & 0.98 & 2.2 & 2.23 & SWASP & ~\\ \hline A2132 & 0.31634 & 0.1 & 0.108 - 0.127 & 5700 & 5953 & 0.95 & 1.18 & 1.22 & TESS & ~\\ \hline A2145 & 0.35385 & 0.075 & 0.099 - 0.116 & 5850 & 5770 & 1.01 & 1.32 & 1.3 & TESS & ~\\ \hline SSS 2213 & 0.36990 & 0.08 & 0.082 - 0.093 & 6100 & 6249 & 1.14 & 1.42 & 1.26 & TESS & ~\\ \hline A2222 & 0.40609 & 0.133 & 0.116 - 0.138 & 5580 & 5853 & 0.9 & 1.38 & 1.49 & TESS & ~\\ \hline A2243 & 0.33535 & 0.085 & 0.091 - 0.104 & 6000 & 6004 & 1.07 & 1.29 & 1.22 & TESS & ~\\ \hline A2250 & 0.31196 & 0.132 & 0.126 - 0.153 & 5400 & 5584 & 0.84 & 1.11 & 1.26 & SWASP & ~\\ \hline A2258-26 & 0.32764 & 0.093 & 0.084 - 0.095 & 6050 & 5955 & 1.12 & 1.29 & 1.17 & SWASP & ~\\ \hline A2258 & 0.30946 & 0.162 & 0.129 - 0.157 & 5300 & 5496 & 0.82 & 1.05 & 1.22 & ASAS-SN & ~\\ \hline NSVS 902 & 0.32510 & 0.108 & 0.104 - 0.122 & 5800 & 5718 & 0.98 & 1.19 & 1.21 & SWASP & ~\\ \hline A2348 & 0.34719 & 0.108 & 0.091 - 0.104 & 5970 & 5807 & 1.07 & 1.29 & 1.22 & SWASP & ~\\ \hline V1222 Tau & 0.29536 & 0.104 & 0.116 - 0.138 & 5600 & 5439 & 0.9 & 1.06 & 1.15 & LIT & \citet{2015PASJ...67...74L}\\ \hline NSVS 431 & 0.25596 & 0.147 & 0.151 - 0.188 & 6000 & 5875 & 0.7 & 0.87 & 1.14 & LIT & \citet{2020NewA...7701352K}\\ \hline A0822 & 0.28005 & 0.11 & 0.087 - 0.099 & 5960 & 6080 & 1.1 & 1.1 & 1.01 & LIT & \citet{2015MNRAS.446..510K}\\ \hline GSC 0341 & 0.27716 & 0.055 & 0.092 - 0.106 & 5870 & 5828 & 1.06 & 1.17 & 1.1 & LIT & \citet{2021ApJ...922..122L}\\ \hline A0832 & 0.31132 & 0.067 & 0.072 - 0.081 & 6300 & 6602 & 1.22 & 1.34 & 1.13 & LIT & \citet{2016AJ....151...69S}\\ \hline NSVS 780 & 0.28120 & 0.098 & 0.141 - 0.173 & 5490 & 5706 & 0.79 & 1.1 & 1.31 & LIT & \citet{2021RAA....21..225P}\\ \hline PZ UMA & 0.26267 & 0.178 & 0.139 - 0.170 & 5430 & 4972 & 0.77 & 0.92 & 1.12 & LIT & \citet{2019PASJ...71...39Z}\\ \hline NSVS 256 & 0.28780 & 0.078 & 0.078 - 0.088 & 6030 & 6100 & 1.17 & 1.19 & 1.04 & LIT & \citet{2018RAA....18..129K}\\ \hline SX Crv & 0.31662 & 0.079 & 0.069 - 0.077 & 6340 & 6160 & 1.25 & 1.32 & 1.09 & LIT & \citet{2004AcA....54..299Z}\\ \hline A1328 & 0.38470 & 0.086 & 0.071 - 0.079 & 6300 & 6319 & 1.23 & 1.49 & 1.25 & LIT & \citet{2021ApJ...922..122L}\\ \hline ZZ PsA & 0.37389 & 0.078 & 0.086 - 0.098 & 6514 & 6703 & 1.213 & 1.42 & 1.24 & LIT & \citet{2021MNRAS.501..229W}\\ \hline \end{tabular} \caption{Summary of the pertinent light curve solution and absolute parameters of potential red nova progenitors. Entries marked with "LIT"have been taken from the literature. They were not identified from examination of the ASAS-SN light curves as described. $q_{inst}$ range is the instability mass ratio from $f=0-1$.} \end{center} \end{table*} The period distribution of low mass contact binary systems ($0.6M_{\odot} < M_1 < 1.4M_{\odot}$) as adopted from \citet{2021ApJS..254...10L} as a whole relative to potential red nova progenitors is illustrated in Figure 3. The median period for the entire sample is in the order of 0.330 days, only marginally less than 0.346 days for potential red nova progenitors. Most of the systems in both groups have periods between 0.25 and 0.5 days although the peak in the distribution around 0.35 days is more pronounced in the potential red nova progenitor sample. There is a hint of possibly some systems with higher periods being more common in the potential red nova progenitors group. The finding is in line with \citet{2022arXiv220201187K} who found that extreme mass ratio systems tended to have a slightly increased frequency of longer periods. \subsection{Average Potential Red Nova Progenitor} Based on the work of \citet{2007MNRAS.377.1635A}, \citet{2021MNRAS.501..229W} the separation at the onset of instability ($A_{inst}$) can be written as: \begin{eqnarray} \label{eq:a-inst} \frac{A_{\mathrm{\scriptscriptstyle inst}}}{R_1} &=& \frac{q\frac{k_2^2}{k_1^2}{P}{Q}}{q \frac{k_2^2}{k_1^2}{P}^2 + \frac{q}{(1+q)k_1^2}}\\ &+&\frac{\sqrt{(q\frac{k_2^2}{k_1^2}{P}{Q})^2 + 3 (1+q\frac{k_2^2}{k_1^2}{Q}^2) (q \frac{k_2^2}{k_1^2}{P}^2 + \frac{q}{(1+q)k_1^2}) }}{q \frac{k_2^2}{k_1^2}{P}^2 + \frac{q}{(1+q)k_1^2}}\nonumber \end{eqnarray} \noindent where $k_{1,2}$ is the gyration radius for the primary and secondary components and \begin{equation} P = \frac{0.49q^{2/3}-3.26667q^{-2/3}(0.27q -0.12q^{4/3})}{0.6q^{2/3} + \ln (1+ q^{1/3})} , \end{equation} \begin{equation} Q = \frac{(0.27q -0.12q^{4/3})({0.6q^{-2/3} + \ln (1+ q^{-1/3})})}{0.15 (0.6q^{2/3} + \ln (1+ q^{1/3}))}. \end{equation} Assuming our sample of potential red nova progenitors as representative we find the median radius ($\pm SD$) of the primary to be 1.285 ($\pm 0.3$) times the corresponding ZAMS equivalent. As noted previously the primary of contact binary systems can be considered as ZAMS so we can estimate the typical radius and hence the instability separation for a typical potential red nova progenitor. We perform this for low mass contact binary systems with low and high degree of contact as described in \citet{2021MNRAS.501..229W} and adopt the mean. Using equations outlined above and mean instability separation we estimate the mean period at the onset of instability ($P_{inst}$) for low mass contact binary systems. The results are summarised in Table 4. From these we derive simple quadratic relations (Figures 4 and 5) linking the mass of the primary with the instability separation and period as follows: \begin{table}[ht] \centering \begin{tabular}{|c|c|c|} \hline Mass ($M_1) (M_{\odot}$) &$A_{inst} (M_\odot)$&$P_{inst} (d)$\\ \hline 0.6&1.571&0.269\\ \hline 0.7&1.738&0.293\\ \hline 0.8&1.899&0.316\\ \hline 0.9&2.054&0.339\\ \hline 1.0&2.202&0.359\\ \hline 1.1&2.339&0.378\\ \hline 1.2&2.459&0.393\\ \hline 1.3&2.556&0.402\\ \hline 1.4&2.620&0.403\\ \hline \end{tabular} \caption{Separation and Period at the onset of instability for modelled potential red nova progenitors} \end{table} \begin{equation} A_{inst} = -0.6766M_1^2 + 2.6932M_1 + 0.1878 \end{equation} \begin{equation} P_{inst}(d) = -0.1446M_1^2 + 0.4645M_1 + 0.0401 \end{equation} From the above we see that a typical one solar mass contact binary system that is near the onset of instability will have a mass ratio near 0.12, maximum light curve amplitude near 0.43 and period near 0.36d. As a means of selecting potential low mass ratio contact binary stars for photometric analysis one could follow the relationships described to select candidates that are more likely to reflect features of orbital instability. Given the inherent scatter and varying cadence of survey photometry we stress the above analysis is only an aid in the selection process and follow up observations would be required to confirm instability criteria that maybe evident on survey photometry. We are however confident given the success of survey photometric analysis compared to dedicated ground based observations in determination of accurate light curve solutions \citep{2020ApJS..247...50S, 2020MNRAS.493.1565D, 2022arXiv220209120W} that if the survey data is chosen to ensure high cadence, low scatter and full phase coverage than the properties deduced from such data would be confirmed on follow-up study and hence are confident that the systems selected represent potential red nova progenitors. \section{Discussion and Conclusions} Although contact binary merger events are predicted to be relatively frequent so far only a single event has been confirmed and that only in retrospect. The linking of orbital stability with the mass ratio of contact binary systems has long been recognised as potential avenue for identifying unstable systems \citep{1995ApJ...444L..41R, 2007MNRAS.377.1635A, 2009MNRAS.394..501A}. The earlier work clearly showed that orbital instability is likely to occur at very low mass ratios and higher mass ratio configurations are likely to be stable. Our theoretical work \citep{2021MNRAS.501..229W} linking the instability mass ratio to the mass of the primary has progressed this further by demonstrating that there exists no global minimum mass ratio at which a system will become unstable rather the instability mass ratio is dependant on the mass of the primary component. In addition, we showed that systems with less massive primaries can have mass ratios higher than 0.2 and still be potentially unstable. When combined with work showing the suitability of survey photometry for light curve analysis \citep{2020ApJS..247...50S, 2020MNRAS.493.1565D, 2022arXiv220209120W} has greatly facilitated the potential for being able to to identify low mass ratio contact binary systems. In this study we enhance this capability by establishing simple light curve and colour parameters that can be used to exclude systems that are likely to have mass ratios above the theoretical instability value. By excluding likely stable systems we greatly increase our chance of identifying potentially unstable systems from the remaining sample. We apply the techniques on bright contact binary systems from the ASAS-SN and identify approximately 50 extreme low mass ratio system (most previously not reported) satisfying the mass ratio criteria for orbital instability. As with almost all low mass contact binary systems \citep{2021ApJS..254...10L} our identified sample of potential merger candidates demonstrate radii that are significantly larger than their main sequence counterparts and secondaries that are considerably hotter than main sequence counterparts of similar mass. We find that relative to the general population of comparable contact binaries those exhibiting signs of orbital instability generally have similar periods although with a more pronounced peak near 0.35d. In addition there is a significant number of systems, relative to the general population, that have longer ($>0.5d$) periods. \citet{2022arXiv220201187K} suggest that it is possible for the period to lengthen to above 0.5d in some cases of extreme low mass ratios and the onset of orbital instability. In this respect our sample of possible unstable systems with long periods represent a good subset study group for future observations. From our sample of potential red nova progenitors we construct theoretical models for low mass potentially unstable contact binary systems which place further constraints on the light curve morphology and timings. We note that for low mass systems there is a narrow period domain from $\approx 0.27d$ to $0.4d$ at which they may become unstable and this increases with the mass of the primary. Those results will further aid in the identification of potential red nova progenitors from large survey samples. We hope to employ all the techniques and modifications described to the VSX database to further identify bright potential red nova candidates. The techniques described in no way ensure all potential red nova progenitors are identified. Limiting our selection to those exhibiting complete eclipses clearly excludes a significant portion of contact binary systems, however, identification of potential systems among these would require both time consuming dedicated observations on modest sized telescopes and high resolution spectroscopic observations. Such requirements are unlikely to be readily available. The light curves of non totally eclipsing systems cannot be reliably analysed due to the high degree of correlation between the three geometric parameters specifically the inclination, mass ratio and degree of contact. The presence of a total eclipse places significant constraints, particularly in the $q/i$, domain allowing for manual search to be performed to find the correct light curve solution. Also, as already noted, survey photometric data although useful in light curve analysis does have limitations given the scatter, particularly with respect to the determining the fill-out fraction \citep{2020MNRAS.493.1565D, 2022arXiv220209120W}. Looking at Equations 1 and 2 above we see that the fill-out can have a significant influence on the instability mass ratio particularly for systems with the primary below one solar mass. The methodology employed places sorting restrictions assuming a high fill-out so it is possible that some of the identified samples may still be in the stable range. The small sample identified offers a good opportunity for dedicated observations with modest optical instruments to further refine the list of potential merger candidates. To this end we have started a programme of dedicated multi-band observations of some of the identified systems with instruments in the 0.5m range with results to be presented progressively. \begin{table*} \centering \begin{tabular}{|l|l|l|} \hline Abbreviation & RA/DEC & Name\\ \hline A0006 & 00 06 50 -35 37 29 & ASASSN-V J000649.98-353729.1\\ \hline LM Psc & 00 34 13 20 52 25 & ASASSN-V J003412.63+205225.4\\ \hline A0346 & 03 46 33.6 41 08 15.8 & ASASSN-V J034633.63+410815.8\\ \hline A0458 & 04 58 14 06 43 09 & ASASSN-V J045813.80+064309.1\\ \hline A0514 & 05 14 59 -73 56 15 & ASASSN-V J051459.38-735615.4\\ \hline NSVS 470 & 07 19 25 41 57 04 & ASASSN-V J071924.64+415705.4\\ \hline V644 Pup & 07 27 29 -50 56 30 & ASASSN-V J072728.92-505631.1\\ \hline A0842 & 08 42 20 -03 03 25 & ASASSN-V J084219.98-030325.3\\ \hline A1037 & 10 37 37 -37 09 30 & ASASSN-V J103736.72-370928.0\\ \hline A1214 & 12 14 30.5 -02 57 04 & ASASSN-V J121430.46-025704.6\\ \hline A1249 & 12 49 08 -29 44 38 & ASASSN-V J124907.83-294437.7\\ \hline A1251 & 12 51 19 -28 08 25 & ASASSN-V J125119.31-280824.8\\ \hline SSS1315 & 13 15 59.6 -37 00 17.7 & ASASSN-V J131559.62-370018.8\\ \hline A1407 & 14 07 13 -30 24 44 & ASASSN-V J140712.93-302443.8\\ \hline A1446 & 14 46 21 -30 04 40 & ASASSN-V J144620.72-300440.9\\ \hline A1517 & 15 17 02 14 10 23 & ASASSN-V J151701.56+141023.3\\ \hline A1531 & 15 31 18 -17 42 36 & ASASSN-V J153118.10-174236.0\\ \hline V396 Lup & 16 03 02 -37 49 21.2 & ASASSN-V J160302.12-374921.2\\ \hline A1629 & 16 29 19.9 35 40 03 & ASASSN-V J162919.96+354003.5\\ \hline A1651 & 16 51 39.4 22 55 44 & ASASSN-V J165139.40+225543.0\\ \hline V565Dra & 17 38 49.82 +57 12 23.2 & ASASSN-V J173849.79+571222.6\\ \hline A1751 & 17 51 10 03 13 20 & ASASSN-V J175109.86+031319.5\\ \hline A1846 & 18 46 43.4 -27 36 29 & ASASSN-V J184643.38-273629.3\\ \hline A1847 & 18 47 37 21 56 06 & ASASSN-V J184737.28+215606.0\\ \hline A1907 & 19 07 28.30 -53 47 24.7 & ASASSN-V J190728.21-534724.9\\ \hline A1928 & 19 28 49 -40 45 54 & ASASSN-V J192848.87-404554.0\\ \hline A1946 & 19 46 45 -04 03 39 & ASASSN-V J194644.82-040339.6\\ \hline A2001 & 20 01 26 07 37 40 & ASASSN-V J200125.92+073739.9\\ \hline A2003 & 20 03 04 -02 56 02 & ASASSN-V J200303.64-025603.3\\ \hline A2044 & 20 44 00 57 52 17 & ASASSN-V J204400.26+575216.7\\ \hline A2044 & 20 44 52 06 22 31 & ASASSN-V J204452.22+062231.3\\ \hline NSVS 114 & 20 45 26 16 59 13 & ASASSN-V J204525.65+165912.7\\ \hline A2048 & 20 48 35 -46 09 42 & ASASSN-V J204835.36-460942.4\\ \hline A2132 & 21 32 19.4 -53 51 33 & ASASSN-V J213219.30-535132.7\\ \hline A2145 & 21 45 37 -58 35 00 & ASASSN-V J214537.35-583459.9\\ \hline SSS 2213 & 22 13 27.3 -44 54 00.5 & ASASSN-V J221327.33-445400.3\\ \hline A2222 & 22 22 17 37 37 41 & ASASSN-V J222217.40+373740.6\\ \hline A2243 & 22 43 19 -73 51 18 & ASASSN-V J224318.80-735718.0\\ \hline A2250 & 22 50 00 -23 16 24 & ASASSN-V J224959.89-231623.1\\ \hline A2258-26 & 22 58 26 -26 03 36 & ASASSN-V J225825.91-260337.8\\ \hline A2258 & 22 58 50 13 49 18 & ASASSN-V J225849.67+134917.7\\ \hline NSVS 902 & 23 19 49 36 03 51 & ASASSN-V J231948.59+360350.6\\ \hline A2348 & 23 48 23 -40 54 41 & ASASSN-V J234823.30-405440.6\\ \hline V1222 Tau & 03 28 26 09 04 24 & V1222 Tau\\ \hline NSVS 431 & 04 59 45 49 25 03 & NSVS 4316778\\ \hline A0822 & 08 22 43 19 26 58 & ASASSN-V J082243.00+192658.5\\ \hline GSC 03415-02229 & 08 27 01 46 28 50 & GSC 03415-02229\\ \hline A0832 & 08 32 41 23 32 26 & ASASSN-V J083240.96+233225.9\\ \hline NSVS 780 & 09 06 43 70 03 29 & NSVS 780649\\ \hline PZ UMA & 09 29 07 49 51 23 & PZ UMA\\ \hline NSVS 256 & 10 10 42.7 67 39 31 & NSVS 2569022\\ \hline SX Crv & 12 40 15 -18 48 01 & SX Crv\\ \hline A1328 & 13 28 29 55 52 45 & ASASSN-V J132829.15+555245.4\\ \hline ZZ PsA & 21 50 35.2 -27 48 35.5 & ZZ PsA\\ \hline \end{tabular} \caption{Abbreviation, coordinates and ASAS-SN designation of low mass contact binary systems with mass ratios suggesting orbital instability.} \end{table*} \section*{Acknowledgements} Based on data acquired on the Western Sydney University, Penrith Observatory Telescope. We acknowledge the traditional custodians of the land on which the Observatory stands, the Dharug people, and pay our respects to elders past and present.\\ BA acknowledges the financial support of the Ministry of Education, Science and Technological Development of the Republic of Serbia through the contract No.~451-03-68/2022-14/200104.\\ During work on this paper, G. Djurasevic and J. Petrovic were financially supported by the Ministry of Education and Science of the Republic of Serbia through contract 451-03-9/2021-14/200002.\\ This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. \vspace{-1em} \bibliography{P1.bib}

Title: Time scales of small body differentiation

Abstract: The petrologic and geochemical diversity of meteorites is a function of the bulk composition of their parent bodies, but also the result of how and when internal differentiation took place. Here we focus on this second aspect considering the two principal parameters involved: size and accretion time of the body. We discuss the interplay of the various time scales related to heating, cooling and drainage of silicate liquids. Based on two phase flow modelling in 1-D spherical geometry, we show that drainage time is proportional to two independent parameters: $\mu_m/R^2$, the ratio of the matrix viscosity to the square of the body radius and $\mu_f/a^2$, the ratio of the liquid viscosity to the square of the matrix grain size. We review the dependence of these properties on temperature, thermal history and degree of melting, demonstrating that they vary by several orders of magnitude during thermal evolution. These variations call into question the results of two phase flow modelling of small body differentiation that assume constant properties.For example, the idea that liquid migration was efficient enough to remove $^{26}$Al heat sources from the interior of bodies and dampen their melting (e.g. Moskovitz and Gaidos, 2011; Neumann et al., 2012) relies on percolation rates of silicate liquids overestimated by six to eight orders of magnitude. In bodies accreted during the first few million years of solar-system history, we conclude that drainage cannot prevent the occurrence of a global magma ocean. These conditions seem ideal to explain the generation of the parent-bodies of iron meteorites. A map of the different evolutionary scenarios of small bodies as a function of size and accretion time is proposed.

https://export.arxiv.org/pdf/2208.10357

\begin{frontmatter} \title{Time scales of small body differentiation} \author[a]{Marc Monnereau} \author[a,b]{Jérémy Guignard} \author[a,c]{Adrien Néri} \author[a]{Michael J. Toplis} \author[a]{Ghylaine Quitté} \address[a]{IRAP, University of Toulouse, CNRS, Toulouse, France} \address[b]{Now at ICMCB, CNRS, Université de Bordeaux, Bordeaux, France} \address[c]{Now at BGI, University of Bayreuth, Bayreuth, Germany} \begin{keyword} Asteroid \sep differentiation \sep thermal history \sep melt migration \end{keyword} \end{frontmatter} \newpage \section{Introduction} The meteoritic record constitutes a well-preserved selection of samples that allow us to shed light on the differentiation processes that occurred on small rocky bodies that accreted early in solar-system history. From the suite of specimens available, there is an obvious link between the overall degree of melting of the parent bodies and the degree of differentiation. Unlike larger planets, the main heat source of these small bodies is the energy produced by the decay of short-lived radionuclides such as $^{26}$Al: the earlier a body accreted, the greater the heating potential and melting degree. The parent bodies of the iron meteorites accreted very early \citep[$< 0.3$\,Myr after CAIs ---CAI stands for Calcium Aluminum Inclusions, the most refractory objects whose condensation defines the age of the solar system---][]{kruijer2014}, with the potential to produce large-scale magma oceans that ensured an efficient metal-silicate differentiation through an iron-rain scenario \cite[e.g.][]{Rubie2003}. Conversely, chondritic parent bodies accreted much later \citep[$>$ 2 Myr after CAIs][]{sugiura2014} such that peak temperatures were not high enough to reach the iron-sulfur or silicate solidi, preventing any differentiation. In between these two differentiation endmembers are the primitive achondrites that accreted early enough $\approx$1.3 Myr after CAIs to be able to cross the silicate solidus but late enough so that their degrees of partial melting did not exceed 20 vol\% \citep{sugiura2014}. Under these conditions, these samples experienced partial differentiation, with the loss of basaltic and sulfur-rich metallic melts. At first sight, a difference in accretion dates thus provides a satisfactory explanation for the different degrees of differentiation found in the meteoritic record. However, body size will play an important role too. Indeed, for the same accretion time, a smaller body will dissipate its heat with greater efficiency than a larger one, thus yielding a different heating potential and a lower peak temperature for the former. Attempts to take this effect into account are scarce and restrained to a handful of samples for which precise anchor-points in the cooling history have been determined (e.g. \citeauthor{Breton2015}, \citeyear{Breton2015} for Tafassasset and \citeauthor{Neumann2018}, \citeyear{Neumann2018} for the acapulcoite-lodranite parent body). The present paper aims at describing the general pattern of evolution and differentiation that allows the fate of a parent body to be predicted as a function of its size and accretion time. An early attempt in this direction was made by \cite{Moskovitz2011} who used simple considerations of the time scales involved to study the consequences of silicate melt migration on the thermal evolution of small bodies. In particular, they popularised the idea that low viscosity melt ($<1$\,Pas) can transport $^{26}$Al heat sources to the surface on time scales shorter than their mean half-live of radioactive decay. They stressed the role of melt viscosity in controlling the degree of melting, mentioning that, above 1\,Pas, melt content in bodies accreted before 1.5\,Myr should exceed the 50\% threshold capable of triggering the generation of a magma ocean --- above about 50\% liquid, the rigid silicate framework dissociates and the matrix becomes a dense suspension of crystals in a liquid; the viscosity of the solid-liquid mixture changes from that of a solid to that of a liquid \citep[e.g.][]{Solomatov2015}. Using the more complex approach offered by two-phase flow numerical modelling, \cite{Lichtenberg2019} found a complementary result. They emphasised the role of grain size, indicating inefficient liquid migration with a matrix grain size below one millimeter (conclusion drawn making the assumption of a melt viscosity of the order of 1 Pas). In fact, the percolation rate is partially controlled by the ratio $\mu_f/a^2$ where $\mu_f$ is the melt viscosity and $a$ is the grain size, such that efficient drainage is conditioned to $\mu_f/a^2 > 10^6$, another way of expressing the result found by \cite{Lichtenberg2019}. For their diagram on small body evolution, \cite{Moskovitz2011} adopted 1\,Pas for the melt viscosity and an ad hoc relationship between grain size and partial melting degree that inevitably boosts the melt migration: they considered an initial grain size of 100\,$\mu$m with an additional 90\,$\mu$m per percent of partial melting, such that the grain size reaches the millimeter scale at 10\% of partial melting. In this respect it is of note that the viscosity of a melt resulting from 10\% of partial melting of an H-chondrite material is in fact close to 1000\,Pas \citep{Dingwell2004} and decreases to 1\,Pas for 50\% of partial melting, the tipping point toward the magma ocean. In this respect, we note that the grain size in natural samples such as lodranites for which the degree of melting reached 20\% is around 500\,$\mu$m \citep{Keil2018}. While the approach developed by \cite{Moskovitz2011} remains not only correct but also elegant, their evolutionary diagram should be reconsidered in the light of more recent knowledge on material properties such as grain growth laws and compositional and temperature effects on melt viscosity. \cite{Lichtenberg2019} have also sketched the possible evolution of small bodies as a function of two parameters: the accretion time, and not their size but a dimensionless number, $\textrm{R}_{\textrm{seg}}$, which depends on almost all the parameters that control the physics of the differentiation except the body radius (fixed to 60\,km in their numerical models). $\textrm{R}_{\textrm{seg}}$ is a function of the ratio of two time scales, one related to the rate of heating by short-lived elements and the other corresponding to melt migration, or more precisely the Darcy flow rate of silicate liquid through the porous matrix that constitutes the unmelted residual rock. At least two other time scales, both explicitly dependent on the radius, are involved in this problem: the first is the time during which the molten silicate persists, which is controlled by the cooling rate of the body, and the second is related to compaction of the matrix. Generally, the latter is considered to be non-limiting and therefore neglected, as in the model developed by \cite{Neumann2012}, because it is assumed to be too fast compared to the Darcy flow, except for very small bodies as we shall see. The two time scales governing fluid migration, Darcy-flow and compaction, are affected, not only by temperature, but also by the thermal history itself via the size of the matrix grains whose growth over time is thermally activated. Therefore, although the Darcy time scale is not explicitly expressed in terms of the size of the body, it also depends on it. To describe the broad possible differentiation pathways for small bodies as a function of their time of accretion and size, we have chosen not to resort to a complete modelling of the thermochemical evolution as in \cite{Mizzon2015} or \cite{Lichtenberg2019}, but rather to follow the approach of \cite{Moskovitz2011} and to treat the thermal aspect and the migration of the fluids separately in order to extract the characteristic time scales from simple modeling. \section{Thermal time scales} Various time scales can be defined to characterize the thermal evolution of small bodies, but some are more relevant than others for the present problem. Basically, three of them are of paramount importance: i) related to the heating rate, ii) related to the cooling rate, and iii) related to the life time of the heating sources, i.e. the short-lived radionuclides. \cite{Lichtenberg2019} introduced the heating time scale as the ratio $\rho C_p \Delta T/ Q$, where $\rho$ is the density, $C_p$ the heat capacity, $Q$ the decay power per unit volume delivered by the short lived elements like $^{26}$Al and $^{60}$Fe which thus depends on the body accretion time (All notations and symbols used are listed in Table\,\ref{table:parameters}). $\Delta T$ is the temperature difference between the solidus and the accretion temperature, so that this time scale represents the time necessary to reach the solidus. This does not depend on the size of the body, despite the fact that a cooling rate that varies as the inverse square of the body radius is also involved. At equilibrium, the temperature of a body of radius $R$ heated by constant internal sources $Q$ reaches a maximum at its centre: $Q R^2/ 6 k_T$, with $k_T$ the thermal conductivity. For instance, at the time of the formation of the solar system, $Q$ is of the order of $6\times 10^{-4}$W/m$^3$ for H-type chondritic material, which implies a maximum temperature of 800\,K and 8\,K for bodies of 10\,km and 1\,km in diameter, respectively. This explains, for example, why dust, too small to retain heat, cannot be heated by the short-lived elements, and also why the elements $^{235,238}$U and $^{40}$K can heat planets, but not small bodies. Of course, these latter elements also heat planets because of their long lifespan. The life time of heat sources thus also plays a critical role. However, rather than using these three time scales (heating time, cooling time and radionuclide lifetime), we prefer to consider two time scales deduced from small body thermal evolution: a) the time necessary to reach the maximum melting degree from the onset of melting, and b) the time span of the existence of melt (i.e. the time interval spent above the solidus). Both of these time-spans are affected by melt migration resulting in the fact that: 1) they cannot be determined analytically and 2) an approach based on dimensionless numbers (e.g. \cite{Lichtenberg2019}) cannot be used. Despite these limitations, relevant time-scales may be deduced from numerical simulations, and it is that approach that is taken here. \subsection{Thermal evolution modelling} The time necessary to reach the maximum melting degree from the beginning of melting and the time span of melt are derived from thermal evolutions of small bodies calculated from a model of pure conduction taking into account a realistic thermodynamic description of the melting of chondritic material. The model, previously used in a study devoted to the determination of the characteristics of the H chondrite parent body \citep[see][for numerical details]{Monnereau2013}, is a classical 1D spherical model based on the resolution of the conservation of heat. The latter can be written as: \begin{equation} \sum_{i=s,m} \rho_i \dfrac{\partial H_i}{\partial t} = \left[ \sum_{i=s,m}\rho_i \dfrac{d H_i}{dT}\right] \dfrac{\partial T}{\partial t} = \Div (k_T \Grad T) +\rho Q, \label{eqn:EnergyC} \end{equation} with $H$ the enthalpy, $k_T$ the thermal conductivity and $Q$ the heat production by radioactive decay, subscipts $s$ and $m$ referring to the silicate and metallic components of the material that has been supposed to have a H chondrite composition \citep{Wasson1988}. The enthalpy of the silicate component is computed using a 1\,bar equilibrium melting simulation along the iron-wüstite (IW) % buffer on the “Rhyolite-Melts" software \citep{Asimow1998, Ghiorso1995, Gualda2012, Ghiorso2015}. For small bodies with a radius less than a thousand kilometers, the energy is essentially provided by the decay of short-lived radionuclides such as $^{26}$Al. $^{60}$Fe is another short-lived radionuclide, with a comparable decay energy, but a half-life almost four times longer \citep{Castillo2009}. \cite{Quitte2010} showed that there were probably important heterogeneities of iron isotopes in the early solar nebula. The initial $^{60}$Fe/$^{56}$Fe ratio of the reservoir from which angrites and eucrites originated could be as low as $\sim 10^{-8}$ \citep{Quitte2011, Tang2012}. For some chondrites, it could be in the range of $4-7 \times 10^{-7}$ \citep[e.g.][]{Mishra2014}, which remains two orders of magnitude lower than the initial $^{26}$Al/$^{27}$Al ratio, so that we chose to neglect this radiogenic heat source. The heating power supplied by $^{26}$Al content is: \begin{equation} Q(t)=X_{\textrm{Al}}Q_0 \exp \left[ -\lambda_{^{26}\textrm{Al}} (t+t_{acc})\right], \label{eqn:heat_sources} \end{equation} where $Q_0$ is the heating rate per mass unit of pure aluminium at CAI condensation, $t_{acc}$ the accretion time, $\lambda_{^{26}\textrm{Al}}$ the $^{26}$Al decay constant and $X_{\textrm{Al}}$ the aluminium mass fraction of the material. \subsection{Melting time scale and melt lifetime} \label{paragraph:thermal time scales} The two thermal time scales relevant to melt migration, i.e. the melting rate time scale and the melt persistence time scale, are presented on Figure\,\ref{Fig:melting_times} as a function of the radius and accretion time of the body. The first of these time-scales is related to heating power. Rather than considering the time required to reach the onset of silicate melting as in \cite{Lichtenberg2019}, the choice is made here to focus on the time span between the onset of melting and the maximum degree of melting. This choice is the result of the fact that we concentrate on the comparison of time intervals during which melt is present. From a practical point of view, given that the temperature evolution at the centre is marked by a plateau before its maximum, we considered the time to reach $T_\textrm{max}-5$K. The temperature to reach 50\% partial melting is another upper bound. As mentioned earlier, it marks the dissociation of the solid matrix and is accompanied by a drop in viscosity of several orders of magnitude. This is the tipping point for the melt migration to the magma ocean. Therefore, Figure\,\ref{Fig:melting_times}.a shows the time required to reach $\min (T_\textrm{max}-5K, T_{50\%})$ since the onset of melting. In this representation, we note that size only plays a role for bodies with a radius of less than 50 km, showing that the cooling time appears to be a limiting factor for melt production only for the smallest bodies. Above this size, the melting time is only a function of the age of accretion. Figure\,\ref{Fig:melting_times}.b displays the time during which at least 1\% of melting is present. As expected, this time increases as the square of the radius. It is also marked by a decrease when the accretion time increases. It should be noted that these times are deduced from the maximum temperatures reached at the centre of the body. They are therefore only representative of the main part of a body larger than 50 km, and only of a very restricted central part of smaller bodies. \section{Melt drainage time scale} The two thermal time scales presented above should be compared with a time scale related to the mobility of silicate liquids. For this purpose, we have chosen to calculate the time required for the separation of the fluid from the solid in an initial homogeneous mixture. These dynamics are described by the compaction equations, first introduced in geosciences by \cite{McKenzie1984}. Here, we used the formalism developed by \cite{Bercovici2001} and \cite{Bercovici2003} that treats both phases as equivalent. This formalism is summarized in \ref{paragraph:Two_phase_flow_formalism}. \subsection{Drainage time for small bodies} The rate of fluid-matrix separation depends on the ability of the matrix to deform and the fluid to flow through the matrix. The two processes have different characteristic times. The former, the compaction time $\tau_C$, can be defined as the ratio of matrix viscosity $\mu_m$ divided by the fluid-matrix difference pressure scale, $\delta \rho g_s R$: \begin{equation} \tau_C = \dfrac{\mu_m}{\delta \rho g_s R} \propto \dfrac{\mu_m}{R^2}, \label{eqn:tauC} \end{equation} with $\delta \rho = \rho_m - \rho_f$, the density contrast between the matrix and the fluid, R the radius of the body and $g_s$ the gravity at its surface. As $g_s$ is proportional to R, $\tau_C$ is inversely proportional to $R^2$. The second characteristic time, the Darcy time $\tau_D$, is the ratio of the body radius $R$ to the filtration velocity: \begin{equation} \tau_D = \dfrac{R \mu_f}{\delta \rho g_s k_0} \propto \dfrac{\mu_f}{a^2}. \label{eqn:tauD} \end{equation} The filtration velocity is proportional to the fluid-matrix difference pressure, to the matrix permeability $k_0$ ---the permeability at the reference porosity $\phi_0$--- and inversely proportional to the fluid viscosity $\mu_f$. As a consequence, this time scale does not depend on the body size. Instead, the relevant length that appears here is the grain size, $a$. Indeed, the permeability is proportional to the square or the cube of the matrix grain size, depending on the pore geometry. Here, we adopted the classic permeability law considered for connected melt tubes : $k(\phi) = a^2\phi^2/(72\pi)$ \citep{maaloe1982permeability}. Other geometries may be considered. For instance, in case of connected films, the permeability law becomes $a^3\phi^3/648$ \citep{Schmeling2000}. Interestingly, the ratio of the two characteristic times corresponds to the square of the ratio of the two natural length scales of the problem, the body radius $R$ and the compaction length $L_c$: $\tau_D/\tau_C = R^2/L_c^2$. The compaction length is the length beyond which the compaction of a constant porosity matrix occurs \citep{McKenzie1984} and only depends on the properties of the matrix and the fluid: \begin{equation} L_c=\sqrt{\dfrac{\mu_m k_0}{\mu_f}}. \label{eqn:Lc} \end{equation} As mentioned above, since the surface gravity of a small body is proportional to its radius, it is also worth noting that the Darcy time does not depend on the size of the body, whereas the compaction time decreases as its square. As a consequence, in the case of porous flow dominated regimes, the radius of the body will not control the drainage characteristic time. On the contrary, when matrix deformation is the slower mechanism, the drainage time will be much longer for smaller bodies than for larger ones. \subsection{Drainage experiments} Drainage experiments consist in solving the flow equations (\ref{eqn:MassC_m}) \& (\ref{eqn:phi_adim}) with no source term, but from an initial constant porosity profile. They were performed for various values of $\phi_0$ and $R/L_c$ ratios. Figure\,\ref{Fig:drainage_time} reports the dependence of the drainage time $\tau$ as a function of both parameters. $\tau$ is arbitrarily defined as the time necessary to reach 90\% of segregation between the fluid and the solid. The segregation is evaluated through the function $s(\phi)$: \begin{equation} s(\phi)= 1 - \dfrac{1}{V \phi_0 (1-\phi_0)}\int \phi (1-\phi) dv, \label{eqn:segragation_1} \end{equation} where $V=\int dv$ is the volume of the body. $s(\phi)$ is also the second central moment of the fluid distribution: \begin{equation} \begin{split} s(\phi)= \dfrac{1}{V \phi_0 (1-\phi_0)} \int (\phi-\phi_0)^2 dv. \end{split} \label{eqn:segragation_2} \end{equation} When the fluid is homogeneously distributed within the solid, its volume fraction is constant and equal to $\phi_0$ everywhere, $s(\phi_0)$ is then zero, while $s(\phi)$ is maximum and equal to 1 when the separation is complete, i.e. when $\phi$ is equal to 0 or 1. Figure\,\ref{Fig:drainage_time} shows that the drainage time is proportional to the compaction time for smaller bodies and to the Darcy time for larger ones. As explained above, in small bodies, the migration of fluid is controlled by the ability of gravitational forces to deform the matrix, whereas in large ones the limiting factor is the friction between the fluid and the matrix. Compaction waves develop in the latter case, while, in the former, the radial distribution of the liquid remains a monotonically increasing function of the radius. We observe the transition between both regimes around $\phi_0 R / L_c \sim 4$. An eye-fitting gives $\phi_0 \tau/\tau_C \simeq 63 \, \phi_0^{0.1} + 3.25 \, \phi_0^2 R^2 / L_c^2$. Since $\phi_0^{0.1}$ varies little for the range of porosity considered, the quantity $\phi_0 \tau/\tau_C$ essentially depends on the characteristic dimensionless number $\phi_0 R / L_c$. This is well illustrated by Figure\,\ref{Fig:drainage_time}b. The drainage time is found to be $\tau \simeq 63 \, \phi_0^{-0.9} \tau_C + 3.25 \, \phi_0 \tau_D$ and varies as function of three independent quantities $\phi_0$, $\mu_m / R^2$ and $\mu_f /a^2$ : \begin{equation} \begin{split} \phi_0 \tau & \simeq \dfrac{3}{4 \pi \delta \rho \bar{\rho} G} \left[ 63 \, \phi_0^{0.1} \dfrac{\mu_m}{R^2} + 3.25 \times 72 \pi \, \dfrac{\mu_f}{a^2} \right] \\ & \approx 4.3 \times 10^{4} \dfrac{\mu_m}{R^2} + 6.3 \times 10^{5} \dfrac{\mu_f}{a^2}, \end{split} \label{eqn:empirical_phi_tau} \end{equation} with $\tau$, lengths and viscosities in SI. A graphical representation of this relationship is given in Figure\,\ref{Fig:drainage_time}c. The line $\phi_0 R/L_c=4$ separating the two regimes is an axis of symmetry for the quantity $\phi_0 \tau$. From any point located on this line, a decrease of $\mu_m/R^2$ or $\mu_f/a^2$ does not affect the drainage time, whilst an increase by a given factor increases the drainage time by the same factor. The blue arrow on Figure\,\ref{Fig:drainage_time}.c depicts what could be the trajectory of a body during its progressive melting, both matrix and fluid viscosity decreases and grains grow. A more precise description of such a trajectory is given in section \ref{sec:drainage_time_variation_during_melting} and the possible extent of the path in the parameter space is explained in the next section. \subsection{The parameters of the drainage time} \cite{Lichtenberg2019} emphasize the key role played by grain size in the differentiation of small bodies, pointing to inefficient percolation below a millimetre grain size. Here, we demonstrated that melt drainage is in fact controlled by two independent quantities, $\mu_m/R^2$ and $\mu_f/a^2$, in which the radius of the body, and the viscosities of the liquid and solid silicate are also involved. \subsubsection{Grain size} \label{paragraph:Grain_size} During the thermal evolution of a body, the quantity $\mu_f/a^2$ is likely to experience a drop by six to seven orders of magnitude, jointly due to a decrease in liquid viscosity and an increase in grain size. First, the thermally activated silicate grain growth will promote a significant increase in grain size. As a matter of fact, the size of the silicate grains observed in meteorites increases with the maximum temperature they have experienced. It ranges from 10 to 100\,$\mu$m in chondrites depending on their metamorphic grade, goes up to 500-700\,$\mu$m in some achondrites like lodranites \citep{Krot2014} where the melting degree reached 20\%, and even to one centimeter for olivines in pallasites for which the melting degree has exceeded the threshold of the matrix disaggregation.% As grains grow more or less quickly depending on the temperature, their size can be computed by integration of the growth law along their thermal history. In general terms, normal grain growth is commonly described by an equation of the form: \begin{equation} a^n-a_0^n = A t, \label{eqn:growth_law} \end{equation} where $a$ is the grain size at time $t$, $a_0$ the initial grain size. A is a thermally activated rate constant \citep{Atkinson1988}: \begin{equation} A = A_0\,e^{-E_a/R_gT}, \label{eqn:growth_rate} \end{equation} with $E_a$ an activation energy, $R_g$ the gas constant and $A_0$ a constant. The exponent, $n$, theoretically an integer, may adopt various values from 2 to 5 depending on the mechanism controlling the grain coarsening \citep[e.g.][]{Brook1976,Atkinson1988, Evans2001}. For the implementation of grain size evolution in their numerical model of small body differentiation, \cite{Neumann2012, Neumann2013, Neumann2014, Neumann2018} adopted a purely theoretical expression of relation (\ref{eqn:growth_rate}). They followed \cite{Taylor1993} who used the Lifshitz-Slyozov-Wagner relationship established for diffusion-controlled coarsening of particles in dilute solutions \citep[e.g.][]{Greenwood1969}. In this case the exponent is $n=3$ and the rate constant $A=8 \mathcal{V}^2 \gamma c D/9R_g T$, with $\mathcal{V}$ the molar volume of the silicate, $\gamma$ the surface free energy of the crystal-liquid interface, $c$ the equilibrium concentration of solute and D the diffusion coefficient. Beyond the fact that all these parameters are considered as constant in \cite{Taylor1993} and \cite{Neumann2012}, which yields a growth rate inversely proportional to temperature contrary to experimental observations, the assigned values (the same in both studies) lead to a rate constant $A=1.6\times 10^{-13}$m$^3$/yr at 1500K. This predicts a grain size of 0.55\,mm after $10^3$ yr, 1.2\,mm after $10^4$ yr, 2.5\,mm after $10^5$ yr and 5.4\,mm after $10^6$ yr; this latter value is higher by one order of magnitude than the size of the most evolved crystals measured in achondrites, whose thermal history is well over a million years. The overestimation of crystal growth places the \citeauthor{Neumann2012}'s model well above the percolation efficiency threshold identified by \cite{Lichtenberg2019} and may affect the conclusions drawn in their papers. This will be discussed later. In addition to the many processes that may govern grain growth, various parameters can also be involved, such as the presence of impurities, secondary phases or melt. Therefore, only an experimental approach can identify those really at work. Several studies have been performed aimed at determining the values of the growth exponent $n$ and the activation energy $E_a$ on peridotitic material. For the first one devoted to olivine, the theoretical growth exponent derived for a pure single phase, n=2 or 3, ($E_a= 520, 600$ kJ/mol, respectively) has been satisfactorily fitted by experimental data run with San Carlos olivine at 0.1\,MPa and 300\,MPa in dry and wet condition \citep{Karato1989}. However, in experiments conducted at 0.1\,MPa on dry aggregates of synthetic olivine Fo$_{91}$, \cite{Nichols1991} determined a value in the range of $n=4$ to 5 (with $E_a=290$\,kJ/mol and 345\,kJ/mol, respectively). They attribute this high value to the control of grain growth by coalescence through surface diffusion of a second phase, the residual porosity, $\sim 5\%$. In sintered samples, porosity is inevitable in samples synthesized at 1 atm but almost disappears at high pressure, which may explain the low value of the exponent in the \cite{Karato1989} experiments at 300\,MPa, but not those at 0.1MPa. \cite{Nichols1991} noticed this disagreement without being able to explain it. Porosity has been measured in H-chondrites at equivalent levels and should have the same effect. Metallic grains, present in the chondritic material, also play the role of a secondary phase. \cite{Guignard2012, Guignard2016} studied a mixture of nickel and forsterite in proportions representative of the metal content of chondrites. In this case, the growth of forsterite grains is limited by that of the metallic particles whose growth process is controlled by diffusion along one dimensional paths due to their location at triple junctions, for which $n=5$. The activation energy was found to be close to 400\,kJ/mol. Moreover, these experimental data were found to be consistent with the size of metal particles measured in H-chondrites at various metamorphic grades \citep{Guignard2016}, indicating that the parameters of this growth law are supported by natural points resulting from growth over several million years. Lastly, in experiments on partially molten olivine aggregates conducted by \cite{Faul2006}, porosity cannot account for a pinning effect. However, the growth exponent $n$ is again measured above 3, close to 4 with an activation energy E=390kJ/mol similar to that measured for forsterite in the absence of melt. Thus, in a single-mineral medium without impurities, the appearance of silicate liquid seems to reduce grain growth, while in poly-mineral or impurity-bearing medium, it has a promoting effect compared to conditions below the melting point. For the present study, we used the relations (\ref{eqn:growth_law}) and (\ref{eqn:growth_rate}) with the parameters determined by \cite{Guignard2016} below the melting point and \cite{Faul2006} above, i.e. $n=5$, $E_a=400$\,kJ/mol, $A_0=10^{-19.04}$\,m$^5$/s and $n=4$, $E_a= 400$\,kJ/mol, $A_0= 10^{-12.02}$\,m$^4$/s, respectively. The initial grain size $a_0$ has been set in our calculations to 1 micron, a characteristic grain size of the matrix in primitive chondrites. Figure\,\ref{Fig:grain_growth} displays the time integration of these growth laws along the thermal history of small bodies. Figure\,\ref{Fig:grain_growth}a describes the time evolution of grain size located at the center of a 100km radius body accreted at various times. A kink marks the appearance of the liquid silicate and its influence on the growth of the grains, when the size increases from a few tens to a hundred microns with the first percent of liquid. After that point in time growth is significant at high temperature, notably during the long temperature plateau that follows the extinction of heat sources. Figure\,\ref{Fig:grain_growth}b shows a map, as a function of accretion time and body radius, of grain size reached just before this temperature plateau or before the tipping point to the magma ocean that occurs at 50\% of partial melting, i.e. at $\min (T_\textrm{max}-5K, T_{50\%})$ (see section \ref{paragraph:thermal time scales} for more detail on this threshold). On this map, the grain size reaches a maximum of 2\,mm for those bodies accreted at around 1.3\,million years, reaching a maximum of 50\% partial melting. For bodies formed earlier, the 50\% partial melting degree threshold is reached more quickly. This faster evolution does not allow the grain to grow as much, reducing the ability of the liquid to migrate during the melting phase. In summary, grain growth can account for a decrease of three or four orders of magnitude in the $\mu_f/a^2$ parameter. \subsubsection{Viscosity of silicate liquid} \label{paragraph:melt_viscosity} The viscosity of liquid silicates varies strongly at the beginning of melting due to a dependence on the content of lattice elements, Si and Al, and notably Al that is only present in plagioclase, one of the earliest phases to melt-out of the matrix, with clinopyroxene. The first percentages of liquid have a viscosity of about $10^4$\,Pas \citep{Collinet2020}, which decreases to about 100\,Pas between 10\% and 15\% melt (basaltic liquids) and to about 1\,Pas above 40\% melt (picritic liquids) \citep{Dingwell2004}. Figure\,\ref{Fig:melt_visco} shows as an example the viscosity of a silicate liquid produced by melting of a H-chondrite composition as a function of temperature. This figure also shows the composition of the residual solid. Composition and viscosity of the liquid have been computed with the "Rhyolite-Melts" thermodynamic calculator \citep{ Ghiorso1995, Asimow1998, Gualda2012, Ghiorso2015}. We note that the reference value taken for the liquid viscosity by \cite{Moskovitz2011}, \citeauthor{Neumann2012} (\citeyear[][and the following papers]{Neumann2012}) and then \cite{Lichtenberg2019} corresponds to a liquid produced by about 50\% of partial melting (dashed line on Figure 4). \subsubsection{The parameter \texorpdfstring{$\mu_f/a^2$}{muf/a2}} \label{paragraph:mu_over_a2} During melting, the variations of the different parameters can lead to a drop of seven to eight orders of magnitude in the $\mu_f/a^2$ ratio, which has a significant impact on the drainage time. For example, the blue arrow in Fig.\,\ref{Fig:drainage_time}c, shows the hypothetical trajectory of a body of radius 100\,km for which the drainage time is almost infinite at the beginning of the fusion ($a=10\,\mu$m, $\mu_m=10^{19}$\, Pas and the viscosity of the first liquids $\mu_f=10^{4}$\,Pas), to reach a state ($a=10^{-3}$\,m, $\mu_m=10^{17}$\,Pas and $\mu_f=1$\,Pas) corresponding to a draining time lower than 100\,kyr. Interestingly, the decrease of the $\mu_f/a^2$ ratio can lead to the region where the drainage is controlled by the compaction (white area on Figure\,\ref{Fig:drainage_time}c) and where its characteristic time then only depends on the viscosity of the matrix, whatever the grain growth or the melt viscosity decrease. Figure\,\ref{Fig:mu_f_over_a2} presents the $\mu_f/a^2$ variations at the centre of a 100\,km radius body during its thermal evolution, computed following the grain growth and melt viscosity laws presented in Figures \ref{Fig:grain_growth} and \ref{Fig:melt_visco} respectively. It highlights this decrease of seven to eight orders of magnitude during melting for bodies accreted before 1.5 million years, and of lesser magnitude for later accretions. \cite{Lichtenberg2019} have shown that melt migration is inefficient for a grain size of less than 1\,mm. Since their calculations were performed with a constant melt viscosity of 1\,Pas, this grain size threshold corresponds to a $\mu_f/a^2$ threshold of $10^6$ Pas/m$^2$. Importantly enough, figure\,\ref{Fig:mu_f_over_a2} shows that, except in some cases and for partial melt degrees above 40\%, the parameter $\mu_f/a^2$ remains above this value of $10^6$ Pas/m$^2$, i.e. in a range where melt migration is inefficient. For comparison, the $\mu_f/a^2$ parameter was also calculated with the grain growth laws used by \cite{Moskovitz2011} (grey dotted line) and by \cite{Neumann2012} (dashed lines), the melt viscosity being constant and fixed at 1\,Pas in both studies. Since the compaction of the matrix was neglected there, the drainage time is simply equal to the Darcy time and thus only proportional to the parameter $\mu_f/a^2$. For both studies, the effective migration threshold of $10^6$ Pas/m$^2$ is crossed for the first percent of liquid, up to 10\% of liquid in the case of \cite{Moskovitz2011}. It is thus of no surprise that these studies have popularised the idea that silicate liquids extract rapidly to the surface taking the $^{26}$Al with them, leading to no further melting at depth. As an example, \cite{Moskovitz2011} showed that the percentage of liquid in the centre of a body formed at 1\,Myr would not exceed 27\%. However, the grain growth laws used in these two studies remain highly questionable. In \cite{Moskovitz2011}, it is an ad hoc dependence on the melt fraction, while the one used in \cite{Neumann2012}, if theoretically correct, adopts inappropriate parameters as discussed in paragraph \ref{paragraph:Grain_size}. This, combined with the very low value chosen for the melt viscosity which does not correspond to the viscosity of the migrating melt, makes their conclusion about the high mobility of silicate liquids in small bodies questionable. On the other hand, the use of realistic laws for both grain growth and melt viscosity tends to show that liquids only start to migrate at melt fractions of several tens of percent, or even do not have time to extract before the rheological limit of a magma ocean is reached. \subsubsection{Matrix viscosity} \label{paragraph:Matrix_viscosity} The drainage time is dependent on a second parameter, $\mu_m/R^2$, solely proportional to the matrix viscosity for a given body. In this respect it is of note that the viscosity of solids depends on the creep mechanism at work. \cite{Lichtenberg2019} assume that the matrix deformation remains in the diffusion creep regime, but it could also be in the dislocation or grain boundary sliding (GBS) regimes, especially when the grain size is large and the stress is low. Here we will consider all of these possibilities, taking the most efficient one at each time step. Whatever the creep mechanism, the rheological behavior of rocks is described by a general power law dependance of strain rate, $\dot{\epsilon}$, on differential stress, $\sigma$ \citep{Hirth2004}, which rewritten here in term of viscosity reads: \begin{equation} \label{eqn:matrix_viscosity} \mu_m = \dfrac{\sigma}{\dot{\epsilon}}=\dfrac{1}{A} \sigma^{1-q} a^n \exp(E_a/R_gT-\alpha \phi), \end{equation} where $A$ is a constant, $q$ the stress exponent, $n$ the grain size exponent, $E_a$ the activation energy and $a$ the grain size. $\alpha$ is a constant corresponding to the dependence on the melt content $\phi$. The presence of melt within an aggregate plays a well identified role on their deformation by favoring the sliding mechanism at grain boundary. \cite{Hirth2004} reported a link between the effective viscosity and the melt fraction that can be approximated to first order by an exponential relationship up to liquid fraction of 12\%. It is not obvious that this law can be extended beyond that. Indeed, the viscosity measured in these experiments is the effective viscosity of the liquid-solid mixture. In the formalism we use, $\mu_m$ is the viscosity of the solid. The value of this parameter is subject to change due to the presence of liquid, notably because the presence of liquid promotes dislocations by accommodation at grain boundaries, resulting in a relaxation of the von Mises criterion. It nevertheless remains possible that this effect does not increase with increasing liquid fraction. For this reason, we will restrict the application of this law to the first 12 percent of liquid ( $\mu_m \propto \exp (-\alpha \min(\phi, 0.12))$), the range over which it has been determined experimentally. This choice has no incidence on our conclusions, as discussed in the next section. The coefficient $\alpha$ depends on the creep mechanism, estimated for lherzolitic material around 20 for diffusion creep and 25 for dislocation and GBS creeps. The other constants or exponents depend on the creep mechanism too. For instance, the stress dependence is linear for diffusion creep ($q=1$), but not for dislocation or GBS creep ($q=3.5$). In addition, there is no dependence on the grain size for dislocation creep, but it is nonlinear for diffusion ($n=3$) and for GBS ($n=2$) \citep{Hirth2004}. The temperature, the liquid fraction and the grain size are calculated throughout the thermal history of the body, but here the stress can only be estimated \textit{a priori}. An upper estimate is the pressure difference at the centre of the body between the fluid and the matrix: \begin{equation} \label{eq:stress} \sigma= \dfrac{2 \pi}{3} \rho_m \delta\rho G R^2, \end{equation} Hence, $ \sigma \approx 1.41\times 10^{-3} R^2$ Pa, ranging from 0.5\,MPa to 50\,MPa for body radii from 30\,km to 300\,km, respectively. The variation of all these parameters during the melting of silicates is likely to result in a variation of the matrix viscosity of six orders of magnitude, a range that is itself likely to shift by six orders of magnitude as a function of the size of the body, mainly because $\mu_m$ is proportional to $\sigma^{2.5}$ and finally to $ R^5$. \subsection{Drainage time variation during melting} \label{sec:drainage_time_variation_during_melting} Figure\,\ref{Fig:tracks_taudrain}a displays the time trajectories for bodies of various sizes and accretion times on the $\phi_0\tau$ map shown on Figure\,\ref{Fig:drainage_time}c. These tracks highlight the magnitude of the variation of $\mu_m/R^2$ and $\mu_f/a^2$, the two independent parameters of the drainage time, and thus underline the importance of not considering material properties as constant in the study of small body differentiation. With the first percent of liquid, the trajectories begin with an increase in matrix viscosity and a decrease in the $\mu_f /a^2$ parameter, both due to the noticeable grain growth boosted by the appearance of silicate liquid, as shown in Figure\,\ref{Fig:grain_growth}a. Then, both parameters decrease while keeping the trajectories in the Darcy domain, such that the drainage time during melting is not very sensitive to variations in the viscosity of the matrix. This is particularly apparent for 100\,km and 300\,km radius bodies for which the variation in matrix viscosity differs greatly without affecting the drainage time. For the 30\,km bodies, this becomes less exact, their evolution pathways progressing into the compaction domain, where the variations in matrix viscosity regain their influence, i.e. where drainage time increases proportionally to the matrix viscosity. Recall here that melt content has been limited to 12\% in the viscosity law. Otherwise, we would observe a greater decrease in the matrix viscosity as melt content increases above this limit. However, this has no effect on the drainage time as long as the evolution remains in the Darcy domain. This would only affect the evolution corresponding to a 30\,km body accreted at 1.4\,Myr in its red part which could have progressed into the Darcy domain. This remains a secondary issue. For each body size, the three trajectories corresponding to the earliest accretions reach the threshold of 50\% degree of fusion ---and have not been extended beyond that for this reason--- indicating a possible evolution into a global magma ocean. However, this end-point is not certain if the drainage time becomes shorter than the remaining time before reaching the threshold. This may be appreciated on Figure\,\ref{Fig:tracks_taudrain}.b that shows the time evolution of the drainage time since the onset of melting. In the case of very early accretion, within the first million years after CAIs and whatever the size of the body, the drainage time never becomes short enough to allow a significant drainage before reaching the threshold. In contrast, the trajectories at 1.3\,Myr show that the time required to reach the threshold remains greater than the drainage time after it falls below 1\,Myr. Thus, despite an accretion time that potentially allows the threshold to a magma ocean regime to be reached, migration of the liquid and heat sources to the surface is very likely, interrupting melting in the centre of the body and leading to shallower liquid accumulation that could itself develop into a shallow magma ocean. This scenario may also concern later accretion as in the case, for example, for 100\,km radius bodies formed between 1.3 and 1.5\,Myr, for which the drainage would be completed before the extinction of heat sources that occurs before 5\,Myr after CAIs. In that case, the concentration of heat sources may induce an overheating of the shallow liquid layer and a fusion of the overlying part \citep{Neumann2014, Lichtenberg2019}. Figure \ref{Fig:tracks_taudrain} also shows that differentiation may take place later, over the long cooling time of the body. This corresponds to accretions occurring between 1.5 and 1.7\,Myr, for bodies of 100\,km radius. Later accretions may lead to moderate partial melting but not to differentiation. The accretion time window for this regime is wider the smaller the body, as smaller bodies experience lower temperatures and faster cooling: between 1.5 and 1.8\,Myr for 30\,km radius against 1.7 to 1.8\,Myr for 300\,km radius. \section{Types of evolution of small bodies } In summary, both representations of Figure\,\ref{Fig:tracks_taudrain} help to distinguish various melt migration regimes. Three have already been described by \cite{Lichtenberg2019}: magma ocean, shallow sills and undifferentiated bodies. We add two more domains here by separating differentiation during melting with heat source transport from differentiation during cooling, and by subdividing undifferentiated bodies into those that have melted and those that have not. These 5 regimes correspond to conditions for which: \begin{enumerate}[label=\textit{\Roman*)}] \item the drainage is unable to prevent the melt fraction from reaching the 50\% rheological threshold and thus the development of a \textbf{global magma ocean} before differentiation; \item the drainage is efficient enough to allow extraction and subsurface accumulation of the melt and the heat sources it contains before their extinction, which may evolve into a \textbf{shallow magma ocean} above a residual harzburgitic or dunitic core; \item the drainage is not efficient enough to extract the melt before the heat source is extinguished, but efficient enough to allow a \textbf{moderate differentiation} of the body during its cooling; \item \textbf{no differentiation} occurs despite moderate partial melting; \item there is no melting, just \textbf{thermal metamorphism}. \end{enumerate} Figure\,\ref{Fig:big_picture} displays a global view of the different possible evolution pathways according to the radius and accretion time of the body. Criteria have been defined to delimit the five types. Type I (in yellow) corresponds to the appearance of a magma ocean. This must satisfy two criteria. Firstly, the maximum possible degree of partial melting, indicated by the red contours, must exceed the 50\% threshold. Secondly, this threshold must actually be reached. This second criterion defines the limit with type II. Type II corresponds to conditions that allow an efficient drainage before the end of melting or before reaching the 50\% melting degree threshold. To define this domain, the moment when the drainage time becomes smaller than the time remaining before reaching $T_\textrm{max} - 5\textrm{K}$ or 50\% of partial melting is inferred for each radius and accretion time based on Figure\,\ref{Fig:tracks_taudrain}. The orange region of Figure\,\ref{Fig:big_picture} represents the area where this criterion is satisfied. Inside, the contour lines indicate the melting degree at which the criterion is satisfied, showing that drainage only becomes effective from about 30\% of fusion. This percentage is not a threshold in itself, but results from several factors such as the viscosity of the liquid and the matrix, or the grain size. The migration of liquids certainly starts before this value, which is more indicative of the degree to which partial melting stops in the center of the body due to the drainage of heat sources. This value also constrains the composition of the unmelted residue remaining in the centre of the body. Outside the type II region, the 50\% partial melting threshold line separates type I, ending into a magma ocean, from types III (green) and IV (blue). The distinction between the latter two types is based on the ratio of melt lifetime, as defined in Figure\,\ref{Fig:melting_times}b, to the drainage time at its minimum value. On Figure\,\ref{Fig:big_picture}, the dashed contour lines plot the value of this ratio. Values less than one indicate an inefficient drainage at the time scale of the body cooling. To these four regions, we have added a fifth one (grey) corresponding to the conditions leading to non-melted bodies, i.e. the parent bodies of chondrites. Naturally, these boundaries between regimes are more progressive than simple lines drawn in Fig.\ref{Fig:big_picture}, for two main reasons. On the one hand, as mentioned earlier, the drainage time is an average quantity of an active process depending strongly on evolving conditions. On the other hand, the criterion used to distinguish these types has been applied to conditions in the centre of the body that differs from those close to the surface. While these conditions may be representative of a large part of a body, a greater fraction as radius increases, the fact remains that its outer parts may have harbored conditions corresponding to another type. This is obvious for the parent bodies of chondrites, which may correspond to the outer parts of bodies of type III or IV. This point is even more striking for very small bodies, those for which the radius is comparable to the thickness of the cooling boundary layer, thus those for which the central temperature is not representative of the body interior. For example, the parent body proposed for the primitive achondrite Tafassasset, whose composition is compatible with a lost of basaltic content, has a radius possibly in the range of 15\,km to 30\,km with an accretion time between 0.5\,Myr and 1.2\,Myr \citep[][]{Breton2015}. This falls in type I region, near the type III border. Details of the radial temperature profile derived from the thermal history modelling to satisfy isotopic dating shows a peak melting degree of 80\% in the center, while the overall composition of the meteorite ---olivine and pyroxene rich with plagioclase traces--- corresponds to a residue from 20\% of partial melting achieved at mid-radius, the outer 40\% of the radius remaining unmelted. Of course, such a description suffers from the lack of deep magma ocean and melt migration modelings and cannot be fully representative of the final evolution of a body accreted under such conditions. \subsection{Did global magma oceans exist on small bodies?} The occurrence of a global magma ocean in the early evolution of small bodies is not an idea that is unanimously accepted. Among the arguments against this hypothesis is the idea that liquid silicates drain efficiently towards the surface as soon as melting begins, either by filtration through the porous matrix resulting from partial melting \citep[e.g.][]{Moskovitz2011,Neumann2012}, or by percolation through a hierarchic network of small and large veins and dikes \citep[e.g.][]{Wilson2012, Wilson2017}. However, this efficiency was deduced from a matrix permeability based on overestimated grain sizes and an underestimated viscosity of percolating liquids (see paragraphs \ref{paragraph:Grain_size} and \ref{paragraph:melt_viscosity} for a detailed discussion), both leading to an overestimation of the percolation velocity by six to eight orders of magnitude compared to a calculation using realistic laws of crystalline growth or melt viscosity (see paragraph \ref{paragraph:mu_over_a2}). In this respect, the theoretical arguments developed by \cite{Wilson2012, Wilson2017} deserve more detailed discussion. Briefly, these ideas are largely inspired by the context of mid-oceanic ridges. The formation of the veins and dikes network is conjectured from the matrix-liquid interplay described by the compaction formalism of \cite{McKenzie1984}, on which those exposed in \cite{Lichtenberg2019} or here in \ref{paragraph:Two_phase_flow_formalism} also rely \citep[see][for a detailed argumention of the vein network development]{Wilson2008}. To put numbers on his analytical developments, McKenzie uses a grain size of 1\,mm and a liquid viscosity of 1\,Pas. Interestingly, these values are those also employed by the other work supporting the ideas of efficient melt extraction \citep{Moskovitz2011, Neumann2012, Wilson2012, Wilson2017}, but they do not correspond to the conditions occurring at the beginning of melting on rocky-bodies in the early solar-system, where the grain size is closer to $100 \mu$m and the viscosity ranges around $10^4$\,Pas. As noted above, the values used in the literature are more relevant to conditions already close to the onset of a magma ocean, with the value of viscosity corresponding to a liquid produced by more than 40\% of partial melting. Hence, the work presented here indicates that extraction of a silicate melt from the interior of a small body is not as effective as believed up to now, and that the development of a global magma ocean is probable for a wide range of early accreted bodies. Indeed, most of the bodies accreted within the first 1.1\,Myr of CAI condensation probably experienced a global magma ocean stage. \subsection{The fate of magma ocean: pallasites, irons and eucrites} \cite{Taylor1993} stated that \textit{“the lack of pyroxene in pallasites suggests that core formation in asteroids was accompanied by $>40\%$ melting of the associated silicate”.} Indeed, pallasites provide simple but compelling evidence of a past magma ocean experienced by their parent body(ies). Aluminum is only present in feldspars, that have been completely removed from the solid residue after a partial melting degree of 15\%. If the silicate liquids are highly mobile and drain heat sources with them, melting cannot continue much further. Alternatively, if the liquids remain in the matrix, nothing can prevent the melting from reaching the rheological threshold, which incidentally corresponds to the disappearance of pyroxenes. With the onset of a magma ocean regime, thermal exchange switches to effective convective cooling. The temperature probably remains buffered at the rheological threshold, with the heat supplied by sources being used to melt and thin the layer above the ocean, up to the surface if sufficient, and then being dissipated outwards. While the demonstration of this point goes beyond the scope of the present study, metal-silicate separation in this context deserves discussion. First, metal is still present in the magma ocean even if it is sometimes assumed to be able to percolate through the silicate matrix as soon as it melts \citep[e.g.][]{Ghosh1998,Sahijpal2007, Neumann2012, Sramek2012}. To segregate, it should form a connected network along the silicate grain boundaries. Due to the high surface tension between the solid silicates and the liquid metal, this only occurs at a metal content of 20\,vol\% \citep{Bagdassarov2009, Neri2020, Solferino2020}, well above the 10\,vol\% present in chondrites. Thus, magma oceans inside planetesimals are a mixture of liquid silicate ($\sim$ 45\,vol\%), olivine crystals ($\sim$ 45\,vol\%) and liquid metal droplets ($\sim$ 10\,vol\%). This context differs from the that envisaged after giants impacts in which the metal-silicate separation is usually considered under the paradigm of iron rain in a fully liquid bath \citep{Rubie2003, Ichikawa2010}. Here, crystals are a significant component of the mixture and interact with the metal liquid. \cite{Neri2021} discussed this situation and showed that metallic drops remain attached to olivine crystals and probably sink as metal-olivine aggregates. The separation finally occurs with the extraction of the interstitial silicate liquid during compaction of the matrix, leading to core formation. This process is similar to the scenario of pallasites formation where olivine crystals issued from 50\% of partial melting float in between the liquid core and the liquid silicate. Iron meteorites are issued from parent bodies larger than 20km and accreted less than 1 Ma after the condensation of CAIs. They are the products of a high degree of partial melting and a complete metal-silicate differentiation; according to Figure\,7, their parent bodies indeed experienced a magma ocean stage. Regarding asteroid Vesta, as it most likely has a metallic core \citep{Ermakov2013}, it must have formed within the first million years after the CAI's condensation. If it accreted later, the migration of liquids would have interrupted its melting and left a harzburgitic (olivine-pyroxene) center hampering metal separation. This conclusion issued from the present model is perfectly in line with cosmochemical data : indeed, based on isotope geochronological data in eucrites, Vesta is known to have accreted very early \citep[e.g.][]{Schiller2011}. \subsection{Incipient differentiation in the absence of a magma ocean} Some other meteorites such as primitive achondrites (winonaites, acapulcoites, lodranites) are characterized by a lower degree of partial melting and incipient differentiation. In more detail, winonaites are thought to have been heated slightly above the Fe-FeS eutectic temperature and potentially slightly above the silicate solidus \citep[e.g.][]{Hunt2017}, acapulcoites show signs of extraction of small amounts of silicate and Fe-Ni-S melts, while lodranites point to a maximum degree of $\sim 20\%$ partial melting and the removal of both a basaltic and a S-rich metallic melts \citep{Mccoy1997}. In all these cases, the melting degree does not reach the threshold required for the onset of a magma ocean. Even so, winonaites, lodranites and accapulcoites might not be representative of the center of their parent bodies, but of the outer parts, as proposed by \cite{Neumann2018}. Combining data inferred from different chronometers together with modeling, they concluded that acapulcoites and lodranites could be issued from a single parent body 260\,km in radius, accreted at 1.7\,Myr after CAIs with burial depth of $\approx$ 7 to 13\,km for acapulcoites and lodranites respectively. Interestingly, they noted that their model is unable to fit the thermo-chronological data with the grain coarsening law and the 1\,Pas melt viscosity that they usually adopt, because the migration of the silicate melt along with the enrichment of $^{26}$Al disturbs the evolution of the temperature. They found a suitable fit, either by inhibiting the partitioning of $^{26}$Al heat sources between matrix and silicate melt, or by keeping small grain size (0.2\,$\mu$m), or by increasing the melt viscosity up to 100\,Pas. We note that a 260\,km radius body formed at 1.7\,Myr lies at the boundary between the green and blue areas of Fig.\ref{Fig:big_picture}, where silicate melt migration is possible, but to slow to allow heat sources redistribution. These regions correspond to regime III and IV, which are compatible with lodranites and acapulcoites resppectively. \section{Key summary points } Contrary to the idea popularised by \cite{Moskovitz2011} and the series of articles published in the following decade, we show here that melt migration is not a process efficient enough to drain and remove $^{26}$Al heat sources from the interiors of small bodies before their heat sources are exhausted. This conclusion is drawn from new considerations regarding grain size and melt viscosity. In summary, melt percolation rate is proportional to the ratio of matrix permeability to melt viscosity, i.e. the ratio of the square of the matrix grain size to the melt viscosity. Literature studies used ad hoc or unsuitable grain growth laws --- that overestimate grain size by more than one order of magnitude---, associated to the canonical melt viscosity value, 1\,Pas, chosen by \cite{McKenzie1984} to assess melt migration rates in the context of MORB generation at mid-oceanic spreading centres. Up to a partial melting degree of 20\%, the relevant values are closer to 100\,$\mu$m and $10^3$\,Pas, corresponding to a migration rate five order of magnitude lower. This finding underlines the importance of not assuming constant values for properties that play a role in magma migration, when modelling the differentiation of small bodies. Other consequences concerning small body evolution are: \begin{enumerate} \item Small bodies accreted within 1.15 million years of CAIs condensation underwent a magma ocean stage which permitted the formation of a metallic core. These bodies are possible parent bodies of pallasites; \item Melt migration and drainage of $^{26}$Al heat sources influenced the differentiation of bodies larger than 30km radius and accreted between 1.15 and 1.5\,Myr after CAIs, leaving an olivine-pyroxene rich core overlain by a shallow magma ocean; \item The parent bodies of winonaites, accapulcoites, and lodranites are consistent with expectations for bodies accreted more than 1.5\,Myr after CAIs condensation. \end{enumerate} \section*{Aknowledgments} This work is part of the PALLAS project funded by the ANR (grant ANR-14-CE33-006-01 to G. Quitté); we also thank the Université Paul Sabatier for its contribution to the PhD grant to A. Néri. \appendix \section{Two phase flow formalism} \setcounter{figure}{0} \label{paragraph:Two_phase_flow_formalism} To describe the migration of the silicate melt within a small body, we follow \cite{Sramek2012} that extended the two-phase flow formalism of \cite{Bercovici2001} and \cite{Bercovici2003} in spherical geometry. Here, we just recall the principal steps of this approach. This formalism consists in averaging on a unit volume the mass and conservation equations written for the fluid and the matrix. This introduces an additional variable that describes the volume fraction of the fluid $\phi$, which will require a closure relation to be solved \citep{Drew1998}. \subsection{Conservation equations} \begin{table*}[t!] \caption{Notations and parameters.} \small \begin{tabular*}{\hsize}{@{\extracolsep{\fill}}llll} \hline Symbol & Quantity & value & unit \\ \hline $\mathcal{A}$ & Avogadro's number & $6.02214076 \times 10^{23}$&atom/mol\\ $E_{^{26}\textrm{Al}}$ & $^{26}$Al decay energy$^a$ & 3.12 &MeV/atom\\ $\lambda_{^{26}\textrm{Al}}$ & $^{26}$Al decay constant & $3.063\times10^{-14}$ & s$^{-1}$\\ $m_{^{26}\textrm{Al}}$ & $^{26}$Al molar mass & 0.026 & kg/mol\\ $X_{\textrm{Al}}$ & Mass fraction of Al in H-type chondrites$^b$ & $11.3\times10^{-3}$ & \\ $[^{26}\textrm{Al}/^{27}\textrm{Al}]^0$ & initial $^{26}\textrm{Al}/^{27}\textrm{Al}$ ratio$^c$ &$5 \times 10^{-5}$&\\ $H$ & enthalpy & & J/kg \\ $k_T$ & thermal conductivity$^d$ & $4\sqrt{Te/T}$ & W/m/K \\ $Q_0$ & initial radiogenic heat power per Al kg & $\left[^{26}\textrm{Al}/^{27}\textrm{Al}\right]^0 \mathcal{A} \lambda_{^{26}\textrm{Al}} E_{^{26}\textrm{Al}} / m_{^{26}\textrm{Al}}\simeq 1.77\times10^{-5}$ & W/kg \\ $Q$ & radiogenic heat power & & W/kg\\ $T$ & temperature& &K\\ $T_e$ &external temperature temperature & $292$ & K\\ $a$ & grain size & & m \\ $a_0$ & initial grain size$^{e}$ & $10^{-6}$ & m \\ $A$ & grain growth rate & & \\ $A_0$ & growth rate constant & & \\ $\vect{g}$, $g_s$ & gravity, surface gravity& & m/s$^2$\\ $E_a$ & activation energy & & J/mol\\ $G$ & gravitational constant& $6.67430 \times 10^{-11}$& m$^3$/kg/s$^2$\\ $k$ & permeability & &m$^2$\\ $k_0$ & reference permeability & &m$^2$\\ $K$ & non dimensional geometrical factor & &\\ $L_c$ & compaction length & &m \\ $P_{f, m}$ & fluid, matrix pressure & & Pa\\ $r$ & radius & & m\\ $R$ & surface radius & & m \\ $R_g$ & gaz constant & 8.314 & J/mole/K \\ $s(\phi)$ & segregation function & & \\ $t$ & time & & s\\ $\vect{v}_{f, m}$ & fluid, matrix velocity& & m/s\\ $\mu_{f, m}$ & fluid, matrix viscosity$^f$,$^g$ & & Pas \\ $\rho$ & average chondritic material density$^h$ & 3800 & kg/m$^3$ \\ $\delta\rho$ & solid-liquid density contrast & 1000 & kg/m$^3$ \\ $\tens{\sigma}_m$ & matrix deviatoric stress tensor & & Pa \\ $\tau_D$ & Darcy characteristic time & & \\ $\tau_C$ & compaction characteristic time & & \\ $\phi$ & fluid volume fraction, matrix porosity& &\\ \hline \end{tabular*} $^a$\cite{Castillo2009}, $^b$\cite{Wasson1988}, $^c$\cite{Macpherson1995}, $^d$\cite{Monnereau2013}, $^{e}$\cite{Krot2014}, $^f$\cite{Dingwell2004}, $^g$\cite{Hirth2004},$^h$\cite{Consolmagno2008} \\ \label{table:parameters} \end{table*} In absence of phase change, the formulation of the mass conservation for the fluid and the matrix are standard (hereinafter, subscripts $f$ and $m$ refers to the fluid and the matrix phases respectively; all notations are summarized in Table\ref{table:parameters}.): \begin{equation} \dfrac{\partial \phi}{\partial t} + \Div \left[\phi \vect{v}_{f} \right] = 0 \label{eqn:MassC_f} \end{equation} and \begin{equation} -\dfrac{\partial \phi}{\partial t} + \Div \left[(1-\phi) \vect{v}_{m} \right] = 0 \label{eqn:MassC_m} \end{equation} whose sum shows that the average velocity $\bar{\vect{v}} = \phi \vect{v}_f +(1-\phi)\vect{v}_m$ is solenoid: \begin{equation} \Div \bar{\vect{v}} = 0. \label{eqn:Div_v} \end{equation} The average and difference quantities of the phases are defined as $\bar{q}=\phi q_f + (1-\phi) q_m$ and $\Delta q = q_m -q_f$. If surface tensions are neglected, the conservation of momentum for the fluid and the matrix phases can be written as \citep[see][for details]{Bercovici2003}: \begin{equation} -\phi \left[ \Grad P_f - \rho_f \vect{g} \right]+ c\Delta\vect{v}=0 \label{eqn:MomC_f} \end{equation} and \begin{equation} -(1-\phi) \left[ \Grad P_m - \rho_m \vect{g} \right]+ \Div \left[ (1-\phi) \tens{\sigma}_m\right] - c\Delta\vect{v}=0. \label{eqn:MomC_m} \end{equation} $P$ is the pressure. The term $\Div \left[ (1-\phi) \tens{\sigma}_m\right] $ is the viscous dissipation, where $\tens{\sigma}_m$, the matrix deviatoric stress tensor, is: \begin{equation} \tens{\sigma}_m = \mu_m\left[ \Grad \vect{v}_m + \left[ \Grad \vect{v}_m \right]^\intercal -\dfrac{2}{3}(\Div \vect{v}_m)\tens{I} \right], \end{equation} $\mu_m$ being the matrix viscosity. Because of the very low fluid viscosity compared to the matrix viscosity ($\mu_f \ll \mu_m$), this dissipation term is neglected in the fluid equation. The term $c\Delta \vect{v}$ is the Darcy term where $c$ is the drag coefficient between both phases that, in case of large viscosity ratio between the matrix and the fluid, reduces to \citep{Bercovici2003}: \begin{equation} c = \dfrac{\mu_f \phi^2}{k(\phi)}. \label{eqn:c_def} \end{equation} $k(\phi)$ is the permeability. The term $\Delta P \Grad \phi$, appearing in the matrix momentum conservation (\ref{eqn:MomC_m}), is the interfacial pressure force acting between both phases. It appears in each momentum equation with a weighting coefficient that is zero for the fluid momentum equation when $\mu_f \ll \mu_m$. The mass conservation equations (\ref{eqn:MassC_f}) \& (\ref{eqn:MassC_m}) and momentum conservation equations (\ref{eqn:MomC_f}) \& (\ref{eqn:MomC_m}) form an incomplete system to solve the five unknowns ($P_f, P_m, \vect{v}_f, \vect{v}_m$ and $\phi$). The conservation of energy and damage allows to write a closure relation that, under the previous approximations, is \citep{Bercovici2001}: \begin{equation} \Delta P= -K \dfrac{\mu_m}{\phi} \Div \vect{v}_m, \label{eqn:Closure} \end{equation} where $K$ is a dimensionless factor related to the geometry of the two-phase mixture, taken equal to 1 in \cite{Bercovici2001}. The combination of momentum equations, $(1-\phi)$(\ref{eqn:MomC_f}) - $\phi$ (\ref{eqn:MomC_m}), yields to the action-reaction equation: \begin{equation} \begin{split} -\Grad \left[ (1-\phi) \Delta P \right] &+ (1-\phi) \Delta \rho \vect{g} \\ &+ \Div \left[ (1-\phi) \tens{\sigma}_m \right] - \dfrac{c \Delta \vect{v}}{\phi} = 0 \end{split} \label{eqn:AR} \end{equation} that gives an equation for the matrix velocity after substitution of $\Delta P$ from (\ref{eqn:Closure}) and of $\Delta \vect{v}$ from (\ref{eqn:Div_v}). Equation (\ref{eqn:Div_v}) indicates that the average velocity is constant. In spherical geometry, it is necessarily null, so that: \begin{equation} \vect{v}_m = \phi \Delta \vect{v}. \label{eqn:matrix_velocity} \end{equation} In 1D spherical geometry, the equation for the matrix velocity is thus (the full development of these equations in spherical geometry can be found in \cite{Sramek2012} or in \cite{Mizzon2015}): \begin{equation} \begin{split} &k(\phi) \frac{\partial}{\partial r} \frac{(1 - \phi)}{r^2} \left[ \dfrac{1}{\phi} + \dfrac{4}{3}\right] \frac{ \partial r^2 v_m}{\partial r} \\ &+\left[ \dfrac{4 k({\phi})}{r} \dfrac{\partial \phi}{\partial r} -\frac{\mu_f}{\mu_m}\right] v_m = \dfrac{k(\phi) (\bar{\rho} - \rho_f)g(r)} {\mu_m}. \end{split} \label{eqn:final_phi} \end{equation} $g(r)$, the radial gravity profile, is obtain by integration of the average density profile, $\bar{\rho}(r)$ which is assumed to be only dependent on the composition ---due to the size of small bodies, the pressure dependency is neglected---: \begin{equation} \label{eqn:gravity} g(r)=\dfrac{4\pi G}{r^2}\int_0^r \bar{\rho}(x) x^2 dx. \end{equation} Solving equations (\ref{eqn:MassC_m}) and (\ref{eqn:final_phi}) allows the calculation of the liquid and solid fraction as a function of depth and time. This is achieved through a finite volume discretization and a second order Runge-Kutta method. More details can be found in \cite{Mizzon2015}. Choosing $R$ and $\tau_{C}$ (cf equation \ref{eqn:tauC}) as length and time scales, the matrix velocity equation (\ref{eqn:final_phi}) reads: \begin{equation} \begin{split} \frac{\partial}{\partial r} \frac{F}{r^2} \frac{ \partial r^2 v_m}{\partial r} + \left[ \dfrac{4 }{r}\dfrac{\partial \phi}{\partial r} - \frac{\phi_0^2 R^2}{\phi^2 L_c^2} \right] v_m = (1-\phi)g \end{split} \label{eqn:phi_adim} \end{equation} where $F=(1-\phi)\left[1/\phi +4/3\right]$. $r$, $v_m$ and $g$ are dimensionless variables. $g$ is the gravity acceleration normalised by the surface gravity $g_s$. A dimensionless number appears in this equation: \begin{equation} \dfrac{\phi_0^2 R^2} {L_c^2} =\dfrac {72 \pi R^2 \mu_f} {a^2 \mu_m}. \end{equation} It is worth noting that it does not depend on $\phi_0$. \subsection{Analytical solutions} The validity of the numerical resolution of equation (\ref{eqn:phi_adim}) was controlled by comparing it to an analytical solution. This is done in the case of a constant porosity profile, for which the matrix velocity equation (\ref{eqn:phi_adim}) reduces to: \begin{equation} F \frac{\partial}{\partial r} \dfrac{1}{r^2} \frac{ \partial r^2 v_m}{\partial r} -\dfrac{R^2}{L_c^2} v_m = (1-\phi_0) r, \label{eqn:vm_bis} \end{equation} whose analytical solution, considering an impermeable surface ($v_m(R)=0$), is: \begin{align} v_m= \dfrac{(1-\phi_0)L_c^2}{R^2}\left[\dfrac{(A-r) e^{\frac{r}{A}}- (A+r)e^{-\frac{r}{A}}}{B r^2} - r\right], \label{eqn:ana_vsol} \end{align} where $A= L_c \sqrt{F}/R$ and $B = {(A-1) e^{1/A} - (A+1)e^{-1/A}}$. Equation (\ref{eqn:vm_bis}) is solved numerically by a tridiagonal inversion method and backward Euler finite difference scheme. As shown in Figure\,\ref{fig:ana_num}, the numerical solution of the velocity equation is solved to a precision of $\sim$ 10$^{-6}$. \subsection{Benchmark}\label{sec:Bench} The numerical solution of liquid transport was benchmarked against two drainage experiments described in \citep{Ricard2001}. The separation between a low viscosity fluid and a highly viscous matrix, under the effect of gravity, is computed for an imposed constant initial liquid fraction. The authors used a slightly different formalism to which we adapted our code to allow the comparison of the experiments \citep[see][for more details]{Mizzon2015}. Figure\,\ref{fig:benchmark_ricard} shows this comparison. The profiles of liquid fractions and the time evolution calculated here appear consistent with the numerical simulation of \cite{Ricard2001}. The dependency of the flow regime on the $R/L_c$ ratio is reproduced. When the size of the system is large compared to the compaction length, compaction waves are observed and when the size of the system is comparable to the compaction length, the liquid fraction is always a monotonically decreasing function of depth. \bibliographystyle{elsarticle-harv} \bibliography{Biblio}

Title: The formation of CO$_2$ through consumption of gas-phase CO on vacuum-UV irradiated water ice

Abstract: [Abridged] Observations of protoplanetary disks suggest that they are depleted in gas-phase CO. It has been posed that gas-phase CO is chemically consumed and converted into less volatile species through gas-grain processes. Observations of interstellar ices reveal a CO$_2$ component within H$_2$O ice suggesting co-formation. The aim of this work is to experimentally verify the interaction of gas-phase CO with solid-state OH radicals above the sublimation temperature of CO. Amorphous solid water (ASW) is deposited at 15 K and followed by vacuum-UV (VUV) irradiation to dissociate H$_2$O and create OH radicals. Gas-phase CO is simultaneously admitted and only adsorbs with a short residence time on the ASW. Products in the solid state are studied with infrared spectroscopy and once released into the gas phase with mass spectrometry. Results show that gas-phase CO is converted into CO$_2$, with an efficiency of 7-27%, when interacting with VUV irradiated ASW. Between 40 and 90 K, CO$_2$ production is constant, above 90 K, O$_2$ production takes over. In the temperature range of 40-60 K, the CO$_2$ remains in the solid state, while at temperatures $\geq$ 70 K the formed CO$_2$ is released into the gas phase. We conclude that gas-phase CO reacts with solid-state OH radicals above its sublimation temperature. This gas-phase CO and solid-state OH radical interaction could explain the observed CO$_2$ embedded in water-rich ices. It may also contribute to the observed lack of gas-phase CO in planet-forming disks, as previously suggested. Our experiments indicate a lower water ice dissociation efficiency than originally adopted in model descriptions of planet-forming disks and molecular clouds. Incorporation of the reduced water ice dissociation and increased binding energy of CO on a water ice surfaces in these models would allow investigation of this gas-grain interaction to its full extend.

https://export.arxiv.org/pdf/2208.13789

\title{The formation of CO$_2$ through consumption of gas-phase CO\\on vacuum-UV irradiated water ice} \titlerunning{CO$_2$ production through gas-phase CO consumption on VUV irradiated water ice} \subtitle{} \author{J. Terwisscha van Scheltinga\inst{1,2,\thanks{Current address: Department of Astronomy, University of Virginia, P.O. Box 400325, Charlottesville, VA 22904, USA}} \and N.F.W. Ligterink\inst{3}\ \and A.D. Bosman\inst{4} \and M. R. Hogerheijde\inst{2,5} \and H. Linnartz\inst{1} } \institute{Laboratory for Astrophysics, Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands\\ \email{jeroentvs@virginia.edu} \and Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands \and Physics Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland \and Department of Astronomy, University of Michigan, 323 West Hall, 1085 S. University Avenue, Ann Arbor, MI 48109, USA \and Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands } \date{Received 8 September 2021 / Accepted 16 August 2022} \abstract {Recent observations of protoplanetary disks suggest that they are depleted in gas-phase CO up to a factor of 100 with respect to predictions from physical-chemical (or thermo-chemical) models. It has been posed that gas-phase CO is chemically consumed and converted into less volatile species through gas-grain processes. Observations of interstellar ices reveal a CO$_2$ component in a polar (H$_2$O) ice matrix, suggesting potential co-formation or co-evolution.} {The aim of this work is to experimentally verify the interaction of gas-phase CO with solid-state OH radicals on the surface of water ice above the sublimation temperature of CO.} {Amorphous solid water (ASW) is deposited in an ultra-high vacuum (UHV) setup at 15~K and irradiated with vacuum-UV (VUV) photons (140--170~nm, produced with a microwave-discharge hydrogen-flow lamp) to dissociate H$_2$O and create OH radicals. Gas-phase CO is simultaneously admitted and only adsorbs with a short residence time on the ASW. Formed products in the solid state are studied in the infrared through Fourier transform infrared spectroscopy and once released into the gas phase with quadrupole mass spectrometry.} {Our experiments show that gas-phase CO is converted into CO$_2$ when interacting with ASW that is VUV irradiated with a conversion efficiency of 7--27\%. Between 40 and 90~K, CO$_2$ production is constant, above 90~K, CO$_2$ production is reduced in favor of O$_2$ production. In the temperature range of 40--60~K, the CO$_2$ remains in the solid state, while at temperatures $\geq$ 70~K the majority of the formed CO$_2$ is immediately released into the gas phase.} {We conclude that gas-phase CO reacts with OH radicals, created on the surface of ASW with VUV irradiation, above its canonical sublimation temperature. The diffusion during the short, but nonzero, residence times of CO on the surface of ASW suggests that a Langmuir-Hinshelwood type reaction is involved. This gas-phase CO and solid-state OH radical interaction could explain (part of) the observed presence of CO$_2$ embedded in water-rich ices when it occurs during the build up of the H$_2$O ice mantle. It may also contribute to the observed lack of gas-phase CO in planet-forming disks, as previously suggested. It should be noted though that our experiments indicate a lower water ice dissociation efficiency than originally adopted in model descriptions of planet-forming disks and molecular clouds. Incorporation of the reduced water ice dissociation and increased binding energy of CO on a water ice surfaces in physical-chemical models would allow investigation of this gas-grain interaction to its full extend.} \keywords{Astrochemistry -- Molecular processes -- Protoplanetary disks -- ISM: clouds -- Methods: laboratory: molecular -- Methods: laboratory: solid state -- Techniques: spectroscopic} \section{Introduction} In typical laboratory astrochemistry experiments, the processes that occur in the solid state and gas phase are investigated independently. However, there are conditions in the interstellar medium where these are intimately intertwined and could affect each other. In this work, we explore experimentally the interaction between gas-phase carbon monoxide (CO) and UV irradiated water (H$_2$O) ice and place the results in astrophysical context. In the study of planet forming disks, CO and its isotopologues are common tracers of the total gas mass, but are often found to be depleted by factors up to 10--100, even after taking into account freeze-out of CO in the coldest disk regions \citep{2016_Ansdell_Lupus_ApJ...828...46A, 2017_Miotello_Lupus_A&A...599A.113M, 2021_Trapman_CO_conv_A&A...649A..95T}. Recent physical-chemical models suggest that gas-phase CO could be converted into CO$_2$ after interaction with a UV-irradiated H$_2$O ice surface, at temperatures just above the CO sublimation temperature \citep{2016_Drozdovskaya_midplanes_MNRAS.462..977D, 2016_Eistrup_ChemEvo_A&A...595A..83E, 2018_Bosman_CO_A&A...618A.182B}. Under realistic disk conditions, this pathway was found by \citet{2018_Bosman_CO_A&A...618A.182B} to convert significant amounts of gas-phase CO into CO$_2$. However, little experimental work exists to confirm this process. If efficient, the UV-irradiated edges of molecular clouds could be another environment where this gas-grain reaction can occur. This possibly explains (part of) the observed presence of CO$_2$ in polar ices \citep[see e.g.,][]{1999_Gerakines_ISO_CO2_ApJ...522..357G, 2008_Pontoppidan_CO2_ApJ...678.1005P}, if gas-phase CO conversion happens on the grain surface during the build up of the H$_2$O ice mantle, adding to contributions from other CO$_2$ formation pathways already studied. The solid-state formation of CO$_2$ has been investigated both theoretically \citep[see e.g.,][]{2008_Goumans_CO2_comp_MNRAS.384.1158G, 2010_Goumans_CO_O_MNRAS.406.2213G, 2013_Arasa_HOCO_JPCA..117.7064A} and experimentally. Several energetic and non-energetic pathways have been experimentally confirmed to form CO$_2$ under astrophysical conditions, such as, ground-state CO reacting with an electronically excited CO* to form CO$_2$ and atomic carbon \citep{1996_Gerakines_UV_ices_A&A...312..289G, 1998_Palumbo_CO2_A&A...334..247P, 2005_Loeffler_CO2_A&A...435..587L, 2006_Jamieson_CO_10K_ApJS..163..184J, 2009_Bennett_CO2_PCCP...11.4210B, 2009_Ioppolo_CO2_A&A...493.1017I}, CO reacting with atomic oxygen to form CO$_2$ \citep{2001_Roser_CO+O_ApJ...555L..61R, 2006_Madzunkov_CO+O_PhRvA..73b0901M, 2011_Raut_CO_O_ApJ...737L..14R, 2013_Ioppolo_RScI...84g3112I, 2013_Minissale_CO_O_A&A...559A..49M}, CO reacting with a hydroxyl (OH) radical to form CO$_2$ and atomic hydrogen \citep{2002_Watanabe_CO2_ApJ...567..651W, 2007_Watanabe_H2O-CO_ApJ...668.1001W, 2009_Ioppolo_CO2_A&A...493.1017I, 2010_Oba_non_OH_CO_ApJ...712L.174O, 2011_Oba_CO_OH_40-60K_PCCP...1315792O, 2011_Ioppolo_CO2_MNRAS.413.2281I, 2011_Noble_CO2_ApJ...735..121N, 2011_Zins_CO+OH_ApJ...738..175Z, 2014_Yuan_ERCO2_ApJ...791L..21Y}, formaldehyde (H$_2$CO) reacting with atomic oxygen and to form CO$_2$ and molecular hydrogen \citep{2015_Minissale_CO2_A&A...577A...2M}, or oxidation of carbonaceous surfaces \citep{2006_Mennella_CO2_carbon_ApJ...643..923M, 2012_Fulvio_CO2_carbon_ApJ...752L..33F, 2012_Raut_CO2_from_ASW_carbongrains_ApJ...752..159R, 2015_Sabri_CO2_carbon_A&A...575A..76S, 2015_Shi_Oxi_Graphite_ApJ...804...24S}. The above pathways were found to significantly produce CO$_2$ in the solid state and only the last two pathways do not include CO. The majority of these solid-state experiments are performed at temperatures below 20~K, representative of dark cloud or disk midplane ($>20$~AU) conditions. This is well below the CO sublimation temperature, which is approximately 20 and 30~K for interstellar and laboratory timescales, respectively \citep[see e.g.,][]{2011_Fayolle_CO_PSD_ApJ...739L..36F, 2016_Schwarz_CO_ApJ...823...91S}. The low temperatures in these experiments ensure that CO stays adsorbed on the surface and is able to react with the other ice constituents. The experimental studies by \citet{2011_Oba_CO_OH_40-60K_PCCP...1315792O} and \citet{2014_Yuan_ERCO2_ApJ...791L..21Y} have investigated the formation of CO$_2$ from CO at substrate temperatures above the sublimation temperature of CO. In the former, CO$_2$ was formed when CO and OH radicals were co-deposited on a substrate in the temperature range from 40 to 60~K. The latter observed formation of CO$_2$ when gas-phase CO interacted with OH radicals produced by UV photons on the surface of water ice at 76~K. Both works show that CO can interact with OH radicals in the solid state above its canonical desorption temperature. In this work, we set out to experimentally investigate the conversion of gas-phase CO into CO$_2$ on the surface of vacuum-UV (VUV) irradiated water ice (40--120~K), and assess the efficiency in astrophysical settings. Specifically, amorphous solid water (ASW) is irradiated at a temperature of $\geq40$~K, which ensures that the majority of the gas-phase CO in our experimental chamber does not freeze out onto our ASW sample. Section~\ref{sec:methods} describes the methods used to investigate this process, and analyze the data. Results are presented in Sect.~\ref{sec:results} and are discussed in Sect.~\ref{sec:disc}. The astrophysical implications are given in Sect.~\ref{sec:astro_imp}, and concluding remarks are summarized in Sect.~\ref{sec:conc}. \begin{table*} \caption{Overview of performed experiments} \label{tab:exp} \centering \begin{tabular}{llccc} \toprule\toprule Series & \multicolumn{1}{c}{Molecules} & Temperature$^a$ & \ch{H2O} column density$^{b, c}$ & Notes\\ & & (K) & (monolayers) & \\ \midrule Main experiments & H$_2${}$^{18}$O (s) + $^{13}$C$^{18}$O (g) & \phantom{1}40\phantom{1} & 57.8 & -- \\ & H$_2${}$^{18}$O (s) + $^{13}$C$^{18}$O (g) & \phantom{1}50\phantom{1} & 56.7 & -- \\ & H$_2${}$^{18}$O (s) + $^{13}$C$^{18}$O (g) & \phantom{1}60\phantom{1} & 57.6 & -- \\ & H$_2${}$^{18}$O (s) + $^{13}$C$^{18}$O (g) & \phantom{1}70\phantom{1} & 61.8 & -- \\ & H$_2${}$^{18}$O (s) + $^{13}$C$^{18}$O (g) & \phantom{1}80\phantom{1} & 62.9 & -- \\ & H$_2${}$^{18}$O (s) + $^{13}$C$^{18}$O (g) & \phantom{1}90\phantom{1} & 57.8 & -- \\ & H$_2${}$^{18}$O (s) + $^{13}$C$^{18}$O (g) & 100\phantom{1} & 56.8 & -- \\ & H$_2${}$^{18}$O (s) + $^{13}$C$^{18}$O (g) & 120\phantom{1} & 58.3 & -- \\ \midrule Control & H$_2${}$^{18}$O (s) & \phantom{1}40\phantom{1} & 70.2 & \multicolumn{1}{l}{water only} \\ & H$_2${}$^{18}$O (s) + $^{13}$CO (g) & \phantom{1}60\phantom{1} & 63.2 & \multicolumn{1}{l}{{$^{13}$C$^{16}$O}} \\ & H$_2${}$^{18}$O (s) + $^{13}$C$^{18}$O (g) & \phantom{1}40\phantom{1} & 66.4 & \multicolumn{1}{l}{no VUV irradiation} \\ \bottomrule \end{tabular} \tablefoot{$^{(a)}$ All ices have been deposited at 15~K. The temperature refers to the value at which the ASW is VUV irradiated. $^{(b)}$ The H$_2$O column density is derived through the integrated area of the OH-stretching mode (boundaries, 3800--2950~cm$^{-1}$) through Eq.~\ref{eq:col_den}, where $A'$ is taken to be $1.5 \times 10^{-16}$~cm molec$^{-1}$ \citep[H$_2${}$^{16}$O,][]{2015_Bouilloud_A'_MNRAS.451.2145B}. $^{(c)}$ It is likely that this value represents a lower limit due to the nonlinearity of RAIRS at column densities above 10~monolayers.} \end{table*} \section{Methods} \label{sec:methods} \subsection{CryoPAD2} All reported laboratory measurements are performed in the Leiden Laboratory for Astrophysics using the Cryogenic Photoproduct Analysis Device 2 \citep[CryoPAD2;][]{2017_Niels_CH3NCO_MNRAS.469.2219L,2018_Niels_CH3OH_A&A...612A..88L}. This setup operates under ultra-high vacuum conditions ($\mathrm{P_{mc}} \sim 5 \times 10^{-11}$~mbar at 15~K). It accommodates a gold-coated substrate which is positioned in the center of a stainless steel chamber and acts as an analogue for an interstellar dust-grain surface. On top of the chamber a closed-cycle helium cryostat is positioned which cools the gold-coated surface down to temperatures of 15~K. A Lakeshore 350 temperature controller sets the temperature of the substrate through PID-controlled Joule heating in the range of 15 to 300~K with an absolute and relative accuracy of $\pm2$ and $\pm1$~K, respectively. In order to further simulate the interstellar environments in which these dust grains reside, a microwave-discharge hydrogen-flow lamp (MDHL) is connected to the chamber. These type of sources generally produce VUV photons at 121.6~nm, Lyman-$\alpha$, and between 140 to 170~nm, which corresponds to photon energies of 7.5 to 10.2~eV. However, in the present experiments a MgF$_2$ window is used which absorbs Lyman-$\alpha$ photons, see Appendix \ref{app:uv_spec} for the VUV spectrum. The flux of the MDHL at the location of the substrate is determined with a NIST-calibrated photodiode (SXUV-100) as $(2.5\pm0.3) \times 10^{14}$~photons s$^{-1}$ cm$^{-2}$. The reactions induced by VUV irradiation under these conditions are diagnosed using infrared spectroscopy and mass spectrometry. The collimated beam of a Fourier-Transform InfraRed Spectrometer (Agilent 660 FTIRS), is used for Reflection Absorption InfraRed Spectroscopy (RAIRS). In this method the incoming FTIR beam is reflected from the substrate under a grazing incidence angle, improving the sensitivity. This in situ diagnostic allows us to probe, qualitatively and quantitatively, the molecular content in the ice adsorbed on the substrate. The infrared spectra are acquired continuously during the experiments to investigate and track the chemical evolution in the solid state under the influence of VUV irradiation. The second diagnostic tool is a Hiden HAL/3F PIC 1000 series quadrupole mass spectrometer (QMS). During VUV irradiation some molecular species desorb from the substrate into the gas phase. The QMS probes the molecular content of the atmosphere in the chamber. This allows for qualitative assignment of species released or produced during the experiments through their characteristic mass-fragmentation patterns. Furthermore, after calibration of the QMS through the procedure described in Sect.~\ref{ssec:QMS_cal}, it is possible to derive the quantitative amount of a species released into the gas phase. After VUV irradiation, the substrate temperature is linearly increased with time in a temperature programmed desorption (TPD) experiment until all adsorbed species have thermally desorbed. During TPD, species are released into the gas phase at their canonical desorption temperature, and are subsequently measured by the QMS. Upon ionization, in our case with 70~eV electrons, molecules fragment into a characteristic fragmentation pattern, which allows for assignment of newly formed species, complementary to the infrared. Rare isotopologue precursors are used to discriminate from background gas contaminations, and to add diagnostic information to the RAIRS and TPD experiments. \subsection{Experimental protocol} The following molecules are used in the experiments: Milli-Q H$_2$O (Type I), H$_2${}$^{18}$O (Sigma-Aldrich, 97\%), regular CO (Linde gas, 99.997\%), and $^{13}$C$^{18}$O (Sigma-Aldrich, 99\% $^{13}$C and 95\% $^{18}$O). The experiment is started by depositing a layer of ASW onto the substrate. The gas-phase H$_2$O enters the chamber roughly one~cm away from the substrate, and is deposited under normal incidence to the substrate through a capillary array. The temperature of the substrate during H$_2$O deposition is set at 15~K and the water ice is deposited for 10 minutes. This ensures that the deposited H$_2$O is porous-ASW. A precision leak valve is used to guarantee consistent column densities of H$_2$O throughout the experiments. Before the experiments continue, the chamber is left to settle for at least 30~minutes. This ensures that the pressure in the main chamber (P$_{\mathrm{mc}}$) is below $2.0 \times 10^{-10}$~mbar and that the amount of residual gas-phase H$_2$O can be neglected. After this, the substrate temperature is slowly increased, with a rate of 2~K min$^{-1}$, to the temperature at which the experiments are performed. Once the desired sample temperature is reached, the MDHL is started and gas-phase CO is admitted into the chamber. The gas-phase CO enters the chamber at roughly 5 cm distance from the substrate, and under 45~degrees with respect to the substrate normal. The precision leak valve is set to have a constant P$_{\mathrm{mc}}$ of $5.0 \times 10^{-8}$~mbar. This translates into the ASW surface being exposed to approximately $5 \times 10^{13}$~CO molecules cm$^{-2}$ s$^{-1}$. The ASW is exposed in total 300~minutes to VUV irradiation (with a total incident fluence of $4.5 \times 10^{18}$~photons) and CO molecules, after which TPD is performed to sublimate parent and newly-formed species. During VUV irradiation, the shutter between the MDHL and vacuum chamber is closed periodically to measure the baseline signals of the chamber without VUV irradiation. The experiments performed in this study are listed in Table~\ref{tab:exp}. \subsection{Data analysis} \label{sec:analysis} \subsubsection{RAIRS vibrational spectroscopy} \label{ssec:RAIRS} The infrared spectra are acquired in RAIRS mode with the FTIR and are subsequently analysed. The column density, $N_{\rm{species}}$, of the probed molecules on the substrate is derived through the following relationship with the measured absorbance \begin{equation} \label{eq:col_den} N_{\rm{species}} = \ln(10)\cdot\frac{\int_{band}\log_{10}\left(\frac{I_0(\tilde{\nu})}{I(\tilde{\nu})}\right) d\tilde{\nu}}{R\cdot A'}, \end{equation} where the absorbance, the ratio of the incoming flux, $I_0(\tilde{\nu})$, and reflected flux, $I(\tilde{\nu})$, is integrated over a range that encompasses the full absorption feature, and $A'$ is the apparent band strength. The apparent band strengths are taken from literature from transmission experiments; for RAIRS these values need to be corrected with a value $R$ in order to retrieve accurate column densities. The RAIRS correction factor (\textit{R}) is empirically determined on CryoPAD2 through isothermal desorption of CO and is found to be 4.5 \citep[see e.g.,][for a description of the used approach]{2009_Oberg_CO_PSD_A&A...496..281O, 2018_Ligterink_peptide_MNRAS.480.3628L}. We assume that the area probed by the infrared beam on our substrate amounts to 1.0~cm$^2$ and thus the amount of molecules, $N_{\rm{species}}$, is also the column density, given in molecules cm$^{-2}$. As stated before and shown in Table~\ref{tab:exp}, all main experiments are performed with $^{13}$C$^{18}$O and H$_2$$^{18}$O resulting in the formation of $^{13}$C$^{18}$O$_2$. To our knowledge the apparent band strength of this specific isotope of CO$_2$ is unknown and thus the apparent band strength of $^{13}$CO$_2$, $6.8 \times 10^{-17}$~cm molecule$^{-1}$ \citep{2015_Bouilloud_A'_MNRAS.451.2145B}, is used to approximate the column density of $^{13}$C$^{18}$O$_2$. As is shown in Sect.~\ref{sec:results}, multiple CO$_2$ features are observed in the infrared. In order to follow the growth of these different CO$_2$ features, the three prominent ones are approximated by fitting a Gaussian profile to each of them in order to deconvolve the spectra. The \textsc{curve\_fit} function from \textsc{SciPy} is used to fit a Gaussian profile to each absorption component through least squares regression \citep{2020SciPy-NMeth}. A 3-Gaussian fit reproduces the integrated absorbance to $\leq10\%$, and suffices as a fit, given the variation in observed profile shapes due to (small) changes in physical or chemical environment and spectroscopic artifacts in the spectra. \subsubsection{QMS calibration} \label{ssec:QMS_cal} RAIRS allows determination of the column density in the solid state, while the QMS allows for quantification of molecules released from the solid state into the gas phase. In order to use the QMS for quantitative purposes, mass signals need to be calibrated and this is realized, through the photodesorption of CO \citep[see e.g.,][]{2013_Fayolle_PD_N2_O2_A&A...556A.122F, 2015_Martin_CO2_A&A...584A..14M}. The loss of solid-state CO is traced with RAIRS and is correlated to the gas-phase CO signal measured simultaneous by the QMS. This calibration allows for the conversion of any measured gas-phase QMS signal, released under VUV irradiation, to a column density from the solid state. However, one needs to correct for the difference in the electron-impact ionization cross section, the fractional fragmentation, and the mass sensitivity of the QMS of the investigated molecule with respect to CO. The following equation is used to quantify the amount of CO$_2$ formed in the solid state and subsequently released into the gas phase \begin{linenomath} \begin{equation} \label{eq:QMS_cal} \frac{N_{\rm{CO_2(ice)}}}{\int S_{\rm{CO_2(gas)}}} = \frac{\sigma_{\rm{CO}}}{\sigma_{\rm{CO_2}}} \cdot \frac{F(\rm{CO^+/CO})}{F(\rm{CO{_2}^{+}/CO_2})} \cdot \frac{M(\rm{CO})}{M(\rm{CO_2})} \cdot \frac{N_{\rm{CO(ice)}}}{\int S_{\rm{CO(gas)}}}, \end{equation} \end{linenomath} where $N_{\rm{CO_2}}$ is the column density of CO$_2$ released from the solid state, $\int S_{\rm{CO_2}}$ the integrated CO$_2$ QMS signal, $\sigma$ the total electron-impact ionization cross section, $F$ the fragmentation fraction of the ionized species, and $M$ the mass sensitivity function of the Hiden QMS on CryoPAD2. The last term of Eq.~\ref{eq:QMS_cal} is the CO calibration experiment, where $N_{\rm{CO}}$ is the amount of CO that photodesorbed from the solid state and $\int S_{\rm{CO}}$ the integrated CO signal measured by the QMS during photodesorption. In a similar fashion the column density of O$_2$ is determined, but with its respective parameters. The electron-impact ionization cross sections ($\sigma$) used in this work for CO, CO$_2$, and O$_2$ are 2.516, 3.521, and 2.441~$\AA^2$ at 70~eV, respectively \citep{EIICS_database}. The fragmentation fractions (\textit{F}) for CO$^+$, CO$_2$$^+$, and O$_2$$^+$ are 0.940, 0.942, and 0.821 respectively, where unity is the normalized summation of all fragmentation fragments. \section{Results} \label{sec:results} In this section, we present the results of the experiments mentioned in Table~\ref{tab:exp}. In general, irradiation of ASW with VUV photons in the presence of gas-phase CO produces CO$_2$. Additionally, in the experiments at the higher end of the temperature range, ($>$ 90~K), formation of molecular oxygen (O$_2$) is observed. To understand the processes that occur on or in the solid state we consider the infrared and QMS results, and how these change with temperature. \subsection{Infrared spectroscopy results} \label{ssec:infrared} We observe the formation of CO$_2$ in the solid state through RAIRS. In Fig.~\ref{fig:diff_spectra} we present five difference RAIRS spectra at experimental temperatures of 40 and 60~K, in the top and bottom panel, respectively. Such spectra are obtained by subtracting the initial ASW spectrum, before VUV irradiation, from the subsequently acquired spectra during irradiation. The spectra shown here are obtained at five different VUV fluence intervals where ASW was simultaneously exposed to gas-phase CO. Although the main isotopes used are $^{13}$C and $^{18}$O, small amounts of $^{12}$C and $^{16}$O are present in our samples. It is apparent that in the wavenumber range 2350--2225~cm$^{-1}$ absorption features grow with increasing fluence. All of the features in this range are attributed to isotopologues of CO$_2$. The three most distinct features are positioned at 2279, 2260, and 2243~cm$^{-1}$ and are attributed to $^{13}$C$^{18}$O$_2$ aggregates on top of the water ice, $^{13}$C$^{16}$O$^{18}$O bound to water, and $^{13}$C$^{18}$O$_2$ bound to water, respectively \citep{1977_Lehmann_CO2_ApPhy..13..153L,2017_Jiao_CO2_ApJ...837...65H}. The lowest ASW temperature at which the experiments are performed is 40~K. This ensures that the majority of the gas-phase CO molecules that enter the vacuum chamber cannot adsorb onto our sample, as it is above the canonical desorption temperature of CO. However, as shown by \citet{2016_Jiao_stick_ApJ...823...56H}, the sticking coefficient of CO on nonporous-ASW (np-ASW) is close to unity at 40~K. Once the ASW is covered with CO, no additional CO freeze-out occurs. This is also seen in our experiment at 40~K through the infrared signal around 2040~cm$^{-1}$ where, preceding VUV-irradiation, the ASW is briefly exposed to gas-phase CO only (Fig.~\ref{fig:diff_spectra}a). In this short 5~minute window, CO adsorbs on top of the ASW with a column density of $\sim1.1$~monolayers, where one monolayer equals $10^{15}$~molecules cm$^{-2}$. This CO ice grows within 60~s and does not further increase. As soon as VUV irradiation starts, this solid-state CO on the surface of the ASW is converted into CO$_2$. At ASW temperatures $\geq50$~K no adsorption of CO is seen, illustrated here for a temperature of 60~K (Fig.~\ref{fig:diff_spectra}b). This is expected as the sticking coefficient of CO on ASW significantly drops at temperatures $\geq50$~K. An upper limit of $\leq0.1$~monolayers is derived for CO adsorbed on top of ASW at temperatures $\geq50$~K. Figure~\ref{fig:exp_40K} shows the combined results of the experiment with ASW at 40~K. The upper panels show infrared data during irradiation (a) and after irradiation upon TPD (b). The lower panels (c and d) show the corresponding gas-phase data recorded with the QMS. Figure~\ref{fig:exp_40K}a shows the growth of each individual solid-state CO$_2$ component during VUV irradiation (shaded areas) as well as the combined results (red curve), while Fig.~\ref{fig:exp_40K}b shows the subsequent decrease during TPD after irradiation. It is evident that the deconvolved components evolve differently from each other. The CO$_2$ component at 2243~cm$^{-1}$ is the first to grow and levels off as the VUV fluence increases. This component is attributed to $^{13}$C$^{18}$O$_2$ that initially forms and is bound to the ASW surface. Sequentially, aggregates start to form on top of the water ice as the column density of CO$_2$ increases, because CO$_2$ does not wet the ASW surface, that is, the binding energy between CO$_2$--CO$_2$ is higher than CO$_2$--H$_2$O \citep{2017_Jiao_CO2_ApJ...837...65H}. These CO$_2$ aggregates absorb infrared light at a different wavenumber, namely 2279~cm$^{-1}$ \citep[cf.][who observed the same features, but shifted by $\sim100$ cm$^{-1}$ due to the different isotopologue used]{2017_Jiao_CO2_ApJ...837...65H}. With increasing temperature of the ASW, the diffusion of CO$_2$ across the ASW increases. This increased diffusion results in earlier formation of CO$_2$ aggregates and the amount of molecules in these aggregates increases. This is confirmed by less CO$_2$ molecules bound to ASW in the 2243~cm$^{-1}$ component with increasing temperature, see Figs.~\ref{fig:exp_40K}a (40~K), \ref{fig:exp_50K}a (50~K), and \ref{fig:exp_60K}a (60~K). The absorption feature at 2260~cm$^{-1}$ is due to $^{13}$C$^{16}$O$^{18}$O, formed from isotope impurities, bound to the ASW surface and is super imposed on top of the two absorption features of $^{13}$C$^{18}$O$_2$. There is a nonzero baseline between the 2243 and 2279~cm$^{-1}$ features due to the range of binding energies on the surface of H$_2$O with CO$_2$. As a result, the fitting of the 2260~cm$^{-1}$ component comprises contributions from both the 2243 and 2279~cm$^{-1}$ features. This makes an unique assignment and quantification of the 2260~cm$^{-1}$ component difficult. A decrease in the CO$_2$ column density is shown in Fig.~\ref{fig:exp_40K}b, following sublimation into the gas phase during TPD. The overall decrease as well as the decrease of the individual components are shown. The CO$_2$ sublimates in two steps, the first desorption event occurs at $\sim80$~K and the second at $\sim155$~K. The former is in line with the canonical desorption temperature of CO$_2$, and the latter with the canonical desorption temperature of H$_2$O. The component at 2279 cm$^{-1}$ drops around $\sim80$~K, which is in line with aggregates of CO$_2$ on top of the water ice. The component at 2243~cm$^{-1}$ gradually drops as the temperature of the ASW increases and disappears with the desorption of H$_2$O at 156~K, which is in agreement with CO$_2$ bound to ASW surface. At ASW temperatures $\geq70$~K, the majority of the formed CO$_2$ is immediately released back into the gas phase (see e.g., Fig.~\ref{fig:exp_70K}). At 70 and 80~K, however, some of the initially formed CO$_2$ remains in the solid state, see Fig.~\ref{fig:exp_70K}a and Appendix~\ref{fig:exp_80K}a, respectively. This CO$_2$ is formed during the initial moments of VUV irradiation and is bound to the deep binding sites on the ASW surface that are able to ``trap'' CO$_2$. The column densities of this CO$_2$ at 70 and 80~K are 0.6 and 0.12~monolayers, respectively, compared to the $\sim3$~monolayers formed at temperatures below 70~K. No solid-state CO$_2$ is detected in the experiments with ASW temperatures $\geq90$~K, and the upper limit of solid-state CO$_2$ is derived to be $\leq0.02$~monolayers. \subsection{QMS results} \label{ssec:QMS} The QMS allows for gas-phase species to be traced in the chamber during VUV irradiation and afterwards during TPD. The signals measured during TPD are only used for identification. In the following two sections we focus first on the QMS analysis of CO$_2$ that remained in the solid state (40--60~K) and then on CO$_2$ released into the gas phase ($\geq70$~K) after formation. \subsubsection{Solid-state CO$_2$ (40--60~K)} The majority of the CO$_2$ formed at ASW temperatures of 40--60~K remains in the solid state, as is found in the infrared experiments. This solid-state CO$_2$ is released into the gas phase during TPD due to thermal desorption, and subsequently measured with the QMS. However, during VUV irradiation there is some gas-phase CO$_2$ signal measured by the QMS. This is illustrated in Fig.~\ref{fig:exp_40K}c (shaded areas) by the signal at mass-to-charge ratio ($m/z$)~=~49, which is associated with the main peak of the $^{13}$C$^{18}$O$_2$ mass spectrum. The increase in this CO$_2$ QMS signal follows approximately the same trend as the growth of the CO$_2$ column density measured in the infrared (Fig.~\ref{fig:exp_40K}a). When the VUV shutter is closed, non-shaded areas, the signal at $m/z$~=~49 drops. We attribute this gas-phase CO$_2$ QMS signal to photodesorbed CO$_2$ from the solid state \citep{2014_Fillion_CO2_PD_FaDi..168..533F}. During TPD, there are two distinct desorption peaks of CO$_2$ with an elevated plateau between them (see e.g., Fig.~\ref{fig:exp_40K}d). The first desorption peak occurs at 78~K, the canonical desorption temperature of CO$_2$. The CO$_2$ molecules that desorb at this temperature, are those in CO$_2$ aggregates. The second desorption peak coincides with the water desorption peak observed at 156~K. Both behave fully in agreement with the deconvolved infrared components at 2279 and 2243~cm$^{-1}$ (see e.g., Fig.~\ref{fig:exp_40K}b). \subsubsection{Gas-phase CO$_2$ ($\geq70$~K)} In the remainder of the experiments, listed in Table~\ref{tab:exp}, and shown in Figs.~\ref{fig:exp_70K}, \ref{fig:exp_80K}--\ref{fig:exp_120K}, the temperature of the ASW ranges from 70--120~K. In the temperature range 70--90~K equal amounts of CO are converted into CO$_2$ as compared to < 70~K. However, the majority of the formed CO$_2$ is immediately released into the gas phase after formation. Similar to Fig.~\ref{fig:exp_40K}, we present the results of ASW at 70~K in Fig.~\ref{fig:exp_70K}. During this experiment $\sim20\%$ of the formed CO$_2$ stays on the surface of ASW, while the remainder is released into the gas phase. The release of CO$_2$ into the gas phase is slightly below the canonical CO$_2$ desorption temperature, 78~K. This is no surprise as the binding energy of CO$_2$ bound to H$_2$O equals 2250~K, while the binding energy between CO$_2$ molecules is higher at 2415~K \citep{2017_Jiao_CO2_ApJ...837...65H}. During the first hour of VUV irradiation, the ASW surface ``traps'' some of the formed CO$_2$ in its deep binding sites, but once these are occupied, most of the subsequently formed CO$_2$ is released into the gas phase. This is reflected by the initial rapid build up of CO$_2$ in the infrared during the first VUV irradiation interval (Fig.~\ref{fig:exp_70K}a). Additionally, in Fig.~\ref{fig:exp_70K}c it is shown that the gas-phase CO$_2$ builds up during the initial VUV interval, where it reaches steady state at the same time when the growth of solid-state CO$_2$ levels off. Lastly, during TPD the two main desorption features appear at approximately 80 and 155~K, but not as prominent as in the 40--60~K experiments. The majority of the CO$_2$ is released during TPD in the ``plateau'' region between 85--145~K, that is, between the canonical desorption of CO$_2$ aggregates and H$_2$O, as is shown in Fig.~\ref{fig:exp_70K}d. In the experiments with ASW temperatures between 90 and 120~K (Figs.~\ref{fig:exp_90K}--\ref{fig:exp_120K}), no solid-state CO$_2$ is observed in the infrared (column density $\leq 2.0\times10^{13}$~cm$^{-2}$). The TPDs in this temperature range, however, do reveal that some CO$_2$ is still bound to the surface of the ASW. Following the trend as seen in the experiments with ASW at 70 and 80~K, the amount of CO$_2$ that remains on the ASW surface decreases with increasing temperature, see Figs.~\ref{fig:exp_90K}--\ref{fig:exp_120K}. This is in line with the decrease in absolute signal of the ``plateau'' during TPD. Interestingly, a significant amount of O$_2$ formation is observed in the 90--120~K temperature range, as shown by the QMS signal at $m/z$~=~36 representing $^{18}$O$_2$. The formation of O$_2$ increases with temperature at the cost of CO$_2$. At 120~K the formation of CO$_2$ is almost completely quenched. Compared to 80~K the raw QMS data at 120~K for CO$_2$ is decreased by over a factor of 10 and the O$_2$ signal increased by over a factor of 50. \subsection{Control experiments} Several control experiments have been performed in order to aid in the investigation of the interaction between gas-phase CO and VUV irradiated water ice. Specifically, a control experiment where a different oxygen isotope is used in CO, an experiment without VUV irradiation, and an experiment where gas-phase CO is omitted, as listed in Table~\ref{tab:exp}. The results of these experiments are presented in Appendix~\ref{fig:exp_isotope}, \ref{fig:exp_no_uv}, and \ref{fig:exp_water_only}, respectively. The experiment with gas-phase $^{13}$C$^{16}$O, instead of the $^{13}$C$^{18}$O isotopologue, provides additional information on the interaction between the gas-phase CO and VUV irradiated water ice (Fig~\ref{fig:exp_isotope}). In the infrared the three main absorption features are shift by approximately 20~cm$^{-1}$, 2297, 2280, and 2262~cm$^{-1}$, with respect to the main experiments, 2279, 2260, and 2243~cm$^{-1}$, respectively. This indicates that these three CO$_2$ features are formed through the same process as each of them shift by approximately the same wavenumber due to the difference in mass of the oxygen isotope. Additionally, the QMS results show that the CO$_2$ is now detected at a $m/z$ that equals 47 ($^{13}$C$^{16}$O$^{18}$O) instead of 49 ($^{13}$C$^{18}$O$_2$). This indicates that both oxygen atoms from the precursors, that is, $^{16}$O from $^{13}$C$^{16}$O and $^{18}$O from H$_2$$^{18}$O, are involved in the formation of CO$_2$. The experiment without VUV irradiation, shown in Fig~\ref{fig:exp_no_uv}, is performed to track the level of CO$_2$ from other sources. A build up of~0.1 ML CO$_2$ is detected during the time that the water ice is exposed to gas-phase CO. Compared to the counterpart experiment with VUV radiation, shown in Fig~\ref{fig:exp_40K}, this is only 4\% of the total CO$_2$ formed when water ice is VUV irradiated. This small contribution is most likely due to contamination from previous experiments or trace amounts of $^{13}$C$^{18}$O$_2$ in our $^{13}$C$^{18}$O gas bottle. Regardless, this amount of contamination is negligible. Lastly, water ice is VUV irradiated without the presence of gas-phase CO in the vacuum chamber. The results of this experiment are presented in Fig~\ref{fig:exp_water_only}. Both the infrared and the QMS show at most trace amounts of CO$_2$. This supports that the main processes through which CO$_2$ in this study is formed through the interaction between gas-phase CO and VUV irradiated water ice. Additionally, the QMS shows the release of O$_2$ into the gas phase during VUV irradiation, which is not seen in the counterpart main experiment with gas-phase CO, as shown in Fig~\ref{fig:exp_40K}. \subsection{CO$_2$ and O$_2$ column densities} \label{ssec:col_dens} For each of the main experiments the column densities of the products, with a total VUV incident fluence of $4.5 \times 10^{18}$~photons, are summarized in Fig.~\ref{fig:tot_col_den}. The column density of solid-state CO$_2$ is derived through the combined integrated absorbance area of the three infrared CO$_2$ features. The CO$_2$ and O$_2$ gas-phase column densities are derived through the calibration of the QMS described in Sect.~\ref{ssec:QMS_cal}. In short, in the temperature range 40--60~K the main product is solid-state CO$_2$. As substrate temperatures at 70~K or above, the CO$_2$ is detected in the gas phase, and at even higher temperatures, that is, $\geq90$~K, O$_2$ formation is observed at the cost of CO$_2$ production. \section{Discussion} \label{sec:disc} It is clear from the presented results that CO$_2$ is formed in our experiments, and that the temperature of the ASW influences the physical appearance of CO$_2$. In the following section we explore the different pathways to CO$_2$, which pathway results in the formation of CO$_2$ in our experiments, and the CO$_2$ production efficiency per absorbed VUV photon. \subsection{Exploring the reaction network} \label{ssec:network} In the introduction, we mentioned several solid state pathways that can form CO$_2$. The formation of CO$_2$ in our experiments is driven by VUV irradiation of ASW that interacts with gas-phase CO. This is different from most earlier studies where CO was embedded and intimately mixed with water ice. Such experiments are relevant for astronomical scenarios in which H$_2$O and CO are mixed in the solid state. However, these conditions are different from those discussed later in this study, that is, protoplanetary disks (Sect.~\ref{ssec:ppds}) and at molecular cloud edges (Sect.~\ref{ssec:mol_clouds}). There are two potential pathways to form CO$_2$ in our experiments, which involve both H$_2$O and CO, and three possible pathways that could lead to the observed formation of O$_2$ at higher temperatures. Figure~\ref{fig:network} gives a schematic overview of these reactions. In general, UV photons dissociate H$_2$O in the solid state through its excited \~{A} and \~{B} states, which mainly lead to the formation of OH radicals and atomic oxygen, \begin{linenomath} \begin{equation} \label{rea:H_OH} \mathrm{H_2O} + h\nu \rightarrow \mathrm{H + OH}, \end{equation} \begin{equation} \label{rea:H2_O1d} \mathrm{H_2O} + h\nu \rightarrow \mathrm{H_2 + O}. \end{equation} \end{linenomath} \citet{1975_Stief_O1D_JChPh..62.4000S} reported quantum efficiencies for both reactions~(\ref{rea:H_OH})~and~(\ref{rea:H2_O1d}) upon irradiation in two different wavelength ranges, namely 145--185~nm and 105--145~nm corresponding with the excited \~{A} and \~{B} states of solid-state H$_2$O, respectively. Water dissociated through the excited \~{A} state has been reported to have quantum efficiencies of 0.99 and $\leq0.01$ for reactions~(\ref{rea:H_OH})~and~(\ref{rea:H2_O1d}), respectively. The dissociation through the excited \~{B} state was reported to have quantum efficiencies of 0.89 and 0.11, respectively. In this study we use a MgF$_2$ window with a cut-off wavelength above the wavelength of Lyman-$\alpha$ photons. This ensures that Lyman-$\alpha$ photons from the MDHL are absorbed, and that the majority of the UV photons are in the 140--170~nm (7.3--8.9~eV) range, see Fig.~\ref{fig:uv_spec}. This, combined with the reported quantum efficiencies, results in the dissociation of H$_2$O only through its excited \~{A} state, producing mainly OH radicals through reaction~(\ref{rea:H_OH}). The removal of Lyman-$\alpha$ photons makes the VUV spectrum less representative of those in interstellar environments. However, it does allow for an in-depth investigation of primary reactions including only OH radicals. The formed H and OH proceed in different ways depending on the depth in the ice at which dissociation occurs. The molecular dynamics calculations by \citet{2008_Andersson_H2O_MD_A&A...491..907A} showed that in the top three monolayers the majority of the photodissocation events results in the desorption of H and trapping of OH. At four monolayers or deeper most of the photodissocation events result in trapping of both species or recombination, reforming H$_2$O through reaction~(\ref{rea:H2O}), \begin{linenomath} \begin{equation} \label{rea:H2O} \mathrm{H + OH \rightarrow H_2O}. \end{equation} \end{linenomath} The desorption of H in the top three monolayers results in an enrichment of OH radicals on the surface. It was found in these calculations that the OH radicals can diffuse up to 60 \r{A} on top of the H$_2$O surface at 10~K. This diffusion occurs on picosecond timescales, and does not include any thermal diffusion on longer timescales. \citet{2009_Hama_OH_desorp_JChPh.131e4508H} showed that the OH radicals produced through reaction (\ref{rea:H_OH}) are hot and have a translational temperature of $1300\pm300$~K. This significant amount of translational energy allows for additional diffusion, and increases the probability of two OH radicals to meet and react with each other. This reaction either forms hydrogen peroxide (H$_2$O$_2$) or H$_2$O and atomic oxygen, see reactions (\ref{rea:H2O2}) and (\ref{rea:H2O_O3p}), respectively. The branching ratio between reactions~(\ref{rea:H2O2}) and (\ref{rea:H2O_O3p}) was found to be 0.8 and 0.2 for two nonenergetic OH radicals reacting with each other at 40--60~K \citep{2011_Oba_CO_OH_40-60K_PCCP...1315792O}, \begin{linenomath} \begin{equation} \label{rea:H2O2} \mathrm{OH + OH \rightarrow H_2O_2}, \end{equation} \begin{equation} \label{rea:H2O_O3p} \mathrm{OH + OH \rightarrow H_2O + O}. \end{equation} \end{linenomath} As the abundance of the H$_2$O$_2$ increases, also the amount of H$_2$O$_2$ dissociated by VUV photons increases through reaction (\ref{rea:OH_OH}), \begin{linenomath} \begin{equation} \label{rea:OH_OH} \mathrm{H_2O_2} + h\nu \rightarrow \mathrm{OH + OH}. \end{equation} \end{linenomath} The translational temperature of these OH radicals was found to be $7500\pm1000$~K, which potentially allows for even further diffusion of OH radicals across the surface \citep{2009_Hama_OH_desorp_JChPh.131e4508H}. These OH radicals on the surface, or those below for that matter, can be subsequently dissociated by VUV photons forming atomic hydrogen and oxygen, see reaction (\ref{rea:H_O3p}), \begin{linenomath} \begin{equation} \label{rea:H_O3p} \mathrm{OH} + h\nu \rightarrow \mathrm{H + O}. \end{equation} \end{linenomath} The OH radicals formed through reactions~(\ref{rea:H_OH}) and (\ref{rea:OH_OH}) are potential candidates for CO$_2$ formation when reacting with gas-phase CO. In the experimental study by \citet{2010_Oba_non_OH_CO_ApJ...712L.174O} the formation of CO$_2$ was observed from co-deposition of nonenergetic OH radicals, cooled to 100~K prior to deposition, and CO molecules at 10 and 20~K. The authors proposed that CO$_2$ forms through reactions~(\ref{rea:trans-HOCO}), (\ref{rea:cis-HOCO}), and (\ref{rea:CO2_H}), \begin{linenomath} \begin{equation} \label{rea:trans-HOCO} \mathrm{CO + OH \rightarrow }\ trans\mathrm{-HOCO}, \end{equation} \begin{equation} \label{rea:cis-HOCO} trans\mathrm{-HOCO \rightarrow}\ cis\mathrm{-HOCO}, \end{equation} \begin{equation} \label{rea:CO2_H} cis\mathrm{-HOCO \rightarrow CO_2 + H}. \end{equation} \end{linenomath} Atomic oxygen formed through reactions~(\ref{rea:H2O_O3p}) and (\ref{rea:H_O3p}) also has the potential to react with CO and form CO$_2$ through reaction~(\ref{rea:CO2}), \begin{linenomath} \begin{equation} \label{rea:CO2} \mathrm{CO + O \rightarrow CO_2}. \end{equation} \end{linenomath} The addition of atomic oxygen to CO has been experimentally shown to work between 5 and 20~K, where CO is adsorbed on a bare substrate \citep{2001_Roser_CO+O_ApJ...555L..61R, 2006_Madzunkov_CO+O_PhRvA..73b0901M, 2011_Raut_CO_O_ApJ...737L..14R, 2013_Ioppolo_RScI...84g3112I}. The same reaction has also been investigated on top of ASW by \citet{2013_Minissale_CO_O_A&A...559A..49M}. These authors show that the CO$_2$ is formed through reaction~(\ref{rea:CO2}) when CO and O are co-deposited on the surface of ASW in the temperature range 10--50~K. The initially formed O, OH, and H$_2$O$_2$ can react with each other to form the hydroperoxyl radical (HO$_2$) and O$_2$. The HO$_2$ radical is formed through subsequent reactions of H$_2$O$_2$ with OH, see reaction~(\ref{rea:H2O2_OH}). Molecular oxygen can be formed through different means, a) an HO$_2$ radical reacts with OH, see reaction~(\ref{rea:HO2_OH}), b) the HO$_2$ radical falls apart, see reaction~(\ref{rea:HO2_dism}), c) atomic oxygen reacts with another atomic oxygen, see reaction~(\ref{rea:O_O}), or d) atomic oxygen reacts with OH, see reaction~(\ref{rea:O_OH}). \begin{linenomath} \begin{equation} \label{rea:H2O2_OH} \mathrm{H_2O_2 + OH \rightarrow HO_2 + H_2O}, \end{equation} \begin{equation} \label{rea:HO2_OH} \mathrm{HO_2 + OH \rightarrow O_2 + H_2O}, \end{equation} \begin{equation} \label{rea:HO2_dism} \mathrm{HO_2 \rightarrow H + O_2}, \end{equation} \begin{equation} \label{rea:O_O} \mathrm{O + O \rightarrow O_2}, \end{equation} \begin{equation} \label{rea:O_OH} \mathrm{O + OH \rightarrow O_2 + H}. \end{equation} \end{linenomath} \subsection{CO$_2$ formation pathway} \label{ssec:pathway} In order to disentangle which of the above formation pathways is active in our experiments, we look into the temperature dependence of the CO$_2$ formation process and which isotopes are incorporated in the produced CO$_2$. We also extensively compare with literature data. Across the temperature range of 40--90~K, the total column density of formed CO$_2$ is constant, but above 90~K the efficiency of CO$_2$ formation decreases due to the competing formation of O$_2$, see Fig.~\ref{fig:tot_col_den}. This temperature dependence contains significant amount of information, which allows us to constrain the formation of CO$_2$ to one pathway. Our experiments show that CO$_2$ is formed through the reaction between gas-phase CO and solid-state OH radicals. These OH radicals are the primary product of H$_2$O dissociation, and are thus most likely to react with CO. This particular reaction pathway to CO$_2$ has been investigated extensively \citep{2002_Watanabe_CO2_ApJ...567..651W, 2007_Watanabe_H2O-CO_ApJ...668.1001W, 2009_Ioppolo_CO2_A&A...493.1017I, 2010_Oba_non_OH_CO_ApJ...712L.174O, 2011_Oba_CO_OH_40-60K_PCCP...1315792O, 2011_Ioppolo_CO2_MNRAS.413.2281I, 2011_Noble_CO2_ApJ...735..121N, 2011_Zins_CO+OH_ApJ...738..175Z, 2014_Yuan_ERCO2_ApJ...791L..21Y}. However, the majority of these studies were performed at temperatures where CO is in the solid state, and mixed with H$_2$O. \citet{2010_Oba_non_OH_CO_ApJ...712L.174O} looked at the formation of CO$_2$ through co-deposition of CO and nonenergetic OH radicals at a temperature of 10 and 20~K. Besides CO$_2$, the authors also observed the intermediate products cis- and trans-HOCO radicals at 1774 and 1812~cm$^{-1}$, respectively. They found that the HOCO absorption features disappear at $T$ > 40~K, which is in line with the experimental work of \citet{1971_Milligan_CO_OH_JChPh..54..927M}. This is most likely the reason why the cis- and trans-HOCO radicals are not detected in the infrared spectra of our experiments (spectra not shown). In a follow-up study, \citet{2011_Oba_CO_OH_40-60K_PCCP...1315792O} investigated the same reactions, but in the temperature range 40--60~K. Formation of CO$_2$ was observed in the infrared, but the efficiency at which CO was converted into CO$_2$ decreased with increasing temperature. The conversion rates were found to be 1.4\%, 0.8\%, and 0.3\% at 40, 50, and 60~K, respectively. This decrease was attributed to the decreasing residence times with increasing surface temperature of both CO and OH. In our experiments this efficiency decrease is not observed, even when only considering solid-state CO$_2$. A possible explanation for this is the different origin of the OH radicals; in our work the radicals are formed in situ with excess energy, whereas in previous studies OH radicals were deposited. The other proposed formation pathway to CO$_2$, that is, atomic oxygen reacting with CO, can be excluded. This is because atomic oxygen can only originate in our experiments as a secondary product through reactions~(\ref{rea:H2O_O3p}) and (\ref{rea:H_O3p}). Additionally, on ASW atomic oxygen and CO have similar binding energies, that is, 1320 and 1350~K, respectively, and thus their residence times on the surface are comparable \citep{2016_minissale_O_N_A&A...585A.146M, 2016_Jiao_bind_energy_ApJ...825...89H}. Because of their similar residence times, no difference should be observed between the formation of CO$_2$ and O$_2$ with experimental temperature. However, O$_2$ is only significantly formed at temperatures $\geq90$~K. This is proof that atomic oxygen is not involved in the formation of CO$_2$. Lastly, \citet{2013_Minissale_CO_O_A&A...559A..49M} investigated the formation of CO$_2$ through co-deposition of CO and atomic oxygen on top of ASW. It was found that the efficiency of CO$_2$ formation peaked at 35~K and dropped to zero at 60~K. As CO$_2$ formation is observed in our experiments up to 120~K, this is again evidence that atomic oxygen is not involved in the formation of CO$_2$ in our experiments. It should be noted that experimental conditions are not identical, as in our experiments the atomic oxygen would be formed in situ instead of co-deposited with CO. However, since atomic oxygen would be in the ground state, for both our work and that of \citet{2013_Minissale_CO_O_A&A...559A..49M}, no clear differences are expected. Formation of CO$_2$ through excited CO* reacting with another CO molecule on the surface of ASW can also be ruled out. This is unlikely to occur, because it would require a CO molecule to be excited during its short, but nonzero, residence time and react with another CO molecule which has an equally short residence time. Additionally, in the control experiment with H$_2${}$^{18}$O and $^{13}$C$^{16}$O, the formed CO$_2$ is measured with the QMS during TPD at $m/z$~=~47, corresponding to $^{13}$C$^{16}$O$^{18}$O, see Appendix~\ref{fig:exp_isotope}. If the CO$_2$ would be formed through excited CO and another CO molecule it would be expected to be detected at $m/z$~=~45, corresponding to $^{13}$C$^{16}$O$_2$. From above results and discussion it is most likely that CO$_2$ is formed through the interaction between CO and OH radicals, formed by UV dissociation of H$_2$O. However, it is not yet clear if CO directly interacts with OH radicals from the gas phase, that is, an Eley-Rideal type reaction, or if CO adsorbs onto the ASW, diffuses, and subsequently reacts with OH radicals, that is, a Langmuir-Hinshelwood type reaction. Additionally, the formation location of CO$_2$ is also not yet clear; is it formed on the surface or embedded in the ASW? Both of these topics are discussed in the following section. \subsection{Formation location of CO$_2$} \label{ssec:location} The average time a species resides on a surface can be estimated at a given temperature starting from the measured or calculated binding energy to that surface. We derive that CO has residence times on ASW of $4.5\times10^{2}$--$7.7\times10^{-8}$~s in the range from 40 to 120~K. This is found through the Arrhenius equation, which can be written as \begin{linenomath} \begin{equation} \label{eq:arrhenius} k = A e^{-\frac{E_{bind}}{T}}, \end{equation} \end{linenomath} where $k$ is the rate constant, $A$ the frequency factor, which is taken to be 10$^{12}$ s$^{-1}$, $E_{bind}$ the binding energy of a species to a specific surface in K, and $T$ the temperature of the surface in K. The residence time is then given by the reciprocal of the rate constant from Eq.~\ref{eq:arrhenius}. For CO on ASW, the binding energy is dependent on the CO surface coverage, ranging from 1000--1700~K at 1--10$^{-3}$~monolayer coverage \citep{2016_Jiao_bind_energy_ApJ...825...89H}. The above residence times are estimated given an average binding energy of 1350~K for CO on ASW. For comparison, the binding energy of CO on the CO--CO interface is $855\pm25$~K \citep{2005_Oberg_CO_bind_ApJ...621L..33O}. Even within these short residence times, some diffusion across the surface is expected. The number of binding sites CO visits on ASW, before desorption occurs, is estimated to be $7.2\times10^{10}$--$4.2\times10^{3}$ in the range 40--120~K. This is also derived through Eq.~\ref{eq:arrhenius}. Specifically, the number of different binding sites a molecule can visit before a species desorbs is approximated by dividing the diffusion rate by the desorption rate. The diffusion rate is estimated by exchanging the $E_{bind}$ term in Eq.~\ref{eq:arrhenius} for the diffusion energy ($E_{diff}$). The rate constant is then a proxy of the number of hops a species makes between different binding sites per second. The diffusion energy for CO on ASW has recently been measured in situ with transmission electron microscopy (TEM), and was found to be $350\pm50$~K \citep{2020_Kouchi_Ediff_CO_CO2_ASW_ApJ...891L..22K}. Given these residence times and amount of binding sites that are ``visited'' before desorption occurs, we conclude that CO spends sufficient time on the surface of ASW to react with OH radicals through a Langmuir-Hinshelwood type reaction. This is different from \citet{2014_Yuan_ERCO2_ApJ...791L..21Y}, who investigated this reaction under similar experimental conditions and attributed it to an Eley-Rideal type of reaction. \citet{2014_Yuan_ERCO2_ApJ...791L..21Y} employed a slightly higher binding energy of 0.125~eV (1450~K) for CO on H$_2$O, which results in a residence time of $\sim 2 \times 10^{-4}$~s at their experimental temperature of 76~K. The residence time is used to calculate the fractional coverage of CO on H$_2$O and was found to be $1 \times 10^{-6}$~ML. The resulting fractional coverage of OH was derived to be 0.05~ML, over four orders of magnitude higher, which led to the conclusion of an Eley-Rideal type of reaction. However, diffusion of CO during this (short) residence time was not considered. At these temperatures, CO visits approximately $10^6$ binding sites during its residence time, and thus, the effective surface scanned by CO is $\sim1$~ML even though the fractional coverage of CO is only $1 \times 10^{-6}$~ML. This supports that the involved mechanism follows a Langmuir-Hinshelwood type reaction. Furthermore, we see no evidence of significant CO diffusion, and subsequent trapping, into the bulk of the H$_2$O ice. However, there is some trapping of CO on the surface or pores of the ASW. This is shown in a control experiment where ASW is exposed to CO molecules, but not to VUV irradiation. During TPD of this control experiment, as is shown in Appendix~\ref{fig:exp_no_uv}, the majority of the CO desorbs at approximately 50~K. Only a small amount of CO ``volcano'' desorbs when the ASW crystallizes. It is most likely that this CO got trapped in ASW due to pore collapse, instead of actually diffusing into the bulk ASW. \subsection{Temperature dependent formation, CO$_2$ vs O$_2$} \label{ssec:CO2vsO2} At ASW temperatures above 90~K, the production of O$_2$ increases at the cost of CO$_2$ formation (see e.g., Fig~\ref{fig:tot_col_den}). An in-depth investigation of O$_2$ formation is beyond the scope of this work, but it is briefly discussed to explain the decrease in CO$_2$ production. For more information on O$_2$ production, we refer to a recent study that quantitatively investigated the production of O$_2$ and H$_2$O$_2$ by VUV irradiation of H$_2$O \citep{2022_Bulak_O2_A&A...657A.120B}. Since O$_2$ formation occurs at the cost of CO$_2$ production, it is likely that both species have a common precursor. In Sect~\ref{ssec:network}, the pathways to O$_2$ are through the OH radical or atomic oxygen. Due to the high temperature and simultaneous increase and decrease in O$_2$ and CO$_2$ abundance, respectively, the O$_2$ is likely formed through OH radicals. These OH radicals form O$_2$ sequentially through H$_2$O$_2$ and HO$_2$, see reactions~(\ref{rea:H2O2_OH}), (\ref{rea:HO2_OH}), and (\ref{rea:HO2_dism}). The formation of O$_2$ involving atomic oxygen is unlikely, as it is shown to not be involved in the formation of CO$_2$, see Sect.~\ref{ssec:pathway}. Additionally, in our experiments atomic oxygen is a secondary product through reactions~(\ref{rea:H2O_O3p}) and (\ref{rea:H_O3p}). The low residence times of this atomic oxygen, $<2.3\times10^{-6}$~s at temperatures above 90~K, combined with low availability makes formation of O$_2$ through OH radicals the dominant pathway. The formation of CO$_2$ and O$_2$ is dependent on the availability of OH radicals and the temperature of the ASW. Given this temperature dependence, it is likely that the increased mobility of the OH radical and reduced residence time of CO at high temperatures holds the answer to why O$_2$ formation is more favourable. The pathway to form O$_2$ at high temperatures is worthy of further investigation. \subsection{Conversion rate of CO into CO$_2$} \label{ssec:CO_into_CO2} In order to demonstrate that the gas-grain pathway to convert CO into CO$_2$ is a process of importance in astrophysical environments, we discuss in this section the conversion rate and limiting factors in our experiments. In total $\sim60$~monolayers of water ice are deposited on the substrate in preparation of our experiments. However, as this conversion of CO into CO$_2$ occurs on the surface, not all of this H$_2$O is available to act as a reacting medium. Classically, the surface of solid-state H$_2$O contains approximately $10^{15}$~molecules per cm$^2$. However, due to the porous nature of our ASW, the available H$_2$O surface for CO to adsorb on is expected to be larger. Additionally, with hydrogen released from the top three monolayers upon UV dissociation of H$_2$O, and the mobility of the OH radicals, we assume that OH radicals formed in the top three monolayers are available to convert CO into CO$_2$ \citep{2008_Andersson_H2O_MD_A&A...491..907A}. These top three monolayers, that is, $3.0\times10^{15}$~H$_2$O molecules cm$^{-2}$, are henceforth the reactive surface. The VUV radiation that impacts the water ice, consists only of photons from molecular H$_2$ emission, as the Lyman-$\alpha$ photons of the MDHL are absorbed by our MgF$_2$ window, see Fig.~\ref{fig:uv_spec}. The average H$_2$O absorption cross section for the VUV photons is taken to be ($1.2\pm0.1$)~$\times10^{-18}$~cm$^2$. This average is derived from the absorption cross section of the two main molecular H$_2$ emission peaks at 157.8 and 160.8~nm in a one-to-one ratio \citep{2014_Cruz-Diaz_VUV_cross_A&A...562A.119C}. Given this cross section and the Beer-Lambert law, $\sim0.4\%$ of the total incident VUV fluence is absorbed by H$_2$O in the reactive surface, and this equals $1.6\times10^{16}$ photons. On the assumption that H$_2$O dissociation is 100\% efficient, this produces an equal amount of OH radicals in the reactive surface. In the temperature range 40--90 K, approximately $2.7\times10^{15}$~CO$_2$ molecules are formed, and thus an equal amount of OH radicals is consumed through reactions (\ref{rea:trans-HOCO}--\ref{rea:CO2_H}). This means that, for the above assumption, only 17\% of the available $1.6\times10^{16}$~OH radicals are involved in the conversion of CO into CO$_2$. The question is, what is the limiting factor that determines this efficiency factor? In our experiments, it is the amount of OH and not CO that is limiting the formation of CO$_2$. In the temperature range of 40--90 K, the formation of CO$_2$ is considered to be constant. However, the residence time of CO is lowered by a factor of $10^{8}$ and the binding sites visited by CO by a factor of $10^5$ from 40 to 90~K. That is to say, once CO finds an OH radical on the surface that is available, the conversion into CO$_2$ is (close to) unity. The relatively small fraction (17\%) of VUV absorption events resulting in the CO$_2$ production may be due to a smaller number of available OH radicals, for example, the recombination probability of H + OH on the surface is high. It is also possible that other products are formed besides CO$_2$ or that UV dissociation of ASW is not unity, which is typically assumed. These are in our view the most logical explanations. Below we argue why from these options the dissociation efficiency of solid-state H$_2$O is most likely the limiting factor in the formation of CO$_2$. It would be surprising if a significant portion of the available OH radicals would not react, because even nonenergetic OH radicals in the ground state are able to form CO$_2$ with CO at 10~K \citep{2010_Oba_non_OH_CO_ApJ...712L.174O}. Formation of other products, such as, H$_2$O$_2$, is also excluded as we do not detect significant amounts of the possible products in the experiments. Assuming that VUV photodissociation of water ice is unity and recombination does not occur in the reactive surface, the reactive surface would be dominated by OH radicals (83\% of the remaining VUV absorption events). This amount of OH radicals should largely find each other and react to form H$_2$O$_2$ and O$_2$. The infrared does not show any of the vibrational modes of H$_2$O$_2$ within our detection limits (spectra not shown), especially not with the expected H$_2$O$_2$ column density of $\sim7$~monolayers, assuming that all of the remaining OH radicals react with each other and form H$_2$O$_2$ through reaction (\ref{rea:H2O2}). In itself this column density is already questionable, as it is approximately twice the amount of available H$_2$O molecules in the reactive surface. Additionally, a control experiment where ASW is irradiated at 40~K and gas-phase CO is omitted does show the formation of H$_2$O$_2$ and O$_2$, see Appendix \ref{fig:exp_water_only}. This shows that in the presence of gas-phase CO the formation of H$_2$O$_2$ and O$_2$ in the reactive surface is quenched, and that in the main experiments all available OH radicals react with CO into CO$_2$. The amount of CO$_2$ produced, and lack of other products, in the experiments points at that either the recombination of H + OH is high in the reactive surface or that not every VUV absorption event results in the dissociation of H$_2$O. Fully investigating this question is beyond the scope of this work, but it leads to a conundrum that requires attention. It is unlikely that recombination to H$_2$O is significantly more efficient than predicted by molecular dynamics calculations. Photodissociation and desorption occur on picosecond timescales after VUV absorption in these simulations, and thus, diffusion of atomic hydrogen within this time window is improbable. It should be noted that in molecular dynamics calculations only a single event is considered per simulation, and thus for H$_2$O recombination to occur the atomic hydrogen needs to find its original OH partner before it desorbs. It could be that in our experiments the atomic hydrogen reforms H$_2$O with previously formed OH radicals due to the high VUV photon fluxes, that is, $2.5\times10^{14}$~photons s$^{-1}$ cm$^{-2}$. Molecular dynamics calculations of H$_2$O photodissociation and recombination with neighbouring available OH radicals are needed to test if this pathway is viable. For now we deem it inefficient due to the extremely short time scales in which desorption occurs after dissociation. A lower dissociation efficiency of water ice upon absorption of a VUV photon, that is, well below unity, seems to be the most likely explanation for our findings, at least based on the processes discussed here. In Fig.~2 of \citet{2008_Andersson_H2O_MD_A&A...491..907A} the fractional probabilities of photodissociation pathways for H$_2$O are given per absorbed photon and sum to (near) unity. However, per design these simulations only follow H$_2$O that is photodissociated. \citet{1990_Schriever_dissociation_H2O_JChPh..93.9206S} investigated the absolute photodissociation quantum yield of H$_2$O in an argon matrix (ratio 1:500). It was found that at 160~nm and 5~K the photodissociation efficiency of isolated H$_2$O in argon equals 20--30\%. It should be noted that in their work the H$_2$O is isolated and trapped in the bulk argon, which cannot be extrapolated to our work. That being said, it does match well with our 17\% efficiency, which suggests that indeed the dissociation of water ice, through the excited \~{A} state, is not unity. Similar results have been found by \citet{2018_Kalvans_eff_dissociation_MNRAS.478.2753K} who investigated how the photodissociation efficiency differs in general for a molecule in the gas-phase and solid-state. They found a best-fit value of 0.3 for the ratio solid-state to gas-phase photodissociation from their 1D astrochemical model in comparison to line-of-sight observations of collapsing interstellar clouds. Even though our findings are not fully conclusive, it seems to be in line with results presented earlier in the literature. \citet{2013_Arasa_HOCO_JPCA..117.7064A} investigated the CO + OH pathway at 10~K with molecular dynamics calculations and found a conversion probability for CO$_2$ of ($3.6\pm0.7$) $\times10^{-4}$ per absorbed photon, which is significantly lower than our 17\%. However, it was found that the formation probability of the intermediate HOCO complex is two orders of magnitude higher with ($3.00\pm0.07)$ $\times10^{-2}$ per absorbed photon. This is explained by the HOCO complex being trapped in the solid state and losing its internal energy to the surrounding molecules, which prevents further reaction to CO$_2$ + H. This does not necessarily align with the findings presented here. In our experiments the HOCO complex is not observed. However, it could very well be that this HOCO complex is briefly present, but transfers into CO$_2$ due to the increased ASW temperature, as seen in \citet{1971_Milligan_CO_OH_JChPh..54..927M} and \citet{2011_Oba_CO_OH_40-60K_PCCP...1315792O}. Even then our formation efficiency is significantly higher than their calculated efficiency, which even includes an assumed solid-state dissociation efficiency of unity. We postulate that this is because of the difference in temperature and that the calculations only consider isolated events, while in the work described here many CO molecules and OH radicals are present at once, which could increase reaction probabilities. A possible source of error in our determination of the formation efficiency are the experimental assumptions. Using Gaussian error propagation, we estimate the error in our formation efficiency of CO$_2$ per absorbed VUV photon in the reactive surface to be 60\%, and thus ranges from 7--27\%. Errors include, but are not limited to, assumptions made on the area which the infrared probes, the apparent band strength of $^{13}$C$^{18}$O$_2$, and the RAIRS correction factor. Although the propagation of all the individual errors results in a large uncertainty on our derived efficiency it is still at least four times smaller than the generally assumed value of unity for UV dissociation of water ice. \section{Astrophysical implications} \label{sec:astro_imp} Figure~\ref{fig:schematic} schematically summarizes the above results and discussion. The formation of CO$_2$ through interaction between gas-phase CO and VUV induced OH radicals on water ice is visualized in the top row of this figure. Details on this process are discussed in Sect.~\ref{sec:disc}. In short, CO has a short, but nonzero, residence time on water ice even though its temperature is above the canonical sublimation temperature of CO. Concurrently, the water ice is irradiated with VUV photons, which results in the production of OH radicals. The diffusion of CO and OH radicals across the surface allows them to react with each other and form CO$_2$ in a Langmuir-Hinshelwood type reaction. Depending on the temperature of the water ice, the CO$_2$ is either mainly bound to the water ice, forms aggregates on top of the water ice, or is released into the gas phase, as is shown in the bottom row of Fig.~\ref{fig:schematic}. A full analysis of the experiments is given in Sect.~\ref{sec:results}, but briefly, in the lower end of the experimental temperature range, that is, 40--60~K, the formed CO$_2$ remains in the solid state. Specifically, at 40~K (Fig.~\ref{fig:schematic}e) the CO$_2$ is not mobile enough and the majority of the CO$_2$ stays bound to the ASW surface. However, in the experiment at 60~K (Fig.~\ref{fig:schematic}f), the CO$_2$ has significant mobility and starts diffusing across the surface, and forms CO$_2$ aggregates on top of the ASW. As shown in Fig.~\ref{fig:schematic}g, the formed CO$_2$ is released into the gas phase at 80~K, and at higher temperatures, the formation of O$_2$ starts competing with CO$_2$, which is shown in Fig.~\ref{fig:schematic}h. In the 40--90~K range, our experiments show that 7--27\% of the absorbed UV photons in the reactive surface, that is, top three monolayers of ASW, result in the conversion of gas-phase CO into CO$_2$. In the following section, we look at two astrophysical environments, that is, protoplanetary disks and molecular clouds, where this pathway could play an important role and may explain observational findings. \subsection{CO conversion in protoplanetary disks} \label{ssec:ppds} In planet forming disks, the gas mass, as derived through CO isotopologues, often comes out factors 10--100 lower than expected based on the dust content. This finding is based on physical-chemical that includes photodissociation and freeze out of CO and its isotopologues or thermo-chemical modeling \citep{2014_Miotello_CO_mass_A&A...572A..96M, 2016_Miotello_CO_mass_A&A...594A..85M, 2017_Miotello_Lupus_A&A...599A.113M, 2021_Calahan_TW_Hya_ApJ...908....8C}. One interpretation is that these disks have already lost a significant fraction of their total gas mass. Another is that some unknown process locks up gas-phase CO on grains. A correct interpretation is essential for models of planet formation that rely on the available gas-mass reservoir as well as on the gas-to-dust ratio. Other gas-mass tracers are problematic, as H$_2$ is undetectable and HD has only been observed in a few sources \citep[see e.g.,][]{2013_Bergin_HD_Natur.493..644B, 2016_McClure_HD_ApJ...831..167M, 2017_Trapman_HD_A&A...605A..69T, 2020_Kama_HD_A&A...634A..88K}, where, incidentally, the HD data support the notion of gas-phase CO being locked up. The CO into CO$_2$ conversion has been proposed as a possible pathway to convert gas-phase CO into a species that is much more difficult to detect. Chemical models by \citet{2018_Bosman_CO_A&A...618A.182B} looked at several pathways through which CO could be converted into less volatile species to explain the low observed CO fluxes. These models are successful in this conversion on timescales shorter than average protoplanetary disk lifetimes, that is, $\sim3$~Myr. It should be noted that it was found in these models that gas-phase CO is in competition with atomic hydrogen for OH radicals on the surface \citep[see][for more details]{2018_Bosman_CO_A&A...618A.182B}. Furthermore, these results depend on the adopted binding energies, reaction rates, and formation of H$_2$. For example, the binding energy of CO in these models is kept constant at 855~K, no matter the environment. However, it has been found that the binding energy of CO on ASW can be as high as 1700~K \citep{2016_Jiao_bind_energy_ApJ...825...89H}. In a similar fashion, \citet{2021_Trapman_CO_conv_A&A...649A..95T} used physical-chemical models to investigate the low CO fluxes in the Lupus star-forming region. Disk regions with $T_{\mathrm{gas}}>35$~K were excluded in these models for gas-phase CO conversion, as verification models showed that CO conversion through grain-surface chemistry is negligible at these temperatures, but again this was tested with a CO binding energy of 855~K. Including the correct binding energy for CO on ASW in these types of models is crucial. This increased binding energy allows gas-phase CO to compete with atomic hydrogen for OH radicals in a larger temperature range, and thus a larger region in protoplanetary disks where CO can be removed from the gas phase. In our experiments, the models by \citet{2018_Bosman_CO_A&A...618A.182B}, and \citet{2021_Trapman_CO_conv_A&A...649A..95T}, the amount of gas-phase CO that is converted into CO$_2$ depends on the availability of OH radicals. The authors assume in their models that once H$_2$O absorbs a VUV photon, dissociation is 100\% efficient. As discussed in Sect.~\ref{ssec:CO_into_CO2}, our experiments show an efficiency for OH production of 7--27\% per absorbed UV photon as opposed to the 100\% adopted in the models. Further modeling is required, which also includes the correct binding energy of CO on ASW, to fully assess the impact of our results on the model predictions about the consumption of gas-phase CO through this gas-grain interaction. \subsection{CO$_2$ formation in edges of molecular clouds} \label{ssec:mol_clouds} Observations of icy grains surrounding young stars suggest that large amounts of the observed solid-state CO$_2$ is in a water-rich environment \citep[see e.g.,][]{2015_Boogert_icy_universe_ARA&A..53..541B}. Here we explore if the conversion of gas-phase CO into solid-state CO$_2$ investigated in this work could (partially) explain the presence of solid-state CO$_2$ embedded in water ice \citep{2008_Pontoppidan_CO2_ApJ...678.1005P}. It is generally assumed that the ice that covers dust grains is composed of two layers: a polar and an apolar layer, where the apolar layer is formed on top of the polar layer . The polar ice layer is dominated by species with larger dipole moments, such as H$_2$O, and the apolar ice layer mainly contains species with smaller or no dipole moments, such as CO and N$_2$. Over the years solid-state CO$_2$ has been detected in a large number of sources. For example, \citet{1999_Gerakines_ISO_CO2_ApJ...522..357G} observed solid-state CO$_2$ with ISO in molecular clouds in a range of different physical environments. Their analysis shows that the majority of the observed solid-state CO$_2$ is found in a polar environment. Additionally, the \textit{Spitzer} ``Cores to Disks'' program similarly showed that in embedded young low-mass stars the majority of the observed solid-state CO$_2$ is mixed in a polar water-rich environment \citep{2008_Pontoppidan_CO2_ApJ...678.1005P}. Given these H$_2$O-dominated environments, it is possible that a substantial fraction of this CO$_2$ is formed through reactions between CO and OH radicals. This would require CO to freeze out during H$_2$O formation in order to be intimately mixed and subsequently to be converted into CO$_2$. Infrared observations show that it is unlikely that CO is mixed in a water-rich environment. The observed CO absorption feature can be deconvolved into three components, namely 4.665~$\mu$m (2143.7~cm$^{-1}$), 4.673~$\mu$m (2139.7~cm$^{-1}$), and 4.681~$\mu$m (2136.5~cm$^{-1}$). The blue component (4.665~$\mu$m) is assigned to CO in an apolar environment, specifically, it is linked to mixtures of solid-state CO and CO$_2$ \citep{2002_Boogert_CO_ApJ...568..761B, 2006_Broekhuizen_CO-CO2_A&A...451..723V} or crystalline CO \citep{2003_Pontoppidan_CO_A&A...408..981P}. The middle component (4.673~$\mu$m) is generally attributed to pure CO \citep[see e.g.,][]{2002_Boogert_CO_ApJ...568..761B}. The red component (4.681~$\mu$m) has a broader ``footprint'' compared to the other two and is linked to CO in a polar environment. This polar environment could be H$_2$O and would set the scene for solid-state formation of CO$_2$ in a polar environment. However, \citet{1988_Sandford_CO_ApJ...329..498S} showed that if CO would reside in H$_2$O, one can expect a feature at 4.647~$\mu$m (2151.9~cm$^{-1}$) due to the dangling OH bond, which has not been seen in interstellar spectra. It is thus unlikely that this CO is mixed with H$_2$O. Mixtures of CO with CH$_3$OH, however, do reproduce the red component in both peak position and width \citep{2011_Cuppen_CO_CH3OH_MNRAS.417.2809C}. A mixture of CO and CH$_3$OH is also more likely, as CH$_3$OH is formed through hydrogenation of CO \citep{1994_Hiraoka_hydro_CPL...229..408H, 2002_Hiraoka_hydro_ApJ...577..265H, 2002_Watanabe_hydro_ApJ...571L.173W, 2004_Hidaka_H2CO_ApJ...614.1124H, 2004_Watanabe_hydro_ApJ...616..638W, 2009_Fuchs_hydro_A&A...505..629F, 2022_Santos_CH3O_H2CO_ApJ...931L..33S}. In order to explain the solid-state CO$_2$ embedded in water-rich environments without invoking the need for CO embedded in H$_2$O ice, we look at the initial build-up of water ice on dust grains. Water can be formed through both gas-phase and solid-state pathways. However, the gas-phase ion-molecule chemistry produces only a fraction of the total observed water abundance, and thus water is mainly formed through the addition of hydrogen to atomic oxygen on the surface of dust grains \citep[see reviews by][and references therein]{2014_vanDishoeck_H2O_prpl.conf..835V, 2015_Linnartz_review_arXiv150702729L}. This formation process of solid-state H$_2$O takes place in the edges of molecular clouds at intermediate extinction ($A_V$). At this extinction, CO is already present, but still resides in the gas phase. For example, in the Taurus molecular cloud water ice is detected at a threshold extinction, that is, the extinction at which a species is detected in the solid state, of $3.2\pm0.1$ $A_V$, while for CO the threshold extinction was determined to be $6.7\pm1.6$~$A_V$ \citep{2001_WHittet_thresAv_H2O_ApJ...547..872W, 2010_Whittet_CO_function_Av_ApJ...720..259W}. Physical-chemical models of molecular clouds show that the dust-grain temperature at the edge of a cloud equals 31 K, with an external VUV field strength of 100~$G_\mathrm{0}$, where $G_\mathrm{0}$ is a scaling factor in multiples of the average local interstellar radiation field \citep{1968_Habing_G0_BAN....19..421H}, which indeed is sufficient to keep CO in the gas phase \citep{2009_Hollenbach_MC_models_ApJ...690.1497H}. For CO$_2$ to be mixed with H$_2$O it has to form simultaneously, and since CO is in the gas-phase during H$_2$O formation, CO$_2$ can be formed through the process described in this work. The OH radicals required for the conversion of CO into CO$_2$ are readily available in this region. They are the intermediate product to H$_2$O formation, and the external UV field is still sufficient at this extinction to photodissociate already existing H$_2$O molecules. The solid-state CO$_2$ that is observed to be embedded in water-rich environment in the ``Cores to Disks'' program, has a relative average abundance w.r.t. H$_2$O of $\sim0.2$. Assuming that roughly equal amounts of atomic oxygen go into CO and H$_2$O, this would imply that approximately 20\% of the gas-phase CO would have to be converted into solid-state CO$_2$ to explain observed abundances. Inclusion of this pathway in physical-chemical models of molecular clouds is required to test how efficient this process is in low-density regions at interstellar timescales and explain the observed solid-state CO$_2$ abundances. \section{Conclusions} \label{sec:conc} The work presented is principally different from many other solid-state astrochemical experiments presented in the past, as it implicitly takes into account gas-grain interactions. A number of systematic measurements have been performed in the temperature range of 40--120~K. Control experiments are performed to narrow down the possible interpretations of the results. Our findings are summarized as follows: \begin{enumerate} \item The interaction between gas-phase CO and vacuum-UV irradiated water ice produces CO$_2$ up to 120~K. Solid-state CO$_2$ is observed in the temperature range 40--60 K. At 70~K or above, the formed CO$_2$ is released into the gas phase. Additionally, above 90 K the formation of O$_2$ is observed at the cost of CO$_2$ production. \item The residence time of CO on water ice is significant, even though it is above the canonical sublimation temperature of CO. In this short, but nonzero, residence time, a CO molecule is able to diffuse up to 7.2 $\times$ 10$^{10}$ different binding sites before desorption occurs. This significant diffusion allows CO to find an OH radical, created by VUV dissociation of H$_2$O, and form CO$_2$ in a Langmuir-Hinshelwood type reaction. \item Given that gas-phase CO can only interact with the surface of the water ice, this includes pores exposed to the vacuum, we derived a conversion efficiency of 7--27\% per absorbed photon in the reactive surface (i.e., top three monolayers). The limiting factor in this conversion rate is the production of OH radicals. \item The VUV dissociation efficiency of solid-state H$_2$O is likely the limiting factor in the above conversion efficiency from gas-phase CO into CO$_2$. \item Understanding this process is important for astrophysical regions, such as planet-forming disks and molecular clouds. In clouds, this process can explain the presence of solid-state CO$_2$ embedded in water-rich ices. In disks, it has been invoked to explain the lack of gas-phase CO. Our results suggest that the process might be more complex then those incorporated in the physical-chemical models. Further theoretical investigation is required to investigate the conversion of gas-phase CO into CO$_2$ to its full extent. \end{enumerate} This work demonstrates the wide temperature efficacy of this gas-grain interaction process. Future work, should focus on further experimental and theoretical exploration of the molecular dynamics which include the effects of high fluxes and neighbouring OH radicals on the reforming of H$_2$O after photodissociation. Additionally, these results should be included quantitatively in models of planet-forming disks and molecular clouds. With this work we show that gas and grain chemistry cannot be considered as fully separate, but that, under the right conditions, interaction of the gas with the icy surface results in observable effects. It is interesting to realize, that similar processes might also at play for other gas-phase molecules that interact with icy grains. \begin{acknowledgements} The authors thank the referee for the constructive feedback. We also thank E. van Dishoeck, C. Eistrup, and C. Walsh for their participation in insightful discussions. This research was funded through the Dutch Astrochemistry II program of the Netherlands Organization for Scientific Research (648.000.025) and NOVA, the Netherlands Research School for Astronomy. JTvS acknowledges recent financial support through the Virginia Initiative on Cosmic Origins (VICO) postdoctoral fellowship program and NL through an SNSF Ambizione grant (\#193453). \end{acknowledgements} \bibliography{JTvS_CO_OH} \bibliographystyle{aa} \begin{appendix} \section{VUV spectrum} \label{app:uv_spec} The spectral energy distribution of the MDHL used in this study is measured in situ with a VUV spectrometer (McPherson Model 234/302), which is mounted opposite to the MDHL on the other side of the main chamber. In this work a MgF$_2$ window was used that does not transmit Lyman-$\alpha$ photons, but does transmit the molecular hydrogen emission lines and continuum between 140--170 nm. The VUV absorption cross section as a function of wavelength is taken from literature \citep{2014_Cruz-Diaz_VUV_cross_A&A...562A.119C}. It is the summation of three Gaussians with the parameters taken from their Table~2. Both the VUV spectrum with which the ASW is irradiated and the wavelength dependent H$_2$O VUV absorption cross section are shown in Fig. \ref{fig:uv_spec}. \section{Additional experiments} \label{app:add_exp} The main experiments at 50, 60, 80, 90, 100, and 120 K and control experiments with $^{13}$CO instead of $^{13}$C$^{18}$O, no VUV irradiation, and no gas-phase CO in the chamber during VUV irradiation are presented in this appendix. In the majority of the experiments there is an artifact in the QMS data during TPD, as can be seen in for example Fig. \ref{fig:exp_50K}d. This is the result of a nonlinear temperature increase in the sample, which is likely caused by a change in thermal conductivity or contact between the sample and heating strip. The LakeShore PID controller compensates for this, but results in a brief period of undershoot followed by overshoot of the sample temperature. This occurs approximately in the temperature range 120--150 K. In the caption of each figure the exact temperature range is given in which the QMS signals are not accurately representing what the TPD should look like if the temperature increase were to be fully linear. \end{appendix}

Title: Reference-star differential imaging on SPHERE/IRDIS

Abstract: Reference-star differential imaging (RDI) is a promising technique in high-contrast imaging that is thought to be more sensitive to exoplanets and disks than angular differential imaging (ADI) at short angular separations (i.e., <0.3"). However, it is unknown whether the performance of RDI on ground-based instruments can be improved by using all the archival data to optimize the subtraction of stellar contributions. We characterize the performance of RDI on SPHERE/IRDIS data in direct imaging of exoplanets and disks. We made use of all the archival data in H23 obtained by SPHERE/IRDIS in the past five years to build a master reference library and perform RDI. In the point-source detection, RDI can outperform ADI at small angular separations (<0.4") if the observing conditions are around the median conditions of our master reference library. On average, RDI has a gain of ~0.8 mag over ADI at 0.15" separation for observations under median conditions. We demonstrate that including more reference targets in the master reference library can indeed help to improve the performance of RDI. In disk imaging, RDI can reveal more disk features and provide a more robust recovery of the disk morphology. We resolve 33 disks in total intensity (19 planet-forming disks and 14 debris disks), and 4 of them can only be detected with RDI. Two disks are resolved in scattered light for the first time. Three disks are detected in total intensity for the first time. The master reference library we built in this work can be easily implemented into legacy or future SPHERE surveys to perform RDI, achieving better performance than that of ADI. To obtain optimal RDI gains over ADI, we recommend future observations be carried out under seeing conditions of 0.6"-0.8".

https://export.arxiv.org/pdf/2208.07915

\title{Reference-star differential imaging on SPHERE/IRDIS} \author{ Chen Xie\inst{\ref{lam}} \and Elodie Choquet\inst{\ref{lam}} \and Arthur Vigan\inst{\ref{lam}} \and Faustine Cantalloube\inst{\ref{lam}} \and Myriam Benisty\inst{\ref{IPAG},\ref{Lagrange}} \and Anthony Boccaletti\inst{\ref{LESIA}} \and Mickael Bonnefoy\inst{\ref{IPAG}} \and Celia Desgrange\inst{\ref{IPAG},\ref{Heidelberg}} \and Antonio Garufi\inst{\ref{INAF}} \and Julien Girard\inst{\ref{STScI}} \and Janis Hagelberg\inst{\ref{Genève}} \and Markus Janson\inst{\ref{Stockholm}} \and Matthew Kenworthy\inst{\ref{leiden}} \and Anne-Marie Lagrange\inst{\ref{LESIA},\ref{IPAG}} \and Maud Langlois\inst{\ref{lyon}} \and François Menard\inst{\ref{IPAG}} \and Alice Zurlo\inst{\ref{Santiago1},\ref{Santiago2},\ref{lam}} } \institute{ Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France \label{lam} \\ \email{\href{mailto:chen.xie@lam.fr}{chen.xie@lam.fr}} \and Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France \label{IPAG} \and Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France \label{Lagrange} \and LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon, France \label{LESIA} \and Max Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany \label{Heidelberg} \and INAF, Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125, Firenze, Italy \label{INAF} \and Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA \label{STScI} \and Départment d’astronomie de l’Université de Genève,Chemin Pegasi 51, 1290 Versoix, Switzerland \label{Genève} \and Department of Astronomy, Stockholm University, 10691, Stockholm, Sweden \label{Stockholm} \and Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands \label{leiden} \and CRAL, UMR 5574, CNRS, Université de Lyon, École Normale Supérieure de Lyon, 46 Allée d’Italie, F-69364 Lyon Cedex 07, France \label{lyon} \and N\'ucleo de Astronom\'ia, Facultad de Ingenier\'ia y Ciencias, Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile \label{Santiago1} \and Escuela de Ingenier\'ia Industrial, Facultad de Ingenier\'ia y Ciencias, Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile \label{Santiago2} } \date{Received --; accepted --} \abstract {Reference-star differential imaging (RDI) is a promising technique in high-contrast imaging that is thought to be more sensitive to exoplanets and disks than angular differential imaging (ADI) at short angular separations (i.e., \textless0.3\arcsec). However, it is unknown whether the performance of RDI on ground-based instruments can be improved by using all the archival data to optimize the subtraction of stellar contributions. } {We characterize the performance of RDI on SPHERE/IRDIS data in direct imaging of exoplanets and disks.} {We made use of all the archival data in $H23$ obtained by SPHERE/IRDIS in the past five years to build a master reference library and perform RDI. To avoid biases caused by limited test targets under specific conditions, 32 targets were selected to obtain the average performances of RDI under different conditions, and we compared the performances with those of ADI.} {In the point-source detection, RDI can outperform ADI at small angular separations (\textless0.4\arcsec) if the observing conditions are around the median conditions of our master reference library. On average, RDI has a gain of $\sim$0.8 mag over ADI at 0.15\arcsec\,separation for observations under median conditions. We demonstrate that including more reference targets in the master reference library can indeed help to improve the performance of RDI. In disk imaging, RDI can reveal more disk features and provide a more robust recovery of the disk morphology. We resolve 33 disks in total intensity (19 planet-forming disks and 14 debris disks), and 4 of them can only be detected with RDI. Two disks are resolved in scattered light for the first time. Three disks are detected in total intensity for the first time. } {RDI is a promising imaging technique for ground-based instruments such as SPHERE. The master reference library we built in this work can be easily implemented into legacy or future SPHERE surveys to perform RDI, achieving better performance than that of ADI. To obtain optimal RDI gains over ADI, we recommend future observations be carried out under seeing conditions of 0.6\arcsec-0.8\arcsec. } \keywords{techniques: high angular resolution - techniques: image processing - planets and satellites: detection - protoplanetary disks } \titlerunning{RDI performance on SPHERE/IRDIS} \section{Introduction} High-contrast imaging is a crucial method for detecting and characterizing wide (\textgreater10~au) giant exoplanets and circumstellar disks around nearby stars. Direct imaging of exoplanets can obtain spectral, orbital, and statistical information of planets to constrain their formation history \citep{Macintosh2015Sci, Nielsen2019, Bowler2020, Vigan2021, Zhang2021Natur}. Direct imaging of disks recovers the disk morphology and surface brightness, which can help to understand the dust grain properties \citep{Milli2017, Chen2020} and potential disk-planet interactions \citep{Ren2020_MWC758}. Surveys of exoplanets found that giant exoplanets 1--10 $M_{\rm Jup}$ are rare beyond 30 au, with an occurrence rate of a few percent \citep{Nielsen2019, Vigan2021}. The two largest imaging surveys are the SpHere INfrared Exoplanets (SHINE) project \citep{Desidera2021_SHINE_I}, conducted with Spectro-Polarimetric High-contrast Exoplanet REsearch \citep[SPHERE;][]{Beuzit2019} and the Gemini Planet Imager \citep[GPI;][]{GPI_Macintosh2014PNAS} Exoplanet Survey \citep[GPIES;][]{Macintosh2018}. Each survey targets 500-600 stars. While these surveys have a completeness above 50\% for companions \textgreater10 $M_{\rm Jup}$ at \textgreater10~au, the detection probability drops to 25\% within 5~au. However, radial velocity (RV) studies found a large number of Neptunes and Jupiters at closer orbits (\textless10~au) and a potential peak in the occurrence rate of giant planets at $\sim$3 au \citep{Fernandes2019}, where the water snow line is located. This encourages the development of instruments and techniques to reach deeper detection limits at closer orbits to detect more planets. The main challenge in the direct imaging of exoplanets and circumstellar disks are the high flux ratios between faint circumstellar objects and bright host stars (i.e., $\sim$$10^{-4}$ arcsec$^{-2}$ for the surface brightness of debris disks and 10$^{-5}$-10$^{-7}$ for giant exoplanets). Dedicated high-contrast imagers can currently suppress the stellar light and reach contrasts of 10$^{-3}$-$10^{-4}$. Then dedicated post-processing methods remove the residual stellar halo and quasi-static features \citep[speckles;][]{Hinkley2007} to detect circumstellar objects. The majority of the post-processing methods are based on reconstructing proper stellar point spread function (PSF) models from a library of references to subtract stellar contributions. Different techniques have been developed depending on how the references were assembled, either from additional reference stars or from the science data themselves. The reference-star differential imaging (RDI) technique collects stellar references by observing reference or calibration stars with no known circumstellar object \citep{Smith1984Sci, Lafreniere2009}. Assembling stellar references from the science images themselves uses dedicated observing strategies that add diversity to the data to disctinguish the astrophysical signals (planets or disks) from stellar components. % With respect to the stellar components, the angular differential imaging \citep[ADI;][]{Marois2006} technique uses the azimuthal motion of the astrophysical signal in the temporal direction. Similarly, the spectral differential imaging \citep[SDI;][]{Racine1999} technique uses the radial motion of the stellar components with respect to the astrophysical signal in the spectral direction. In addition, the polarimetric differential imaging \citep[PDI;][]{Kuhn2001} technique uses the fact that scattered light from the dusty disk is polarized, but not that of the starlight, which makes PDI efficient in disk imaging. Because the astrophysical signal can be distinguished from the stellar component, it is possible to model and subtract stellar contributions while retaining the astrophysical signal. Polarimetric differential imaging is the typical imaging technique for acquiring good contrast for disk detection at small angular separations \citep{Hashimoto2011, Quanz2011, Muto2012}. However, a drawback of PDI is that it can only detect polarized light because any unpolarized flux becomes self-subtracted in differential imaging. Hence, disks with a low polarization fraction can be difficult to detect with PDI. Directly imaging such disks in total intensity is a natural solution. Furthermore, we can derive the polarization fraction from the obtained total and polarized intensities, which is an important quantity for understanding the dust properties in the disk. % Angular differential imaging is the primary imaging technique for the SHINE and GPIES surveys in searching for exoplanets and imaging disks in total intensity. However, ADI has limitations at small angular separations (e.g., \textless0.3\arcsec~on 8m class telescopes) mainly due to the self-subtraction effect \citep{Marois2006}. The astrophysical signal (i.e., planets) needs to move a certain arc-length to be distinguishable from stellar components to avoid subtracting itself, which is the so-called self-subtraction effect. For details of ADI limitations (e.g., timing constraints, limited sky coverage, limited sensitivity around the inner working angle (IWA), or sharping azimuthal features of extended sources), we refer to the introduction of \cite{Ruane2019}. As a result, the self-subtraction effect degrades the sensitivity of ADI in detecting exoplanets at small angular separations and limits the accurate recovery of disk morphology \citep{Milli2012}. Therefore, the sensitivity of current instruments using ADI at \textless0.3\arcsec is limited to massive objects and does not include the bulk of lower-mass planets expected from models or RV studies. A similar effect also limits the performance of SDI on current instruments because it also uses the science data themselves to build the stellar model. To avoid the self-subtraction effect and improve the sensitivity at short separations (\textless0.3\arcsec), we can use RDI as an alternative technique for the direct imaging of exoplanets and disks in total intensity. The key aspect of RDI is to assemble proper stellar references. There are different strategies to assemble references, which can be divided into two types. The first strategy type is to collect stellar references similar to science images, which can be achieved by 1) observing binaries \citep{Kasper2007, BDI_Rodigas2015}, 2) using the stars observed under similar conditions \citep{YSES_Bohn2020MNRAS} or at the same night \citep{Weinberger1999, Xuan2018}, and 3) observing science and reference stars nearly simultaneously \citep{Wahhaj2021}. The second strategy type is to use all the archival data obtained with a stable instrument and then down-select those that are most similar to the science data \citep{Lafreniere2009, Soummer2011, ALICE_Choquet2014SPIE}. Space-based instruments usually have more stable PSFs than ground-based instruments. Hence, RDI was successfully applied to space-based instruments \citep[e.g.,][]{Schneider2014, Schneider2016, Schneider2018} and became their primary imaging technique, including \textsl{Hubble Space Telescope} (\textsl{HST})/NICMOS and \textsl{HST}/STIS \citep{ALICE_Choquet2014SPIE, Hagan2018, Ren2021}. Recently, \cite{Sanghi2022} compared the performances of RDI and ADI based on the \textsl{HST}/WFC3 observation of \object{PDS\,70} \citep{Zhou2021}. They found that the performance of RDI is worse than that of ADI, which might be attributed to the lack of a high-quality reference library. In ground-based instruments, RDI is not commonly used because of the rapid atmospheric turbulence. The recent developments of RDI have achieved some gains at short angular separations with ground-based instruments equipped with adaptive optics (AO) system, such as Keck/NIRC2 \citep{Xuan2018, Ruane2019} and the star-hopping mode on SPHERE \citep{Wahhaj2021}. Both studies adopted the first type of strategy that collects stellar references similar to science images. It remains unknown whether the performance of RDI can be improved by using all the archival data to optimize the subtraction of stellar contributions. \cite{Gerard2016} assembled a reference library of 207 reference targets from GPIES and applied RDI on GPI data of \object{51\,Eri}. However, no significant improvement of RDI over ADI was found for \object{51\,Eri} data. Nevertheless, it is worth exploring whether the RDI performance improves with the number of reference targets. More importantly, the study was established on a single test case (51~Eri), which was observed under specific observing conditions with specific observation settings. It is possible that RDI may achieve different performances for observations obtained in different observing and instrument conditions. More studies are needed to avoid potential bias caused by limited test cases to obtain a general performance of RDI on ground-based instruments. % In this paper, we present a thorough analysis of the RDI performance on SPHERE using all the archival $H23$ data observed in the past five years, including $\sim$$2.6\times10^{5}$ images from more than 1000 observations. We explore the RDI performance in the detection of point sources and disk imaging. In Sect.~\ref{sec:method} we describe the data used in this work and our RDI and ADI approaches. In Sect.~\ref{sec:point_srouce_detection} we present the RDI performances in the detection of point sources under different conditions and compare them with that of ADI. In Sect.~\ref{sect:disk_imaging} we present the advantages of RDI in disk imaging and show the disks detected by SPHERE in total intensity. Finally, we give the conclusions and discuss our results in Sect.~\ref{sec:conclusions}. \section{Methods} \label{sec:method} \subsection{Data} The SPHERE instrument \citep{Beuzit2019} on the Very Large Telescope (VLT) has delivered high-quality images since first light in May 2014. SPHERE has three science instruments: the infrared dual-band imager and spectrograph \citep[IRDIS;][]{IRDIS_Dohlen2008SPIE}, the integral field spectrograph \citep[IFS;][]{IFS_Claudi2008SPIE}, and the Zurich imaging polarimeter \citep[ZIMPOL;][]{ZIMPOL_Schmid2018A&A}. Both IFS and IRDIS work in the near-infrared and can provide high Strehl ratios (e.g., \textgreater75\% in $H$), while ZIMPOL works in the visible and provides lower Strehl ratios (e.g., \textless50\% in $R$). IRDIS is a dual-band imager \citep[DBI;][]{Vigan2010} that produces simultaneous images at two nearby wavelengths that were selected around expected features in the spectrum of young giant exoplanets. Both IFS ($Y$-$J$) and IRDIS ($H$) have a similar spectral resolution of $\sim$30, but IFS provides 39 spectral channels. To quantify the performance of RDI on ground-based high-contrast imagers such as SPHERE, we used IRDIS due to its high Strehl ratio in the infrared and similar spectral resolution as IFS, but is less complex than IFS data. The diversity of the reference images can be used to build the proper stellar model only with a large number of reference images. The most commonly used coronagraph is the apodized pupil Lyot coronagraph in its $\texttt{N\_ALC\_YJH\_S}$ configuration \citep{Carbillet2011, Guerri2011}, with a mask diameter of 185~mas. The three most commonly used filter pairs are $\texttt{DB\_H23}$, $\texttt{DB\_K12}$, and $\texttt{BB\_H}$, with $\sim$$1.3 \times 10^{5}$, $\sim$$8.7 \times 10^{4}$, and $\sim$$5.8 \times 10^{4}$ exposures in the public archive as of 2021 January 1, respectively. % To achieve optimal performance, we adopted $\texttt{DB\_H23}$ in this work because this filter pair has the highest number of exposures. The total number of coronagraphic images from two bands (dual-band mode; $H2$:1.593~$\mu$m and $H3$:1.667~$\mu$m) is about $2.6 \times 10^{5}$. The raw data were processed using the $\texttt{vlt-sphere}$\footnote{\url{https://github.com/avigan/SPHERE}, version 1.4.2} pipeline \citep{SPHERE_pipeline_Vigan2020} to produce calibrated data cubes and stellar PSF images without the coronagraph mask. SPHERE real-time computer data (also called SPARTA files) were also processed by the $\texttt{vlt-sphere}$ pipeline to provide the observing conditions used in the performance analyses. Because real-time computer data and science exposures were not synchronous, we associated each science image with the observing condition that was closest in time. % \subsection{Reference-star differential imaging} \label{subsec:RDI} \subsubsection{Construction of the master reference library} \label{subsec:building_ref_lib} Building the master reference library is the key element in RDI. This was done in two steps, the image alignment and the identification of poor reference stars. Throughout the paper, the master reference library refers to all the available reference images, and the reference library refers to a selected subset of the master reference library for the subtraction of stellar contributions in the individual science image. The selection of a subset of the master reference library is described in Sect.~\ref{subsection:image_select_psf_sub}. % The image alignment is important because the diffraction features and speckles must be aligned in all the images and to optimize the subtraction of stellar contributions. Furthermore, it can maximize the signal of any astrophysical object in the combined image. SPHERE typically uses satellite spots in the first and last images of an observation to locate the star center behind the coronagraph. During the entire science observation, SPHERE relies on the differential tip-tilt sensor control to maintain the star at the same position behind the coronagraphic mask. However, the differential tip-tilt sensor loop runs at 1 Hz, so that some residual jitter of the images can occur at a faster rate. This may therefore induce some small shift. We discuss the pointing stability of SPHERE/IRDIS in Appendix~\ref{subsec:pointing_stability}. The first step of the image alignment is to choose a reference template image to which all the images will be aligned. A coronagraphic image with a high signal-to-noise ratio (S/N) can be a good reference template to improve the accuracy of the image alignment. We arbitrarily adopted the bright star (\citealp[$H$$\sim$3.7 mag;][]{2MASS_Skrutskie2006}) \object{HD\,121156} (program 098.C-0583, PI: Pantoja), which was observed on 2017 Febuary 5 under excellent conditions (seeing: 0.33\arcsec, $\tau_{0}$:16.89 ms, and Strehl:93\%) with a clear correction ring (see, Fig.~\ref{Fig:frame_registration}). The correction ring marks the boundary of the AO-corrected area (inside) and the seeing-limited area (outside). For the \texttt{DB\_H23} filter pair, the control radius (inner edge of the correction ring) is about 0.7\arcsec-0.8\arcsec. An image mask (see the magenta circles in Fig.~\ref{Fig:frame_registration}) was used to only focus on the correction ring in the image alignment. Before performing the image alignment, we removed the low spatial frequencies of the stellar halo \citep[e.g., wind-driven halo; ][]{Cantalloube2020} using a median filter with a size of 15 $\times$ 15 pixels. The stellar halo was only removed with the median filter in the image alignment to obtain the offsets, not to perform the subtraction of stellar contributions. Each image was only aligned to the selected reference template by minimizing the loss function $L$ as \begin{equation} \label{equ:loss_function} \arg \min_{a,\ x,\ y} L \ =\ \log \left( \sum^{N_{\text{pix} }}_{i=1} \left( \left( a\ I_{i}\left( x,y\right) \ -\ T_{i}\right) M_{i}\right)^{2} \right), \end{equation} where $a$, $x$, $y$, $i$, and $N_{\rm pix}$ are the intensity scaling factor, the offset on the x-axis, the offset on the y-axis, the image pixel index, and the total number of pixels, respectively. $T$, $M$, and $I(x,y)$ are the reference template, the image mask, and a given image after being shifted by given offsets on the x- and the y-axis, respectively. % All the images in our database were aligned to the same reference template of the same band after being shifted by derived offsets. $H2$ and $H3$ images were processed separately. % To ensure that the images in different bands were also aligned, we aligned the reference templates in $H2$ and $H3$ beforehand. This was done by scaling the reference template in $H3$ according to its wavelength and aligning it with the template in $H2$. % Throughout the paper, we excluded the images that failed in the image alignment, which could be caused by bright sources around the correction ring, failed coronagraphic images (i.e., star outside the coronagraph mask or fail AO corrections), short exposures (shorter than a few seconds), and poor observing conditions. About 12\% of the images failed in the image alignment (see Appendix~\ref{appendix:frame_registration} for details). Then, as a second step, we analyzed each target in order to identify poor reference stars. Any reference image that contained point sources and/or extended sources was considered to be a poor reference in RDI, which results in an incorrect starlight subtraction \citep{ALICE_Choquet2014SPIE}. After the image alignment, we performed the subtraction of stellar contributions using ADI and then RDI (see, Sects.~\ref{subsec:ADI} and~\ref{subsec:RDI} for the details) to exclude poor reference stars by visual inspections. Most of the points and extended sources can be identified after ADI. Then we performed our RDI using the master reference library obtained after ADI to further remove poor references. After the image alignment and the identification of bad reference stars, we assembled the master reference library, which contains about $7\times10^{4}$ images per band from 725 observations. Throughout the paper, we define an observation as a complete observing sequence of a target that consisted of successive frames observed in the pupil-stabilized mode. A given target may contain multiple observations obtained at different nights under different observing conditions. \subsubsection{Image selection and subtraction of stellar contributions} \label{subsection:image_select_psf_sub} Some level of image selection is necessary to find the optimal reference images that can effectively improve the subtraction performance of stellar contributions \citep{Ruane2019}. It is not necessary to use the whole master library because it contains reference images that were obtained in very different observing conditions that are significantly different from a given science image. The best two methods tested in \cite{Ruane2019} are the mean square error (MSE) and structural similarity index metric (SSIM). MSE and SSIM provided similar correlations and performances. However, MSE required fewer computation resources, which is suitable for applying it to a very large number of images. In the image selection, we first normalized both science and reference images with a robust scaler\footnote{The robust scaler was defined as $I_{\rm scaled} = (I_i - {\rm med}(I))/(Q_3(I) - Q_1(I))$, where $I$, $i$, ${\rm med}(I)$, $Q_3$, and $Q_1$ are the image, the pixel index, the sample median, the third quartile, and the first quartile, respectively.} that removed the median and scaled the data according to the interquartile range. Then we computed the MSE values between the science image and each of the frame in the master reference library, which was given by \begin{equation} \label{equ:MSE} \text{MSE}^{\left( k\right)} \ =\ \frac{1}{N_{\text{pix} }} \sum^{N_{\text{pix} }}_{i=1} \left( R^{\left( k\right)}_{i}-S_{i}\right)^{2}, \end{equation} where $k$, $N_{\rm pix}$, $R$, and $S$ are the frame index in the master reference library, the number of pixels, a reference image, and a given science image, respectively. An image mask was used with an inner mask of 8 pixels ($\sim$0.1\arcsec) in radius and an outer mask of 60 pixels (0.735\arcsec) in radius. The size of the outer image mask was adopted to only focus on the AO-corrected area. For each science image, we selected a reference library to reconstruct a stellar coronagraphic image using principal component analysis \citep[PCA;][]{Soummer2012}. Multiple sizes of reference libraries and numbers of subtracted principal components (PCs) were used in the main analyses, such as making contrast curves (see the main text in Sect.~\ref{subsec:contrast_curve}) and disk imaging (see the main text in Sect.~\ref{sec:disk_imaging}). To identify poor reference stars, we used 500 and 2000 images as the sizes of the reference library to perform RDI. The number of PCs to be subtracted are 2\%, 5\%, 10\%, 15\%, 20\%, and 50\% of the reference library. % After the subtraction of stellar contributions, the residual science cubes were derotated and mean combined to form the residual images in $H2$ and $H3$ for each science target. We also created a final residual image by combining the two residual images in $H2$ and $H3$. \subsection{Angular differential imaging} \label{subsec:ADI} We systematically processed all the targets after the image alignment using ADI. The aims were to exclude the poor reference stars from the master reference library and provide a standard for the comparison with RDI. % After the image alignment (see, Sect.\ref{subsec:building_ref_lib}), a stellar coronagraphic image was built for each science image using PCA. In the identification of poor reference stars, the number of PCs to be subtracted were 2\%, 5\%, 10\%, 15\%, 20\%, and 50\% of the total number of science images. Then, each science image was subtracted by the corresponding stellar image to remove the stellar contributions. $H2$ and $H3$ bands were processed separately. Similar to RDI, residual cubes were derotated and mean combined to form the residual images in $H2$ and $H3$ bands. A final residual image was also obtained by combining two residual images in $H2$ and $H3$. \subsection{Contrast curves} \label{subsec:contrast_curve} In the detection of point sources, the performances of RDI and ADI can be quantified and compared via their contrast curves. To calculate the 5$\sigma$ contrast curve $C$, we followed \begin{equation} \label{equ:contrast} C=\left( 5t_{\rm{s} tu}\sigma +f_{\rm{r} es}\right) T^{-1}_{\rm{i} ns}f^{-1}_{\rm{s} tar}, \end{equation} where $t_{\rm stu}$, $\sigma$, $f_{\rm res}$, $T_{\rm ins}$, and $f_{\rm star}$ are the correction factor for small sample statistics \citep{Mawet2014}, noise, the residual flux after the PCA subtraction, the throughput of the PCA subtraction, and the stellar flux observed without the coronagraph, respectively. All terms are radial dependent from the star center and sampled by increasing the radial separation in steps of one full width at half maximum (FWHM), except for the stellar flux. % The noise and residual flux were estimated in the residual image as fluxes integrated within the resolution elements. To do this, we sampled each annulus at a given radial separation with apertures with a diameter of 1~FWHM. The residual flux and the noise were estimated as the mean and standard deviation of the fluxes integrated within these apertures, respectively. By fitting the stellar PSF observed by SPHERE without the coronagraph mask, we obtained the size of the aperture (1~FWHM) and the stellar flux. The throughput of the PCA subtraction was estimated by injecting a simulated planet. The artificial planet was created based on the stellar PSF with a flux scaled to 20 times the noise. We show in Appendix~\ref{appendix:Validation_throughput_estimation} that the level of injected flux has a negligible impact on the estimation of the throughput. To reduce the computation time, eight artificial planets were injected each time with a radial separation of 2 FWHM and azimuthal separation of 90$^{\circ}$ to avoid biasing the throughput estimation. The throughput at injected locations was estimated via the ratio of recovered and injected flux. We used eight different position angles (each separated by 45$^{\circ}$) to cover the field of view. Finally, we sampled each radial separation with eight spatial locations and obtained an azimuthally averaged $C$. A few parameters affect the performances of RDI and ADI, such as the number of subtracted PCs and the size of the reference library. To explore the parameter space, a grid of parameters was adopted for RDI with sizes of the reference library of 200, 500, 1000, 2000, 3000, 5000, and 10000 images and numbers of PCs of 2\%, 5\%, 10\%, 15\%, 20\%, 50\%, and 70\% of the reference library. We used the same parameters for ADI, except for the size of the reference library, which was the number of science images for each observation. Both RDI and ADI use the same approach to estimate the contrast curve for the detection of point sources. The only difference is the dataset that was used in PCA to calculate PCs. We finally adopted the best contrast value that we achieved with our grid of parameters at each angular separation to form the optimal contrast curve. The concept of optimal contrast was used by \cite{Xuan2018} to describe the best achievable contrast at a given angular separation. Throughout the paper, we use the optimal contrast curve as the contrast curve we achieved for each target. \subsection{Sample selections} \label{subsec:sample_selection} The aim of this paper is to illustrate the general performance of RDI using the diversity of SPHERE/IRDIS data and compare it with that of ADI. The atmospheric conditions have strong impacts on the performance of RDI and ADI. Furthermore, ADI is affected by the self-subtraction effect due to the limited parallactic angle (PA) rotations of the science data. Therefore, we specifically study these two effects independently in Sects.\ref{subsec:RDI_vs_seeing} and \ref{subsec:RDI_vs_PA}. To obtain general RDI performances for different observing and instrument conditions, we selected two groups of targets from the master reference library. The selected targets are free from the contamination of celestial sources, which may otherwise bias the performance analysis. Ideally, the same observing conditions should result in the same stellar halo and speckle patterns if the instrument were perfectly stable. Therefore, RDI observations under similar observing conditions may lead to a larger number of well-matching reference images, which was the idea for multiple surveys designed to use RDI \citep[e.g.,][]{YSES_Bohn2020MNRAS}. To explore the impact of the observing conditions, we adopted seeing condition bins of 0.2\arcsec, 0.4\arcsec, 0.6\arcsec, 0.8\arcsec, and 1.0\arcsec. Only fewer than 5\% of observations in our master reference library have a seeing higher than 1.0\arcsec. In addition to the seeing condition, we adopted the field rotation, wind speed, Strehl ratio, $H$ band magnitude, and AO loop frequency as control variables. We selected targets with values within $m \pm 0.2m$, where $m$ is the median value of the control variable. Here we adopted an arbitrary fraction of 20\%, which is a trade-off between having similar conditions and obtaining enough targets in each bin to perform a minimalist statistical analysis. The distributions of observing conditions in our master reference library and the corresponding median values are shown in Fig.~\ref{Fig:distribution_ob_condition}. The selected sample of targets with different seeing conditions is listed in Table~\ref{table:sample_delta_Seeing}. The main limitation of ADI is the self-subtraction effect that degrades the sensitivity of point-source detection at small angular separations (i.e., \textless0.3\arcsec) and modifies any extended structure \citep{Marois2006, Milli2012}. For example, a required PA rotation is $\sim$28$^{\circ}$ at a separation of 0.1\arcsec~in order to move the PSF of a point source by an arc-length of 1~FWHM (50~mas with SPHERE). However, our master reference library of over 700 observations has a median PA rotation of $\sim$29$^{\circ}$. A significant fraction of SPHERE/IRDIS data suffers the self-subtraction effect at small angular separations. In contrast, RDI does not have the self-subtraction effect because it builds stellar models from references that have been vetted to be free from companions. We adopted PA rotation bins of 5$^{\circ}$, 20$^{\circ}$, 30$^{\circ}$, 40$^{\circ}$, 60$^{\circ}$, and 80$^{\circ}$ to study its impact on the performance. Only fewer than 8\% of observations in our master reference library have PA rotations higher than 80$^{\circ}$. The rest of the conditions (seeing, wind speed, Strehl ratio, H band magnitude, and AO loop frequency) are about the median values within 20\% of their median values. The selected sample of targets with different PA rotations is listed in Table~\ref{table:sample_delta_PA}. Each PA bin contains four to five targets, except for the last bin of 60$^{\circ}$ - 80$^{\circ}$ , which has only two targets. \section{RDI performance for point-source detection} \label{sec:point_srouce_detection} \subsection{RDI performance as a function of seeing conditions} \label{subsec:RDI_vs_seeing} We selected 19 targets out of 725 observations in our master reference library based on the criteria described in Sect. 2.5, listed in Table~\ref{table:sample_delta_Seeing}. Each observation has similar observing conditions (i.e., PA rotation, wind speed, Strehl ratio, $H$-band magnitude, and AO loop frequency) that are about the median values shown in Fig.~\ref{Fig:distribution_ob_condition}. Only the seeing conditions are significantly different and in a range of 0.2\arcsec-1.0\arcsec. % We estimated the contrast curves of RDI and ADI on selected targets (see Sect.~\ref{subsec:contrast_curve} for the details). In each bin of the seeing conditions, we took the mean of the contrast curves of the test targets. We estimated the uncertainty by taking the standard deviation of the results for the targets in each bin, representing the scatter of RDI gains over ADI. Fig.~\ref{Fig:RDI_vs_ADI_delta_Seeing} shows the contrast curves of RDI and ADI as a function of angular separations in each seeing bin. The RDI and ADI contrasts improve with better seeing conditions, which is expected because a better AO performance can be achieved with better seeing. The improvement of RDI over ADI is also shown in Fig.~\ref{Fig:RDI_vs_ADI_delta_Seeing}. RDI outperforms ADI at small separations (\textless 0.4\arcsec) for observations under seeing conditions of about 0.4\arcsec-0.6\arcsec. These seeing conditions are about the median value of our master reference library, which is 0.53\arcsec\,(see, Fig.~\ref{Fig:distribution_ob_condition}). At a separation of 0.15\arcsec, RDI reaches its peak efficiency with an average gain of $0.8 \pm 0.3$ mag over ADI for observations with seeing conditions ranging from 0.4\arcsec-0.6\arcsec. In slightly worse seeing conditions of 0.6\arcsec-0.8\arcsec, RDI and ADI have a similar performance. In excellent seeing conditions of \textless0.4\arcsec, ADI is very efficient at large separations, but RDI still outperforms ADI at 0.15\arcsec\,separation. However, ADI outperforms RDI for observations under extreme conditions (i.e., 0.2\arcsec-0.4\arcsec~or 0.8\arcsec-1.0\arcsec) for most separations. The degradation of RDI performance in extreme seeing conditions can be explained by the fact that there are fewer well-matching references for observations under extreme conditions. A larger number of well-matching reference images can only be found when the observing conditions are similar (i.e., about median values in Fig.~\ref{Fig:distribution_ob_condition}), which leads to better PCA subtractions and deeper contrasts. Consequently, increasing the size of the master library reference over time will statistically increase the number of matching reference images for all conditions and improve the overall performance of RDI. We further explore the impact of the size of the master reference library in Sect.\ref{subsect:impact_of_m_size}. The uncertainty shown in Fig.~\ref{Fig:RDI_vs_ADI_delta_Seeing} is affected by multiple factors. The small sample size directly affects the statistic. The scatter of observing conditions among the targets may also lead to scatter in RDI gains over ADI. % Therefore, we selected targets with similar observing conditions. However, we need to balance being selective to have targets under similar conditions and having enough targets in each bin. % Overall, the scatter in RDI gains over ADI justifies the necessity of selecting a sample to generate an average performance for certain observing conditions. Evaluating the RDI performance based on only a few targets under different conditions will lead to a biased result. Fig.~\ref{Fig:RDI_vs_ADI_delta_Seeing_throughput_noise} shows the throughput of the PCA subtraction and noise in the residual image after the processes of RDI and ADI (see Sect.~\ref{subsec:contrast_curve} for the details), which correspond to the contrasts shown in Fig.~\ref{Fig:RDI_vs_ADI_delta_Seeing}. The differences in throughput and noise after the reductions of RDI and ADI are also shown in Fig.~\ref{Fig:comparison_TT_NN}. Noise in the RDI and ADI images decreases with the increase in angular separations. Higher noise at short separations for both RDI and ADI is expected because both speckle noise and photon noise are stronger at short separations. Moreover, better conditions lead to a better AO performance, resulting in less noise for all separations. Both RDI and ADI throughputs increase from about 5\% to 50\% with angular separations for all bins of seeing conditions. The RDI technique was expected to have higher throughput than the ADI technique. However, completely removing the speckle noise requires RDI to subtract a large number of PCs, which may lead to some levels of oversubtraction and hence lower its throughput. Therefore, the RDI throughput is not higher than that of ADI in Fig.~\ref{Fig:RDI_vs_ADI_delta_Seeing_throughput_noise} (see also Fig.~\ref{Fig:comparison_TT_NN}). As a comparison, the self-subtraction effect in ADI not only lowers the throughput, but also contributes to removing the speckles, thus lowering the noise. Overall, the optimal contrast was the balance between higher throughput and lower noise, as shown in Eq.~\ref{equ:contrast}. \subsection{RDI performance as a function of PA rotations} \label{subsec:RDI_vs_PA} The previous analysis was performed on targets with PA rotations of $28.9^{\circ}~\pm~5.8^{\circ}$. However, the PA rotation has a strong impact on the ADI performance. To illustrate the impacts of PA rotations, we selected 20 targets out of 725 observations in our master reference library based on the criteria described in Sect.~\ref{subsec:sample_selection} and listed in Table~\ref{table:sample_delta_PA}. Each observation has similar observing conditions (i.e., seeing, wind speed, Strehl ratio, $H$-band magnitude, and AO loop frequency) that are about the median conditions shown in Fig.~\ref{Fig:distribution_ob_condition}. Only the PA rotations are significantly different and in a range of 5$^{\circ}$ to 80$^{\circ}$, which is divided into five bins. % As in Sect.~\ref{subsec:RDI_vs_seeing}, we estimated the contrast curves for the selected targets and took the mean of the contrast curves of the test targets in each bin of PA rotation, shown in Fig.~\ref{Fig:RDI_vs_ADI_delta_PA}. The corresponding improvement of RDI over ADI as a function of PA rotations is also shown in Fig.~\ref{Fig:RDI_vs_ADI_delta_PA}. The corresponding uncertainty was estimated by taking the standard deviation of results for targets in each bin at the given separation. % As shown in Fig~\ref{Fig:RDI_vs_ADI_delta_PA}, both RDI and ADI show that deeper contrasts are achieved for observations under large PA rotations. Observations with large PA rotations usually have a relatively long exposure time. Furthermore, the combination of frames with large PA rotations whitens the noise \citep[i.e., removes the correlated component of the speckle and causes the residual noise to become closer to a Gaussian distribution;][]{Marois2008, Mawet2014} and thus improves the contrast. Similar to the results shown in Sect.~\ref{subsec:RDI_vs_seeing}, RDI can outperform ADI at small angular separations (i.e., \textless0.4\arcsec) for observations with PA rotations of 5$^{\circ}$-60$^{\circ}$. \cite{Wahhaj2021} also reported that RDI outperforms ADI at separations \textless0.4\arcsec~based on the SPHERE star-hopping data. At a separation of 0.15\arcsec, RDI reaches a peak gain of $0.8 \pm 0.2$ mag over ADI for PA rotations between 30$^{\circ}$-40$^{\circ}$. % RDI shows no or limited gain over ADI for PA rotations of 60$^{\circ}$-80$^{\circ}$, which is expected as the self-subtraction effect can be mitigated by large PA rotations. However, the largest RDI gain over ADI is achieved for PA rotations of 30$^{\circ}$-40$^{\circ}$, not 5$^{\circ}$-20$^{\circ}$\footnote{The median PA rotation is 14.3$^{\circ}$ for targets in a PA bin of 5$^{\circ}$-20$^{\circ}$.}. Their difference is within the scatter of the sample, however. Fig.~\ref{Fig:RDI_vs_ADI_delta_PA_throughput_noise} shows the throughput of the PCA subtraction and noise in the residual image after the processes of RDI and ADI, which correspond to the contrast curves shown in Fig.~\ref{Fig:RDI_vs_ADI_delta_PA}. Both RDI and ADI throughputs increase from about 5\% to 45\% as increasing angular separations for all bins of PA rotations, which are very similar to observations with different seeing conditions in Sect.~\ref{subsec:RDI_vs_seeing}. For most of the angular separations, the throughput of ADI tends to be higher if an observation has a larger PA rotation, which is a direct demonstration of the self-subtraction effect. Moreover, no significant trend between the RDI throughput and the PA rotation can be found because RDI does not have the self-subtraction effect. The throughput and noise in RDI are coupled, as shown in Eq.~\ref{equ:contrast}, and are only affected by oversubtraction. As expected, noise in the RDI and ADI images decreases with the increase in angular separations because of the properties of speckle noise and photon noise. \subsection{Impact of the size of the master reference library} \label{subsect:impact_of_m_size} To optimize the subtraction of stellar contributions, the master reference library should contain enough coronagraphic images to match given science images obtained under all conditions. Space-based instruments (e.g., \textsl{HST}/STIS or \textsl{HST}/NICMOS) usually have stable PSFs and require only thousands of images in the master reference library \citep{ALICE_Choquet2014SPIE}. Conversely, for ground-based instruments, the master reference library is expected to be much larger than that of space-based instruments. This is because the rapid atmospheric turbulence makes RDI less effective in providing representative stellar coronagraphic images. As demonstrated in Fig.~\ref{Fig:RDI_vs_ADI_delta_Seeing}, RDI can outperform ADI for observations at about median conditions of the master reference library. This indicates that more well-matching reference images can improve the PCA subtraction and lead to a deeper contrast. However, the impact of the size of the master reference library remains unclear. In other words, designers of an RDI observation or survey need to know whether they need more reference images to further improve their detection limit. The large size of our master reference library (725 observations with $\sim$$7\times10^{4}$ frames per band) enables us to investigate the impact of the size of the master reference library by dividing our library into subsets. To estimate the RDI performance with a smaller master reference library, we randomly extracted 1/12, 1/6, 1/3, and 2/3 of observations from our master reference library, yielding four subsets that contained 60, 120, 241, and 483 observations. The corresponding numbers of reference images per band are about $9\times10^{3}$, $1.2\times10^{4}$, $2.6\times10^{4}$, and $4.4\times10^{4}$, respectively. A medium-sized RDI survey may contain about 60 reference targets (e.g., \citealp[SHARDDS;][]{Milli2017_SHARDDS} and \citealp[YSES;][]{YSES_Bohn2020MNRAS}). % We adopted the selected sample from Sect.~\ref{subsec:RDI_vs_seeing}, but only focused on seeing conditions within 0.4\arcsec-0.6\arcsec, yielding a subsample of 8 targets in this test (see also Table~\ref{table:sample_delta_Seeing}). Then we estimated the optimal contrast curves for each target as described in Sect. \ref{subsec:contrast_curve}. The only difference was the size of the master reference library. For the down-selected reference library, we adopted sizes ranging from 200 to 10000 (if possible) images, as described in Sect.~\ref{subsec:contrast_curve}. The final contrast curve of RDI for a given size of a master reference library was formed by averaging over the contrasts of the selected 8 targets. The corresponding uncertainty at each separation was estimated by taking the standard deviation of the contrasts from the selected 8 targets. Fig.~\ref{Fig:RDI_vs_m_size} shows the RDI performance with the different sizes of the master reference library. We achieved a gain of $\sim$1 mag at 0.15\arcsec\,separation and gains of 0.5-0.8 mag between separations of 0.20\arcsec-0.65\arcsec \ when we increased the size of the mater reference library from 1/12 to full size. This indicates that the performance of an RDI survey with 60 reference targets can be further improved by about 1 mag when we include all the archival data. Although the uncertainty is about 0.5 mag, the RDI performance has systematic gains across all separations (0.15\arcsec-0.65\arcsec) with increasing size of our mater reference library from only 1/12 to full size. % A larger master reference library statistically provides more well-matching frames, which in turn improves the accuracy of the PCA subtraction. Despite the fluctuation of gains at larger separations (\textgreater0.35\arcsec), RDI shows a systematic gain at short separations (\textless0.35\arcsec). This indicates that a better RDI performance at short separations can be expected with the accumulation of SPHERE data in the future. Overall, Fig.~\ref{Fig:RDI_vs_m_size} proves the necessity of a large number of references in the archive to obtain good RDI performance. \subsection{RDI performance as a function of the down-selection of the reference library} \label{subsect:RDI_vs_size} The size of the reference library can also affect the RDI performance. In our RDI reduction, we down-selected references from the master reference library to form the reference library. By selecting the most correlated references to each science image, we optimized the PCA subtraction of the stellar contributions. Unlike Sect.~\ref{subsect:impact_of_m_size}, here we used the full size of the master reference library for the down-selection. % Using two target samples from Sects.~\ref{subsec:RDI_vs_seeing} and \ref{subsec:RDI_vs_PA}, we investigated the average performance of RDI for a given size of the reference library. The adopted size of the reference library ranges from very small and selective (200 images, i.e., 0.3\% of all the references) to large and more diverse (10000 images, i.e., 14.5\% of all the references). For each observation at each angular separation, we compared the improvement of RDI contrast with respect to the RDI contrast obtained with a library size of 200 images. Then we averaged over all the targets in each sample to obtain the average RDI gain for a given angular separation. Fig.~\ref{Fig:RDI_average_gain_ref_size} shows the average RDI gain in contrast as a function of reference library size, derived from targets with different seeing conditions and different PA rotations listed in Tables~\ref{table:sample_delta_Seeing} and \ref{table:sample_delta_PA}. The performance of RDI systematically improves with the size of libraries and reaches the plateau of contrast gain where the library size is larger than about 3000-5000 images (4.3\% --7.2\% of all the references). This suggests that including more good references can indeed optimize the PCA subtraction. This is due to the nature of the reconstruction of the stellar model using PCA. An infinite number of references is needed to reconstruct the given science image exactly \citep{Soummer2012}. However, without additional observations, adding more references can only include references that are less correlated with a given science image. Therefore, we reach a plateau of contrast gain after using roughly 5\% of most correlated images from the master reference library. As a comparison, \textsl{HST}/NICMOS achieved its optimal performance with 30\% to 80\% of most correlated images from its master reference library \citep{ALICE_Choquet2014SPIE}. Furthermore, increasing the size of the reference library provides higher gains at large separations (up to 1.5 mag at 0.65\arcsec) than at short separations (about 0.5 mag at 0.15\arcsec), as shown in Fig.~\ref{Fig:RDI_average_gain_ref_size}. Less improvement at short angular separations may be caused by the large variation of speckles, which cannot be completely removed by RDI using archival data without oversubtraction. Observing references and science targets nearly simultaneously can help to remove these speckles and obtain better RDI performance at short angular separations, as demonstrated by the star-hopping mode on SPHERE \citep{Wahhaj2021}. In contrast, the speckles at large separations are relatively stable and hence can be more easily modeled and subtracted when enough good references are available. Interestingly, both samples show similar behaviors. This indicates that the RDI gains as a function of reference library size that we observed do not depend on seeing conditions or PA rotations. \section{Circumstellar disk imaging in total intensity with RDI} \label{sec:disk_imaging} \label{sect:disk_imaging} The RDI technique can not only outperform the ADI technique at short separations for point-source detection but also has advantages for disk imaging. Currently, ADI is adopted as the primary technique for most observations and surveys of disks in total intensity. It is well known, however, that ADI is limited by the self-subtraction effect that may lead to poor recoveries of disk morphology and surface brightness \citep{Milli2012}. The ALICE project \citep{Soummer2014, ALICE_Choquet2014SPIE} demonstrated that applying RDI on space-based instruments can provide a gain in disk imaging. In this section, we demonstrate the capability of SPHERE in the detection of circumstellar disks by using RDI and compare it with that of ADI. \subsection{Robust recovery of disk features with RDI} To illustrate the performance of RDI in disk imaging, we selected targets with known circumstellar disks that were not or poorly detected by ADI, as shown in Fig.~\ref{Fig:disk_ADI_vs_RDI}. % The ADI images of selected six targets\footnote{For more details about the targets, we refer to the following papers: \object{HD\,169142}: \cite{Gratton2019}; \object{UX\,Tau}: \cite{Menard2020}; \object{HD\,97048}: \cite{Ginski2016}; \object{HD\,100546}: \cite{Rameau2017}; \object{V1094\,Sco}: \cite{van_Terwisga2018}; and \object{2MASS\,J16042165-2130284} (hereafter J1604): \cite{Pinilla2018}.} show no or limited disk features due to the self-subtraction effect, while RDI images reveal more features. To avoid aggressive post-processing in ADI, we only used 5\% of the number of temporal frames as the number of subtracted PCs for UX\,Tau\,A and V1094\,Sco. For the rest of the targets in Fig.~\ref{Fig:disk_ADI_vs_RDI}, we adopted a fraction of 2\% for subtracted PCs in ADI reductions. For RDI, we used 20\% of the number of reference images as the number of PCs in RDI for all the targets in Fig.~\ref{Fig:disk_ADI_vs_RDI}. The parameters used in the RDI reduction are listed in Table~\ref{table:disk_list}. The ADI image is sensitive to the number of subtracted PCs due to the self-subtraction effect. Even when only 2\% of the PCs are subtracted, the self-subtraction effect still strongly affects the surface brightness of disks detected by ADI, reducing the disk emission (see HD\,100546 and 2MASS\,J16042165-2130284 in Fig.~\ref{Fig:disk_ADI_vs_RDI}). For example, all the disk signals of 2MASS\,J16042165-2130284 (hereafter J1604) become too faint to be considered as a detection when the subtracted PCs are larger than 10\% using ADI. The disk signal of HD\,100546 also has similar behavior that decreases very fast when the subtracted PCs change from 2\% to 10\%. The remaining images show no significant disk signal when no more than 5\% of the PCs are subtracted, probably due to the self-subtraction effect caused by small PA rotations (\textless25$^{\circ}$). In contrast, the disks in RDI images remain visible with consistent features as the PCs increase from 2\% to 50\%, providing more robust recoveries of disk features. \cite{Ruane2019} analyzed the RDI performance on Keck/NIRC2 data and found consistent results: the ADI image was sensitive to the number of subtracted PCs, while the RDI image was not. The insensitivity to subtracted PCs can help avoid the bias in post-processing that is caused by the fine-tuning of the parameters. This increases the robustness of the RDI detection. The comparisons between ADI and RDI in Fig.~\ref{Fig:disk_ADI_vs_RDI} show the improvement in disk imaging after the removal of the self-subtraction effect. In addition to reducing the surface brightness of disks, the self-subtraction effect can also sharpen the azimuthal features \citep{Milli2012}. For example, the disk signal of J1604 is self-subtracted after ADI, which alters the real disk features such as disk shadowing \citep{Pinilla2018} by introducing artifacts (e.g., more dips). For HD\,100546 in Fig.~\ref{Fig:disk_ADI_vs_RDI}, the disk features revealed in our ADI image are similar to the ADI image in \cite{Garufi2016} and \cite{Sissa2018} that used the same data set. Our RDI image is similar to the GPI $H$ band images that were reduced with RDI by \cite{Rameau2017}. Assessing the confidence of recovered disk features after the PCA subtraction requires disk modeling, which is beyond the scope of this paper. The difference between ADI and RDI images shows one of the main limitations in ADI in precisely recovering geometric parameters and photometry. One of the solutions might be applying disk forward modeling in ADI \citep{Pueyo2016, DiskFM_Mazoyer2020SPIE}. Alternatively, RDI could be a natural solution because it does not have the self-subtraction effect. Although both ADI and RDI suffer oversubtraction caused by the PCA subtraction, most of the disk signal is self-subtracted in ADI with no more than 5\% of PCs. Moreover, oversubtraction is limited in RDI images even at short angular separations, as shown in Fig.~\ref{Fig:disk_ADI_vs_RDI}. It is worth noting that the disk features are clearly visible even when 50\% of PCs are subtracted. This mild oversubtraction effect in PCA can be corrected for by forward modeling \citep{Pueyo2016} to accurately characterize the surface brightness. Providing the throughput estimation is very difficult for disk imaging due to the complex disk morphology and requires detailed disk modelings. No throughput correction was made in the disk images presented in this work. We will cover the disk imaging with detailed disk modeling and analysis in future works. In summary, applying RDI on SPHERE/IRDIS data can provide a more robust recovery of disk features in total intensity than ADI. \subsection{Higher disk detection rate with RDI than ADI} After processing archival SPHERE/IRDIS data in the $H23$ band with ADI and RDI, we detected 33 circumstellar disks in total intensity, including 19 planet-forming disks and 14 debris disks. The RDI detection of the disks in this work is shown in Fig.~\ref{Fig:RDI_disks} and summarized in Table~\ref{table:disk_list}. The size of the reference library and the number of PCs subtracted in the RDI reduction are also listed in Table~\ref{table:disk_list}. The circumstellar disks of \object{DG\,Tau\,A} and HD\,131488 are resolved in scattered light for the first time. The detailed analysis of the disk in HD\,131488 will be presented in Pawellek~et~al.~(in~prep.). To our best knowledge, three disks (\object{V1094\,Sco}, \object{UX\,Tau\,A}, and \object{SZ\,Cha}) are detected in total intensity for the first time. The detailed analysis of the disk SZ\,Cha will be presented in Hagelberg~et~al.~(in~prep.). In addition to new detections, RDI recovers more disk features than previous ADI results. We detect the inner ring~1 at $\sim$0.25\arcsec~around HD\,97048 in total intensity, which was not or only marginally visible in the ADI image \citep{Ginski2016}. In HD\,141569, we detect more disk emission in the west side of the ring R3 than the ADI detection in \cite{Singh2021}. We also detect a large fraction of the full ring including the backside for AK Sco, while the previously published ADI result \citep{Janson2016} was only able to see the forward-scattered side. The detailed analysis of the disks around HD\,110058, HD\,111520, and HD\,120326 will be presented in Stasevic~et~al.~(in~prep.) and Desgrange~et~al.~(in~prep.). In addition to recovering more disk features, RDI also has a higher detection rate of disks than ADI. Of these 33 disks, 4 disks (\object{TW\,Hya}, V1094\,Sco, UX\,Tau\,A, and HD\,169142) are only detected in the RDI reductions. All the disks with a nondetection in ADI have low inclinations (\textless50$^{\circ}$), which may lead to severe self-subtraction. The debris disk survey from GPIES also found that an increasing disk inclination can improve the capability of ADI to detect disks in total intensity \citep{Esposito2020}. Our work shows that RDI potentially has a higher detection rate than ADI in disk imaging, especially for low-inclination disks. We note that our field of view is 0.735\arcsec~in radius. It is possible that disk features can be detected outside our field of view and are not listed in Table~\ref{table:disk_list}. \section{Discussions and conclusions} \label{sec:conclusions} We have presented a thorough analysis of the RDI performance on SPHERE/IRDIS in the direct imaging of exoplanets and circumstellar disks. We made use of all public archival data observed in the past 5 years by IRDIS in the $H23$ band with an unprecedented size of reference images to perform RDI, including about $1.4 \times 10^{5}$ reference images from 725 observations. The results of averaged RDI performance on SPHERE/IRDIS are summarized below. \begin{enumerate} \item Reference-star differential imaging can outperform ADI at small angular separation (\textless0.4\arcsec) if the observing conditions are approximately the median conditions of our master reference library (i.e., seeing in 0.4\arcsec-0.6\arcsec). At a separation of 0.15\arcsec, RDI outperforms ADI by $0.8 \pm 0.3$ mag on average for observations under median conditions. Furthermore, our target sample shows that the RDI performance is better than or equal to that of ADI within separations of 0.4\arcsec~for observations with PA rotations lower than 60$^{\circ}$. \item In the point source detection, we demonstrated that including more observations (i.e., more references) in the master reference library indeed helps to improve the performance of RDI at separations of 0.15\arcsec-0.65\arcsec. An average gain of $\sim$1 mag can be achieved at 0.15\arcsec\,separation by increasing the number of observations in the master reference library from 60 (typical survey size) to 725 (full archive). \item In the point source detection, we find that increasing the number of reference images to reconstruct the stellar image with PCA help to improve the average performance of RDI for all tested angular separations. The average performance of RDI shows no or limited improvement when more than 3000 reference images are used. This suggests that a typical library size for obtaining an optimal RDI reduction is about 3000 - 5000 images for a search for point-like sources. \item In disk imaging, RDI does not have the self-subtraction effect and reveals more features than ADI. The disk features in RDI images are insensitive to subtracted PCs. Hence, RDI provides a more robust recovery of the disk morphology. \item We systematically processed and presented 33 circumstellar disks in total intensity obtained by SPHERE/IRDIS in $H23$. The circumstellar disks of DG\,Tau\,A and HD~131488 are resolved in scattered light for the first time. To our best knowledge, three disks (V1094\,Sco, UX\,Tau\,A, and SZ\,Cha) are detected in total intensity for the first time. \item Reference-star differential imaging has a better capability of detecting disks than ADI, especially for low-inclination (\textless 50$^{\circ}$) disks. Four of the 33 disks detected in this work are only detected in RDI images, not in ADI images. \end{enumerate} Our successful application of RDI on SPHERE data shows that RDI can be applied to ground-based surveys of exoplanets with better performances than ADI at short separations. More importantly, our RDI strategy does not require additional observation of calibration or reference stars. Therefore, it can easily be adopted into legacy or future SPHERE surveys. Furthermore, for a medium-size RDI survey with 60 reference targets, implementing our master reference library can further improve the RDI performance by $\sim$1 mag at 0.15\arcsec\,separation, as demonstrated in Sect.~\ref{subsect:impact_of_m_size}. We showed that the self-subtraction effect on disks that were previously processed with ADI can be overcome in the disk imaging. No additional observations are needed. The self-subtraction effect limits the detection of low-inclination disks in total intensity \citep{Esposito2020} and sharpens azimuthal features. However, RDI does not have such issues, as shown in Fig.\ref{Fig:disk_ADI_vs_RDI}, which is expected. Future observations or surveys can adopt our master reference library and perform RDI in their disk observations to obtain disk images in total intensity. In general, we recommend future observations be carried out under median observing conditions, as shown in Fig.~\ref{Fig:distribution_ob_condition}, to achieve the optimal RDI gain over ADI. However, in practice, observers can only specify the required seeing conditions\footnote{SPHERE defined six turbulence categories for users to choose from, which consist of a few combinations of seeing and coherence time.} to meet their observational goal. % Since the upgrade in April 2016, the Paranal Astronomical Site Monitoring (ASM) has the new MASS-DIMM system \citep{Sarazin1990, Kornilov2007}, which provides the seeing conditions at zenith for SPHERE to decide whether an observation can be executed. In Fig.~\ref{Fig:seeing_vs_seeing} we compare the line-of-sight seeing measured by the SPHERE real-time computer system (SPARTA) and the seeing measured by the MASS-DIMM system. % Although the two systems measured the turbulence at different spatial locations with different ground effects (i.e., with and without a dome), the correlation is still tight, shown in \cite{Milli2017_AO4ELT5}. The median DIMM seeing for our reference library is 0.7\arcsec. Excellence seeing conditions (i.e., DIMM seeing \textless0.6\arcsec) lead to deeper contrasts for both RDI and ADI, but they are also an expensive request. As a technique alternative to ADI, RDI has optimal gains over ADI for DIMM seeing conditions of 0.6\arcsec-0.8\arcsec (see also Fig.\ref{Fig:RDI_vs_ADI_delta_Seeing}). Therefore, we recommend future observations be carried out under seeing conditions of 0.6\arcsec-0.8\arcsec. Such seeing conditions correspond to the turbulence categories of 20\% and 30\%. In addition, using our reference library to perform RDI requires the same coronagraph settings as for the coronagraph in the $\texttt{N\_ALC\_YJH\_S}$ configuration and the $\texttt{DB\_H23}$ filter pair. We will present the RDI library for the $\texttt{DB\_K12}$ filter pair in future works. By adopting RDI in the survey of SPHERE or other ground-based instruments, we can avoid several limitations introduced by ADI. As a result, the design of a survey or observation can be more flexible, covering a larger declination range without the time constraints caused by the self-subtraction effect. % We studied the impact of the reference library based on the images processed with PCA. Other methods are available to build and subtract the stellar contribution, such as non-negative matrix factorization \citep[NMF;][]{NMF_Ren2018}. PCA removes the mean of the image. If a bright astrophysical signal (i.e., a bright disk) is present, PCA creates negative regions (i.e., dark features around the disk), which clearly is an artifact. Although such an artifact will not affect the general comparisons between ADI and RDI in Sect.~\ref{sec:disk_imaging}, it may prevent the further application of RDI in characterizing disks. However, this issue can be largely resolved with NMF. % Furthermore, to avoid overfitting in RDI, \cite{Ren2020_DIsNMF} proposed an improved method, which is data imputation using sequential non-negative matrix factorization (DI-sNMF). In future work, we will use DI-sNMF to demonstrate the performance of our RDI approach in disk imaging (Xie~et~al.~2022,~in~prep.). SPHERE+ is the proposed upgrade of SPHERE. It is equipped with a pyramid infrared wavefront sensor, an optimized coronagraph, and noncommon path aberrations compensation \citep{SPHERE+_Boccaletti2020}. With a faster AO correction ($\sim$3kHz), SPHERE+ is expected to regularly obtain the same image quality that can currently only be obtained with SPHERE in the best 5\% of observing conditions \citep{SPHERE+_Boccaletti2020}. Better AO correction can provide a more stable PSF, which enhances the performance of RDI. The prime aim of SPHERE+ is to image a young Jupiter down to the snow line at $\sim$3 au, bridging the gap with indirect techniques \citep{SPHERE+_Boccaletti2020}. At short angular separations around the IWA of SPHERE+ (\textless100 mas) like this, ADI will be limited by the severe self-subtraction effect. Based on current SPHERE data, we demonstrated that RDI is more sensitive than ADI in searching for planets at short separations (\textless0.4\arcsec). With a more stable PSF and a smaller IWA, it is necessary to consider using RDI as one of the imaging strategies on the future SPHERE+, especially to image the expected peak of gain in the exoplanet population at 3-5~au. \begin{acknowledgements} We thank the anonymous referee for comments that improved the clarity of this work. We thank Dr. David Mary for the beneficial discussion. This research made use of Astropy\footnote{\url{http://www.astropy.org}}, a community-developed core Python package for Astronomy \citep{astropy:2013, astropy:2018}. The performance analyses are based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programmes 095.C-0298, 095.C-0346, 095.C-0549, 095.C-0607, 096.C-0241, 097.C-0060, 097.C-0079, 097.C-0826, 097.C-0864, 097.C-0865, 097.C-0949, 097.C-1019, 097.C-1042, 098.C-0739, 099.C-0693, 0100.C-0543, 0101.C-0753, 0104.C-0183, 1100.C-0481, 198.C-0209, and 295.C-5034. We thank all the principal investigators and their collaborators who prepared and performed the observations with SPHERE. Without their efforts, we would not be able to build the master reference library to enable our RDI technique. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. AV acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No.~757561). \end{acknowledgements} \bibliographystyle{aa} \bibliography{ms} \begin{appendix} \section{Pointing stability of SPHERE/IRDIS} \label{subsec:pointing_stability} Image alignment is one of the key steps in RDI. The ALICE program used the diffraction pattern of the telescope struts to perform the alignment \citep{ALICE_Choquet2014SPIE}. GPI used the positions of satellite spots to align all of the images \citep{Gerard2016}. SPHERE can also generate satellite spots on coronagraph images as GPI by introducing a 2D periodic modulation on the high-order deformable mirror \citep{Beuzit2019}. However, SPHERE usually turns on the satellite spots only at the beginning and/or the end of the science observation to obtain so-called star center images. There is no additional calibration to obtain the star center behind the coronagraphic mask during the science observations and observers usually rely on the pointing stability of SPHERE. % Even if the instrument were very stable for a given observation, we still need to align such a data set with the master reference library. This was done with our image alignment approach, in which we aligned all the data to a common template image. We examined the pointing stability of SPHERE/IRDIS during the science observations based on the position offsets we measured in our image alignment process. In this analysis, we used all the data in the master reference library, but excluded the observations with failed star center images. We therefore only focused on completed observations, which yielded a total number of 654 observations. Each temporal frame in a given observation has offsets in the $x$ and $y$ directions with respect to the template we used to align all the images. We took the standard deviations of the offsets in $x$ and $y$ directions to trace the pointing stability of IRDIS in the given observation. % \FloatBarrier Fig.~\ref{Fig:pointing_stability_IRDIS} shows the pointing stability of IRDIS in $H$2 in 654 science observations. The median values of the standard deviation of pointing offsets in the $x$ and $y$ directions are about 0.06 pixel and 0.07 pixel, respectively. Assuming the offsets of a given observation follow a Gaussian distribution, a full width at tenth maximum (FWTM) of 1 pixel requires the standard deviation to be smaller than 0.233 pixel. As shown in Fig.~\ref{Fig:pointing_stability_IRDIS}, over 94\% of observations have both $\sigma_{x}$ and $\sigma_{y}$ smaller than 0.233 pixel, which demonstrates the stability of IRDIS at the subpixel level. Furthermore, the standard deviation of \textless~0.1 pixel corresponds to an FWTM of \textless~0.43 pixel or \textless~5.3 mas on average, assuming the offsets of a given observation follow a Gaussian distribution. For the $H$3 band, we obtain a similar stability, which is expected because $H$2 and $H$3 images were observed simultaneously. \section{Image alignment} \label{appendix:frame_registration} As mentioned in Sect.~\ref{subsec:building_ref_lib}, we adopted the bright star HD~121156 as our reference template in the image alignment. The median-filtered $H2$ image of HD~121156 is shown in Fig.~\ref{Fig:frame_registration}. The image mask used in the image alignment is indicated by two magenta circles in Fig.~\ref{Fig:frame_registration}. The image mask was linearly scaled to adopt the larger size of the correction ring in the $H3$ band. We evaluated the alignment by using Eq.~\ref{equ:MSE}. We compared the reference template with the aligned images that were shifted by offsets provided by Eq.~\ref{equ:loss_function}. The image mask shown in Fig.~\ref{Fig:frame_registration} was used to only focus on the regions of the correction ring in the reference template $S$ and a given image $R$. % We set a threshold of 90\% of the MSE value of the reference template subtracting an empty image to identify images that failed the image alignment. An empty image indicates an extreme case in which the image was entirely shifted outside the FoV. Such a 90\% threshold was determined by balancing low S/N images and really failed images due to reasons mentioned in Sect.~\ref{subsec:building_ref_lib}. \FloatBarrier \section{Validation of the throughput estimation} \label{appendix:Validation_throughput_estimation} We estimated the throughput of the PCA subtraction using the simulated planet injection. To validate the amount of injected planet flux, we injected the planet flux at 20 or 5 times the noise. We tested this on the same target sample used in Sect.~\ref{subsect:impact_of_m_size}, eight targets in total. We estimated the average throughput and corresponding uncertainty by taking the mean and standard deviation of the results from the selected eight targets. Fig.~\ref{Fig:validation_thoughput_estimation} shows the difference in throughput with different amounts of injected planet flux. Neither RDI nor ADI show a difference in throughput with different injected fluxes at separations from 0.15\arcsec-0.65\arcsec. \FloatBarrier It is expected that different amounts of injected planet flux lead to almost the same throughput because the projection of the planet signal on PCs is a linear calculation \citep{Soummer2012}. Thus the small change in planet flux will not significantly change the throughput of the PCA subtraction. In the first step of PCA, we need to subtract the average values of reference and science images to ensure they have zero mean. However, the injection of simulated planets has a minor impact on the mean value of the image. This is because we only injected planet flux in a small fraction of the image (i.e., \textless1\%) compared to the entire FoV. The injected flux was also low, at the planetary rather than stellar level. \section{Selected samples} Tables~\ref{table:sample_delta_Seeing} and \ref{table:sample_delta_PA} list the observations of targets in our selected samples (see Sect.~\ref{subsec:sample_selection} for details).\nopagebreak[4] \begin{table*} \caption{Targets with different seeing conditions.} % \label{table:sample_delta_Seeing} % \centering % \begin{tabular}{l c c c c c c c c c} % \hline\hline % Name & Program ID & $n_{\rm DIT} \times t_{\rm DIT}^{(a)}$ & $t_{\rm tot}$ & Seeing & Field rotation & Windspeed & Strehl & $H$ & $F_{\rm AO}$ \\ % & & & (s) & (arcsec) & (degree) & (m s$^{-1}$) & & (mag) & (Hz) \\ \hline % \object{HD\,207204} & 097.C-1042 & 80$\times$32 & 2560 & 0.29 & 28.0 & 7.4 & 0.87 & 6.7 & 1380 \\ \object{HD\,133954} & 0100.C-0543 & 144$\times$16 & 2304 & 0.30 & 27.2 & 7.9 & 0.78 & 7.7 & 1380 \\ \object{HD\,11506} & 096.C-0241 & 36$\times$64 & 2304 & 0.37 & 28.1 & 6.4 & 0.82 & 6.3 & 1380 \\ \object{HD\,90884} & 1100.C-0481 & 34$\times$96 & 3264 & 0.39 & 29.8 & 7.2 & 0.79 & 6.5 & 1380 \\ \object{*\,tau01\,Aqr} & 095.C-0549 & 192$\times$8 & 1536 & 0.43 & 28.3 & 8.0 & 0.82 & 5.8 & 1200 \\ \object{V889\,Her} & 198.C-0209 & 85$\times$64 & 5440 & 0.46 & 30.7 & 5.8 & 0.82 & 5.9 & 1380 \\ \object{LQ\,Hya} & 097.C-0864 & 64$\times$32 & 2048 & 0.47 & 33.8 & 6.0 & 0.84 & 5.6 & 1380 \\ \object{HD\,97244} & 1100.C-0481 & 48$\times$96 & 4608 & 0.51 & 27.4 & 7.8 & 0.79 & 5.8 & 1380 \\ \object{HD\,24966} & 097.C-1042 & 75$\times$32 & 2400 & 0.51 & 34.5 & 5.7 & 0.85 & 6.9 & 1380 \\ \object{HD\,8558} & 096.C-0241 & 64$\times$64 & 4096 & 0.53 & 28.2 & 6.0 & 0.66 & 6.9 & 1200 \\ \object{HD\,207575} & 095.C-0298 & 80$\times$64 & 5120 & 0.54 & 32.2 & 5.9 & 0.85 & 6.1 & 1200 \\ \object{BD+20\,1790} & 198.C-0209 & 80$\times$64 & 5120 & 0.58 & 27.3 & 7.4 & 0.70 & 7.0 & 1380 \\ \object{HD\,204277} & 1100.C-0481 & 64$\times$96 & 6144 & 0.68 & 34.0 & 6.4 & 0.84 & 5.5 & 1380 \\ \object{HD\,156751} & 1100.C-0481 & 48$\times$96 & 4608 & 0.69 & 31.3 & 8.4 & 0.81 & 6.3 & 1380 \\ \object{HD\,1466} & 096.C-0241 & 64$\times$64 & 4096 & 0.69 & 25.0 & 8.3 & 0.62 & 6.2 & 1200 \\ \object{HD\,1466} & 097.C-0865 & 80$\times$64 & 5120 & 0.69 & 31.1 & 5.9 & 0.81 & 6.2 & 1380 \\ \object{HD\,104125} & 096.C-0241 & 64$\times$64 & 4096 & 0.73 & 28.4 & 6.0 & 0.78 & 6.3 & 1380 \\ \object{HD\,45270} & 096.C-0241 & 256$\times$16 & 4096 & 0.91 & 28.4 & 6.8 & 0.79 & 5.2 & 1380 \\ \object{BD+21\,1764} & 198.C-0209 & 70$\times$64 & 4480 & 1.0 & 23.7 & 5.7 & 0.69 & 6.2 & 1380 \\ \hline% \end{tabular} \tablefoot{ \tablefoottext{a}{$n_{\rm DIT}$ is the number of image frames and $t_{\rm DIT}$ is exposure time per image frame.} The median values of seeing, PA rotation, wind speed, Strehl ratio, $H$ magnitude, and AO loop frequency ($F_{\rm AO}$) in our master reference library are 0.53\arcsec, 28.9$^{\circ}$, 7.1~m~s$^{-1}$, 0.75, 6.6 mag, and 1380 Hz. } \end{table*} \begin{table*} \caption{Targets with different PA rotations.} % \label{table:sample_delta_PA} % \centering % \begin{tabular}{l c c c c c c c c c c} % \hline\hline % Name & Program ID & $n_{\rm DIT} \times t_{\rm DIT}^{(a)}$ & $t_{\rm tot}$ & Field rotation & Seeing & Windspeed & Strehl & $H$ & $F_{\rm AO}$ \\ % & & & (s) & (degree) & (arcsec) & (m s$^{-1}$) & & (mag) & (Hz) \\ \hline % \object{HD\,81485B} & 095.C-0346 & 48$\times$32 & 1536 & 8.0 & 0.46 & 7.6 & 0.75 & 7.6 & 1380 \\ \object{HD\,208233} & 097.C-0826 & 144$\times$16 & 2304 & 13.7 & 0.59 & 5.9 & 0.62 & 6.9 & 1380 \\ \object{HD\,73267} & 096.C-0241 & 16$\times$64 & 1024 & 14.3 & 0.48 & 8.0 & 0.72 & 7.1 & 1380 \\ \object{HD\,223340} & 1100.C-0481 & 16$\times$96 & 1536 & 16.3 & 0.53 & 6.2 & 0.80 & 7.2 & 1380 \\ \object{HD\,16743} & 097.C-1019 & 64$\times$32 & 2048 & 17.8 & 0.50 & 5.8 & 0.87 & 6.0 & 1380 \\ \object{HD\,77825} & 198.C-0209 & 20$\times$64 & 1280 & 20.1 & 0.45 & 5.7 & 0.78 & 6.5 & 1380 \\ \object{HD\,105690} & 098.C-0739 & 48$\times$64 & 3072 & 22.9 & 0.44 & 8.1 & 0.77 & 6.6 & 1380 \\ \object{BD+20\,1790} & 198.C-0209 & 80$\times$64 & 5120 & 27.3 & 0.58 & 7.4 & 0.70 & 7.0 & 1380 \\ \object{HD\,97244} & 1100.C-0481 & 48$\times$96 & 4608 & 27.4 & 0.51 & 7.8 & 0.79 & 5.8 & 1380 \\ \object{HD\,8558} & 096.C-0241 & 64$\times$64 & 4096 & 28.2 & 0.53 & 6.0 & 0.66 & 6.9 & 1200 \\ \object{V889\,Her} & 198.C-0209 & 85$\times$64 & 5440 & 30.7 & 0.46 & 5.8 & 0.82 & 5.9 & 1380 \\ \object{HD\,207575} & 095.C-0298 & 80$\times$64 & 5120 & 32.2 & 0.54 & 5.9 & 0.85 & 6.1 & 1200 \\ \object{LQ\,Hya} & 097.C-0864 & 64$\times$32 & 2048 & 33.8 & 0.47 & 6.0 & 0.84 & 5.6 & 1380 \\ \object{HD\,24966} & 097.C-1042 & 75$\times$32 & 2400 & 34.5 & 0.51 & 5.7 & 0.85 & 6.9 & 1380 \\ \object{BD+01\,2063} & 1100.C-0481 & 44$\times$96 & 4224 & 41.3 & 0.46 & 7.2 & 0.78 & 6.2 & 1380 \\ \object{HD\,75519} & 198.C-0209 & 64$\times$64 & 4096 & 50.4 & 0.55 & 6.7 & 0.79 & 6.3 & 1380 \\ \object{HD\,212658} & 1100.C-0481 & 48$\times$96 & 4608 & 51.3 & 0.58 & 6.2 & 0.70 & 6.6 & 1380 \\ \object{HD\,119152} & 097.C-1042 & 144$\times$16 & 2304 & 53.7 & 0.56 & 8.2 & 0.78 & 6.8 & 1380 \\ \object{HD\,219246} & 1100.C-0481 & 30$\times$96 & 2880 & 77.7 & 0.50 & 6.5 & 0.82 & 7.2 & 1380 \\ \object{HD\,199443} & 097.C-0865 & 64$\times$64 & 4096 & 79.6 & 0.56 & 6.3 & 0.87 & 5.5 & 1380 \\ \hline% \end{tabular} \tablefoot{ \tablefoottext{a}{$n_{\rm DIT}$ is the number of image frames and $t_{\rm DIT}$ is exposure time per image frame.} The median values of seeing, PA rotation, wind speed, Strehl ratio, $H$ magnitude, and AO loop frequency ($F_{\rm AO}$) in our master reference library are 0.53\arcsec, 28.9$^{\circ}$, 7.1~m~s$^{-1}$, 0.75, 6.6 mag, and 1380 Hz. } \end{table*} \FloatBarrier \section{Comparisons of throughput and noise}\label{appendix:comparison_throughput_noise} The seeing condition affects the performances of RDI and ADI. To better compare the throughput and noise after the RDI and ADI reductions shown in Fig.~\ref{Fig:RDI_vs_ADI_delta_Seeing_throughput_noise}, we calculated their differences and present them in Fig.~\ref{Fig:comparison_TT_NN}. \FloatBarrier \section{Disk detections} Table~\ref{table:disk_list} lists the detection of disks in this work with the parameters of the RDI reductions. \begin{table*} \caption{Detected disks in this work.} % \label{table:disk_list} % \centering % \begin{tabular}{l c c c c c c c c} % \hline\hline % Name & $H^{a}$ & Program ID & Filed rotation & ADI detection & RDI detection & PC mode$^{b}$ & Reference size$^{b}$\\ % & (mag) & & (degree) & & & & \\ \hline \multicolumn{8}{c}{Planet-forming disks} \\ \hline \object{V4046\,Sgr} & 7.4 & 095.C-0298 & 83.8 & Yes & Yes & 100 & 500 \\ \object{PDS\,70} & 8.8 & 095.C-0298 & 52.0 & Yes & Yes & 100 & 500 \\ \object{T\,Cha} & 7.9 & 095.C-0298 & 28.9 & Yes & Yes & 50 & 500 \\ \object{RX\,J1615.3-3255} & 8.8 & 095.C-0298 & 74.2 & Yes & Yes & 250 & 500 \\ \object{RY\,Lup} & 7.7 & 097.C-0865 & 71.0 & Yes & Yes & 100 & 500 \\ \object{UX\,Tau\,A} & 8.0 & 097.C-0865 & 16.4 & No & Yes & 40 & 200 \\ \object{V1094\,Sco} & 9.0 & 099.C-0693 & 15.4 & No & Yes & 40 & 200 \\ \object{SZ\,Cha} & 8.4 & 198.C-0209 & 24.7 & Yes & Yes & 100 & 500 \\ \object{J1604} & 9.1 & 295.C-5034 & 91.9 & Yes & Yes & 100 & 500 \\ \object{HD\,34282} & 8.5 & 096.C-0241 & 53.5 & Yes & Yes & 250 & 500 \\ \object{MWC\,758} & 6.6 & 1100.C-0481 & 29.1 & Yes & Yes & 100 & 200 \\ \object{HD\,97048} & 6.7 & 096.C-0241 & 24.5 & Yes$^{c}$ & Yes & 100 & 500 \\ \object{HD\,100453} & 6.4 & 096.C-0241 & 31.3 & Yes & Yes & 20 & 200 \\ \object{DG\,Tau\,A} & 7.7 & 0104.C-0183 & 10.1 & Yes & Yes & 100 & 200 \\ \object{HD\,100546} & 6.0 & 095.C-0298 & 34.7 & Yes & Yes & 100 & 500 \\ \object{TW\,Hya} & 7.6 & 095.C-0298 & 76.7 & No & Yes & 20 & 200 \\ \object{RY\,Tau} & 6.1 & 096.C-0241 & 19.5 & Yes & Yes & 40 & 200 \\ \object{HD\,169142} & 6.9 & 095.C-0298 & 15.7 & No & Yes & 40 & 200 \\ \object{AK\,Sco} & 7.1 & 097.C-0079 & 38.5 & Yes & Yes & 75 & 500 \\ \hline \multicolumn{8}{c}{Debris disks} \\ \hline \object{NZ\,Lup} & 6.4 & 198.C-0209 & 60.4 & Yes & Yes & 100 & 500 \\ \object{HD\,141569} & 6.9 & 095.C-0298 & 42.1 & Yes & Yes & 100 & 500 \\ \object{HD\,131835} & 7.6 & 095.C-0298 & 72.6 & Yes & Yes & 100 & 500 \\ \object{HD\,106906} & 6.8 & 095.C-0298 & 30.1 & Yes & Yes & 100 & 500 \\ \object{HR\,4796A} & 5.8 & 1100.C-0481 & 64.1 & Yes & Yes & 100 & 500 \\ \object{HD\,129590} & 7.9 & 097.C-0949 & 36.9 & Yes & Yes & 100 & 200 \\ \object{HD\,15115} & 5.9 & 096.C-0241 & 29.6 & Yes & Yes & 250 & 500 \\ \object{HD\,120326} & 7.6 & 097.C-0060 & 36.3 & Yes & Yes & 100 & 500 \\ \object{HD\,115600} & 7.4 & 095.C-0298 & 27.3 & Yes & Yes & 100 & 500 \\ \object{HD\,111520} & 7.8 & 097.C-0060 & 36.2 & Yes & Yes & 100 & 500 \\ \object{HD\,110058} & 7.6 & 095.C-0607 & 10.7 & Yes & Yes & 100 & 500 \\ \object{HD\,131488} & 7.8 & 0101.C-0753 & 28.9 & Yes & Yes & 100 & 500 \\ \object{GSC\,07396-00759} & 8.8 & 198.C-0209 & 112.8 & Yes & Yes & 140 & 200 \\ \object{AU\,Mic} & 4.8 & 095.C-0298 & 118.7 & Yes & Yes & 250 & 500 \\ \hline% \end{tabular} \tablefoot{Here we show the data we used to make Fig.~\ref{Fig:RDI_disks}. We note that some targets may have multiple observations. For consistency in the paper, we fixed our FoV (\textless~0.735\arcsec). It is possible to increase the FoV to detect more extended disks using RDI. The ADI and RDI detections only indicate the detection within our adopted FoV. We note that this table should not be treated as a complete result of disk detections by SPHERE/IRDIS in the dual-band mode using the $H23$ filter. because our FoV is limited and we only included archival data released before 2021 January 1. \tablefoottext{a}{The $H$-band magnitudes are adopted from \cite{2MASS_Skrutskie2006}} \tablefoottext{b}{The parameters used in the RDI reduction. The number of PCs used in the PCA subtraction and the number of reference images used in the reference library.} \tablefoottext{c}{No clear disk structure was detected in the ADI image using a FoV of 0.735\arcsec~in radius. However, we indeed detected similar disk structures as the ADI-PCA image reported in \cite{Ginski2016} when we adopted a larger FoV of 1.2\arcsec~in the ADI-PCA subtraction. } } \end{table*} \end{appendix}

Title: Cascading Dark Energy

Abstract: The standard cosmological model is in the midst of a stress test, thanks to the tension between supernovae-based measurements of the Hubble constant $H_{0}$ and inferences of its values from Cosmic Microwave Background (CMB) anisotropies. Numerous explanations for the present-day cosmic acceleration require the presence of a new fundamental scalar field, as do Early Dark Energy (EDE) solutions to the Hubble tension. This raises the possibility that \textit{multiple} fields cooperatively contribute to the dark energy component in bursts throughout cosmic time due to distinct initial conditions and couplings. Here, this Cascading Dark Energy (CDE) scenario is illustrated through a realization that effectively reduces to a two-field model, with two epochs in which dark energy is cosmologically significant. The model is compared to measurements of the CMB, baryon acoustic oscillations, and observations of Type-Ia supernovae. It is found that this scenario ameliorates the Hubble tension, improving over purely late-time models of dark energy, and improves agreement between the related Rock `n' Roll EDE scenario and galaxy survey measurements of baryon acoustic oscillations.

https://export.arxiv.org/pdf/2208.07631

\title{Cascading Dark Energy} \author{ K. Rezazadeh$^{1}$\footnote{kazem.rezazadeh@ipm.ir}, A. Ashoorioon$^{1}$\footnote{amjad@ipm.ir}, and D. Grin$^{2}$\footnote{dgrin@haverford.edu}} \affiliation{ \small{$^{1}$School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran}\\ \small{$^{2}$Department of Physics and Astronomy, Haverford College, 370 Lancaster Avenue, Haverford, PA 19041, United States}\\ } \date{\today} \pacs{98.80.Cq, 04.50.+h} \keywords{cosmological parameters -- cosmology: theory -- dark energy -- large-scale structure of Universe} \section{Introduction} \label{section:introduction} Measurements of the present-day Hubble parameter (the Hubble constant $H_{0}$) from supernovae \cite{Riess:2016jrr, Riess:2018byc} and lensing time delays \cite{Bonvin:2016crt, Birrer:2018vtm} disagree with the value inferred from the Cosmic Microwave Background (CMB) data \cite{Planck:2018vyg, Planck:2019nip, Planck:2018lbu} fit to the $\Lambda$CDM model. More specifically, CMB power spectra determined by the \textit{Planck} collaboration yield a best fit value of $H_{0}=67.4\pm0.6\,{\rm km\,s^{-1}\,Mpc^{-1}}$ \cite{Planck:2018vyg}. Similarly, measurements of the acoustic horizon by the Dark Energy Survey (DES), combined with constraints to the baryon density from Big-Bang Nucleosynthesis (BBN) abundances, yield the value $H_{0}=67.4_{-1.2}^{+1.1}\,{\rm km\,s^{-1}\,Mpc^{-1}}$ \cite{DES:2017txv}. In contrast, observations of Type-Ia supernovae, tethered to a distance ladder obtained using Hubble Space Telescope (HST) measurements of $70$ long-period Cepheids in the Large Magellanic Cloud, imply a substantially different value, $H_{0}=74.03\pm1.42 \,\mathrm{km\ s^{-1}Mpc^{-1}}$ \cite{Riess:2019cxk}. This is known as the ``Hubble tension''. Although this discrepancy might be caused by systematic effects in the data (though none seem sufficient so far), it could alternatively herald exciting new physics beyond the $\Lambda$CDM concordance model. Many proposals have been suggested to resolve this tension (see, e.g. Refs. \cite{Umilta:2015cta, Karwal:2016vyq, Poulin:2018dzj, Poulin:2018cxd, Pandey:2019plg, Agrawal:2019lmo, Vagnozzi:2019ezj, Smith:2019ihp, Davari:2019tni, Sola:2020lba, Smith:2020rxx, Krishnan:2020vaf}), some of which (late-time resolutions) invoke modifications to $\Lambda$CDM which become important near the current cosmological epoch, others of which cause modifications to the cosmic budget at early times (early-time resolutions, which modify cosmic evolution around or before matter-radiation equality). Among the proposed late-time resolutions are phantom-like Dark Energy (DE) \cite{DiValentino:2016hlg, DiValentino:2017zyq}, a vacuum phase transition \cite{DiValentino:2017rcr}, interacting DE \cite{Kumar:2016zpg, DiValentino:2017iww}, and modified theories of gravity \cite{Barreira:2014jha, Umilta:2015cta, Ballardini:2016cvy, Renk:2017rzu, Belgacem:2017cqo, Nunes:2018xbm, Lin:2018nxe}. These scenarios \cite{DiValentino:2017zyq, DiValentino:2017iww, Addison:2017fdm} and more model-independent generalizations of them \cite{Bernal:2016gxb, Zhao:2017cud, Poulin:2018zxs} are highly constrained by the data, especially by measurements of the Baryon Acoustic Oscillations (BAO) \cite{Beutler:2011hx, Ross:2014qpa, BOSS:2016wmc} in galaxy surveys. Late-time resolutions to the Hubble tension usually suffer from some fundamental shortcomings, such as being a worse fit to CMB data than $\Lambda$CDM, fine-tuning issues, inappropriate use of an $H_{0}$ prior \cite{Efstathiou:2021ocp}, and conflicts with the ages of globular clusters \cite{Jimenez:2019onw, Valcin:2020vav, Valcin:2021jcg}, as discussed in Ref. \cite{Bernal:2021yli}. In most realizations of early-time solutions, the sound horizon is reduced by introducing additional radiation energy density to the matter-energy content of the Universe. Such scenarios are also constrained by BAO and the high-$\ell$ CMB power spectrum \cite{Poulin:2018cxd, Smith:2019ihp, Karwal:2021vpk, Poulin:2021bjr, Murgia:2020ryi}. For example, in Early Dark Energy (EDE) scenarios, the Universe contains a component (typically a scalar field) whose behavior is like a cosmological constant prior to a critical redshift preceding matter-radiation equality, and dilutes as fast or faster than radiation \cite{Poulin:2018cxd, Smith:2019ihp} subsequently. Aside from the Hubble tension, there is another disagreement between cosmological data sets, known as the $S_8$ tension. In particular, the value of $S_{8}\equiv\sigma_{8}\sqrt{\Omega_{\rm m}/0.3}$ (where $\Omega_{\rm m}$ is the today's matter density and $\sigma_8$ denotes the variance of matter perturbations within $8\mathrm{Mpc}/h$ today) implied by the CMB (when fit by the $\Lambda$CDM model) does not agree with the value inferred from measurements of the amplitude of matter density fluctuations in the late-time Universe \cite{DES:2017myr, Hildebrandt:2018yau, HSC:2018mrq, KiDS:2020suj, Hildebrandt:2017qln, DES:2021wwk, DES:2021epj}. Results from Dark Energy Survey (DES) 3-year data yield the constraint $S_{8}=0.797_{-0.013}^{+0.015}$ (68\% CL) \cite{DES:2021epj}, in contrast with the value $S_{8}=0.832\pm0.013$ (68\% CL) implied by the best fit value for the amplitude of the scalar density power spectrum from Planck 2018 TT,TE,EE +lowE+lensing data, assuming ${\Lambda}$CDM \cite{Planck:2018vyg}. Although EDE models can alleviate the $H_0$ tension, they tend to exacerbate the $S_8$ tension \cite{Hill:2020osr, Ivanov:2020ril, Murgia:2020ryi} and worsen fits to BAO data (see, e.g. \cite{Poulin:2018cxd, Smith:2019ihp, Murgia:2020ryi}). It is interesting to consider the possibility that these tensions (and the required fine tuning of EDE models) could be alleviated by a richer dark energy sector, for example, if there are multiple epochs of cosmic acceleration driven by one field \cite{Niedermann:2019olb,Niedermann:2021vgd,Freese:2021rjq,Allali:2021azp}, or if many scalar fields acting over time could yield better concordance between cosmological data sets \cite{Sabla:2021nfy}. In this paper, we explore if a resolution for the Hubble tension can be found in the Cascading Dark Energy (CDE) scenario, in which multiple scalar fields contribute to dark energy, analogously to the assisted inflationary scenario \cite{Liddle:1998jc}. CDE is motivated by recent developments in string theory, such as the swampland conjecture \cite{Vafa:2005ui, Ooguri:2006in, Obied:2018sgi}. CDE reduces the Hubble tension primarily by altering the early-time sound horizon, like the standard EDE scenario. Some of the fields, however drop out of sync from the others due to their initial conditions - that is to say that they no longer roll slowly (with nearly constant-energy density) even as other fields jointly continue to behave as dark energy. The evolution of each field becomes significant after the Hubble parameters drop below some specific value which depends on the effective mass of the field as well as the background energy density. After that, the field begins to oscillate around the local minimum of its potential and loses its energy accordingly. In the simplest realization, our model will reduce to two fields, allowing us to treat the dynamics of each field separately without resorting to an effective one-field approximation, as was done in, e.g. Ref. \cite{Sabla:2021nfy}. Multi-field models for dark energy are well motivated by considerations from string theory, such as the axiverse scenario \cite{Arvanitaki:2009fg}, in which a broad mass spectrum of ultra light axions could contribute to both dark matter and dark energy \cite{Hlozek:2014lca,Marsh:2015xka}. They may contribute to explaining the ``why now" question for the late-time dark energy driving present day cosmic acceleration \cite{Kamionkowski:2014zda,Emami:2016mrt}. At earlier times, multiple-field scenarios could help reduce the fine tuning needed for EDE models to succeed. Here, we consider the possibility that some dark energy fields are relevant near equality/recombination, while others are more relevant today. We investigate the behavior of the Hubble parameter and the field configuration in our setup. Both fields couple to gravity minimally, and their kinetic terms are assumed to be canonical. The potentials of both fields in the simplest realization are assumed to be quartic, although one can assume that they are different, as we will explain later. In our work, we check the consistency of the CDE scenario with the existing data, including the CMB \cite{Planck:2018vyg, Planck:2019nip, Planck:2018lbu}, Pantheon SN \cite{Pan-STARRS1:2017jku}, BAO \cite{BOSS:2016wmc, Ross:2014qpa, Beutler:2012px}, and Riess et al. (2019) \cite{Riess:2019cxk} measurements. We use the publicly available CosmoMC code \cite{Lewis:2002ah} in our investigation to constrain model parameters in light of the experimental data. Using the CosmoMC code, we calculate the $\chi^2$ values for our model for different data sources. To analyze the CosmoMC chains, we apply the GetDist package \cite{Lewis:2019xzd} and a burn-in fraction of $0.3$. We compare the CDE scenario with the concordance $\Lambda$CDM model, as well as with a single-field canonical scalar field that couples minimally to gravity and has a quartic potential. Furthermore, we compare our model to the Rock `n' Roll quartic model \cite{Agrawal:2019lmo}, where there is a cosmological constant and an oscillating scalar field acting as EDE. It is known that in the case of a full $(1-\cos{\theta})^{n}$ potential (as considered in Refs. \cite{Poulin:2018cxd, Smith:2019ihp}), anharmonic deviations from quadratic behavior are important in driving perturbative mode evolution towards behavior that more optimally address the $H_{0}$ tension than the Rock `n' Roll scenario. Nonetheless, we compare our CDE scenario to the Rock `n' Roll realization of EDE, as it provides a useful foil for comparing single and multi-field models with similar potentials. We compare the result of our two-field CDE scenario for $H_0$ with the results of the $\Lambda$CDM model, the single-field DE model, and also the Rock `n' Roll scenario. We want to know if the $H_0$ tension can be resolved via the CDE framework, and if so, whether it has any advantages. We compare the implied value of $S_{8}$ for the empirically allowed parameter space of our model with that of these other models. We find that the two-field CDE model fits the observational data better than the $\Lambda$CDM, single-field DE, and Rock `n' Roll models. The result of our two-field CDE model is more consistent with the Riess et al. (2019) measurement in comparison with the predictions of the $\Lambda$CDM and single-field DE models, and hence our model can ameliorate the $H_0$ tension existing among the cosmological data from different sources. Due to the resemblance of our two-field model to the Rock `n' Roll model with $n=2$, we contrast our setup with that model too. We find that the predictions of quartic two-field CDE are very close to the Rock `n' Roll model with $n=2$, although the late-time evolution of the dark energy at late times in our model yields a modestly better fit to the BAO data. The rest of this paper is structured as follows: In Sec. \ref{section:motivation}, we introduce the CDE model and explain its theoretical motivation. Then, in Sec. \ref{section:setup}, we explore the two-field realization of the CDE model and present its equations of motion. Subsequently, in Sec. \ref{section:numerical}, we conduct a Monte Carlo Markov Chain (MCMC) analysis to test a two-field CDE scenario using cosmological data. We present our conclusions in Sec. \ref{section:conclusions}, where we also put forward avenues to expand and further test the CDE scenario. \section{Setup of Cascading Dark Energy} \label{section:motivation} We consider $N+1$ scalar fields with quartic monomial potentials, with $1\ll N$, $V(\phi_i)=\frac{\lambda}{4} \phi_i^4$, $i=1\ldots N+1$, with the Lagrangian \begin{equation}\label{Lag1} S=\int \sqrt{-g} d^4 x \left[\sum_{i=1}^{N+1}\left(\frac{1}{2}\partial_{\mu} \phi_i \partial^{\mu} \phi_i-\frac{\lambda}{4} {\phi}_i^4\right) \right] \, . \end{equation} For simplicity, we have assumed that the quartic couplings of all the fields are the same. In principle, these scalar fields can have different initial conditions. We assume the swampland distance conjecture, under which these fields can at most transverse $M_P$ in the field space before a tower of massless species appears. We thus assume that the initial conditions of all the fields are sub-Planckian. Following the de-Sitter swampland conjecture, we also assume that the relative slope of the potential should not be very flat, yielding the constraint that \begin{equation}\label{desitter-swampland} M_P\frac{V'}{V}\gtrsim c=\mathcal{O}(1)\,. \end{equation} Let us assume that all the first $N$ fields have the same initial conditions, which is different from that of the $(N+1)$-th field, \begin{eqnarray}\label{initcond} &&\phi_1=\phi_2=\ldots=\phi_N=\phi_0\,,\nonumber \\ &&\phi_{N+1}= \chi_0\,. \end{eqnarray} Then the effective Lagrangian of $\phi_0$ and $\chi_0$ can be written as, \begin{align} S= & \int d^{4}x\sqrt{-g}\bigg(\frac{N}{2}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{0}+\frac{1}{2}\partial_{\mu}\chi_{0}\partial^{\mu}\chi_{0} \nonumber \\ & -N\frac{\lambda}{4}\phi_{0}^{4}-\frac{\lambda}{4}\chi^{4}\bigg) \,. \label{Lag2} \end{align} We introduce the new effective fields, $\phi$ and $\chi$, \begin{align} \phi &\equiv \sqrt{N} \phi_0\,,\nonumber\\ \chi &\equiv \chi_0\,, \label{field-dressing} \end{align} to make the kinetic term of the $\phi_0$ field in the Lagrangian canonical, which leads to the Lagrangian \begin{align} S= & \int\sqrt{-g}d^{4}x\bigg(\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi \nonumber \\ & -\frac{\lambda_{\phi}}{4}\phi^{4}-\frac{\lambda_{\chi}}{4}\chi^{4}\bigg) \, , \label{Lag3} \end{align} where \begin{eqnarray}\label{field-dressing2} \lambda_{\phi}&\equiv& \frac{\lambda}{N},\nonumber\\ \lambda_{\chi}&\equiv& \lambda\,, \end{eqnarray} Although the fields $\phi_i$, $i=1\ldots N+1$, cannot be super-Planckian due to swampland conjecture \cite{Ooguri:2006in}, the fields $\phi$ can be super-Planckian due to the large dressing factor $\sqrt{N}$, if $N\gg 1$. With these initial conditions, the fields can act as dark energy components in our setup. The other notable thing is that if $N \gg 1$, the quartic couplings of the $\chi$ field become much larger than the $\phi$ field, whereas the initial condition for the $\chi$ field becomes smaller and in fact sub-Planckian compatible with the swampland conjecture. Due to these, the $\chi$ field can play the role of the cascade field in our setup which starts oscillating around its minimum after the Hubble parameter drops below its mass, $\partial_{\chi}^2 V(\chi)$ and its energy density becomes a substantial part of the background energy density. This will lead to a sudden drop of the comoving sound horizon before the decoupling, which enhances the Hubble parameter respectively today if the angular $\theta_{\rm MC}$ parameter is fixed by the CMB experiments. A similar model could be constructed in the context of multi-giant matrix inflation \cite{Ashoorioon:2009sr,Ashoorioon:2009wa, Ashoorioon:2011ki, Ashoorioon:2014jja}) that uses a concentric multiple stack of D3-branes. In that model, the matrix structure of the coordinates perpendicular to the stack of D3-branes, and the ansatz of the SU(2) generator for three of the orthogonal directions perpendicular to the stacks of D3-branes are used. To exploit the model to describe the late time Universe, with the string coupling $g_{{}_S}\sim 1$, one has to use a large number of D3-branes, $N\sim 10^{40}$, and then one has to worry about to backreaction effects of the D3-branes on the background geometry. Alternatively, one can assume that that $g_{{}_S}$ itself is extremely small, say $g_{{}_S}\sim 10^{-100}$ and further suppression of the quartic couplings of the fields to the observed value to explain the dark energy vacuum density, $\lambda_{\phi,\chi}\sim 10^{-120}$ is achieved via the multiplicity of the D3-branes. Recently in Ref. \cite{Ashoorioon:2019kcy}, some of us showed that considering the scalar field non-minimally coupled to gravity with a moderate value of non-minimal coupling, one can reduce the required number of D3-branes to achieve the required number of D3-branes to a reasonable number during inflation. This is something that we hope to pursue in the context of cascading dark energy in the future, but for the purpose of this work, how the bare coupling has taken such small values is out of the scope of this work. Instead, we focus on two minimally coupled scalar fields where their couplings, $\lambda$, are already small. \section{The Two-Field Setup} \label{section:setup} In this work, we focus on the two-field realization of the CDE model, which consists of two dynamical scalar fields with canonical kinetic terms. The first Friedmann equation for a flat FRW Universe in this setup takes the following form \begin{equation} \label{H} H^{2}=\frac{1}{3M_{P}^{2}}\left(\rho_{\mathrm{\rm m}}+\rho_{\mathrm{\rm r}}+\rho_{\phi}+\rho_{\chi}\right), \end{equation} where $H=\dot{a}/a$ is the Hubble parameter and $M_{\rm P}\equiv1/\sqrt{8\pi G}$ is the reduced Planck mass. Furthermore, $\rho_{\rm m}$ and $\rho_{\rm r}$ denote the energy densities of matter and radiation, respectively. The energy densities of the scalar field $\phi$ and $\chi$ are respectively denoted by $\rho_{\phi}$ and $\rho_{\chi}$. It should be noted that in our work, we assume that the neutrinos to be massless, and therefore their contribution is included in the energy density of radiation component. The more general treatment of this setup requires also the inclusion of the massive neutrinos, and this possibility may be taken into account in the future extensions of our scenario. The energy densities of matter and radiation vary with scale factor as follows \begin{align} \label{rhom} \rho_{\rm m} &= \rho_{{\rm m}i}\left(\frac{a_{i}}{a}\right)^{3},\\ \label{rhor} \rho_{\rm r} &= \rho_{{\rm r}i}\left(\frac{a_{i}}{a}\right)^{4}, \end{align} where $\rho_{{\rm m} i}$ and $\rho_{{\rm r}i}$ are the energy densities of matter and radiation, respectively, at the initial scale factor $a_i$ that we take it deep inside in the radiation dominated era. We normalize these quantities as follows \begin{align} \label{rhomti} \tilde{\rho}_{{\rm m}i} &\equiv \frac{\rho_{{\rm m}i}}{M_{\rm P}^{2}H_{0}^{2}},\\ \label{rhorti} \tilde{\rho}_{{\rm r}i} &\equiv \frac{\rho_{{\rm r}i}}{M_{\rm P}^{2}H_{0}^{2}}\,, \end{align} where $H_0$ is the Hubble parameter today. We express the scale factor of the Universe in terms of the number of $e$-folds as \begin{equation} \label{a} a=a_{i}e^{N}\,, \end{equation} and hence from Eqs. \eqref{rhom} and \eqref{rhor}, we find \begin{align} \label{rhom-N} \rho_{\rm m} &= M_{\rm P}^{2}H_{0}^{2}\tilde{\rho}_{{\rm m}i}e^{-3N},\\ \label{rhor-N} \rho_{\rm r} &= M_{\rm P}^{2}H_{0}^{2}\tilde{\rho}_{{\rm r}i}e^{-4N}. \end{align} The energy densities of the two canonical scalar fields are given by \begin{align} \label{rhophi} \rho_{\phi} &= \frac{1}{2}\dot{\phi}^{2}+V_{\phi}(\phi),\\ \label{rhochi} \rho_{\chi} &= \frac{1}{2}\dot{\chi}^{2}+V_{\chi}(\chi). \end{align} We take the potential of both the scalar fields in the following quartic forms \begin{align} \label{Vphi} V_{\phi}(\phi) &= \frac{1}{4}\lambda_{\phi}\phi^{4},\\ \label{Vchi} V_{\chi}(\chi) &= \frac{1}{4}\lambda_{\chi}\chi^{4}, \end{align} where $\lambda_{\phi}$ and $\lambda_{\chi}$ are respectively the self-interaction coupling constants for the $\phi$ and $\chi$ fields. Applying the continuity equations for the energy densities Eqs. \eqref{rhophi} and \eqref{rhochi}, we obtain the equations of motion for $\phi$ and $\chi$, respectively, as \begin{align} \label{phiddot} \ddot{\phi}+3H\dot{\phi}+\frac{dV_{\phi}(\phi)}{d \phi} &= 0\,,\\ \label{chiddot} \ddot{\chi}+3H\dot{\chi}+\frac{dV_{\chi}(\chi)}{d \chi} &= 0\,. \end{align} Now, following Ref. \cite{Rezazadeh:2020zrd}, we introduce the following normalized quantities \begin{align} & \tilde{H}\equiv\frac{H}{H_{0}},\qquad\tilde{\phi}\equiv\frac{\phi}{M_{P}},\qquad\tilde{\chi}\equiv\frac{\chi}{M_{P}}, \nonumber \\ & \tilde{\lambda}_{\phi}\equiv\frac{M_{\rm P}^{2}}{H_{0}^{2}}\lambda_{\phi},\qquad\tilde{\lambda}_{\chi}\equiv\frac{M_{\rm P}^{2}}{H_{0}^{2}}\lambda_{\chi} \, . \label{normalization} \end{align} As a result, from Eq. \eqref{H}, we get \begin{equation} \label{Ht} \tilde{H}^{2}=\frac{4\left(\tilde{\rho}_{{\rm m}i}e^{-3N}+\tilde{\rho}_{{\rm r}i}e^{-4N}\right)+\tilde{\lambda}_{\phi}\tilde{\phi}^{4}+\text{\ensuremath{\tilde{\lambda}_{\chi}}}\tilde{\chi}^{4}}{2\left(6-\tilde{\phi}'^{2}-\tilde{\chi}'^{2}\right)}, \end{equation} where prime denotes the derivative with respect to the $e$-fold number $N$. If we take the derivative of both sides of the above equation with respect to $N$, we obtain \begin{align} \tilde{H}'= & -\frac{1}{\tilde{H}\left(6-\tilde{\phi}'^{2}-\tilde{\chi}'^{2}\right)}\bigg[3\tilde{\rho}_{{\rm m}i}e^{-3N}+\tilde{\rho}_{{\rm r}i}4e^{-4N} \nonumber \\ & -\tilde{H}^{2}\left(\tilde{\phi}'\tilde{\phi}''+\tilde{\chi}'\tilde{\chi}''\right)-\tilde{\lambda}_{\phi}\tilde{\phi}^{3}\tilde{\phi}'-\tilde{\lambda}_{\chi}\tilde{\chi}^{3}\tilde{\chi}'\bigg] \, . \label{dHt-d2phit-d2chit} \end{align} Applying the normalized quantities \eqref{normalization} in \eqref{phiddot} and \eqref{chiddot}, we also reach \begin{align} \label{d2phit-dHt} \tilde{\phi}'' &= -\frac{\tilde{H}\left(3\tilde{H}+\tilde{H}'\right)\tilde{\phi}'+\tilde{\lambda}_{\phi}\tilde{\phi}^{3}}{\tilde{H}^{2}}\,,\\ \label{d2chit-dHt} \tilde{\chi}'' &= -\frac{\tilde{H}\left(3\tilde{H}+\tilde{H}'\right)\tilde{\chi}'+\tilde{\lambda}_{\chi}\tilde{\chi}^{3}}{\tilde{H}^{2}}\,. \end{align} Inserting these into Eq. \eqref{dHt-d2phit-d2chit}, and then solving the resulting equation for $\tilde{H}'$, we arrive at \begin{equation} \label{dHt} \tilde{H}'=-\frac{1}{6\tilde{H}e^{4N}}\left[3\tilde{H}^{2}\left(\tilde{\phi}'^{2}+\tilde{\chi}'^{2}\right)e^{4N}+3\tilde{\rho}_{{\rm m}i}e^{N}+4\tilde{\rho}_{{\rm r}i}\right]. \end{equation} To eliminate $\tilde{H}'$ in Eqs. \eqref{d2phit-dHt} and \eqref{d2chit-dHt}, we use the above equation and hence we will have \begin{align} \tilde{\phi}'' = & -\frac{1}{6\tilde{H}^{2}}\bigg[e^{-4N}\text{\ensuremath{\tilde{\phi}}}'\bigg(3e^{4N}\tilde{H}^{2}\left(6-\tilde{\chi}'^{2}-\tilde{\phi}'^{2}\right) \nonumber \\ & -3\tilde{\rho}_{{\rm m}i}e^{N}-4\tilde{\rho}_{{\rm r}i}\bigg)+6\tilde{\lambda}_{\phi}\tilde{\phi}^{3}\bigg] \, , \label{d2phit} \\ \tilde{\chi}'' = & -\frac{1}{6\tilde{H}^{2}}\bigg[e^{-4N}\text{\ensuremath{\tilde{\chi}}}'\bigg(3e^{4N}\tilde{H}^{2}\left(6-\tilde{\chi}'^{2}-\tilde{\phi}'^{2}\right) \nonumber \\ & -3\tilde{\rho}_{{\rm m}i}e^{N}-4\tilde{\rho}_{{\rm r}i}\bigg)+6\tilde{\lambda}_{\chi}\tilde{\chi}^{3}\bigg] \, . \label{d2chit} \end{align} Eqs. \eqref{dHt}, \eqref{d2phit}, and \eqref{d2chit}, are basic equations that we will solve in our work to find the background dynamics. To determine the initial conditions, we assume the evolution of the two scalar fields start from the slow-roll regime. Therefore, the first term in Eqs. \eqref{phiddot} and \eqref{chiddot} can be neglected before the other terms, and these equations simplify as follows \begin{align} \label{phidot} 3H\dot{\phi}_i+\frac{dV_{\phi_i}(\phi_i)}{d\phi_i} &\approx 0,\\ \label{chidot} 3H\dot{\chi}_i+\frac{dV_{\chi_i}(\chi_i)}{d\chi_i} &\approx 0\,, \end{align} which in turn can be written in terms of $e$-folds number as \begin{align} \label{dphit} \tilde{\phi}' &\approx -\frac{\tilde{\lambda}_{\phi}\tilde{\phi}^{3}}{3\tilde{H}^{2}},\\ \label{dchit} \tilde{\chi}' &\approx -\frac{\tilde{\lambda}_{\chi}\tilde{\chi}^{3}}{3\tilde{H}^{2}}. \end{align} Besides, in the slow-roll regime, the kinetic terms of $\phi$ and $\chi$ are negligible in comparison with their potentials, and therefore the Friedmann equation \eqref{H} can be approximated as \begin{equation} \label{H-sr} H^{2}\approx\frac{1}{3M_{P}^{2}}\left[\rho_{\rm m}+\rho_{\rm r}+V_{\phi}(\phi)+V_{\chi}(\chi)\right]. \end{equation} If we substitute $\rho_{\rm m}$ and $\rho_{\rm r}$ from Eqs. \eqref{rhom} and \eqref{rhor}, respectively, into this equation, and then use Eqs. \eqref{rhom-N}, \eqref{rhor-N}, \eqref{Vphi}, \eqref{Vchi}, and \eqref{normalization}, we reach \begin{equation} \label{Ht-sr} \tilde{H}^{2}\approx\frac{1}{12}\left[4\left(\tilde{\rho}_{{\rm m}i}e^{-3N}+\tilde{\rho}_{\mathrm{ r} i}e^{-4N}\right)+\tilde{\lambda}_{\phi}\tilde{\phi}^{4}+\tilde{\lambda}_{\chi}\tilde{\chi}^{4}\right]. \end{equation} This equation now can be inserted into Eqs. \eqref{dphit} and \eqref{dchit} to give the initial values of derivative of the two scalar fields with respect to $N$ as \begin{align} \label{dphiti} \tilde{\phi}_{i}' &\approx -\frac{4\tilde{\lambda}_{\phi}\tilde{\phi}_{i}^{3}}{4\left(\tilde{\rho}_{{\rm m}i}+\tilde{\rho}_{{\rm r}i}\right)+\tilde{\lambda}_{\phi}\tilde{\phi}_{i}^{4}+\tilde{\lambda}_{\chi}\tilde{\chi}_{i}^{4}},\\ \label{dchiti} \tilde{\chi}_{i}' &\approx -\frac{4\tilde{\lambda}_{\chi}\tilde{\chi}_{i}^{3}}{4\left(\tilde{\rho}_{{\rm m}i}+\tilde{\rho}_{{\rm r}i}\right)+\tilde{\lambda}_{\phi}\tilde{\phi}_{i}^{4}+\tilde{\lambda}_{\chi}\tilde{\chi}_{i}^{4}}. \end{align} In this equation, $\tilde{\phi}_{i}$ and $\tilde{\chi}_{i}$ refer to the initial values of the scalar fields at the $e$-fold number $N_i = 0$. It should be noted that we cannot use Eq. \eqref{Ht-sr} as the initial condition for $\tilde{H}$, because it leads to self-inconsistency of the differential equations. Instead, in order to prevent this problem, we apply the following relation which follows from Eq. \eqref{Ht}, \begin{equation} \label{Hti} \tilde{H}_{i}^{2} = \frac{4\left(\tilde{\rho}_{{\rm m}i}+\tilde{\rho}_{{\rm r}i}\right)+\tilde{\lambda}_{\phi}\tilde{\phi}_{i}^{4}+\tilde{\lambda}_{\chi}\tilde{\chi}_{i}^{4}}{2\left(6-\tilde{\phi}_{i}'^{2}-\tilde{\chi}_{i}'^{2}\right)}. \end{equation} For $\tilde{\phi}'$ and $\tilde{\chi}'$ in this equation, we substitute their values from the slow-roll equations \eqref{dphiti} and \eqref{dphiti}, respectively. To integrate the background equations \eqref{dHt}, \eqref{d2phit}, and \eqref{d2chit}, we used the 8th-order Runge-Kutta algorithm. Our modified version of \textsc{Camb}, (which we use with CosmoMC to obtain constraints to the CDE model), is available online \footnote{https://github.com/krezazadeh/CAMB-CDE-two-field}. In order to ensure that the Universe always remains flat in our code for each set of input parameters, we use the parameter $\tilde{\lambda}_\phi$ as a derived parameter. To determine this parameter numerically, we note that the total density parameter at the present epoch is equal to unity for a flat Universe, \begin{equation} \label{Omegatotal} \Omega_{\rm m0}+\Omega_{\rm r0}+\Omega_{\phi0}+\Omega_{\chi0}=1, \end{equation} where the subscript ``0'' refers to the present time. From this equation, we find \begin{equation} \label{lambdaphit} \tilde{\lambda}_{\phi}=\frac{12-\tilde{\lambda}_{\phi}\tilde{\chi}_{0}^{4}-2\tilde{\phi}_{0}'^{2}-2\tilde{\chi}_{0}'^{2}-12\Omega_{\rm m0}-12\Omega_{\rm r0}}{\tilde{\phi}_{0}^{4}}. \end{equation} We use a shooting method in our numerical code that tests different values for $\tilde{\lambda}_\phi$ in the above equation for each set of free parameters. After several steps, the code finally finds a suitable value for this parameter that satisfies this equation with enough precision. As a result the parameter $\tilde{\lambda}_\phi$ is treated as a derived parameter in our numerical analysis. By requiring the $\phi$ field to provide the energy density needed required for a flat universe today, we target scenarios in which the data require $\chi$ to act as an EDE field and $\phi$ to be the present-day DE. \section{Numerical analysis} \label{section:numerical} Here, we obtain observational constraints to the CDE model at the level of background dynamics, using recent cosmological data. We use the publicly available CosmoMC computational package \cite{Lewis:2002ah}. In this work, we use the July 2019 version of CosmoMC. This code uses a Markov Chain Monte Carlo (MCMC) simulation to explore the parameter space of the model, using the Metropolis-Hastings algorithm \cite{Lewis:2002ah}. Our parameter space consists of $\{ \Omega_b h^2, \Omega_c h^2, \theta_{\rm MC}, \tau, A_s, n_s, \tilde{\phi}_i, \tilde{\chi}_i, \tilde{\lambda}_{\chi} \}$, where $\Omega_b$ and $\Omega_c$ denote the present-day density parameters for baryon and cold dark matter, $h$ is the dimensionless Hubble constant $h\equiv H_{0}/(100~{\rm km}{\rm s}^{-1}~{\rm Mpc}^{-1})$, $\theta_{\rm MC}$ refers to the ratio of the comoving sound horizon at decoupling to the comoving angular diameter distance to the surface of last scattering, $\tau$ indicates the optical depth, $A_s$ implies the amplitude of the primordial scalar power spectrum, and $n_s$ is the scalar spectral index. In order to obtain well-behaved sampling as described in Refs. \cite{Planck:2013pxb} (with helpful formulae in Ref. \cite{Hu:1995en}), the parameter $\theta_{\rm MC}$ is varied in CosmoMC. A bisection root finding method is then used to obtain the appropriate $H_{0}$ value within the $\Lambda$CDM model. Observables are properly computed using our modified version of \textsc{Camb}. The parameters $\tilde{\phi}_i$ and $\tilde{\chi}_i$ denote the value of the scalar fields at scale factor $a_i$ taken to be deep inside the radiation-dominated era. The coupling constant $\tilde{\lambda}_{\chi}$ that is used for the coupling constant of the $\chi$ scalar field, is treated as a free parameter in our MCMC analysis, while the parameter $\tilde{\lambda}_{\phi}$ is a derived parameter, as explained earlier. The CosmoMC package computes the likelihood of cosmological parameters by including the observational data from various sources. We include the combination of CMB, SNe Ia, BAO, and Riess et al. (2019) data sets in our work, and so multiplying the separate likelihoods for these data sets, the total likelihood will be $\mathcal{L}\propto e^{-\chi_{{\rm total}}^{2}/2}$, where $\chi_{{\rm total}}^{2}=\chi_{\mathrm{CMB}}^{2}+\chi_{\mathrm{SN}}^{2}+\chi_{\mathrm{BAO}}^{2}+\chi_{\mathrm{Riess2019}}^{2}$ encodes the deviation between the observational and theoretical results. Following Refs. \cite{Poulin:2018cxd, Smith:2019ihp, Poulin:2021bjr, Murgia:2020ryi}, we terminate our MCMC analysis when the Gelman-Rubin convergence criterion \cite{Gelman:1992zz} fulfills $R-1<0.1$. For the statistical analysis of the MCMC chains generated by CosmoMC, we use the publicly available GetDist package \cite{Lewis:2019xzd}. To establish priors for CDE parameters in our MCMC as well as an initial guess for the best fit value, we begin by finding rough initial guesses for $\tilde{\chi}_{i}$, $\log{(\tilde{\phi}_{i})}$, and $\log{(\tilde{\lambda}_{\chi})}$ that reproduce published Rock `n' Roll \textit{Planck} values for $H_{0}$, redshift of peak CDE energy-density fraction $z_{c}$, peak CDE energy density-fraction $f(z_{c})$, and $\Omega_{\Lambda}$, when numerically integrated using the equations in Sec. \ref{section:setup}. We then obtain an initial estimate of best-fit parameters for $\Lambda$CDM + CDE parameters using a simple random-walk simulation. We begin by choosing standard (but relatively broad) flat priors for the usual cosmological parameters, centered around Planck 2018 best fit values \cite{Planck:2018vyg}, as well as a trial range for CDE parameters: $\Omega_{b}h^{2}\in\left[0.022,0.023\right],$ $\Omega_{c}h^{2}\in \left[0.115,0.124\right]$, $\theta_{\rm MC}\in\left[1.02,1.06\right]$, $\tau\in\left[0.02,0.09\right]$, $n_{s}\in\left[0.959,0.974\right]$, $\ln{A}_{s}\in \left[3.019,3.075\right]$, $\tilde{\chi}_{i}\in\left[0.4,0.8\right]$, and $\tilde{\lambda}_{\chi}\in \left[14.5,15.5\right]$. An initial guess is made for CDE parameters $\log{\tilde{\phi}}_{i}$, $\tilde{\chi}_{i}$, $\log{\tilde{\lambda}}_{\chi}$ as well as $\Lambda$CDM parameters, and used to compute the likelihood for the full data set (see below) computed within CosmoMC. Random guesses are subsequently made for all $8$ parameters, but kept only if they improve the model likelihood, with a maximum of $100$ iterations. The results are insensitive to the values/ranges initially chosen for $\log{\tilde{\phi}}_{i}$ and $\log{\tilde{\lambda}_{\chi}}$, and indicate a preferred value for $\tilde{\chi_{i}}\simeq 0.48$. We then use the same flat priors on $\Lambda$CDM parameters and final simulation values to initialize a proper likelihood minimization within CosmoMC, whose initial values are used for the subsequent MCMC. A similar procedure was used for the Rock `n' Roll and Single-field DE models considered. We verified that the posterior probability distributions for $\log{\tilde{\phi}}_{i}$ and $\log{\tilde{\lambda}_{\chi}}$ are nearly as flat as the priors, justifying their use without loss of generality. The initial field value $\tilde{\chi}_{i}$ is well constrained in our MCMC and contained in the assumed prior. We incorporate the Planck 2018 CMB data for the temperature and polarization at small (TT,TE,EE) and large (lowl+lowE) angular scales \cite{Planck:2018vyg, Planck:2019nip}. We additionally take into account the CMB lensing potential power spectrum measured in the multipole range $40\leq\ell\leq400$ \cite{Planck:2018lbu}. The acoustic peaks are affected by the physics of the decoupling epoch, and their locations are sensitive to physical processes occurring between the decoupling epoch and today. Type Ia supernovae are standardizable candles that have approximately the same absolute magnitude, once corrections for the width of their light curve are applied. Therefore, they are a powerful tool that can be used to probe the expansion history of the Universe. In our MCMC analysis, we use the Pantheon SN sample \cite{Pan-STARRS1:2017jku}, which consists magnitude measurements for 1048 SNe Ia with redshifts $0.01 < z < 2.3$. The baryonic acoustic oscillation standard ruler provides a measurement of the angular diameter distance as a function of the cosmological redshift. BAO data can be used to constrain dark energy models. The pressure waves arising from the cosmological inhomogeneities in the baryon-photon primordial plasma affect CMB anisotropies and observations of the Large-Scale Structure (LSS) of the galaxy density field. The peak appearing in measurements of the large-scale galaxy correlation function is caused by BAOs. In our analysis, we use BAO measurements from the the Baryon Oscillation Spectroscopic Survey (BOSS) \cite{BOSS:2016wmc} ($z\simeq 0.15$), the SDSS Main Galaxy Sample \cite{Ross:2014qpa} ($z\simeq 0.15$), and the 6dFGS \cite{Beutler:2012px} ($z\simeq 0.11$). Finally, we include the Riess et al. (2019) determination of $H_0 = 74.03 \pm 1.42\,{\rm km\,s^{-1}\,Mpc^{-1}}$ \cite{Riess:2019cxk} for the Hubble constant, based on $70$ Cepheid observations in the LMC and observations of nearby Type-Ia supernovae. This determination is an independent constraint of the expansion rate of the local Universe in our computations. \begin{table*}[!ht] \caption{The best fit values and 68\% CL constraints for the parameters of the investigated models.} \centering \scalebox{0.9}{ \begin{tabular}{|c|c c|c c|c c|c c|} \hline \multirow{2}{*}{Parameter} & \multicolumn{2}{c|}{$\Lambda$CDM} & \multicolumn{2}{c|}{Single-field DE} & \multicolumn{2}{c|}{Rock `n' Roll} & \multicolumn{2}{c|}{Two-field CDE}\tabularnewline & best fit & 68\% limits & best fit & 68\% limits & best fit & 68\% limits & best fit & 68\% limits\tabularnewline \hline $\Omega_{b}h^{2}$ & $0.0226089$ & $0.02251\pm0.00013$ & $0.0225312$ & $0.02251\pm0.00013$ & $0.0228385$ & $0.02280\pm0.00015$ & $0.0227861$ & $0.02282\pm 0.00016$\tabularnewline $\Omega_{c}h^{2}$ & $0.11827$ & $0.11849\pm0.00090$ & $0.118691$ & $0.11851\pm0.00087$ & $0.121385$ & $0.1222_{-0.0014}^{+0.0011}$ & $0.122667$ & $0.1222^{+0.0012}_{-0.0015}$\tabularnewline $100\theta_{\rm MC}$ & $1.04116$ & $1.04116\pm0.00029$ & $1.04108$ & $1.04113\pm0.00029$ & $1.03951$ & $1.03942_{-0.00042}^{+0.00060}$ & $1.0396$ & $1.03939^{+0.00063}_{-0.00043}$\tabularnewline $\tau$ & $0.0596178$ & $0.0568\pm0.0071$ & $0.0572416$ & $0.0566_{-0.0072}^{+0.0064}$ & $0.054335$ & $0.0529\pm0.0071$ & $0.0492794$ & $0.0532\pm 0.0072$\tabularnewline ${\rm ln}(10^{10}A_{s})$ & $3.04763$ & $3.046\pm0.014$ & $3.04411$ & $3.046_{-0.014}^{+0.013}$ & $3.04998$ & $3.046\pm0.014$ & $3.04385$ & $3.046\pm 0.014$\tabularnewline $n_{s}$ & $0.968599$ & $0.9690\pm0.0037$ & $0.969799$ & $0.9688\pm0.0036$ & $0.968158$ & $0.9688\pm0.0037$ & $0.967781$ & $0.9691\pm 0.0037$\tabularnewline $\log(\tilde{\phi}_{i})$ & $-$ & $-$ & $1.81705$ & $2.56_{-0.57}^{+1.3}$ & $-$ & $-$ & $3.13425$ & $> 1.76$\tabularnewline $\tilde{\chi}_{i}$ & $-$ & $-$ & $-$ & $-$ & $0.480162$ & $<0.523$ & $0.484908$ & $0.502^{+0.033}_{-0.096}$\tabularnewline $\log(\tilde{\lambda}_{\chi})$ & $-$ & $-$ & $-$ & $-$ & $15.3182$ & $-$ & $15.4771$ & $-$\tabularnewline \hline $H_{0}$ & $68.7567$ & $68.60\pm0.41$ & $68.5669$ & $68.60\pm0.41$ & $70.9546$ & $70.94_{-0.84}^{+0.58}$ & $70.5298$ & $70.95^{+0.61}_{-0.85}$\tabularnewline $\Omega_{\rm m}$ & $0.297999$ & $0.7003\pm0.0052$ & $0.300383$ & $0.2997\pm0.0051$ & $0.286467$ & $0.2883\pm0.0057$ & $0.292401$ & $0.2883\pm 0.0058$\tabularnewline $\Omega_{\rm DE}$ & $0.702001$ & $0.2997\pm0.0052$ & $0.699617$ & $0.7003\pm0.0051$ & $0.713533$ & $0.7117\pm0.0057$ & $0.707599$ & $0.7117\pm 0.0058$\tabularnewline $\sigma_{8}$ & $0.820063$ & $0.8208\pm0.0060$ & $0.821043$ & $0.8208\pm0.0058$ & $0.839236$ & $0.8417_{-0.0087}^{+0.0072}$ & $0.841436$ & $0.8415^{+0.0075}_{-0.0087}$\tabularnewline $S_{8}$ & $0.817324$ & $0.820\pm0.010$ & $0.821567$ & $0.8204\pm0.0098$ & $0.820089$ & $0.825\pm0.010$ & $0.830711$ & $0.825\pm 0.010$\tabularnewline ${\rm Age}/{\rm Gyr}$ & $13.7303$ & $13.741\pm0.019$ & $13.7235$ & $13.739\pm0.020$ & $13.4444$ & $13.43_{-0.049}^{+0.10}$ & $13.4573$ & $13.43_{-0.049}^{+0.11}$\tabularnewline $\log(\tilde{\lambda}_{\phi})$ & $-$ & $-$ & $-6.34384$ & $-9.3_{-4.8}^{+4.1}$ & $-$ & $-$ & $-11.6081$ & $-8.5\pm 3.7$\tabularnewline \hline \end{tabular} } \label{table:parameters} \end{table*} \begin{table*}[!ht] \caption{The best fit value of $\chi^2$ for each model and each data set. The $\chi^2$ values corresponding to different CMB measurements including the Planck lensing power spectrum reconstruction ($\chi^2_{\rm lensing}$), baseline high-$\ell$ Planck power spectra (plik cross-half-mission, $30\leq\ell\leq2508$) ($\chi^2_{\rm plik}$), low-$\ell$ Planck temperature ($2\leq\ell\leq29$) ($\chi^2_{\rm lowl}$), and low-$\ell$ HFI EE polarization ($2\leq\ell\leq29$) ($\chi^2_{\rm simall}$), are also presented in the table. The table furthermore includes the values of $\chi^2_{\rm tot} $ and $\Delta \chi^2 = \chi^2_{\rm Model} -\chi^2_{\rm \Lambda CDM}$.} \centering \scalebox{0.9}{ \begin{tabular}{|c|c c|c c|c c|c c|} \hline \multirow{2}{*}{Parameter} & \multicolumn{2}{c|}{$\Lambda$CDM} & \multicolumn{2}{c|}{Single-field DE} & \multicolumn{2}{c|}{Rock `n' Roll} & \multicolumn{2}{c|}{Two-field CDE}\tabularnewline & best fit & 68\% limits & best fit & 68\% limits & best fit & 68\% limits & best fit & 68\% limits\tabularnewline \hline $\chi^2_{\rm lensing}$ & $8.68279$ & $9.10\pm 0.55$ & $9.10419$ & $9.08\pm 0.50$ & $9.44935$ & $9.89\pm 0.95$ & $9.83209$ & $9.88\pm 0.94$\tabularnewline $\chi^2_{\rm plik}$ & $2348.84$ & $2359.6\pm 5.8$ & $2352.99$ & $2359.7\pm 6.0$ & $2350.28$ & $2360.8\pm 5.8$ & $2348.35$ & $2360.8\pm 5.8$\tabularnewline $\chi^2_{\rm lowl}$ & $23.041$ & $22.91\pm 0.76$ & $22.028$ & $22.93\pm 0.73$ & $22.768$ & $23.34\pm 0.82$ & $23.4205$ & $23.29\pm 0.80$\tabularnewline $\chi^2_{\rm simall}$ & $397.156$ & $397.2\pm 1.9$ & $395.741$ & $397.2\pm 1.8$ & $395.778$ & $396.6\pm 1.3$ & $395.745$ & $396.7\pm 1.3$\tabularnewline \hline $\chi_{{\rm CMB}}^{2}$ & $2777.72$ & $2788.9\pm5.9$ & $2777.19$ & $2788.9\pm6.0$ & $2777.13$ & $2790.7\pm6.1$ & $2777.35$ & $2790.7\pm 6.0$\tabularnewline $\chi_{{\rm SN}}^{2}$ & $1034.74$ & $1034.792\pm0.083$ & $1034.74$ & $1034.80\pm0.12$ & $1035.07$ & $1035.05\pm0.29$ & $1034.82$ & $1035.05\pm 0.32$\tabularnewline $\chi_{{\rm BAO}}^{2}$ & $5.77561$ & $5.89\pm0.81$ & $5.44215$ & $5.86\pm0.82$ & $8.63314$ & $8.4\pm2.2$ & $6.61057$ & $8.4\pm 2.2$\tabularnewline $\chi_{{\rm Riess2019}}^{2}$ & $13.7907$ & $14.7\pm2.2$ & $14.8014$ & $14.7\pm2.2$ & $4.69061$ & $5.0\pm2.1$ & $6.07593$ & $5.0\pm 2.2$\tabularnewline \hline $\chi_{\mathrm{total}}^{2}$ & $3832.03$ & $-$ & $3832.17$ & $-$ & $3825.52$ & $-$ & $3824.86$ & $-$\tabularnewline $\Delta\chi^{2}$ & $0.0$ & $-$ & $0.14$ & $-$ & $-6.51$ & $-$ & $-7.17$ & $-$\tabularnewline \hline \end{tabular} } \label{table:chi2} \end{table*} The resulting best fit values and 68\% confidence limit (CL) constraints for the parameters of the investigated models are shown in Table \ref{table:parameters}. In the table, we see that the results for our single-field DE model are very close to $\Lambda$CDM, and the results of the two-field CDE model are very close to those of the Rock `n' Roll scenario. We also see that both the $\Lambda$CDM and single-field DE models return the 68\% CL constraint for the present-day Hubble parameter as $H_{0}=68.60\pm0.41~\mathrm{km\ s^{-1}Mpc^{-1}}$, which disagrees with the value $H_{0}=74.03\pm1.42~\mathrm{km\ s^{-1}Mpc^{-1}}$ measured by Riess et al. (2019) \cite{Riess:2019cxk}. The Rock `n' Roll and two-field CDE models yield a best fit value for $H_0$ of $H_{0}=70.94_{-0.84}^{+0.58}~\mathrm{km\ s^{-1}Mpc^{-1}}$ and $H_{0}=70.95_{-0.85}^{+0.61}~\mathrm{km\ s^{-1}Mpc^{-1}}$, respectively. These ranges are similar to one other and also in much better agreement with the Riess et al. (2019) measurement than models with no EDE, reducing the $H_0$ tension appreciably. Despite this, we see in Table \ref{table:parameters} that the Rock `n' Roll and two-field CDE scenarios predict higher values for the $S_8$ parameter compared with the $\Lambda$CDM and single-field DE frameworks, and hence their results show more deviations from the DES 3-year observations \cite{DES:2021epj}, which indicate that $S_{8}=0.832^{+0.015}_{-0.013}$. Therefore, the Rock `n' Roll and two-field CDE scenarios worsen the $S_8$ tension compared with the $\Lambda$CDM and single-field DE setups, a common shortcoming of EDE models \cite{Poulin:2018cxd, Smith:2019ihp, Karwal:2021vpk, Poulin:2021bjr, Murgia:2020ryi}. In Table \ref{table:parameters}, we also show results for the age of Universe in all the models considered here. The Rock `n' Roll and two-field CDE models result in lower values for the age of Universe than the $\Lambda$CDM and single-field DE models, agreeing more closely with the value $t_{\mathrm{U}}=13.5\pm0.15\,(\mathrm{stat.})\,\pm0.23\,(\mathrm{syst.})$ ($\pm0.27$ by regarding statistical and systematic uncertainties in quadrature) inferred using a sample of old globular clusters \cite{Jimenez:2019onw, Valcin:2020vav, Valcin:2021jcg}. In Table \ref{table:chi2}, we present $\chi^2$ values for different models with respect to each data set. We see that the single-field DE model performs comparably to $\Lambda$CDM. The Rock `n' Roll and two-field CDE models fit the CMB data better than the standard cosmological model. The Rock `n' Roll model fits the CMB data slightly better than the two-field CDE model, but the two-field scenario performs better in fitting the PANTHEON data. An interesting feature of the two-field CDE model is that it provides a better fit to the BAO data in comparison with the Rock `n' Roll model which implies that BAO data, showing some preference for late-time evolution in the DE component. Altogether, the table shows that the two-field CDE model provides the minimum value of $\chi_{\mathrm{total}}^{2}$. From Table \ref{table:chi2}, we conclude that in our two-field CDE scenario, $\Delta\chi^{2}$ gets reduced relative to $\Lambda$CDM, single-field DE, and Rock `n' Roll by $7.17$, $7.31$, and $0.66$, respectively. Results for 1D likelihoods and 2D contour plots in the 68\% and 95\% CL regions are shown in Fig. \ref{figure:2D}. The graph indicates that the results of the single-field DE model are very close to those of the $\Lambda$CDM model, and also the results of the two-field CDE model are very close to the Rock `n' Roll results. The two-field CDE and Rock `n' Roll models provide higher values for the $H_0$ parameter relative to the $\Lambda$CDM and single-field DE models, and therefore mitigate the Hubble tension. The 2D contour plots of the two-field CDE and Rock `n' Roll models which include the $H_0$ parameter, are separated thoroughly from the contour plots of the $\Lambda$CDM and single-field DE models. This reinforces the fact that the Hubble tension can be alleviated in models which invoke some type of energy injection before recombination in the past evolution of the Universe. Nevertheless, the two-field CDE and Rock `n' Roll scenarios yield greater values for the $S_8$ parameter, and so these models worsen the LSS tension in comparison with the $\Lambda$CDM and single-field DE settings, as already noted. The value of $\tau$, the optical depth parameter, decreases with respect to the $\Lambda$CDM and single field DE scenario in our CDE model. In the two-field CDE model, the value of this parameter gets reduced slightly more. In contrast to Ref. \cite{Agrawal:2019lmo}, the amplitude of the best fit scalar power spectrum is enhanced in the Rock `n' Roll mode in comparison with the $\Lambda$CDM, while the scalar spectral index gets reduced. In the two-field CDE though, both the amplitude of power spectrum and scalar spectral index are enhanced with respect to $\Lambda$CDM. We see in Fig. \ref{figure:2D} that the two-field CDE and Rock `n' Roll models prefer smaller values for $\Omega_m$ than the $\Lambda$CDM and single-field DE scenarios, a common property of EDE scenarios \cite{Poulin:2018cxd, Smith:2019ihp, Karwal:2021vpk, Poulin:2021bjr, Murgia:2020ryi}. Figure \ref{figure:2D} additionally shows that the likelihood is quite insensitive to $\tilde{\phi}_i$ and $\tilde{\lambda}_{\chi}$, as evidenced by the flatness of the posterior contours in these parameters. Despite this, there is a preferred value for $\tilde{\chi}_i$ in the two-field CDE and Rock `n' Roll models. This fact implies that our numerical analysis prefers a non-vanishing contribution of EDE to the cosmic energy density. In Fig. \ref{figure:H}, we used the best fit values of the parameters listed in Table \ref{table:parameters} to plot the evolution of the Hubble parameter against of redshift, showing that the Hubble parameter of the single-field DE model is very similar to $\Lambda$CDM. Also, the results of the Rock `n' Roll and two-field CDE models are very close to each other. Nevertheless, at earlier times, the Hubble parameter of the two-field CDE model is above the one for the Rock `n' Roll scenario, but at late times, the Rock `n' Roll Hubble parameter takes larger values, so this model gives a greater $H_0$. Smaller values of the Hubble parameter at lower redshifts give better compatibility of the two-field CDE model with BAO data, compared to Rock `n' Roll. Simultaneous agreement with PANTHEON and BAO data causes the $\phi$ field to be over-constrained by the data, limiting the ability of the CDE model to better fit BAO data. It may be possible to go beyond this limitation by adding additional scalar fields. We have also developed a modified version of \textsc{Camb} that has \textbf{three} dynamical fields ($\chi$, $\phi$, and a field $\xi$ that rolls at an intermediate redshift). By trial and error, we have found that there are combinations of initial field values and coupling constants that provide a significantly better fit to lower-$z$ BAO data (e.g. as in Fig. \ref{figure:H}). In future work, we will extend our MCMC analysis to this richer scenario. In Fig. \ref{figure:phi}, we show the evolution of the normalized scalar field $\tilde{\phi}$ in the single-field DE and two-field CDE models as a function of scale factor. Aside from a small interval at late times, the scalar field $\tilde{\phi}$ tends to remain constant. But, around the present epoch, it becomes dynamical. The fractional variation of $\tilde{\phi}$ in the (best-fitting) single-field scenario is substantially greater than in the (best-fitting) two-field CDE scenario - this is because simultaneously allowing late and early-time dark energy allows the model to fit both the late-time acceleration of the Universe and provide the required early-time reduction in the sound horizon needed to resolve the Hubble tension. In some sense, demanding less of the $\phi$ field allows its late-time behavior to more closely resemble $\Lambda$CDM. In Fig. \ref{figure:chi}, the evolution of the scalar field $\tilde{\chi}$ as a function of scale factor is plotted for the Rock `n' Roll and two-field CDE setups. Here we have used the best fit values given in Table \ref{table:parameters}. The figure displays that the scalar field $\tilde{\chi}$ is almost constant at earlier times in these models, and after a period of time, it begins to oscillate around the potential minimum at $\tilde{\chi} = 0$. The initial value of $\tilde{\chi}$ is bigger in the two-field CDE model compared to the Rock `n' Roll model, and also its oscillations occur in this model earlier than in the Rock `n' Roll model. At late times, the amplitude of the $\tilde{\chi}$ oscillations becomes very small, and accordingly, its contribution to the matter-energy content of the Universe becomes negligible. In Fig. \ref{figure:fEDE}, we show the fraction of the EDE energy density relative to the total energy density, that is, $f_{\rm EDE}\equiv\rho_{\chi}/\rho_{\mathrm{total}}$. We see that $f_{\rm EDE}$ is negligible at the initial times, but after a while, it grows sharply and reaches a peak at the critical redshift $z_c$, and then it drops again and becomes very small at the late-times. For the Rock `n' Roll model, the peak appears at the critical redshift $z_{c}=2.44\times10^{4}$ with the maximum value $f_{\rm EDE}=0.069$. The peak of the two-field CDE model appears at $z_{c}=2.34\times10^{4}$ with the maximum amplitude $f_{\rm EDE}=0.057$. Figure \ref{figure:wDE} shows the evolution of the equation of state parameter of dark energy, $w_{\rm DE}\equiv p_{\rm DE}/\rho_{\rm DE}=\left(p_{\phi}+p_{\chi}\right)/\left(\rho_{\phi}+\rho_{\chi}\right)$, in terms of cosmological redshift. At high redshifts, $w_{\rm DE}$ begins at $-1$, and subsequently oscillates around zero, with $w_{\rm DE}$ varying between $-1$ and $1$. Eventually, it converges to $-1$ and remains very close to this value until the present time. The oscillations of $w_{\rm DE}$ start sooner in the two-field CDE model than in the Rock `n' Roll scenario, although the oscillations last longer for Rock 'n Roll. We present the residuals of CMB power spectra in Fig. \ref{figure:Cls_residuals}. In the figure, we have also shown Planck 2018 data points \cite{Planck:2018vyg, Planck:2019nip, Planck:2018lbu} for comparison. We see in the figure that the results of the single-field DE are indistinguishable from $\Lambda$CDM results, while the Rock `n' Roll and two-field CDE models show deviations from $\Lambda$CDM, exceeding cosmic variance in some cases. To roughly estimate the improvement in constraining power of future CMB experiments, we compute the statistic \cite{2010PhRvD..81h3005G} \begin{equation} \mathcal{Z}\equiv \sqrt{\sum_{\ell}\frac{\Delta C_{\ell}^{2}}{\sigma_{\ell}^{2}}}, \end{equation}where the sum over the binned spectrum is evaluated using the deviation $\Delta C_{\ell}$ of the model of interest, with binned cosmic variance errors used for $\sigma_{\ell}$. We separately compute this sum for temperature and $EE$ polarization. A full forecast requires a proper analysis of T and E covariances, or better yet, a full Fisher matrix analysis that properly accounts for parameter degeneracies including nuisance parameters). Nonetheless, this rough estimate can give us some sense of whether or not the CMB alone can detect deviations between the models we consider, with future data. Roughly speaking, $\mathcal{Z}$ is the number of sigmas at which two scenarios can be distinguished. Using temperature, we find that future data could distinguish between the singe and two-field scenarios with $\sim 3\sigma$ significance, which improves to a $\sim 20\sigma$ potential detection in the limit that TE covariances vanish. Of course, this is not the case, and so a full Fisher matrix forecast is likely to show an answer closer to $\sim 7\sigma$ (ie, the geometric mean of the two extreme cases). The predictions of the Rock `n' Roll model match the CMB data better than the other models, and provide the best fit to CMB data of the models considered in this work, as also shown in Table \ref{table:chi2}. The better consistency of this model with the observational data points is more apparent at low $\ell$s, $\ell \sim 700$, $\ell \sim 1000$, and also high $\ell$s. The two-field CDE model displays the next best agreement with CMB data. \section{Conclusions} \label{section:conclusions} We studied the cascading dark energy model, in which many scalar fields contribute to the dark energy component of the Universe. In this setup, a large number of canonical scalar fields with sub-Planckian field excursions and steep potentials cooperate to provide a super-Planckian excursion and a relatively flat potential that can induce the late acceleration of the Universe as well as an early dark energy epoch. In the example we considered, the discordant initial conditions between fields cause some to cascade, drop out of the ensemble, and start oscillating around their minima. We restricted our attention to a single cascade involving an interplay reducing to an effective theory of two fields, $\phi$, and $\chi$. We choose the initial conditions such that $\phi$ plays the role of the late dark energy, while $\chi$ field behaves as EDE, subsequently cascading and decaying quickly. We used MCMC simulations to test CDE and related scenarios using CMB, BAO, PANTHEON observations of Type-Ia supernovae, and Riess et al. (2019) data. Our work shows that $\Delta\chi^{2}$ gets reduced relative to $\Lambda$CDM, single-field DE, and Rock `n' Roll by $7.17$, $7.31$, and $0.66$, respectively. The two-field CDE model yields a fit of $H_0 = 70.95_{-0.85}^{+0.61}~\mathrm{km\ s^{-1}Mpc^{-1}}$, substantially higher than $\Lambda$CDM ($H_0 = 68.60\pm0.41~\mathrm{km\ s^{-1}Mpc^{-1}}$) and single-field DE ($H_0 = 68.60\pm0.41~\mathrm{km\ s^{-1}Mpc^{-1}}$) values, while very close to the Rock `n' Roll values ($H_0 = 70.94_{-0.84}^{+0.58}~\mathrm{km\ s^{-1}Mpc^{-1}}$). The CDE and Rock `n' Roll versions of the EDE scenario thus reduce the Hubble tension between supernovae ($H_{0}=74.03\pm1.42~\mathrm{km\ s^{-1}Mpc^{-1}}$) and other methods. Our analysis also shows that the two-field CDE model ($\chi_{{\rm BAO}}^{2} = 6.61057$) provides a modestly improved fit to BAO data compared with the Rock `n' Roll model ($\chi_{{\rm BAO}}^{2} = 8.63314$) . There are a number of important avenues needed to expand and critically test our results. The analysis shown here did not follow the linear perturbations of either DE component - as noted, these do have a statistical impact on inferences about EDE properties, and in the future, we will generalize our analysis to include CDE clustering in the linear regime. We will also explore the possibility of rich resonant non-linear phenomenology, as discussed in Refs. \cite{Smith:2019ihp}. Another area for future work is the exploration of a broader range of potential energy functions, motivated by a variety of considerations. Since the effective super-Planckian field samples the potential at relatively large values of the scalar field, the form of the potential at such values of the field can effectively be different from the part of the potential that only samples the sub-Planckian values of the field. For example, the late field can effectively be on the parts of the potential that is just a cosmological constant. In this limit, the predictions and statistical significance of the model should be comparable approaches to the Rock `n' Roll model. However, in principle, the late and early dark energy fields can have different potentials and this affects the predictions of the model. This is one interesting research avenue that is worth considering in the future. Even with monomial potentials for both the late and early dark energy fields, one can assume non-renormalizable forms for the potentials. For example with a sixth-order monomial potential, $\phi^6$, we expect that the model can achieve larger values of $H_0$. The scalar fields also may have non-canonical kinetic terms. In particular, we can consider the DBI kinetic terms which have well-based theoretical motivations. In addition, although in this paper, we assumed that the late dark energy and the cascading fields couple to the Einstein gravity minimally, one can consider their non-minimal couplings with gravity too. These possibilities are left for future investigations. \section*{Acknowledgments} D.~G. acknowledges support in part by NASA ATP Grant No. 17-ATP17-0162, and the provost's office of Haverford College. We thank Tristan Smith for useful conversations. \bibliographystyle{aip}

Title: Fan beamed X-ray emission from 1 keV to above 130 keV from the ultraluminous X-ray pulsar RX J0209.6-7427 in the Small Magellanic Cloud

Abstract: We present detailed timing and spectral analyses of the transient X-ray pulsar RX J0209.6$-$7427 in the Small Magellanic Cloud during its 2019 giant outburst. With a better known distance than most galactic X-ray pulsars, its peak luminosity is determined to be $(1.11\pm0.06)\times 10^{39}\, \rm erg\ s^{-1}$; it is thus a {\it bonda fide} pulsating ultraluminous X-ray source (PULX). Owing to the broad energy band of \textit{Insight}-HXMT, its pulsed X-ray emission was detected from 1 keV up to the 130$-$180 keV band, which is the highest energy emission detected from any PULXs outside the Milky Way. This allows us to conclude that its main pulsed X-ray emission is from the "fan beam" of the accretion column, and its luminosity is thus intrinsic. We also estimate its magnetic field of (4.8$-$8.6)$\times10^{12}$ G or (1.7$-$2.2)$\times10^{13}$ G, from its spin evolution or transition in the accretion column structure during the outburst; we suggest that the two values of the magnetic field strength correspond to the dipole and multipole magnetic fields of the neutron star, similar to the recent discovery in the Galactic PULX Swift J0243.6+6124. Therefore, the nature of the neutron star and its ULX emission can be understood within the current theoretical frame of accreting neutron stars. This may have implications for understanding the nature of those farther away extragalactic PULXs.

https://export.arxiv.org/pdf/2208.14785

. \begin{document} \title{Fan beamed X-ray emission from 1 keV to above 130 keV from the ultraluminous X-ray pulsar RX J0209.6$-$7427 in the Small Magellanic Cloud} \correspondingauthor{X. Hou, S.N. Zhang} \email{xhou@ynao.ac.cn, zhangsn@ihep.ac.cn} \author[0000-0003-0933-6101]{X. Hou} \affiliation{Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China} \affiliation{Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, China} \author[0000-0002-2749-6638]{M.Y. Ge} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China} \author[0000-0001-9599-7285]{L. Ji} \affiliation{School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China} \author[0000-0001-5586-1017]{S.N. Zhang} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China} \author{Y. You} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China} \author[0000-0002-2705-4338]{L. Tao} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China} \author{S. Zhang} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China} \author[0000-0002-4622-796X]{R. Soria} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China} \author[0000-0001-7584-6236]{H. Feng} \affiliation{Department of Astronomy, Tsinghua University, Beijing 100084, China} \author{M. Zhou} \affiliation{Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China} \affiliation{Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, China} \author{Y.L. Tuo} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China} \author{L.M. Song} \affiliation{Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China} \author{J.C. Wang} \affiliation{Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China} \affiliation{Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, China} \affiliation{University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, China} \keywords{stars: neutron --- pulsars: individual: RX J0209.6-7427 --- X-rays: binaries --- galaxies: individual: SMC} \section{Introduction} \label{sec:intro} Ultraluminous X-ray sources (ULXs) are objects first detected in nearby galaxies with apparent luminosity $\gtrsim 10^{39}\, \rm erg\ s^{-1}$, above the Eddington limit for a stellar-mass black hole (BH) ($\sim 10M_{\rm \odot}$) \citep{Kaaret2017}, but which are not supermassive BHs. The recent discoveries of coherent pulsations in ULXs unambiguously established that accreting neutron stars (NSs) can be the central engines of ULXs \citep{Bachetti2014,Furst2016,Israel2017a,Israel2017b,Carpano2018}. The radiative mechanisms of pulsating ULXs (PULXs) is still under debate. Different scenarios have been proposed to explain their super-Eddington luminosity, involving NSs with strong magnetic dipole fields or multipolar fields close to that of magnetars \citep{Mushtukov2015a,Chashkina2017,Israel2017a,Chashkina2019}, or NSs with standard magnetic fields ($10^{12}-10^{13}$ G) but whose emission is highly collimated rather than quasi-isotropic \citep{King2009,Kluzniak2015,King2016,Koliopanos2017,Pintore2017,King2017,Middleton2017,Walton2018,King2019,King2020}. Until recently, PULXs have mostly been found in extragalactic galaxies at a distance of a few Mpc. Therefore, detailed study on PULXs is hampered by the limited number of PULXs detected and limited X-ray observations. RX J0209.6$-$7427 is a transient accreting X-ray pulsar \citep[see][for a recent review on X-ray pulsars]{Mushtukov2022} in the outer wing of the SMC discovered during its outburst in 1993 \citep{Kahabka2005}. During its recent 2019 giant outburst, a spin period of 9.2\,s and a peak luminosity of $\sim 10^{39} \, \rm erg\ s^{-1}$ in the energy range of 0.2$-$12 keV was first reported using \textit{NICER} data \citep{Iwakiri2019,Vasilopoulos2020}. Pulsations were then detected in 8$-$50 keV with the \textit{Fermi}/GBM data, in 3$-$70 keV with the \textit{NuSTAR} data \citep{Vasilopoulos2020} and in 3$-$80 keV with the \textit{AstroSat} data \citep{Chandra2020}. RX J0209.6$-$7427 is thus established as a new PULX. Optical observations classified its companion star to be a Be type \citep{Coe2020}. Compared to the extragalactic PULXs, this source has the advantage of proximity to allow detailed observational diagnoses. Compared to the first Galactic PULX Swift J0243.6+6124 \citep{Wilson-Hodge2018}, whose distance has rather large uncertainty ranging from 5 kpc to 8.9 kpc \citep{van2018}, RX J0209.6$-$7427 has a better determined distance of 55 kpc and much smaller relative distance uncertainty of 5\%\footnote{The mean distance modulus of the SMC is $18.89\pm0.04 \, \rm (stat) \pm0.10 \, \rm(syst)$, which corresponds to a distance of $60\pm 2.8 \, \rm (stat+syst)$ kpc \citep{Harries2003}. However, a smaller modulus of 18.7 corresponding to 55 kpc is usually adopted for sources located at the SMC wing, but no uncertainty was reported \citep{Cignoni2009}. We thus adopted the same uncertainty of 2.8 kpc as for the SMC mean distance.}, hence less uncertainty on its luminosity. Therefore, RX J0209.6$-$7427 is an ideal source to gain insight into the PULX emission properties. In this work, we analyze \textit{Insight}-HXMT data and more \textit{NICER} data than previously reported \citep{Vasilopoulos2020} to characterize the spectral and temporal properties of RX J0209.6$-$7427, so as to investigate its ultraluminous emission origin. The paper is organized as follows: observations and data reduction method are described in Section \ref{obs}, and the detailed data analysis results are presented in Section \ref{result}. We discuss our results in Section \ref{discuss} and conclude in Section \ref{conclude}. \section{Observations and data reduction} \label{obs} \subsection{\textit{NICER}} \textit{NICER}, launched on 2017 June 03, is an International Space Station payload devoted to the study of NSs through high sensitivity X-ray timing in the soft (0.2$-$12 keV) X-ray band \citep{Gendreau2016}. Its X-ray Timing Instrument (XTI) is an aligned collection of 56 X-ray concentrator optics (XRC) and silicon drift detector (SDD) pairs. Each XRC collects X-rays over a large geometric area from a roughly 30 arcmin$^{2}$ region of the sky and focuses them on to a small SDD. The SDD detects individual photons, recording their energies with good spectral resolution and their detection times to a $\sim$100 nanoseconds RMS relative to the Universal Time. We used \textit{NICER} observations from 2019 November 21 to 2020 March 20 (58808-58928 MJD) covering both the rising and decaying parts of the ourburst. \textit{NICER} data reduction is performed using \texttt{HEASOFT} (version 6.28). For the timing analysis, the good time intervals (GTIs) are selected according to the following criteria: \textit{NICER} not in the South Atlantic Anomaly region, source elevation $>10^{\circ}$ above the Earth limb ($>20^{\circ}$ above the bright Earth), pointing offset $\lesssim 54^{\prime}$, and magnetic cut-off rigidity $>1.5$ GeV/c. We further correct the arrival time of every event to barycentre via the \texttt{barycorr} tool and the JPL-DE405 planetary ephemeris. \subsection{\textit{Insight}-HXMT} Launched on 2017 June 15, the Hard X-ray Modulation Telescope \citep{Zhang2020} (\textit{Insight}-HXMT) is the first Chinese X-ray astronomy satellite. There are three scientific payloads onboard the satellite: the Low Energy X-ray telescope (LE, 1$-$10 keV) \citep{Chen2020}, the Medium Energy X-ray telescope (ME, 5$-$30 keV) \citep{Cao2020}, and the High Energy X-ray telescope (HE, 20$-$250 keV) \citep{Liu2020}. \textit{Insight}-HXMT observed RX J0209.6-7427 from 2019 December 10 to December 15 (58827-58832 MJD) around the outburst peak, for a total exposure of 100 ks. The data reduction is performed using the \textit{Insight}-HXMT data analysis software package \texttt{HXMTDAS} v2.04\footnote{http://www.hxmt.org/software.jhtml} following the standard procedure in the \texttt{HXMTDAS} user guide\footnote{http://www.hxmt.org/SoftDoc/67.jhtml}. Guide lines of the procedure is as following: (1) Use the commands \texttt{hepical}, \texttt{mepical} and \texttt{lepical} to calibrate the photon events from the raw data according to the Calibration Database (CALDB) of \textit{Insight}-HXMT. (2) Select the GTIs using the commands \texttt{hegtigen}, \texttt{megtigen} and \texttt{legtigen} for calibrated events. (3) Extract the good events basing on the GTIs using the commands \texttt{hescreen}, \texttt{mescreen} and \texttt{lescreen}. (4) Generate spectra for the selected events using the commands \texttt{hespecgen}, \texttt{mespecgen} and \texttt{lespecgen}. (5) Generate the background spectra basing on the emission detected by blind detectors using the commands \texttt{hebkgmap}, \texttt{mebkgmap} and \texttt{lebkgmap}. (6) Generate the response matrix files required for spectral analysis using the commands \texttt{herspgen}, \texttt{merspgen} and \texttt{lerspgen}. In the timing analysis, the arrival times of all the cleaned events are further barycenter corrected via the \texttt{hxbary} tool and the JPL-DE405 planetary ephemeris. In the spectral analysis, only the small field of view (FOV) detectors from the LE and ME instruments are used. The spectral fit is performed using \texttt{XSPEC} v12.11.0 \citep{1996ASPC..101...17A}. Uuncertainties are reported at the 68\% confidence interval and are computed using Markov Chain Monte Carlo simulations (MCMC, available through \texttt{XSPEC}) of length $10^{5}$. \section{Data analysis and results} \label{result} \subsection{Long-term flux and spectral evolution} \label{lcHID} There are 89 \textit{NICER} observations with the exposure of $>$ 100\,s during the outburst of RX J0209.6$-$7427 in 2019. We fit the spectra with a phenomenological model \texttt{Tbabs*(bb+powerlaw)} that describes the data well, and calculate the flux ($F$) in the energy range of $0.5-10$\,keV. We note that our aim here is to estimate the flux only, and therefore the selection of other alternative models has little influence on our results. Then we translate $F$ to the ``bolometric" X-ray luminosity (0.5$-$70\,keV) assuming an isotropic radiation as $L=4\,\pi\,C_{\rm bol} F\,D^2 $, where $D$=55\,kpc is the distance to the source \citep{Harries2003,Cignoni2009} and $C_{\rm bol}$ is a conversion factor. The $C_{\rm bol}$ at the outburst peak is determined using the broadband \textit{Insight}-HXMT observations, and is taken from the previous {\it NuSTAR} report \citep{Vasilopoulos2020} when the source is fainter. In practice, considering the variation of the spectral shape with luminosity, we estimate this factor using linear interpolation as \[C_{\rm bol} = \left\{ \begin{array}{l} C_{\rm N}, {\rm when}\ F<F_{{\rm N}}\\ C_{\rm N} + (F-F_{\rm N}) \frac{C_{\rm H}-C_{\rm N}}{F_{\rm H}-F_{\rm N}}, {\rm when}\ F\geq F_{{\rm N}} \end{array} \right.\] where $F_{\rm H}$ and $F_{\rm N}$ represent fluxes observed with \textit{Insight}-HXMT and {\it NuSTAR} in the energy range of 0.5$-$10\,keV, and $C_{\rm H}$ and $C_{\rm N}$ are the corresponding conversion factors, respectively. We show the long-term evolution of the bolometric luminosity in Figure~\ref{fig:nicerlc}. We find the bolometric luminosity is $\sim$ $\rm 10^{39} \, erg\,s^{-1}$ at the outburst peak, making the source a PULX as previously reported. The Hardness-Intensity Diagram (HID) presents the luminosity and hardness ratio relation of RX J0209.6$-$7427 (Figure~\ref{fig:hardness}), where the hardness is defined as the count rate ratio of 2$-$10\,keV to 0.5$-$2\,keV observed with \textit{NICER}. We fitted the HID with broken lines, which results in two turning points at (3.02$\pm$0.02) and (0.57$\pm$0.01) $\rm \times 10^{38}\,erg\ s^{-1}$, respectively. Implication of the luminosity turning will be discussed in Section \ref{statetrant}. \subsection{Timing analysis} \subsubsection{Pulsation search} We use the partially phase-coherent analysis to search for pulsations in \textit{NICER} data and generate ephemeris describing the spin and binary parameters of the source \citep{2015ApJ...812...95F}. In practice, we estimated the frequency by folding events on a trial period to obtain an averaged pulse profile for each observation. We then folded 200\,s segments on the same period and cross-correlated the resulting pulse profile with the averaged one to get the time-of-arrival (TOA) of pulsations for each segment. The frequency $\nu$ was then calculated based on these TOAs using the software {\rm {\texttt{Tempo2}} \citep{Hobbs2006}}. The frequency derivative $\nu_{1}$ is estimated from two adjacent observations of \textit{NICER} with the same method. During the 2019 outburst of RX J0209.6$-$7427, a spin-up trend is significantly found, which is probably caused by the accretion process (Figure~\ref{fig:spin_evol}). On the other hand, there is a frequency variation that shows approximately sinusoidal modulations superposed on the spin-up evolution, which is likely due to the Doppler effect of the binary motion. Thus, the frequency can be described as \begin{eqnarray} \nu(t) = \nu(t)_{\rm in} - \nu(t)_{\rm Do} \, , \\ \nu(t)_{\rm in}=\nu_0 + \sum_{n=1}^{6}\frac{1}{n!}\nu_{n}(t-t_0)^n \, , \\ \nu(t)_{\rm Do} = \frac{2\pi\nu_0 a\,{\rm sin}\,i}{P_{\rm orb}} ({\rm cos}\,l + e\,{\rm sin}\,\omega\,{\rm sin}\,2l + e\,{\rm cos}\,\omega\,{\rm cos}\,2l) \, , \end{eqnarray} where $\nu(t)_{\rm in}$ and $\nu(t)_{\rm Do}$ represent the intrinsic frequency of the source and the frequency shift caused by the Doppler effect, respectively. $\nu_0$ is the frequency at the reference time $t_0$, $a\,{\rm sin}\,i$ is the projected orbital semi-major axis in units of light-travel time, $P_{\rm orb}$ is the orbital period, $e$ is the eccentricity, $\omega$ is the longitude of periastron, and $l$ is the mean longitude. The higher order derivatives $\nu_{n}$ are utilized to fit the spin evolution in order to fit the binary parameters. We obtained an acceptable fit ($\chi^2$=0.992 with 76 $dof$) using the model mentioned above, and show the results in Figure~\ref{fig:rate_evol} and Table~\ref{table:lcfit}. We search independently the spin periods for the five \textit{Insight}-HXMT observations. Due to the short time of \textit{Insight}-HXMT in each observation, spin period derivatives and binary parameters have been ignored in the search, which have in fact negligible effect on the result. We used the cross-correlation technique (See Appendix \ref{highestE}) to determine the highest energy pulsation. \subsubsection{Pulse profiles} We folded events using the resulted timing model as described above to obtain the pulse profile for each \textit{NICER} observation in the energy range of 0.5$-$10\,keV (Figure~\ref{fig:nicerprofiles}, left panel). We found that the morphology of the pulse profile is related to time or the flux: Epoch I, when time $<$ 58880\,MJD, i.e., around the outburst peak, the pulse profile shows one main peak and one minor peak; Epoch II, when 58880\,MJD $<$ time $<$ 58920\,MJD, there is only one broad peak found; Epoch III, when time $>$ 58920\,MJD, i.e., in the faint state of the source, the minor peak appears again. We show averaged pulse profiles using \textit{NICER} data during these three epochs in the right panel of Figure~\ref{fig:nicerprofiles}. The pulsation is detected in all three instruments of \textit{Insight}-HXMT, covering a combined energy range of 1$-$250 keV, as shown in Figure~\ref{fig:hxmtprofiles}. Before 58880 MJD, the pulse profile of RX J0209.6$-$7427 exhibits one main peak and one minor peak separated by $\sim$0.5 in phase. Its main peak is detected above 130 keV and up to 180 keV at a 4.3$\sigma$ confidence level using the cross-correlation technique, the highest energy pulsation detected from this PULX. However, the minor peak becomes weaker with increasing energy and is eventually not prominent above around 27 keV. In comparison, the standard $\chi^2$-test method gave only 1.7$\sigma$ for the 130$-$180 keV profile; this demonstrates that the cross-correlation method is more advantageous since it makes use the lower energy pulse profile, which has very high signal-to-noise ratio, as a template in searching for the higher energy pulsed signals. To quantify the shape of pulse profiles, we computed the pulse fraction (PF)\footnote{We verified that the root-mean-squared PF formula gave consistent result.} in the energy range of 0.5$-$10\,keV for the \textit{NICER} profile and 1$-$180\,keV for the \textit{Insight}-HXMT profile, respectively: \begin{equation} PF=(F_{\rm max}-F_{\rm min})/(F_{\rm max}+F_{\rm min}) \, , \end{equation} where $F_{\rm max}$ and $F_{\rm min}$ are the maximum and minimum fluxes in the pulse profile, respectively. We show the result in Figure~\ref{fig:profiles_PF}. The errors were estimated using Monte-Carlo simulations. It is clear that the \textit{NICER} PF has a maximum point around 6 $\rm \times 10^{38}\,erg\ s^{-1}$. In addition, we found the slope of the PF evolution changes around the critical luminosity inferred from the hardness variation (Figure~\ref{fig:hardness}). Implications of the PF turning will be discussed in Section \ref{statetrant}. \subsection{Insight-HXMT spectral analysis} We fitted the 5-observation combined phase-averaged spectra using different phenomenological models available at \texttt{XSPEC} and using the standard \textit{Insight}-HXMT background model. The spectra were best fitted with the \texttt{Tbabs*(cutoffpl+bbodyrad+Gaussian)} model. The \texttt{Tbabs} model is used to account for the photoelectric absorption by the interstellar medium, and the hydrogen column density has been fixed to the Galactic value of $1.58\times 10^{21} \, \rm cm^{-2}$ in the direction of the SMC as was done in previous studies\footnote{We verified that letting the column density free has negligible effect on our result.} \citep{Vasilopoulos2020}. The continuum is composed of a power law with exponential cutoff and a blackbody component. The emission line at $\sim$ 6.4 keV that originates from neutral Fe is modeled by a Gaussian function, but the limit of the \textit{Insight}-HXMT spectral resolution does not allow more detailed modeling of the emission line to probe its physical origin. The best fitting parameters are presented in Table~\ref{table:fitave} and the spectra are shown in Figure~\ref{fig:avespec_hxmtbkg}. The broad band (1$-$150 keV) luminosity is $(1.11\pm 0.06)\times 10^{39}\, \rm erg\ s^{-1}$. The phase-resolved spectra using the definition of ON (pulsed) and OFF (unpulsed) ranges in the pulse profile for both the main and minor peaks (Figure~\ref{fig:hxmtprofiles}) were created using standard \textit{Insight}-HXMT data analysis procedure and corrected for exposure in different phases. Both the standard \textit{Insight}-HXMT background model \citep{Liao2020} and the OFF spectrum have been used to estimate the background emission. We tried different phenomenological models available in \texttt{XSPEC} to fit the main and minor peaks. When using the standard \textit{Insight}-HXMT background model, the best-fit models for the main and minor peaks are \texttt{Tbabs*(cutoffpl+bbodyrad+bbodyrad+Gaussian)} and \texttt{Tbabs*(cutoffpl+bbodyrad+Gaussian)}, respectively. When using the OFF spectrum as background, the best-fit models for the main and minor peaks are \texttt{Tbabs*(cutoffpl+bbodyrad)} and \texttt{Tbabs*cutoffpl}, respectively. The blackbody component for the main peak, about 1.4\% of the total flux, has a temperature of $kT \sim 2$\,keV and a size of $\sim 3$\,km, possibly accounting for the thermal emission from a hot spot on the NS surface. When using the \textit{Insight}-HXMT background model, an additional thermal component ($kT_{\rm bb} \sim 0.2$\,keV, $R_{\rm BB} \lesssim 10^3$\,km), about 0.13\% of the total flux, is required for the spectra of both the main and minor peaks. By considering a magnetospheric radius $R_{\rm m} \sim$1000 km with a dipole magnetic field of 4.8$\times10^{12}$ G (see Section \ref{statetrant} for details), this thermal component is likely from the inner part of the accretion disk truncated by the magnetosphere. The inner disk is illuminated and heated up by the fan beam emission of the accretion column (AC), and then can emit thermal X-ray photons. Moreover, since the disk component appears equally in each phase, it should be eliminated when using the OFF spectrum as background, consistent with our spectral fitting results. The best-fit parameters are presented in Tables~\ref{table:fitbigpeak}-\ref{table:fitsmallpeak} and the spectra are shown in Figure~\ref{fig:bigspec}-\ref{fig:smallspec}. \section{Discussion} \label{discuss} \subsection{Emission pattern} \label{pattern} In accreting X-ray pulsars, the accretion disk is truncated at the magnetospheric radius ($R_{\rm m}$) where the ram pressure of the matter in the disk is balanced by the magnetic pressure. After that, the matter is guided by the magnetic field lines and funnelled on to polar caps of the NS to generate X-rays. It is believed that the radiation structure near the surface of the NS depends on luminosity and accretion rate \citep{Basko1975, Basko1976,Becker2007, Becker2012, Mushtukov2015b}. When the source is faint (subcritical accretion), most of the radiation escapes along the magnetic field lines forming a ``pencil beam" pattern \citep{Basko1975,Burnard1991,Nelson1993,Becker2012}. When the source is brighter, an optically thick AC begins to rise above the polar cap, and the radiation escapes mostly from the wall of the AC, i.e., perpendicular to the magnetic field, forming a ``fan'' beam pattern \citep{Davidson1973,Basko1976,Becker2012,Mushtukov2015b}. The separation of the two accretion regimes is defined as the so-called ``critical luminosity" ($L_{\rm crit}$). However, the observed emission properties like pulse profiles, spectral shapes and cyclotron resonance scattering feature (CRSF) depend on luminosity, energy, the emission geometry, etc., thus implying a much more complex emission pattern than this simple pencil/fan beam scenario. As we have demonstrated (see Appendix \ref{ACmodel}), lower energy photons can contribute to both beam patterns, while higher energy photons will preferentially escape in the form of fan beam. Therefore, profiles at higher energies are more useful to discriminate the emission patterns. Applying the commonly used model of AC to RX J0209.6$-$7427, we can demonstrate that the $\sim$130 keV pulsation during the peak of its outburst can only originate from the fan beam of the AC (see Appendix \ref{ACmodel}), which does not have significant effect of beaming toward certain direction. This indicates that the observed luminosity of RX J0209.6$-$7427 is intrinsic. Therefore, the value of $L_{\rm crit}$ we identified from the profile, hardness and PF evolution is robust (see Section \ref{statetrant}), which in turn can result in a robust magnetic field estimation (see Section \ref{mag_estimate}). In addition, we identified a rapid decline of PF using \textit{Insight}-HXMT data with increasing energy above 50 keV (Figure~\ref{fig:profiles_PF}). Similar trend of PF decrease has been reported in the \textit{AstroSat} analysis of RX J0209.6$-$7427 as well \citep{Chandra2020}. We propose that this is mostly likely due to the increased fraction of reflection \citep{Poutanen2013} of higher energy photons off the NS surface. The reflected photons will smear/broaden the pulse peak and thus reduce the PF at high energies. A recent Monte-Carlo simulation shows that the reflected flux off a NS surface increases rapidly with energy and reaches a maximum just before the CRSF \citep{Kylafis2021} (see Fig. 2 in their paper, where the cyclotron energy was taken as around 25 keV). Therefore the reflection fraction increases with energy (below CRSF) and then at some points, it becomes dominant to reduce the observed PF. For RX J0209.6$-$7427, the CRSF energy is probably at around 200 keV or above (see Section~\ref{mag_estimate} for magnetic field estimates), though a CRSF has not been detected yet, most likely due to the limited sensitivity of the current X-ray telescopes above 100 keV. Therefore the PF decrease with energy above about 50 keV is consistent with the increased reflection fraction with increasing energy before reaching the CRSF energy. Actually, the simulations in \cite{Kylafis2021} did not consider the energy dependence of the emission geometry of the AC, as shown in Figure~\ref{fig:geometry}, where the emission region of higher energy photons in the form of fan beam is closer to the surface of the NS and thus can illuminate the NS surface even more easily. The rapid PF decrease with energy above around 50 keV (and below the possible CRSF energy) is likely due to the combination of the reflection physics simulated in \cite{Kylafis2021} and the geometrical effect illustrated in Figure~\ref{fig:geometry}. Further detailed modeling and simulation are required to fully explain the broad-band PF evolution with energy, which may provide further confirmation of the fan-beam nature of the high energy pulsed emission. \subsection{State transition} \label{statetrant} With the change of accretion rate, significant transitions of the spectral shape, pulse profile and cyclotron line evolution have been expected around $L_{\rm crit}$ in theory and observed in a large number of sources \citep{Sasaki2012,Reig2013,Postnov2015,Doroshenko2017,Wilson2018,Doroshenko2020,Ji2020,Doroshenko2020,Kong2020}. We have investigated the temporal and spectral properties of RX J0209.6$-$7427 and found significant changes of its hardness around 3.02$\pm$0.02 and (0.57$\pm$0.01)$\rm \times 10^{38}\,erg\ s^{-1}$ (Figures \ref{fig:hardness}, \ref{fig:nicerprofiles} and \ref{fig:profiles_PF}) during the outburst. Following the arguments in \cite{Reig2013}, the first turning luminosity of (3.02$\pm$0.02) $\times 10^{38}\,{\rm erg\ s}^{-1}$ which is the separation of the two branches in the HID may correspond to the transition from the supercritical to the subcritical regimes with the disappearance of the AC, similar to what was found in the Galactic PULX Swift J0243.6+6124 \citep{Doroshenko2020,Kong2020}. This tuning luminosity is therefore considered to be $L_{\rm crit}$. In addition, around this luminosity it seems that the evolution of the PF changes as well (Figure~\ref{fig:profiles_PF}), which is consistent with theoretical expectations \citep[e.g.,][]{Basko1976}. Under this interpretation, it is clear that the source was already in the supercritical state when \textit{NICER} started the series of observations. We propose that the second turning luminosity represents the transition from the subcritical regime to the state when the gas shock disappears or becomes non-dominant \citep{Becker2012}, while this was not found in Swift J0243.6+6124. On the other hand, we found a turning point where the \textit{NICER} PF evolution changes from increasing to decreasing (Figure \ref{fig:profiles_PF}, left panel) around 6 $\rm \times 10^{38}\,erg\ s^{-1}$. We suggest that this may correspond to the transition between the radiation pressure dominated (RPD) disk and the gas pressure dominated (GPD) disk as theoretically proposed \citep{Shakura1973,Mushtukov2015a}, similar to the case of Swift J0243.6+6124 \citep{Doroshenko2020,Kong2020}. Following the methodology presented in \cite{Doroshenko2020}, we can compare the magnetospheric radius \citep{Andersson2005,Mushtukov2015a,Campana2018,Monkkonen2019} \begin{equation} R_{\rm m} = 2.6\times 10^8\, k\, m^{1/7}\,R_{6}^{10/7}\,B_{12}^{4/7}\,L_{37}^{-2/7} \,\,\, \rm cm \end{equation} with the boundary between the GPD and RPD zones \citep{Monkkonen2019} \begin{equation} R_{\rm AB} = 10^7\, m^{1/3}\,\dot{M}_{17}^{16/21}\,\alpha^{2/21} \,\,\, \rm cm \, , \end{equation} where $m$, $R_{6}$, $B_{12}$, $L_{37}$ and $\dot{M}_{17}$ are the mass, radius, magnetic field, luminosity and accretion rate in units of $M_{\rm \odot}$, $10^6$ cm, $10^{12}$ G, $10^{37} \,\rm erg\ s^{-1}$ and $10^{17}\, \rm g\ s^{-1}$, respectively. We assume a typical value of $\alpha=1$. The parameter $k$ is model dependent and is defined as the ratio between $R_{\rm m}$ and the Alfv\'en radius ($R_{\rm A}=(\frac{\mu^4}{GM\dot{M}^2})^{1/7}$). The transitional luminosity from the GPD to the RPD disk is \citep{Andersson2005,Monkkonen2019} \begin{equation} L_{\rm AB} = 3\times 10^{38}\, k^{21/22}\, \alpha^{-1/11}\, m^{6/11}\, R_{6}^{7/11}\,B_{12}^{6/11} \,\,\, \rm erg\ s^{-1} \,\, . \end{equation} In this work, we consider that the NS has a radius of $R=10^6\,{\rm cm}$ and a mass of $1.4\,\rm M_{\odot}$. Using the dipole magnetic field of about 4.8$\times10^{12}$ G estimated from the GL model \citep{Ghosh1979} with $k=0.52$ (Section \ref{mag_estimate}), we got $R_{\rm AB}\sim R_{\rm m} \sim$ 1000 km which is indeed the condition for the transition, and $L_{\rm AB}=5.5\times10^{38} \,\rm erg\ s^{-1}$. This transitional luminosity is consistent with that inferred from the PF evolution. This interpretation should be, however, considered with caution, given that in Swift J0243.6+6124, both pulse profile and power spectrum changes have been observed to accompany the transition, while in RX J0209.6$-$7427 the power spectrum seems to be consistent with a single power law throughout the outburst and no breaks occurred. In comparison, 2S 1417$-$624 was proposed to transit between GPD and RPD based on the profile changes, but no significant power spectrum change has been detected either \citep{Ji2020b}; while in GRO J1744$-$28, a transition from the GPD disk to the RPD disk was proposed to explain the observed power spectrum change which differs dramatically from the canonical shape for a continuous GPD disk \citep{Monkkonen2019}. The power spectrum change of GRO J1744$-$28 is, however, different from what was found in Swift J0243.6+6124. Such transitions appear to be quite complicated and the origins of which are still not well known. Investigating the properties of the RPD disk in detail is beyond the scope of the paper. \subsection{Magnetic field estimates} \label{mag_estimate} Different approaches have been adopted to estimate the magnetic field of the NS in RX J0209.6$-$7427. (1) Relation between the magnetic field and $L_{\rm crit}$. In theory $L_{\rm crit}$ depends on the magnetic field. It is thus important to measure the value of $L_{\rm crit}$ accurately in order to estimate the magnetic field, which requires an accurate measurement of the distance. This is difficult for sources in the Milky Way but for RX J0209.6$-$7427, its known distance and much smaller relative distance uncertainty are a big advantage. In literature, there are several attempts to compute $L_{\rm crit}$ \citep{Wang1981b, Becker2012, Mushtukov2015b}. For example, \cite{Becker2012} proposed that $L_{\rm crit}\sim 1.49\times 10^{37}B_{12}^{16/15}{\rm erg\,s^{-1}}$. This suggests that the magnetic field of this source is $(1.68\pm0.01$)$\times10^{13}$\,G if $L_{\rm crit}$ is ($3.02\pm0.02$)$\times 10^{38}\,\rm erg\ s^{-1}$ as we determined in Section \ref{lcHID}. On the other hand, according to the model proposed by \cite{Mushtukov2015b} which takes into account different polarization modes, this $L_{\rm crit}$ value requires $E_{{\rm cyc}}>$200 keV, which results in a surface magnetic field strength of higher than 2.2$\times10^{13}$ G \citep[see Figure 5 in][private communication with Alexander Mushtukov]{Mushtukov2015b}. This is similar to the findings in SMC X-3 \citep{Tsygankov2017}. (2) The magnetic field of the source can be investigated according to the spin-up during the outburst. Assuming the GL model \citep{Ghosh1979}, the spin-up rate of the NS can be written as: \begin{equation} \dot{\nu} = 2^{-15/14}k^{1/2}\mu^{2/7}(GM)^{-3/7}(I\pi)^{-1}R^{6/7}L^{6/7}n(\omega)\ {\rm Hz\,s^{-1}} \, , \end{equation} where $I=\frac{2}{5}MR^2$, $\mu=\frac{1}{2}BR^3$ and $k$ are the moment of inertia, the magnetic dipole moment, and the dimensionless constant introduced previously. We note that $n(\omega)$ is related to the fastness parameter $\omega \equiv (R_{\rm m}/R_{\rm co})^{3/2}$, where $R_{\rm co} \equiv(GMP^2/4\pi^2)^{1/3}$ is the co-rotation radius: \begin{equation} n(\omega) \approx1.39\frac{1-\omega \left[4.03(1-\omega)^{0.173}-0.878\right]}{1-\omega} . \end{equation} $k$ is usually assumed to be $0.5-1$ for a geometrically thin accretion disk. However, different values have also been proposed \citep{Bozzo2018,Monkkonen2019,Chashkina2019, Doroshenko2020,Ji2020}. We show the fitting result in Figure~\ref{fig:fit}, which indicates that the GL model can well describe the spin evolution of the source ({$\chi^2=110.7, 82\ dof$}) and the inferred magnetic field is ($5.1\pm0.2$)$\times10^{12}\,(\frac{k}{0.5})^{-7/4}$\,G. Using $k=0.52$ as proposed in the GL model, $B$ is then ($4.8\pm0.2)\times10^{12}$ G which is close to an order of magnitude lower than that derived from $L_{\rm crit}$. In literature, there are several calculations on the fastness parameter under different conditions \citep{Ghosh1979,Wang1987,Wang1995,Kluzniak2007}, which however do not have a significant influence on the conclusion for a slow rotator ($\omega\sim0$) like RX J0209.6$-$7427. In comparison, the \cite{Wang1995} model has the spin-up rate as: \begin{equation} \dot{\nu} = \dfrac{n(\omega)}{2\pi I}\dot{M}(GM R_{\rm m})^{1/2} \, , \label{torqueWang} \end{equation} with $n(\omega)\approx 7/6$. If taking the same definitions of $R_{\rm A}$ and $\mu$, then the \cite{Wang1995} model will predict a magnetic field a factor of 1.8 larger than the GL model, no matter what value of $k$ is used. Unlike the Galactic X-ray pulsars whose distances usually have large uncertainties, the distance of RX J0209.6$-$7427 is well known and with small relative uncertainty of 5\%, thus resulting in a relative uncertainty on the measured luminosity of 10\%. This will cause a relative uncertainty of 9\% on the estimated magnetic field using $L_{\rm crit}$, and of 30\% using the spin evolution approach. On the other hand, the measurement uncertainty on the luminosity from our analysis is negligible, compared to the uncertainties induced by different estimation methods and from the distance. In summary, the magnetic field strength inferred from $L_{\rm crit}$ is $(1.7-2.2)\times10^{13}$ G, and from different torque models is $(4.8-8.6)\times10^{12}$ G. Even after taking into account uncertainties of both methods and those from the distance and the measurements, we still got inconsistent magnetic field strength for RX J0209.6$-$7427. Similar discrepancy between different approaches has also been reported for SMC X-3 and Swift J0243.6+6124. Possible existence of multipole magnetic fields was suggested for SMC X-3 \citep{Tsygankov2017} and Her X-1 \citep{Monkkonen2022}. For Swift J0243.6+6124, \cite{Doroshenko2020} proposed that either a small magnetosphere size with $k\sim 0.1-0.2$ or the presence of multipole strong magnetic field can explain the discrepancy. The recent discovery of a spin phase-dependent CRSF with high statistical significance at energies around 120$-$146 keV from Swift J0243.6+6124 has lead to a magnetic field estimate of $\sim 1.6\times 10^{13}$ G, which is an order of magnitude higher than those estimated from other methods, such as the spin evolution, but is in excellent agreement with that estimated from $L_{\rm crit}$. The authors claimed that the observed CRSF traces the multipole component of the field which dominates the field in the vicinity of the NS’s surface \citep{Kong2022} thus supports the latter scenario in \cite{Doroshenko2020}. Although the \textit{Insight}-HXMT background model uncertainty does not allow us to perform reliable spectral fit for the much fainter RX J0209.6$-$7427 at above 50 keV to investigate the possible CRSF at around 100 keV, which may have caused the dip at phase around 0.55 in the pulse profile of 80$-$130 keV band, the magnetic fields we estimated for RX J0209.6$-$7427 using the above two methods are very similar to Swift J0243.6+6124. Therefore, multipole field may also be a solution to the discrepancy between different magnetic field estimation methods for RX J0209.6$-$7427. It is thus intriguing to ask this question: Are multipole fields common on the surfaces of NSs? Indeed, for some magnetars there are large differences between their magnetic fields measured from their observed spin-down rates and that inferred from their magnetar-unique behaviours (e.g., prolific glitches, soft gamma-ray flares, including some rare giant flares, etc.); these are commonly interpreted as due to the existence of multipole magnetic fields. Among the more distant extragalactic PULXs, a multipole strong magnetic field was suggested to interpret the properties of NGC 5907 ULX-1 \citep{Israel2017a}. Further observational and theoretical studies on the different aspects of the pulsational properties of accreting X-ray pulsars, especially those with high luminosity, are needed to fully understand the topology of the magnetic fields of these NSs. \section{Conclusion} \label{conclude} RX J0209.6$-$7427 is a new transient PULX in the SMC, whose ULX nature was identified during its super-Eddington outburst in 2019 November. The analysis of the broad band \textit{Insight}-HXMT data revealed its pulsed emission up to 180 keV and certainly above 130 keV, and we demonstrated that this emission is from the fan beam pattern of the accretion column. This is the highest energy pulsation detected so far from all PULXs not in the Milky Way. With the more accurately determined distance than that of the Galactic X-ray pulsars, we show that its peak luminosity of ($1.1\pm 0.06)\times 10^{39}\, \rm erg\ s^{-1}$ is intrinsic rather than highly beamed toward a certain direction. Moreover, the longer span \textit{NICER} monitoring data suggested a state transition from subcritical to supercritical accretion regime occurred at around $3\times10^{38}\, \rm erg\ s^{-1}$, which allowed us to estimate the surface magnetic field of RX J0209.6$-$7427 to be $(1.7-2.2)\times10^{13}$ G However, from its spin evolution we obtained a magnetic field about one order of magnitude lower. We interpreted the two values of magnetic fields as the dipole and multipole magnetic fields of the neutron star, similar to what has been proposed for the Galactic PULX Swift J0243.6+6124. Although there are still some detailed differences between the two closest PULXs, the striking similarity between them may suggest a common nature of their neutron stars and their ultraluminous emissions. This may shed light on better understanding of the nature of more distant extragalactic PULXs. \acknowledgments We thank Alexander Mushtukov for discussions on estimating the NS surface magnetic field from the observed very high critical luminosity, as well as the scenario of multipole surface magnetic field. This work made use of the data from the \textit{Insight}-HXMT mission, a project funded by the China National Space Administration (CNSA) and the Chinese Academy of Sciences (CAS). This research also made use of data obtained with the \textit{NICER}, a NASA experiment placed on the International Space Station (ISS). The \textit{Insight}-HXMT team gratefully acknowledges the support from the National Program on Key Research and Development Project (grant No. 2021YFA0718500) from the Ministry of Science and Technology of China (MOST). The authors are thankful for support from the National Natural Science Foundation of China under grants U1938103, 12041303, U1938109, U1838202, U1838201, U1838115, U1838104, 12073029, U1838107, U1938201 and U2038101. X.H. is supported by the Light of West China Program of the CAS. L.J. is supported by the Guangdong Major Project of Basic and Applied Basic Research (Grant No. 2019B030302001). \appendix \section{Determine the highest energy pulsation} \label{highestE} We apply the cross correlation technique to determine the highest energy pulsation of RX J0209.6$-$7427 using \textit{Insight}-HXMT data. We consider HE profiles in the energy range of 20$-$250 keV. We first calculate the cross correlation of a given HE profile with the 1$-$10 keV LE profile. Then we use Monte-Carlo method to simulate the cross correlation distribution between two non-correlated profiles: \begin{itemize} \item The first profile is generated by performing Poisson sampling of a flat profile; the mean of the simulated profile is taken as that of each observed HE profile. \item The second profile is generated by performing Poisson sampling for each bin of the 1$-$10 keV LE profile. \item Calculate the cross correlation for the two profiles. \item The simulation has been repeated for $10^6$ times to obtain the distribution of the cross correlation. \end{itemize} Figure~\ref{fig:simuCross} shows a few examples of the cross correlation of HE profiles and the distribution of the cross correlation from simulated profiles. To reduce the window effect, both the data and simulated profiles have been extended to 3 spin periods (nbins in Figure~\ref{fig:simuCross}). The number of bins was chosen arbitrary, but we have tested different binning of 10, 20, 30, 40, 50, 80, 100 per periods, all resulting in similar significance of the given HE pulse profile. The blue line in the plot of distribution indicates the cross correlation of a given HE profile. The significance of such cross correlation can then be evaluated by calculating the proportion of the part on the right of the blue line, i.e., the chance probability ($p$-value) that the result is obtained by chance given that the null-hypothesis (the given HE profile is flat) is true. We show the simulated cross correlations and chance probabilities in Figure~\ref{fig:flatproba}. We find that the pulsation is detected up to the 130$-$180 keV band with a significance of 4.3$\sigma$. In addition, we note that the phase lag (upper panel of each example in Figure~\ref{fig:simuCross}) of each HE profile relative to the 1$-$10 keV LE profile is close to zero, which means that the broad band pulse profiles of RX J0209.6$-$7427 are all in phase. \section{Model of accretion column and application to RX J0209.6$-$7427} \label{ACmodel} We now demonstrate that the $\sim$130 keV pulsation of RX J0209.6$-$7427 can only originate from the fan beam model which does not have significant beaming effect; this conclusion is robust, since it is less dependent on the exact assumptions of the AC and NS properties. In the AC, the factor which determines whether a photon can escape from the AC is the optical depth: \begin{equation} \tau = n_{\rm e} \sigma l \, , \end{equation} with $n_{\rm e}$ the electron number density, $\sigma$ the scattering cross section between photons and electrons and $l$ the photon propagation length. The scattering cross section depends on the photon energy, the magnetic field, the polarization state and the photon direction momentum. We are interested in photons propagating parallelly and perpendicularly to the magnetic field. We have \citep{Arons1987}: \begin{enumerate} \item $E<E_{\rm c}$ \begin{itemize} \item O-Mode \begin{equation} \begin{aligned} \sigma_{\parallel} = \sigma_{\rm T} (E/E_{\rm c})^2 \, , \\ \sigma_{\perp} = \sigma_{\rm T} \, , \end{aligned} \end{equation} \item X-Mode \begin{equation} \sigma_{\rm X} = \sigma_{\rm T} (E/E_{\rm c})^2 \, , \end{equation} \end{itemize} \item $E \geq E_{\rm c}$ \begin{equation} \sigma_{\rm O} = \sigma_{\rm X} = \sigma_{\rm T} \, , \end{equation} \end{enumerate} where O-mode is the ordinary polarization mode with the electric vector in the plane of the magnetic field $\boldsymbol{B}$ and the photon momentum $ \boldsymbol{\hbar k}$, X-mode is the extraordinary polarization mode with the electric vector perpendicular to the plane of ($\boldsymbol{B}, \boldsymbol{k}$), $\sigma_{\rm T} $ is the Thomson scattering cross section, and $E_{\rm c}=11.6B_{12}/(1+z)$ is the CRSF line energy. $B_{12}$ is the magnetic field in units of $10^{12}$ G and $z$ is the gravitational redshift, usually between 0 and 0.3 in the case of NSs. For $n_{\rm e}$ and $l$, we make use of the analytical expressions in \cite{Becker2012}. In the supercritical high luminosity case, a radiation-dominated shock is formed above the NS whose height increases with increasing luminosity. $n_{\rm e}$ is calculated from the mass conservation relation (Eq. 12) and by assuming that the inflow velocity in the AC equals to the post-shock velocity (Eq. 4) which is $1/7$ of the free-fall velocity of the accreting matter approaching the top of the radiation-dominated shock, and that all the matter are accreted on to the NS ($L_{X} = GM\dot{M}/R$). The photon propagation length $l$ in the direction parallel to the magnetic field can be estimated by the height of the radiation-dominated shock $H$ (Eq. 16). In the direction perpendicular to the magnetic field, $l$ is estimated by the AC radius $r_{0}$ (Eq. 23). In our calculations, we took typical values for involved parameters as shown in \cite{Becker2012}: $M=1.4\,M_{\rm \odot}$, $R=10$ km, $\xi = 0.01$, $\Lambda=0.1$ and $\tau_{*}=20$. The critical luminosity can be calculated as $L_{\rm crit} = 1.49\times 10^{37} B_{12}^{16/15}$. We consider different magnetic field strengths of $10^{11}$, $10^{12}$ and $10^{13}$ G and assuming $z=0$ (we verified that taking $z=0.3$ does not affect our conclusion). The corresponding $L_{\rm crit}$ is then $1.5\times 10^{36}$, $1.5\times 10^{37}$ and $1.5\times 10^{38} \, \rm erg \, s^{-1}$, respectively. From our analysis, RX J0209.6$-$7427 is in the supercritical state before 58880 MJD for all three values of the magnetic field, thus it is appropriate to apply the equations aforementioned to calculate the optical depth. We consider the case of peak luminosity of $10^{39} \, \rm erg \, s^{-1}$. The results are shown in Table~\ref{table:optdepth}. Three conclusions can be drawn from the results in the case of supercritical high luminosity state (Figure~\ref{fig:geometry}): (1) For photons of 130 keV, no matter what kind of polarization mode they have, the optical depth in the direction parallel or perpendicular to the magnetic field is always $\gg1$, thus they can not escape directly from the core of the AC. However, it is more appropriate to consider a thin shell in the direction perpendicular to the magnetic field, i.e., the wall of the AC. Setting $\tau_{\perp,\rm wall}=1$ then gives the shell thickness $dr$ for photons to be able to just escape from the wall. We can see that $dr$ is far less than $H$ in the direction parallel to the magnetic field and the AC radius $r_{0}$. So the photons of 130 keV can, and can only escape from the wall of the AC, forming a fan beam. (2) For photons of 1 keV, if it is O-mode, then they can escape from the wall of the AC, similar to the 130 keV photons since $\tau_{\perp}$ is independent of photon energy. If it is X-mode, in the case of a strong magnetic field as $10^{13}$ G, the cross section in the direction perpendicular to the magnetic field is largely reduced, so that the optical depth $\tau_{\perp,\rm X}\ll 1$ and the photons can also escape without problem. This can be inferred as well from the shell thickness $dr$ which in this case is already larger than the column radius $r_{0}$. This implies that the AC is optically thin and photons from the center of the column can escape. So the 1 keV photons can form a fan beam as the 130 keV photons do. On the other hand, the optical depth $\tau_{\parallel}$ for O-mode photons is still larger than unity with a strong magnetic field of $10^{13}$ G, so the photons can not escape. Nevertheless, $\tau_{\parallel}$ can be reduced to less than unity if the magnetic field is increased by two times. Besides, the ratio of $\tau_{\parallel}$ to $\tau_{\perp}$ is $\ll 1$ for $B\sim 10^{12-13}$ G, suggesting that photons will escape more easily from the direction parallel to the magnetic field. In this case, the 1 keV photons can also escape from the top of the AC along the magnetic field lines in a pencil beam pattern. (3) For photons with intermediate energy like 40 keV, they can escape from the wall of the AC as explained for the 130 keV and 1 keV photons to form a fan beam. However, for O-mode photons, $\tau_{\parallel}$ is comparable to $\tau_{\perp}$ with a ratio of 7 for a magnetic field of $10^{13}$ G. In this case, pencil beam can not be excluded. Considering the simplification of the model, low and intermediate energy photons probably can also escape from the pencil beam for $B\sim 10^{12-13}$ G even at $L\sim 10^{39}\, \rm erg\ s^{-1}$, though with significant attenuation. Before 58880 MJD when RX J0209.6$-$7427 was in the supercritical luminosity state, its pulse profile exhibits one main peak and one minor peak at lower energies separated by $\sim$0.5 in phase, while at above 27 keV, only the main peak is prominent reaching up to 180 keV and certainly above 130 keV. We have demonstrated that the 130 keV photons can only originate from the fan beam, therefore the main peak is fan beam in origin. It is not surprising that the main peak is also visible at lower energies, since we have demonstrated that both 130 keV and 1 keV photons can form a fan beam. \bibliography{RXJ0209.bib}{} \bibliographystyle{aasjournal} \begin{table}[ht!] \small \caption{The orbit and spin parameters of RX J0209.6$-$7427.} \label{table:lcfit} \begin{center} \begin{tabular}{l l l c c} \hline \hline Parameters & Value \\ \hline $P_{\rm orb}$ (day) & $47.97\pm0.16$ \\ $a\rm sin$ $i$ (light-sec) & $168\pm3$ \\ $e$ & $0.317\pm0.007$ \\ $\omega$ & $-98.5\pm3.8$ \\ $T_{\omega}$ (MJD) & $58782.7\pm0.2$\\ \hline PEPOCH (MJD) & $58800$\\ $\nu_{0}\, \rm{(Hz)} $ & $0.107557(8)$ \\ $\nu_{1}\, (10^{-10}\, \rm Hz\, s^{-1}$) & $-1.66\pm0.15$ \\ $\nu_{2}\, (10^{-16}\, \rm Hz\, s^{-2}$) & $3.14\pm0.20$ \\ $\nu_{3}\, (10^{-22}\, \rm Hz\, s^{-3}$) & $-2.25\pm0.18$ \\ $\nu_{4}\, (10^{-29}\, \rm Hz\, s^{-4}$) & $1.01\pm0.11$ \\ $\nu_{5}\, (10^{-35}\, \rm Hz\, s^{-5}$) & $-2.68\pm0.38$ \\ $\nu_{6}\, (10^{-42}\, \rm Hz\, s^{-6}$) & $3.28\pm0.65$ \\ \hline $\chi^{2}$ & 0.992(76)\\ \hline \end{tabular} \end{center} \end{table} \begin{table}[ht!] \scriptsize \begin{center} \begin{threeparttable} \caption{\insight{} best-fit spectral results for the 5-observation combined phase-averaged spectrum.} \label{table:fitave} \begin{tabular}{lcc} \toprule Component & Parameters & \insight{} background model \\[0.8ex] \midrule Tbabs & $N_{\rm H} \, (10^{22} \, \rm cm^{-2})$ & $0.158$ \\[0.8ex] cufoffpl & $\Gamma$ & $0.72^{+0.01}_{-0.01}$ \\[0.8ex] & $E_{\rm cut}$ (keV) & $11.38^{+0.12}_{-0.11}$ \\[0.8ex] & norm ($10^{-2}$) & $9.74^{+0.09}_{-0.09}$ \\[0.8ex] bbodyrad & $kT$ (keV) & $0.18^{+0.01}_{-0.01}$ \\[0.8ex] & norm & $18400^{+3810}_{-3050}$ \\[0.8ex] & $R_{\rm BB}$ (km)$^{a}$ & $746^{+340}_{-304}$ \\[0.8ex] Gaussian & $E_{\rm Fe}$ (keV) & $6.58^{+0.03}_{-0.03}$ \\[0.8ex] &$\sigma_{\rm Fe}$ (keV) & $0.24^{+0.05}_{-0.05}$ \\[0.8ex] & norm ($10^{-3}\, \rm cm^{-2} \, s^{-1}$) & $1.11^{+0.15}_{-0.14}$ \\[0.8ex] \hline & $\chi^{2}/d.o.f$ & $1464/1323$ \\ & $F_{\rm X}\,(10^{-9} \, \rm cm^{-2} \, s^{-1})$ & $3.06^{+0.02}_{-0.02}$ \\[0.8ex] & $L_{\rm X}\,(10^{38} \, \rm erg \, s^{-1})^{b}$ & $11.08^{+0.06}_{-0.06}$ \\ \bottomrule \end{tabular} \begin{tablenotes} \scriptsize \item{}{$^{a}$ The black body component radius was estimated from the normalization of the model assuming a distance to the SMC of 55 kpc (i.e., $D_{10}=5.5$).} \item{}{$^{b}$ Luminosity was calculated in the energy band of 1$-$150 keV.} \vspace{0.5cm} \item {}{\textit{Notes.} The fits are performed in the energy bands of $1-9$ keV (LE), $8-30$ keV (ME) and $28-50$ keV (HE). The best-fit model is \texttt{Tbabs*(cutoffpl+bbodyrad+Gaussian)}. All the errors are given at the 68\% confidence level.} \end{tablenotes} \end{threeparttable} \end{center} \end{table} \begin{table}[ht!] \scriptsize \begin{center} \begin{threeparttable} \caption{\insight{} best-fit spectral results for the 5-observation combined main peak. } \label{table:fitbigpeak} \begin{tabular}{lccc} \toprule Component & Parameters & OFF spectrum as background & \insight{} background model \\[0.8ex] \hline Tbabs & $N_{\rm H} \, (10^{22} \, \rm cm^{-2})$ & $0.158$ & $0.158$ \\[0.8ex] cufoffpl & $\Gamma$ & $0.64^{+0.03}_{-0.03}$ & $0.71^{+0.01}_{-0.01}$ \\[0.8ex] & $E_{\rm cut}$ (keV) & $12.48^{+0.39}_{-0.37}$ & $11.70^{+0.16}_{-0.14}$ \\[0.8ex] & norm ($10^{-2}$) & $5.78^{+0.16}_{-0.16}$ & $11.24^{+0.08}_{-0.09}$ \\[0.8ex] bbodyrad & $kT$ (keV) & $2.17^{+0.48}_{-0.36}$ & $1.47^{+1.80}_{-1.16}$ \\[0.8ex] & norm & $0.22^{+0.25}_{-0.15}$ & $0.36^{+0.74}_{-0.07}$ \\[0.8ex] & $R_{\rm BB}$ (km)$^{a}$ & $2.6^{+2.7}_{-2.1}$ & $3.3^{+4.7}_{-1.4}$ \\[0.8ex] bbodyrad & $kT$ (keV) & ... & $0.16^{+0.01}_{-0.01}$ \\[0.8ex] & norm & ... & $28500^{+9090}_{-6600}$ \\[0.8ex] & $R_{\rm BB}$ (km)$^{a}$ & ... & $929^{+524}_{-447}$ \\[0.8ex] Gaussian & $E_{\rm Fe}$ (keV) & ... & $6.61^{+0.04}_{-0.04}$ \\[0.8ex] & $\sigma_{\rm Fe}$ (keV) & ... & $0.21^{+0.06}_{-0.05}$ \\[0.8ex] & norm ($10^{-3}\, \rm cm^{-2} \, \rm s^{-1}$) & ... & $1.14^{+0.13}_{-0.13}$ \\[0.8ex] \hline & $\chi^{2}/d.o.f$ & $1461/1395$ &$1477/1325$ \\ & $F_{\rm X}\,(10^{-9}\, \rm cm^{-2} \,s^{-1})$ & $2.51^{+0.04}_{-0.09}$ & $3.68^{+0.02}_{-0.02}$ \\[0.8ex] & $L_{\rm X}\,(10^{38}\, \rm erg \, s^{-1})^{b}$ & $9.08^{+0.15}_{-0.32}$ & $13.31^{+0.08}_{-0.09}$ \\ \bottomrule \end{tabular} \begin{tablenotes} \scriptsize \item{}{$^{a}$ The black body component radius is estimated from the normalization of the model assuming a distance to the SMC of 55 kpc (i.e., $D_{10}=5.5$).} \item{}{$^{b}$ Luminosity is calculated in the energy band of 1$-$150 keV.} \vspace{0.5cm} \item{}{\textit{Notes.} The fits are performed in the energy bands of $1-9$ keV (LE), $8-30$ keV (ME) and $28-150$ keV (HE) when using OFF spectrum as the background model, and in the energy bands of $1-9$ keV (LE), $8-30$ keV (ME) and $28-50$ keV (HE) when using the \insight{} background model. The best-fit models for the 5-observation combined spectra using the OFF spectrum and \insight{} background model are \texttt{Tbabs*(cutoffpl+bbodyrad)} and \texttt{Tbabs*(cutoffpl+bbodyrad+bbodyrad+Gaussian)}, respectively. All the errors are given at the 68\% confidence level.} \end{tablenotes} \end{threeparttable} \end{center} \end{table} \begin{table}[ht!] \scriptsize \begin{center} \begin{threeparttable} \caption{\insight{} best-fit spectral results for the 5-observation combined minor peak.} \label{table:fitsmallpeak} \begin{tabular}{lccc} \toprule Component & Parameters & OFF spectrum as background & \insight{} background model \\[0.8ex] \hline Tbabs & $N_{\rm H} \, (10^{22} \, \rm cm^{-2})$ & $0.158$ & $0.158$ \\[0.8ex] cufoffpl & $\Gamma$ & $1.15^{+0.09}_{-0.10}$ & $0.80^{+0.02}_{-0.02}$ \\[0.8ex] & $E_{\rm cut}$ (keV) & $22.17^{+12.95}_{-6.29}$ & $10.86^{+0.30}_{-0.28}$ \\[0.8ex] & norm ($10^{-2}$) & $2.78^{+0.21}_{-0.20}$ & $7.42^{+0.19}_{-0.19}$ \\[0.8ex] bbodyrad & $kT$ (keV) &... & $0.19^{+0.01}_{-0.01}$ \\[0.8ex] & norm &... & $12400^{+3820}_{-2740}$ \\[0.8ex] & $R_{\rm BB}$ (km)$^{a}$ &... & $612^{+340}_{-288}$ \\[0.8ex] Gaussian & $E_{\rm Fe}$ (keV) & ... & $6.48^{+0.07}_{-0.07}$ \\[0.8ex] &$\sigma_{\rm Fe}$ (keV) & ... & $0.33^{+0.09}_{-0.07}$ \\[0.8ex] & norm ($10^{-3}\, \rm cm^{-2} \, s^{-1}$) & ... & $1.17^{+0.27}_{-0.24}$ \\[0.8ex] \hline & $\chi^{2}/d.o.f$ & $1380/1314$ &$1362/1309$ \\ & $F_{\rm X}\,(10^{-9} \, \rm cm^{-2} \, s^{-1})$ & $0.61^{+0.08}_{-0.14}$ & $1.84^{+0.02}_{-0.03}$ \\[0.8ex] & $L_{\rm X}\,(10^{38} \, \rm erg \, s^{-1})^{b}$ & $2.21^{+0.29}_{-0.51}$ & $6.67^{+0.06}_{-0.10}$ \\ \bottomrule \end{tabular} \begin{tablenotes} \scriptsize \item{}{$^{a}$ The black body component radius is estimated from the normalization of the model assuming a distance to the SMC of 55 kpc (i.e., $D_{10}=5.5$).} \item{}{$^{b}$ Luminosity is calculated in the energy band of 1$-$150 keV.} \vspace{0.5cm} \item{}{\textit{Notes.} The fits are performed in the energy bands of $1-9$ keV (LE) and $8-30$ keV (ME). The best-fit models for the 5-observation combined spectra using the OFF spectrum and \insight{} background model are \texttt{Tbabs*cutoffpl} and \texttt{Tbabs*(cutoffpl+bbodyrad+Gaussian)}, respectively. All the errors are given at the 68\% confidence level.} \end{tablenotes} \end{threeparttable} \end{center} \end{table} \begin{table}[ht!] \begin{center} \begin{threeparttable} \caption{Optical depth calculation for different polarization modes, magnetic fields and photon energies. } \label{table:optdepth} \begin{tabular}{lccccccc} \toprule \hline $E \ \rm (keV)$ & $B_{12}$ & $\tau_{\parallel}$ & $\tau_{\perp}$ &$\tau_{\parallel}/\tau_{\perp}$ &$r_{0} \ \rm (cm)$ & $dr \ \rm (cm)$ & $dr/r_{0}$ \\ \midrule \multicolumn{8}{c}{O-mode} \\ \hline $130$ & $0.1$ & $5.5\times10^{3}$ & $3.4\times10^{2}$ & $16$ & $7\times10^{5}$ & $2\times10^{3}$ &$0.003$ \\ & $1$ & $2\times10^{4}$ & $6.6\times10^{2}$ & $30$ & $4\times10^{5}$ & $6\times10^{2}$ &$0.002$ \\ & $10$ &$7.6\times10^{4}$ & $1.3\times10^{3}$ & $58$ & $2\times10^{5}$ & $2\times10^{2}$ &$0.001$ \\ $40$ & $0.1$ & $5.5\times10^{3}$ & $3.4\times10^{2}$ & $16$ & $7\times10^{5}$ & $2\times10^{3}$ &$0.003$ \\ & $1$ & $2\times10^{4}$ & $6.6\times10^{2}$ & $30$ & $4\times10^{5}$ & $6\times10^{2}$ &$0.002$ \\ & $10$ &$9\times10^{3}$ & $1.3\times10^{3}$ & $7$ & $2\times10^{5}$ & $2\times10^{2}$ &$0.001$ \\ $1$ & $0.1$ & $4\times10^{3}$ & $3.4\times10^{2}$ & $12$ & $7\times10^{5}$ & $2\times10^{3}$ &$0.003$ \\ & $1$ & $1.5\times10^{2}$ &$6.6\times10^{2}$ & $0.2$ & $4\times10^{5}$ & $6\times10^{2}$ &$0.002$ \\ & $10$ & $5.6$ & $1.3\times10^{3}$ & $0.004$ & $2\times10^{5}$ & $2\times10^{2}$ &$0.001$ \\ \hline \multicolumn{8}{c}{X-mode} \\ \hline $130$ & $0.1$ & $5.5\times10^{3}$ & $3.4\times10^{2}$ & $16$ & $7\times10^{5}$ & $2\times10^{3}$ &$0.003$ \\ & $1$ & $2\times10^{4}$ & $6.6\times10^{2}$ & $30$ & $4\times10^{5}$ & $6\times10^{2}$ &$0.002$ \\ & $10$ & $7.6\times10^{4}$ & $1.3\times10^{3}$ & $58$ & $2\times10^{5}$ & $2\times10^{2}$ &$0.001$ \\ $40$ & $0.1$ & $5.5\times10^{3}$ & $3.4\times10^{2}$ & $16$ & $7\times10^{5}$ & $2\times10^{3}$ &$0.003$ \\ & $1$ & $2\times10^{4}$ & $6.6\times10^{2}$ & $30$ & $4\times10^{5}$ & $6\times10^{2}$ &$0.002$ \\ & $10$ & $9\times10^{3}$ & $1.5\times10^{2}$ & $60$ & $2\times10^{5}$ & $1\times10^{3}$ &$0.007$ \\ $1$ & $0.1$ & $4.1\times10^{3}$ & $2.6\times10^{2}$ & $16$ & $7\times10^{5}$ & $3\times10^{3}$ &$0.004$ \\ & $1$ & $1.5\times10^{2}$ & $5$ & $30$ & $4\times10^{5}$ & $8\times10^{4}$ &$0.203$ \\ & $10$ & $5.6$ & $0.1$ & $56$ & $2\times10^{5}$ & $2\times10^{6}$ &$10.492$ \\ \bottomrule \end{tabular} \begin{tablenotes} \scriptsize \item \textit{Notes.} The photon propagation length $H$ in the direction parallel to the magnetic field lines depends only on luminosity and is $1\times10^{7}$ cm for $10^{39} \, \rm erg \, s^{-1}$ considered here. \end{tablenotes} \end{threeparttable} \end{center} \end{table}

Title: DESI Observations of the Andromeda Galaxy: Revealing the Immigration History of our Nearest Neighbor

Abstract: We present DESI observations of the inner halo of M31, which reveal the kinematics of a recent merger - a galactic immigration event - in exquisite detail. Of the 11,416 sources studied in 3.75 hours of on-sky exposure time, 7,438 are M31 sources with well measured radial velocities. The observations reveal intricate coherent kinematic structure in the positions and velocities of individual stars: streams, wedges, and chevrons. While hints of coherent structures have been previously detected in M31, this is the first time they have been seen with such detail and clarity in a galaxy beyond the Milky Way. We find clear kinematic evidence for shell structures in the Giant Stellar Stream, the NE Shelf and Western Shelf regions. The kinematics are remarkably similar to the predictions of dynamical models constructed to explain the spatial morphology of the inner halo. The results are consistent with the interpretation that much of the substructure in the inner halo of M31 is produced by a single galactic immigration event 1 - 2 Gyr ago. Significant numbers of metal-rich stars are present in all of the detected substructures, suggesting that the immigrating galaxy had an extended star formation history. We also investigate the ability of the shells and Giant Stellar Stream to constrain the gravitational potential of M31, and estimate the mass within a projected radius of 125 kpc to be ${\rm log_{10}}\, M_{\rm NFW}(<125\,{\rm kpc})/M_\odot = 11.78_{-0.10}^{+0.13}$. The results herald a new era in our ability to study stars on a galactic scale and the immigration histories of galaxies.

https://export.arxiv.org/pdf/2208.11683

\begin{CJK*}{UTF8}{gbsn} \title{DESI Observations of the Andromeda Galaxy: Revealing the Immigration History of our Nearest Neighbor} \input{author_list} \keywords{Andromeda Galaxy, Galaxy mergers, Galaxy evolution, Galaxy dynamics, Stellar kinematics, Redshift surveys, Radial velocity, Catalogs} \section{Introduction} \label{sec:introduction} The histories of galaxies have much in common with that of the United States: in both cases, waves of immigration (of stars, people) have added to the existing inhabitants. In the process of galaxy assembly, smaller galaxies are expected to fall into larger galaxies and disperse their stars in a hierarchical merging process \citep{bullock2001,bullock_johnston_2005,Cooper2010}. How do we know this? In the case of immigration to the US, numerous documents, such as government records, can be used to reconstruct the historical movements of individuals and therefore large-scale migration patterns. Although no such records are available for galaxies, we can nevertheless reconstruct their immigration histories from the motions of their individual stars. Migrating stars merge into galaxies on cosmic timescales and we can expect to observe stars on their migration paths today; the record of their immigration ancestry preserved in phase space even for migration events that began billions of years ago. Discerning migration events (i.e., to identify coherent structure in the positions and motions of stars on galactic scales) requires measurements of large stellar samples over large areas. Previously prohibitive, such studies are now straightforward with the advent of highly multiplexed multi-object spectroscopy on telescopes with wide fields of view. M31, our closest large galactic neighbor, has a mass comparable to that of the Milky Way. Our location in the Milky Way offers a fortuitous vantage point from which to observe galactic migration in action in M31. While the Milky Way gives us an up-close (``on-stage'') view of the dynamics of a large spiral galaxy, our position within the disk of the Milky Way obscures large portions of the Galaxy from our view. In contrast, with our external (``upper balcony'') perspective on M31, it is straightforward to survey the entire galaxy for clues to its immigration history. The expected observational signatures of galactic migration include debris streams, shells, rings, and plumes, the expected outcomes of merger interactions between large galaxies and their companions \citep[e.g.,][]{bullock2001,bullock_johnston_2005,McConnachie2009,Cooper2010,Martinez-Delgado2010,Pop2018}. The detailed study of these features can help us reconstruct the assembly history of a galaxy as well as enable dynamical measurements of its mass distribution \citep[e.g.,][]{Merrifield1998,Ibata2004}. Both the Milky Way and M31 show signs of mergers. Photometric and kinematic studies of the Milky Way reveal complex substructure suggesting that the vast majority of the stars in the halo may have been accreted in past mergers \citep{bell2008,schlaufman_etal_2012,Naidu_etal_2020} with the inner halo dominated by a single merger event with a massive {\it Gaia}-Enceladus-Sausage galaxy \citep{Belokurov2018,Helmi2018} that happened around 8-11 Gyr ago \citep{Helmi2018,Gallart2019,Bonaca2020,XiangRix2022}. In addition, the Milky Way is currently in the process of assimilatibe the Sagittarius Dwarf Galaxy, a merger which had its first passage through the Milky Way disk about 5.7~Gy ago \citep[e.g.,][]{Ibata1994,Ruiz-Lara2020}. Similarly, photometric observations of the M31 stellar halo suggest that our large neighbor has had a complex merger history: its halo shows a high degree of asymmetry, with spatially and chemically coherent structures spread out over its entire extent \citep[e.g.,][]{Ibata2004,Ferguson2016,McConnachie2018}. In particular, the inner halo of M31 contains prominent tidal features, including the Giant Stellar Stream \citep[GSS;][]{Ibata2001,Ibata2004}, which extends 100~kpc to the southeast, and the Northeast and Western Shelves---diffuse but sharp-edged, fan-shaped extensions to the Northeast and West of the center of M31 respectively \citep[e.g.,][]{Ferguson2016}, structures that have been interpreted as tidal debris from a companion galaxy that merged with M31 relatively recently \citep[e.g.,][]{Ibata2004, font2006, Fardal2006, Fardal2007, Fardal2008, Fardal2012, Fardal2013, Mori2008, Sadoun2014, Hammer2018, dsouza2018, Kirihara2017a,Milosevic2022}. Spectroscopy of individual stars can greatly enhance our ability to identify migration patterns through the measurement of radial velocities and metallicities. The disk and halo of M31 have been the focus of numerous spectroscopic studies, especially over the last two decades. Most studies of the M31 halo have used the DEIMOS instrument \citep{DEIMOS} on the Keck II Telescope for pencil-beam surveys in various regions, catching tantalizing glimpses of complex kinematic structure. These studies have determined that the GSS is a relatively metal rich \citep[{[Fe/H]$\approx-0.8$}; e.g.,][]{Gilbert2019,Gilbert2020,Escala2021}, kinematically cold feature \citep[velocity dispersion $11\pm3$~km/s; e.g.,][]{Ibata2004} within a larger metal poor halo, and have revealed an additional cold velocity structure in the region of the GSS \citep{Gilbert2007}. The kinematics of the GSS have been used to estimate an enclosed total galaxy mass of $7.5\times 10^{11} M_\odot$ within 125~kpc \citep{Ibata2004}. The average metallicity of the M31 halo appears to decrease with radius \citep{Kalirai2006, Ibata2014,Gilbert2020,Escala2021}, suggesting that much of the inner stellar halo is a mixture of relatively more metal rich accreted satellite galaxies into the underlying, more metal poor halo. The kinematically cold substructures like the GSS are found to be more metal-rich than the surrounding dynamically hot stellar population \citep{Gilbert2019}, which can be understood if they are produced by fairly massive (and therefore metal-rich) progenitors. The DESI instrument \citep{DESI_Instr_Overview2022} on the Mayall 4m telescope at KPNO provides a unique opportunity to advance our understanding of the M31 system. DESI's $3.2^\circ$ diameter field of view and high multiplex capability ($\approx$5000 fibers) are well matched to the density on the sky of the brightest constituents of M31: its Asymptotic Giant Branch (AGB) stars, those at the tip of the Red Giant Branch (TRGB), and luminous blue stars, stellar clusters, HII regions, and planetary nebulae. Here we present new DESI observations of $\sim$11,000 stars towards M31 which clearly demonstrate that high-quality stellar kinematics can be acquired efficiently over the wide field of view needed to provide unique insights into the migration history of this galaxy. This paper is organized as follows. In \S~\ref{sec:data} we describe the M31 observations and the pipeline reductions. In \S~\ref{sec:results} we present the position-velocity data for the observed sources. In \S~\ref{sec:discussion} we compare our results to those of previous studies and model predictions, discuss the nature of the progenitor galaxy responsible for the observed kinematic substructure and the constraints we can place on the mass of M31 from these data. We present our conclusions in \S~\ref{sec:conclusion}. The Appendices present tables of the redshifts of non-M31 sources, i.e., higher redshift galaxies and Milky Way stars, measured by our DESI observations. Throughout this paper we adopt the M31 line-of-sight velocity of $-$300~\kms\ (based on the value of $-$300$\pm$4~\kms\ reported by the Revised Catalog of Bright Galaxies, \citealt{rc3}), and a distance to M31 of 785$\pm$25~kpc \citep{McConnachie2005}, which results in a scale of $\approx13.7$~kpc/deg. We assume that the galaxy disk is centered at (RA,Dec) = (10.6847$^\circ$, 41.26875$^\circ$) and viewed at an inclination of 77$^\circ$ to the line of sight and at a sky position angle of PA=$38^\circ$ \citep[see, e.g.,][]{WalterbosKennicutt1987,mackey2019}. % We define the ellipse containing the disk of M31 to have semi-major and semi-minor axes of 1.5$^\circ$ and 0.337$^\circ$ respectively. While heliocentric velocities are presented in the tables, in all figures and discussion we convert all velocities to the Galactic Standard of Rest (GSR) and also reference velocities to a M31-centric frame by adding 113.656~km~s$^{-1}$ (i.e., the equivalent of adding 300~\kms\ to their heliocentric velocities). \section{Observations and Data} \label{sec:data} \subsection{Target Selection} \label{sec:targetselection} The goal of this initial short M31 campaign with DESI, a fiber-fed spectrograph on a 4m diameter telescope, was to determine whether the instrument was capable of measuring stellar radial velocities and metallicities for M31 halo stars. Since the DESI Legacy Imaging Surveys \citep[hereinafter LS;][]{Dey2019} in the South Galactic Cap only extends south of Dec $\lesssim33^\circ$ and does not include the region around M31, our primary target selection was based on the source catalogs from the PAndAS survey \citep{McConnachie2018}, a 2-color $g$- and $i$-band survey covering a $>400$~deg$^2$ region around M31 and M33, which we cross matched with the {\it Gaia} DR2 \citep{gaia,GaiaDR2summary}, and CatWISE2020 \citep{catwise2021} catalogs. While the PAndAS data contain $>10\sigma$ photometry for stars to $g\approx25, i\approx24$, our target selection was restricted to stars brighter than $z=21.5$~mag to ensure measurements of sufficient signal-to-noise ratio in about 90~min of effective exposure time with DESI. Since the PAndAS catalog does not include $z$-band measurements, we constructed an estimate of the DESI Legacy Imaging Surveys $z$-band magnitude using the following relation: \begin{align} (g-z) \,= \,0.15\,{\rm max}((g-i) - 1.8, 0)\, +\, 2.21\, + \, 1.27 \, ((g-i)-1.8) \end{align} This relation was derived by cross-matching point sources in PAndAS and LS DR9 in the region where they overlap and fitting a broken linear function to $(g-z)$ vs $(g-i)$. The [16,84] percentiles of the residuals in ($g-z$) are [$-0.05,0.11$] mag for sources with $i\le 21$~mag. As M31 is centered at a Galactic latitude of $b=-21.6^\circ$ and this work targets relatively bright stars with $z<21$, the main contamination to stellar target samples is from Milky Way disk and halo stars. Prior spectroscopic surveys have primarily selected targets using colors in the region spanned by the Red Giant Branch (RGB) isochrones at the distance of M31, and using photometry in the DDO51 intermediate band filter to separate M31 red giants from Milky Way dwarf stars \citep[e.g.,][]{Guhathakurta2006}. The resulting samples tend to have $(V-I) \lesssim 2$, which is generally appropriate given the location of the metal poor isochrones that define the bulk of the M31 halo. Our target selection for M31 stars took a different approach, where we primarily focused on maximizing the number of targeted M31 stars with the help of machine learning-driven classification. We constructed separate selections based on Random Forest classifications optimized for: \begin{itemize} \item a bright ($z<19$~mag) M31 disk selection (M31 Disk Bright); \item a faint ($19\le z\lesssim 21.5$~mag) M31 disk selection (M31 Disk Faint); and \item a faint ($z\lesssim 21.5$~mag) halo selection, tuned to select targets in the Giant Stellar Stream (M31 Stream Faint). \end{itemize} The Random Forest classification \citep{Breiman2001} approach uses an ensemble of decision trees constructed from training data. Our classification relies on the following inputs: $g$ and $i$ photometry from the PAndAS catalog; the proper motion (PMRA, PMDEC, PMRA\_ERROR, PMDEC\_ERROR), parallax (PARALLAX, PARALLAX$\_$ERROR), and photometric (PHOT\_G\_MEAN\_MAG, PHOT\_BP\_MEAN\_MAG, PHOT\_RP\_MEAN\_MAG) data from {\it Gaia} DR2; % along with the {\it WISE} W1 and W2 photometry (W1MPRO, W2MPRO) from the CatWISE2020 catalogs. When these quantities were unavailable (i.e., for sources too faint for {\it Gaia} or {\it WISE}), placeholder values were used (i.e., 99.99). We did not use the PAndAS morphology flags in the Random Forest selection. Each classifier is trained on a set of stars labeled as either an M31 member or a background/foreground star. Since we do not have an unambiguous classification for every star (as an M31 member or non-member) we use a statistical decontamination approach. Specifically we consider two areas around M31, one centered on an object of interest (i.e., the disk or the GSS), and another far enough away that it would not have many M31 stars. We then remove the (likely) MW contaminants from the first field by picking a nearest neighbor in data-space for each star in the background field (with appropriate scaling to the areas of the field). We are left with a list of objects that are quite likely M31 members in the first field, and background stars in the second field. This provides us with the training set for the random forest. We use a standard cross-validation technique to choose the best tuning parameters of the random forest classifier (such as the tree depth and minimum leaf sizes) and obtain the probabilities for each star that it belongs to M31, $P_{M31}$. We then select targets with $P_{M31}>P_{cut}$, where the minimum probability $P_{cut}$ is chosen to ensure a high-enough target density to match the DESI fiber density. While the Random Forest results in a fairly complex selection, most of the faint ($z>19$ mag) targets are approximately bounded by the polygon defined by the points $((g-i),i)$=([2.0, 2.4, 3.05, 4.0, 4.0, 2.0], [22.0, 21.67, 21.67, 20.8, 22.9, 22.0]). The resulting samples for the halo and disk are shown by the blue filled circles in Figure~\ref{fig:colormagsel}. Our selection is biased to redder regions in $(g-i)$ relative to the selections used by the previous Keck/DEIMOS campaigns. We therefore sample primarily the metal-rich and older RGB and redder AGB stars and do not sample the metal-poor regions well. Despite this bias, the Random Forest approach is ``optimal'' in the sense of minimizing the contamination by Milky Way stars and background galaxies. Figure~\ref{fig:colormagsel1} shows the density of the selected sources on the sky as well as color-magnitude distributions for the 3 selections (M31 Disk Faint, M31 Disk Bright, and M31 Stream Faint). In the outer regions of the M31 halo, the Random Forest selection results in a target density that underfills the DESI fibers. Hence, we supplemented the Random Forest selection with a simple selection to define backup, or filler targets: $$z\le 21.5\quad {\rm\ and}\quad 20.5 \le i\le 24.5 $$ $$[23.5-(g-i)] \le i \le [14.5+5(g-i)]$$ $$(g-i) \le 5.0$$ In the disk field (which was originally selected for DESI first light observations), the filler targets included known bright targets---HII regions, planetary nebulae (PNe), globular clusters, luminous blue variables (LBVs)---many of which have spectroscopic information from past studies and can be used as a check on the DESI radial velocities. HII region and PNe sources were selected from the compilation of \citet{Sanders2012}. Globular cluster candidate sources were selected from the compilation of \citet{mackey2019} and from Version 5 of the Revised Bologna Catalog \citep[RBCv5;][]{galleti2007,galleti2014cat}. A small number of bright variable sources identified in the Zwicky Transient Survey Catalogs using the ANTARES time-domain event broker \citep{Matheson2021} were also included, as were bright sources from the SPLASH survey \citep{Guhathakurta2006,Dorman2012,Dorman2015}. In the halo fields, the existing spectroscopy at magnitudes DESI can reach ($z\lesssim 21.5$~mag) is more limited, but we included all known cluster and variable sources as potential targets. Finally, we complemented the list of M31 targets with background QSO candidates selected using data from the {\it WISE} and {\it Gaia} satellites. Background QSOs are invaluable probes of the interstellar and circumgalactic medium around galaxies, and all prior studies have only yielded confirmed redshifts for $\sim100$ QSOs. We used a simple {\it WISE} selection (described in Appendix~\ref{appendix:qso}) to select bright ($G\le20.5$~mag) QSOs (with a sky surface density of $\approx1.8~{\rm deg}^{-2}$) around M31. We vetted this selection using spectroscopically confirmed QSOs from the study of \cite{massey2019} and the LAMOST surveys \citep{Huo2010,Huo2013,Huo2015}. All these targets were prepared were assigned unique TARGETIDs and prepared for inclusion in the DESI Secondary Target Program. The technical details of DESI target selection, such as the unique TARGETID associated with a target, the different phases of DESI targeting, and how targeting bits can be used to isolate targets from different DESI programs are described in \citet{myers22a}. \subsection{Observations} \label{sec:observations} The Dark Energy Spectroscopic Instrument (DESI) is a wide-field, fiber-fed multi-object spectroscopic instrument mounted on the Mayall 4m Telescope of the Kitt Peak National Observatory. With a 3.2$^\circ$ diameter field of view populated by 5020 robotically positioned fibers, DESI offers an unprecedented (and currently unmatched) capability for wide-field astrophysical surveys. Details of the DESI instrument, operational plan, and science mission are presented in \citet{DESI_Tech_FDR,DESI_Science_FDR} and \citet{DESI_Instr_Overview2022}. Briefly, the $\approx$1.5~arcsec diameter DESI fibers feed ten 3-arm spectrographs which provide continuous coverage over the wide wavelength range 3600\AA\ to 9800\AA\ with a resolving power $R\equiv\lambda/\Delta\lambda$ varying from $\approx$2000 in the blue to 5500 in the red. The three spectrograph arms span the wavelength ranges 3600--5930\AA\ (blue or B), 5600--7720\AA\ (red or R), and 7470--9800 (NIR or Z). DESI is very efficient: its total system throughput varies from 20\% at 3800\AA\ to nearly 50\% at 8500\AA\ (not incuding fiber aperture losses or atmospheric extinction) and has an overhead of less than 2~minutes between exposures \citep[for details see][]{DESI_Instr_Overview2022}. Technical details of DESI operations, such as the unique TILEID associated with a tile, and how DESI observations are planned and proceed, are detailed in \citet{schlafly22a}. DESI observations of M31 were obtained in 2021 January (TILEIDs 80713 and 80715, covering the optical disk of the galaxy) and 2022 January (TILEID 82634, positioned on the Giant Stellar Stream; and 82635, targeting the NE Shelf; see Table~\ref{tab:desiobs}). The 2021 January data (on M31's disk) were taken during the early Survey Validation phase of DESI observations, when the instrument was not fully operational and observing procedures were being tested. Tile 80713 was observed on the night of 2021 January 10 in mediocre observing conditions for an effective exposure time\footnote{The DESI effective exposure time corresponds to the time required to reach the observed signal-to-noise ratio under the ``standard’’ observing conditions of a dark sky with ideal transparency and median seeing of 1.1\arcsec\ at an airmass of 1.0 (i.e., at zenith). See \citet{guy22a} for details.} of $t_{\rm eff} = 758$~sec, but the bulk of the fibers were not positioned correctly due to a bug and the observation resulted in usable spectra for only 730 targets. The tile was redesigned (with the same targets) as TILEID 80715, and successfully observed on the night of 2021 January 15 for $t_{\rm eff}=1906$~sec. During these observations, Petal \# 3 (i.e., the 36$^\circ$ pie-shaped focal-plane wedge containing 500 fibers spanning the position angle range $270^\circ<PA<306^\circ$) was non-functional. As a result, no data were obtained on a portion of the Western Shelf region of the M31 inner halo during these observations. DESI observed the tile centered on the Giant Stellar Stream (TILEID=82634) on the night of 2022 January 3. These observations were obtained under excellent conditions: dark, clear skies with seeing of 1\arcsec, and an effective exposure time of 1.5 hours was reached in 63~minutes. The tile centered on the NE Shelf (TILEID=82635) was observed on the nights of 2022 January 21 and 2022 January 27, under somewhat poorer conditions. In summary, DESI observed a total of three tiles with a total effective exposure time of $\approx 3.75$~hours. The tiles 82634 and 82635 were each observed for an effective time of $\approx$1.5~hours. \begin{deluxetable}{cccccccl} \tablecaption{DESI Observations of M31} \label{tab:desiobs} \tablehead{ \colhead{Obs Date} & \colhead{Tile ID} & \colhead{RA$_{\rm cen}$} & \colhead{DEC$_{\rm cen}$} & \colhead{Exposure Time} & \colhead{$t_{\rm eff}$} & \colhead{$N_{\rm targ}$} & \colhead{Comments} } \startdata 20210110 & 80713 & 10.170 & +41.380 & 2700 & 758\tablenotemark{1} & 730 & Petal 3 non-functional; limited fiber reach \\ 20210115 & 80715\tablenotemark{2} & 10.170 & +41.380 & 2700 & 1906 & 3130 & Petal 3 non-functional; limited fiber reach\\ 20220103 & 82634 & 11.185 & +38.768 & 3809 & 5400 & 4215 & \\ 20220121 & 82635 & 11.700 & +42.100 & 2960 & 1800 & 4263 & \\ 20220127 & 82635 & 11.700 & +42.100 & 5003 & 3600 & & \\ \enddata \tablenotetext{1}{The bulk of fibers in these observations did not reach their targets because of an error in the fiber assignment file.} \tablenotetext{2}{Tile ID 80715 is a duplicate of 80713, with the errors in the 80713 assignment file corrected.} \end{deluxetable} \subsection{Data Reduction} \label{sec:datareduction} The data were processed using the standard initial data reduction pipeline corresponding to the internal data release ``Fuji'' \citep{Guy2022}. There were however several modifications required to process the M31 data. Initially the targeting for tile 80713 did not have correctly identified flux calibration standards as it was located outside the LS footprint. As a result, it could not be processed with the default DESI pipeline parameters, and we therefore manually identified a set of stars as flux standards through color-magnitude selection in {\it Gaia} G/BP/RP bands and provided the TARGETIDs of these new flux standards to the spectroscopic pipeline. Subsequent to the first observations of the 80713 tile, DESI targeting is now able to correctly deal with fields outside the LS footprint and the standards are selected purely through {\it Gaia} photometry, with no custom flux calibration standards needed. We visually inspected the spectra using the ``Prospect'' tool\footnote{\url{https://github.com/desihub/prospect}} created by E. Armengaud \citep[for further details please see][]{alexander22a,lan22a}. An initial visual inspection (VI) revealed that spectra with low quality flags (i.e., $0\le {\rm VI\_QUALITY} \le 2$) are located near the disk of the galaxy where the sky subtraction is poor due to the sky fibers being contaminated by emission lines and continuum light from the M31 disk. While DESI observations typically reserve 50-100 fibers for sky observations (``sky fibers''), the pipeline can successfully subtract the sky with minimal additional noise or systematic issues using as few as 10 sky fibers. We therefore examined each of the sky fibers, identified ones with the lowest median flux\footnote{We selected sky fibers which satisfied $(\bar{s}_i - \bar{s}_{<70})/\bar{s}_{<70} \le 0.2$, where $\bar{s}_i$ is the median value of the sky in sky fiber $i$ measured in the wavelength region $\lambda\lambda6000-7000$\AA, and $\bar{s}_{<70}$ is the similarly measured median sky value measured across all the sky fibers after rejecting the 30\% of the fibers with the highest skies. This procedure resulted in $\ge10$ sky fibers per petal which could be used for sky subtraction.}, and then reran the pipeline reductions using this subset. This re-reduction corrected the bulk of the problems with the sky subtraction. After the initial pipeline data reduction, the data were then processed through the redshift and stellar radial velocity/parameters pipelines. The initial catalog of redshifts was obtained with the Redrock package\footnote{\url{https://github.com/desihub/redrock}} \citep{Bailey_Redrock:inprep} which estimates redshifts by fitting a set of eigenspectra to the DESI spectra. The eigenspectra are constructed from star, galaxy, and QSO templates and are optimized for determining the velocities of galaxies over a wide range in redshift (from $-1100$~\kms\ to $z=6$). To determine the radial velocities and stellar parameters, we also used the Radial Velocity pipeline (RVS) that is built on the RVSpecFit code\footnote{\url{https://github.com/segasai/rvspecfit}} \citep{Koposov2011,rvspecfit} and is used by the DESI Milky Way Survey (MWS). Details about the RVS pipeline and its outputs are provided in the MWS overview paper \citep{Cooper2022}, while here we provide a brief summary. The stellar models for the fitting are built using the interpolated PHOENIX stellar atmosphere models (spanning effective temperatures $2300\le T_{\rm eff} \le 15000$K) from \citet{Phoenix_2013} convolved to DESI resolution. These models are fit simultaneously to all 3 arms of the DESI spectra by optimizing the combined $\chi^2$. The spectra are not continuum normalized; instead we fit the spectra directly with functions of the form $T(\lambda)P(\lambda)$ where $T(\lambda)$ is the interpolated stellar template from the PHOENIX models and $P(\lambda)$ is a polynomial that takes care of potential flux calibration and/or normalization differences between the data and the model. The model fit provides estimates of the stellar atmospheric parameters $\log g$, $T_{\rm eff}$, [Fe/H] and [$\alpha$/Fe] together with radial velocities in the range $\vert V_{\rm rad}\vert\le 1500$~\kms. For each DESI target, we therefore have two velocity estimates, one from Redrock and the other from the RVS pipeline. For stars, the two pipelines agree extremely well: the median radial velocity difference is 0.05~\kms\ and the RMS scatter is 3~\kms. The accuracy of the stellar parameter determination by the RVS pipeline is discussed in the MWS overview paper, although the M31 data, especially the observation of the outer halo, represent a very different regime than most of the main MWS, as the majority of the M31 targets are very faint cool giants, where the dominant spectral information comes from the molecular absorption bands. We found that the surface gravity estimates are particularly useful to identify M31 members and separate them from the Milky Way contaminants. For the nominal effective exposure time of 90 min (achieved for tiles 82634 and 82635), the majority of the $z<21.5$~mag stars have velocity uncertainties $\sigma_V < 5$~\kms\ (Figure~\ref{fig:velocityaccuracy}). We expect that the estimates of [Fe/H] should be accurate to $\sim 0.2$ dex except for the faintest objects \citep{Cooper2022}. The estimates of [$\alpha$/Fe] are more uncertain and require better calibration datasets for comparison; the discussion of the [$\alpha$/Fe] measurements is therefore postponed to a future study. Four of the authors (GM, JJZ, JN, AD) visually inspected \NVI\ of the spectra using the ``Prospect'' spectral inspection (VI) tool. We inspected all spectra for which Redrock returned a SPECTYPE of GALAXY or QSO, found a redshift of $z>0.001$, or where the target was selected to be a QSO. In addition, we visually inspected the spectra of all targets selected from previous catalogs (i.e., the globular cluster, planetary nebulae, and variable star candidates). % Further, we also visually inspected all the well-measured \citep[i.e., RVS\_WARN=0; see][for details]{cooper22a} sources for which the Redrock- and RV-measured velocities differed by more than 50~\kms. Spectra were visually classified according to 3 broad types (STAR, GALAXY, and QSO) and assigned a quality flag (varying from 0 = `No useful data' to 4 = `robust redshift and spectral type') based on the reliability of the redshift estimate. To create a final catalog, we retained only sources for which a velocity could be determined, i.e., sources with quality flags of 3 or 4 (which only excludes 6.3\% of the VI-ed sources). The catalog reports a ``best'' velocity, selected from among the velocity measured by VI and those reported by the analysis pipelines. If the VI velocity was within 100~\kms\ of either the corresponding RVS or Redrock values, both of which % were determined with greater precision than the VI value, we selected the value closer to the VI value. Conversely, if the VI velocity was $>100$~\kms\ away from the RVS and Redrock values, we selected the VI velocity. \section{Spectroscopic Results} \label{sec:results} \subsection{Measurements} \label{sec:measurements} The DESI observations resulted in spectra of \Ntargets\ unique astronomical targets. Of these, \Ngalaxies\ are confirmed as galaxies and \NQSO\ as QSOs (see Appendix A). \Nstars\ of these are sources within M31 or foreground stars in the Milky Way. As shown in Figure~\ref{fig:logg_RV_selection} we can effectively isolate a robust sample of the M31 sources using the following combined criteria: $$ {\rm RVS\_WARN} = 0 $$ $$ \sigma(V_r) \le 20 {\ \rm km\,s^{-1}}$$ $$ \log g \le 4\quad {\rm or}\quad V_r \le -150 {\ \rm km\,s^{-1}} $$ For the subset of sources that were visually inspected, we excluded those sources with VI\_SPECTYPE = GALAXY or QSO or 0 $\le$ VI\_QUALITY $\le$ 2. We note that this is not a 100\% complete selection, as there a few objects ($\sim$ 100) that seem to belong to M31 based on the RV but have a measured ${\log g}>4$. These criteria result in a final sample of % \NAndStars\ stars, \NAndHIIPN\ HII regions or planetary nebulae, and \Nclusters\ open or globular clusters. Of the 9266 targets selected using the Random Forest algorithm, 8416 have reliable radial velocity measurements (i.e., no processing errors and $\sigma(V_{\rm r})\le 10$~\kms), and of these 6768 (73\% of all targeted) are M31 stars. This high success fraction demonstrates the efficiency of the Random Forest selection. For the backup selection, 213 of the 562 targets (38\%) are M31 stars. In this paper, we present results based on the stars in the M31 halo, i.e., the region outside the ellipse encompassing the disk (e.g., see Figure~\ref{fig:spatialveldistribution}. The spectra of M31 disk sources will be discussed in a separate publication. The measured velocities and positions of the 6,436 confirmed M31 stellar sources are presented in Table~\ref{tab:stars}. The list of spectroscopically confirmed cluster, HII region and planetary nebula candidates is presented in Table~\ref{tab:HIIPNGC}. Digital versions of the complete tables are available online. The columns are: (1) a running index; (2,3) RA and Dec in J2000; (4,5) $V_{\rm los}$, the line-of-sight heliocentric velocity and its formal uncertainty; (6,7) the spectroscopic estimate of [Fe/H] and its formal uncertainty; (8,9) $T_{\rm eff}$, the effective temperature and its formal uncertainty; (10,11) the surface gravity (log g) and its formal uncertainty; (12) the {\it Gaia} DR2 G-band flux, if available; (13) the PAndAS $g$ and $i$ magnitude, if available; (14) an alternate name for the target (i.e., from {\it Gaia} DR2 or PAndAS). The online table also includes the {\it Gaia} DR3 Source ID, matched within a radius of 1\arcsec. \begin{rotatetable} \begin{deluxetable}{ccccccccccccccl} \movetabledown=3mm \movetableright=0.1mm \tablecaption{M31 Stars\tablenotemark{1}} \tablecomments{Table 3 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.} \tablehead{ \colhead{\bf ID} & \colhead{\bf RA ($^\circ$)} & \colhead{\bf Dec ($^\circ$)} & \colhead{${\bf V_{\rm los}}$} & \colhead{${\bf \sigma(V_{\rm los})}$} & \colhead{\bf [Fe/H]} & \colhead{\bf $\sigma$([Fe/H])} & \colhead{${\bf T_{\rm eff}}$} & \colhead{${\bf \sigma(T_{\rm eff})}$} & \colhead{\bf log(g)} & \colhead{\bf $\sigma$(log(g))} & \colhead{\bf $G_{\rm DR2}$} & \colhead{\bf $g_{\rm PAndAS}$} & \colhead{\bf $i_{\rm PAndAS}$} & \colhead{\bf Alternate Name} } \startdata 1 & 9.7371301 & 40.2178233 & -472.5 & 6.7 & -1.85 & 0.06 & 3643.0 & 5.0 & 2.46 & 0.28 & NaN & 24.90 & 21.89 & PANDAS 170264 \\ 2 & 9.9989009 & 40.3012761 & -520.7 & 5.5 & 0.44 & 0.19 & 4389.0 & 84.4 & 2.99 & 0.02 & NaN & 25.48 & 22.20 & PANDAS 96604 \\ 3 & 10.1620093 & 40.4871788 & -479.8 & 10.2 & -0.05 & 0.05 & 4162.9 & 20.9 & 3.04 & 0.02 & NaN & 22.42 & 19.74 & PANDAS 5541 \\ 4 & 10.2713676 & 40.3688261 & -521.1 & 4.4 & 0.42 & 0.06 & 3870.8 & 7.2 & 2.12 & 0.01 & NaN & 25.53 & 21.77 & PANDAS 176484 \\ 5 & 9.9539718 & 40.3950344 & -509.8 & 5.0 & -0.70 & 0.05 & 4058.6 & 15.9 & 1.94 & 0.01 & NaN & 25.85 & 22.20 & PANDAS 96676 \\ 6 & 9.7809551 & 40.2289816 & -489.6 & 4.6 & 0.21 & 0.12 & 4055.0 & 69.2 & 3.21 & 0.01 & NaN & 25.40 & 22.09 & PANDAS 54369 \\ 7 & 9.8708426 & 40.2325316 & -413.9 & 6.0 & 0.16 & 0.09 & 4067.7 & 39.8 & 2.82 & 0.28 & NaN & 25.86 & 21.92 & PANDAS 171130 \\ 8 & 10.1081926 & 40.4273066 & -503.8 & 4.6 & -0.62 & 0.04 & 4130.8 & 19.5 & 2.47 & 0.19 & NaN & 24.74 & 21.76 & PANDAS 176049 \\ 9 & 10.0735259 & 40.3537205 & -454.4 & 4.9 & -1.08 & 0.07 & 3779.1 & 5.8 & 0.74 & 0.00 & NaN & 25.24 & 21.78 & PANDAS 96643 \\ 10 & 9.9964951 & 40.4045608 & -524.8 & 11.5 & 0.50 & 0.07 & 9800.0 & 6.2 & 1.51 & 0.00 & 19.99 & 0.00 & 0.00 & Gaia DR2 381120004684395520 \\ 11 & 9.9106488 & 40.2671799 & -158.2 & 1.8 & -1.12 & 0.07 & 4576.7 & 11.9 & 4.24 & 0.01 & 20.42 & 0.00 & 0.00 & Gaia DR2 369107977589805824 \\ 12 & 9.7966509 & 40.2791149 & -395.6 & 11.1 & 0.88 & 0.00 & 4139.0 & 45.8 & 4.00 & 0.01 & NaN & 25.03 & 21.69 & PANDAS 96533 \\ 13 & 10.0611343 & 40.2721011 & -452.8 & 5.2 & -0.88 & 0.07 & 3788.7 & 5.9 & 0.21 & 0.00 & NaN & 24.92 & 21.91 & PANDAS 174999 \\ 14 & 10.2674759 & 40.5024538 & -461.4 & 9.7 & -0.44 & 0.04 & 4067.8 & 15.3 & 1.84 & 0.01 & NaN & 24.69 & 21.85 & PANDAS 179680 \\ 15 & 10.2065926 & 40.4144844 & -501.4 & 5.7 & -2.12 & 0.04 & 3371.2 & 3.7 & 0.51 & 0.00 & NaN & 24.90 & 21.82 & PANDAS 176513 \\ 16 & 9.7446926 & 40.3032594 & -485.0 & 4.2 & 0.07 & 0.11 & 5758.0 & 103.1 & 3.00 & 0.01 & NaN & 23.08 & 21.59 & PANDAS 54401 \\ 17 & 9.8513801 & 40.3184038 & -493.3 & 4.0 & -0.82 & 0.05 & 3945.2 & 7.2 & 1.69 & 0.01 & NaN & 24.29 & 21.72 & PANDAS 174596 \\ 18 & 10.1946551 & 40.3319622 & -482.1 & 3.9 & 0.46 & 0.06 & 3759.3 & 5.8 & 0.97 & 0.01 & NaN & 25.71 & 22.13 & PANDAS 176249 \\ 19 & 10.0397343 & 40.4225761 & -499.6 & 4.7 & 0.09 & 0.03 & 4230.9 & 18.6 & 2.60 & 0.13 & NaN & 25.76 & 21.83 & PANDAS 96749 \\ 20 & 9.9798933 & 40.4267977 & -506.4 & 0.4 & -0.39 & 0.02 & 3851.9 & 3.2 & -0.50 & 0.00 & 20.12 & 21.62 & 19.35 & PANDAS 5270 \\ 21 & 10.0378426 & 40.3165594 & -493.0 & 5.7 & 0.41 & 0.07 & 4143.4 & 35.1 & 2.78 & 0.22 & NaN & 24.67 & 21.81 & PANDAS 175068 \\ 22 & 10.1195551 & 40.3864344 & -436.8 & 9.9 & -0.37 & 0.09 & 4044.9 & 32.2 & 2.84 & 0.26 & NaN & 25.02 & 21.90 & PANDAS 175857 \\ 23 & 9.9955093 & 40.3262261 & -478.3 & 3.5 & 0.36 & 0.07 & 4153.6 & 32.8 & 2.54 & 0.20 & NaN & 24.80 & 21.87 & PANDAS 175062 \\ 24 & 9.9753384 & 40.2584122 & -482.7 & 6.0 & 0.07 & 0.09 & 4270.8 & 48.8 & 3.67 & 0.01 & NaN & 25.13 & 21.93 & PANDAS 174942 \\ \enddata \tablenotetext{1}{See Online Version for complete Table} \label{tab:stars} \end{deluxetable} \end{rotatetable} \begin{deluxetable}{cccccccccl} \tablecaption{M31 HII, PNe, and GC Targets} \tablecomments{Table 4 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.} \tablehead{ \colhead{ID} & \colhead{RA ($^\circ$)} & \colhead{Dec ($^\circ$)} & \colhead{$V_{\rm los}$} & \colhead{$\sigma(V_{\rm los})$} & \colhead{\bf TargetClass} & \colhead{\bf $G_{\rm DR2}$} & \colhead{\bf $g_{\rm PAndAS}$} & \colhead{\bf $i_{\rm PAndAS}$} & \colhead{\bf Alternate Name} } \startdata 30 & 10.6240000 & 41.0584000 & -438.9 & 7.0 & H2PN & 0.00 & 20.94 & 22.10 & HII+PNe PN184 \\ 31 & 10.4541000 & 41.0738000 & -513.6 & 1.4 & H2PN & 0.00 & 0.00 & 0.00 & HII+PNe PN129 \\ 32 & 10.4348000 & 41.0364000 & -489.7 & 3.3 & H2PN & 0.00 & 21.42 & 22.63 & HII+PNe PN118 \\ 33 & 10.6293000 & 40.8846000 & -184.8 & 4.5 & H2PN & 0.00 & 0.00 & 0.00 & HII+PNe PN185 \\ 34 & 10.4922000 & 41.1361000 & -470.6 & 2.9 & H2PN & 0.00 & 21.47 & 26.18 & HII+PNe PN144 \\ 35 & 10.3843000 & 41.0033000 & -486.6 & 3.8 & H2PN & 0.00 & 21.80 & 22.02 & HII+PNe PN102 \\ 36 & 9.8293724 & 40.3661097 & -51.3 & 8.8 & GC & 0.00 & 0.00 & 0.00 & GC3106,SH06 ,SH06 \\ 37 & 9.9358057 & 40.2355292 & -475.7 & 1.0 & GC & 0.00 & 0.00 & 0.00 & GC308,B314 ,B314-G037 \\ 38 & 9.8905182 & 40.5207430 & -507.2 & 0.2 & GC & 0.00 & 0.00 & 0.00 & GC305,B311 ,B311-G033 \\ 39 & 10.1869141 & 40.8855375 & -536.7 & 2.0 & GC & 0.00 & 0.00 & 0.00 & GC3710,BH10 ,BH10 \\ 40 & 10.2529516 & 39.9317236 & -235.2 & 0.3 & GC & 0.00 & 0.00 & 0.00 & GC332,B339 ,B339-G077 \\ 41 & 10.2205474 & 40.5888014 & -552.0 & 1.1 & GC & 0.00 & 0.00 & 0.00 & GC9074,KHM31-74 ,KHM31-74 \\ \enddata \tablenotetext{1}{See Online Version for complete Table} \label{tab:HIIPNGC} \end{deluxetable} \subsubsection{Comparison to Previous Work} \label{sec:previouswork} M31 has been the target of several spectroscopic campaigns over many decades. A search of the SIMBAD database \citep{SIMBAD2000} resulted in a total of 139,078 entries (for 35,374 sources with unique names) within 5$^\circ$ of M31, of which 73,090 (representing 14,617 unique sources) have reported radial velocities. In addition, the \href{https://oirsa.cfa.harvard.edu/signature_program/}{CFA Optical/Infrared Science Archive} \citep[][and references therein]{Sanders2012, Caldwell2016, Bhattacharya2019} consolidates the many years of MMT/Hectospec and Hectochelle campaigns in M31. Of the 10,322 sources in the Archive that are within 5$^\circ$ of M31, 5064 have measured radial velocities and 2,099 are also included in the SIMBAD list. In summary, there are 17,582 sources with published radial velocities in this region. The bulk of the literature radial velocities are foreground (Milky Way Galaxy) stars, and only 6,939 sources have radial velocities typical of M31 ($<-100$~\kms). The bulk of these stars lie within the projected area of the M31 main disk and, unlike the DESI data, do not sample the M31 inner halo well. Thus the new DESI radial velocities presented here only have a small overlap with the published radial velocity measurements. Only 145 DESI targets have matches (within 1\arcsec) to sources with radial velocities in the literature. Where there is overlap, the DESI radial velocities agree well: they have a median offset of $\approx 2.8$~\kms\ and an rms scatter of $\approx 14$~\kms. For the matched sources, the median velocity uncertainty of the measurement quoted in the literature is $\sim15$~\kms. The DESI data provide more precise radial velocities, with $\approx88$\% of the sources having velocity uncertainties $\le 10$~\kms. While most of the spectroscopy to date of individual stars in M31 has been carried out with 6.5-m to 10-m class telescopes \citep[e.g.,][and references therein]{Ibata2004, Bhattacharya2019, Caldwell2016, Guhathakurta2006, Kalirai2006, Gilbert2007, Gilbert2009a, Gilbert2020}, the present results illustrate the science potential of highly multiplexed spectrometers on smaller aperture telescopes. A caveat here is that although several campaigns by different groups have targeted the fainter M31 halo populations, the data have not been published along with the papers reporting the results. These campaigns (primarily with Keck/DEIMOS) have targeted primarily giants and horizontal branch stars in M31 in a number of pencil beams scattered across the region. These prior studies typically reach targets significantly fainter than our DESI observations, and have the advantage of higher signal-to-noise ratio measurements and better constraints on metallicity \citep[e.g.,][]{Ibata2004,Kalirai2006,Gilbert2020,Kirby2020,Escala2020a,Escala2020b,Escala2022}. However, the advantage DESI offers is the ability to (approximately) uniformly sample large spatial regions of the M31 halo both quickly and efficiently: a total DESI on-sky exposure time of $\approx 3.75$ hours yielded \Ngoodspec\ velocities, {\NAnd} of which are M31 sources with well-measured line-of-sight velocities with uncertainties $\sigma_V\le 10$~\kms. \subsection{Position-Velocity Diagrams} \label{sec:PVdiagram} The left panel of Figure~\ref{fig:spatialveldistribution} shows the density distribution of sources in the % inner halo of M31 selected from the PAnDAS catalog \citep[]{McConnachie2018} in the region covered by the DESI spectroscopy, with the unWISE coadded W3/W4 image superposed on the central galaxy \citep{unWISEcoadd_2014}. The distribution of inner halo sources shows the previously identified morphological features: the GSS to the SSE; the SE and NE Shelves; and the Western Shelf \citep[see][for details]{Ferguson2016}. To create the image of the inner halo, we selected catalog sources from the $i$ vs $(g-i)$ color-magnitude diagram that lie within the polygon defined by [$(g-i)$,$i$]=[[0.9, 1.8, 5.0, 5.0, 2.2, 2.0], [23, 21, 22, 22.5, 22.5, 23]] and used a Gaussian kernel density estimator to adaptively smooth the spatial point distribution of sources. The ellipse separating the inner halo and central galaxy (with semi-major axis $a_e=1.5^\circ$, semi-minor axis $b_e=0.337^\circ$, and PA=38$^\circ$ and centered at (RA, DEC)=(10.6847$^\circ$, +41.26875$^\circ$)) denotes the disk of M31 and roughly traces the ring of star formation so clearly visible in young stars and mid-infrared observations of the galaxy \citep[e.g.,][]{barmby2006,lewis2015}. In the left panel, radial dashed lines demarcate the zones in which we explore the position-velocity distributions of the observed sources. The zonal boundaries are chosen to overlap known overdensities and to distinguish these from each other and the M31 disk. Zone 1 is dominated by the GSS; Zone 2 contains the SE Shelf (a portion of which begins in Zone 1) and more than half of the NE Shelf; Zone 3 includes the NE Shelf and the blobby feature located at $(\xi,\eta)\approx(0.8,1.8)$; Zone 4 contains the inner halo region just north of the M31 disk; and Zone 5 is dominated by the Western Shelf. The remaining range of azimuth % does not contain much DESI spectroscopy beyond the boundary of the disk. The right panel of Figure~\ref{fig:spatialveldistribution} shows the positions of the measured M31 sources color coded by line-of-sight velocity. There is a clear red-blue asymmetry along the major axis of the galaxy, with an apparent strong flaring and/or a warp near [$\xi,\eta$] $\approx$ [$+0.7^\circ,+1.8^\circ$], also observed in the stellar density distribution. We can examine the kinematics of each zone by plotting the line-of-sight radial velocity ($V_{\rm los}$) as a function of projected distance from the center of M31 ($R_{\rm proj}$) for the sources in each sector, as shown in Figures \ref{fig:zoneposvela} and \ref{fig:zoneposvelb}. In each panel, stars at velocities $\sim 300$\,\kms relative to M31 are primarily foreground Milky Way stars. Figure~\ref{fig:allfeaturesannotated} shows the line-of-sight positions and velocities for stars in all zones and summarizes the linear features in position-velocity space detected in each zone. The kinematic features are also tabulated in Table~\ref{tab:kinematicfeatures} and identified by the following convention: a number for the zone, a letter index to distinguish multiple features in the same zone, and a ``b'' or ``r'' based on whether the feature is blue- or redshifted relative to the M31 systemic velocity. For example, the GSS feature is labelled as ``1ab'', meaning that it is feature ``a'' in Zone 1, and is blue-shifted relative to the M31 systemic velocity. % We refer to the GSS and other linear features in Figures~\ref{fig:zoneposvela} and \ref{fig:zoneposvelb} as ``streams” based on the previous use of the term in naming the GSS. These ``streams'' are only-redshifted or only-blueshifted, mostly linear features, in contrast to the features we refer to as ``shells”, which have the morphology of chevrons or wedges and typically have both red- and blue-shifted components in Figures~\ref{fig:zoneposvela} and \ref{fig:zoneposvelb}. In contrast to true narrow Galactic streams such as GD1 \citep[e.g.,][]{Koposov2010}, the structures we call ``streams" may be more accurately described as ``one-sided shells”, with their member stars possibly spanning a range of total energies. The position-velocity diagram for {\bf Zone 1} reveals at least three main features. Most prominent is the GSS (labeled `1ab' in Figure \ref{fig:zoneposvela}), which appears as a tight band of blueshifted stars whose average velocity varies smoothly with distance from $\sim -300$~\kms at 0.5$^\circ$ to $\sim -50$~\kms at 4$^\circ$ separation from M31. These stars are also highlighted in blue in Figure \ref{fig:zone2feh}. Our data cover most, but not all, of the entire visible extent of the GSS. Extrapolating linearly, we expect the kinematic structure to cross the zero velocity line at a projected distance of $\approx 5.0^\circ$ from M31. To determine the velocity dispersion of the GSS, we fit the observations with a two-component model: $$P(V|R) = f N(V|V_{\rm str}(R),\sigma) + (1-f) S_{\rm bg}(V-V_{\rm str}(R)),$$ where $N(V)$ is a Gaussian density in projected radial velocity that represents the GSS and $S_{\rm bg}(x),$ which represents the foreground component, is an appropriately normalized piece-wise linear function of the form $\min(\max(x,0),1)$. The quantity $f$ is the mixing fraction between the stream and the foreground, $V_{\rm str}(R)$ is the radial velocity of the stream as a function of distance parameterized by a cubic spline with 9 knots, and $\sigma$ is the velocity dispersion of the GSS. The model has 14 parameters in total and was fitted to the sample of stars between the grey lines shown in the left panel of Figure~\ref{fig:Sergey_GSS_RV}. We use in the fit only stars at projected distances between 1$^\circ$ and 3.8$^\circ$ from M31 and at position angles between 147$^\circ$ and 175$^\circ$. The posterior of the parameters is sampled using the {\tt dynesty} nested sampling code \citep{dynesty,speagle2020}, and the fitted velocity dispersion is determined to be $10.80 \pm 0.75$ \kms. % This velocity dispersion is lower than most of the measurements by \citet{Gilbert2009b} of the primary GSS component in the GSS core and envelope region, but is more consistent with their measurement in the ``m4" field of $11.4^{+5.2}_{-4.1}$~\kms\ centered on Stream C. This may be due to the better resolution and better sampling of the DESI study, which results in the ability to cleanly isolate the primary GSS kinematic structure (1ab) from the background. Given the multiple structures that make up this region of the halo, measuring a single velocity dispersion for all the components together is not physically meaningful. The fitted position-velocity locus of the GSS is provided in Table~\ref{tab:gss_rv_track} and used in the mass estimate analysis presented in Section~\ref{sec:gssmassest}. Zone 1 also includes another band of blueshifted stars that runs parallel to the GSS in the position-velocity diagram. Less blueshifted by $\sim 100$~\kms\ than the GSS, this kinematically cold component (labeled `1bb' in Figure \ref{fig:zoneposvela}) has a velocity dispersion similar to that of the GSS (see also green points in Figure \ref{fig:zone2feh}), and is more limited in length, extending to $\sim 2.7^\circ$ from the center of M31. As shown in Figure \ref{fig:zone2feh}, feature `1bb' is also spatially offset from the GSS, extending outward from the center of M31 at a different mean angle than the GSS stars. Feature 1bb was previously identified in spectroscopy carried out in pencil-beam surveys of discrete portions of the M31 halo \citep{Kalirai2006,Gilbert2009b}. Our results are consistent with the velocities previously reported and illustrate the spatially continuous nature of the structure and its spatial offset from the GSS. We also see in Zone 1 a hint of a more compact feature: a {\bf chevron} pattern, i.e., a concentration of stars along a triangular-shaped edge (its blue- and redshifted edges labeled `1cb' and `1cr' in Figure~\ref{fig:zoneposvela}), similar to the general shape expected for radial shells \citep{Merrifield1998}. The chevron extends to $\sim 1.3^\circ$ in projected distance and reaches an apex at a velocity % within $\sim 30$~\kms\ of M31 (Figure~\ref{fig:zoneposvela}; red points in Figure~\ref{fig:zone2feh}). Higher density sampling is needed to confirm this feature and define its kinematic structure. \begin{table}[h] \centering \caption{Approximate Parameters of Kinematic Features} \begin{tabular}{ccllrcl} \hline {\bf Zone} & {\bf Angular Range\tablenotemark{a}} & {\bf Feature} & {\bf $R_{\rm max}$} & {\bf d$V_{\rm los}$/d$R$} & {\bf $M_{\rm enc}$}\tablenotemark{c} & Type\\ & & & (deg) & (\kms/deg) & ($10^{11} M_\odot$) & \\ \hline 1 & $85^\circ-130^\circ$ & 1ab (GSS) & 5.057\tablenotemark{b} & 58 & & Stream \\ % && 1bb & 2.70 & 88 & & Stream? \\ % && 1cb & 1.25 & 182 & 2.0 & Shell \\ % && 1cr & 1.25 & -224 & 3.1 & Shell \\ % 2 & $130^\circ-230^\circ$& 2ar & 2.70 & -114 & 8.0 & Shell \\ % && 2br & 1.44 & -191 & 3.4 & Shell, related to 1cr \\ % 3 & $230^\circ-255^\circ$ & 3ar & 2.70 & -108 & 7.3 & Shell, related to 2ar \\ % && 3br & -- & -202 & & Short linear feature \\ && 3ab & 3.15\tablenotemark{b} & 115 & & Stream? \\ % 4 & $255^\circ-315^\circ$ & 4ab & 3.15\tablenotemark{b} & 122 & & Stream? related to 3ab \\ % && 4bb & 2.00 & 140 & 5.0 & Shell \\ % && 4br & 2.00 & -150 & 5.7 & Shell \\ % 5 & $315^\circ-30^\circ$ & 5b & 2.00 & 150 & 5.7 & Shell, related to 4bb \\ % && 5r & 2.00 & -170 & 7.3 & Shell, related to 4br \\ % \hline \end{tabular} \tablenotetext{a}{Angle is measured clockwise from the $\eta=0$ axis; i.e., $\theta=270^\circ-$PA} \tablenotetext{b}{$R_{\rm max}$ for features 1ab (GSS), 3ab, and 4ab are determined from the linear extrapolation to $V_{\rm los}=0$.} \tablenotemark{c}{Enclosed masses $M_{\rm enc}$ for shells estimated from Merrifield \& Kuijken (1998).} \label{tab:kinematicfeatures} \end{table} {\bf Zones 2, 3, and 4} include the NE Shelf, which extends out to $\sim 2.5^\circ$ from M31 (Figure \ref{fig:spatialveldistribution}) and has a shell-like morphology. The position-velocity diagram for Zone 2, which samples the portion of the NE Shelf south of the M31 disk, shows a large, prominent triangular {\bf ``wedge'' shape (a filled chevron)}, with an apex at $\sim 0$~\kms\ relative to M31 at a distance of $\sim 2.5^\circ$ and extending to $\pm 300$~\kms\ at $\sim 0.5^\circ$. The redshifted edge of the wedge (labeled `2ar' in Figure~\ref{fig:zoneposvela}) is better defined than the blueshifted edge, and the interior of the wedge is more populated at redshifted velocities. Within this feature, a smaller wedge-shaped feature also appears to be present, with an apex at $\sim 1.5^\circ$ distance and extending to $\sim 150$~\kms\ at 0.5$^\circ$ distance (feature `2br' in Figure~\ref{fig:zoneposvela}). The position-velocity plots for Zones 3 and 4, which are radially opposite from Zone 1, show a narrow blueshifted feature with kinematics similar to that of the GSS in Zone 1 (feature `3ab' in Figure~\ref{fig:zoneposvela} and `4ab' in Figure~\ref{fig:zoneposvelb}. The stars comprising the feature are widely distributed spatially across both zones. Perhaps these are stars that were once in the GSS and have passed back through M31 to the northern side of the galaxy. Such features do appear in merger simulations (e.g., that discussed in Section~\ref{sec:cosmo}). Zone 3 also includes a hint of a narrow redshifted feature (feature `3ar' in Figure~\ref{fig:zoneposvela}), which is likely related to feature 2ar in Zone 2. Unlike the wedge associated with 2ar, the wedge associated with 3ar is mostly empty. A striking feature of Zone 3 is the presence of a group of about 40 stars that define a short ``stub'' in the position-velocity diagram, defined as 3br in Figure~\ref{fig:zoneposvela}. These stars appear to be at the northern eastern and southwestern edges, respectively, of the overdensities defined as the Northern Spur and the North East Clump by \citet{mackey2019}. The 3br feature is notable for its small velocity dispersion of $4.6\pm1.5$~\kms\ despite its stars covering the width of the zone. Apart from features 3ab, 3ar and 3br, the rest of the stars in Zones 3 are preferentially redshifted, as in Zone 2, and scattered across position velocity space. Zone 4 also shows a preference for redshifted stars. The major feature in Zone 4 is a more completely filled wedge bordered by 4br and 4bb. Finally, the position-velocity plot for {\bf Zone 5} shows a chevron pattern (i.e., the outline of a wedge-like shape; labeled 5b and 5r), similar to the shape expected for radial shells \citep{Merrifield1998}. A similar feature was reported for the Western Shelf region by \citet[see their Figure 8]{Fardal2012} based on spectroscopy of stars in a narrow strip along the minor axis of the M31. Here, the stars that make up the chevron pattern are broadly distributed across the Western Shelf feature in Zone 5. The stars that make up the red- and blue-shifted edges (5b and 5r) spatially overlap each other as expected for an umbrella-like fan viewed tangentially \citep{Merrifield1998}. The 5b/5r chevron pattern overlaps the edge of the filled wedge bordered by 4bb/4br (Figure~\ref{fig:allfeaturesannotated}). \subsection{Comparison to Planetary Nebulae from the Literature} Figure~\ref{fig:pnlit} compares the spatial and velocity distributions of M31 stars with those of planetary nebulae (PNe) reported in the literature. The angular zones shown are the same as those shown in Figures~\ref{fig:zoneposvela} and \ref{fig:zoneposvelb}, with the exception that Zones 3 and 4 are combined. The PNe shown were identified using SIMBAD and the MMT/Hectospec archive, and are the result of a large body of work by many authors % \citep[see][and references therein]{Merrett2006,Yuan2010,Sanders2012,Bhattacharya2019}. The comparison shows that the known PNe that lie beyond the main disk of the galaxy trace the same kinematic structures visible in the DESI data. The similarity is apparent in all spatial regions of M31, but most strikingly in the regions shown in the bottom two panels. In the angular range $230^\circ < \theta \le 315^\circ$ (zones 3+4 in Figures~\ref{fig:zoneposvela} and \ref{fig:zoneposvelb} covering the northern portion of the NE shelf), the PNe are preferentially redshifted, echoing the distribution of the stars, and roughly demarcate the two wedges visible in the stellar data. At $315^\circ < \theta \le 30^\circ$ in the Western Shelf (bottom panel, zone 5 in Figure~\ref{fig:zoneposvelb}), the PNe trace the red- and blue-shifted edges of the chevron. \citet{Fardal2007} previously pointed out how the PNe in the Western Shelf preferentially fall near the boundary of a triangular region in position-velocity space. The present comparison shows how the PNe distribution echoes the more densely sampled stellar distribution over much of the inner halo, as expected. \subsection{Metallicities} \label{sec:metallicities} Photometric studies have demonstrated that the M31 halo shows a wide range of stellar metallicities with much of the substructure being metal rich \citep{Ibata2001,Brown2006,Conn2016}. Spectroscopy from Keck/DEIMOS has not only found evidence for a low-metallicity halo component that is detectable both in the inner regions and at large distances, but also confirmed that the stars associated with some of the kinematic substructure are metal rich \citep[e.g.,][]{Guhathakurta2006,Kalirai2006,Ibata2007,Gilbert2009a,Ibata2014,Escala2020a,Escala2020b}. The metallicity of the Western Shelf is the same as that of the GSS \citep{Fardal2012, Tanaka2010}, with a typical metallicity of [Fe/H] = --0.7 for the satellite debris and --1.2 for the spheroid component of M31. Since the selection of targets for the DESI observations presented here is biased toward redder colors (and thus higher metallicity populations) and does not sample the metal-poor RGB populations, we cannot use the DESI data to infer directly the metallicity distributions in the different kinematic components. However, we do find significant numbers of metal-rich stars across all regions surveyed. For the stars in the region of the GSS, Figure~\ref{fig:zone2feh} shows that we measure similar median metallicities in the three different kinematic components (the median metallicities in 1ab, 1bb, and 1cb are $-0.33$, $-0.26$ and $-0.32$, respectively, with all the observed stars in this zone -- represented by the dashed line -- showing a median metallicity of $-0.37$). The overall distribution of metallicities is remarkably similar to that presented by \citet[][see their Figure~11]{Fardal2012}, showing a skewed distribution with a tail to lower metallicities. This similarity is surprising given that our target selection is biased toward the high metallicity regions of the color-magnitude diagram. The presence of lower metallicity stars in our sample may result from photometric scatter in the PAndAS data (i.e., with the more metal-poor stars scattering into our selection region). Nevertheless, assuming that the DESI data only sample the high metallicity tail of the distribution, the measurements suggest [Fe/H]$\lesssim-0.4$ is a strong upper limit to the median metallicity in these regions. There is weak evidence that the metallicity distribution in the compact wedge component in Zone 1 (1cb; shown by the red points in Figure~\ref{fig:zone2feh}) is flatter (i.e., stretching to higher metallicities) than the main 1ab (GSS) and 1bb components. However, this component may also be contaminated by stars from the inner spheroid and M31 disk. We see no significant variation in the metallicities in the region of the GSS (Figure~\ref{fig:zone2fehvsposn}) either along the radial direction (left panel) or with azimuthal angle (right panel). Previous photometric studies have reported spatial variations of the metallicity: \citet{Conn2016} finds that the metallicity in the GSS region increases from [Fe/H]~$\approx-0.7$ to about $-0.2$ near $R_{\rm proj}\approx2.8^\circ$, and then decreases steadily to [Fe/H]~$\approx -1$ at $R_{\rm proj}\approx5.9^\circ$. The pencil-beam spectroscopic metallicity estimates by \citet{Escala2021} find a gradient of $-0.25$ dex/degree, even stronger than those reported by \citet{Conn2016}. While the DESI measurements in Figure~\ref{fig:zone2fehvsposn} show a high mean value of the metallicity, they also show large scatter with no statistically significant systematic trends. However, we caution that these results may be due to our biased selection of targets, and a more comprehensive study of the metallicity variations would require a more complete sampling of the low-metallicity portions of the RGB (by future observations) and a careful accounting of the selection function. \section{Comparison to Simulations} \label{sec: simulations} \subsection{Comparison to Galaxy Formation Simulations in a cosmological context} \label{sec:cosmo} Simulations of galaxy formation in a cosmological context illustrate how mergers can generate complex, organized structure similar to that observed in M31. To illustrate this point, we show in Figure 13 an example of a system like M31 which experienced a fairly massive merger in the last few Gyr. This example is not meant to replicate M31 in any detail, but is provided only to illustrate how streams and shells emerge naturally in cosmological simulations. The example is taken from the TNG-50 simulation \citep{Pillepich2018,Pillepich2019}, which simulates a large cosmological volume (51.7 Mpc on a side) with high resolution (300\,pc softening length for the collisionless particles), enabling an analysis of the detailed kinematics of merger debris. To identify this system within the simulation, we began by selecting systems with properties similar to that of M31 \citep{Ibata2014,dsouza2018,dsouza2021}, i.e., systems with stellar masses 5$\times 10^{10}\,M_{\odot}$ to 15$\times 10^{10}\,M_{\odot}$. We selected galaxies that have a prominent disk by requiring that more than 40\% of stars are on orbits that have a circularity $\epsilon = J_z/J(E) > 0.7$, where $J_z$ is the specific angular momentum of a particle around the angular momentum axis of the stellar body of a galaxy, and $J(E)$ is the maximum angular momentum of the 100 particles with the most similar total binding energies (see also \citealt{genel2015}). In addition, we required that the galaxy have a total accreted mass of at least $3 \times 10^9\,M_{\odot}$ and have had an encounter with a massive satellite ($M_{\rm sat} > 10^{10}\,M_{\odot}$) that fell into the system 2 Gyr to 8 Gyr ago. We then examined recent snapshots of these systems for visual analogs of M31's giant stream and shells. The best match is subhalo ID 482155, which has a present-day dark halo mass of 2.2$\times 10^{12}\,M_{\odot}$ and a present-day stellar mass of 1.2$\times 10^{11}\,M_{\odot}$, and is in the process of accreting a large satellite (stellar mass $10^{10}\,M_{\odot}$) that experienced first infall 6.7 Gyr ago. As the merger is still underway, the dissipating satellite retains a compact core of mass $1.8 \times 10^9\,M_{\odot}$ located $\approx$30\,kpc from the center of the primary galaxy (see Figure \ref{fig:tng50}, left panel); the median metallicity of all the particles from this massive satellite is nearly solar, with ${\rm [Fe/H]} = -0.07$. The satellite is no longer star-forming, but underwent star formation as recently as 2.5\,Gyr ago. Fig.\ \ref{fig:tng50}, which shows three different projections of this system, illustrates how the merger of a single progenitor galaxy can generate a stream, multiple shells, and nested wedges in phase space, similar to those seen in M31. The left panel shows the projected stellar mass density in greyscale with logarithmic scaling. The giant stream analog is clearly visible, as are shell structures and the compact core of the satellite (seen as the dark dot near x$\approx$30, z$\approx-$2.5). The center panel shows the overall kinematic structure of the stellar particles from the infalling satellite using the conventional (simulation) visualization of radial velocity (centered in the frame of the M31 analog) as a function of radius (in 3D rather than projected coordinates), in direct analogy to e.g., Fig.~10a from \citet{Fardal2007} or \citet{Pop2018}. Particles are color-coded by the time when they were last part of the satellite's subhalo (prior to their tidal stripping), showing a clear progression in which the outermost tidal debris arises from earlier episodes of stripping (e.g., red and green points), and material near the still-bound core of the satellite is the most recently stripped (darker blue). % Finally, the right panel shows the line-of-sight velocity as a function of projected radius for stars in the angle wedge containing the giant stream analog, color-coded by metallicity. Although the contributions from all merged satellites are shown in this panel (not just that of the most recent merger as in the center panel), the earlier accreted satellites are all low mass and they merged long enough ago that they no longer contribute fine-scale kinematic structure \citep[e.g.,][]{Beraldo_e_Silva2019}. As a result, the most recent merger completely dominates the properties of the inner halo. Indeed, {\it all} of the substructure in this particular halo is metal-rich, and arises from the stripping of this most massive satellite. We discuss this topic further in Section~\ref{sec:discussion}. \subsection{Comparison to an N-Body Model} \label{sec:nbody_model} The DESI observations can be compared in greater detail to simulations that have been customized to replicate the structure of M31. In order to understand whether a single encounter could account for much of kinematic structure observed in our data, we constructed a simple model informed by the results of previous studies. Previous modeling efforts in this field are described in greater detail in Section~\ref{sec:discussion}. We also publish all our good velocity measurements to aid future modeling attempts. It is our hope that these observations, insights from the cosmological models, and the comparisons presented here can inform future modelers in their efforts to reproduce more of the density and phase space structure of M31's halo. The model consists of a single component Plummer sphere \citep{Plummer1911} describing the progenitor of the GSS, which has a mass of $\sim 2 \times 10^8\,M_\odot$, a half-mass radius of 1 kpc, and is represented by 300,000 particles. M31 is represented by a static analytic potential that consists of the disk, halo, and a bulge, where we use parameters similar to those employed in previous modeling efforts \citep{Fardal2006,Kirihara2017a}. The bulge is a Hernquist bulge \citep{Hernquist1990} with a mass of $3.2\times 10^{10}\,M_\odot$ and size $a=0.5$\, kpc. The dark matter halo is described by an NFW profile \citep{NavarroFrenkWhite1997} with $V_{\rm max}=215$\,\kms\ and a scale radius of $r=7.63$\,kpc. The disk is assumed to have an exponential scale length of $r_d=5.4 $\,kpc, vertical height $h=0.6$\,kpc, and a total mass of $M_{\rm disk} = 3.7\times 10^{10} M_\odot$; it is represented as a linear combination of 3 Miyamoto-Nagai disks, following the prescription of \citet{smith2015}. Further details regarding the simulation are provided in Appendix B. Figure~\ref{fig:Sergey_model} shows results 791 Myr after the start of the simulation. The initial pericentric passage of the GSS progenitor occurred at 188 Myr. While our simulation has not been tuned to match the data perfectly, it does provide a heuristic interpretive guide to the complex kinematic structures in M31. In the bottom panels of Figure~\ref{fig:Sergey_model}, a random selection of 0.1\% of particles are color-coded by their total energy (i.e., kinetic + potential), with the color range extending from red representing particles with the least negative total energy, to blue representing the most negative (i.e., most tightly bound) particles. Unlike the higher mass merger in Section~\ref{sec:cosmo} that retains a bound remnant to the present day, this progenitor is fully disrupted on the first apocentric passage; thus the resulting set of shells can be understood as the debris from one % disruption event, arranged according to energy \citep{DongPaez2022}. The GSS-like southern stream (orange-red points) and the nested shells (green, cyan, and navy points) are cleanly separated in energy. Specifically the stars with the least negative energy have not yet had a second pericentric passage after being stripped, % while the particles with the most negative energies and therefore much shorter orbital period (the cyan points), have already had multiple pericentric passages after the initial stripping episode. Comparing Figures~\ref{fig:Sergey_model} and \ref{fig:allfeaturesannotated}, we see that the main part of the GSS (feature 1ab) is similar to the orange-red points in the simulation; and that the shells denoted by the structures 2a, 4b+5 and 1c+2b (i.e., the NE Shelf, Western Shelf, and SE Shelf) are similar to the simulation points shown in green, cyan, and navy, respectively, in Figure~\ref{fig:Sergey_model}. Stars with the most negative energies (navy-colored points), located in the southeastern sector of M31, are the stars from the leading part of the debris. Figure~\ref{fig:kcc_rvr} shows only the radial velocities of particles in the southeast sector, using the same energy color-mapping scheme as in Figure~\ref{fig:Sergey_model}. Interestingly, the leading particles in position-velocity space do not occupy a full chevron, but instead primarily trace out a locus at negative velocities, because many of these stars have not yet experienced a turnaround at apocenter. The 1bb feature in M31 may have a similar origin. Similar to the situation shown in Figure~\ref{fig:kcc_rvr}, 1bb appears in the same sector as the GSS (equivalent to the orange points in the Figure), is blueshifted, and has no companion redshifted feature that would create a chevron-like pattern. The results suggest that the multiple structures observed in M31 could arise from a single encounter, with the various nested structures produced by subsequent wraps (i.e., different pericentric passages) of stripped stars from the same progenitor. Importantly, in our simulation, the progenitor is fully disrupted in the encounter, so all of the shells that result are essentially a single set of stars wrapping around the galaxy. If the progenitor instead preserves some mass for a second pericentric encounter, as in the cosmological model discussed in Section~\ref{sec:cosmo}, an additional set of shells would be created; these are unaccounted for in our current simulation. Simulating the shell system from a possible second pericentric encounter would be somewhat more challenging, as the dynamical friction of the progenitor likely would need to be taken into account. This second shell system is probably needed to explain some of the smaller chevrons observed in our data. Future observations that more densely sample the structures in the inner halo will provide a unique opportunity to constrain the mass and orbit of the progenitor. \smallskip \section{Constraints on the Mass of M31} \subsection{Shell Kinematics} \label{sec:ShellKinematics} The shell-shaped tidal signatures of galaxy mergers that we observe also offer the opportunity to measure the gravitational potential of the host galaxy, with nested shells probing the gravitational potential as a function of galactocentric distance \citep{Merrifield1998, Sanderson2013}. Our current sample measures radial velocities for stars in multiple shells spanning a range of distances and thereby offers a rare opportunity to constrain the dynamical mass of the galaxy as a function of galactocentric distance using this technique. As described by \citet{Merrifield1998}, for a shell oriented in the plane of the sky, the projected velocity of the shell has a distinctive triangular shape as a function of projected distance (a filled `wedge' or empty `chevron' shape), and the slope of the projected velocity near the outer edge of the shell can be used to infer the gravitational potential. That is, for a spherical shell of radius $r_s$ with a projected velocity $v_{\rm los}$ that increases with decreasing projected distance $R,$ the velocity gradient $d v_{\rm los}/dR = -\Omega,$ where $\Omega$ is the circular frequency at $r_s.$ \citet{Sanderson2013} derived a related expression that includes the effect of the outward velocity of the shell (their eq.~23) and argued that the simpler \citet{Merrifield1998} method will tend to overestimate the enclosed mass. The multiple shell structures observed in M31 allow us to explore these ideas. To explore how well the simple \citet{Merrifield1998} prescription recovers the expected gravitational potential of M31, we measured (by eye) the velocity gradient of the red- and blue-shifted edges of the wedges and chevrons seen in the NE Shelf, the Western Shelf, and in the region of the GSS. These features have approximate projected extents of $\sim 1.3^\circ$ (Features 1cb/1cr and 2br), $\sim 2^\circ$ (Features 2ar and 3ar), and $\sim 2.7^\circ$ (Features 4bb/4br and 5b/5r), which correspond to projected distances of $\sim 19$ kpc, $\sim 28$ kpc and $\sim 38$\,kpc. The lines shown in the insets of Figures~\ref{fig:zoneposvela} and \ref{fig:zoneposvelb} show the regions used to estimate the velocity gradients. Slopes are better defined for features that are densely populated (e.g., 2ar). Feature crowding, the possibility that features overlap each other or are embedded in a distributed background halo, can make it difficult to define the slope of a feature (e.g., 2br resides within 2ar). Higher density spectroscopy of the M31 halo can potentially mitigate these challenges. The measured slopes correspond to circular velocities of 230~\kms\ to 340~\kms\ at the shell radius (i.e., the apex of the wedge or chevron) and imply enclosed masses of $2\times 10^{11}\,M_\odot$ to $8\times 10^{11}\,M_\odot$ over this range of distances (Figure~\ref{fig:vcirc_Me}). The circular velocities are similar to, or larger than, the velocity of the \ion{H}{1} rotation curve of M31 measured over the same range of radii (horizontal line in left panel of Figure~\ref{fig:vcirc_Me}). The rotation curve, which is roughly flat at $\sim 250$~\kms\ from 10 kpc to 40 kpc, implies an enclosed mass that is $4.7\times 10^{11}\,M_\odot$ within 38 kpc \citep{Chemin2009} and declines toward smaller radii as $1/r$ (dashed line, right panel). In comparison, the circular velocity and enclosed mass derived from the properties of the smallest shell (radial extent $\sim 1.3^\circ$) are close to the values inferred from the \ion{H}{1} rotation curve at the same distance. The values for the larger shells (radial extents of 2.0$^\circ$ and 2.7$^\circ$) are larger than the corresponding values from the \ion{H}{1} rotation curve. These results are consistent with the findings of \citet{Sanderson2013}, that the \citet{Merrifield1998} prescription can correctly recover the enclosed mass in some cases, but that it often overestimates enclosed mass by a factor of 2 to 3. A similar result was reported by \citet{Escala2022} in their analysis of the kinematics of stars in a portion of the NE Shelf. In summary, mass estimates from the observed velocity gradients of shells rely on the assumption of shells of stars oriented in the plane of the sky and do not account for complexities introduced by geometry, angular momentum, or the details of the interaction. Consequently, while the overall idea of using shells to estimate the mass of M31 is potentially useful, it is clear that more sophisticated modeling and more extensive spectroscopic samples will be necessary to reach an interesting level of accuracy. \subsection{Kinematics of the GSS} \label{sec:gssmassest} As a complementary approach, stellar streams like the GSS also probe the galactic potential and can plausibly be interpreted using a more detailed dynamical model that is driven by a few simple assumptions. The shells from a single pericentric passage represent a group of stars on a sequence of orbits ordered by energy \citep[see a detailed exposition of shell formation in][]{DongPaez2022}. Stars with more negative energies have shorter orbital periods, while more positive energy particles have longer orbital periods. Thus the shell system will have an energy gradient. Although the energy gradient makes the analysis of the shells more cumbersome, as we cannot rely on the constant energy assumption that approximately works for thin tidal streams \citep{Koposov2010}, we can still effectively use the assumption that the energy changes monotonically along the structure due to energy sorting in the shell. The strength of the energy gradient in the shell is itself limited by the total energy spread in it, which in turn is determined by the energy spread of stars at the pericentric passage of the progenitor i.e $\delta E \sim \frac{1}{2} (V_{\rm peri}+\sigma)^2-(V_{\rm peri} -\sigma)^2 = V_{\rm peri} \sigma$, where $\sigma$ is the velocity dispersion of the progenitor and $V_{\rm peri}$ is the velocity of the progenitor at the pericenter \citep[see][for more details]{DongPaez2022}. It turns out that these basic principles, together with a few assumptions about the GSS geometry, can help us model the radial velocity vs.\ distance behaviour observed in the GSS and constrain the M31 gravitational potential. To define the model we begin by defining a coordinate system $x,y,z$ in which the $z$-direction is oriented along the line connecting the Sun and M31 pointing away from the Sun, the $x$-direction is aligned with the East, and the y-direction points North. Projected on the sky, the GSS forms an essentially linear structure, with position angle $\phi_{GSS}\sim $155$^\circ$. We assume that the GSS is also a linear structure in 3D, defined by the unit vector ${\bf \hat k} = [k_x,k_y,k_z]$ and that the line defined by this vector intersects the projected center of M31 (i.e., $x,y=(0,0)$) at a (small) distance $z_{\rm off}$ from the center of M31. We then assume that stars in the GSS move along this vector $ {\bf \hat k}$. Thus the GSS stars are assumed to be on nearly radial orbits. As discussed earlier, we expect an energy gradient along the GSS and therefore we assume that the total energy (potential and kinetic) of stars in the GSS can be approximated by a linear gradient along the stream (see, e.g., Appendix B and Figure~\ref{fig:sergey_energy_plot}), i.e., $ E(R) = E_0 + \frac{dE}{dR}(R-R_0)$ where $R$ is a projected distance along the stream and $E_0$ is the energy at $R_0,$ the projected distance at the same point (i.e., the middle of the stream). We assume that the energy gradient is positive and limited by the maximum range of energies along the stream $\delta E_{\rm max}$ $0<\frac{dE}{dR} (R_2-R_1)<\delta E_{\rm max}$, where $R_1$ and $R_2$ are the projected distances that limit the observed portion of the GSS. The reason for the assumption of the positive energy gradient is that this is exactly what we expect for the trailing part of the shell. The GSS shell stars are currently falling back to M31 (from the first pericentric encounter of the GSS progenitor) and the most distant stars have the least negative energies (and therefore longest orbital periods). See for example the bottom left panel of Figure~\ref{fig:Sergey_model} showing the positive energy gradient in the GSS. The adopted upper bound on the energy spread is $\delta E_{\rm max}$ = 30 $\times$ 500 (\kms)$^2$, the energy spread resulting from a progenitor with an initial pericentric velocity of 500~\kms\ and velocity dispersion of 30~\kms. Because the GSS is only the very end of the shell system in M31, and the middle and leading part of the shell system are likely responsible for the NE Shelf and Western Shelf respectively, we expect that the actual energy spread for stars in the GSS is much smaller than $\delta E_{\rm max}$. The final assumption is that energies $E(R)$ are always negative along the stream, i.e., all the GSS stars are bound. While we make several assumptions here (e.g., that the energy gradient is linear as a function of projected radius and that the stars within the GSS are moving on primarily radial orbits), we have verified that in the fiducial N-body model of the disruption of the GSS progenitor presented in Section~\ref{sec:nbody_model} and Appendix~B that these assumptions are satisfied. The geometric assumptions that the stream is linear and the stars move along it tell us that the 3D velocity should be changing as a function of projected distance $R$ along the stream as \begin{equation} {\bf V}(R) = a(R) {\bf \hat k} \label{eqn:v_ar} \end{equation} where $a(R)$ is an unknown function. Since the (line-of-sight) radial velocity is simply a projection of the 3D velocity along the $z$-axis, \begin{equation} V_{\rm los}(R) = a(R) k_z. \label{eqn:vrad} \end{equation} Under the assumption of a linear change of energy with radius along the GSS we can write \begin{equation} \frac{V^2(R)}{2}+ \Phi({\bf X}) = E_0 + (R-R_0) \frac {dE}{dR}, \label{eqn:energy} \end{equation} where ${\bf X}$ is the 3D position along the stream \begin{equation} {\bf X}=(R {k_x},\ R {k_y},\ z_{\rm off}+R k_z/\sqrt{1-k_z^2} ) \label{eqn:definitionX} \end{equation} corresponding to a projected distance $R$ along the stream and $\Phi({\bf X})$ is the gravitational potential. Combining Eq.~\ref{eqn:v_ar} and \ref{eqn:energy} allows us to write an expression for $a(R)$: \begin{equation} a(R) =\sqrt{2 \left( E_0 +\left[\frac{dE}{dR} (R-R_0) \right] - \Phi( {\bf X})\right)} \label{eqn:ar} \end{equation} which gives us the expression for $V_{\rm los}(R)$ through Eq.~\ref{eqn:vrad} if we know $E_0$, $\frac{dE}{dR},{\bf \hat k},$ and the gravitational potential. Essentially we now can write the likelihood for the radial velocity as a function of projected distance $P(V_{\rm los}|R,{\bf \hat k},\Phi(R),E_0, \frac{dE}{dR}) $ that we can fit to the velocity track of the GSS. While the number of parameters is potentially quite large, we can adopt informative priors on many of them. We have previously described our constraints on energy and energy gradients $E_0$ and $\frac{dE}{dR}$. Furthermore, the GSS orientation parametrized by ${\bf \hat k}$ is well constrained by its projected orientation on the sky and the measured distance gradient of 20\,kpc\,deg$^{-1}$ along its 6$^\circ$ extent \citep{Conn2016}. We therefore adopt a uniform prior for the distance gradient to be between 15\,kpc\,deg$^{-1}$ and 25\,kpc\,deg$^{-1}$. A simple algebraic equation for the distance gradient provides a prior on $k_z$. For the gravitational potential we adopt a typical bulge/disk/halo decomposition with the bulge and disk models to be Hernquist and Miyamoto-Nagai models respectively with fixed parameters from \citet{Kirihara2017a}. We assume a disk inclination angle of 77$^\circ$ and a position angle of the major axis of 38$^\circ$. The dark matter halo component is modeled as an NFW \citep{NavarroFrenkWhite1997}, where the halo mass $M_{\rm halo}$ and scale-length $r_s$ are to be determined. We adopt a log-uniform prior on the mass $10^8\,M_\odot <M_{\rm halo}<10^{14}\,M_\odot$ and scale-length $1\,{\rm kpc}<r_s < 100\,{\rm kpc}$. This completes the definition of our model likelihood and parameter priors. For the locus of the GSS, we used the result of the two-component fit described in Section~\ref{sec:PVdiagram}. The radial velocity measurements in 9 positions together with their uncertainties along the GSS were then fit by the $V(R)$ model as described in Eq.~\ref{eqn:v_ar} and \ref{eqn:ar}. The posterior was sampled with the {\tt dynesty} nested sampler. The model had 6 parameters in total: the halo mass and scale length, the distance gradient, energy and its gradient $E_0$ and $dE/dR$, and the offset of the stream from pointing directly at the M31 center $z_{\rm off}$. The posterior on these parameters is shown in Figure~\ref{fig:sergeymassest}. To avoid the typical mass-size degeneracy, we show the posterior for the mass inside 125 kpc rather than the total halo mass. Multiple parameters are unconstrained (such as the distance gradient, where we are purely driven by the prior), which is not very surprising given the limited data available. We also note that the offset of the GSS from pointing directly at the M31 center ($z_{\rm off}$) is consistent with zero, confirming that the orbits are very close to radial. We also see that the energy gradient prefers significantly lower values than our threshold, which is reasonable, given that we expect the GSS to be only a small (trailing) part of the shell. We find the halo mass within 125 kpc to be $\log_{10}\, M_{\rm NFW}(<125\,{\rm kpc})/M_\odot = 11.78_{-0.10}^{+0.13}$ or if we include the disk and the bulge $\log_{10}\,M_{\rm total}(<125\,{\rm kpc})/M_\odot = 11.83_{-0.10}^{+0.13}$. As the method we employed makes significant assumptions, we have also applied exactly the same fitting procedure to the sample of stars from the simulation presented in Section~\ref{sec:nbody_model} and obtained the halo mass with the bias of $\log_{10} M_{\rm halo,fit} - \log_{10} M_{\rm halo,true}\approx 0.1$ which is within our uncertainty. Our mass estimate of $\log_{10}\, M_{\rm NFW}(<125\,{\rm kpc})/M_\odot = 11.78_{-0.10}^{+0.13}$ is consistent with estimates from the literature of the enclosed mass at this distance \citep[e.g., graphical summary in][]{Kafle2018}. In particular, our result is similar to that of \citet{Ibata2004}, who carried out the first kinematic study of the GSS, measuring the velocities of 184 stream stars and using the velocity gradient along the stream to estimate a halo mass of $M_{125}=7.6\pm1.2\times 10^{11} M_\odot$ for a logarithmic halo and $M_{125}=6.4\pm1.3\times 10^{11} M_\odot$ for an NFW halo. \section{Discussion} \label{sec:discussion} As described in the previous sections, DESI spectroscopy reveals intricate, coherent spatial-velocity structure in the inner halo of M31, including nested chevrons and wedge-shaped structures (Figures~\ref{fig:zoneposvela}, \ref{fig:zoneposvelb}), with a spatial and kinematic clarity never-before observed in an extragalactic source (Section~\ref{sec:results}). The DESI results affirm earlier ``pencil beam'' spectroscopy carried out in restricted portions of the inner halo. % The observed structures are consistent with the expected kinematic signatures of shells and streams produced in galaxy mergers (\S~\ref{sec:cosmo}, \ref{sec:nbody_model}) and suggest that most, if not all, of the structure observed in M31 arises from a single merger event (\S~\ref{sec:nbody_model}). We illustrated how the kinematics of the structure induced by the merger---the shells and the GSS---can dynamically probe the mass distribution of M31 as a function of galactocentric distance (\S~\ref{sec:ShellKinematics}, \ref{sec:gssmassest}). In this section we situate our results in the context of prior work and turn to the question of the nature of the progenitor that produced the observed substructure. \subsection{Comparison to Previous M31 Merger Models} \label{sec:PrevObsandModels} Many previous studies have explored and advanced a picture in which much of the inner halo substructures of M31 are tidal debris from a single companion galaxy that encountered M31 on a nearly radial orbit \citep[e.g.,][]{Ibata2004, font2006, Fardal2006, Fardal2007, Fardal2008, Fardal2012, Fardal2013, Mori2008, Sadoun2014, Kirihara2017a,Milosevic2022}. These simulations have explored a wide range of parameters, and found that a wide range of progenitor stellar masses can reproduce the observed morphologies. Several studies have suggested that the visible debris is the result of a minor merger ($\sim$ 1:10 to 1:5), with the stellar mass of the companion in the range 1--5$\times 10^9\,M_\odot$ \citep[e.g.,][]{Fardal2013, Kirihara2017a,Sadoun2014}. In contrast, a few studies have suggested that the observational data suggest a major merger (i.e., $\sim$1:4) with a progenitor of stellar mass $>10^{10}\,M_\odot$ \citep{dsouza2018,Hammer2018}. \citet{dsouza2018} advocated for a major merger based on the mass, metallicity, and star formation history (SFH) of the halo. They also hypothesized that M32, M31's compact satellite, could be the core of the disrupting satellite based on metallicity and star-formation history. M32 is located within the debris field close to where the GSS meets the M31 disk and has a very different velocity from the GSS, indicating that M32 would be at a very different phase of its orbit than the GSS material. Other studies predict that the progenitor lies elsewhere in the debris or may be completely disrupted. \citet{Hammer2018} additionally note that the 2$-$4~Gy-old star-formation episode and significant thick disk of M31 might both be the product of the interaction with a massive progenitor. Although the morphology of the debris appears to be insensitive to the mass of the progenitor \citep[e.g.,][]{Hammer2018,Boldrini2021}, the simulations reveal that it is sensitive to the orbital parameters of the encounter. Dynamical models which attempt to account for both the observed spatial distribution of the debris and the radial velocities available to date generally infer an initial pericentric passage within a few kpc of the center of M31 within the last 1--2 Gyr. In many of the models tailored to M31 (including our own from \S~\ref{sec:nbody_model}), a companion galaxy plunges into M31 and its stars are pulled out on the far side of M31 to form the GSS following the first pericenter passage. The DESI observations do not detect any outward moving stars following the first pericentric passage, but they do detect the infalling stream of stars on their way back toward M31 after their first apocentric passage. The NE Shelf is produced as the second wrap of the orbit, and the Western Shelf constitutes the third wrap \citep{Fardal2007,Fardal2008,Fardal2012,Fardal2013}. In the models of \citep{Fardal2013, Kirihara2017a}, the core of the progenitor, if it has survived tidal disruption, is predicted to reside somewhere in the NE Shelf. All of the models are successful in accounting for the general spatial morphology of the GSS, NE Shelf, and Western Shelf, as well as spectroscopic observations of the GSS and the Western Shelf available to date. The anticlockwise-rotating thick-disk model of \citet{Kirihara2017a} better reproduces the edge-brightening observed on the eastern side of the GSS. In addition, some models \citep[e.g.,][]{Milosevic2022,Kirihara2017a} also reproduce the metallicity variations observed in the GSS by \citet{Conn2016}. Previous studies have compared their simulations with the velocities of either small numbers of PNe or larger numbers of RGB stars measured with Keck/DEIMOS in pencil beams located at a few radial positions within the inner halo \citep[e.g.,][]{Fardal2007,Fardal2013}. In particular, the model of \citet{Fardal2007}, which was designed to replicate the observed substructure in photometric imaging studies of M31, is remarkable in capturing many of the observed features in the DESI radial velocity data. Since these model data were not available to us, we made simple comparisons of our observations plotted in the same way as the simulations, comparisons which corroborate many of the features predicted by \citet{Fardal2007}. In comparing to earlier data, \citet{Fardal2007} showed how 11 PNe from \citet{Merrett2003,Merrett2006} (which have kinematics classified as ``stream" or ``stream?'') trace out the predicted locus of the blue-shifted edge of the NE Shelf, the large wedge in the position-velocity diagram at $-500 \lesssim V_{\rm los} \lesssim -100$~\kms\ (see the right panel of Figure~3 of \citet{Fardal2007}). The DESI observations overlap with the PNe and chart out the wedge-like structure more completely and in greater detail on both the red- and blue-shifted edges and show that the structure extends to slightly larger projected distances than predicted by the \citet{Fardal2007} model (see also Figure~\ref{fig:pnlit}). Earlier studies also compared the positions of PNe from \citet{Merrett2003,Merrett2006} and stars along the minor axis of M31 with the simulation predictions in the Western Shelf region \citep[e.g.,][]{Fardal2007,Fardal2013}. The DESI data clearly trace out the shell-like nature of the kinematic structure in the Western Shelf over a large spatial extent, also showing only minor deviations, especially at projected radii $>1.5^\circ$. While much of the structure observed with DESI is roughly consistent with the dynamical models published to date, some features remain unaccounted for. Notably, the blueshifted feature 1bb in Zone 1 is not reproduced, as has been previously noted \citep[e.g.,][]{vanderMarel2012}. The DESI data also reveal new features. For example, in the Zone 2 region of the NE Shelf, the DESI data show a second, smaller wedge extending out to 1.3$^\circ$ and highlighted in Figure~\ref{fig:zoneposvela}, which is not predicted by any of the published simulations. In contrast, the compact chevron in Zone 1 that is bounded by $|V_{\rm los}| \lesssim 150$~\kms\ does appear to be present in one simulation shown in \citet{Fardal2013}. Their Figure 6 shows a compact component in Zone 1 that extends to a similar distance from M31 as the observed structure. In the model, the component arises from NE Shelf stars on the near side of M31 that overlap the GSS. Finally, the 3br feature in Zone 3 is also not present in the models. These initial DESI results represent a significant advance by covering large areas more uniformly and revealing the kinematic structures in unprecedented clarity. These data inform future modeling efforts to understand the merger history responsible for the complex inner halo substructure of M31. \subsection{Clues to Nature of the Progenitor} \label{sec:NatureofProgenitor} As simulations have demonstrated that the morphology of the debris is relatively insensitive to the mass of the progenitor \citep[e.g.,][]{Boldrini2021}, other information is needed to constrain the nature of the progenitor. Previous studies have attempted to infer the nature of the progenitor using the metallicity (a wide range in metallicity, reaching more than 1/3 solar in the inner parts of the debris; \citealt{Gilbert2014,Ibata2014}) and star formation history (showing star formation until around 2-3\,Gyr ago; \citealt{Brown2006}) implied by measurements of the inner stellar halo and substructures \citep{dsouza2018}. Here we contribute to this topic by commenting on (1) the metallicity of the stellar debris, (2) the number of \textcolor{cyan}{dwarf galaxies and} globular clusters potentially associated with the progenitor, and (3) whether there is any evidence for a surviving progenitor galaxy. Since our target selection introduces a bias towards metal-rich RGB stars (\S~\ref{sec:metallicities}), we cannot use the current DESI data to reliably measure the metallicity distribution of the accreted stars. However, we do find that significant numbers of metal rich stars are present across all regions surveyed, suggesting that the progenitor responsible for these structures is relatively high mass, high enough to have stars up to solar metallicity. Future DESI observations that target stars more metal poor that those studied here can better characterize the metallicity distribution of the progenitor. Given that the progenitor was probably massive (i.e., $> 10^9\,M_{\odot}$), it is possible that the merger event will have delivered star clusters and dwarf galaxies to M31. Figures~\ref{fig:dwarfgals} and \ref{fig:gclit} show the distribution of dwarf galaxies and (massive) star clusters with measured velocities from the literature compared to the stars measured with DESI. The dwarf galaxy measurements are from the compilation of \citet{McConnachie2012} and the star cluster measurements are primarily from LAMOST spectroscopic surveys of \citet{Chen2015,Chen2016}. The clusters include globular clusters and massive young clusters spanning a range of ages \citep[for details see][]{Chen2016}. The dwarf galaxy Andromeda I closely overlaps the GSS (Figure~\ref{fig:dwarfgals}), as does at least one globular cluster at a distance of 3.5$^\circ$ (Figure~\ref{fig:gclit}), suggesting a possible physical association. The association of this globular cluster, LAMOST-1, with the GSS has been previously noted by \citet{Chen2015,Chen2016}. LAMOST-1's metallicity ([Fe/H] =$-0.4$) and age (9.2 Gyr) are consistent with an association with the GSS progenitor. In Zones 2 through Zone 5, the distribution of star clusters is similar to that of the stars in the inner halo. In particular, they populate the interior of the wedge in Zone 2 ($\theta$ = 130$^\circ$--230$^\circ$) and the small wedge in Zones 3+4 ($\theta$ = 230$^\circ$--315$^\circ$). Thus, interestingly, many of the clusters are potentially associated with the wedge structures within 2$^\circ$ of M31, while relatively few clusters overlap the GSS. These results are roughly consistent with expectations from merger simulations. For example, considering the representative galaxy merger in the Illustris TNG-50 simulation (Fig.\ \ref{fig:tng50}, larger solid points in the right-hand panel), one could very crudely subsample particles from the dominant merger companion that are `old' (9-12\,Gyr ago), mirroring the epoch of globular cluster formation in the Milky Way. These early-forming star particles tend to be more centrally concentrated and kinematically hotter than the bulk of the progenitor stars, and are not as clearly confined to kinematically cold substructures. While more detailed model predictions that follow globular cluster formation in galaxies and their expected distribution among the tidal debris are clearly needed to fully interpret the observations and obtain robust constraints on the nature of the progenitor, this exercise tentatively suggests that it may be challenging to accurately attribute globular clusters to the progenitor solely on the basis of their clustering into kinematic substructures. Future studies might explore the metallicities and orbits of M31 globular clusters to infer their association with the progenitor galaxy. Finally, with its ability to map out stellar velocity structure over large areas, DESI offers the opportunity to locate the remnant core of the progenitor galaxy. \citet{Fardal2013} predicted that if the progenitor survives, stars from the core of the progenitor will populate a fairly compact structure in phase space located at a projected distance of $1^\circ \lesssim R_{\rm proj} \lesssim 2^\circ$ and a line-of-sight velocity of $\approx$ 0 to $-200$~\kms (i.e., in Zone 2 and blue-shifted relative to the M31 systemic velocity). \citet{Kirihara2017a} predict that the stripped bulge of the progenitor lies in the eastern shell and in front of the disk of M31, compact in phase space and at a location of ($\xi,\eta,V_{\rm los})\approx (1.1^\circ,0.5^\circ,-200$~\kms). No such structures are detected in the DESI data, although this may yet be due to the sparseness of our current sampling. We do find that the velocities in Zone 2 show a preferential {\it redshift}, rather than the blueshift predicted by \citet{Fardal2013}, and they show evidence of multiple shells rather than a component that is compact in phase space. Future DESI observations could place stronger constraints on (or possibly identify) a surviving progenitor galaxy. More densely sampled spectroscopy will permit quantitative assessment of the possibility that M32 (D’Souza \& Bell 2018) or another existing galaxy is the progenitor, or perhaps identify a remnant or disrupting core in the inner halo of M31. M31 and the Milky Way show a remarkable parallel, in that the inner halos of both galaxies are dominated by debris from a single accretion event. The Milky Way's inner halo is dominated by the {\it Gaia}-Sausage-Enceladus structure, a radial accretion event of mass $>10^{10}~M_\odot$ nearly 8-11~Gyr ago \citep[e.g.,][]{Belokurov2018,Helmi2018}. The inner halo of M31 is also dominated by the single radial accretion event that produced the GSS and the intricate kinematic structures studied here, but which began only 1-2 Gyr ago. If the M31 shell system progenitor is indeed as massive as suggested based on its total stellar luminosity and stellar metallicities, M31 may provide a glimpse of what the Milky Way looked like several Gyr ago. Future spectroscopic surveys of the M31 inner halo will be able to explore this exciting possibility in greater detail. \section{Summary and Conclusions} \label{sec:conclusion} We have obtained spectra, in three DESI pointings, of \Ntargets\ targets in the direction of M31. Using these observations, we have measured accurate radial velocities of \Nstars\ stellar sources, of which \NAnd\ are members of the M31 system. These include radial velocities for \NAndHIIPN\ HII regions and Planetary Nebulae, and \Nclusters\ M31 clusters. We have also identified \NQSO\ QSOs and \Ngalaxies\ galaxies behind M31, which can provide unique probes of the gas associated with the GSS progenitor and other circumgalactic and interstellar material associated with M31. While most of the earlier spectroscopy of individual stars in M31 had been carried out with 6.5-m to 10-m class telescopes, a few hours of spectroscopy with DESI has added significantly to our knowledge of the stellar kinematics of the M31 halo. These data represent a $>3$-fold increase in the number of known M31 stars in the region outside the M31 disk, and provide a much more uniform sampling of the inner halo than any previous spectroscopic study. The rapid advance is due to (1) DESI's wide field of view, high multiplex, and high observing efficiency; (2) the use of selection criteria that efficiently select M31 stars with limited contribution from foreground Milky Way stars; (3) the strong molecular bands in the late-type spectra of the M31 sample, which enables reasonable radial velocity accuracy ($< 10$ \kms) on faint stars ($z$ = 21.5 AB mag); and (4) the good match of DESI's fiber density to the stellar target density of M31. The DESI spectra reveal intricate coherent kinematic structure in the positions and velocities of individual stars in the inner halo of M31: streams, wedges, and chevrons that provide evidence of a recent merger, i.e., a galactic migration event. While hints of these structures have been glimpsed in earlier spectroscopic studies of M31, this is the first time wedges and chevrons have been {\it mapped} with such detail and clarity in a galaxy beyond the Milky Way. We find evidence for multiple coherent structures in the vicinity of the GSS and clear kinematic evidence for shell structures in the W Shelf and NE Shelf regions. In particular, we identify % 750 stars in the largest kinematic component (feature 1ab) of the GSS and measure a narrow velocity dispersion of $10.80\pm0.75$~\kms. The DESI data also reveal new structures not predicted by existing merger simulations. The kinematic structures seen in the stellar distribution of M31 halo stars are echoed in the position-velocity distribution of known M31 PNe. Dynamical models from the literature that were constructed to explain the spatial morphology of the GSS and other inner halo features, as well as the models presented here, predict position-velocity structures that are remarkably similar to those observed. The results suggest that much of the substructure in the inner halo of M31 is produced by a single merger event with a companion galaxy a few Gyr ago. Taken together, the richness of the observed structure demonstrates that large spectroscopic samples can place valuable constraints on the recent merger history of M31 and that such samples are within the grasp of the Mayall/DESI system. We find significant numbers of metal-rich stars across all of the detected substructures, suggesting that the progenitor galaxy (or galaxies) had an extended star formation history, one perhaps more representative of more massive galaxies. Known populations of stellar clusters in the halo of M31 appear to be more closely associated with the inner wedge structures (within 2$^\circ$ of M31) than the spatially extended GSS. The difference seems plausible if the clusters are predominantly older systems that originated in a kinematically hotter component in the progenitor galaxy. The shell structures and the GSS also offer an opportunity to constrain the gravitational potential of M31 as a function of galactocentric distance. Using the simple prescription of \citet{Merrifield1998}, we obtained from the velocity gradients of the nested shell structures galaxy mass estimates ranging from $2\times 10^{11}M_\odot$ to $8\times 10^{11}\, M_\odot$ at projected distances between 17 and 38~kpc. These estimates are within a factor of 2 of the enclosed mass inferred from the \ion{H}{1} rotation curve of M31 at distances of $\sim 20$~kpc to $\sim 40$~kpc. A more detailed dynamical model fit to the GSS velocities implies a dark matter mass of $6.0^{+2.1}_{-1.2} \times 10^{11}\,M_\odot$ within 125 kpc, in good agreement with estimates from the literature \citep[e.g., ][]{Ibata2004}. M31 is remarkably similar to the Milky Way in that the inner halos of both galaxies are dominated by stars from a single accretion event. Indeed, a recent study of the kinematics of Milky Way stars near the Sun reports chevron-shaped kinematic substructures % \citep{belokurov2022a} that are reminiscent of those reported here. If the progenitor of the M31 shell system studied here is $\gtrsim 10^{10}\,M_\odot,$ M31 may provide a close analogue to what our own galaxy looked like several Gyr ago. More extensive DESI studies of M31 can explore this possibility by: (1) better characterizing kinematic substructures (shells, etc.) with higher sampling density; (2) extending our study of the metal-rich halo population to a characterization of the metal-poor population; (3) identifying the dwarf galaxies and globular clusters potentially associated with the progenitor; and (4) searching for evidence for a surviving progenitor galaxy. Although here we identified shells by eye---which was appropriate given the limited data available---with higher density sampling, we can measure the shells more accurately. By combining these more precise measurements with a detailed dynamical model customized to M31, we can place better constraints on the orbit of the progenitor and the mass and shape of the gravitational potential of M31. Characterizing the metal-poor population will allow us to better determine the metallicity of the progenitor and constrain its star formation history and total mass, as well as explore the more virialized (dynamically older) halo of M31. Extending over a large fraction of the galaxy’s volume, the delicate chevrons we observe are also sensitive to the gravitational perturbations from substructure within the M31 halo, such as satellites and dark matter subhaloes. More refined mapping of the chevrons may be able to provide constraints on the number of such substructures. Finally, future work can also % examine the structure and kinematics of the disk and the nature of the circumgalactic and interstellar media probed by the background QSOs and galaxies. The observations presented here, obtained in just three DESI pointings with effective exposure times of $\le90$~min, demonstrate the remarkable ability of DESI, on the Mayall 4-m telescope, to efficiently map out the large scale kinematic structure of M31. Given DESI’s efficiency, we can extend these studies to a larger volume and probe the outer halo of M31 and its interaction with its galactic neighbors (M33 and others). Photometric imaging studies of this region show streams and other structures. A future targeted survey could cover a significant fraction of M31's stellar halo with about 25 tiles. Such a survey would potentially increase the number of M31 halo stars by over an order of magnitude and reveal its structure and immigration history in unprecedented detail. \begin{acknowledgments} We thank Chien-Hsiu Lee, Monica Soraisam, Amanda Quirk, and Raja Guhathakurta for help in selecting filler targets for the original M31 DESI first light tile and for useful conversations regarding our remarkable neighbor. We also thank Nelson Caldwell for generously providing early access to the CFA Optical/Infrared Science Archive which records the many MMT/Hectospec spectroscopic campaigns on M31. We thank Vasily Belokurov for stimulating discussions and detailed comments on the manuscript. AD and JN's research activities are supported by the NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. AD and JN's research is also supported in part by Fellowships from the John Simon Guggenheim Memorial Foundation and by the Institute for Theory and Computation at the Harvard-Smithsonian Center for Astrophysics. JN also acknowledges support from the Harvard Radcliffe Fellowship Program of the Radcliffe Institute for Advanced Study at Harvard University. JN, GM, and JJ-Z acknowledge support from the Harvard University's Radcliffe Research Partners Program of the Radcliffe Institute for Advanced Study. EFB is grateful for support from the National Science Foundation through grant NSF-AST 2007065. LBS acknowledges NASA-ATP award 80NSSC20K0509 and Science Foundation AAG grant AST-2009122. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This research is supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No. DE–AC02–05CH11231, and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract; additional support for DESI is provided by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to the NSF’s National Optical-Infrared Astronomy Research Laboratory; the Science and Technologies Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; the French Alternative Energies and Atomic Energy Commission (CEA); the National Council of Science and Technology of Mexico (CONACYT); the Ministry of Science and Innovation of Spain (MICINN), and by the DESI Member Institutions: \url{https://www.desi.lbl.gov/collaborating-institutions}. The authors are honored to be permitted to conduct scientific research on Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the Tohono O’odham Nation. This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement. This paper made use of the Whole Sky Database (wsdb) created by Sergey Koposov and maintained at the Institute of Astronomy, Cambridge by Sergey Koposov, Vasily Belokurov and Wyn Evans with financial support from the Science \& Technology Facilities Council (STFC) and the European Research Council (ERC). This research also used the PAndAS data hosted by the facilities of the Canadian Astronomy Data Centre, which is operated by the National Research Council of Canada with the support of the Canadian Space Agency. For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission. % This work was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. This work was partially supported by a grant from the Simons Foundation. \facility{KPNO:Mayall (DESI), WISE, Gaia, CFHT:Megacam} \software{{\tt pyyaml} (\url{https://github.com/desihub}), Astropy \citep{astropy:2013, astropy:2018}, sqlutilpy \citep{sqlutilpy}, dynesty \citep{speagle2020,dynesty}, Chainconsumer \citep{Hinton2016}, gala \citep{gala,adrian_price_whelan_2020_4159870}, Q3C \citep{Q3C}, gyrfalcon \citep{dehnen2014}} \end{acknowledgments} \bibliographystyle{apj} \bibliography{bibliography,bibliog_overview} \appendix \section{Quasars and Galaxies Behind M31} \label{appendix:qso} QSO candidates were selected using a combination of the {\it Gaia} DR2 \citep{GaiaMission2016,GaiaDR2summary} and the deep combined imaging data from the unWISE catalogs \citep{unWISEcoadd_2014,unWISE3_2017,unWISE1_2017,unWISE5_2019,unWISE5cat_2019} using the following criteria: \begin{itemize} \item $\pi-\sigma(\pi) \le 0.1$ \item $\vert\mu_\alpha-2\sigma(\mu_\alpha)\vert \le 0.1$ and $\vert\mu_\delta-2\sigma(\mu_\delta)\vert \le 0.1$ \item ($G<19$ and AEN$<10^{0.5}$) or ($G\ge 19$ and AEN$< 10^{0.5+0.2(G-19)})$ \item $(W1-W2) > 0.5$ \item $(W1-W2) > (1.0 - 0.125*(G-W1))$ \item $G \le 26.46 - 5.991*(BP-RP) + 1.313*(BP-RP)^2 - 0.07856*(BP-RP)^3$ \end{itemize} where $\pi$, $\mu_\alpha$, $\mu_\delta$ $\sigma(\pi)$, $\sigma(\mu_\alpha)$, $\sigma(\mu_\delta)$ are the parallax, proper motion, and associated uncertainties from the {\it Gaia} DR2 catalog; $G$, $BP$, $RP$ are the {\it Gaia} DR2 mean photometric magnitudes; $W1$, $W2$ are the {\it WISE} channel 1 and 2 magnitudes from the unWISE catalogs; and AEN is the Astrometric Excess Noise parameter from the {\it Gaia} DR2 catalog. The first three criteria are used to distinguish QSOs from Milky Way stars on the basis of parallaxes and proper motions consistent with zero in the {\it Gaia} DR2 catalog; these are generalized versions of the criteria used by \citet{vanderMarel2019} to select stars in M31 from the {\it Gaia} catalog. The AEN criterion is the same used by the DESI program to separate point sources from extended sources (i.e., galaxies) for {\it Gaia} DR2\footnote{See \url{https://github.com/desihub/desitarget/blob/2.5.0/py/desitarget/gaiamatch.py\#L207-L210}}. The $(W1-W2) > 0.5$ criterion is a more relaxed version of the {\it WISE} AGN selection discussed in \citet{SternWISEQSO2012}. The {\it Gaia} - {\it WISE} criteria were determined based on identifying the known spectroscopically confirmed QSOs \citep[from][]{massey2019,Huo2010,Huo2013,Huo2015} in $G-W1-W2$ color-color space. Finally, the {\it Gaia} $G-BP-RP$ color criterion is an attempt to avoid stars from the M31 RGB in the QSO selection (see Figure~\ref{fig:QSOs}). 183 QSO candidates were targeted successfully (i.e., without fiber positioning errors) on the three DESI tiles discussed in this paper, 172 of which were spectroscopically confirmed as QSOs. The remaining 11 includes 8 stars and 3 galaxies. This represents a $\approx$94\% success rate in the QSO selection criteria. In addition, 12 of our M31 stellar candidates turned out to be QSOs, and \Ngalaxies\ were background galaxies. The spectroscopically confirmed QSOs and galaxies are presented in Tables~\ref{tab:qsos} and \ref{tab:galaxies} respectively. Figure~\ref{fig:qsogalzs} shows the redshift distribution of the extragalactic sources and the sky distribution of the QSOs. These targets are useful probes of the interstellar and circumgalactic media of M31, and in particular provide a way of investigating any gas that may be associated with the various kinematic structures traced by the stellar debris \cite[e.g.,][]{Koch2015}. \begin{deluxetable}{cccccccl} \tablecaption{QSOs Behind M31 \label{tab:qsos}} \tablecomments{Table 5 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.} \tablehead{ \colhead{ID} & \colhead{RA ($^\circ$)} & \colhead{Dec ($^\circ$)} & \colhead{Redshift} & \colhead{\bf $G_{\rm DR2}$} & \colhead{\bf $g_{\rm PAndAS}$} & \colhead{\bf $i_{\rm PAndAS}$} & \colhead{\bf Alternate Name} } \startdata 1 & 10.0373918 & 40.1050291 & 2.197 & 19.92 & 19.68 & 19.46 & Gaia DR2 369102106371697792 \\ 2 & 10.3592920 & 40.8907780 & 1.158 & 0.00 & 19.87 & 19.27 & \\ 3 & 10.0770974 & 39.8989696 & 0.284 & 19.77 & 19.90 & 19.54 & Gaia DR2 368710061756016896 \\ 4 & 10.4851362 & 39.9701449 & 1.834 & 19.57 & 19.84 & 19.42 & Gaia DR2 369044622529595520 \\ 5 & 10.0417886 & 39.7983967 & 0.675 & 19.98 & 0.00 & 0.00 & Gaia DR2 368707720998989184 \\ 6 & 11.2855672 & 37.7439449 & 1.934 & 19.90 & 20.38 & 20.00 & Gaia DR2 367495681227400192 \\ 7 & 10.8090715 & 37.6107886 & 2.489 & 19.69 & 19.98 & 19.86 & Gaia DR2 367443454424848768 \\ 8 & 11.0832708 & 37.6063001 & 2.533 & 19.02 & 19.48 & 19.19 & Gaia DR2 367490218028868352 \\ 9 & 11.2844517 & 37.5931928 & 1.287 & 20.30 & 20.74 & 20.23 & Gaia DR2 367492142174203904 \\ 10 & 10.8512314 & 37.7872685 & 2.199 & 19.11 & 0.00 & 0.00 & Gaia DR2 367543651717272064 \\ \enddata \end{deluxetable} \begin{deluxetable}{ccccccl} \tablecaption{Galaxies Behind M31 \label{tab:galaxies}} \tablecomments{Table 6 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.} \tablehead{ \colhead{ID} & \colhead{RA ($^\circ$)} & \colhead{Dec ($^\circ$)} & \colhead{Redshift} & \colhead{\bf $g_{\rm PAndAS}$} & \colhead{\bf $i_{\rm PAndAS}$} & \colhead{\bf Alternate Name} } \startdata 1 & 10.3558474 & 40.5148375 & 0.236 & 0.00 & 0.00 & GC7191,SK078B ,SK078B \\ 2 & 9.8651259 & 39.8292622 & 0.749 & 24.41 & 21.92 & PANDAS 95102 \\ 3 & 10.8841125 & 37.6855167 & 0.560 & 23.81 & 21.52 & PSUPP 20989 \\ 4 & 11.0291593 & 37.6711566 & 0.676 & 24.29 & 21.83 & PANDAS 90320 \\ 5 & 11.0145968 & 37.7888733 & 0.765 & 24.67 & 22.08 & PANDAS 90419 \\ 6 & 11.0623134 & 37.7701455 & 0.674 & 24.99 & 21.89 & PANDAS 90457 \\ 7 & 11.1578093 & 37.8240122 & 0.801 & 24.54 & 22.03 & PANDAS 90472 \\ 8 & 11.2559801 & 37.7954344 & 0.698 & 24.54 & 21.57 & PANDAS 164999 \\ 9 & 10.9724426 & 37.7146899 & 0.080 & 18.34 & 17.02 & PANDAS 164935 \\ 10 & 10.8359593 & 37.7258511 & 0.701 & 25.09 & 22.24 & PANDAS 90323 \\ \enddata \end{deluxetable} \section{N body simulation details} \label{sec:nbodysimdetails} For the simulations described in Section~\ref{sec:PrevObsandModels}, we use a coordinate system that is aligned with the disk of M31, such that the $x_{\rm M31}$ axis is in the plane of sky pointing towards the north, $z_{\rm M31}$ is perpendicular to the M31 disk plane and is pointing southward on the sky and slightly away from the observer. The transformation between the M31 aligned coordinate system and the sky-oriented coordinate system such that $x$ is pointing east, $y$ pointing north, and $z$ pointing away from us can be done with the matrix $M$ (constructed assuming the position angle of the M31 line of nodes of 37$^\circ$ and and inclination of 77$^\circ$) $$M=\begin{pmatrix} 0.60181502 & -0.1796539 & -0.77816653 \\ 0.79863551 & 0.13537892 & 0.58639054\\ 0. & -0.97437006 & 0.22495105 \end{pmatrix}$$ We start the simulation with the progenitor at $X_{M31}=(-5.44,22.5,35.25)$\,kpc with velocity $V_{M31}=(19.66,-28.79,64.68)$ {\kms} (in the coordinate system aligned with the disk), where the initial coordinates and velocities are taken from \citep{Kirihara2017a} and rotated using the matrix $M$. We run the model for 977 Myr using the GyrFalcon integrator \citep{dehnen2000,dehnen2014} from the NEMO software package \citep{Teuben1995} using the following command: \begin{verbatim} mkplum - 300000 r_s=1 seed=1 mass=9000 | snapshift rshift=-5.44,22.5,35.25 vshift=19.66,-28.79,-64.68 in=- out=- | gyrfalcON - out.snp accname=nfw,miyamoto,miyamoto,miyamoto,hernquist accpars='0,7.63,215;10.68,.72,3.07e5;22.99,.72,-2e5;3.49,0.72,.329e5;0,1.39e5,.6' tstop=1 kmax=18 eps=0.1 step=0.001 \end{verbatim} Some 1000 snapshots of the simulation made are provided on zenodo at \url{https://doi.org/10.5281/zenodo.6977494}. After running the simulations, we convert the outputs back into the space of observables, i.e., the coordinate system aligned with the sky by applying the inverse rotation matrix and assuming that M31 is at a distance of 750 kpc. We also compute the energies of each particle using the {\tt gala} package \citep{gala}. Figure~\ref{fig:sergey_energy_plot} shows the resulting energy as a function of projected distance for the particles in the simulation associated with the structure which matches M31's Giant Stellar Stream. In the range between 1$^\circ$ and 4$^\circ$, where we fit for the GSS in \S~\ref{sec:gssmassest}, the total energy is approximately linear with radius. \begin{table}[h] \centering \caption{GSS radial velocity measurements} \begin{tabular}{ccc} \hline { R } & { $V_{GSR}$ - $V_{GSR, M31}$} & { $\sigma_V$} \\ { deg } & {km/s} & {km/s} \\ \hline 1.00& -269.26 & 6.87\\ 1.35& -225.72 & 4.22\\ 1.70& -197.85 & 2.79\\ 2.05& -169.80 & 2.74\\ 2.40& -148.71 & 2.53\\ 2.75& -132.18 & 2.52\\ 3.10& -109.85 & 2.18\\ 3.45& -92.11 & 2.23\\ 3.80& -71.21 & 3.64\\ \hline \end{tabular} \label{tab:gss_rv_track} \end{table} \end{CJK*}

Title: Stochastic gravitational wave background phenomenology in a pulsar timing array

Abstract: Pulsar timing offers an independent avenue to test general relativity and alternative gravity theories. This requires an understanding of how metric polarizations beyond the familiar transverse tensor ones imprint as a stochastic gravitational wave background and correlate the arrival time of radio pulses from a pair of millisecond pulsars. In this work, we focus on an isotropic stochastic gravitational wave background and present a straightforward, self-contained formalism for obtaining the power spectrum and the overlap reduction function, the relevant physical observable in a pulsar timing array, for generic gravitational degrees of freedom featuring both transverse and longitudinal modes off the light cone. We additionally highlight our consideration of finite pulsar distances, which we find significant in two ways: first, making all the modes well defined, and second, keeping the small scale power that is contained by pulsars of subdegree separations in the sky. We discuss this for tensor, vector, and scalar polarizations, for each one focusing on the angular power spectrum and the overlap reduction function for an isotropic stochastic gravitational wave background. Our results pave the road for an efficient numerical method for examining the gravitational wave induced spatial correlations across millisecond pulsars in a pulsar timing array.

https://export.arxiv.org/pdf/2208.12538

\title{Stochastic gravitational wave background phenomenology in a pulsar timing array} \author{Reginald Christian Bernardo} \email{rbernardo@gate.sinica.edu.tw} \affiliation{Institute of Physics, Academia Sinica, Taipei 11529, Taiwan} \author{Kin-Wang Ng} \email{nkw@phys.sinica.edu.tw} \affiliation{Institute of Physics, Academia Sinica, Taipei 11529, Taiwan} \affiliation{Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 11529, Taiwan} \section{Introduction} \label{sec:intro} The groundbreaking direct discovery of gravitational waves by the LIGO/Virgo collaborations was probably the last decade's most significant scientific breakthrough \cite{LIGOScientific:2016aoc}. This has ushered in the era of gravitational wave astronomy, opening up a novel observational window into the Universe and letting in new independent opportunities to test our fundamental understanding of nature. The first few years alone have already made an outstanding impact, telling us that the graviton cannot be heavier than $10^{-22}$ electron volts and that gravitational waves -- spacetime distortions that displace masses on its path -- should practically be as fast as light in vacuum \cite{LIGOScientific:2017vwq, LIGOScientific:2021djp, LIGOScientific:2021sio}. These observations in the hundred to kilo hertz gravitational wave band will soon be supported by space based observations, relieved of the terrestrial restrictions, that aim to probe the millihertz frequencies \cite{LISA:2022kgy}. The science that gravitational wave observations promises, from learning their sources to the physics behind them, is simply astonishing and captures the interest and imagination of both the scientific community and the public alike. At the same time, pulsar timing brings in another piece into the picture, providing gravitational wave observations in the nanohertz frequency band. This is done by observing the arrival time of radio pulses by millisecond pulsars which should be spatially correlated due to the stochastic gravitational wave background \cite{Burke-Spolaor:2018bvk}. The targeted sources here are phase transitions in the early universe, cosmic strings, and supermassive binary black holes, all of which carry information about the cosmological history \cite{Romano:2019yrj}. Recently, pulsar timing array efforts by the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) \cite{NANOGrav:2020bcs}, the Parkes Pulsar Timing Array (PPTA) \cite{Shannon:2015ect}, the European Pulsar Timing Array (EPTA) \cite{Lentati:2015qwp}, or jointly the International Pulsar Timing Array (IPTA) which take in years of observation of about a hundred millisecond pulsars in the sky have presented strong evidence for a common spectrum process. This teases an intriguing departure from the quadrupolar dominated spatial correlation sourced by the transverse tensor modes expected in general relativity \cite{Hellings:1983fr, NANOGrav:2020bcs, Chen:2021wdo, Chen:2021ncc}. Granted, the uncertainties in the present data set are quite large due to the limitations in the optimal statistic analysis, but this also advances alternative scenarios where nontensorial spatial correlations weigh in the stochastic gravitational wave background. Perhaps tensor modes off the light cone, say, by a dispersive gravitational wave \cite{deRham:2012az}, or additional gravitational degrees of freedom such as the ones that come in modified gravity theories \cite{Chamberlin:2011ev, Liang:2021bct, Tachinami:2021jnf} can explain this correlation. Unequivocally, the science we have yet to learn about the nanohertz gravitational wave sky is rich and exquisite \cite{Tasinato:2022xyq, Chu:2021krj, Bernardo:2022vlj}. In this work, we study an isotropic stochastic gravitational wave background, utilizing a Sachs-Wolfe line of sight integral, in the setting of a pulsar timing array. We do so by directly constructing the power spectrum through which the overlap reduction function (ORF), describing the cross correlated power across pulsars in the sky, the main physical observable of interest, can be obtained \cite{Dai:2012bc, Qin:2018yhy, Qin:2020hfy, Liu:2022skj, Ng:2021waj}. We argue this alternative to the so called real space formalism \cite{NANOGrav:2021ini, Boitier:2021rmb, Hu:2022ujx} is numerically efficient for the present data set, requiring only the first few multipoles since each pulsar pair has at least about a ten degree separation. We further take into consideration finite pulsar distances and arbitrary propagation velocities, which make \textit{all} of the metric polarizations well defined unlike their infinite distance and luminal counterparts. To the best of our knowledge, this is the most general setup that take finite pulsar distances and arbitrary velocities in the literature. Our main results are presented in Section \ref{sec:summary} for future data analysis work, whereas the rest of the paper is spent on the derivation and a discussion of the phenomenological signatures of an isotropic stochastic gravitational wave background in a pulsar timing array. After the summary (Section \ref{sec:summary}), we lay down the formalism we follow in calculating the stochastic gravitational wave background's power spectrum and overlap reduction function (Section \ref{sec:ptobservables}). This includes a brief review of the real space formalism which we also use to confirm our power spectrum calculations. Then, we derive the power spectrum and the overlap reduction function for tensor (Section \ref{sec:tensor_pols}), vector (Section \ref{sec:vector_pols}), and scalar (Section \ref{sec:scalar_pols_finite_dist}) metric polarizations, and for each one study their phenomenology. We discuss the advantages and disadvantages of the power spectrum calculation and how it can be improved (Section \ref{sec:discussion}). We remark on several future work including possible anisotropies in the stochastic gravitational wave background (Section \ref{sec:conclusions}). In the appendices, we explicitly write down the polarization tensors considered in our analysis (Appendix \ref{sec:gw_pol_tensors}) and list down identities for spherical harmonics relevant to the calculations in the main text (Appendix \ref{sec:3Y3j}). We also briefly review another approach to obtaining the overlap reduction function (Section \ref{sec:another}) and outline some of the well known analytical results in the literature (Section \ref{sec:orfs_review}). We work with the mostly plus metric signature $(-, +, +, +)$ and geometrized units $c = 8 \pi G = 1$. Also, we list down the symbols often mentioned in this paper in Table \ref{tab:symbols}. \begin{table}[h!] \centering \caption{Description of the common symbols appearing throughout this paper.} \begin{tabular}{|c|c|} \hline symbol & description \\ \hline \hline $\gamma_{ab}(\zeta)$ & overlap reduction function \\ \hline $\zeta$ & angular separation \\ \hline $\gamma_{ab}(0)$ & overlap reduction function at $\zeta = 0^\circ$ (`zero lag') \\ \hline $\gamma_{aa}$ & autocorrelation function \\ \hline $l$ & multipole number/index \\ \hline $C_l$ & power spectrum multipole \\ \hline $D_a$ & distance to pulsar $a$ \\ \hline $f$ & gravitational wave frequency \\ \hline $v$ & group velocity \\ \hline $P_l(x)$ & Legendre polynomial \\ \hline $j_l(x)$ & spherical Bessel function \\ \hline $Y_{lm}\left(\hat{k}\right)$ & spherical harmonics \\ \hline $\, _s Y_{lm}\left(\hat{k}\right)$ & spin weighted spherical harmonics \\ \hline $h_{ij}$ & gravitational wave \\ \hline $\vec{k} = k \hat{k}$ & wave vector \\ \hline $\varepsilon_{ij}$ & gravitational wave polarization basis tensor \\ \hline $\hat{e}_a$ & unit vector from earth to pulsar $a$ \\ \hline $d^{ij}$ & detector tensor $\hat{e}^i \otimes \hat{e}^j$ \\ \hline $z(t, \hat{e})$ & redshift fluctuation \\ \hline $r(t, \hat{e})$ & pulsar timing residual \\ \hline $F_a^A \left(\hat{k}\right)$ & antenna pattern for polarization $A$ \\ \hline $D_{m'm}^l(-\alpha,-\theta,-\phi)$ & Wigner D matrix \\ \hline $\Gamma(x)$ & gamma function \\ \hline $\, _2F_1(a, b;c;x)$ & hypergeometric function \\ \hline $\, _2\tilde{F}_1(a,b;c;x)$ & regularized hypergeometric function $\, _2F_1(a, b;c;x)/\Gamma(c)$ \\ \hline $\, _2F_2(a, b;c, d;x)$ & hypergeometric function \\ \hline $\, _2\tilde{F}_2(a,b;c, d;x)$ & regularized hypergeometric function $\, _2F_1(a, b;c, d;x)/\left( \Gamma(c) \Gamma(d) \right)$ \\ \hline \end{tabular} \label{tab:symbols} \end{table} \section{Summary of main results} \label{sec:summary} We summarize the main results of this paper intended for future work with spatial correlation data from a pulsar timing array. The overlap reduction function, or rather the cross correlated power between pulsars in the sky, sourced by an isotropic stochastic gravitational wave background is given by \begin{equation} \gamma_{ab}\left( \zeta \right) = \sum_{l} \dfrac{2l + 1}{4\pi} C_l P_l \left( \cos \zeta \right) \,, \end{equation} where $\zeta$ is the angle between a pulsar pair and the $C_l$'s are the power spectrum multipoles. The gravitational degrees of freedom determine which polarizations contribute to $C_l$. We write this formally as \begin{equation} \label{eq:Clgen} C_l^A = \dfrac{J_l^A\left(f D_a\right) J_l^{A*}\left(f D_b\right)}{\sqrt{\pi}} \,, \end{equation} where the superscript index $A$ determines the mode contributing to the gravitational wave, $f$ is the gravitational wave frequency, and $D_i$ is the distance to the pulsars. This is an efficient formula in practice since only a small number of multipoles need to be considered. To elucidate on this point, consider for example, the smallest pulsar pair separation in NANOGRAV's 12.5 year data set which is about ten degrees. In this case, even the first thirty multipoles, resolving angular separations $\zeta > 180^\circ/l_\text{max} = 6^\circ$, are sufficient for the analysis. We enumerate in what follows the $J_l^A(x)$'s appearing in \eqref{eq:Clgen} for modes propagating at a velocity $v$. For the tensor polarizations, the $J_l^A(x)$'s are given by \begin{equation} \label{eq:JT} J^\text{T}_l\left(fD\right) = \sqrt{2} \pi i^l \sqrt{\dfrac{(l + 2)!}{(l - 2)!}} \int_0^{2\pi fDv} \dfrac{dx}{v} \ e^{ix/v} \dfrac{j_l(x)}{x^2} \,, \end{equation} while for the vector polarizations these are \begin{equation} \label{eq:JV} J^\text{V}_l\left(fD\right) = 2 \sqrt{2} \pi i^l \sqrt{l(l+1)} \int_0^{2\pi fDv} \dfrac{dx}{v} \ e^{ix/v} \dfrac{d}{dx} \left( \dfrac{j_l(x)}{x} \right) \,, \end{equation} where $j_l(x)$ is the spherical Bessel function of the first kind. On the other hand, for the scalar transverse polarization, this is given by \begin{equation} \label{eq:JST} J^{\text{ST}}_l\left( fD \right) = 2 \sqrt{2} \pi i^l \int_0^{2\pi f D v} \dfrac{dx}{v} \ e^{ix/v} \left( j_l''(x) + j_l(x) \right) \,, \end{equation} and for the scalar longitudinal polarization, this is \begin{equation} \label{eq:JSL} J^{\text{SL}}_l\left( fD \right) = - 2 \pi i^l \int_0^{2\pi f D v} \dfrac{dx}{v} \ e^{ix/v} j_l(x) \,. \end{equation} The above integrals admit analytical expressions in the infinite distance limit (see for instance \cite{Qin:2020hfy}). In general, for finite pulsar distances, $f D_i < \infty$, and subluminal velocities $v < 1$, these integrals can be evaluated using computer algebra systems such as python and Mathematica, leading to the power spectrum, and the overlap reduction function. A simple algorithm for spatial correlation data analysis is shown below: \begin{enumerate} \item Select a gravitational wave mode $A$ and provide its velocity $v$ and frequency $f$; \item Calculate the power spectrum \eqref{eq:Clgen} given the set of pulsar distances $D_i$; \item Calculate the overlap reduction function $\gamma_{ab}^A(\zeta)$ using \eqref{eq:orf_general}; \item Obtain the normalized overlap reduction function $\Gamma_{ab}^A(\zeta) = 0.5 \times \gamma_{ab}^A(\zeta)/\gamma_{ab}^\text{HD}(0)$ where $\gamma^\text{HD}_{ab}(0) = \gamma_{ab}^\text{T}\left(\zeta = 0^\circ\right)|_{fD \rightarrow \infty, v \rightarrow 1}$; \item Compare the curve $A_\text{GW}^2 \Gamma_{ab}^A(\zeta)$ with the observed $\left[ A_\text{GW}^2 \Gamma_{ab}^A(\zeta) \right]_\text{PTA}$ from pulsar timing array, where $A_\text{GW}$ is the characteristic gravitational wave strain. \end{enumerate} These steps are easy to implement and, as we advocate for the current data set, requires only about thirty multipoles, making the computation very fast \cite{Bernardo:2022vlj}. We mention that the normalization (step 4) is an aesthetic choice. Overlap reduction functions used for data analysis are normalized relative to the traditional Hellings-Downs curve as $\Gamma^A_{ab}(0)$, such that $\Gamma_{ab}^\text{HD}(0) = 0.5$. We spend the bulk of this paper deriving the above equations in detail and studying their phenomenology in a pulsar timing array. We emphasize that this was not the first time the power spectrum approach was marketed for pulsar timing array analysis (see \cite{Dai:2012bc, Qin:2018yhy, Qin:2020hfy, Liu:2022skj}). Instead we present the most general output from this, keeping the pulsars at finite distances throughout and the propagation velocities arbitrary. Our derivation is in addition self contained and arguably pedagogical, requiring only a few textbook material on spherical harmonics and Bessel functions \cite{Weinberg, Arfken}. \section{Pulsar timing observables} \label{sec:ptobservables} We briefly discuss the pulsar timing residual and overlap reduction function \cite{Liu:2022skj, NANOGrav:2021ini}. \subsection{Pulsar timing and gravitational waves} \label{subsec:pulsar_timing_gws} We consider a gravitational wave propagating along the $\hat{k}$ direction in a mixture of various polarizations $A$. In this context, for concreteness, `polarization' means the various independent ways a gravitational wave displaces masses on its path -- scalar transverse, scalar longitudinal, transverse vector, and transverse-traceless $(+, \times)$ tensor modes -- that is induced by the propagating degrees of freedom. In symbols, we represent this as a typical plane wave superposition, \begin{equation} \label{eq:gw_general} h_{ij}\left(\eta, \vec{x}\right) = \sum_A \int_{-\infty}^\infty df \int_{S^2} d\hat{k} \ h_A\left(f, \hat{k}\right) \varepsilon_{ij}^A e^{-2\pi i f \left( \eta - v \hat{k} \cdot \vec{x} \right)} \,, \end{equation} where $\varepsilon_{ij}^A$ are basis polarization tensors (Appendix \ref{sec:gw_pol_tensors}) and $v = d\omega/dk$ is the group velocity. Now, the main observable in a pulsar timing experiment is the timing residual $r(t)$. Our goal is to single out the influence of a gravitational wave on this observable. We proceed to do this through the power spectrum. To start, in terms of the redshift space fluctuation $z(t)$, we write down the timing residual as \begin{equation} \label{eq:timing_residual} r\left(t\right) = \int_0^t dt' \ z\left(t'\right) \,, \end{equation} where $t$ is the duration of an observation. For a passing gravitational wave $h_{ij}\left(\eta, \vec{x}\right)$, the redshift fluctuation considering a photon emitted at time $\eta_e$ and received by the detector at time $\eta_r$ is given by \begin{equation} \label{eq:z_swolf} z\left(t', \hat{e}\right) = - \dfrac{1}{2} \int_{t' + \eta_e}^{t' + \eta_r} d\eta d^{ij} \partial_\eta h_{ij} \left( \eta, \vec{x} \right) \,, \end{equation} where $d^{ij} = \hat{e}^i \otimes \hat{e}^j$ is the detector tensor with $\hat{e}$ being a unit vector pointing toward the pulsar from earth, in words, the projections along the pulsar's line of sight. Substituting the gravitational wave \eqref{eq:gw_general}, we have \begin{equation} r\left(t, \hat{e}\right) = \int_0^t dt' \left( - \dfrac{1}{2} \right) \int_{t' + \eta_e}^{t' + \eta_r} d \eta \ d^{ij} \sum_{A} \int_{-\infty}^\infty df \int_{S^2} d\hat{k} \ h_A \left(f, \hat{k}\right) \varepsilon_{ij}^A\left(\hat{k}\right) \left(-2\pi i f\right) e^{-2\pi i f \left( \eta - v \hat{k} \cdot \vec{x} \right)} \,. \end{equation} We move forward by expanding the plane wave in terms of spherical harmonics, $Y_{lm}\left(\hat{e}\right)$, \begin{equation} e^{2\pi i f v \hat{k}\cdot\vec{x}} = 4\pi \sum_{lm} i^l j_l\left(2\pi fv|\vec{x}|\right) Y_{lm}^*\left(\hat{k}\right) Y_{lm}\left(\hat{e}\right) \,, \end{equation} such that \begin{equation} \begin{split} r\left(t, \hat{e}\right) = & \int_0^t dt' \left( - \dfrac{1}{2} \right) \int_{t' + \eta_e}^{t' + \eta_r} d \eta \ d^{ij} \sum_{A} \int_{-\infty}^\infty df \int_{S^2} d\hat{k} \\ & \ \ \times h_A \left(f, \hat{k}\right) \varepsilon_{ij}^A\left(\hat{k}\right) \left(-2\pi i f\right) e^{-2\pi i f \eta} \ 4\pi \sum_{lm} i^l j_l\left(2\pi fv \left( t' + \eta_r - \eta \right) \right) Y_{lm}^*\left(\hat{k}\right) Y_{lm}\left(\hat{e}\right) \,, \end{split} \end{equation} where $\vec{x} = |\vec{x}| \hat{e} = \left( t' + \eta_r - \eta \right) \hat{e}$ is the position vector to the pulsar at time $t'$ and $j_l(x)$ is the spherical Bessel function of the first kind. We consider the following integral identities to simplify this: \begin{equation} \int_0^t dt' \int_{t' + \eta_e}^{t' + \eta_r} d\eta' e^{-2\pi i f \eta'} W \left( t' + \eta_r - \eta' \right) = \left( \dfrac{1 - e^{-2\pi i f t}}{2\pi i f} \right) \int_{\eta_e}^{\eta_r} d\eta \ e^{-2\pi i f \eta} W\left( \eta_r - \eta \right) \end{equation} and \begin{equation} \int_{\eta_e}^{\eta_r} d\eta e^{-2\pi i f \eta} j_l \left( 2\pi f v \left(\eta_r - \eta\right) \right) = \left( \dfrac{e^{-2\pi i f \eta_r}}{2\pi f v} \right) \int_0^{2\pi f D v} dx \ e^{i x/v} j_l\left(x\right) \,. \end{equation} These can be proven by a dummy variable change. The timing residual simplifies to \begin{equation} \begin{split} r\left(t,\hat{e}\right) = \ & 2 \pi \sum_A \int_{-\infty}^\infty df \int_{S^2} d\hat{k} \ \left( 1 - e^{-2\pi i f t} \right) \left( \dfrac{e^{-2\pi i f \eta_r}}{2\pi f} \right) \\ & \ \ \ \ \times h_A\left(f, \hat{k}\right) \int_0^{2\pi f Dv} \dfrac{dx}{v} e^{ix/v} \left[ d^{ij} \varepsilon_{ij}^A \right] \sum_{lm} i^l j_l\left(x\right) Y^*_{lm}\left(\hat{k}\right) Y_{lm}\left(\hat{e}\right) \,. \end{split} \end{equation} We proceed to use this to calculate the timing residual power spectrum and the overlap reduction function. \subsection{Timing residual power spectrum and overlap reduction function} \label{subsec:timing_and_orf} We expand the timing residual in spherical harmonics, \begin{equation} r\left(t, \hat{e}\right) = \sum_{l,m} a_{lm} Y_{lm} \left( \hat{e} \right) \,. \end{equation} Following the previous calculation, in the presence of a stochastic gravitational wave background comprised of a set of polarizations $A$, it can be shown that the two-point function is given by \begin{equation} \langle r\left(t_a, \hat{e}_a\right) r\left(t_b, \hat{e}_b\right) \rangle = \sum_{l_1, m_1} \sum_{l_2, m_2} \langle a_{l_1 m_1} a^*_{l_2 m_2} \rangle Y_{l_1 m_1}\left( \hat{e}_a \right) Y^*_{l_2 m_2}\left( \hat{e}_b \right) \,, \end{equation} where \begin{equation} \label{eq:two_point_lm} \langle a_{l_1 m_1} a^*_{l_2 m_2} \rangle = \int_{-\infty}^\infty \dfrac{df}{\left(2\pi f\right)^2} \left( 1 - e^{-2\pi i f t_a} \right) \left( 1 - e^{2\pi i f t_b} \right) \sum_{A_1, A_2} \int_{S^2} d\hat{k} \ P_{A_1 A_2}\left( f, \hat{k} \right) J^{A_1}_{l_1 m_1} \left( f D_a, \hat{k} \right) J^{A_2 *}_{l_2 m_2} \left( f D_b, \hat{k} \right) \end{equation} and \begin{equation} \label{eq:Jlm_def} J_{lm}^A \left( fD, \hat{k} \right) = \int_0^{2\pi f D v} \dfrac{d x}{v} \ e^{i x/v} \sum_{LM} 2 \pi i^L Y^*_{LM} \left( \hat{k} \right) j_L(x) \int_{S^2} d\hat{e} \ d^{ij} \varepsilon_{ij}^A\left(\hat{k}\right) Y_{LM}\left( \hat{e} \right) Y_{lm}^*\left(\hat{e}\right) \,. \end{equation} In \eqref{eq:two_point_lm} and \eqref{eq:Jlm_def}, we remind that $d^{ij} = \hat{e}^i \otimes \hat{e}^j$ is the detector tensor, $\varepsilon_{ij}^A\left(\hat{k}\right)$ is the polarization basis tensor of a metric polarization $A$ propagating toward $\hat{k}$, and $P_{A_1 A_2} \left(f, \hat{k}\right)$ is the frequency space amplitude of the gravitational wave two point function, i.e., \begin{equation} \langle h_A\left(f, \hat{k}\right) h_B^* \left( f', \hat{k}'\right) \rangle = \delta \left( f - f' \right) \delta \left( \hat{k} - \hat{k}' \right) P_{AB} \left(f, \hat{k}\right) \,. \end{equation} Focusing on an isotropic stochastic gravitational wave background, such that $P_{AB} = \delta_{AB }P_{AA}\left(f\right)$, that is no directional dependence, the overlap reduction function measuring the angular correlation between pulsar pairs in harmonic space can be shown to be \footnote{The anisotropic case can be tackled by the replacement $Y_{00}\left(\hat{k}\right) \rightarrow Y_{lm}\left(\hat{k}\right)$ in the integral. We shall discuss this elsewhere.} \begin{equation} \label{eq:orf_general} \gamma_{ab}^A \left( \zeta, f D_i \right) = \sum_{l_1, m_1} \sum_{l_2, m_2} Y_{l_1 m_1}\left( \hat{e}_a \right) Y^*_{l_2 m_2} \left( \hat{e}_b \right) \int_{S^2} d\hat{k} \ Y_{00}\left(\hat{k}\right) J^A_{l_1 m_1} \left( f D_a, \hat{k} \right) J^{A*}_{l_2 m_2} \left( f D_b, \hat{k} \right) \,, \end{equation} where $\zeta$ is the angular separation between the pulsars, i.e., $\hat{e}_a \cdot \hat{e}_b = \cos \zeta$, and $Y_{00}\left(\hat{k}\right) = 1/\sqrt{4\pi}$. This is the physical observable we intend to calculate throughout this work. \subsection{Real space formalism} \label{subsec:realspace} It is useful to confirm that our calculations agree with the real space formalism \cite{Chu:2021krj, NANOGrav:2021ini}. In this direction, starting with the gravitational wave \eqref{eq:gw_general} and the redshift fluctuation \eqref{eq:z_swolf}, but this time evaluating the time integrals explicitly, we find the pulsar timing residual to be \begin{equation} \label{eq:timing_realspace} r\left(t, \hat{e}\right) = \dfrac{1}{4\pi} \sum_A \int_{-\infty}^\infty \dfrac{df}{f} \int_{S^2} d\hat{k} \ \left(1 - e^{-2\pi i f t} \right) e^{-2\pi i f \eta_r v \hat{k} \cdot \hat{e}} h_A\left(f, \hat{k}\right) d^{ij} \varepsilon_{ij}^A\left(\hat{k}\right) \left[ 1 - e^{2\pi i fD \left( 1 + v\hat{k}\cdot\hat{e} \right)} \right] \left( \dfrac{i}{1 + v \hat{k}\cdot\hat{e} } \right) \,. \end{equation} The first term in the square brackets is referred to as the `earth' term whereas the second one is the `pulsar' term, which becomes a highly oscillatory function in $\hat{k}$. Nonetheless, for the angular scales relevant in current observations, the pulsar term may be safely neglected for simplicity. Even so, we always keep the pulsar term in the formalism. We calculate the two point function, following \cite{Chu:2021krj}, for an isotropic stochastic gravitational wave background, \begin{equation} \label{eq:twopoint_realspace} \langle r\left(t_a, \hat{e}_a\right) r\left(t_b, \hat{e}_b\right) \rangle = \sum_A \int_{-\infty}^\infty df \ \left( 1 - e^{-2\pi i f t_a} \right) \left( 1 - e^{2\pi i f t_b} \right) \dfrac{8 P_{AA}(f)}{f^2} \times \gamma_{ab}^A\left( \zeta, f D_i \right) \,, \end{equation} from which we identify the overlap reduction function to be \begin{equation} \label{eq:orf_realspace} \gamma_{ab}^A\left( \zeta, f D_i \right) = \int_{S^2} \dfrac{d \hat{k}}{\sqrt{4\pi}} \ U_a \left(f D_a, \hat{k} \right) U_b^*\left(f D_b, \hat{k} \right) F_a^A \left( \hat{k} \right) F_b^{A*} \left( \hat{k} \right) \,, \end{equation} where \begin{equation} \label{eq:antenna_functions} F_a^A\left(\hat{k}\right) = \dfrac{d^{ij} \cdot \varepsilon_{ij}^A \left( \hat{k} \right)}{2\left( 1 + v \hat{k} \cdot \hat{e} \right)} \end{equation} and \begin{equation} \label{eq:Uadef} U_a\left(f D_a, \hat{k}\right) = 1 - e^{2 \pi i f D_a \left( 1 + v \hat{k} \cdot \hat{e} \right)} \,. \end{equation} The quantity $F_a^A\left(\hat{k}\right)$ are the so called antenna pattern functions. The quantity $H(f) = 8 \pi P_{AA}(f)$ is the one sided power spectral density of the gravitational wave background \cite{Chu:2021krj, NANOGrav:2021ini}, and is related to the fractional energy density $\Omega_{\text{GW}}(f)$ via $H(f) = \left( 3H_0^2 / (2\pi^2)\right) \times \left(\Omega_{\text{GW}}(f)/ f^3\right)$, where $H_0$ is the Hubble constant and $\Omega_\text{GW}(f) = \left( d \rho_\text{GW}/ d \ln(f) \right)/\rho_c$ for the critical energy density $\rho_c$ and gravitational wave energy density $\rho_\text{GW}$. We utilize the real space formalism to confirm the power spectrum calculations particularly for the autocorrelation function $\gamma_{aa}^A$. This physical quantity takes in the small scale power encoded in the stochastic gravitational wave background, and so must involve at least a few thousand multipoles. By this standard, it is an incredible assessment tool for the power spectrum calculation. Our computations are presented in Table \ref{tab:GaafD100} for $fD = 100$ or $D \sim 30$ parsecs, showing the agreement between the canonical real space formalism and the power spectrum method. It is worth noting that at nonrelativistic speeds the low multipoles ($l \lesssim 100$) are enough to calculate the small scale gravitational wave power for each metric polarization. In Appendix \ref{sec:another}, we present an alternative real space formalism that is also often considered in the literature but which starts with an explicit decomposition of the redshift fluctuation into earth and pulsar terms. It can be confirmed that this leads to the same overlap reduction function, apart from an overall factor. \section{Tensor polarizations} \label{sec:tensor_pols} We derive the power spectra and overlap reduction functions for the tensor polarizations and discuss their phenomenology. \subsection{Calculation of $J_{lm}$} \label{subsec:Jlm_tensor} We follow \cite{Liu:2022skj}. For convenience, we simply point the $\hat{k}$ direction to the $\hat{z}$ direction and take the magnitude to proceed. We make use of the right and left handed complex circular polarization basis tensors: \begin{equation} \varepsilon^\text{R} = \dfrac{\varepsilon^+ + i \varepsilon^\times}{\sqrt{2}} \ \ \ \ \ \text{and} \ \ \ \ \varepsilon^\text{L} = \dfrac{\varepsilon^+ - i \varepsilon^\times}{\sqrt{2}} \,. \end{equation} The contraction of the detector tensor with the basis tensors give \begin{equation} d^{ij} \varepsilon_{ij}^\text{R, L} = \sqrt{\dfrac{16\pi}{15}} Y_{2 \pm 2} \left( \hat{e} \right) \,, \end{equation} where the helicity R (L) takes on $m = + 2$ ($-2$). Substituting this into \eqref{eq:Jlm_def}, and noting that \begin{equation} Y_{LM}\left(\hat{k}\right) = \sqrt{\dfrac{2L+1}{4\pi}} \delta_{M0} \, \end{equation} since $\hat{k} = \hat{z}$, we obtain \begin{equation} J_{lm}^\text{R,L} \left( fD, \hat{k} \right) = \int_0^{2\pi f D v} \dfrac{d x}{v} \ e^{i x/v} \sum_{L} 4 \pi i^L \sqrt{\dfrac{2L+1}{15}} j_L(x) \int_{S^2} d\hat{e} \ Y_{2\pm2}\left(\hat{e}\right) Y_{L0}\left( \hat{e} \right) Y_{lm}^*\left(\hat{e}\right) \,. \end{equation} The triple spherical harmonics integral (Appendix \ref{sec:3Y3j}) vanishes unless $m = \pm 2$ and $L = l - 2, l, l + 2$. The nonvanishing integrals are \begin{eqnarray} \int_{S^2} d\hat{e} \ Y_{2\pm 2}\left(\hat{e}\right) Y_{(l - 2) 0}\left(\hat{e}\right) Y^*_{l \pm 2}\left(\hat{e}\right) &=& \sqrt{\dfrac{15}{32\pi}} \left( \dfrac{(l-1)l(l+1)(l+2)}{(2l-3)(2l-1)^2(2l+1)} \right)^{1/2} \,, \\ \int_{S^2} d\hat{e} \ Y_{2\pm 2}\left(\hat{e}\right) Y_{l0}\left(\hat{e}\right) Y^*_{l \pm 2}\left(\hat{e}\right) &=& - \sqrt{\dfrac{15}{8\pi}} \left( \dfrac{(l-1)l(l+1)(l+2)}{(2l-1)^2(2l+3)^3} \right)^{1/2} \,, \\ \int_{S^2} d\hat{e} \ Y_{2\pm 2}\left(\hat{e}\right) Y_{(l + 2) 0}\left(\hat{e}\right) Y^*_{l \pm 2}\left(\hat{e}\right) &=& \sqrt{\dfrac{15}{32\pi}} \left( \dfrac{ (l-1)l(l+1)(l+2)}{(2l+1)(2l+3)^2(2l+5)} \right)^{1/2} \,. \end{eqnarray} Subsitutting into the last expression, we obtain \begin{equation} J_{lm}^\text{R,L}\left(fD, \hat{z}\right) = -\delta_{m\pm 2} 2 \pi i^l \sqrt{ \dfrac{2l+1}{8\pi} \dfrac{(l + 2)!}{(l - 2)!} } \int_0^{2\pi fDv} \dfrac{dx}{v} \ e^{ix/v} \left( \dfrac{j_{l-2}(x)}{(2l-1)(2l+1)} + \dfrac{2j_l(x)}{(2l-1)(2l+3)} + \dfrac{j_{l+2}(x)}{(2l+1)(2l+3)} \right) \,. \end{equation} Then, through the recursion relation \begin{equation} \label{eq:bessel_id1} \dfrac{j_l(x)}{x} = \dfrac{j_{l-1}(x) + j_{l+1}(x)}{2l+1} \,, \end{equation} we are able to compactify the last expression to \begin{equation} J_{lm}^\text{R,L}\left(fD, \hat{z}\right) = - \delta_{m\pm 2} \sqrt{\dfrac{2l+1}{4\pi}} \left( \sqrt{2} \pi i^l \sqrt{ \dfrac{(l + 2)!}{(l - 2)!} } \int_0^{2\pi fDv} \dfrac{dx}{v} \ e^{ix/v} \dfrac{j_l(x)}{x^2} \right) \,. \end{equation} We note that the factor $\sqrt{(2l + 1)/4\pi}$ corresponds to an arbitrary rotational degree of freedom. The magnitude we are interested in for an isotropic stochastic gravitational wave background is thus the one enclosed in the parenthesis in the above result. To generalize the result, we merely rotate the $\hat{z}$ axis into the $\hat{k} = (\theta,\phi)$ direction. This way, we obtain \begin{equation} J^A_{lm}\left( fD, \hat{k} \right) = \sum_{m'} D_{m'm}^{l*} \left(-\alpha,-\theta,-\phi\right) J^A_{lm'}\left( fD, \hat{z} \right) \,, \end{equation} where $D_{m'm}^l(-\alpha,-\theta,-\phi)$ is the Wigner-D matrix given by \begin{equation} D_{m'm}^l(-\alpha,-\theta,-\phi) = \sqrt{\dfrac{4\pi}{2l + 1}} \, _{-m'}Y_{lm}\left(\theta,\phi\right) e^{i m' \alpha} \,. \end{equation} Above, $\,_s Y_{lm}\left(\hat{e}\right)$ is a spin weighted spherical harmonic (Appendix \ref{sec:3Y3j}). Rotating the $\hat{z}$ to an arbitrary direction $\hat{k}$, we obtain \begin{equation} \label{eq:Jlm_tensor} J_{lm}^\text{R,L}\left(fD, \hat{k}\right) = - _{\mp 2}Y_{lm}^* \left( \hat{k} \right) e^{\mp 2i \alpha} \left( \sqrt{2}\pi i^l \sqrt{ \dfrac{(l + 2)!}{(l - 2)!} } \int_0^{2\pi fDv} \dfrac{dx}{v} \ e^{ix/v} \dfrac{j_l(x)}{x^2} \right) \,, \end{equation} where the upper (lower) signs belong to R (L). We remind that the factor $e^{i m' \alpha}$ is a redundant phase owing to a remaining rotational degree of freedom about the $\hat{k}$ axis. It does not enter the observables we are interested in. In the infinite distance limit, we may confirm that the integral admits an analytical expression: \begin{equation} \int_0^{\infty} \dfrac{dx}{v} \ e^{ix/v} \dfrac{j_l(x)}{x^2} = i \sqrt{\pi } 2^{-(l+1)} (i v)^{l-2} \Gamma (l-1) \, _2\tilde{F}_1\left(\frac{l-1}{2},\frac{l}{2};l+\frac{3}{2};v^2\right) \,, \end{equation} where $_2\tilde{F}_1\left( a, b; c; x \right) = _2 F_1\left( a, b; c; x \right)/\Gamma(c)$ is a regularized hypergeometric function. With $v = 1$, this simplifies further to \begin{equation} \int_0^\infty dx \ e^{ix} \dfrac{j_l(x)}{x^2} = 2 i^{l-1} \dfrac{(l - 2)!}{(l + 2)!} \,, \end{equation} which can be used to get to the Hellings-Downs correlation. \subsection{ORF and power spectra} \label{subsec:power_spectra_tensor} Inserting the result into \eqref{eq:orf_general}, using the spherical harmonics addition theorem \begin{equation} \label{eq:addition_theorem} P_l \left( \hat{e}_a \cdot \hat{e}_b \right) = \dfrac{4\pi}{2l+1} \sum_{m} Y_{lm}\left(\hat{e}_a\right) Y_{lm}^*\left(\hat{e}_b\right) \,, \end{equation} and adding the contributions of the right and left handed helicity contributions, we obtain the overlap reduction function \begin{equation} \gamma_{ab}\left( \zeta, f D_i \right) = \sum_l \dfrac{2l + 1}{4\pi} C_l P_l \left( \cos \zeta \right) \,, \end{equation} where $\hat{e}_a \cdot \hat{e}_b = \cos \zeta$, and the tensor power spectrum multipoles are given by \begin{equation} C_l^\text{T} = \dfrac{J^\text{T}_l\left(f D_a\right) J^{\text{T}*}_l\left(f D_b\right)}{\sqrt{\pi}} \, \end{equation} with \begin{equation} J^\text{T}_l\left(fD\right) = \sqrt{2} \pi i^l \sqrt{\dfrac{(l + 2)!}{(l - 2)!}} \int_0^{2\pi fDv} \dfrac{dx}{v} \ e^{ix/v} \dfrac{j_l(x)}{x^2} \,. \end{equation} For luminal tensor degrees of freedom ($v = 1$) and large pulsar distances, $fD_{a,b} \gg 1$, this reduces to the Hellings-Downs power spectrum, that is, \begin{equation} C_l^\text{T} \sim \dfrac{8\pi^{3/2}}{(l + 2)(l + 1)l(l - 1)} \,. \end{equation} We check the power spectrum calculation compared with the real space formalism through the autocorrelation function. This requires the antenna pattern functions for the tensor $+$ and $\times$ modes which are given by \begin{equation} F^+_a \left( \hat{k} = (\theta, \phi) \right) = \dfrac{\cos(2\phi) \sin^2\theta}{2 \left( 1 + v \cos \theta \right)} \end{equation} and \begin{equation} F^\times_a \left( \hat{k} = (\theta, \phi) \right) = \dfrac{\sin(2\phi) \sin^2\theta}{2 \left( 1 + v \cos \theta \right)} \,. \end{equation} The tensor autocorrelation is then given by the integral \begin{equation} \label{eq:gammaaa_T} \gamma_{aa}^\text{T} = \int_0^\pi \dfrac{d\theta}{\sqrt{4\pi}} \left( \frac{2 \pi \sin ^5 \theta \sin ^2(\pi fD (1 + v \cos \theta))}{(1 + v \cos \theta )^2} \right) \,. \end{equation} \subsection{Phenomenology} \label{subsec:tensor_phenomenology} We view the power spectrum multipoles and the ORF for various velocities and pulsar distances in Figure \ref{fig:ClT}. This is displayed together with the Hellings-Downs correlation ($v = 1$ and $fD \rightarrow \infty$) for reference. In the near luminal ($v \sim 1$) tensor case (Figures \ref{fig:ClT}(a-b)), we reflect the small angle modifications highlighted in \cite{Ng:2021waj, Liu:2022skj} when finite pulsar distances are considered. The trend is, instead of continuously dropping as $l$ increases, the multipoles $C_l$ for finite $fD$ feature a slight growth about some $l \sim 30-50$, corresponding to an angular resolution of about $\theta \sim 3.6-6^\circ$. Also, notably, the further the pulsars are, the higher $l$ becomes to exhibit this partial sustenance. The difference is inconceivable as can be seen in the overlap reduction function where it is only the Hellings-Downs curve ($v = 1$) that is visually distinguishable from the $v = 0.99$ cases. This could be of course expected as the power spectrum in all cases are dominated by the quadrupole. This canonical picture gradually changes as the modes go further away from the light cone. At half the speed of light (Figures \ref{fig:ClT}(c-d)), it can be seen that the finite distance modification approaches much larger scales, now with $l \sim 10-20$ being the multipole number where the $C_l$'s sustain itself. In both the finite and infinite distance case, it is most noteworthy that the spectrum becomes almost completely dominated by the quadrupole at this velocity. This manifests as an enhanced difference between the $v = 1/2$-ORF and the Hellings-Downs curve, although the finite distance cases remain indistinguishable from the infinite distance limit. This no longer holds for extreme subluminal, nonrelativistic, modes (Figure \ref{fig:ClT}(e-f)). In this limit, the infinite distance case can practically be considered a pure quadrupole. On the other hand, the finite distance cases showcase an increase at low $l$ up to some maximum power that is competitive to the quadrupole, giving a nonquadrupolar dominated power spectrum, quite like the Hellings-Downs curve but exhibiting angular oscillations beginning at the peak $l > 2$. As the peak of the power spectrum depends on the distance, the overlap reduction function also becomes distinguishable by shape depending on the distance. This clearly manifests in the overlap reduction function at this extreme subluminal velocity. Understandably, current gravitational wave astronomy constraints in the $\sim 100$ hertz band indicate that the tensor degrees of freedom propagate on the light cone, with very little wiggle room for uncertainty. This may change in a different frequency band, as is allowed in effective field theory. However, even if it does not, we can be conservative and take the tensor modes to just be on the light cone in all frequencies, and find the analogous modifications for non tensor polarizations that could hint at modified gravity. We interpret the enhanced small angle correlation due to the finite distance as the pulsars' perhaps interacting by some physical mechanism. Of course, such would not be case in the infinite distance limit, since the pulsars would be too far apart regardless of the size of their angular separation in the sky. This is also exhibited by the vector and scalar polarizations, as we are about to see. \section{Vector polarizations} \label{sec:vector_pols} We derive the power spectra and the overlap reduction functions for the vector polarizations and discuss their phenomenology. \subsection{Calculation of $J_{lm}$} \label{subsec:Jlm_vector} We simplify the calculation considerably by pointing the gravitational wave to the $\hat{z}$ direction and taking the magnitude of the result. This is sufficient for an isotropic stochastic gravitational wave background analysis. As with the tensor polarizations, we rely on right and left handed helicity basis tensors, \begin{equation} \varepsilon^\text{VR} = \dfrac{\varepsilon^x + i \varepsilon^y}{\sqrt{2}} \ \ \ \ \ \text{and} \ \ \ \ \varepsilon^\text{VL} = \dfrac{\varepsilon^x - i \varepsilon^y}{\sqrt{2}} \,, \end{equation} to derive the transverse vector power spectrum. The contraction of the detector tensor with the basis tensors gives \begin{equation} d^{ij} \varepsilon_{ij}^\text{VR, VL} = \mp \sqrt{\dfrac{16\pi}{15}} Y_{2 \pm 1} \left( \hat{e} \right) \,, \end{equation} where the upper (lower) signs belong to VR (VL). The relevant spherical harmonics integrals are \begin{equation} \int d\hat{e} \ Y_{21}\left(\hat{e}\right) Y_{L0}\left(\hat{e}\right) Y_{lm}\left(\hat{e}\right) = \sqrt{\dfrac{15}{2 \pi }} \dfrac{ (-l+L+1) \sqrt{l (l+1) (2 l+1) (2 L+1)} \left(-(l-1) (l+2)+L^2+L\right)}{(-l+L-2) (l+L-1) (l+L+1) (l+L+3) (l-L)! (-l+L+2)!} \,, \end{equation} which holds for $m = -1, l \geq 1, l - 2 \leq L \leq l + 2$ and $L + l \geq 2$ and \begin{equation} \int d\hat{e} \ Y_{2-1}\left(\hat{e}\right) Y_{L0}\left(\hat{e}\right) Y_{lm}\left(\hat{e}\right) = (-1)^{L + l} \int d\hat{e} \ Y_{21}\left(\hat{e}\right) Y_{L0}\left(\hat{e}\right) Y_{lm}\left(\hat{e}\right) \,, \end{equation} which holds for $m = 1, l \geq 1, l - 2 \leq L \leq l + 2$ and $L + l \geq 2$. Since $L + l$ is even, the two integrals become equal except with $m = \mp 1$. We write this compactly as \begin{equation} \begin{split} \int d\hat{e} \ Y_{2\pm 1}\left(\hat{e}\right) Y_{L0}\left(\hat{e}\right) Y_{lm}\left(\hat{e}\right) = \delta_{m \mp 1} \bigg[ & \delta_{l1} \delta_{L1} \left( - \sqrt{\dfrac{3}{20\pi}} \right) + \delta_{l1} \delta_{L3} \sqrt{ \dfrac{9}{140\pi} } \\ & + \Theta\left( l - 2 \right) \bigg[ \delta_{L(l-2)} \left( - \sqrt{\dfrac{15}{2 \pi }} \dfrac{(l-1) \sqrt{l \left(4 l^3-7 l-3\right)}}{2 (2 l-3) (2 l-1) (2 l+1)} \right) \\ & \phantom{ggggggggggg} + \delta_{Ll} \left( - \sqrt{\dfrac{15}{2 \pi }} \dfrac{ \sqrt{l (l+1) (2 l+1)^2}}{2 (2 l-1) (2 l+1) (2 l+3)} \right) \\ & \phantom{ggggggggggg} + \delta_{L(l+2)} \left( \sqrt{\dfrac{15}{2 \pi }} \dfrac{ l (l+1) (l+2)}{2 (2 l+3) \sqrt{l (l+1) (2 l+1) (2 l+5)}} \right) \bigg] \bigg] \,, \end{split} \end{equation} where $\Theta(x)$ is the step function. Substituting this into \eqref{eq:Jlm_def}, we get to \begin{equation} \begin{split} J_{lm}^\text{VR,VL} \left( fD, \hat{z} \right) = & \int_0^{2\pi fD v} \dfrac{dx}{v} \ e^{i x/v} \sum_{LM} 2 \pi i^L Y^*_{LM}\left( \hat{k} \right) j_L(x) \int_{S^2} d\hat{e} \ \left( d^{ij} \varepsilon_{ij}^\text{VR, VL} \left( \hat{k} \right) \right) Y_{LM}\left(\hat{e}\right) Y_{lm}^*\left(\hat{e}\right) \\ = & \mp \sqrt{ \dfrac{16\pi}{15} } \int_0^{2\pi f D v} \dfrac{dx}{v} \ e^{ix/v} \sum_L 2 \pi i^L \sqrt{\dfrac{2L+1}{4\pi}} j_L(x) \int_{S^2} d\hat{e} \ Y_{2\pm 1} \left(\hat{e}\right) Y_{L0}\left(\hat{e}\right) Y_{lm}^* \left(\hat{e}\right) \\ = & \mp \sqrt{ \dfrac{16\pi}{15} } \int_0^{2\pi f D v} \dfrac{dx}{v} \ e^{ix/v} \delta_{m \pm 1} \bigg[ - \delta_{l1} \dfrac{3i}{2 \sqrt{5}} \left(j_1(x) + j_3(x)\right) \\ & + \Theta(l - 2) \sqrt{\dfrac{15}{2}} \dfrac{i^l}{2} \bigg[ \dfrac{(l-1) \sqrt{l (l+1) (2 l+1)} j_{l-2}(x)}{4 l^2-1} \\ & \phantom {ggggggggggggggggg} -\frac{2 \sqrt{l (l+1) (2 l+1)} j_l(x)}{8 l (l+1)-6} -\dfrac{2 (l+2) \sqrt{l (l+1) (2 l+1)} j_{l+2}(x)}{8 l (l+2)+6} \bigg] \bigg] \,. \end{split} \end{equation} The last expression simplifies to \begin{equation} \begin{split} J_{lm}^\text{VR,VL} \left( fD, \hat{z} \right) = \mp \sqrt{ \dfrac{16\pi}{15} } \delta_{m \pm 1} \bigg[ & - \delta_{l1} \dfrac{3i}{2 \sqrt{5}} \int_0^{2\pi f D v} \dfrac{dx}{v} \ e^{ix/v} \left(j_1(x) + j_3(x)\right) \\ & + \Theta(l - 2) \sqrt{\dfrac{15}{2}} \dfrac{i^l}{2} \sqrt{l (l + 1)(2l + 1)} \int_0^{2\pi f D v} \dfrac{dx}{v} \ e^{ix/v} \dfrac{d}{dx} \left( \dfrac{j_l(x)}{x} \right) \bigg] \,. \end{split} \end{equation} Now, the $l = 1$ piece above can be continued to give the same expression as the $l \geq 2$ pieces. We therefore have \begin{equation} \begin{split} J_{(l \geq 1)m}^\text{VR,VL} \left( fD, \hat{z} \right) = \mp \delta_{m \pm 1} \sqrt{\dfrac{2l + 1}{4\pi}} \left( 2 \sqrt{2} \pi i^l \sqrt{l (l + 1)} \int_0^{2\pi f D v} \dfrac{dx}{v} \ e^{ix/v} \dfrac{d}{dx} \left( \dfrac{j_l(x)}{x} \right) \right) \,. \end{split} \end{equation} We take the magnitude above aside from the rotation factor $\sqrt{(2l + 1)/4\pi}$ to compute the overlap reduction function of an isotropic gravitational wave background. We rotate the $\hat{z}$ direction to an arbitrary $\hat{k}$. This leads to \begin{equation} \begin{split} J_{(l \geq 1)m}^\text{VR,VL} \left( fD, \hat{k} \right) = \mp \, _{\mp 1} Y_{lm}^*\left(\hat{k}\right) e^{\mp i \alpha} \left( 2\sqrt{2} \pi i^l \sqrt{l (l + 1)} \int_0^{2\pi f D v} \dfrac{dx}{v} \ e^{ix/v} \dfrac{d}{dx} \left( \dfrac{j_l(x)}{x} \right) \right) \,. \end{split} \end{equation} By integration by parts, it is useful to note that the integral can be written as \begin{equation} \label{eq:Jlm_vec} \int_0^{2\pi f D v} \dfrac{dx}{v} \ e^{ix/v} \dfrac{d}{dx} \left( \dfrac{j_l(x)}{x} \right) = - \dfrac{i}{v} \int_0^{2\pi f D v} \dfrac{dx}{v} \ e^{ix/v} \dfrac{j_l(x)}{x} + \dfrac{e^{2\pi i fD}}{v} \dfrac{j_l\left(2\pi f D v\right)}{2\pi f D v} - \dfrac{\sqrt{\pi } 2^{-(l + 1)}}{v \Gamma \left(l+(3/2)\right)} \epsilon^{l - 1}|_{\epsilon \rightarrow 0^+} \,. \end{equation} The phase factor $e^{\pm i \alpha}$ corresponds to an arbitrary rotational degree of freedom along the $\hat{k}$ direction. This drops out in the physical observables of interest in this work. We also put attention to the boundary terms (second and third terms in the right hand side) in \eqref{eq:Jlm_vec}. The first one comes from the finite distance modification, as is clear this vanishes when $fD \rightarrow \infty$. The second one vanishes for $l > 1$ but reduces to a constant ($\sim 1/v$) for the dipole $l = 1$. We note that the integral admits an analytical expression for the infinite distance case: \begin{equation} \int_0^{\infty} \dfrac{dx}{v} \ e^{ix/v} \dfrac{j_l(x)}{x} = \sqrt{\pi } 2^{-(l+1)} i^l v^{l - 1} \Gamma (l) \, _2\tilde{F}_1\left(\frac{l}{2},\frac{l+1}{2};l+\frac{3}{2};v^2\right) \,. \end{equation} In the luminal limit, this further simplifies to \begin{equation} \int_0^{\infty} dx \ e^{ix} \dfrac{j_l(x)}{x} = i^l \dfrac{(l - 1)!}{(l + 1)!} \,, \end{equation} which can be used to derive the analogous Hellings-Downs correlation ($v = 1$ and $fD \rightarrow \infty$) for the vector modes. \subsection{ORF and power spectra} \label{subsec:power_spectra_vector} Inserting the result to \eqref{eq:orf_general}, using the addition theorem, and taking in the contributions from the left and right handed helicity vector polarizations, we obtain the overlap reduction function for a vector sourced isotropic stochastic gravitational wave background: \begin{equation} \gamma_{ab} \left( \zeta, f D_i \right) = \sum_l \dfrac{2l + 1}{4\pi} C_l P_l \left( \cos \zeta \right) \,, \end{equation} where the vector power spectrum is \begin{equation} C_l^\text{V} = \dfrac{J^\text{V}_l\left(f D_a\right) J^{\text{V}*}_l\left(f D_b\right)}{\sqrt{\pi}} \, \end{equation} with the function $J^\text{V}_l(fD)$ given by \begin{equation} J^\text{V}_l\left(fD\right) = 2 \sqrt{2} \pi i^l \sqrt{l(l+1)} \int_0^{2\pi fDv} \dfrac{dx}{v} \ e^{ix/v} \dfrac{d}{dx} \left( \dfrac{j_l(x)}{x} \right) \,. \end{equation} As with the tensor, we validate the power spectrum calculation by comparing it with the real space formalism through the calculation of the autocorrelation function. The antenna pattern functions for the vector $x$ and $y$ modes are \begin{equation} F^x_a \left( \hat{k} = (\theta, \phi) \right) = \dfrac{\sin(2\theta) \cos \phi}{2 \left( 1 + v \cos \theta \right)} \end{equation} and \begin{equation} F^y_a \left( \hat{k} = (\theta, \phi) \right) = \dfrac{\sin(2\theta) \sin \phi}{2 \left( 1 + v \cos \theta \right)} \,. \end{equation} The vector autocorrelation reduces to the integral \begin{equation} \label{eq:gammaaa_V} \gamma_{aa}^\text{V} = \int_0^\pi \dfrac{d\theta}{\sqrt{4\pi}} \left( \frac{8 \pi \sin ^3 \theta \cos ^2 \theta \sin ^2(\pi fD (1 + v \cos \theta ))}{(1 + v \cos \theta )^2} \right) \,. \end{equation} \subsection{Phenomenology} \label{subsec:vector_phenomenology} Figure \ref{fig:ClV} presents the power spectra multipoles and the corresponding ORFs for vector polarizations of various velocities and distances. This is shown together with the Hellings-Downs correlation for comparison. For the near luminal vector (Figures \ref{fig:ClV}(a-b)), it can be seen that throughout the vector power spectrum drops differently compared with the Hellings-Downs correlation. It is noteworthy that the dipolar power is also suppressed compared to the quadrupole and the succeeding multipoles such as the octupole and so on, regardless of the pulsars' distances. This shows up as a difference in the overlap reduction function at large angles for the vector induced correlation compared with the Hellings-Downs curve. As with the tensor modes, the power spectrum starts to sustain itself at some $l \sim 30 - 50$, corresponding to an angular resolution $3.6-6^\circ$. However, this difference is realizable only at small angles, irrelevant for the current pulsar timing array data, which reflects as the indistinguishability of the overlap reduction functions for various pulsar distances and the infinite distance limit. We realize an angular dependence due to the pulsar distance to be more pronounced for subluminal velocities. At half the speed of light (Figures \ref{fig:ClV}(c-d)), we find that the dipole becomes more relevant, but still supressed compared to the quadrupole and the octupole. This is reflected in the power spectrum and the overlap reduction function which appears to be shaped more like the Hellings-Downs curve compared with the luminal vector case. The difference can be attributed due to the quadrupole, being too dominant in the vector power spectrum. This time, the multipole number at which higher modes start to sustain themselves becomes lower, $l \sim 5-10$, making the small angle departure more realizable provided sufficient sensitivity. Yet, as the power drop per multipole becomes steeper for subluminal velocity, it remains to distinguish between the finite and infinite pulsar distance cases. This picture drastically changes for the nonrelativistic vector (Figures \ref{fig:ClV}(e-f)), where low multipoles other than the dipole and the quadrupole contribute significantly to the power spectrum. In particular, in Figure \ref{fig:ClV}(f), the overlap reduction function for the vector with $fD = 100$ becomes strikingly similar to the Hellings-Downs curve, while for the $fD = 500$ and infinite distance cases, it is not. This can also be realized in the vector power spectrum multipoles for $fD = 100$ where it can be seen that the dipole is suppressed at this extreme subluminal velocity while the low multipoles beginning with the quadrupole and the octupole follow the trend of the Hellings-Downs correlation. The picture changes further for more distant pulsars, at $fD = 500$, in which case the higher multipoles $l \sim 20-30$ can be seen to even be as significant as the quadrupole. This changes the shape of the vector overlap reduction function at all angles while still being dominated by the quadrupole, as it manifests visually. The above results tease a degeneracy in the tensor and vector degrees of freedom, particularly with the luminal tensor and nonrelativistic vector, in the overlap reduction function and the present data set. Nonetheless, this can be settled by resolving small angular separations, which may be realizable in upcoming pulsar timing array missions. \section{Scalar polarizations} \label{sec:scalar_pols_finite_dist} We derive the power spectra and the overlap reduction functions for the scalar polarizations and study their phenomenology. \subsection{Calculation of $J_{lm}$} \label{subsec:calcu_Jlm} As we did with the tensor and vector cases, to calculate the overlap reduction function for an isotropic gravitational wave background, we simply choose $\hat{k} = \hat{z}$ direction and pick up the magnitude to take in \eqref{eq:orf_general}. The contraction $d^{ij} \varepsilon_{ij}$ of the detector tensor and the polarization basis for the scalar transverse and scalar longitudinal modes becomes \begin{equation} \label{eq:deps_ST} d^{ij} \varepsilon_{ij}^{\text{ST}} = \sin^2 \theta \end{equation} and \begin{equation} \label{eq:deps_SL} d^{ij} \varepsilon_{ij}^{\text{SL}} = \sqrt{2} \cos^2 \theta \,. \end{equation} Since these appear in \eqref{eq:Jlm_def} together with two more spherical harmonics $Y_{lm}\left(\hat{e}\right)$ in an integral, it is useful to express the above contractions as a spherical harmonic series: \begin{equation} d^{ij} \varepsilon_{ij}^{\text{ST}} = \dfrac{4\sqrt{\pi}}{3} Y_{00}\left(\hat{e}\right) - \dfrac{4}{3} \sqrt{\dfrac{\pi}{5}} Y_{20}\left(\hat{e}\right) \end{equation} and \begin{equation} d^{ij} \varepsilon_{ij}^{\text{SL}} = \sqrt{2} \left( \dfrac{2\sqrt{\pi}}{3} Y_{00}\left(\hat{e}\right) + \dfrac{4}{3} \sqrt{\dfrac{\pi}{5}} Y_{20}\left(\hat{e}\right) \right) \,. \end{equation} We perform the summation and integration in \eqref{eq:Jlm_def} for each of the scalar polarizations. After simplification, a rotation is then acted on the result to generalize it to a gravitational wave propagating in an arbitrary direction $\hat{k}$. We start with the scalar transverse polarization. The only term which survives the sum over ${L,M}$ in \eqref{eq:Jlm_def} is $M = 0$. Consequently, the three spherical harmonics integrals we need are \begin{equation} \label{eq:IntY3_Scalar1} \int d \hat{e} \ Y_{00}\left(\hat{e}\right) Y_{L0}\left(\hat{e}\right) Y_{l m}\left(\hat{e}\right) = \dfrac{\delta_{m0}\delta_{lL}}{\sqrt{4\pi}} \end{equation} and \begin{equation} \label{eq:IntY3_Scalar2} \int d \hat{e} \ Y_{20}\left(\hat{e}\right) Y_{L0}\left(\hat{e}\right) Y_{l m}\left(\hat{e}\right) = \dfrac{ \delta_{m0} }{2} \sqrt{\dfrac{5}{\pi}} \dfrac{(L-l +1)^2 (L+l ) (L+l +2) \sqrt{(2 L+1) (2 l +1)} \Gamma (-L+l +3)}{(-L+l +2)^2 (L+l -1) (L+l +1) (L+l +3) \Gamma\left(l -L+1\right)^2 \Gamma (L-l +3)} \,, \end{equation} where \eqref{eq:IntY3_Scalar2} holds provided $l - 2 \leq L \leq l + 2$ and $L + l \geq 2$; otherwise, it is zero. Now, in performing the sum over $L$, we note that $l_1 + l_2 + l_3$ in the Wigner-$3j$ symbol must be an even integer for our purposes since $m_1 = m_2 = 0$ (thus consequently setting up $m_3 = 0$). This leaves three terms corresponding to $L = l - 2, l, l + 2$. Also, from the $L = l + 2$ contribution, we may pull out $l = 0, 1$ terms. Likewise, from the $L = l$ contribution, we may pull out $l = 1$. In this way, we can add the terms for $l \geq 2$ coming from all $L = l - 2, l, l + 2$ terms. This way, we are able to write down the last integral as \begin{equation} \label{eq:IntY3_Scalar2b} \int d \hat{e} \ Y_{20}\left(\hat{e}\right) Y_{L0}\left(\hat{e}\right) Y_{l m}\left(\hat{e}\right) = \begin{cases} \dfrac{3\delta_{m0}}{4} \sqrt{\dfrac{5}{\pi }} \dfrac{ (l-1) l}{ \sqrt{2 l-3} (2 l-1) \sqrt{2 l+1}} &, \ \ \ \ L = l-2 , l \geq 2 \phantom{\dfrac{\dfrac{1}{2}}{\dfrac{1}{2}}} \\ \delta_{m0} \sqrt{\dfrac{5}{\pi }} \dfrac{ l (l+1)}{2(2l - 1)(2l+3)} &, \ \ \ \ L = l , l \geq 1 \phantom{\dfrac{\dfrac{1}{2}}{\dfrac{1}{2}}} \\ \dfrac{3\delta_{m0}}{4}\sqrt{\dfrac{5}{\pi }} \dfrac{ (l+1) (l+2)}{(2 l+3) \sqrt{(2 l+1) (2 l+5)}} &, \ \ \ \ L = l+2 , l \geq 0 \phantom{\dfrac{\dfrac{1}{2}}{\dfrac{1}{2}}} \,. \end{cases} \end{equation} Using the above identities, and carefully performing the sum over $L$, we get to \begin{equation} \label{eq:J00_STc} \begin{split} J_{lm}^\text{ST} \left( fD, \hat{z} \right) = \ & \delta_{m0} \delta_{l0} \dfrac{2\sqrt{ \pi}}{3} \int_0^{2\pi fDv} \dfrac{d x}{v} \ e^{i x/v} \left( j_0(x) + j_2(x) \right) + \delta_{m0}\delta_{l1} \dfrac{2 \sqrt{3\pi} i}{5}\int_0^{2\pi fDv} \dfrac{d x}{v} \ e^{i x/v} \left( j_1(x) + j_3(x) \right) \\ & \ \ - \delta_{m0} \Theta\left(l - 2\right) 4\pi i^l \sqrt{\dfrac{2l+1}{4\pi}} \int_0^{2\pi fDv} \dfrac{d x}{v} \ e^{i x/v} \left[ \dfrac{d}{dx} \left( \dfrac{j_l(x)}{x} \right) -\dfrac{(l-1)(l+2)}{2} \dfrac{j_l(x)}{x^2} \right] \,. \end{split} \end{equation} By using the spherical Bessel function differential equation, \begin{equation} x^2 j_l''(x) + 2xj_l'(x) + \left( x^2 - l(l+1) \right) j_l(x) = 0 \,, \end{equation} and \eqref{eq:bessel_id1}, we simplify this further to \begin{equation} \label{eq:J00_STd} \begin{split} J_{lm}^\text{ST} \left( fD, \hat{z} \right) = \delta_{m0} \sqrt{\dfrac{2l+1}{4\pi}} \left( 2\pi i^l \int_0^{2\pi fDv} \dfrac{d x}{v} \ e^{i x/v} \left( j_l''(x) + j_l(x) \right) \right) \,. \end{split} \end{equation} The factor we need for the isotropic stochastic gravitational wave background is enclosed in the parenthesis. Making use of the observation $J_{lm}^\text{ST}(fD, \hat{z}) \propto \delta_{m0}$, and performing a three dimensional rotation, we finally get to \begin{equation} \label{eq:JlmSTfinal} \begin{split} J^\text{ST}_{lm}\left( fD, \hat{k} \right) = Y^*_{lm}\left(\hat{k}\right) \left( 2\pi i^l \int_0^{2\pi fDv} \dfrac{d x}{v} \ e^{i x/v} \left( j_l''(x) + j_l(x) \right) \right) \,. \end{split} \end{equation} Now, moving onto the scalar longitudinal polarization, starting with the contraction \eqref{eq:deps_SL} and performing the sum over $L$ with the same spherical harmonics identities, we end up with \begin{equation} \label{eq:J00_SLc} \begin{split} \dfrac{J_{lm}^\text{SL} \left( fD, \hat{z} \right)}{\sqrt{2}} = \ & \delta_{m0} \delta_{l0} \dfrac{2\sqrt{ \pi}}{3} \int_0^{2\pi fDv} \dfrac{d x}{v} \ e^{i x/v} \left( \dfrac{j_0(x)}{2} - j_2(x) \right) + \delta_{m0}\delta_{l1} \dfrac{2 \sqrt{3\pi} i}{5}\int_0^{2\pi fDv} \dfrac{d x}{v} \ e^{i x/v} \left( \dfrac{3 j_1(x)}{2} - j_3(x) \right) \\ & + \delta_{m0} \Theta\left(l - 2\right) 4\pi i^l \sqrt{\dfrac{2l+1}{4\pi}} \int_0^{2\pi fDv} \dfrac{d x}{v} \ e^{i x/v} \left[ \dfrac{d}{dx} \left( \dfrac{j_l(x)}{x} \right) - \dfrac{(l-1)(l+2)}{2} \dfrac{j_l(x)}{x^2} + \dfrac{j_l(x)}{2} \right] \,. \end{split} \end{equation} By using the Bessel function differential equation and identity, we are further able to express this simply as \begin{equation} \label{eq:J00_SLd} \begin{split} \dfrac{J_{lm}^\text{SL} \left( fD, \hat{z} \right)}{\sqrt{2}} = - \delta_{m0} \sqrt{\dfrac{2l+1}{4\pi}} \left( 2\pi i^l \int_0^{2\pi fDv} \dfrac{d x}{v} \ e^{i x/v} j_l''\left(x\right) \right) \,. \end{split} \end{equation} Rotating the $\hat{z}$ axis to a general $\hat{k}$ direction, we get to \begin{equation} \label{eq:JlmSLfinal} \begin{split} \dfrac{J^\text{SL}_{lm}\left( fD, \hat{k} \right)}{\sqrt{2}} = - Y^*_{lm}\left(\hat{k}\right) \left( 2\pi i^l \int_0^{2\pi fDv} \dfrac{d x}{v} \ e^{i x/v} j_l''\left(x\right) \right) \,. \end{split} \end{equation} We proceed to calculate the isotropic stochastic gravitational wave background's overlap reduction function using the magnitude in the parenthesis. As with the tensor and vector polarizations, we evaluate these integrals numerically to compute the scalar power spectrum. We note that the scalar transverse integral can be recast as a total boundary for $v = 1$. This can be realized by writing \begin{equation} \label{eq:JS_boundary} e^{ix/v} \left( j_l''(x) + j_l(x) \right) = \dfrac{d}{dx} \left[ e^{ix/v} \left( j_l'(x) - \dfrac{i}{v} j_l(x) \right) \right] + \dfrac{v^2 - 1}{v^2} e^{ix/v} j_l\left(x\right) \,, \end{equation} which reduces to a boundary term if $v = 1$. Therefore by noting the asymptotic expansion \begin{equation} e^{ix/v} \left( j_l'(x) - \dfrac{i}{v} j_l(x) \right) \sim \dfrac{\sqrt{\pi } 2^{-(l+1)}}{\Gamma \left(l+(3/2)\right)} x^l \left( \dfrac{l}{x} + \dfrac{i}{v}(l - 1) + O\left( x \right) \right) \ \ , \ \ \ \ x \rightarrow 0^+ \end{equation} and the integral identity \begin{equation} \int_0^{r} d x \ e^{i x} j_l\left(x\right) = 2^l r^{l+1} \Gamma (l+1)^2 \, _2\tilde{F}_2(l+1,l+1;l+2,2 l+2;2 i r) \ \ , \ \ \ \ \text{Re}(l) > -1 \,, \end{equation} we may obtain the following analytical expressions for finite $fD$ and $v = 1$: \begin{equation} \int_0^{2\pi fD} d x \ e^{i x} \left( j_l''(x) + j_l(x) \right) = e^{2\pi i fD} \left[ j_l'\left(2\pi fD\right) - i j_l\left(2\pi fD\right) \right] \end{equation} and \begin{equation} \begin{split} \int_0^{2\pi fD} d x \ e^{i x} j_l''(x) = \, & e^{2\pi i fD} \left[ j_l'\left(2\pi fD\right) - i j_l\left(2\pi fD\right) \right] \\ & \ \ - 2^l \left(2\pi fD\right)^{l+1} \Gamma (l+1)^2 \, _2\tilde{F}_2 \left( l+1,l+1;l+2,2 l+2;4\pi i fD \right) \,, \end{split} \end{equation} where $\, _2\tilde{F}_2 \left( a, b; c, d; x \right) = \, _2{F}_2 \left( a, b; c, d; x \right)/\left( \Gamma(c) \Gamma(d) \right)$ is a regularized hypergeometric function. These help significantly to reduce the numerical evaluation time of the power spectra, at least for $v = 1$. Analytical expressions for arbitrary $v$ in the infinite distance limit may also be obtained by utilizing \eqref{eq:JS_boundary} and noting that \begin{equation} \int_0^\infty \dfrac{dx}{v} \ e^{ix/v} j_l(x) = \sqrt{\pi } 2^{-(l+1)} (i v)^{l+1} \Gamma (l+1) \, _2\tilde{F}_1\left(\frac{l+1}{2},\frac{l+2}{2};l+\frac{3}{2};v^2\right) \,. \end{equation} This may be used to speed up the numerical integration for the infinite distance limit. \subsection{ORF and power spectra} \label{subsec:power_spectra_scalar} We take the calculated magnitudes in the previous section to compute the overlap reduction function for an isotropic stochastic gravitational wave background \eqref{eq:orf_general}. This way, doing it separately for the scalar transverse and scalar longitudinal polarizations, using the addition theorem \eqref{eq:addition_theorem}, we get to the result \begin{equation} \gamma_{ab} \left( \zeta, fD_i \right) = \sum_l \dfrac{2l+1}{4\pi} C_l P_l\left( \cos \zeta \right) \,, \end{equation} where the scalar power spectrum multipoles $C_l$ are given by \begin{equation} C_l = \dfrac{32 \pi^2 F_l\left(f D_a \right) F_l^*\left( fD_b\right)}{\sqrt{4\pi}} \, \end{equation} with the quantity $F_l(fD)$ being \begin{equation} F_l(fD) = - \dfrac{i}{2} \int_0^{2\pi fDv} \dfrac{dx}{v} \ e^{ix/v} R_l\left(x\right) \,. \end{equation} In the above expression, $R_l^\text{SL}(x) = j_l''(x)$ for the scalar longitudinal polarization and $R_l^\text{ST}(x) = -\left( R_l^\text{SL}(x) + j_l(x) \right)/\sqrt{2}$ for the scalar transverse polarization. These $R_l(x)$ functions can be confirmed to be same ones singled out in \cite{Qin:2020hfy} in their Appendix A. The quantity $F_l(\infty)$ are the projection factors considered in \cite{Qin:2020hfy} such that $C_l \propto 32 \pi^2 F_l(\infty) F_l^*(\infty)$. We highlight the main difference to be that the upper limit of the integral is finite which keeps the power spectra defined for either polarizations. In the limit $fD \rightarrow \infty$ and $v \rightarrow 1$, for the scalar transverse monopole and dipole, it can be checked that $F_0^\text{ST}(\infty) = -1/\left(2\sqrt{2}\right)$ and $F_1^\text{ST}(\infty) = -i/\left(6\sqrt{2}\right)$. In the same limit, the scalar longitudinal monopole and dipole projection factors become undefined. On the other hand, in the infinite pulsar distance limit, $fD \rightarrow \infty$, but with arbitrary group velocity $v$, the higher order multipoles $l \geq 2$ become constrained as $F^\text{ST}_l\left(\infty\right) + \left( 1 - v^2 \right) \left( F^\text{SL}_l\left(\infty\right)/\sqrt{2} \right) = 0$. All these agree with \cite{Qin:2020hfy}. We calculate the autocorrelation using the power spectrum and the real space formalism to assess the validity of our power spectrum calculation. The antenna pattern functions for the scalar transverse and longitudinal modes are \begin{equation} F^\text{ST}_a \left( \hat{k} = (\theta, \phi) \right) = \dfrac{\sin^2 \theta}{2 \left( 1 + v \cos \theta \right)} \end{equation} and \begin{equation} F^\text{SL}_a \left( \hat{k} = (\theta, \phi) \right) = \dfrac{\cos^2\theta}{\sqrt{2} \left( 1 + v \cos \theta \right)} \,. \end{equation} The scalar autocorrelation reduces to the integrals \begin{equation} \label{eq:gammaaaST} \gamma_{aa}^\text{ST} = \int_0^\pi \dfrac{d\theta}{\sqrt{4\pi}} \left( \frac{2 \pi \sin ^5 \theta \sin ^2 ( \pi fD (1 + v \cos \theta ))}{(1 + v \cos \theta)^2} \right) \end{equation} and \begin{equation} \label{eq:gammaaaSL} \gamma_{aa}^\text{SL} = \int_0^\pi \dfrac{d\theta}{\sqrt{4\pi}} \left( \frac{4 \pi \sin \theta \cos ^4 \theta \sin ^2 \left(\pi fD (1 + v \cos \theta ) \right)}{(1 + v \cos \theta )^2} \right) \,. \end{equation} It is interesting that $\gamma_{aa}^\text{ST}$ coincides with the transverse tensor $\gamma_{aa}^T$. \subsection{Phenomenology} \label{subsec:scalar_phenomenology} We present the power spectra and resulting overlap reduction functions individually for each of the scalar polarizations. Figure \ref{fig:ClST} shows this for the scalar transverse polarization with $v \sim 1$, $v = 1/2$, and $v = 10^{-2}$ at various pulsar distances. We include the Hellings-Downs signal in the plots for reference. This again echoes the important physical difference between the finite and infinite pulsar distance cases. In the infinite case \cite{Qin:2018yhy, Qin:2020hfy}, the power spectra drops continuously at large $l$, or small angles, as $C_l \sim 1/l^k$ for some positive $k$. This implies that the correlations vanish for pulsars that are infinitesimally separated in the sky. However, in a real setting, pulsars are separated at a finite line of sight distance from the observer (`Us'). Figure \ref{fig:ClST}(a) shows that at some $l \sim 20 - 50$ with a nearly luminal scalar degree of freedom ($v \sim 1$), the power spectra ceases to drop and instead sustains a slow increase. This tells that correlations at small angular separations are strengthened for nearby pulsars, which makes sense for neighboring astrophysical sources \cite{Bernardo:2022vlj}. We recognize this partial sustenance in the transverse-traceless tensor as well as the vector polarizations for finite pulsar distances in the previous sections. The corresponding overlap reduction function is shown in Figure \ref{fig:ClST}(b). This presents the scalar transverse signal resembling what looks like a dipole \cite{NANOGrav:2020bcs} understandably because its power spectrum is dominated by the dipole. In this near luminal limit, we also find that the finite pulsar distance curves by themselves are not so much distinguishable, as in the tensor and vector cases. However, the finite and infinite pulsar distance cases are visually distinguishable, unlike their tensor and vector counterparts. Figures \ref{fig:ClST}(c-d) show the multipoles and the overlap reduction functions when the scalar modes propagate at half the speed of light. In this case, we find that the power spectra multipoles feature an overall decrease in magnitude, and comes with an even sharper drop at small $l$. This is reflected in the overlap reduction function which tends to flatter values, obviously being dominated by the monopole, as compared with the near luminal scalar case. Take note that the overlap reduction functions for large angles relevant for pulsar timing array for the finite pulsar distance cases are visually indistinguishable as displayed by their multipoles (Figures \ref{fig:ClST}(c)). It is worth noting that the monopole in this velocity in the infinite distance limit diverges. In Figure \ref{fig:ClST}(d), the infinite distance curve was as a matter of fact divided by twenty in order to be shown together with the other signals. This divergence will be even more drastic at lower velocities. When the velocity is decreased further to the nonrelativistic limit and infinite distance limit, the power spectrum multipoles drop sharper beginning with the monopole, and so the overlap reduction function reduces to practically a flat horizontal line. However, for the finite distance cases, at some point, we find that the dipole, and even the quadrupole, becomes suppressed compared to the succeeding low multipoles until a peak of the power spectrum appears. This manifests itself as an oscillation in the overlap reduction function, at an angle $\zeta = 180^\circ/l_\text{peak}$ defined by the multipole number $l_\text{peak}$ at which the power spectrum peaks. Figures \ref{fig:ClST}(e-f) explicitly show this with $v = 0.01$. As alluded, for this case, the low multipoles for the finite distance cases can now be distinguished, as contributions beyond the monopole and dipole become significant, and this manifests in the overlap reduction function at large angular separations. A drastic technical difference in the overlap reduction function between the finite and infinite pulsar distance case with $v \ll 1$ can also be realized. The overlap reduction function for the infinite distance limit in Figure \ref{fig:ClST}(f) was divided by $2 \times 10^8$ to be comparable with the other curves. Further, whereas the infinite distance limit signal reduces to a monopole (a mere horizontal line), the finite distance cases present oscillations owing to the fact that pulsars are astrophysical objects in an observable universe. Figure \ref{fig:ClSL} shows the power spectra multipoles and the overlap reduction functions for the scalar longitudinal polarization with the previous choices for the velocity. We present this together with the Hellings-Downs correlation. First off, we recall that the scalar longitudinal mode is undefined with $v = 1$ and infinite pulsar distances. For this alone, we realize the practical advantage of keeping the pulsars at finite distances. In \cite{Bernardo:2022vlj}, we even find compelling statistical evidence of scalar longitudinal polarization in pulsar timing array data. In the nearly luminal scalar case (Figures \ref{fig:ClSL}(a-b)), we find the power spectrum to be dominated by the low multipoles aside from the dipole and the quadrupole. This translates to the particular shape of the overlap reduction function at low angles, as shown. As with the scalar transverse polarization, the power drop is eventually disrupted at some sufficiently large $l$, or small angles, where real, neighboring pulsars may be correlated in their history. However, in the infinite pulsar distance case, the dipole drops compared to the monopole and others close by, resulting in the overlap reduction function being distinguishable for large angles compared to the finite pulsar distance cases. This large angular separation distinction between the finite and infinite pulsar distance cases also manifests at other velocities. We find the similar behavior in the scalar transverse case where the monopolar and dipolar powers numerically diverge as seen in the power spectrum plots. Again, as in the scalar transverse polarization, at half the speed of light, the scalar longitudinal power spectra feature a steeper drop beginning with the quadrupole as $l$ increases (Figures \ref{fig:ClSL}(c-d)). However, it can be seen that in the infinite distance case, the dipolar contribution is larger than the quadrupole, while for the finite distance cases, the dipolar power is otherwise suppressed. This clearly reflects in the overlap reduction functions where the infinite distance curve can be seen to be shaped like a dipole, while the finite distance curves look like a quadrupole. We may mention that the monopole also starts to contribute significantly in the infinite distance case at subluminal velocities \cite{Qin:2020hfy}. The infinite distance curve in Figure \ref{fig:ClSL}(d) was divided by forty to be comparable with the other curves. In this case, the infinite distance correlation becomes mainly dominated by the monopole and the dipole, and so the corresponding overlap reduction function shapes appear like the dipole at half the speed of light. The finite pulsar distance cases, being more dominantly sourced by the quadrupole, instead show a significant departure from the infinite distance limit, as now their shape resembles the Hellings-Downs curve. This time, also, the overlap reduction functions for the finite pulsar distance cases very much coincide. We find that the situation changes in the nonrelativistic scalar ($v = 1/100$), in Figures \ref{fig:ClSL}(e-f). In this case, in the infinite distance limit, the power spectrum is significantly dominated by the monopole, which is reflected in the overlap reduction function being practically a horizontal line \cite{Qin:2020hfy}. As we have also alluded a while ago, the monopole numerically diverges at subluminal velocities, such that we had to divide the infinite distance curve in Figure \ref{fig:ClSL}(f) by a factor $O\left(10^8\right)$. The situation also manifested in the scalar transverse case. On the other hand, for finite pulsar distances, all computations remain well at hand. In this case, we find that most of the low multipoles continue to contribute to the power spectrum, giving a nontrivial angular correlation, and so producing an overlap reduction function that is distinguishable compared with the infinite distance limit. It is notable that the overlap reduction functions for the finite distance cases are also distinguishable between themselves. This is teased by the scalar power spectrum, which drops significantly beginning at $l \sim 10$ for $fD = 100$ and beginning at $l \sim 30$ for $fD = 500$. The overlap reduction function of the scalar longitudinal polarization remains completely nontrivial in all cases for finite distance and subluminal velocities. \section{Discussion} \label{sec:discussion} We have presented a power spectrum method for calculating the overlap reduction function in a pulsar timing array. We argue that this is fast and efficient, particularly with the present data set in which the pulsar pairs have about at least ten degrees angular separation, thus requiring only the first few of the power spectrum multipoles. This is its main advantage over the real space formalism which is challenged numerically by integration over a pole of the integrand. The two methods lead to the same result regardless, as captured by the autocorrelation computations in Table \ref{tab:GaafD100}. \begin{table}[h!] \centering \caption{Autocorrelation $\gamma_{aa}$ calculated using the power spectrum ($l \leq l_\text{max}$) and the real space formalism (RSF) with $fD = 100$. The modes T, V, ST, and SL stand for `tensor', `vector', `scalar transverse', and `scalar longitudinal', respectively.} \begin{tabular}{|c|c|c|c|c|} \hline mode & $\phantom{\dfrac{1}{1}}$ $v$ $\phantom{\dfrac{1}{1}}$ & $\phantom{gg}$ $\gamma_{aa}^{l \leq 30}$ $\phantom{gg}$ & $\phantom{gg}$ $\gamma_{aa}^{l \leq 1000}$ $\phantom{gg}$ & $\phantom{gg}$ $\gamma_{aa}^{\text{RSF}}$ $\phantom{gg}$ \\ \hline \hline \multirow{3}*{\phantom{ggg} T \phantom{ggg}} & $\phantom{\dfrac{1}{1}}$ $0.99$ $\phantom{\dfrac{1}{1}}$ & $1.08$ & $2.16$ & $2.17$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.50$ $\phantom{\dfrac{1}{1}}$ & $0.53$ & $1.06$ & $1.06$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.01$ $\phantom{\dfrac{1}{1}}$ & $0.97$ & $0.97$ & $0.97$ \\ \hline \multirow{3}*{\phantom{ggg} V \phantom{ggg}} & $\phantom{\dfrac{1}{1}}$ $0.99$ $\phantom{\dfrac{1}{1}}$ & $7.92$ & $15.7$ & $15.7$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.50$ $\phantom{\dfrac{1}{1}}$ & $0.67$ & $1.34$ & $1.34$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.01$ $\phantom{\dfrac{1}{1}}$ & $1.20$ & $1.20$ & $1.20$ \\ \hline \multirow{3}*{\phantom{ggg} ST \phantom{ggg}} & $\phantom{\dfrac{1}{1}}$ $0.99$ $\phantom{\dfrac{1}{1}}$ & $1.08$ & $2.17$ & $2.17$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.50$ $\phantom{\dfrac{1}{1}}$ & $0.53$ & $1.06$ & $1.06$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.01$ $\phantom{\dfrac{1}{1}}$ & $0.97$ & $0.97$ & $0.97$ \\ \hline \multirow{3}*{\phantom{ggg} SL \phantom{ggg}} & $\phantom{\dfrac{1}{1}}$ $0.99$ $\phantom{\dfrac{1}{1}}$ & $81.8$ & $151$ & $151$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.50$ $\phantom{\dfrac{1}{1}}$ & $0.65$ & $1.26$ & $1.26$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.01$ $\phantom{\dfrac{1}{1}}$ & $0.40$ & $0.40$ & $0.40$ \\ \hline \end{tabular} \label{tab:GaafD100} \end{table} The autocorrelation can be interpreted as the correlation of a pulsar with itself and embodies the small scale power in a pulsar timing array. To compute this using the power spectrum formalism requires at least a few thousand multipoles to guarantee a degree of numerical convergence. This is shown in Table \ref{tab:GaafD100} for $fD = 100$, or $D \sim 30$ parsecs, where it can be seen that the large scale multipoles ($l \leq 30$ or $\zeta \geq 6^\circ$), those which we have utilized in the overlap reduction functions in the previous sections, only capture nearly half of the total power in small scales ($ l \leq 1000$ or $\zeta \geq 0.18^\circ$). We confirm that the real space formalism gets to the same numbers. Understandably this is because the real space formalism involves both earth and pulsar terms, thereby incorporating the small and large scale power at the same time. The situation deviates in the nonrelativistic modes ($v \ll 1$) where nearly the full power is captured by the low multiples ($l \lesssim 100$), but this can be explained quite easily by looking at the power spectrum. The power peaks and drops sharply within the first hundred multipoles for nonrelativistic polarization modes ($v \ll 1$). Figures \ref{fig:ClT}, \ref{fig:ClV}, \ref{fig:ClST}, and \ref{fig:ClSL} support this point that the low multipoles by themselves effectively capture the full power at nonrelativistic speeds. We repeat the calculation for pulsars as far as $fD = 500$ or $D \sim 150$ parsecs, and can confirm similar observations. At this larger distance, however, we find that the power spectrum method becomes quite challenged by the extremely rapid oscillations in the integrand for large multipole numbers. The computation slows down considerably and the accuracy becomes less reliable, in addition to needing more multipoles, this time with $l \leq 3000$, for numerical convergence. This shows where the real space formalism, providing autocorrelation functions instantly, takes the advantage over the power spectrum method. The power spectrum even overestimates the autocorrelation function, which we associate to a breakdown of precision that the computation suffers with highly oscillating integrands. Nonetheless, the power spectrum method works well for nearby pulsar distances and that the real space formalism is always there to backup our calculations in this regime. We make some final remarks on the drawbacks of the infinite distance limit, where the pulsars are at unreachable distances from the observer, and the power spectrum method. Table \ref{tab:GaafDinf} shows the autocorrelation calculated using the power spectrum and the real space formalism. \begin{table}[h!] \centering \caption{Autocorrelation $\gamma_{aa}$ calculated using the power spectrum ($l \leq l_\text{max}$) and the real space formalism (RSF) in the infinite pulsar distance limit, $fD \rightarrow \infty$. The modes T, V, ST, and SL stand for `tensor', `vector', `scalar transverse', and `scalar longitudinal', respectively.} \begin{tabular}{|c|c|c|c|c|} \hline mode & $\phantom{\dfrac{1}{1}}$ $v$ $\phantom{\dfrac{1}{1}}$ & $\phantom{gg}$ $\gamma_{aa}^{l \leq 30}$ $\phantom{gg}$ & $\phantom{gg}$ $\gamma_{aa}^{l \leq 1000}$ $\phantom{gg}$ & $\phantom{gg}$ $\gamma_{aa}^{\text{RSF}}$ $\phantom{gg}$ \\ \hline \hline \multirow{3}*{\phantom{ggg} T \phantom{ggg}} & $\phantom{\dfrac{1}{1}}$ $0.99$ $\phantom{\dfrac{1}{1}}$ & $1.08$ & $1.08$ & $2.17$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.50$ $\phantom{\dfrac{1}{1}}$ & $0.53$ & $0.53$ & $1.06$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.01$ $\phantom{\dfrac{1}{1}}$ & $0.47$ & $0.47$ & $0.95$ \\ \hline \multirow{3}*{\phantom{ggg} V \phantom{ggg}} & $\phantom{\dfrac{1}{1}}$ $0.99$ $\phantom{\dfrac{1}{1}}$ & $7.77$ & $7.77$ & $15.5$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.50$ $\phantom{\dfrac{1}{1}}$ & $0.67$ & $0.67$ & $1.34$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.01$ $\phantom{\dfrac{1}{1}}$ & $0.47$ & $0.47$ & $0.95$ \\ \hline \multirow{3}*{\phantom{ggg} ST \phantom{ggg}} & $\phantom{\dfrac{1}{1}}$ $0.99$ $\phantom{\dfrac{1}{1}}$ & $1.24$ & $1.24$ & $2.17$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.50$ $\phantom{\dfrac{1}{1}}$ & $15.4$ & $15.4$ & $1.06$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.01$ $\phantom{\dfrac{1}{1}}$ & $8.86 \times 10^7$ & $8.86 \times 10^7$ & $0.95$ \\ \hline \multirow{3}*{\phantom{ggg} SL \phantom{ggg}} & $\phantom{\dfrac{1}{1}}$ $0.99$ $\phantom{\dfrac{1}{1}}$ & $66.1$ & $66.2$ & $158$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.50$ $\phantom{\dfrac{1}{1}}$ & $31.0$ & $31.0$ & $1.26$ \\ \cline{2-5} & $\phantom{\dfrac{1}{1}}$ $0.01$ $\phantom{\dfrac{1}{1}}$ & $1.77 \times 10^8$ & $1.77 \times 10^8$ & $0.71$ \\ \hline \end{tabular} \label{tab:GaafDinf} \end{table} As expected, as the power spectrum profile merely drops sharply as the multipole number increases, the infinite distance power spectrum calculation misses about half of the total power that is contained in small angular scales, regardless of how many multipoles are included. This is reflected quite clearly for the tensor and vector modes in Table \ref{tab:GaafDinf} where the $\gamma_{aa}^{l \leq 30}$ and $\gamma_{aa}^{l \leq 1000}$ are the same at this reasonable level of precision. For the scalar polarizations, the situation worsens, now that the monopole and dipole spoils the power spectrum calculation through their dominance at low velocities. In this limit, in fact, it can be shown that the monopole and dipole behaves as $1/v^4$ and $1/v^2$, respectively, at low speeds $v \ll 1$ \cite{Qin:2020hfy}. Exactly, a straightforward integration for the monopole in the limit $fD \rightarrow \infty$ leads to $C_0^{\text{ST}}/(4\pi) = \sqrt{\pi}/\left(2 v^4\right) \sim 8.86 \times 10^7$ and $C_0^{\text{SL}}/(4\pi) = \sqrt{\pi}/v^4 \sim 1.77 \times 10^8$ as $v \sim 1/100$. This alone explains the unphysical numbers in Table \ref{tab:GaafDinf} for the scalar modes at half the speed of light and the nonrelativistic cases. A similar analytical calculation with the dipole can be done in this limit. Just the same, it appears that only the real space formalism is trustworthy for the autocorrelation in the infinite distance limit. We take these considerations to add more case to factoring in finite pulsar distances in pulsar timing array analysis. In a future work, we look forward to a merging of the power spectrum method and the real space formalism for an accurate calculation of the overlap reduction function in a pulsar timing array, in the same way post Newtonian gravity and numerical relativity are utilized in studying gravitational waves from compact binaries. This will become relevant as pulsar pairs of subdegree separation are observed, something which we anticipate in future pulsar timing array missions. \section{Conclusions} \label{sec:conclusions} We have derived the overlap reduction functions of an isotropic stochastic gravitational wave background sourced by tensor, vector, and scalar metric polarizations, for finite pulsar distances and subluminal velocities. This reveals for one the importance of keeping the pulsars at realistic finite distances, particularly, in letting all modes be well defined and keeping the power in small scales, and in future pulsar timing array data where millisecond pulsar pairs may be of subdegree separations. Our technical results prepare the general data analysis of the various possible metric polarizations which may be anchored in the nanohertz gravitational wave sky, such as we have demonstrated in \cite{Bernardo:2022vlj}. However the results may be, this would be complementary to the picture provided by ground and space based gravitational wave observatories about our Universe and the matter that lives, and lived, within. The power spectrum calculation is an efficient way of obtaining the pulsar timing array observables, in addition to providing an independent check of the real space formalism which is known to be technically challenged by poles in numerical integration over an angular domain. Even so, the power spectrum calculation and the real space formalism for obtaining the overlap reduction function can be complimentary, particularly in future data sets with pulsar pairs of subdegree separation, which will challenge the power spectrum calculation but not quite the real space formalism. We have shown this for the autocorrelation function, describing pulsar pairs along the same line of sight or the correlation of a pulsar with itself, where at least a few thousand multipoles were needed to ensure the numerical convergence of the sum in the power spectrum calculation but which was a quick calculation using the real space formalism. A merger of these two methods, much in the same way gravitational waveform analyses rely on post Newtonian and numerical relativity calculations, would be something to look forward to in the pulsar timing community. This work sets up several future directions. First, it would be interesting to see whether the tensor modes by themselves, if they are freed from the light cone, could match statistically the present pulsar timing array data. If so, this hints at dispersive gravitational waves that only so happen to be luminal in the ground based detectors' frequency band of about a hundred hertz. Second, following on \cite{Qin:2020hfy, Bernardo:2022vlj}, in theories, the various metric polarizations are expected to be constrained by the theory parameters rather than just independently contributing to the overall signal. This calls on alternative gravity theorists to setup these theoretical constraints which would also reduce the parameter space and lift potential degeneracy of the models for data analysis. Lastly, building on \cite{Liu:2022skj}, it remains to setup the general formalism for scalar and vector polarizations, as well as tensors off the light cone, for studying the anisotropies in the stochastic gravitational wave background. It may take a while for pulsar timing science to mature to this level of sensitivity, but this takes to a different level, hinting at natural clocks and nonGaussianities in the primordial universe \cite{Pol:2022sjn, Bodas:2022zca, Dimastrogiovanni:2022afr}. \acknowledgments This work was supported in part by the Ministry of Science and Technology (MOST) of Taiwan, Republic of China, under Grant No. MOST 110-2112-M-001-036. \appendix \section{Gravitational wave polarization basis} \label{sec:gw_pol_tensors} For a gravitational wave propagating along the $\hat{\Omega}$ direction, the polarization basis tensors can be expressed as \cite{Boitier:2021rmb} \begin{eqnarray} \varepsilon^{+} &=& \hat{m} \otimes \hat{m} - \hat{n} \otimes \hat{n} \,, \\ \varepsilon^{\times} &=& \hat{m} \otimes \hat{n} + \hat{n} \otimes \hat{m} \,, \\ \varepsilon^{x} &=& \hat{m} \otimes \hat{\Omega} + \hat{\Omega} \otimes \hat{m} \,, \\ \varepsilon^{y} &=& \hat{n} \otimes \hat{\Omega} + \hat{\Omega} \otimes \hat{n} \,, \\ \varepsilon^{\text{ST}} &=& \hat{m} \otimes \hat{m} + \hat{n} \otimes \hat{n} \,, \\ \varepsilon^{\text{SL}} &=& \sqrt{2} \hat{\Omega} \otimes \hat{\Omega} \,, \end{eqnarray} where $\left( \hat{m}, \hat{n}, \hat{\Omega} \right)$ for an orthonormal basis. The $\left(\varepsilon^{+}, \varepsilon^{\times} \right)$ stand for the transverse-traceless tensor modes, $\left(\varepsilon^{x}, \varepsilon^{y} \right)$ for the vector modes, and $\left(\varepsilon^{\text{ST}}, \varepsilon^{\text{SL}} \right)$ for the scalar modes. In practice, it is useful to orient $\hat{\Omega}$ along the $\hat{z}$-direction. In this case, the orthonormal basis $\left( \hat{m}, \hat{n}, \hat{\Omega} \right)$ may be written as \begin{eqnarray} \hat{m} &=& \cos \varphi \ \hat{x} + \sin \varphi \ \hat{y} \,, \\ \hat{n} &=& -\sin \varphi \ \hat{x} + \cos \varphi \ \hat{y} \,, \\ \hat{\Omega} &=& \hat{z} \,. \end{eqnarray} With the Cartesian basis $\left( \hat{x}, \hat{y}, \hat{z} \right)$, the polarization tensors can be identified to be \begin{equation} \varepsilon^{+} = \left( \begin{array}{ccc} \cos(2\varphi) & \sin(2\varphi) & 0 \\ \sin(2\varphi) & -\cos(2\varphi) & 0 \\ 0 & 0 & 0 \end{array} \right) \,, \end{equation} \begin{equation} \varepsilon^{\times} = \left( \begin{array}{ccc} -\sin(2\varphi) & \cos(2\varphi) & 0 \\ \cos(2\varphi) & -\sin(2\varphi) & 0 \\ 0 & 0 & 0 \end{array} \right) \,, \end{equation} \begin{equation} \varepsilon^{x} = \left( \begin{array}{ccc} 0 & 0 & \cos \varphi \\ 0 & 0 & \sin \varphi \\ \cos \varphi & \sin \varphi & 0 \end{array} \right) \,, \end{equation} \begin{equation} \varepsilon^{y} = \left( \begin{array}{ccc} 0 & 0 & -\sin \varphi \\ 0 & 0 & \cos \varphi \\ -\sin \varphi & \cos \varphi & 0 \end{array} \right) \,, \end{equation} \begin{equation} \varepsilon^{\text{ST}} = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right) \,, \end{equation} and \begin{equation} \varepsilon^{\text{SL}} = \sqrt{2} \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right) \,. \end{equation} \section{Triple spherical harmonics integral and the Wigner-3j symbol} \label{sec:3Y3j} We put down identities on spin weighted spherical harmonics $\,_s Y_{lm}\left(\hat{n}\right)$ \cite{Weinberg, Arfken}. First, for $s = 0$, $\,_s Y_{lm}\left(\hat{n}\right)$ reduces to the spherical harmonic $ Y_{lm}\left(\hat{n}\right)$. In general, $\,_s Y_{lm}\left(\hat{n}\right)$ satisfies the orthogonality relation \begin{equation} \int_{S^2} d\hat{n} \,_s Y_{lm}^*\left(\hat{n}\right) \,_s Y_{l'm'}\left(\hat{n}\right) = \delta_{ll'} \delta_{mm'} \,, \end{equation} and completeness relation \begin{equation} \sum_{lm} \,_s Y^*_{lm}\left(\hat{n}\right) \,_s Y_{lm}\left(\hat{n}' \right) = \delta^{(2)}\left( \hat{n} - \hat{n}' \right) = \delta\left(\phi - \phi'\right) \delta\left(\cos \theta - \cos \theta' \right) \,. \end{equation} This also satisfies the conjugate identity \begin{equation} \,_s Y_{lm}^*\left( \hat{n} \right) = (-1)^{s + m} \,_{-s}Y_{l-m}\left( \hat{n} \right) \,. \end{equation} We progress in the text with triple spherical harmonics identity \begin{equation} \int d \hat{e} \,_{s_1} Y_{l_1 m_1}\left(\hat{e}\right) \,_{s_2}Y_{l_2 m_2}\left(\hat{e}\right) \,_{s_3}Y_{l_3 m_3}\left(\hat{e}\right) = \sqrt{\dfrac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \left( \begin{array}{ccc} l_1 & l_2 & l_3 \\ -s_1 & -s_2 & -s_3 \end{array} \right) \left( \begin{array}{ccc} l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \end{array} \right) \,, \end{equation} where $\left(\begin{array}{ccc} a & b & c \\ d & e & f \end{array}\right)$ is the Wigner-3j symbol. The 3j symbol vanishes unless $|l_1 - l_2| < l_3 < l_1 + l_2$ and $m_1 + m_2 + m_3 = 0$. Further, if $m_1 = m_2 = m_3 = 0$, then $l_1 + l_2 + l_3$ must be an even integer. It also satisfies a reflection property \begin{equation} \left( \begin{array}{ccc} l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \end{array} \right) = (-1)^{l_1 + l_2 + l_3} \left( \begin{array}{ccc} l_1 & l_2 & l_3 \\ -m_1 & -m_2 & -m_3 \end{array} \right) \,. \end{equation} \section{Another real space formalism} \label{sec:another} Provided a passing gravitational wave in the direction $\hat{k}$, the redshift fluctuation is also often considered as the Shapiro time delay \cite{NANOGrav:2021ini}: \begin{equation} \label{eq:redshift_realspace} z(t) = \dfrac{\hat{e}^i \otimes \hat{e}^j}{2\left( 1 + v \hat{k} \cdot \hat{e} \right)} \left( h_{ij}^e - h_{ij}^p \right) \,, \end{equation} where $h_{ij}^e = h_{ij}\left(t, \vec{0}\right)$, the earth term, is the metric perturbation evaluated on earth where the pulse is received, and $h_{ij}^p = h_{ij}\left( t - D , D \hat{e} \right)$, the pulsar term, is evaluated at the pulsar during emission. Substituting \eqref{eq:gw_general}, the redshift fluctuation for a pulsar $a$ can be simplified as \begin{equation} \label{eq:redshift_altreal} z_a(t) = \sum_A \int_{-\infty}^\infty df \int_{S^2} d\hat{k} \ \tilde{h}_A\left(f, \hat{k}\right) F_a^A \left( \hat{k} \right) e^{-2\pi i f t} U_a\left( f, \hat{k} \right) \,, \end{equation} where $F_a$ are the antenna pattern functions \eqref{eq:antenna_functions} and $U_a$ is given by \eqref{eq:Uadef}. Substituting \eqref{eq:redshift_altreal} into \eqref{eq:timing_residual}, one gets to the two point pulsar timing residual correlation function \eqref{eq:twopoint_realspace} and the overlap reduction function \eqref{eq:orf_realspace}. \section{A brief review of the overlap reduction function} \label{sec:orfs_review} We briefly review the well established results about the overlap reduction function (see e.g. \cite{NANOGrav:2021ini}). We start with the standard one, that is, due to the transverse traceless tensor polarizations predicted by general relativity. The overlap reduction function is given by \begin{equation} \Gamma_{ab}^{\text{TT}} = \Gamma_{ab}^+ + \Gamma_{ab}^\times \approxeq \dfrac{\delta_{ab}}{2} + C\left(\zeta_{ab}\right) \,, \end{equation} where $\zeta_{ab}$ is the angular separation of two pulsars, and $C\left(\zeta_{ab}\right)$ is the Hellings-Downs curve \cite{Hellings:1983fr}: \begin{equation} C\left(\zeta_{ab}\right) = \dfrac{3}{2} \left( \dfrac{1}{3} + \left( \dfrac{1 - \cos \zeta_{ab}}{2} \right) \left[ \ln \left( \dfrac{1 - \cos \zeta_{ab}}{2} \right) - \dfrac{1}{6} \right] \right) \,. \end{equation} Provided tensor modes propagating at the speed of light, we expect the stochastic gravitational wave background signal to be given by the Hellings-Downs correlation. There are also phenomenological ones, that are not necessarily due to gravitational degrees of freedom, but nonetheless present a competitive signal to noise ratio in the current data \cite{NANOGrav:2021ini}. These are the gravitational wave like monopole and dipole: \begin{equation} \Gamma_{ab}^{\text{GW mon}} = \dfrac{\delta_{ab}}{2} + \dfrac{1}{2} \,, \end{equation} and \begin{equation} \Gamma_{ab}^{\text{GW dip}} = \dfrac{\delta_{ab}}{2} + \dfrac{\cos \zeta_{ab}}{2} \,. \end{equation} The gravitational wave monopole particularly has a significant signal to noise ratio, compared with the gravitational wave dipole and the Hellings-Downs correlation, in the present data set. We move on to non-Einsteinian polarization modes on the light cone, and considering infinite pulsar distances. The scalar transverse polarization (also often referred to as the `breathing' mode) leads to the overlap reduction function \cite{Chamberlin:2011ev} \begin{equation} \Gamma_{ab}^{\text{ST}} \approx \dfrac{\delta_{ab}}{2} + \dfrac{1}{8} \left( 3 + \cos \zeta_{ab} \right) \,. \end{equation} On the other hand, for the scalar longitudinal modes, the overlap reduction function cannot be evaluated analytically for arbitrary pulsar pair angular separations. But the more prominent issue is that this cannot be defined for infinite pulsar distances. Keeping the pulsars at a finite distance $fD \gg 1$, from the observer, the autocorrelation function can be shown to be linearly divergent \cite{Chamberlin:2011ev}, \begin{equation} \Gamma_{aa}^\text{SL} \sim \dfrac{3\pi^2}{4} fD - 3 \ln \left( 4\pi fD \right) + \dfrac{37}{8} - 3 \gamma_\text{E} \,, \end{equation} where $\gamma_\text{E}$ is Euler's constant. A similar situation arises for the vector modes, whereas the overlap reduction function can be determined to be \cite{2008ApJ...685.1304L} \begin{equation} \Gamma_{ab}^\text{V} = \Gamma_{ab}^{(\text{V})_x} + \Gamma_{ab}^{(\text{V})_y} \approx 3 \log \left( \dfrac{2}{1 - \cos \zeta_{ab}} \right) - 4 \cos \zeta_{ab} - 3 \,, \end{equation} the autocorrelation function \cite{2008ApJ...685.1304L} \begin{equation} \Gamma_{aa}^\text{V} \sim 6 \ln \left( 4 \pi fD \right) - 14 + 6 \gamma_\text{E} \, \end{equation} becomes undefined, diverges logarithmically, in the infinite distance limit, albeit not as strongly as the scalar longitudinal polarization. \bigskip

Title: The Atacama Cosmology Telescope: limits on dark matter-baryon interactions from DR4 power spectra

Abstract: Diverse astrophysical observations suggest the existence of cold dark matter that interacts only gravitationally with radiation and ordinary baryonic matter. Any nonzero coupling between dark matter and baryons would provide a significant step towards understanding the particle nature of dark matter. Measurements of the cosmic microwave background (CMB) provide constraints on such a coupling that complement laboratory searches. In this work we place upper limits on a variety of models for dark matter elastic scattering with protons and electrons by combining large-scale CMB data from the Planck satellite with small-scale information from Atacama Cosmology Telescope (ACT) DR4 data. In the case of velocity-independent scattering, we obtain bounds on the interaction cross section for protons that are 40\% tighter than previous constraints from the CMB anisotropy. For some models with velocity-dependent scattering we find best-fitting cross sections with a 2$\sigma$ deviation from zero, but these scattering models are not statistically preferred over $\Lambda$CDM in terms of model selection.

https://export.arxiv.org/pdf/2208.08985

\title{The Atacama Cosmology Telescope: limits on dark matter--baryon interactions from DR4 power spectra} \author{Zack Li\altaffilmark{1}, Rui An\altaffilmark{2}, Vera Gluscevic\altaffilmark{2}, Kimberly K.~Boddy\altaffilmark{3}, J~Richard~Bond\altaffilmark{1}, Erminia Calabrese\altaffilmark{4}, Jo Dunkley\altaffilmark{5,6}, Patricio~A.~Gallardo\altaffilmark{7}, Yilun Guan\altaffilmark{8}, Adam Hincks\altaffilmark{9,10}, Kevin~M.~Huffenberger\altaffilmark{11}, Arthur Kosowsky\altaffilmark{12}, Thibaut Louis\altaffilmark{13}, Mathew S.~Madhavacheril\altaffilmark{14}, Kavilan~Moodley\altaffilmark{15,16}, Lyman~A.~Page\altaffilmark{5}, Bruce Partridge\altaffilmark{17}, Frank~J.~Qu\altaffilmark{18}, Maria Salatino\altaffilmark{19,20}, Blake Sherwin\altaffilmark{18}, Crist\'obal Sif\'on\altaffilmark{21}, Cristian Vargas\altaffilmark{22}, Edward J.~Wollack\altaffilmark{23} } \altaffiltext{1}{Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, ON, Canada M5S 3H8} \altaffiltext{2}{Department of Physics \& Astronomy, University of Southern California, Los Angeles, CA, 90007, USA} \altaffiltext{3}{Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA} \altaffiltext{4}{School of Physics and Astronomy, Cardiff University, The Parade, Cardiff, CF24 3AA, UK} \altaffiltext{5}{Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544} \altaffiltext{6}{Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544, USA} \altaffiltext{7}{Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA} \altaffiltext{8}{Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George St., Toronto, ON M5S 3H4, Canada} \altaffiltext{9}{David A. Dunlap Department of Astronomy \& Astrophysics, University of Toronto, 50 St. George St., Toronto, ON M5S 3H4, Canada} \altaffiltext{10}{Specola Vaticana (Vatican Observatory), V-00120 Vatican City State} \altaffiltext{11}{Department of Physics, Florida State University, Tallahassee FL, USA 32306} \altaffiltext{12}{Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA} \altaffiltext{13}{Universit\'e Paris-Saclay, CNRS, Institut d'astrophysique spatiale, 91405, Orsay, France.} \altaffiltext{14}{Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo ON N2L 2Y5, Canada} \altaffiltext{15}{Astrophysics Research Centre, University of KwaZulu-Natal, Westville Campus, Durban 4041, South Africa} \altaffiltext{16}{School of Mathematics, Statistics \& Computer Science, University of KwaZulu-Natal, Westville Campus, Durban 4041, South Africa} \altaffiltext{17}{Department of Physics and Astronomy, Haverford College, Haverford, PA, USA 19041} \altaffiltext{18}{DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UK} \altaffiltext{19}{Department of Physics, Stanford University, Stanford, California 94305, USA} \altaffiltext{20}{Kavli Institute for Particle Astrophysics and Cosmology, Stanford, CA 94305, USA} \altaffiltext{21}{Instituto de F\'isica, Pontificia Universidad Cat\'olica de Valpara\'iso, Casilla 4059, Valpara\'iso, Chile} \altaffiltext{22}{Instituto de Astrof\'isica and Centro de Astro-Ingenier\'ia, Facultad de F\'isica, Pontificia Universidad Cat\'olica de Chile, Av. Vicu\~na Mackenna 4860, 7820436 Macul, Santiago, Chile} \altaffiltext{23}{NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA} \section{Introduction} \setcounter{footnote}{0} Measurements of the cosmic microwave background (CMB) anisotropies let us view the early Universe as a high-energy gravitational laboratory. In the standard model of cosmology, the dynamics of the early Universe were dominated by scattering between radiation and baryonic matter, as well as gravitational interactions from dark matter, leading to the oscillations that generated the famous acoustic features in the CMB power spectrum. Precise measurements of these features have helped build a successful, predictive model for the contents, geometry, and evolution of the early Universe. The lack of deviations from this standard model of cosmology has provided stringent constraints on extensions that would change the CMB acoustic features, such as physics arising from neutrinos and axions \citep{boyarsky2009, brust2013, marsh2016, green2019, brinckmann2019}, interactions in the dark sector \citep{cyrracine2014, hlozek2017, buenabad2018}, and models of dark energy. Scattering between dark matter and baryons (DM-baryon scattering) is an example of an extension of the standard model of cosmology that alters the dynamics of the early Universe, leaving fingerprints on the acoustic oscillations seen in the CMB \citep{chen2002, DvorkinBlum, BoddyGluscevic, GluscevicBoddy, Xu189710, 2021PhRvD.104j3521N,2022PhR...961....1B}. Although a diverse set of astrophysical observations suggests that the cold dark matter in the Universe interacts only gravitationally, the possibility of a coupling between dark matter and baryons still presents a tantalizing target towards understanding the particle nature of dark matter. Direct detection experiments \citep[e.g.,][]{sensei2018, darkside2018, supercdms2019, xenonnt2020} have made substantial progress in searching for such a coupling, typically through searching in the laboratory for nuclear recoils from scattering with dark matter. Astrophysical constraints on dark matter can complement direct detection experiments, in particular delivering competitive sensitivities towards models with dark matter particles masses below approximately 1\,GeV for nuclear recoils and near to 1 MeV for electronic recoils. The most powerful astrophysical constraints on pre-recombination scattering between DM and baryons come from the Milky Way satellite abundance measurements \citep{Nadler:2020,2021PhRvL.126i1101N, 2021ApJ...907L..46M,2021PhRvD.104j3521N} and Lyman-$\alpha$-forest measurements \citep{2021arXiv211004024H, 2021arXiv211110386R}, while the CMB provides the most competitive bounds for post-recombination scattering \citep{SlatyerWu,2018PhRvD..98l3506B}. At the same time, constraints from the CMB data have very different sources of uncertainty than other observational probes, do not require modeling of baryonic physics arising from galaxy formation, and probe the same physics at different physical scales. Notably, small-scale polarization measurements of the CMB have less contamination from astrophysical foregrounds relative to temperature anisotropies, and can carry valuable information about DM physics. The features induced in CMB power spectra by scattering effects can vary, depending on how the momentum-transfer cross section $\sigma_{\mathrm{MT}}$ varies with relative velocity $v$; following the literature, we parameterize this dependence as a power law with index $n$, such that $\sigma_{\mathrm{MT}}= \sigma_0 \, v^n$ in natural units, for either scattering with protons, or electrons. In this work, the choice of $n$ amounts to the choice of the scattering model at hand. DM-baryon scattering with power law index $n < 0$ tends to produce progressively stronger relative suppression of power in the CMB at small angular scales, as shown in Figure \ref{fig:models}, and represents scattering that takes place in the post-recombination universe (after $v$ redshifts due to the universal expansion), affecting the degree of lensing of the CMB. DM-baryon scattering with non-negative $n$ tends to also exhibit a substantial increase in power at intermediate scales ($\ell \sim 1000-2000$), by effectively increasing the mass of the baryons, and is dominated by scattering in the pre-recombination universe (where $v$ is driven by thermal velocities that are large at early times). The \planck\, satellite measured the temperature anisotropies to the cosmic variance limit for scales up to $\ell \sim 2000$, but current and upcoming ground-based experiments such as the Atacama Cosmology Telescope \citep[ACT,][]{thornton/2016}, the South Pole Telescope \citep[SPT,][]{benson/etal:2014}, the Simons Observatory \citep[SO,][]{so_forecast:2019}, and CMB-S4 \citep{cmbs4:2019} promise to push the cosmic variance limit to $\ell \sim 3000 - 4000$ in both temperature and polarization. As an example, the addition of ACT DR2 and SPT data improved the \planck{}-2015 constraint by a factor of two for some interaction models \citep{SlatyerWu}. Previous CMB constraints were driven primarily by data from the \planck{} satellite \citep{DvorkinBlum, BoddyGluscevic, BoddyGluscevicPoulin, Xu189710, 2021PhRvD.104j3521N}. In this work, we use the ACT DR4 data \citep{choi2020, aiola2020}, collected during 2013$-$2016, to search for DM-baryon scattering. In combination with the 2018 \planck{} data, we use these data to improve the CMB constraints on DM-baryon scattering. In Sec.~\ref{sec:models}, we review how observables predicted by the Einstein-Boltzmann equations change in the presence of DM-proton scattering. In Sec.~\ref{sec:data}, we describe the data we use for our analysis and detail the fitting procedure. We present our main results in Sec.~\ref{sec:results} and conclude in Sec.~\ref{sec:conclusions}. \section{Scattering model observables} \label{sec:models} Within the power-law parameterization for the momentum-transfer cross-section, the index $n=0$ arises in the simplest example of a spin-independent or spin-dependent contact interaction; millicharged DM exhibits Coulomb-like interaction with $n=-4$; $n=2$ corresponds to DM with an electric dipole moment \citep{Fitzpatrick:2010br}; In this study, we consider models with $n\in\{-4,-2,0,2,4,6\}$, as described in e.g., \cite{DvorkinBlum}. We separately constrain elastic scattering with protons and elastic scattering with electrons. % Different values of $n$ lead to a different redshift evolution of the rate of momentum transfer $R_\chi$ between DM and baryons, affecting matter perturbations at different cosmological times: for $n=-4$, scattering is more important as thermal particle velocities decay, later on in cosmic history; for $n\geq-2$, scattering mainly occurs prior to recombination, at high redshift when thermal velocities are large \citep{BoddyGluscevic, BoddyGluscevicPoulin}. However, all forms of scattering interactions considered here affect the matter distribution in the universe through collisional damping of small-scale perturbations. The resulting suppression of the matter transfer function is captured in the CMB temperature, polarization, and lensing power spectra. The main effect of the interactions is suppression of power at small angular scales. Secondary effects include small shifts in the acoustic peaks, as well as the increase in power on large angular scales. The latter is particularly prominent in models where DM couples strongly to the baryon-photon fluid prior to recombination, producing an effective ``baryon-loading'' effect and increasing power at low multipoles \cite{BoddyGluscevic,BoddyGluscevicPoulin}. To accurately model the effects of DM-proton scattering on the CMB primary power spectra, we use a modified Boltzmann code CLASS \citep{class} developed for previous studies \citep{BoddyGluscevic} and publicly released with the work of \cite{2021PhRvD.104j3521N}. This code includes scattering interactions and their effects on the matter transfer function and the thermal history.\footnote{\url{https://github.com/kboddy/class\_public/tree/dmeff}} % \section{Data and methodology} \label{sec:data} \subsection{Cosmological Model} \label{sec:model} We use Markov Chain Monte Carlo (MCMC) chains to sample the standard six parameters of the $\Lambda$CDM model, plus one or two extension parameters. The six $\Lambda$CDM parameters are \begin{equation} \left\{ n_s, \, \log \left( 10^{10}A_s \right), \, \tau_{\mathrm{reio}}, \, \Omega_b h^2, \, \Omega_c h^2, \, 100 \theta_s \right\}. \end{equation} for the scalar spectral index, scalar amplitude, optical depth to reionization, baryon density, cold dark matter density, and CMB peak position respectively. The extension parameters are $\sigma_0$, the DM-baryon interaction cross section, and $m_{\chi}$, the DM particle mass. We treat each DM-baryon interaction cross section velocity dependence, $n$, individually as separate phenomenological models for analysis. For each choice of $n$, we perform one analysis in which scattering is limited only to protons, and a second with only electron scattering. To report confidence limits on cross section constraints, we sample the six $\Lambda$CDM parameters in addition to $\sigma_0$, fixing $m_\chi$ at seven different values between 1 MeV and 1 TeV, and for $n\in\{-4,-2,0,2,4,6\}$. In these cases we impose a uniform prior on the cross section. For exploration purposes we also estimate parameters for an eight-parameter model: $\Lambda$CDM plus $\log(\sigma_0)$ and $\log(m_\chi)$, in this case imposing no preference on the order of magnitude of the cross section and mass. Sampling both parameters simultaneously has not been done before in CMB analyses of DM-baryon scattering. For each mass, we choose a lower prior on $\log(\sigma_0)$ that is several decades below the lowest limit obtained from sampling $\sigma_0$ at fixed mass. This choice of prior will have a small effect on numerical results like the 95-percentile upper bound, as it removes a small region of parameter space close to zero. We avoid masses below 1 MeV for numerical stability. We include an approximate treatment of neutrinos and other light relics by setting a massless light relic density of $N_{\mathrm{ur}} = 2.0328$ and including a single massive neutrino with mass $m_{\mathrm{ncdm}} = 0.06$ eV. In this work we replace all of the DM density in the universe with a component that interacts with baryons. However, we do retain a small tracer component of standard non-interacting cold dark matter (CDM) at the level of $10^{-12}$, in order to allow for numerical computation in the synchronous gauge of this component. Existing formulae like {\tt halofit} \citep{halofit} are derived from N-body simulations which do not include DM-baryon scattering, and thus are unreliable for predicting effects of nonlinear growth on the late-time matter power spectrum in our extension cosmologies. We found {\tt halofit} produces unrealistic nonlinear matter power spectra when used in conjunction with dark matter scattering models, even for cross sections that result in only modest deviations from $\Lambda$CDM. Throughout this work, we include only linear $P(k)$ computations. Nonlinear growth tends to amplify power at small scales, which would tend to amplify CMB power spectra at scales most sensitive to dark matter scattering with baryons. Incorporating nonlinear growth thus amplifies the scattering signal relative to the instrumental noise. Thus, we argue that the bounds we present in this analysis are \emph{conservative} bounds derived from linear cosmology. This has particular impact on the lensing of the CMB, and may affect parameter constraints. We leave the treatment of nonlinear structure formation within dark matter-baryon interaction cosmologies for a later work, which we expect would provide even tighter constraints from the larger matter power signal. \cite{hill2021} showed that the ACT DR4 data were sufficiently precise to have a difference in cosmological parameters of order $0.2\sigma$ due to Boltzmann code precision settings; we also leave this implementation to future work. \subsection{Data and Sampling} \label{subsec:data} In our main analysis, we use a combination of \planck{} 2018 and the Atacama Cosmology Telescope Data Release 4 (DR4). We use the foreground- and nuisance-marginalized versions of these likelihoods, representing the best estimates of the CMB bandpowers provided by these experiments. These marginalized likelihoods are Gaussian and each have one remaining nuisance parameter. We thus additionally sample over \begin{equation} \left\{ A_{\mathrm{Planck}}, \, y_p \right\}, \end{equation} representing the \planck{} absolute calibration and ACT polarization efficiency respectively. We perform this likelihood analysis using the sampling framework $\texttt{cobaya}$ \citep{cobaya}. We include $\texttt{planck\_2018\_highl\_plik.TTTEEE\_lite}$ and $\texttt{planck\_2018\_lowl.TT}$ for \planck{}, and $\texttt{pyactlike.ACTPol\_lite\_DR4}$ for ACT. We do not use the CMB lensing data from ACT. Following \cite{aiola2020}, we exclude $\ell < 1800$ data in $TT$ when combining the ACT data with \planck{}, in order to avoid double-counting the same sky measured at the cosmic variance limit. We also include a Gaussian prior on $\tau_{\mathrm{reio}} = 0.065 \pm 0.015$ to replace the large-scale polarization likelihood. We also experimented with the addition of some other common cosmological datasets (\planck{} low-$\ell$ polarization, \planck{} lensing, and BAO from SDSS DR12 \citep{sdss12}), but found these do not improve constraints on DM-baryon scattering. The lack of improvement when including the \planck{} lensing is consistent with previous analyses with the \planck{} data \citep[e.g.,][]{BoddyGluscevic}, but we expect this to change with next-generation surveys \citep{LiGluscevic2018}. \section{Results} \label{sec:results} \subsection{Velocity-Independent Constraints} For the fiducial model of velocity-independent ($n=0$) scattering with a 1\,GeV DM particle, we find the inclusion of the ACT DR4 data reduces the upper limit on the cross section for proton scattering by $\sim 40$\%, with 95-percentile upper limits of \begin{equation} \sigma_0^{\text{GeV, }n=0} < \left\{ \begin{array}{ll} 4.7 \times 10^{-25} \,\mathrm{cm}^2 \quad \text{\textit{(Planck)}} \\ 2.9 \times 10^{-25} \,\mathrm{cm}^2 \quad \text{(\textit{Planck} + ACT DR4)}. \end{array} \right. \end{equation} We illustrate these results in Figure \ref{fig:marginalized1GeV}, showing the marginalized 1D posterior of the DM scattering cross section. We find almost no correlation of this parameter with the $\Lambda$CDM parameters. For this model the data show no evidence for a nonzero cross section. This model demonstrates the constraining power of the ACT DR4 data, which provide improved measurements of the CMB damping tail and additional acoustic peaks in TE and EE, cutting the space of allowed cross sections compared to \planck{} alone. For other masses, and for the case of electron scattering, we provide upper limits derived from the \planck{} and ACT data in Appendix Table \ref{table:constraints}. Since the cross section parameter has a positive prior we check if the improved upper limit is compatible with expectation. In the Appendix we perform a Fisher matrix analysis, finding an expected $\sim$30\% improvement in errors from adding the ACT data to \planck{} for the $n=0$ model, consistent with our findings with the real data. Both the \planck{} and ACT likelihoods used in this analysis include spectra and covariances that have been marginalized over models of foreground parameters. The effect of DM-proton and DM-electron scattering is imprinted in the CMB and is frequency-independent, but could still be biased by astrophysical foregrounds. ACT DR4 contains both additional small-scale information in temperature and polarization, but we expect the foreground contamination to primarily affect the temperature spectrum. The foregrounds in temperature primarily affect small-scale measurements, so we expect our analysis with the ACT DR4 temperature power spectra to be more susceptible to foreground contamination than previous work with lower resolution Planck data. However, we find that the $\Delta \chi^2$ arising from TT spectra at $\ell > 2000$ between the best-fit $\Lambda$CDM theory and DM-baryon scattering extension is less than half of the total $\Delta \chi^2$ arising from TT. % We also confirm that the scattering cross section is not correlated with the \planck{} and ACT calibration nuisance parameters. In Figure \ref{fig:freemass} we show constraints from the eight-parameter model for proton scattering, where we simultaneously sample both $\log(m_{\chi})$ and $\log(\sigma_0)$, with $n=0$ shown in the upper right panel. We see a strong correlation between mass and the cross-section upper limit. \subsection{Velocity-Dependent Constraints} \label{subsec:veldepconstr} We report results for the 7-parameter model ($\Lambda$CDM+$\sigma_0$) in Appendix Table \ref{table:constraints} for the suite of masses and model indices.\footnote{Our choice to report cross sections for a linear rather than logarithmic prior can change constraints by up to a factor of two, which affects comparisons with previous work, e.g., \cite{GluscevicBoddy}.} When adding the ACT data we find posterior densities for proton scattering which have nonzero best-fitting scattering cross sections for $n=2$ and $n=4$. % This is shown in Figure \ref{fig:linearMeV} for the 1 MeV case, for the cross section and the primordial tilt, $n_s$, which is most degenerate with the cross section. We find that the combined \planck{} and ACT posterior does not directly shrink inwards from the \planck{} constraints, but rather shifts upwards in cross section altogether by $\sim 1\sigma$. The difference in goodness-of-fit for the ACT DR4 likelihood between a $\Lambda$CDM model and the best-fitting $n=2$, $m_{\chi}=1$ MeV model is driven primarily by the ACT temperature data, with \begin{equation} (\chi^2_{\Lambda\text{CDM}} - \chi^2_{n=2, \text{1 MeV}})_{\mathrm{ACT}} = \left\{ \begin{array}{llr} 5.3 & & \text{\emph{(TT, TE, EE)}} \\ 4.0 & & \text{\emph{(TT)}} \\ 0.9 & & \text{\emph{(TE)}} \\ -0.3 & & \text{\emph{(EE)}}. \end{array} \right. \end{equation} For two extra parameters applied to more than 100 degrees of freedom, this is not a significant improvement. Overall from a model selection viewpoint we find no evidence for DM-baryon scattering in any of the six models considered in this work ($n \in \{-4, -2, 0, 2, 4, 6\}$). Approximately one in three datasets would randomly exhibit a similar $\sim 2 \sigma$ statistical fluctuation, when testing six models like $n \in \{-4, -2, 0, 2, 4, 6\}$, and for two additional parameters (mass and cross section). In testing the impact that ACT DR4 has on parameter constraints, we show in the Appendix that ACT improves on \textit{Planck} uncertainties by less than 10\% for $n<0$ models. For $n \ge 0$ models the improvement on the uncertainty is 30--50\%, with these models benefiting more from the smaller scale data. This also confirms that ACT would have been expected to improve constraints on $\sigma_0$ for these models, if not for a presumably statistical fluctuation towards nonzero best-fit values. We show the derived posterior densities for the combined \planck{} and ACT data in Figure \ref{fig:freemass}, for the eight-parameter model for proton scattering. These samples are presented with a flat prior in the logarithm of the cross section, instead of the flat prior in the cross section used in \ref{fig:linearMeV}. The approach of sampling in the logarithms of the DM particle mass and cross section is useful for exploring the space of allowed models. The CMB constraints exhibit a significant degeneracy between DM particle mass and cross section, and the allowed cross sections vary by several orders of magnitude as the mass changes from 1 MeV to 1 TeV. In the limit where the DM mass is much greater or much less than the mass of the scattering target, this degeneracy becomes a true power law. Although there are some masses for which the best-fitting cross-section is nonzero, we have no reason to believe that the DM particle mass takes on any particular value from 1 MeV to 1 TeV. CMB-only constraints struggle to break the degeneracy between DM particle mass and the scattering cross section. Bayesian analysis would marginalize over mass. Any such marginalization would erase a preference for nonzero cross section, as one can infer from Figure~\ref{fig:freemass}. We also present constraints on DM-electron scattering in Table \ref{table:constraints}. These constraints exhibit behavior similar to the DM-proton scattering, with some modest 1$-$2$\sigma$ best-fit deviations from zero cross-section, but still consistent with $\Lambda$CDM. We test the effect of including additional \planck{} large-scale polarization in place of a prior on $\tau_{\text{reio}}$, as well as including \planck{} lensing and BAO constraints from SDSS DR12 \citep{sdss12}. The constraints on the DM-baryon interaction cross section are virtually unchanged with the inclusion of these data. There are the expected shifts for the optical depth to reionization, the amplitude $A_s$, and the DM density from these additional data sources, but these are not correlated with the DM-baryon scattering parameters. \section{Conclusions and Discussion} \label{sec:conclusions} We have used new measurements of the CMB, particularly at small scales and in polarization, to look for evidence of elastic scattering between DM and baryons (protons and electrons). Compared to previous work, the inclusion of the ACT DR4 data provides more precise measurements of the high-$\ell$ acoustic peaks and damping tail in TE and EE. Relative to a $\Lambda$CDM model, the scattering models affect mostly the small scales, so the inclusion of the ACT DR4 dataset is especially suited to investigating this physics. Indeed, although the addition of the ACT DR4 likelihood does not significantly improve constraints on the standard $\Lambda$CDM parameters \citep{aiola2020}, we find that for the fiducial model of velocity-independent dark matter scattering with a 1\,GeV dark matter particle, the combination of ACT DR4 and \planck{} improves the upper limit on the scattering cross section by $\sim 40\%$. The combined \planck{} and ACT likelihood yields posteriors consistent with statistical fluctuations about the non-scattering $\Lambda$CDM model, for all models considered in this work. However, many of the $n \neq 0$ models do exhibit a mild $< 2 \sigma$ deviation from zero cross section for many masses, as shown in Figures~\ref{fig:freemass} and \ref{fig:linearMeV}. This preference arises primarily from the ACT temperature data, but is not statistically significant. Since DM-baryon scattering reduces power at small scales, we expect that new high-resolution ground-based data, particularly measurements of the TE and EE correlations at high-$\ell$, will provide noteworthy improved constraints in the near future. Existing instruments like ACT and SPT, as well as future instruments like the Simons Observatory and CMB-S4, will provide as much as an order of magnitude in improvement for the scattering cross section. % \begin{acknowledgments} VG and RA acknowledge the support from NASA through the Astrophysics Theory Program, Award Number 21-ATP21-0135 and from the National Science Foundation under Grant No. PHY-2013951. KB acknowledges support from the NSF under Grant No.\ PHY-2112884. EC acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 849169). JD is supported by NSF grants AST-1814971 and AST-2108126. ADH acknowledges support from the Sutton Family Chair in Science, Christianity and Cultures and from the Faculty of Arts and Science, University of Toronto. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Industry Canada and by the Province of Ontario through the Ministry of Colleges and Universities. Computations were performed on Della as part of Princeton Research Computing resources at Princeton University. Support for ACT was through the U.S.~National Science Foundation through awards AST-0408698, AST-0965625, and AST-1440226 for the ACT project, as well as awards PHY-0355328, PHY-0855887 and PHY-1214379. Funding was also provided by Princeton University, the University of Pennsylvania, and a Canada Foundation for Innovation (CFI) award to UBC. ACT operates in the Parque Astron\'omico Atacama in northern Chile under the auspices of the Agencia Nacional de Investigaci\'on y Desarrollo (ANID). The development of multichroic detectors and lenses was supported by NASA grants NNX13AE56G and NNX14AB58G. Detector research at NIST was supported by the NIST Innovations in Measurement Science program. \end{acknowledgments} \bibliographystyle{yahapj} \bibliography{biblio.bib} \appendix \FloatBarrier \label{appendix} \begin{table}[ht] \centering \centering \begin{tabular}{ccccc} \hline\hline $n$ & $m_\chi$ & \planck{} & ACT + \planck{} & A+P electron \\ & [GeV] & [95\%, cm$^2$] & [95\%, cm$^2$] & [95\%, cm$^2$] \\ \hline $-4$ & 0.001 & $1.3 \times 10^{-41}$ & $1.8 \times 10^{-41}$ & $1.4 \times 10^{-37}$ \\ $-4$ & 0.01 & $1.4 \times 10^{-41}$ & $1.9 \times 10^{-41}$ & $1.4 \times 10^{-36}$ \\ $-4$ & 0.1 & $1.5 \times 10^{-41}$ & $2.0 \times 10^{-41}$ & $9.1 \times 10^{-36}$ \\ $-4$ & 1.0 & $2.7 \times 10^{-41}$ & $3.7 \times 10^{-41}$ & $9.1 \times 10^{-35}$ \\ $-4$ & 10.0 & $1.6 \times 10^{-40}$ & $2.1 \times 10^{-40}$ & $8.0 \times 10^{-34}$ \\ $-4$ & 100.0 & $1.4 \times 10^{-39}$ & $1.9 \times 10^{-39}$ & $8.4 \times 10^{-33}$ \\ $-4$ & 1000.0 & $1.4 \times 10^{-38}$ & $1.9 \times 10^{-38}$ & $5.3 \times 10^{-32}$ \\ \hline $-2$ & 0.001 & $1.8 \times 10^{-33}$ & $2.3 \times 10^{-33}$ & $7.0 \times 10^{-32}$ \\ $-2$ & 0.01 & $1.8 \times 10^{-33}$ & $2.2 \times 10^{-33}$ & $5.2 \times 10^{-31}$ \\ $-2$ & 0.1 & $2.0 \times 10^{-33}$ & $2.4 \times 10^{-33}$ & $4.9 \times 10^{-30}$ \\ $-2$ & 1.0 & $3.6 \times 10^{-33}$ & $4.6 \times 10^{-33}$ & $5.7 \times 10^{-29}$ \\ $-2$ & 10.0 & $2.1 \times 10^{-32}$ & $2.5 \times 10^{-32}$ & $5.6 \times 10^{-28}$ \\ $-2$ & 100.0 & $1.9 \times 10^{-31}$ & $2.4 \times 10^{-31}$ & $4.9 \times 10^{-27}$ \\ $-2$ & 1000.0 & $1.9 \times 10^{-30}$ & $2.4 \times 10^{-30}$ & $5.8 \times 10^{-26}$ \\ \hline 0 & 0.001 & $5.7 \times 10^{-26}$ & $3.5 \times 10^{-26}$ & $6.5 \times 10^{-27}$ \\ 0 & 0.01 & $1.1 \times 10^{-25}$ & $6.4 \times 10^{-26}$ & $4.0 \times 10^{-26}$ \\ 0 & 0.1 & $1.9 \times 10^{-25}$ & $1.1 \times 10^{-25}$ & $6.4 \times 10^{-25}$ \\ 0 & 1.0 & $4.7 \times 10^{-25}$ & $2.9 \times 10^{-25}$ & $4.2 \times 10^{-24}$ \\ 0 & 10.0 & $2.9 \times 10^{-24}$ & $1.9 \times 10^{-24}$ & $5.7 \times 10^{-23}$ \\ 0 & 100.0 & $2.9 \times 10^{-23}$ & $1.8 \times 10^{-23}$ & $1.1 \times 10^{-21}$ \\ 0 & 1000.0 & $2.9 \times 10^{-22}$ & $1.8 \times 10^{-22}$ & $3.7 \times 10^{-21}$ \\ \hline \end{tabular} \quad \begin{tabular}{ccccc} \hline\hline $n$ & $m_\chi$ & \planck{} & ACT + \planck{} & A+P electron \\ & [GeV] & [95\%, cm$^2$] & [95\%, cm$^2$] & [95\%, cm$^2$] \\ \hline 2 & 0.001 & $3.9 \times 10^{-21}$ & $5.3 \times 10^{-21}$ & $1.5 \times 10^{-22}$ \\ 2 & 0.01 & $5.2 \times 10^{-20}$ & $7.0 \times 10^{-20}$ & $9.3 \times 10^{-22}$ \\ 2 & 0.1 & $6.5 \times 10^{-19}$ & $8.5 \times 10^{-19}$ & $1.1 \times 10^{-20}$ \\ 2 & 1.0 & $8.7 \times 10^{-18}$ & $7.3 \times 10^{-18}$ & $1.1 \times 10^{-19}$ \\ 2 & 10.0 & $9.2 \times 10^{-17}$ & $6.3 \times 10^{-17}$ & $1.4 \times 10^{-18}$ \\ 2 & 100.0 & $9.5 \times 10^{-16}$ & $6.0 \times 10^{-16}$ & $1.2 \times 10^{-17}$ \\ 2 & 1000.0 & $9.7 \times 10^{-15}$ & $5.8 \times 10^{-15}$ & $9.2 \times 10^{-17}$ \\ \hline 4 & 0.001 & $1.1 \times 10^{-16}$ & $1.9 \times 10^{-16}$ & $1.4 \times 10^{-18}$ \\ 4 & 0.01 & $1.4 \times 10^{-14}$ & $2.2 \times 10^{-14}$ & $1.8 \times 10^{-17}$ \\ 4 & 0.1 & $1.4 \times 10^{-12}$ & $2.3 \times 10^{-12}$ & $1.9 \times 10^{-16}$ \\ 4 & 1.0 & $7.3 \times 10^{-11}$ & $1.1 \times 10^{-10}$ & $1.7 \times 10^{-15}$ \\ 4 & 10.0 & $1.3 \times 10^{-9}$ & $1.5 \times 10^{-9}$ & $1.7 \times 10^{-14}$ \\ 4 & 100.0 & $1.3 \times 10^{-8}$ & $1.6 \times 10^{-8}$ & $1.5 \times 10^{-13}$ \\ 4 & 1000.0 & $8.9 \times 10^{-10}$ & $3.5 \times 10^{-10}$ & $1.3 \times 10^{-12}$ \\ \hline 6 & 0.001 & $2.3 \times 10^{-12}$ & $3.8 \times 10^{-12}$ & $7.4 \times 10^{-15}$ \\ 6 & 0.01 & $2.5 \times 10^{-9}$ & $4.3 \times 10^{-9}$ & $7.3 \times 10^{-14}$ \\ 6 & 0.1 & $2.2 \times 10^{-6}$ & $4.0 \times 10^{-6}$ & $1.2 \times 10^{-12}$ \\ 6 & 1.0 & $5.7 \times 10^{-4}$ & $9.7 \times 10^{-4}$ & $1.8 \times 10^{-11}$ \\ 6 & 10.0 & $1.3 \times 10^{-2}$ & $1.6 \times 10^{-2}$ & $1.2 \times 10^{-10}$ \\ 6 & 100.0 & $8.2 \times 10^{-3}$ & $8.4 \times 10^{-3}$ & $1.1 \times 10^{-9}$ \\ 6 & 1000.0 & $1.6 \times 10^{-3}$ & $3.8 \times 10^{-4}$ & $1.1 \times 10^{-8}$ \\ \hline \end{tabular} \caption{Upper 95\% limits on DM cross sections, sampled with linear prior, for models with velocity dependence $n$ and DM mass $m_{\chi}$, for proton scattering (column 3 \& 4) and electron scattering (column 5). Many of the models with $n \neq 0$ do not show an improvement in the upper bound over Planck with the addition of ACT DR4, due to the nonzero best-fitting cross sections discussed in Section \ref{subsec:veldepconstr}.} \label{table:constraints} \end{table} In this appendix we estimate how much statistical power ACT contributes to the ACT+\emph{Planck} combination, relative to \emph{Planck} alone. The constraining power is distinct from the upper bound, since the upper bound is sensitive to statistical fluctuations which change the peaks of the respective likelihoods. We perform a simple Fisher matrix analysis with the likelihoods directly.\footnote{ \url{https://xzackli.github.io/realfisher/fisheranalysis.jl.html}.} The Fisher matrix for parameters $\{\theta_i\}$ is the expectation value for the Hessian of the log-likelihood $\mathcal{L}$, \begin{equation} F_{ij} = - \left\langle\frac{ \partial^2 \mathcal{L}}{\partial \theta_i \, \partial \theta_j} \right\rangle_{\theta}. \end{equation} We approximate this expectation value with the value at the peak of the likelihood. To compute the Hessian of the log-likelihoods, we use forward-mode automatic differentiation (AD) in the \texttt{ForwardDiff.jl} package \citep{forwarddiff} within the Julia language \citep{bezanson2017julia}. Although it is possible to derive an analytic Hessian, the ACT likelihood has complexities (binning, nontrivial bandpower window functions, deep and wide patches) which would make this tedious. We re-implement the ACT and \emph{Planck} likelihood in Julia, to enable the use of this AD package. Our new implementation reproduces the ACT DR4 likelihood \citep{aiola2020} to numerical precision. We use the compressed high-$\ell$ likelihood provided in \cite{princedunkley} for \emph{Planck} 2018, and we reproduce this likelihood to numerical precision as well. We obtain gradients of the model spectra using finite differences. After computing the negative Hessian of the log-likelihood numerically with AD, we invert it to obtain the covariance matrix via the Cramer-Rao bound, $C_{ij} = F_{ij}^{-1}$. To represent the combined likelihoods, we add the Fisher matrices corresponding to ACT alone and \emph{Planck} alone. We also impose a prior on the error of the optical depth to reionization, $\sigma(\tau_{\mathrm{reio}}) = 0.015$, to replace the large-scale polarization data in \emph{Planck} by adding $(0.015)^2$ to the diagonal element of the Fisher matrix corresponding to $\tau_{\mathrm{reio}}$. This Fisher analysis configuration reproduces the $\Lambda$CDM constraints in \cite{aiola2020} fairly well, with only 10$-$20\% differences for each of the parameter errors in each configuration (\emph{Planck}, ACT, ACT and \emph{Planck} combined). For parameter $\theta_i$, we then compute the marginalized error $\sigma(\theta_i) = \sqrt{C_{ii}}$. We also define an overall figure of merit (FoM) that describes the full extension model, $\mathrm{FoM} = 1 / \sqrt{\det F}$. We present ratios of these quantities in Table \ref{table:fisher} for the \emph{Planck} likelihood alone (P), with respect to ACT and \emph{Planck} combined (AP). We use the ACT and \emph{Planck} combined best-fit as the fiducial model. We perform this analysis for the 1\,GeV case, as the degeneracy between mass and cross section results in similar results for other masses. Although the overall figure of merit improves by roughly a factor of two across all models, the ACT data improves the constraint on $\sigma_{0}$ by 30$-$40\% for $n\geq0$ models. For $n<0$ models, we see that the ACT data contributes little constraining power. \begin{table}[t] \centering \caption{Statistical constraining power estimates from \planck{} alone (P), and ACT and \planck{} combined (AP), for $m_{\chi} = 1 \text{ GeV}$.} \label{table:fisher} \begin{tabular}{ccc} \hline\hline $n$ & $\sigma^{\rm P}(\sigma_{0}) / \sigma^{\rm AP}(\sigma_{0})$ & ${\rm FoM}^{\rm P} / {\rm FoM}^{\rm AP}$ \\ \hline -4 & 1.07 & 1.64 \\ -2 & 1.06 & 1.62 \\ 0 & 1.34 & 2.06 \\ 2 & 1.49 & 2.29 \\ 4 & 1.32 & 2.04 \\ 6 & 1.40 & 2.15 \\ \hline \end{tabular} \end{table}

Title: How do the dynamics of the Milky Way - Large Magellanic Cloud system affect gamma-ray constraints on particle dark matter?

Abstract: Previous studies on astrophysical dark matter (DM) constraints have all assumed that the Milky Way's (MW) DM halo can be modelled in isolation. However, recent work suggests that the MW's largest dwarf satellite, the Large Magellanic Cloud (LMC), has a mass of 10-20$\%$ that of the MW and is currently merging with our Galaxy. As a result, the DM haloes of the MW and LMC are expected to be strongly deformed. We here address and quantify the impact of the dynamical response caused by the passage of the LMC through the MW on the prospects for indirect DM searches. Utilising a set of state-of-the-art numerical simulations of the evolution of the MW-LMC system, we derive the DM distribution in both galaxies at the present time based on the Basis Function Expansion formalism. Consequently, we build $J$-factor all-sky maps of the MW-LMC system in order to study the impact of the LMC passage on gamma-ray indirect searches for thermally produced DM annihilating in the outer MW halo as well as within the LMC halo standalone. We conduct a detailed analysis of 12 years of Fermi-LAT data that incorporates various large-scale gamma-ray emission components and we quantify the systematic uncertainty associated with the imperfect knowledge of the astrophysical gamma-ray sources. We find that the dynamical response caused by the LMC passage can alter the constraints on the velocity-averaged annihilation cross section for weak scale particle DM at a level comparable to the existing observational uncertainty of the MW halo's density profile and total mass.

https://export.arxiv.org/pdf/2208.03312

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} Galaxy: evolution -- Galaxy: halo -- Galaxy: kinematics and dynamics -- Galaxy: structure -- Magellanic clouds -- gamma-rays: galaxies\end{keywords} \section{Introduction} \label{sec:intro} The Large Magellanic Cloud (LMC) is believed to be on its first approach to the Milky Way (MW) since the early Universe \citep{Besla+2007}. This is supported by many lines of evidence which show that it still hosts a massive dark matter (DM) halo, in line with expectations from abundance matching, $\sim 2\times10^{11}~{\rm M}_\odot$ \citep[e.g.][]{Behroozi+2013,Moster+2013}. First, the nearby presence of the Small Magellanic Cloud (SMC) requires an LMC mass of $\sim10^{11}~{\rm M}_\odot$ in order to remain bound to the LMC \citep{Kallivayalil+2013}. Similar analyses show that the recently discovered Magellanic satellites also require similarly high LMC masses, $\sim (1-2)\times 10^{11}~{\rm M}_\odot$ \citep{Erkal+2020,Patel+2020} in order to have originally been bound to the LMC. The LMC is massive enough that it induces a strong reflex motion in the MW \citep{Gomez+2015}, as evidenced by its effect on the timing argument with Andromeda and the nearby Hubble flow, which require a mass of $\sim 2.5\times10^{11}~{\rm M}_\odot$ \citep{Penarrubia+2016}. This reflex motion is also seen in the MW's stellar halo, both in its kinematics \citep{Erkal+2021,Petersen+2021} and density \citep{Belokurov+2019,Conroy+2021}, all requiring a mass $>\sim10^{11}~{\rm M}_\odot$. Finally, its effect has been seen and characterized in many stellar streams around the MW, giving masses of $(1.3-1.8)\times10^{11}~{\rm M}_\odot$ \citep{2019MNRAS.487.2685E,Koposov+2019,Shipp+2019,Shipp+2021,Vasiliev+2021}. Such a massive LMC halo is $\sim(10-20)\%$ of the MW's mass \citep[e.g.][]{Wang+2020}, suggesting that this merger should have a substantial effect on the dark matter haloes of both galaxies. This has been studied with simulations \citep[e.g.][]{Laporte+2018,Garavito-Camargo+2019,Garavito-Camargo+2021,Petersen+2020} which have shown that substantial DM deformations are expected. Several works have also explored the observational consequences of these deformations. \cite{Vasiliev+2021} and \cite{2022arXiv220501688L} showed that these deformations can affect the Sagittarius and Orphan-Chenab stream, respectively. These effects are larger than current observational uncertainties, suggesting that in the future it will be possible to measure these deformations. \cite{Conroy+2021} claimed the detection of the MW-LMC haloes' deformations through a positive correlation between giant halo stars in \textit{Gaia} and \textit{WISE} data and the simulations from~\cite{Garavito-Camargo+2019}. Such deformations can also alter the velocity distribution of DM. \citet{2019JCAP...11..013B} and \citet{Donaldson+2022} studied the impact of the MW-LMC dynamics on the local DM velocity distribution, finding an enhanced reach of direct DM detection experiments. In this work, for the first time, we will explore another avenue for characterising these deforming haloes more directly: gamma-ray searches for DM signals. In the standard paradigm, 80\% of the matter content of the Universe, i.e.~the DM, can be explained by new particles beyond the ones in the Standard Model of particle physics. In particular, weak scale DM particle candidates may be thermally produced in the early universe through interactions with Standard Model particles. These so-called Weakly Interacting Massive Particles (WIMPs) are still present today, and can self-annihilate or decay into final stable products contributing to the fluxes of cosmic gamma rays and charged cosmic rays~(see, e.g., the review on this topic in \cite{Cirelli:2012tf}). The spatial distribution of gamma rays from DM annihilation or decay in the sky depends on the distribution of DM in the target of interest, through the integral along the line of sight (l.o.s) of the DM density (squared in the case of annihilation). Therefore, any change in the expected DM spatial density, $\rho$, affects the large-scale {\it morphology} of the DM signal. In this respect, a perfectly legitimate question to ask is: How does the MW-LMC dynamics affect gamma-ray searches for DM? We quantify the answer to this question by analysing all-sky data of the Fermi Large Area Telescope (\Fermi-LAT) and search for DM at high Galactic latitudes. This approach follows traditionally performed searches for DM with \Fermi-LAT data~\citep{2012ApJ...761...91A, Zechlin:2017uzo, Chang:2018bpt}, and builds on the modelling and optimisation of the astrophysical background and foreground components of the gamma-ray high-energy sky. In addition, we also study the impact of our DM spatial model on the constraints derived from gamma-ray observations of a region centred around the LMC, to be compared with previous \Fermi-LAT analyses~\citep{Buckley:2015doa}, and the recent spatial model from~\cite{Regis:2021glv}. The main novelties of the present work are: (i) the modelling of the DM gamma-ray all-sky signal based on state-of-the-art simulations of the MW-LMC interaction~\citep{Donaldson+2022,2022arXiv220501688L}; and (ii) the quantification of the uncertainty on the DM limits issued by one of the most accurate models of the MW potential and its associated uncertainties~\citep{2017MNRAS.465...76M}, based on which we can properly assess the impact of the MW-LMC dynamics in gamma-ray DM searches. As a result, our limits on DM at high latitude represent the most up-to-date and robust constraints from \Fermi-LAT gamma-ray observations. The paper is organised as follows: In Sec.~\ref{sec:simulation}, we describe the modelling of the DM signal, namely its spatial distribution, based on the outcome of simulations of the MW-LMC interaction, and spectrum. We dedicate Sec.~\ref{sec:fermianalysis} to explaining the details of the \Fermi-LAT analysis, statistical framework and fitting procedure for the high-latitude sky and LMC regions. This section is complemented by Appendix~\ref{app:astro_comp} about the modelling of the astrophysical fore- and back-ground gamma-ray components. In Sec.~\ref{sec:upperlimits}, we present the new constraints on the particle DM parameter space from the high-latitude and LMC regions. We discuss the impact of varying the interstellar emission model, i.e.~the dominant source of background modelling systematic uncertainties in Appendix~\ref{app:iem_limits}. We draw our conclusions in Sec.~\ref{sec:conclusions}. \section{Milky Way -- Large Magellanic Cloud dark matter distribution} \label{sec:simulation} \subsection{Simulation of the Milky Way -- Large Magellanic Cloud dynamics} \label{sec:mw-lmc-sim} In order to explore the deformations of the MW and LMC, we use a suite of $N$-body simulations run with the \textsc{exp} code \citep{Petersen+2022}. Unlike other $N$-body codes which evaluate forces with a hierarchical tree \citep[e.g.][]{Appel1985,Barnes+1986} or a particle mesh \citep[e.g.][]{Klypin+1983,White+1983}, this code evolves $N$-body particles by using a basis function expansion (BFE). In particular, \textsc{exp} uses biorthogonal basis functions for the potential and density which satisfy the Poisson equation. The angular structure of each model is described by spherical harmonics, with harmonic indices $\ell$ and $|m|\le\ell$ ($\ell=0$ is the monopole, $\ell=1$ is the dipole, $\ell=2$ is the quadrupole, and so on), while the radial structure is described by $n$ basis functions per harmonic order derived from the Poisson equation \citep{Weinberg1999}. At each timestep, the coefficients of each basis function are estimated by summing each potential term in the expansion over the location of the particles \citep[see eq. 5 in][]{Petersen+2022}. With these coefficients, the forces can be readily computed by differentiating the potential. This technique has several advantages for our study (see \citealt{Petersen+2022} for more details). First, it is computationally efficient, scaling as $\mathcal{O}\propto N$ where $N$ is the number of particles used, allowing for more particles at a reduced computational cost. Second, the forces are less noisy than standard gravity solvers, allowing us to study the subtleties of the MW and LMC deformations without worrying about noise. Lastly, the BFE of the density allows us to quickly determine the density when modelling gamma-ray signals. In this work, we make use of two sets of simulations of the ongoing MW-LMC merger. First, we use the simulation of \cite{2022arXiv220501688L}. Their MW and LMC system is based on the results of \cite{2019MNRAS.487.2685E} who measured the MW and LMC potentials with the Orphan stellar stream. In particular, the MW is initialized as a Miyamoto-Nagai disc \citep{Miyamoto+1975} with a mass of $6.8\times10^{10}~{\rm M}_\odot$, a scale radius of $3$ kpc, and a scale height of $0.28$ kpc, a Hernquist bulge \citep{hernquist90} with a mass of $5\times10^{9}~{\rm M}_\odot$ and a scale radius of $0.5$ kpc, and an Navarro-Frenk-White (NFW) halo \citep{NFW+1996} with a mass of $7.92\times10^{11}~{\rm M}_\odot$, a scale radius of $12.8$ kpc, and a concentration of 15.3. The LMC is modelled as a Hernquist DM halo with a mass of $1.25 \times10^{11}~{\rm M}_\odot$ and a scale radius of $14.9$ kpc. These values all come from the best-fit model of \cite{2019MNRAS.487.2685E} assuming a spherical DM halo for the MW. As a result, we dub this simulation the `Erkal19' model. \cite{2022arXiv220501688L} show that the MW and LMC DM haloes experience strong deformations, most notably in the dipole of the MW and in the quadrupole of the LMC. Second, we use the simulation suite of \cite{Donaldson+2022} which uses the same \textsc{exp} technique and considers four different MW-LMC models. These simulations are built to roughly match the rotation curve and total mass constraints of the MW \citep{Eilers+2019,Eadie+2019} and LMC \citep{vanderMarel:2013jza,Penarrubia+2016,2019MNRAS.487.2685E}. The MW model consists of an exponential disc with a mass of $5\times10^{10}~{\rm M}_\odot$, a scale radius of 3 kpc, and a sech$^2$ scale height of 0.6 kpc, and a dark matter halo with a profile given by $\rho(r) = \rho_0(r)\tilde{r}^{-1}(1+\tilde{r})^{-\alpha}T(r)$ with scaled radius $\tilde{r}=r/R_s$ where $R_s$ is the scale radius, and the truncation function $T(r)=0.5\left(1-{\rm erf}\left[(r-r_{\rm trunc})/w_{\rm trunc}\right]\right)$ with $r_{\rm trunc}=430$ kpc and $w_{\rm trunc}=54$ kpc. One can set this profile to be either an NFW ($\alpha=2$) or Hernquist ($\alpha=3$) profile dark matter halo. We require a similar mass enclosed at 50 kpc by tuning the respective scale radii of the MW models: $R_{s,{\rm NFW MW}}=15$ kpc and $R_{s,{\rm Hernquist MW}}=44$ kpc. The total mass of the NFW halo is $1\times10^{12}~{\rm M}_\odot$, while the total mass of the Hernquist halo is $0.94\times10^{12}~{\rm M}_\odot$. We build two LMCs, modelled as a dark matter halo only, again following the same truncated halo profile as above, with $R_{s,{\rm NFW LMC}}=33.8$ kpc and $R_{s,{\rm Hernquist LMC}}=63$ kpc. The total masses are set to be $2.5\times10^{11}~{\rm M}_\odot$ for the NFW LMC and $2.35\times10^{11}~{\rm M}_\odot$ for the Hernquist LMC. The simulation suite is constructed as a grid of four models by mixing the MW and LMC halo profiles. That is, one simulation is an NFW MW and NFW LMC, one is an NFW MW and Hernquist LMC, one is a Hernquist MW and NFW LMC, and one is a Hernquist MW and Hernquist LMC. We label this second set of simulations by referring to the halo profiles of the MW and the LMC at the stage of simulation initialisation, i.e.~either an NFW or a Hernquist (HERN) profile. When analyzing the gamma-ray signals expected from these models, we take an approach inspired by \cite{2022arXiv220501688L} and consider the full multipole expansions as well as the monopoles for comparison. The monopole terms describe the spherical representation of the MW and LMC haloes, but changes over time as the coefficients of the monopole radial basis functions vary. Due to the relatively short travel times of gamma rays through the Milky Way ($\sim1$ Myr to travel 300 kpc) compared to the timescales over which the basis function changes substantially \citep[$\sim 50-100$ Myr, see fig. 1 of][]{2022arXiv220501688L}, we only consider the coefficients at the present-day. By comparing these two, we can see how much the deformations affect the predicted gamma-ray signal. We consider this difference as the most robust result of this work. Indeed, as a word of caution, we notice that, while the initial models of \cite{2022arXiv220501688L} and \cite{Donaldson+2022} were consistent with the MW gravitational potential constraints, the full consistency with the MW potential has not been {\it a posteriori} checked for the finally deformed models. That is, the rotation curve and mass enclosed constraints originally imposed may no longer be met. Therefore, in order to quantify the impact of the absolute value of the new constraints for gamma-ray DM searches, in addition to these two sets of models, we also consider a static MW model. For this potential, we use the results of \cite{2017MNRAS.465...76M} who modelled the MW with a bulge, four discs (thin, thick, H\textsc{i}, and H$_2$), and an NFW DM halo. \cite{2017MNRAS.465...76M} fit this model to a range of data and constraints: maser data, the solar velocity, terminal velocity curves, the vertical force near the plane of the disc, and a mass constraint based on satellite kinematics (see Sec.~3 of \citealt{2017MNRAS.465...76M} for more details). While these constraints are primarily within the plane of the MW disc, it represents one of the most accurate models of the MW potential. To explore how the uncertainties in the MW potential affect the predicted gamma-ray signal, we sample over the posterior chains from \cite{2017MNRAS.465...76M}. This allows us to (a) quantify the systematic uncertainties on the high-latitude DM limits from the MW gravitational potential, and (b) properly assess the impact of the variations of the limits induced by the MW-LMC dynamics. \subsection{Dark matter-induced gamma-ray signal} In this work, we consider the gamma-ray emission resulting from pair-annihilating thermally produced DM in the MW and LMC halo. We restrict ourselves to the prompt gamma-ray component of these interactions neglecting potential secondary or tertiary contributions from particle cascades triggered by the primary annihilation products. The expected (prompt) differential gamma-ray flux $\textrm{d}\Phi_{\gamma}/\textrm{d}E_{\gamma}/\mathrm{d}\Omega$ at the top of the Earth's atmosphere reads (see, e.g., \citealt{Cirelli:2010xx, Bringmann:2012ez}) \begin{equation} \label{DMflux} \frac{d\Phi_{\gamma}}{d\Omega\, dE_\gamma} (E_\gamma,\psi) = \left(\vphantom{\frac{dN_\gamma^{f}}{dE_\gamma}\sum_f}\frac{1}{4\pi} \int_\mathrm{l.o.s} d\ell(\psi) \rho_\chi^2(\bm{r})\right) \left({\frac{\langle\sigma v\rangle_\mathrm{ann}}{2S_\chi m_{\chi}^2} \sum_f B_f\frac{dN_\gamma^{f}}{dE_\gamma}}\right) \,, \end{equation} where $\psi$ refers to the direction of the line-of-sight in Galactic coordinates and $E_{\gamma}$ quantifies the gamma rays' energy. The DM-induced gamma-ray flux factors into two contributions under the assumption of velocity-independent annihilation cross section -- the so-called $s$-wave annihilation process. The term in the first parenthesis is commonly referred to as \Jf-factor while the term in the second parenthesis includes and describes the particle physics model chosen for the DM candidate under study. In what follows, we provide further details about the ingredients required to compute both contributions to the DM gamma-ray signal. \subsubsection{\Jf-factor all-sky maps for the Milky Way} \label{sec:jfactor_maps} In order to produce \Jf-factor maps of the MW that incorporate deformations induced by the dynamics of the MW-LMC system, we take the density from the present-day snapshots of the simulations in \cite{2022arXiv220501688L} and \cite{Donaldson+2022}, and measure the square of the DM density $\rho_{\chi}$ along lines of sight on a HEALPix grid \citep{2005ApJ...622..759G} with $N_{\rm side}=64$, resulting in 49,152 lines of sight. Along these lines of sight, the density is evaluated in the centre of 1 kpc-sized bins out to 100 kpc. This choice of resolution is not a limitation of the BFE, which can faithfully reproduce structure on $\sim100$ pc scales, but rather motivated by the chosen resolution for the \Fermi-LAT gamma-ray data set (see Sec.~\ref{sec:lat_data_selection}). For these mock observations, the Sun is placed at a distance of $8.249$ kpc from the Galactic centre \citep{GravityCollaboration2020} in the plane of the MW disc. As discussed in Section~\ref{sec:mw-lmc-sim}, we create \Jf-factor maps for the full BFE as well as solely the monopole term for the MW standalone or the combined MW-LMC system. The outlined extraction scheme allows us to directly perform the l.o.s.~integral of $\rho_{\chi}^2$ where the l.o.s.~direction is given by the central coordinates of a particular HEALPix pixel. To this end, we linearly interpolate the density values at the extracted positions and numerically integrate the result from 0 to 100 kpc. In contrast, we derive two-dimensional all-sky \Jf-factor maps of the static MW model of \cite{2017MNRAS.465...76M} with the publicly available software \textsc{CLUMPY} (version 3) \cite{charbonnier2012clumpy, bonnivard2016clumpy, 2019CoPhC.235..336H}. We draw 200 realisations from the posterior distributions of the parameters characterising the NFW profile \citep{NFW+1996, Navarro:1996gj} adopted for the MW: \emph{(i)} distance of the Sun to the Galactic centre, $R_{\odot}$, \emph{(ii)} scale radius $r_s$ of the NFW profile, \emph{(iii)} virial radius $r_{200}$ of the MW and \emph{(iv)} the DM density $\rho_{\odot}$ at the position of the Sun. To illustrate the expected deformations induced by the MW-LMC dynamics, we provide a direct comparison in terms of the Erkal19 simulation suite in Fig.~\ref{fig:jfactor_comparison}. The panels display the outer MW halo in the southern hemisphere of the sky at $b \leq -20^{\circ}$, which turns out be the optimal part of the sky to perform the \Fermi-LAT gamma-ray analysis (see Sec.~\ref{sec:fitting_opti}). The left panel shows the \Jf-factor map associated to the monopole term of the BFE, which has been used to decompose the DM density distribution in the MW, whereas the central panel displays the \Jf-factor map of the MW halo derived from the full BFE of the Erkal19 simulation. Confronting the monopole contribution with the full BFE yields the most faithful assessment of the expected deformations since the initial conditions are the same for both scenarios. To better highlight the differences, we show the relative ratio of the two quantities in the right panel. \subsubsection{\Jf-factor all-sky maps for the Large Magellanic Cloud} \label{sec:jfactor_maps_LMC} To study the impact of the MW-LMC dynamics on the prospects of indirect searches for DM in the LMC, we repeat the approach outlined in Sec.~\ref{sec:jfactor_maps} for the LMC halo, which results in all-sky maps of the LMC \Jf-factors including the full BFE or only its monopole term for the five simulation suites. Although the DM density $\rho_{\chi}$ has been extracted at a time slice that corresponds to the current stage of the evolution of the MW-LMC system, the position of the LMC differs between the simulations. Moreover, it is not necessarily aligned with the astronomically determined centre of this galaxy, which itself is debated in the literature \citep{vanderMarel:2002kq, 1998ApJ...503..674K, vanderMarel:2013jza}. For definiteness, we define the LMC centre to be located at $\left(\ell, b\right) = \left(280.54^{\circ}, -32.51^{\circ}\right)$ which is the favoured rotational centre derived from stellar kinematics \citep{vanderMarel:2002kq}. All HEALPix \Jf-factor maps are rotated such that the pixel with the largest value coincides with this position thus aligning the DM halo with the stellar centre of the LMC. To enable a comparison with the expectations from a static LMC DM halo profile, we adopt the parametrization of the NFW and Hernquist profile from Tab.~1 in \cite{Regis:2021glv}. The authors of the latter study have examined the radio emission from the centre of the LMC, and obtained constraints on thermally produced WIMP DM under the assumption of different LMC halo shapes whose parameters have been determined via a fit to rotation curve data \citep{1998ApJ...503..674K}, i.e.~the inner parts of the LMC. We derive all-sky \Jf-factor maps from these two static profiles (NFW and Hernquist) with \textsc{CLUMPY}\footnote{Since the virial radius of the LMC is larger than the distance of the Solar System to this object, we use the `galactic' mode of \textsc{CLUMPY} by defining the LMC centre as the new reference Galactic centre.}. The thereby generated all-sky \Jf-factor maps are by default centred on the Galactic centre. We thus rotate the \textsc{CLUMPY} output on the stellar centre of the LMC as before. In full analogy to Fig.~\ref{fig:jfactor_comparison}, we visualise the degree of deformations of the LMC DM halo profiles via the Erkal19 simulation in Fig.~\ref{fig:jfactor_comparison_LMC}. The left panel displays the LMC \Jf-factor map taking only into account the monopole term of the BFE whereas the central panel shows a dynamically perturbed LMC halo according to the full BFE of the Erkal19 simulation. The right panel complements both profiles by highlighting the relative differences between the selected cases. Especially the latter panel illustrates the forward wake in the northern hemisphere induced by the LMC passage. In comparison to spherically symmetric, static LMC density profiles fitted to rotation curve data -- as done, e.g., in \cite{Regis:2021glv} -- we find that the central part of the LMC is predicted less dense. We stress that the reduced central \Jf-factor is related to the initial conditions of the simulation: the LMC models in both \cite{2022arXiv220501688L} and \cite{Donaldson+2022} are dark matter only and do not include a stellar component for the LMC, which will affect the central dark matter distribution. The halo parameters in \cite{Regis:2021glv} have been fixed via fits to stellar rotation curves and thus incorporate information about the small-scale behaviour around the centre of the LMC while the initial LMC haloes in both \cite{2022arXiv220501688L} and \cite{Donaldson+2022} aim to match the enclosed LMC mass at larger radii, $\gtrsim 9$ kpc. Since the total mass is rather insensitive to the innermost part of a DM halo, differences between simulated and static LMC profiles may occur. \subsubsection{Annihilation spectra and gamma-ray flux} \label{sec:DM_gamma_flux} In this work, we are referring to a generic thermally produced particle DM candidate at the weak scale, such as those belonging to the class of WIMPs. In addition, we assume that the WIMP candidate is a Majorana fermion ($S_{\chi} = 1$) annihilating into a single Standard Model particle species $f$, i.e.~the branching ratio for this process reads $B_f \equiv 1$. We consider a single annihilation channel, namely $\chi\chi \rightarrow b\bar{b}$ to present our results. The associated differential gamma-ray spectrum $\mathrm{d}N_{\gamma}^f/\mathrm{d}E_{\gamma}$, stating the expected average number of photons with energy $E_{\gamma}$ per annihilation event, is taken from \href{http://www.marcocirelli.net/PPPC4DMID.html}{PPPC}~\citep{Cirelli:2010xx}. Eventually, the DM gamma-ray flux depends on two parameters, the DM mass $m_{\chi}$ as well as the velocity-weighted, thermally averaged annihilation cross section $\langle\sigma v\rangle_\mathrm{ann}$ that determines the intensity of the signal, which is the parameter we ultimately want to constrain. We stress that the main purpose of our study is to analyse the impact of deformations of the MW halo due to the passage of the LMC on indirect gamma-ray DM searches. These dynamically induced deformations are -- at least for $s$-wave annihilation processes -- merely affecting the spatial morphology of the DM signal, and their impact is the same no matter what the chosen spectrum is. Thus, the choice of the exact particle DM model plays a minor role on our results and discussion. Consequently, we focus only on one annihilation channel in the main text and provide the results for the $\tau^+ \tau^-$-channel in Appendix~\ref{app:tau_results}. \section{\Fermi-LAT data analysis} \label{sec:fermianalysis} \subsection{Data selection} \label{sec:lat_data_selection} The analysis is based on $\sim$12 years of \Fermi-LAT data (Pass8 reconstruction standard) taken between the 4th August 2008 and the 3rd September 2020. The considered reconstructed gamma-ray energies range from 500 MeV to 500 GeV while we focus on those photons classified as belonging to the \texttt{ULTRACLEANVETO} event class and \texttt{FRONT+BACK} event type. We apply further cuts on the selected sample of photons via the requirement of \texttt{DATA\_QUAL==1 \&\& \\ LAT\_CONFIG==1} and restricting the event zenith angle to $<90^{\circ}$, which reduces the contamination of this sample by Earth limb photons. All work that requires selection, manipulation or simulation of \Fermi-LAT data has been conducted via use of the Fermi Science Tools\footnote{\url{https://github.com/fermi-lat/Fermitools-conda}} (version 2.0.8). We perform a binned log-likelihood analysis for which we bin the selected data as all-sky maps in the HEALPix format \citep{2005ApJ...622..759G} with $N_{\mathrm{side}}=64$ -- resulting in a mean pixel spacing of $\approx 0.9^{\circ}$ -- as well as 30 logarithmically spaced energy bins. Due to the scarcity of photon events at the highest energies of the LAT's sensitivity range, we rebin the LAT data above 7 GeV as well as all astrophysical background and signal all-sky maps (see Sec.~\ref{sec:bkg_selection}) into larger macro bins. Hence, the high-energy range is included within the analysis framework by creating the following two macro bins: $\left[7, 20\right]$ GeV and $\left[20, 500\right]$ GeV.\footnote{The generation of these maps respects the initial fine energy binning in order to properly account for the LAT's instrument response functions.} \subsection{Statistical framework} \label{sec:stat_framework} We employ a template-based analysis to constrain DM annihilation in the MW halo, a well-known procedure for \Fermi-LAT gamma-ray analyses. To this end, we seek to describe the processed LAT data map via a set of astrophysical background templates $\{B_j\}_{j\in J}$, which is supposed to capture as best as possible the expected different types of gamma-ray sources in and outside of the MW. To these background components we add the DM signal $S$ and evaluate whether it is preferred by the data and if such a preference is not statistically significant (see Eq.~\ref{eq:TS_discovery} in Appendix \ref{app:signal_recovery}), we set upper limits on the strength of the DM contribution.\\ The foundation of the statistical framework to conduct this kind of analysis is the binned Poisson likelihood function \begin{equation} \label{eq:poisson_L} \mathcal{L\!}\left(\left.\bm{\mu}\right|\bm{n}\right)=\prod_{i,p} \frac{\mu_{ip}^{n_{ip}}}{\left(n_{ip}\right)!}e^{-\mu_{ip}} \end{equation} where $p$ denotes the spatial pixels of the all-sky map and $i$ the energy bins of the templates. The linear combination of our set of background and signal templates $\bm{\mu}$ is called \emph{the model} whereas $\bm{n}$ represents the data map to which the model is fitted by maximising the value of the likelihood function. In detail, this linear combination of components is defined as follows: \begin{equation} \bm{\mu}=N^{\mathrm{DM}} \bm{S} + \sum_{j\in J}\sum_{i}N_{i}^{B_j}B_{j,i}\rm{,}\label{eq:model_eq} \end{equation} where, again, the index $i$ denotes the energy bins used in this analysis. Such a model definition relies on two kinds of normalisation parameters: \begin{itemize} \item Background normalisation parameters $\left\{N_{i}^{B_j}\right\}_{i, j\in J}$ for each background component and energy bin rendering it possible to incorporate spectral fluctuations present in the experimental data and to mitigate potential deviations of the model from reality. Such imperfections of the utilised astrophysical models are expected and thus taken care of. Similar approaches have been employed in the context of template-based analyses, for instance~\cite{TheFermi-LAT:2017vmf, Macias:2019omb}. \item A single signal normalisation parameter, $N^{\mathrm{DM}}$, which enables us to exploit both the spectral and spatial shape of the signal component (more details concerning the morphology of the signal are given in Sec.~\ref{sec:jfactor_maps}). \end{itemize} We modify the standard Poisson likelihood function in Eq.~\ref{eq:poisson_L} in two directions: \emph{(i)}, we work with the logarithm of this function, thus turning the statistical inference into a series of function minimizations which are numerically more accessible with well-tested algorithms and software packages; \emph{(ii)}, we employ a pixel weighing scheme to incorporate the impact of instrumental systematic uncertainties. To this end, we adopt the weighted Poisson log-likelihood prescription developed by the \Fermi-LAT collaboration in \cite{Fermi-LAT:2019yla}, which reads \begin{equation} \label{eq:weighted_likelihood} \ln\mathcal{L}_{w}\left(\left.\bm{\mu}\right|\bm{n}\right)=\sum_{i,p}w_{ip}\left(n_{ip}\ln\mu_{ip}-\mu_{ip}\right). \end{equation} Per energy bin, each template pixel is assigned a weight $w_{ip}$ whose value we obtain via the Fermi Science Tools calling its routines \texttt{gteffbkg}, \texttt{gtalphabkg} and \texttt{gtwtsmap}. These weights are essentially obtained via integration in space and energy of the provided model or \Fermi-LAT data\footnote{The technical aspects of these routines' implementation is provided at \url{https://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/weighted_like.pdf} or Appendix B of \cite{Fermi-LAT:2019yla}.}. Since we aim to incorporate the effect of instrumental systematic uncertainties, we rely on a ``data-driven'' weight calculation approach, that is, we utilise the LAT data themselves to derive the weights. Thus, pixels are penalised in particularly bright parts of the gamma-ray sky (point-like/extended sources, diffuse emission along the MW's disc) taking into account potential sources of systematic errors like contamination of the data sample by charged cosmic-ray events, calibration of the instrument's point spread function (PSF) or spectral mismodelling of large-scale diffuse sources. Throughout the analysis, we keep the level of systematic uncertainties to $3\%$ (for all energy bins) following the approach of the \Fermi-LAT collaboration \cite{Fermi-LAT:2019yla}. Hypothesis testing -- the procedure to discriminate between competing, alternative descriptions of reality, here represented by our model in Eq.~\ref{eq:model_eq} -- is implemented via the log-likelihood ratio test statistic, which in the relevant case of setting upper limits reads \begin{widetext} \begin{equation} \label{eq:TS_stat_reach} \qquad\qquad\qquad\qquad\qquad\qquad\textrm{TS}\!\left(N^{\mathrm{DM}}\right)= \begin{cases} -2\min_{\{N_i^{B_j}\}}\left(\ln\!\left[\frac{\mathcal{L}_w\!\left(\left.\bm{\mu}(N^{\mathrm{DM}},N_i^{B_j}) \right|\bm{n}\right)}{\mathcal{L}_w\!\left(\left.\bm{\hat{\mu}}\right|\bm{n}\right)}\right]\right)\, & N^{\mathrm{DM}} \geq \hat N^{\mathrm{DM}}\\ 0 & N^{\mathrm{DM}} < \hat N^{\mathrm{DM}}\rm{.} \end{cases} \end{equation} \end{widetext} This construction relies on the profiled likelihood function (that is, treating the background normalisations $\left\{N_{i}^{B_j}\right\}_{i, j\in J}$ as nuisance parameters) as discussed in \cite{Cowan:2010js}. Model parameters marked with $\hat{\cdot}$ denote the best-fit values found via minimization of the log-likelihood function. The test statistic in Eq.~(\ref{eq:TS_stat_reach}) only depends on the DM normalisation. Moreover, possible values of $N^{\mathrm{DM}}$ smaller than the best-fit value are discarded, thus, the TS-distribution follows a half-$\chi^{2}$-distribution with one degree of freedom (see Sec.~3.6 of \cite{Cowan:2010js}). Therefore, we set an 95\% confidence level (C.L.) upper limit on $N^{\mathrm{DM}}$ where the test statistic attains a value of 2.71. All log-likelihood minimization steps are performed with the \textsc{iminuit} python package \cite{iminuit} (version 1.5.3). \subsection{Astrophysical background model selection} \label{sec:bkg_selection} The gamma-ray sky as seen by the LAT can be decomposed into a combination of a multitude of distinct contributions, which either originate in Galactic or extragalactic sources. To constrain the extended signal such as gamma rays from DM annihilation in the outer MW halo, we incorporate the same astrophysical contributions considered in~\cite{Calore:2021hhn}. We summarise the components below and refer the interested reader to Appendix~\ref{app:astro_comp} for more details. \begin{itemize} \item The interstellar emission (IE) originates from cosmic-ray interactions with gas and low-energy ambient photon fields. Among different IE models and based on the findings of~\cite{Calore:2021hhn}, we consider as a baseline IE model the henceforth called \emph{Lorimer I}, taken from the set of realisations considered in the ``1st Fermi LAT Supernova Remnant Catalog''\footnote{The model files have been made public by the \Fermi-LAT collaboration at: \url{https://fermi.gsfc.nasa.gov/ssc/data/access/lat/1st_SNR_catalog/}.} \citep{Acero:2015prw}. Three alternative models utilised in \cite{Calore:2021hhn} -- called \emph{foreground model A, B} and \emph{C} -- are adopted from \cite{Ackermann:2014usa}. More details about the IE models are given in Appendix~\ref{app:astro_comp}. \item The isotropic diffuse gamma-ray background (IGRB) is a large-scale contribution to the gamma-ray sky which is spatially isotropic and believed to originate from the superposition of many, sub-threshold, sources~\citep{Fornasa:2015qua}. \item The modelling of the resolved point-like and extended sources is based on a 10-year data set, i.e.~the so-called 4FGL-DR2 \citep{Fermi-LAT:2019yla,2020arXiv200511208B}. \item We also model other large-scale extended gamma-ray emissions from the Fermi Bubbles (FB), Loop I, the Sun and the Moon, following standard practice in LAT data analyses. \end{itemize} Passing from these background (and signal) models to templates containing the expected photon counts per sky direction is achieved by dedicated routines of the Fermi Science Tools. For our statistical tests, we generate ``infinite statistics'' or \emph{Asimov} realisations \citep{Cowan:2010js} of the background and signal models via \textit{gtmodel}, which internally processes the full convolution of the input model files with the LAT's instrument response functions associated with the selected gamma-ray data set (c.f.~\ref{sec:lat_data_selection}). We include the LAT's energy dispersion for all background and signal components by adding the parser argument \texttt{edisp\_bins=-1}. \subsection{Fitting procedure and region-of-interest optimization} \label{sec:fitting_opti} We employ the following general analysis rundown and reasoning to statistically soundly and robustly assess the implications on DM pair-annihilation in the MW halo from \Fermi-LAT data: \begin{enumerate} \item Generating a \emph{baseline gamma-ray sky model} from a fit of the full set of astrophysical background templates to the all-sky data. \item Including the signal component, preparing the MW halo analysis by shrinking the total region of interest (ROI) to a smaller fraction of the sky that yields a good agreement between the TS-distribution (Eq.~\ref{eq:TS_stat_reach}) with respect to LAT data and baseline model as input data $\bm{n}$. \item Setting upper limits on the DM pair-annihilation cross section with respect to the optimised ROI and a particular scenario of signal templates. \end{enumerate} \subsection*{Deriving a baseline model of the gamma-ray sky} The importance of a baseline model entirely composed of the background templates considered in this analysis lies in its utility for all future statistical inference. Such a model may be used as an alternative data map that is guaranteed to contain only known gamma-ray emitters, which is not necessarily true for the experimentally observed gamma-ray data. Hence, whenever the TS-distribution as a function of the signal normalisation (Eq.~\ref{eq:TS_stat_reach}) shows a comparable behaviour with respect to baseline model and real LAT data, the selected sky region can reliably be described via the set of background and signal templates at hand. Since the baseline model is a combination of ``infinite statistics'' templates, we can draw Poisson realisations to quantify the expected statistical scatter of the eventually reported upper limits. Since we employ the same astrophysical gamma-ray emission components as in \cite{Calore:2021hhn}, we repeat the prescription presented therein to derive the baseline model. In short, the algorithm consists of an iterative fitting scheme that splits the entire sky into three disjoint parts defined as follows: (a) \textit{high-latitude} -- $|b|>30^{\circ}$ and neglecting the ``patch''-region (c.f.~\citealt{Fermi-LAT:2019yla}), which is located at $-105^{\circ}\leq\ell\leq60^{\circ}$, (b) \textit{outer galaxy} -- $|b|\leq30^{\circ}$, $|\ell|\geq90^{\circ}$ and (c) \textit{inner galaxy} -- $|b|\leq30^{\circ}$, $|\ell|\leq90^{\circ}$. The normalisation constants $N_i^{B_j}$ of a particular gamma-ray emitter are only fit in the part of the sky where it mainly contributes to while these parameters are held fixed at the thereby obtained best-fit values when the other regions are addressed. The exact details about the assignment of sky regions to particular background components are provided in Sec.~IV B of \cite{Calore:2021hhn}. We run the iterative fit for 100 times to eventually derive a baseline fit to the all-sky LAT data. Besides this general scheme, there are a few technical modifications necessary to incorporate the IE models foreground model A, B and C -- henceforth abbreviated as FGMA, FGMB and FGMC -- into this framework as well as the wealth of sources in 4FGL-DR2, which deviates from the original receipe in \cite{Calore:2021hhn}.\\ \emph{Treatment of IE models:} All five IE models feature a single template quantifying the gamma-ray emission from inverse Compton (IC) scattering processes, which we split into three sub-templates whose boundaries coincide with the definitions of the `high-latitude', `outer galaxy' and `inner galaxy' regions of the iterative fit. The same procedure is applied to the gamma-ray emission associated with the gas maps in FGMA-C. After the iterative fit, all IE-related components are multiplied by their best-fit normalisation parameters and added to form an optimised IE template that is utilised in the subsequent analysis parts.\\ \emph{Treatment of 4FGL-DR2 sources:} Combining all sources listed in 4FGL-DR2 into a single template causes the brightest sources in the template to drive the best-fit value of the template's normalisation parameter. We weaken the impact of bright sources by separating sources with an energy flux of $E_{100} < 4\cdot10^{-10}\;\left[\mathrm{MeV}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\right]$ (integrated from 100 MeV to 100 GeV) from those above this threshold. A source that surpasses this threshold is fitted individually during the all-sky fit (d) of an iteration step. We list these bright sources and some of their properties in Tab.~\ref{tab:bright_4fgl} in App.~\ref{app:bright_4FGL}. All others are combined in a single 4FGL template. As for the IE templates, all 4FGL-DR2 templates are combined after the fit according to their best-fit values, hence creating an optimised total 4FGL template to be used in all steps that follow. \subsection*{Optimising the analysis region of interest} The strategy to perform an ROI optimization before conducting any statistical inference is heavily based on the approach presented in \cite{Zechlin:2017uzo}. In order to search for an ROI that yields statistically reliable upper limits on the DM annihilation cross section, we resort to the southern hemisphere -- thereby circumventing most of LoopI's contribution to the gamma-ray sky -- and investigate the distribution of the test statistic in Eq.~\ref{eq:TS_stat_reach} for both the true LAT data and the baseline model as data input vector $\bm{n}$. As the signal spectrum depends on the chosen DM parameters and annihilation channel, we construct a model-independent DM signal -- that still exhibits the currently assumed spatial morphology of the MW halo -- by exchanging the generic WIMP spectra with a power law of spectral index -2. Hence, the signal's spectrum is featureless and yields non-zero photon counts throughout the entire energy range considered in this work. The initial normalisation of the power law is somewhat arbitrary since there is no connection to a physical DM model. We hence choose the normalisation such that the expected counts in the first energy bin are maximally of order unity per pixel. The optimisation is carried out by systematically shrinking the ROI boundaries from $\ell\in\left[-180^{\circ},180^{\circ}\right]$ to $\ell\in\left[-90,90^{\circ}\right]$ (symmetrically) with $b\in\left[-90^{\circ},-20^{\circ}\right]$. In addition, we mask the FBs by setting all pixels to zero whose counterpart in the FB template predicts non-zero counts. We introduce the requirement of $b\geq20^{\circ}$ to reduce the impact of IE along the Galactic disc. For each ROI, we scan the TS-distribution with respect to the LAT data and baseline model for $\mathrm{TS}\in\left[0, 25\right]$ as a function of $N^{\mathrm{DM}}$ and use the $\ell^2$-metric to quantify their mutual compatibility. We select those Galactic longitude and latitude ROI boundaries for which this metric is minimal. We stress that this optimisation step must be repeated for each of the five considered IE models. \emph{Treatment of the optimised 4FGL-DR2 template:} A special note concerns the treatment of 4FGL catalogue sources. In contrast to the iterative fit, we now mask a circular region around the central position of each source. The mask radius is energy-dependent and corresponds to the $95\%$ containment radius of the LAT's PSF\footnote{See \url{https://www.slac.stanford.edu/exp/glast/groups/canda/lat_Performance.htm} for details.} for the chosen LAT event class and type. For the first three energy bins, however, we reduced the mask radius to $90\%$ of the PSF size. The latter exception has been introduced to ensure a reasonable number of pixels to be non-zero so that a template-based analysis remains feasible. We illustrate the TS-distribution comparison in Fig.~\ref{fig:McMillan_example_ROI} of Sec.~\ref{sec:ULIM_McMillan2017} for a particular combination of spatial DM distribution and IE model. In what follows, we will always report the selected optimal ROI. Moreover, we test our analysis pipeline in terms of its capability to recover an injected signal in simulated data. The results of this sanity check are described and discussed in Appendix \ref{app:signal_recovery}. \subsection{Dedicated analysis of the Large Magellanic Cloud region} \label{sec:lat_analysis_LMCroi} Since the LMC passage through the MW halo does not only induce a response in the latter but also in the LMC DM halo itself, we aim to analyse the surroundings of the LMC in a dedicated gamma-ray study. To this end, we utilise the same LAT data selected and described in Sec.~\ref{sec:lat_data_selection} but restrict the ROI to a maximal size of $30^{\circ}\times30^{\circ}$ centred on the stellar position of the LMC at $(\ell, b) = \left(280.54^{\circ}, -32.51^{\circ}\right)$ in agreement with the centre of the LMC \Jf-factor maps discussed in Sec.~\ref{sec:jfactor_maps_LMC}. The pixel size of the binned data is set to $0.1^{\circ}\times0.1^{\circ}$ while the energy binning is adopted from the all-sky data set. The general analysis and fitting strategy is completely analogous to the scheme outlined in the previous section, with the addition of emission model components of the LMC itself. A decisive characteristic of the dedicated LMC analysis is the need for an additional astrophysical background component quantifying the expected gamma-ray emission due to cosmic-ray interactions with gas and radiation fields in the LMC. To this end, we include four separate templates adopted from the set of extended templates being part of the 4FGL-DR2 catalogue. The components LMC-Galaxy, LMC-North, LMC-FarWest and LMC-30DorWest, representing the cosmic-ray induced gamma-ray emission of the LMC, have been derived in a previous study of the \Fermi-LAT collaboration \citep{Fermi-LAT:2015bpm} based on a six-year data set. These models are the result of a convolution of gas column density maps of the LMC reported in \cite{2012ApJ...756....5L} and a data-driven approach to extract regions of significant extended gamma-ray emission associated with the LMC. Due to their data-driven nature, these models may already incorporate a contribution from an exotic gamma-ray emitter like DM if taken at face value. The authors of \cite{Buckley:2015doa} comment on this point by analysing the correlation between the DM component and the LMC-related templates. They find a particularly sizeable correlation with the ``LMC-Galaxy'' template, which is centred on the stellar position of the LMC and, at the same time, the most extended among the considered ones. Such a correlation might artificially boost the constraining power of our template-based approach. We mitigate this effect by including the four LMC templates as unmasked, independent model components whose normalisation parameters are a subset of all the model's nuisance parameters that are profiled over to set upper limits on the DM annihilation cross section. Moreover, our ROI is considerably larger than the $10^{\circ}\times10^{\circ}$ ROI adopted in \cite{Fermi-LAT:2015bpm} and \cite{Buckley:2015doa}. Consequently, the spatial extension of the DM component beyond the size of the four LMC templates partially brakes the degeneracy with these templates. We anticipate that the later derived upper limits that we find are indeed comparable to the results reported in \cite{Buckley:2015doa}. The remaining differences to the scheme in Sec.~\ref{sec:fitting_opti} occur in the derivation of a baseline fit for a particular IE model and the optimisation of the analysis ROI. In details, these changes are: \begin{itemize} \item[(a)] All additional point-like gamma-ray sources within $3^{\circ}$ of the stellar position of the LMC are fitted separately by means of a template for each individual source. \item[(b)] All detected gamma-ray sources, which already were reported in the 3FGL catalogue \citep{Fermi-LAT:2015bhf} and which are at a distance of $3^{\circ} < r \leq 40^{\circ}$ from the stellar centre of the LMC are cast into a single 3FGL template. \item[(c)] All remaining 4FGL-DR2 point-like sources that are neither in 3FGL nor within $3^{\circ}$ of the ROI centre, are cast into another template. \item[(d)] Regarding the IE contribution, we consider the following models: We adopt FGMA as baseline model but keep Lorimer I as an alternative test case. This change is motivated by the improved performance of FGMA in the LMC ROI compared to Lorimer I. Regarding the latter, we only fit its IC, ring 3 CO component as well as the HI templates of ring 2 to 4 -- the emission associated to the remaining ones is almost zero in the LMC ROI. FGMA and FGMC are treated as outlined in Sec.~\ref{sec:fitting_opti}. Lastly, we add the Galactic diffuse background model of the \Fermi-LAT collaboration in our list of viable IE models. The reasons for excluding this particular model from the MW halo study do not apply to the LMC region. For example, the data-driven nature of this model is not hampering our efforts because the relevant parts of the sky do not fall into the chosen LMC ROI. \item[(e)] The FBs as well as the gamma-ray emission from the Sun and Moon are neglected because their contribution is almost zero in the chosen ROI. \item[(f)] All astrophysical background components are fitted in the full ROI at the same time without specifying disjoint fit regions to derive a baseline fit to the gamma-ray data in the ROI. \item[(g)] After the baseline fit, the IE components are summed with their best-fit normalisation to an optimised IE model. The localised point-like sources in 4FGL-DR2 are treated analogously except for the LMC-related templates, which we keep as individual templates even in the stage for setting upper limits. \item[(h)] The ROI optimisation translates to symmetrically shrinking the width and height of the ROI up to the point where the optimal correspondence between expected TS-distribution and observed TS-distribution (with respect to the auxiliary DM signal template featuring a power law spectrum) is achieved. \item[(i)] To explore the TS-distributions and to set upper limits, we mask all detected point-like sources in the same way as outlined before except for the positions of the LMC-related emitters. \end{itemize} \section{Constraints on particle dark matter} \label{sec:upperlimits} In this section, we present the results of the constraints on DM pair-annihilation processes in the MW and LMC haloes with the analysis framework described in Sec.~\ref{sec:fermianalysis}. As mentioned in Sec.~\ref{sec:simulation}, the characterisation of the gravitational potential of the MW comes with a non-negligible uncertainty. The dynamical impact of the LMC passage may be regarded as a second-order effect that adds to the inherent uncertainties of the available astronomical observations. Consequently, we first explore the expected scatter of constraints on the DM parameter space induced by observational uncertainties of the MW's DM halo while in the following subsections, we shed light on the significance of including the MW's response to the LMC passage for the outcome of indirect searches for DM. The following results for the MW outer halo have been obtained with our benchmark IE model, Lorimer I, unless stated otherwise, whereas the benchmark for the LMC analysis is FGMA. We investigate the impact of the chosen IE model on the DM constraints in Appendix~\ref{app:iem_limits} -- we anticipate that limits at high latitude are mildly affected by the IE choice, and are robust in this respect. \subsection{Impact of the uncertainty of the Milky Way's gravitational potential} \label{sec:ULIM_McMillan2017} We utilise the assessment of the MW's gravitational potential and mass distribution in \citet{2017MNRAS.465...76M} to explore the impact of their uncertainty on DM indirect searches in the outer MW DM halo. To re-iterate, the author of that study assumes a standard NFW profile (inner slope parameter $\gamma = 1$) to describe the MW DM halo and following this fundamental assumption derives, among others, posterior distributions for $ r_s, R_{\odot}, \rho_{\odot}$, and $R_{200}$ to scale and normalise the NFW profile in accordance with the observational constraints. From these posterior distributions we randomly draw 200 points and generate the associated all-sky \Jf-factor map of the MW. For each of these MW realisations, we derive \Fermi-LAT upper limits on the DM pair-annihilation cross section following the scheme outlined in Sec.~\ref{sec:fitting_opti}. As the first step of this analysis pipeline, we search for an optimal ROI in the gamma-ray sky by successively shrinking the considered fraction of the southern hemisphere in Galactic longitude and latitude. We illustrate the statistical performance of the thus obtained optimised ROI via one particular realisation from the McMillan posterior distributions characterised by the tuple of parameters $(r_s, R_{\odot}, \rho_{\odot}, r_{200}) = (12.1\unit{kpc}, 8.3\unit{kpc}, 0.4\unit{GeV}\unit{cm}^{-3},199.0\unit{kpc})$ in Fig.~\ref{fig:McMillan_example_ROI}. We find the best-suited analysis ROI to be defined by $\ell\in\left[-167^{\circ}, 167^{\circ}\right]$ and $b\in\left[-90^{\circ}, -35^{\circ}\right]$, which ensures a reasonable compromise between the constraining power of the remaining gamma-ray sky and the statistical robustness of the resulting upper limits. Both panels in Fig.~\ref{fig:McMillan_example_ROI} have been obtained from 200 Poisson realisations of the baseline fit. On one hand, the left panel of this figure shows that the TS-distribution on LAT data stays within the 68$\%$ containment band of the scatter of the baseline fit's TS-distribution. On the other hand, the width of its median's parabola deviates from its analogue on LAT data. This qualitatively indicates that the chosen ROI may contain further gamma-ray emission components which are not or not fully accounted for in the selected set of astrophysical background contributors. Hence, this ROI may feature further emission components that are degenerate with the DM signal. Consequently, the $95\%$ C.L.~upper limits fall within the 68$\%$ containment band derived from mock data for most of the scanned DM masses except for light DM below $\lesssim 10$ GeV. Here, the constraints are slightly stronger (at the $2\sigma$ level) than expected from the baseline fit. We have made sure that the chosen ROI is suitable for all of the 200 random parameter tuple $(r_s, R_{\odot}, \rho_{\odot}, R_{200})$. In fact, the observed moderate fluctuation for light DM is a common feature among all of these realisations of the MW halo. From a qualitative point of view, the consistency between the TS-distribution of mock and real data in this work is similar to the benchmark scenario studied and described in Fig.~1 of \cite{Chang:2018bpt}. There, the authors find the same slight deviation for light DM but they also show a more pronounced deviation for DM around the TeV scale where our TS-distributions differ only at the $1\sigma$ level. However, the selected optimal ROIs in both analyses are largely disjoint since we exclude the position of the FBs. The optimised ROI in \cite{Zechlin:2017uzo} (c.f.~Fig.~5 therein) shows a larger overlap with our analysis ROI. The reported accordance of the statistical expectations from their baseline fit and the corresponding performance on real data in their Fig.~4 is well in line with the results presented in this work, i.e.~consistency at the $1\sigma$ level for most of the tested parameter space. The comparison with both literature results corroborates that our optimal ROI provides statistically sound upper limits from DM annihilation processes in the outer MW halo. The left panel of Fig.~\ref{fig:McMillan_limits} summarises the set of constraints derived from these 200 random realisations of the MW DM halo assuming WIMP DM pair-annihilating into $b\bar{b}$ final states that eventually generate a prompt gamma-ray flux via further processes. The median $95\%$ C.L.~upper limits obtained within the optimised ROI and with respect to LAT data are denoted by a black dashed line whereas the corresponding $1\sigma/2\sigma$ containment bands are depicted as dark grey/light grey shaded bands. The impact of the observational uncertainty of the MW's gravitational potential is less than a factor of two at the $2\sigma$ level across the entire range of DM masses considered in our analysis. This finding may seem astonishing at first glance. It is well known that the uncertainty of the shape of the MW's DM halo has a much larger effect on indirect searches towards the Galactic centre since the DM distribution in this region of the Galaxy can be peaked, core-like or even be completely devoid of DM \citep[see, for example,][]{2011JCAP...11..029I, 2015NatPh..11..245I, 2017PDU....15...90I, 2019JCAP...09..046K, Benito:2020lgu}. However, high Galactic latitudes as inspected by us are dominated by the outer MW DM halo, that is, we probe a much larger volume of the total halo with guaranteed DM presence in order to stabilise the Galactic rotation curve of the MW. Hence, small-scale uncertainties of the MW's gravitational potential, for instance in the Galactic centre, are washed out by the fact that we investigate a large volume of the MW DM halo and probe its cumulative gravitational imprint. Results for the $\tau^+ \tau^-$ DM annihilation channel are provided in Appendix~\ref{app:tau_results}. \subsection{Impact of the perturbation of the Milky Way's dark matter halo caused by the Large Magellanic Cloud's passage} \label{sec:ULIM_MW+LMC} As discussed in Sec.~\ref{sec:simulation}, the passage of the LMC through the MW halo induces dynamical responses, which add to the already discussed uncertainty of the MW's gravitational potential. The respective responses in the form of wakes are particularly present in the outer MW halo trailing the LMC orbit or developing in front of its current orbital direction. Hence, this dynamical effect is supposed to be detectable in the southern hemisphere, as visualised in Fig.~\ref{fig:jfactor_comparison}. We quantitatively examine the importance of the LMC passage via the simulations of the MW-LMC system described and discussed in Sec.~\ref{sec:mw-lmc-sim}. To this end, we confront in Fig.~\ref{fig:LMC_impact_on_MW} the $95\%$ C.L.~upper limits on the DM pair-annihilation cross section ($\chi\chi\longrightarrow b\bar{b}$) obtained from three different simulations with the previously derived uncertainty of the same limits due to observational uncertainty of the MW's gravitational potential. We distinguish constraints for the cases of taking into account solely the MW DM halo's monopole term from the BFE (black lines) and the corresponding full BFE (red lines) according to the respective simulation. As concerns the chosen simulations, we have selected two models from the \citet{Donaldson+2022} suite -- assuming an initial NFW (solid)/HERN (dotted) profile for both the MW and the LMC -- and the Erkal19 simulation (dashed). We have checked that the optimised ROI discussed in Sec.~\ref{sec:ULIM_McMillan2017} can also be applied to these signal DM profiles. We emphasize that the utilised \Jf-factor maps are the sum of both the MW and LMC halo, although the additional boost due to the LMC halo is marginal. A comparison of the black curves reveals that the different initial conditions in terms of MW halo profile are well within the $2\sigma$ range of the uncertainty reported by \citet{2017MNRAS.465...76M} and thus plausible representations of the physically realised halo of the MW. If we compare the results in red that incorporate the full influence of the LMC as a perturber of the MW halo with those that disregard this effect, we see that the impact on the DM constraints depends on the respective simulation. While both simulations from \cite{Donaldson+2022} predict an almost negligible improvement of the upper limits, the Erkal19 simulation yields a much stronger response that induces an improvement for light DM as large as the $1\sigma$ uncertainty of the MW's gravitational potential. The difference of the obtained bounds is not related to the total \Jf-factor of the full MW halo -- which is the highest for the NFW-NFW simulation -- but rather correlated with the appearance of deviations from the static, spherically symmetric DM halo scenario. Local over- and under-densities induced by the dynamical response of the MW halo are most pronounced in the Erkal19 simulation, which explains the prominent effect in Fig.~\ref{fig:LMC_impact_on_MW} compared to the remaining simulation suites. The difference is thus a manifestation of the initial conditions of each model simulation (i.e. assumptions about extent, profile, mass, internal structure/deformation and components) of the MW-LMC system. We discuss these properties and their interplay in Sec.~\ref{sec:discussion_systematics_MW}. \subsection{Indirect searches towards the Large Magellanic Cloud} \label{sec:ULIM_LMC} We assess the constraining power of the LMC as a target for indirect DM searches and the importance of including the altered shape (and mass) of the LMC DM halo due to its gravitational interaction with the MW halo. This analysis follows the outline given in Sec.~\ref{sec:lat_analysis_LMCroi}. We stress again that we select FGMA as the benchmark IE model because it shows a better performance than Lorimer I in this particular ROI. We discuss the systematic uncertainty due to the chosen IE model for this dedicated LMC analysis in Appendix \ref{app:iem_limits} and Fig.~\ref{fig:iem_uncertainty_main} therein. In Fig.~\ref{fig:LMC_ROI_study}, we summarise our findings regarding the analysis of the LMC's DM halo. In the left panel, we compare the constraints on the DM pair-annihilation cross section (channel: $\chi\chi\rightarrow b\bar{b}$) using either the full BFE (red) or only its monopole term for the LMC halo (black) of a particular simulation of the evolution of the MW-LMC system. For each of these sets of halo profiles, we optimised the square ROI centred on the LMC position in terms of its size. The optimal ROI sizes are stated in Tab.~\ref{tab:lmc_rois}. We observe a mild impact of the dynamical response of the LMC on the final upper limits for the four simulations from Ref.~\cite{Donaldson+2022} whereas -- and as we had already noticed in the case of the outer MW halo -- the Erkal19 simulation suggests a more pronounced effect, which may be as large as a factor of two. We discuss the observed simulation-dependent variations in more detail in Sec.~\ref{sec:discussion_systematics_LMC}. The right panel of Fig.~\ref{fig:LMC_ROI_study} puts the results from the perturbed LMC haloes in the context of the current state-of-the-art in the field. We confront a subset of the DM upper limits from the left panel (red curves) to constraints derived with standard DM haloes following the profiles from an NFW or Hernquist profile with parameters adopted from~\cite{Regis:2021glv} presented in Sec.~\ref{sec:jfactor_maps_LMC} (orange lines). The latter set of upper limits hence represent a static LMC halo that does not incorporate deviations from spherical symmetry. While both types of LMC haloes agree on the strength of the constraints for light DM particles, they differ for larger DM masses. In fact, haloes from~\cite{Regis:2021glv} feature a more pronounced cusp towards the LMC's centre whereas our simulated haloes appear less peaked and smoother in general. The discussion is continued in Sec.~\ref{sec:discussion_systematics_LMC}. Representative results of the \Fermi-LAT collaboration’s search for DM in the LMC \citep{Buckley:2015doa} with five years of data are displayed with different shades of blue. We comment on the comparison to our bounds in Sec.~\ref{sec:conclusions}. \begin{table*} \begin{centering} \begin{tabular}{c c c} \hline \multirow{3}{*}{simulation/DM profile} & only monopole & multipole expansion\tabularnewline \cline{2-3} \cline{3-3} & ROI size & ROI size\tabularnewline & \Jf-factor {[}GeV$^2$cm$^{-5}${]} & \Jf-factor {[}GeV$^ 2$cm$^{-5}${]}\tabularnewline \hline \hline \multirow{2}{*}{Erkal+, `19} & $29.9^{\circ}\times29.9^{\circ}$ & $29.8^{\circ}\times29.8^{\circ}$\tabularnewline & $3.39\times10^{20}$ & $3.55\times10^{20}$\tabularnewline \hline \multirow{2}{*}{MW: NFW, LMC: NFW} & $29.9^{\circ}\times29.9^{\circ}$ & $30.0^{\circ}\times30.0^{\circ}$\tabularnewline & $1.46\times10^{20}$ & $1.70\times10^{20}$\tabularnewline \hline \multirow{2}{*}{MW: NFW, LMC: HERN} & $29.8^{\circ}\times29.8^{\circ}$ & $29.7^{\circ}\times29.7^{\circ}$\tabularnewline & $1.53\times10^{20}$ & $1.83\times10^{20}$\tabularnewline \hline \multirow{2}{*}{MW: HERN, LMC: NFW} & $29.5^{\circ}\times29.5^{\circ}$ & $29.5^{\circ}\times29.5^{\circ}$\tabularnewline & $1.45\times10^{20}$ & $1.70\times10^{20}$\tabularnewline \hline \multirow{2}{*}{MW: HERN, LMC: HERN} & $29.8^{\circ}\times29.8^{\circ}$ & $29.7^{\circ}\times29.7^{\circ}$\tabularnewline & $1.54\times10^{20}$ & $1.85\times10^{20}$\tabularnewline \hline \multirow{2}{*}{NFW \citep{Regis:2021glv} } & / & $27.5^{\circ}\times27.5^{\circ}$\tabularnewline & / & $1.07\times10^{20}$\tabularnewline \hline \multirow{2}{*}{Hernquist \citep{Regis:2021glv}} & / & $27.5^{\circ}\times27.5^{\circ}$\tabularnewline & / & $0.98\times10^{20}$\tabularnewline \hline \end{tabular} \par\end{centering} \caption{Summary of the optimised ROI sizes for the dedicated study of the LMC environment. The optimisation has been performed under the assumption of the FGMA IE model. For each simulation and basis function expansion scenario we state the total \Jf-factor contained within the reported ROI to facilitate better comparison between the different cases. \label{tab:lmc_rois}} \end{table*} \section{Discussion} \label{sec:discussion} \subsection{Systematic uncertainties affecting the study of the outer Milky Way halo} \label{sec:discussion_systematics_MW} In this work, we have demonstrated that accounting for the deformed dark matter haloes of the Milky Way and LMC is crucial for getting accurate cross section constraints from gamma ray searches. Indeed, Figure~\ref{fig:LMC_impact_on_MW} shows that the change to the constraint from including the deformations (i.e. the difference between the black and red curves) is comparable to the uncertainty on the constraint due to uncertainties in the Milky Way's mass profile (i.e. the grey band). In addition to the effect of accounting for deformations, we also see that precise constraints also depend on how we model the Milky Way and LMC system. The reason for this is that these simulations span a wide range of Milky Way masses and concentrations which affect the strength of the dark matter deformations. In particular, the simulations from \cite{Donaldson+2022} have more massive and concentrated Milky Way haloes which deform less than the simulated Milky Way halo in \cite{2022arXiv220501688L}. For reference, we note that the model in \cite{2022arXiv220501688L} appears similar to models in the literature of the Milky Way-LMC interaction \citep{Garavito-Camargo+2019,Rozier2022}. Given this range of possibilities, we argue that the deformations can be considered as a source of systematic uncertainty on the inferred cross section until they are better characterized. We note that there are also other physical effects which have altered the Milky Way's dark matter halo and could affect the cross section constraints in similar ways. For example, the Gaia-Sausage/Enceladus \citep[GSE,][]{Belokurov+2018,Helmi+2018} merger likely brought a substantial amount of dark matter into the inner Milky Way which may still not be phase-mixed \citep[e.g.][]{Naidu+2021,Han+2022}. Accounting for this dark matter would likely also lead to changes in the cross section constraints. \subsection{Systematic uncertainties affecting the study of the LMC} \label{sec:discussion_systematics_LMC} In Sec.~\ref{sec:lat_analysis_LMCroi} we cautioned that the use of the data-driven astrophysical templates for the LMC may artificially drive the DM bounds towards tighter contraints, which we aimed to avoid via the design of the analysis pipeline and the size of the chosen ROI. The \Fermi-LAT collaboration's search for DM in the LMC \citep{Buckley:2015doa} with five years of data mitigated this effect in a different manner. To exemplify the results of this study, we show a selection of upper limits in the right panel of Fig.~\ref{fig:LMC_ROI_study} displayed in three shades of blue representing different choices for the static LMC DM profile. These profiles are tuned to fit the rotation curve of the LMC as traced by stars and gas. The profile dubbed \textsc{nfw-med} is very similar to the initial LMC profiles in the NFW, HERN and Erkal19 simulations on which we base our work. In fact, the cyan line in Fig.~\ref{fig:LMC_ROI_study} is comparable to the results from the dynamical LMC halo (red lines) but also to the bounds from the static DM halo profiles from \cite{Regis:2021glv}. The deviation of the latter bounds can be explained with the enlarged ROI compared to \cite{Buckley:2015doa}. Thus, we find corroborating evidence that our constructed analysis pipeline is not severely affected by a bias due to the astrophysical LMC templates. The left panel of Fig.~\ref{fig:LMC_ROI_study} illustrates and quantifies the level of the expected induced variation of upper limits on thermal DM when deformations of the LMC DM halo are included. The implications for the profiles of the obtained upper limits on the annihilation cross section vary between individual simulation suites. Including the LMC halo's deformations may either improve the constraints or weaken them by up to a factor of two. As pointed out in Sec.~\ref{sec:discussion_systematics_MW}, the simulations themselves exhibit an intrinsic uncertainty regarding initial conditions and the definition of the MW/LMC morphology. This uncertainty consequently translates into a range of the \Jf-factor maps of the LMC compatible with the simulated MW-LMC dynamics. Regarding the obtained upper limits, however, the effect of deviations from spherical symmetry (a defining feature of the monopole of the BFE) must be understood on a case-by-case basis since the DM signal is degenerate with some of the astrophysical background gamma-ray components in the LMC ROI. Deformations of the LMC halo result in asymmetries of the \Jf-factor maps, which may help to break (or worsen) these degeneracies. Considering, for example, the almost spatially uniform isotropic background it is clear that asymmetries in the \Jf-factor maps greatly reduce its degeneracy with the DM template. On the flip-side, it is also conceivable that the morphology of a deformed LMC halo increases existing degeneracies as is the case in the NFW+NFW simulation from \cite{2022arXiv220501688L}. The degeneracies with the astrophysical background templates also explain the differences between the results for static NFW profiles and simulated LMC haloes displayed in the right panel of Fig.~\ref{fig:LMC_ROI_study}. On one hand, the simulation results exhibit less peaked DM density profiles towards the centre of the LMC than the profiles from \cite{Regis:2021glv}, thus reducing the features that may break degeneracies with background templates. On the other hand, the spectral shape of the DM signal is relevant too in order to improve the constraining power of the analysis. For light DM, for instance, the degeneracy can be broken by the information from the DM gamma-ray annihilation spectrum that shows a cutoff around the DM mass. This cutoff falls within the sweet spot of the LAT's sensitivity. Heavy DM with $m_{\chi} > 100\unit{GeV}$, in contrast, features a break in the spectrum at tens of GeV where the LAT sensitivity starts to decrease. Hence, in this particular case, the spectral shape of the annihilation signal does not contribute as much to breaking the degeneracy with the background. \section{Conclusions} \label{sec:conclusions} In the present work, we used state-of-the-art simulations of the MW encounters with the LMC to assess how the deformations of the MW and LMC dark matter haloes affect indirect detection for DM. First, we focused on high Galactic latitudes and performed a search for a DM annihilation signal in twelve years of \Fermi-LAT data. Since no significant signal was found (regardless of the DM spatial distribution adopted), we set 95\% C.L.~constraints on the DM annihilation cross section. In particular, we modelled the DM distribution in the Galactic halo following a recent MW mass modelling~\citep{2017MNRAS.465...76M}. The mass modelling of the MW comes with non-negligible uncertainties. We propagated this uncertainty on our final limits, and provided an uncertainty band which reflects it. Moreover, we verified that the systematic uncertainties from IE modelling are indeed mild as they induce a variance of the derived bounds by at most a factor of two for DM masses at the light and heavy end of the probed mass range. The optimal ROI sizes for each of the employed IE models are largely overlapping rendering a direct comparison sensible (see Appendix~\ref{app:iem_limits} for more details). The limits in Fig.~\ref{fig:McMillan_limits} represent the most up-to-date and robust limits on DM at high latitudes derived with \Fermi-LAT data. Our high-latitude DM limits are stronger than the results obtained by \cite{Zechlin:2017uzo} from their most constraining energy band from 1 to 2 GeV (c.f.~orange line in the left panel of Fig.~\ref{fig:McMillan_limits}). As noted in Sec.~\ref{sec:ULIM_McMillan2017}, the authors performed their analysis in an ROI overlapping with ours but of reduced size. Thus, our enlarged ROI improves the constraining power allowing us to exclude thermal DM for masses $m_{\chi} \lesssim 40$ GeV, while \cite{Chang:2018bpt} report a slighter stronger bound for DM below 200 GeV and an exclusion for $m_{\chi} \lesssim 70$ GeV. To this end, the authors have derived an ROI that is rather disjoint with ours. It is closer to the Galactic centre and it takes into account gamma-ray data from the position of the FBs. The latter two differences can easily explain the increased constraining power compared to our study. In the right panel of Fig.~\ref{fig:McMillan_limits}, we place our MW outer halo bounds in the context of existing constraints on the parameter space of thermal DM derived from different targets and cosmic-ray channels. The light purple line indicates gamma-ray constraints from the observation of dwarf spheroidal galaxies with space-borne and ground-based instruments. The results of this joint analysis \citep{Hess:2021cdp} are the most state-of-the-art constraints from dwarf spheroidal galaxies using traditional inference techniques. While this set of exclusion limits outperforms our bounds over the entire probed mass range, they are affected by non-negligible systematic uncertainties due to both modelling of the DM distribution in these systems and the background modelling in and around dwarf spheroidal galaxies. As for the latter, the authors of \cite{Calore:2018sdx,Alvarez:2020cmw} designed an analysis that incorporates background modelling systematic effects, which weaken the upper limits on the DM annihilation cross section of about a factor of three. These limits (based only on classical dwarf spheroidal galaxies) are shown as a purple line. In this case, constraints from the outer MW halo are stronger than the gamma-ray limits from dwarf spheroidal galaxies. The variance induced by the uncertainty of the outer MW halo profile is hence less pronounced than the impact of background modelling uncertainties in dwarf spheroidal galaxy studies. We stress that we assumed a very conservative approach in the selection of the region of interest for the analysis by requiring a strict compatibility between the statistical expectations derived from Poisson realisations of the baseline fit and the true \Fermi-LAT data. This method yields reliable and robust limits but reduces the potential constraining power of the full data set and limits the accessible dynamically generated features of the MW-LMC system. As indicated by the right panel of Fig.~\ref{fig:jfactor_comparison_LMC}, the northern sky may be indeed a more promising target to explore the signatures of the MW-LMC interaction. In this part of the sky, the LMC is expected to provoke a response in the forward direction of its orbit. However, finding a good agreement between astrophysical model and data is more challenging and needs better theoretical refinements. For example, if we relax the strict constraint of considering only data at high latitudes, we find that adding to the optimal ROI in Sec.~\ref{sec:upperlimits} a counterpart in the northern hemisphere defined by $-102^{\circ} < \ell < 102^{\circ}$ and $b > 16^{\circ}$ (excluding the FBs region) yields the best accordance -- although being far from perfect -- between the expectations derived from the baseline fit and the true \Fermi-LAT data. The improvement we can achieve in this way is as large as a factor of 3 for either a static or a simulated MW halo profile (see details in Appendix~\ref{app:newROI_limits}). In contrast, constraints (green band) from radio searches towards the LMC as found in \cite{Regis:2021glv} show a larger intrinsic uncertainty than ours. Even in a conservative scenario, however, the bounds on thermal DM from the LMC's emission in radio light are stronger over the considered mass range. It should be noted that these results are derived assuming a static LMC DM profile while dynamical effects -- as we have studied in this work -- may alter the picture. The authors of \cite{DiMauro:2021qcf} have derived a set of upper limits (dark blue) comparable to the radio LMC bounds considering the latest AMS-02 antiproton data release. Anti-proton constraints seem robust and only mildly affected by modelling uncertainties (DM halo profile, comsic-ray propagation). In the future, these bounds can be further strengthened with a re-calibrated prediction of secondary cosmic rays to achieve a better agreement with new data, which is further discussed in~\cite{Calore:2022stf}. Thanks to the uncertainty band from the MW gravitational potential modelling, we quantified the significance of the MW-LMC dynamics' impact on indirect DM detection. For the set of simulations adopted in this work, we found that the MW-LMC interaction does not strongly affect the upper limits on thermal DM. The obtained variations are within a factor of 1.3, which we find in the Erkal19 simulation that generally yields the most pronounced dynamical responses. The MW-LMC interactions also affect the mass distribution in the LMC itself -- and the ensuing DM annihilation signal. Therefore, we derived bounds on DM annihilation from the region around the LMC as well, with dedicated modelling of the astrophysical backgrounds. The limits derived from the LMC ROI are largely consistent with the previous bounds derived by the \Fermi-LAT collaboration for the case of a static NFW profile as discussed in Sec.~\ref{sec:discussion_systematics_LMC}. The level of systematic uncertainty of the constraints on thermal DM caused by the dynamical deformation of the LMC halo is not of the same order of magnitude as the one caused by the allowed range of static DM halo profiles that can reproduced the stellar rotation curve of the LMC as illustrated by the range of the blue-shaded lines in Fig.~\ref{fig:LMC_ROI_study}. However, we note that the initial states of the LMC in all simulations utilised in this work did not aim at bracketing the full margin of inner halo profiles consistent with stellar data. We can thus not quantify how the LMC's internal structure is affected by its passage through the MW in case of an aggressive assumption like the \textsc{sim-med} parametrization in \cite{Buckley:2015doa}. Refined simulations with a wider range of inner DM halo slopes may be warranted to assess the full implications of the MW-LMC dynamics for indirect searches towards the LMC. Eventually, the effect may even alter the prospects of radio searches in the central region of the LMC which consequently relax or tighten the already stringent bounds on thermal DM reported by \cite{Regis:2021glv}. In conclusions, we have shown that high Galactic latitudes have the potential to be the leading target for DM searches in gamma rays in the future. A crucial step in this direction will be the optimisation of interstellar emission models thanks to, for example, machine learning techniques and verification schemes, see e.g.~\cite{Storm:2017arh,Mishra-Sharma:2021oxe, Caron:2021wmq} in the context of background model optimisation in the inner Galaxy. If we focus on the LMC region, a leap forward in the understanding of the astrophysical emission (and therefore a better modelling of the LMC astrophysical templates) is expected thanks to the up-coming observations of the Cherenkov Telescope Array, CTA, of this particular region as outlined in \cite{CTAConsortium:2017dvg}. Looking forward, future work with stellar streams and other tracers will allow the community to robustly measure the dark matter deformations of the Milky Way and LMC dark matter haloes through their gravitational effects. Indeed, \cite{Shipp+2021} show that the streams have their closest approaches with the LMC at different times, giving hope to the idea that the time dependence can be measured. Once these deformations are measured, they can be folded into the analysis, as we have done in this work, to derive the most accurate annihilation cross sections. On the flip side, if annihilating dark matter is detected in gamma rays then we will be able to directly measure the deforming dark matter haloes of the Milky Way and LMC. \section*{Acknowledgements} \addcontentsline{toc}{section}{Acknowledgements} The work of C.E. is supported by the ``Agence Nationale de la Recherche'', grant n. ANR-19-CE31-0005-01 (PI: F. Calore). M.S.P is partially supported by grant Segal ANR-19-CE31-0017 of the French Agence Nationale de la Recherche (https://secular-evolution.org). For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.\\ \emph{Software:} \textsc{astropy} \citep{2013A&A...558A..33A, 2018AJ....156..123A,TheAstropyCollaboration2022}, \textsc{clumpy} (version 3) \cite{charbonnier2012clumpy, bonnivard2016clumpy, 2019CoPhC.235..336H}, \textsc{fermi science tools} \citep{2019ascl.soft05011F}, \textsc{healpy} \citep{2005ApJ...622..759G, Zonca2019}, \textsc{iminuit} \citep{iminuit}, \textsc{jupyter} \citep{soton403913}, \textsc{matplotlib} \citep{4160265}, \textsc{numpy} \citep{2020Natur.585..357H}, \textsc{scipy} \citep{2020NatMe..17..261V} \section*{Data Availability} \addcontentsline{toc}{section}{Data Availability} A python interface to integrate orbits and access the expansion model for the Erkal19 simulation can be found here: \url{https://github.com/sophialilleengen/mwlmc}. \bibliographystyle{mnras} \bibliography{MWLMC.bib} % \clearpage \appendix \section{Astrophysical fore- and back-ground components} \label{app:astro_comp} We here provide more details about the astrophysical fore- and back-ground components used to fit the gamma-ray sky. \begin{itemize} \item \textbf{interstellar emission (IE)}: A diffuse Galactic gamma-ray source emerging due to very-high energy, charged cosmic rays impinging on particles of the MW's interstellar medium, dust and radiation fields. The dominant processes that create gamma rays are $\pi^0$-decay, Bremsstrahlung and IC scattering. The modelling of this contribution is subject to many uncertainties so that we include five different models that aim to quantitatively characterise the IE. Two IE models (henceforth called \emph{Lorimer I} and \emph{Lorimer II}) are taken from the set of realisations considered in the ``1st Fermi LAT Supernova Remnant Catalog''\footnote{The model files have been made public by the \Fermi-LAT collaboration at: \url{https://fermi.gsfc.nasa.gov/ssc/data/access/lat/1st_SNR_catalog/}.} \cite{Acero:2015prw}. While a detailed description of the models' preparation and their respective properties are given in the cited catalogue publication, we restrict ourselves to a brief summary of their main characteristics relevant to this work: The distribution of primary cosmic rays is linked to the distribution of pulsars as analysed and discussed in \cite{Lorimer:2006qs}. They are confined in a volume with propagation height $z=10$ kpc whereas the spin temperature of the interstellar medium is assumed to be $T_s = 150/1\cdot10^5$ K (Lorimer I/II). The interstellar medium's gas content is split in atomic hydrogen \ce{H} and carbon monoxide \ce{CO} maps. The latter serves as a proxy for the distribution of molecular hydrogen \ce{H2}. The total volume of the gas maps is decomposed into four Galactocentric annuli with respective extensions: 0-4 kpc: ``ring 1'', 4-8 kpc: ``ring 2'', 8-10 kpc: ``ring 3'' and 10-30 kpc: ``ring 4''. Since the IE from different rings contributes to different parts of the gamma-ray sky, we make use of this decomposition in our fitting strategy outlined in Sec.~\ref{sec:fitting_opti}. We note that the properties of IE model Lorimer I are comparable to those of the official diffuse background model of the \Fermi-LAT derived in connection with the 4FGL catalogue \cite{Fermi-LAT:2019yla,2020arXiv200511208B}.Therefore, we declare Lorimer I our benchmark IE model.\linebreak The three remaining models -- called \emph{foreground model A, B} and \emph{C} -- are adopted\footnote{All model files are stored in the \Fermi-LAT collaboration's public data archive: \url{https://www-glast.stanford.edu/pub_data/845/}.} from a careful and detailed study of the diffuse extragalactic gamma-ray background conducted by the \Fermi-LAT collaboration \cite{Ackermann:2014usa}. In contrast to Lorimer I and II, these IE instances exhibit the advantage to be directly prepared for the study of a large-scale emission component at high Galactic latitudes -- a feature aligned with the aim of our analysis. Again, we refer to the cited publication to learn more about the exact composition of the models. \item \textbf{isotropic diffuse gamma-ray background (IGRB)}: This large-scale contribution to the gamma-ray sky is spatially isotropic, hence following the spatial structure of the LAT's exposure, and generated by the collective emission of distant extragalactic gamma-ray emitters too faint to be resolved individually. We adopt the IGRB spectrum associated with the selected LAT data's event class and type (see Sec.~\ref{sec:lat_data_selection}) provided by the Fermi Science Tools\footnote{The IGRB spectrum files are also provided at \url{https://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html}}. We stress that these spectra are only valid in combination with the official diffuse background model of the \Fermi-LAT collaboration. Since our fit model (Eq.~\ref{eq:model_eq}) renormalises each background component per energy bin, this restriction is, however, irrelevant for our cause. \item \textbf{resolved point-like and extended gamma-ray sources}: Based on a 10-year data set, the \Fermi-LAT collaboration has published an extensive catalogue, 4FGL-DR2, \citep{Fermi-LAT:2019yla,2020arXiv200511208B} of all resolved and localised point-like or extended gamma-ray emitters in or outside of the MW. We include all of these sources with their respective properties reported in the catalogue in our analysis. A detailed description of the treatment of these sources in each analysis step is given in the following Sec.~\ref{sec:fitting_opti}. \item \textbf{Fermi Bubbles (FB)}: An extended, hourglass-shaped diffuse gamma-ray source, which is present above and below the Galactic disc up to high Galactic latitudes. We adopt the spatial morphology of the FBs as derived in \cite{TheFermi-LAT:2017vmf} whereas its spectrum is taken to be a log-parabola $\frac{\mathrm{d}N}{\mathrm{d}E}=F_{0}\left(\frac{E}{E_{0}}\right)^{-\alpha-\beta\ln\!{\left(E/E_{0}\right)}}$ described by the parameters $F_{0}=5\times10^{-10}\;\mathrm{ph}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{MeV}^{-1}$, $\alpha$ = 1.6, $\beta$ = 0.09 and $E_{0}$ = 1 GeV as reported in \cite{Herold:2019pei}. \item \textbf{LoopI}: A large-scale, loop-like structure exhibiting a diffuse gamma-ray emission mainly concentrated in the northern hemisphere above the Galactic disc. We include in our model the spatial and spectral characterisation of LoopI given in \cite{Wolleben:2007pq}. This particular model also features a non-vanishing contribution in the southern hemisphere of the gamma-ray sky. \item \textbf{gamma-ray emission induced by the Sun and Moon}: Since the LAT is in constant observation mode, both the Sun and the Moon cross the field of view of the telescope. Along their orbital trajectory they contribute a non-negligible gamma-ray emission. We make use of dedicated routines within the Fermi Science Tools\footnote{A technical description of the software developed for this task is given at: \url{https://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/solar_template.html}} to derive a flux model for the gamma-ray emission of these two celestial bodies tailored towards the selected LAT data (see \cite{2013ICRC...33.3106J}). \end{itemize} \section{Selected bright 4FGL-DR2 sources} \label{app:bright_4FGL} We report in Tab.~\ref{tab:bright_4fgl} all sources in 4FGL-DR2 whose energy flux (100 MeV - 100 GeV) is above the threshold of $4\cdot10^{-10}\;\left[\mathrm{MeV}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\right]$. These sources are considered as individual templates during the iterative fit (see Sec.~\ref{sec:fitting_opti}) designed to derive a baseline model of the gamma-ray sky solely comprised of the astrophysical background components introduced in Sec.~\ref{sec:bkg_selection}. We state their name in 4FGL-DR2, their position in Galactic coordinates, the nominal energy flux and the source class using abbreviations from \cite{Fermi-LAT:2019yla}. \begin{table*} \centering \begin{tabular}{l l c c} \hline 4FGL source name & $\left(\ell, b\right)\;\left[^{\circ}\right]$ & $E_{100}\;\left[\mathrm{MeV}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\right]$ & source class\\ \hline \hline 4FGL J0835.3-4510 & $\left(263.6, -2.8\right)$ & $9.4\cdot10^{-9}$ & PSR\\ 4FGL J0633.9+1746 & $\left(195.1, 4.3\right)$ & $4.2\cdot10^{-9}$ & PSR\\ 4FGL J0534.5+2200 & $\left(184.6, -5.8\right)$ & $1.4\cdot10^{-9}$ & PSR\\ 4FGL J1709.7-4429 & $\left(343.1, -2.7\right)$ & $1.4\cdot10^{-9}$ & PSR\\ 4FGL J2028.6+4110e & $\left(79.6, 1.4\right)$ & $1.1\cdot10^{-9}$ & SFR\\ 4FGL J2253.9+1609 & $\left(86.1, -38.2\right)$ & $1.0\cdot10^{-9}$ & FSRQ\\ 4FGL J2021.5+4026 & $\left(78.2, 2.1\right)$ & $8.4\cdot10^{-10}$ & PSR\\ 4FGL J1836.2+5925 & $\left(88.9, 25.0\right)$ & $6.2\cdot10^{-10}$ & PSR\\ 4FGL J2021.1+3651 & $\left(75.2, 0.1\right)$ & $5.3\cdot10^{-10}$ & PSR\\ 4FGL J2232.6+1143 & $\left(77.4, -38.6\right)$ & $4.9\cdot10^{-10}$ & FSRQ\\ 4FGL J1855.9+0121e & $\left(34.7, -0.4\right)$ & $4.9\cdot10^{-10}$ & SNR\\ 4FGL J0240.5+6113 & $\left(135.7, 1.1\right)$ & $4.7\cdot10^{-10}$ & HMB\\ 4FGL J1256.1-0547 & $\left(305.1, 57.1\right)$ & $4.5\cdot10^{-10}$ & FSRQ\\ 4FGL J0617.2+2234e & $\left(189.0, 3.0\right)$ & $4.5\cdot10^{-10}$ & SNR\\ 4FGL J1809.8-2332 & $\left(7.4, -2.0\right)$ & $4.4\cdot10^{-10}$ & PSR\\ 4FGL J0007.0+7303 & $\left(119.7, 10.5\right)$ & $4.3\cdot10^{-10}$ & PSR\\ 4FGL J1104.4+3812 & $\left(179.8, 65.0\right)$ & $4.2\cdot10^{-10}$ & BLL\\ 4FGL J1745.6-2859 & $\left(359.95, -0.04\right)$ & $4.2\cdot10^{-10}$ & spp\\ 4FGL J1512.8-0906 & $\left(351.2, 40.1\right)$ & $4.2\cdot10^{-10}$ & FSRQ\\ 4FGL J0534.5+2201i & $\left(184.6, -5.8\right)$ & $4.1\cdot10^{-10}$ & PWN\\ \hline \end{tabular} \caption{Summary table listing bright 4FGL-DR2 sources with an energy flux $E_{100} \geq4\cdot10^{-10}\;\left[\mathrm{MeV}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\right]$ that are fit individually during the iterative fit procedure aimed to derive a baseline model of the gamma-ray sky. The table states the source's name in 4FGL-DR2, its position in Galactic longitude $\ell$ and latitude $b$ in degree, the nominal energy flux $E_{100}$ and the source class using abbreviations from \protect\cite{Fermi-LAT:2019yla}.\label{tab:bright_4fgl}} \end{table*} \section{Injected dark matter signal recovery} \label{app:signal_recovery} In order to test the performance of the analysis pipeline outlined in Sec.~\ref{sec:fitting_opti} and to verify its trustworthiness, we conduct a simple signal recovery exercise: Utilising the baseline fit with IE model Lorimer I on the all-sky gamma-ray data, we inject a DM signal into the model and try to detect it with the chosen inference method, namely the log-likelihood ratio test statistic. The test statistic in Eq.~\ref{eq:TS_stat_reach} is not suited for this task since it is designed to constrain an alternative hypothesis when the data seems to prefer the background-only hypothesis. Thus, we modify the test statistic for the case of determining the detection significance according to \citep{Cowan:2010js}: \newpage \begin{widetext} \begin{equation} \label{eq:TS_discovery} \qquad\qquad\qquad\qquad\qquad\qquad\textrm{TS}_{\mathrm{discovery}}= \begin{cases} -2\min_{\{N_i^{B_j}\}}\left(\ln\!\left[\frac{\mathcal{L}_w\!\left(\left.\bm{\mu}(N^{\mathrm{DM}} = 0,N_i^{B_j}) \right|\bm{n}\right)}{\mathcal{L}_w\!\left(\left.\bm{\hat{\mu}}\right|\bm{n}\right)}\right]\right)\, & \hat N^{\mathrm{DM}} \geq 0\\ 0 & \hat N^{\mathrm{DM}} < 0\rm{,} \end{cases} \end{equation} \end{widetext} where we keep the notation established in Sec.~\ref{sec:stat_framework}. Succinctly put, this test statistic assumes that the preferred hypothesis contains a positive signal with a positive best-fit normalisation $\hat N^{\mathrm{DM}}$. The alternative hypothesis, in contrast, becomes the background-only hypothesis, that is, finding the best-fit background parameters under the assumption that $N^{\mathrm{DM}} = 0$. The likelihood ratio thus quantifies the significance of the detected signal and for a given threshold one may claim a detection at the $\cdot\sigma$ level. We have applied this prescription to ascertain that no significant signal is present in the selected LAT data for all scenarios provided in the main text. Hence, deriving upper limits on the WIMP DM pair-annihilation cross section is justified. For the purpose of our pipeline check, we conduct the following adapted approach: \begin{itemize} \item For the monopole and full BFE MW + LMC haloes of the Erkal19 simulation, we prepare a signal template featuring a DM particle with mass $m_{\chi} = 100\unit{GeV}$ annihilating into $b\bar{b}$ pairs. \item For discrete points in the range of annihilation cross section $N^{\mathrm{DM}} = \langle\sigma v \rangle \in \left[10^{-28}, 10^{-23}\right]\unit{cm}^3\unit{s}^{-1}$, we inject the signal template with normalisation $N^{\mathrm{DM}} $ into the baseline fit of Lorimer I. \item Drawing 200 Poisson realisations of the baseline fit + signal gamma-ray sky, we perform a maximum likelihood fit with respect to this mock data and save the retrieved best-fit signal strength $\hat N^{\mathrm{DM}}$. \item Again, using 200 Poisson realisations of the baseline fit + signal gamma-ray sky, we derive the upper limit on the DM annihilation cross section that we may set given the presence of a signal in the utilised mock data. \end{itemize} The results of this sanity check are displayed in Fig.~\ref{fig:signal_recovery_100GeV}, which shows the median/scatter of the recovered signal normalisation (red solid/shaded band) and the median/scatter of the upper limits (green solid/shaded band) as a function of the injected signal's strength. To guide the eye, we also denote the upper limit on this particular particle DM model derived from the baseline fit without injected signal as orange line; its $68\%$ containment band is given as an orange-shaded band. The general expectation for this kind of pipeline check is to recover the injected signal strength; the higher the signal strength the higher the confidence of recovering the signal, i.e.~the smaller the observed scatter of the best-fit parameter. Moreover, for extremely small annihilation cross sections the upper limits with respect to mock data that contain the signal should asymptotically approach the corresponding upper limits with respect to the baseline fit standalone. Both of these features we can confirm with our sanity check. In addition, we find that the solid red line approaches the dashed black line for cross section values below the nominal upper limit marked by the vertical orange line. This suggests that the pipeline is capable of theoretically detecting a signal less luminous than the obtained upper limit. At the same time, the solid green line starts to deviate from the horizontal orange line when the injected signal strength can be recovered. We can hence conclude that the constructed analysis pipeline works as intended and it is suitable to perform the envisaged tasks. \section{Dark matter constraints for different interstellar emission models} \label{app:iem_limits} The main results in Sec.~\ref{sec:upperlimits} of our analysis of the MW-LMC system rely on the choice of a particular benchmark IE model. However, it is well known that the characterisation of the interstellar emission is not perfect and a great deal of data- and observation-driven as well as simulated models have been put forward to achieve a sufficiently good agreement between reality and theoretical understanding plus observational data. On one hand, since our study mostly focuses on the high-latitude in the southern hemisphere of the gamma-ray sky, our results are less severely affected by the IE directly related to the emission originating from the interstellar medium along the Galactic disc. On the other hand, gamma-ray emission due to IC scattering events is particularly hard to model and exhibits large uncertainties so that the impact of the IE model on our results is certainly non-negligible. As motivated in Sec.~\ref{sec:bkg_selection}, we aim to assess the systematic uncertainty of our results due to the modelling of the IE via alternative models that supplement the benchmark choices for either the outer MW halo study or the dedicated study of the LMC surroundings. In Fig.~\ref{fig:iem_uncertainty_main}, we provide a set of $95\%$ C.L.~upper limits on the DM pair-annihilation cross section (again, with respect to the channel $\chi\chi\rightarrow b\bar{b}$) for different choices of the IE model. The left panel of this figure concerns the impact of the IE on the study of the outer MW halo, where we exemplify the performance of all five IE models with the signal morphology according to the \Jf-factor map of the Erkal19 simulation including all terms of the BFE for both DM haloes. We checked that the choice of the simulation is not important to quantify the impact of the IE uncertainty; the other four simulations as well as the static MW halo from \cite{2017MNRAS.465...76M} yield very similar results. Using a different IE model than Lorimer I as in the main text, requires us to re-perform the ROI optimisation, which yields optimal ROI sizes detailed in Tab.~\ref{tab:iem_rois} for the respective case. As concerns the comparison of upper limits derived from real data and the baseline fit, we find consistency for almost the entire probed DM mass range at the $2\sigma$ level and even better for FGMA to FGMC; an expected behaviour since these three models were initially created to facilitate searches at high latitudes. A slight fluctuation to stronger constraints for light DM with masses below $\lesssim 20\unit{GeV}$ is a common feature among all probed IE models. Overall, the numerical values of the derived constraints are comparable to the corresponding upper limits for the same simulation set in Fig.~\ref{fig:LMC_impact_on_MW} but IE model Lorimer I. The apparent differences certainly partially arise because of the varying optimal ROI sizes that reduce or increase the total \Jf-factor. The right panel of Fig.~\ref{fig:iem_uncertainty_main} addresses the question of the IE model's impact on the study of the LMC region. As explained in Sec.~\ref{sec:lat_analysis_LMCroi}, we include for this particular task the Galactic diffuse background model of the \Fermi-LAT collaboration. Besides, we do not show the upper limits for FGMB, as it appears to behave exactly like FGMC, and Lorimer II, which does not provide a good fit to the data in general. In this case, we use the simulation based on a Hernquist profile for the MW and an NFW profile for the LMC as initial conditions from Ref.~\cite{Donaldson+2022}. This example suffices to quantify the impact of the IE modelling as the remaining LMC halo models exhibit similar behaviour. Although we display the set of upper limits for each particular IE model on the same plot, the underlying optimised ROI sizes are different and all applied values are given in Tab.~\ref{tab:lmc_rois_iem}. However, the differences in those sizes are not noticeably altering the total \Jf-factor we are probing so that a common plot is justified. As it turns out, varying the IE has a remarkably small effect on the resulting constraints on WIMP DM. The only exception being the IE model Lorimer I, which induces a deterioration of the limits by a factor of $\sim2$ compared to the other three IE model instances. Hence, the results of the dedicated LMC study are robust against variations in the IE modelling. \begin{table*} \begin{centering} \begin{tabular}{c c c} \hline \multirow{3}{*}{IE model} & only monopole & multipole expansion\tabularnewline \cline{2-3} \cline{3-3} & ROI size & ROI size\tabularnewline & \Jf-factor {[}GeV$^2$cm$^{-5}${]} & \Jf-factor {[}GeV$^2$cm$^{-5}${]}\tabularnewline \hline \hline \multirow{2}{*}{Lorimer I} & $27.2^{\circ}\times 27.2^{\circ}$ & $27.2^{\circ}\times27.2^{\circ}$ \tabularnewline & $1.38\times10^{20}$ & $1.62\times10^{20}$\tabularnewline \hline \multirow{2}{*}{\Fermi-LAT IE model (v07) } & $28.4^{\circ}\times28.4^{\circ}$ & $29.0^{\circ}\times29.0^{\circ}$\tabularnewline & $1.42\times10^{20}$ & $1.68\times10^{20}$\tabularnewline \hline \multirow{2}{*}{FGMA} & $29.5^{\circ}\times29.5^{\circ}$ & $29.5^{\circ}\times29.5^{\circ}$\tabularnewline & $1.45\times10^{20}$ & $1.70\times10^{20}$\tabularnewline \hline \multirow{2}{*}{FGMC} & $29.4^{\circ}\times29.4^{\circ}$ & $29.4^{\circ}\times29.4^{\circ}$\tabularnewline & $1.45\times10^{20}$ & $1.79\times10^{20}$\tabularnewline \hline \end{tabular} \par\end{centering} \caption{ Summary of the optimised ROI sizes for the dedicated study of the LMC environment by varying the underlying IE model. The optimisation has been performed under the assumption of the LMC halo shapes according to the simulation in Ref.~\protect\cite{Donaldson+2022} using an Hernquist (MW) and NFW (LMC) profile as initial conditions. \label{tab:lmc_rois_iem}} \end{table*} \begin{table*} \begin{center} \begin{tabular}{lc} \hline IE model & optimal ROI size \\ \hline \hline Lorimer I & $\ell\in\left[-167^{\circ},167^{\circ}\right]$ and $b\in\left[-90^{\circ},-35^{\circ}\right]$ \\ Lorimer II & $\ell\in\left[33^{\circ},327^{\circ}\right]$ and $b\in\left[-90^{\circ},-30^{\circ}\right]$ \\ FGMA & $\ell\in\left[-165^{\circ},165^{\circ}\right]$ and $b\in\left[-90^{\circ},-35^{\circ}\right]$ \\ FGMB & $\ell\in\left[21^{\circ},339^{\circ}\right]$ and $b\in\left[-90^{\circ},-35^{\circ}\right]$ \\ FGMC & $\ell\in\left[25^{\circ},335^{\circ}\right]$ and $b\in\left[-90^{\circ},-35^{\circ}\right]$ \\ \hline \end{tabular} \caption{Summary of the optimised ROI sizes for the five IE models used in the analysis of the outer MW halo. These optimised ROIs are derived with respect to the spatial morphology of the MW-LMC system found in the Erkal19 including the dynamical response of both DM haloes. \label{tab:iem_rois}} \end{center} \end{table*} \section{Dark matter constraints for an alternative annihilation channel} \label{app:tau_results} In this appendix, we present our results for an alternative spectral DM model. Instead of the rather soft $b\bar{b}$-channel, we here assume 100\% annihilation into $\tau^+ \tau^-$ pairs that yield a hard annihilation spectrum. We repeat the analysis with respect to the optimised ROIs to re-derive the main results that have been shown in the main text in Fig.~\ref{fig:LMC_impact_on_MW} for the MW-LMC system and in Fig.~\ref{fig:LMC_ROI_study} for the LMC standalone. The corresponding constraints for the $\tau^+\tau^-$-channel are provided in Fig.~\ref{fig:results_tautau}. As a general observation, the upper limits for DM particles with masses above a few tens of GeV are weaker compared to the $b\bar{b}$-channel. This result is reasonable given the fact that the $\tau^+ \tau^-$ annihilation spectrum is harder, which shifts the most constraining part of the spectrum to higher energies. At energies of a few GeV, \Fermi-LAT analyses tend to be statistics limited. As a consequence, their constraining power regarding hard spectra peaking above $\mathcal{O}(10)$ GeV is reduced. \section{Dark matter constraints for an enlarged region of interest} \label{app:newROI_limits} In the main text of this work, we focused on determining an ROI in the southern hemisphere with optimal properties regarding the compatibility of the statistical expectations derived from our baseline fit and the true \Fermi-LAT data set. On one side, the dynamics of the MW-LMC system generate anisotropies that are located in this particular fraction of the entire sky as illustrated by Figs.~\ref{fig:jfactor_comparison} and \ref{fig:jfactor_comparison_LMC}. On the other side, such deformations -- mostly associated with the LMC halo -- are likewise present in the northern hemisphere. Their impact on indirect gamma-ray searches for DM in the outer MW halo remains unprobed. To investigate the potential of an extended ROI that incorporates patches in both hemispheres of the gamma-ray sky, we fix the optimal southern ROI derived for the benchmark IE model Lorimer I and re-perform the optimisation routine in the northern sky. With respect to the example of a static MW halo referenced in Fig.~\ref{fig:McMillan_example_ROI}, we obtain the best match between expectation from mock data and performance on real \Fermi-LAT data for an ROI defined by $-102^{\circ} < \ell < 102^{\circ}$ and $b > 16^{\circ}$. The quality of this accordance is, however, not as good as for the southern hemisphere standalone. We quantify the suitability of the obtained extended ROI in Fig.~\ref{fig:McMillan_example_extended_ROI} by confronting the DM upper limits from true \Fermi-LAT data (red) with the corresponding constraints (black) and their scatter from mock data. The region in the parameter space where we can exclude thermal DM enlarges by a factor of about two compared to Fig.~\ref{fig:McMillan_example_ROI}. However, the derived bounds are not within the $2\sigma$ containment band of the upper limits derived from the baseline fit in a broad range of the probed DM parameter space. This reduces the credibility of this set of upper limits and it furthermore indicates that our baseline astrophysics model is not entirely describing reality in such an extended sky region. In order to exploit the full potential of searches towards the outer MW halo it is thus essential to improve the current state-of-the-art of astrophysical models at high latitudes; in particular, the gamma-ray emission of extended diffuse components as the FBs or loop-like structures and the residual IE. As concerns the sensitivity of such indirect searches to deformations of the MW and LMC haloes, we find ROIs of similar extensions in the northern hemisphere for the Erkal19 simulation. The statistical robustness that these sky regions provide is comparable to the case shown in Fig.~\ref{fig:McMillan_example_extended_ROI}. Translated to constraints on the DM annihilation cross section, we do not observe an enhanced difference between a signal morphology following the monopole term or the full BFE of the Erkal19 model. The effect of the MW-LMC dynamics remains at the level displayed in Fig.~\ref{fig:LMC_impact_on_MW}. To stress it again, the reliability of this statement is suffering from the lack of a good fit to reality. Improving the astrophysical background modelling at high latitudes may also improve the sensitivity to the dynamical response of the LMC passage through the MW. \bsp % \label{lastpage}

Title: The entropy of galaxy spectra: How much information is encoded?

Abstract: This paper approaches the inverse problem of extracting the stellar population content of galaxy spectra from a basic standpoint based on information theory. By interpreting spectra as probability distribution functions, we find that galaxy spectra have high entropy, caused by the high correlatedness in wavelength space. The highest variation in entropy is unsurprisingly found in regions that have been well studied for decades with the conventional approach. Therefore, we target a set of six spectral regions that show the highest variation in entropy - the 4,000 Angstrom break being the most informative one. As a test case with real data, we measure the entropy of a set of high quality spectra from the Sloan Digital Sky Survey, and contrast entropy-based results with the traditional method based on line strengths. The data are classified into star-forming (SF), quiescent (Q) and AGN galaxies, and show - independently of any physical model - that AGN spectra represent a transition between SF and Q galaxies, with SF galaxies featuring a more diverse variation in entropy. The high level of entanglement complicates the determination of population parameters in a robust, unbiased way, and affect traditional methods that compare models with observations, as well as machine learning and deep learning algorithms that rely on the statistical properties of the data to assess the variations among spectra. Therefore, caution must be exercised when retrieving detailed population parameters or even star formation histories from galaxy spectra.

https://export.arxiv.org/pdf/2208.05489

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} methods: data methods -- methods: statistical -- techniques: spectroscopic -- galaxies: stellar content \end{keywords} \section{Introduction} \label{sec:intro} \label{Sec:Intro} Stars and gas constitute the most fundamental observables in extragalactic astrophysics. While the gaseous component, in its different guises, mostly reflects the ongoing physical state, the stellar populations encode valuable information about the past formation history, both through their collisionless nature -- that keep track of the past dynamical history -- and their age and chemical composition -- that trace the past star formation history. The standard approach to study the underlying stellar population focuses on comparisons of photo-spectroscopic observables with population synthesis models \citep[to name a few,][]{BC93,Wo:94,BC03,Pegase,Maraston:05,Vazdekis:10,FSPS:10}, that combine our understanding of stellar formation and evolution, along with a determination (empirical, theoretical or mixed) of the stellar atmospheres \citep[see, e.g.][for a general view of these models]{Walcher:11,Conroy:13}. There is a large number of papers developing methods of extracting information from the observations focusing on the determination of stellar age, metallicity and targeted abundance ratios \citep[see, e.g.,][]{JJG:93,Wo:94b,Trager:00,Thomas:05,Gallazzi:05,Graves:09,FLB:13}. These models range from approaches based on the concept of a simple stellar population (uniquely defined by a stellar initial mass function and age, along with a fixed chemical composition) to complex mixtures spanning a range of those parameters. The most detailed models invoke comparisons of targeted line strengths (such as the Lick system, and variations thereof) or full spectral fitting \citep[e.g.][]{SL:05,STECMAP,pPXF:12}. However, all these methods are hampered by the so-called age-metallicity-dust degeneracy whereby changes in age can mimic the effects on the photometric and spectroscopic information of a change in chemical composition \citep{Wo:94}. Most notably, this degeneracy is present at all scales regarding spectral resolution, with similar behaviour in broadband photometry and spectroscopy \citep{FCS:99}. It is in fact a major paradox that the amount of information that can be gathered either from a few broadband colours, or from thousands of fluxes in a spectrum is rather comparable, reflecting the strong correlation of spectral features due to the underlying astrophysics. Alternatively to model comparison methods, other works adopted multivariate techniques aimed at disentangling the information via principal component analysis \citep{Ronen:99,HCGPCA:06,Rogers:07}, factor analysis \citep{Nolan:07}, independent component analysis \citep{Kaban:05}, Fisher information matrices \citep{HJL:00,MOPED:03}, or the Information Bottleneck \citep{IB:01,IF:12}. While these methods only depend on the input data and are not affected by potential systematics of the synthesis models, the results are rather similar in the sense that only very generic constraints can be retrieved regarding the star formation histories. More recently, similar results have been obtained with deep learning methods based on convolutional neural networks \citep[see, e.g.,][]{Lovell:19,Portillo:20,CLing:21,Teimoorinia:22}, where the algorithms are only capable of robustly deriving a few, general properties from galaxy spectra. This paper performs a fundamental theoretical analysis focused on the interpretation of galaxy spectra as an information carrier, by redefining a spectrum as a probability distribution function, with its associated entropy, following the standard approach of information theory. Entropy encodes the amount of 'surprise' in the outcome of a given system. In spectra, this can be naturally defined when the acquisition of a spectrum is interpreted as a photon counting experiment. In this case, the simplest scenarios are represented by a laser beam and a white light source. In the former, the entropy will be very low, given its highly monochromatic nature. In contrast, 'white' light will produce maximum entropy. Throughout this paper we often link entropy to ``information content'', which was the traditional approach of \citet{Shannon:53}. This is the most fundamental way of assessing information, as in, e.g., determining the number of bits needed for a full representation of the data. Subsequent definitions of information content focus instead on cross-entropy \citep[see, e.g.][]{IB:01,MacKay:03} involving an additional set of data or a classification scheme. Extraction of population parameters from galaxy spectra depends on diverse practical issues: spectral resolution, modeling line widths, etc, that vary between data sets and plague model comparisons. Our proposal to use entropy content provides the most fundamental approach possible for assessing the information content of galaxy spectra and the ultimate limits attainable for inferring the parametric nature of the stellar populations. We compare the results from models of population synthesis with actual galaxies from the Sloan Digital Sky Survey (SDSS). We stress that our aim here is not to suggest new spectral regions or to supersede the traditional approach based on model comparisons, but to explore the inverse problem of extracting stellar population parameters from galaxy spectra from a standpoint based on information theory. As we will see, the analysis illustrates the inherent entanglement that explains why highly sophisticated approaches, such as Deep Learning, can only produce generic descriptions of the stellar content. We emphasize that any methodology aimed at the derivation of detailed properties about the stellar populations from galaxy spectra will be subject to this fundamental limit concerning information content. The paper has a concise structure, and is organised as follows: the basic methodology based on entropy is presented in Sec.~2, followed by an application to real galaxies from the SDSS in Sec.~3. Finally, our conclusions are given in Sec.~4 followed by an epilogue that presents a similar outlook with a more standard treatment based on covariance. \section{The entropy of galaxy spectra} \label{Sec:Hglx} The best way to illustrate the connection between spectroscopy and information theory is to consider the measurement of a spectrum as a photon counting process. The specific flux density can then be interpreted as a conditional probability distribution. For a galaxy $g$, the flux at wavelength $\lambda$ is thus: \begin{equation} \Phi_g(\lambda) \quad \Longrightarrow \quad p(\lambda | g). \end{equation} This formalism was introduced by \citet{IB:01} to motivate a classification algorithm based on the so-called ``information bottleneck'', whereby a sample of galaxy spectra is subject to an agglomerative binning procedure based on the evolution of entropy, as spectra are progressively binned into classes. The mutual information contained in the class representation of the original set serves as a target to achieve a description of the data with the maximum amount of information encoded into the smallest number of classes. In this paper, we do not follow this approach. However, this interpretation allows us to exploit the concept of entropy in galaxy spectra as a fundamental way to quantify the way information is stored across the spectral window. The Shannon definition of entropy for a probability distribution is \citep{SW:75}: \begin{equation} H(\Phi) \equiv -\sum_i p(\lambda_i)\log p(\lambda_i). \label{eq:H1} \end{equation} Hereafter, we denote the entropy with the letter $H$, following standard notation. This methodology allows us, for instance, to quickly identify the language of a given text, by use of the frequency of the different letters of the alphabet, leading to the definition of the entropy of a language \citep{Shannon:51}. In our case, the goal is to identify the most likely star formation history corresponding to a given spectrum. In contrast with the traditional methods based on spectral fitting or targeted line strengths \citep[see, e.g.,][]{Walcher:11}, we focus here on the information content as quantified by the entropy. We begin with a simple test determining $H$ in a set of well-known and tested population synthesis models. These models incorporate our knowledge of the formation and evolution of stars, as well as the radiative properties of stellar atmospheres that ultimately give rise to spectra \citep{Conroy:13}. These models are typically produced for a reduced set of variables, mainly stellar initial mass function, age, total chemical composition (metallicity), and possibly a number of non-solar abundance ratios, most notably [Mg/Fe]. At present, state of the art models rely on a reduced set of libraries of stellar spectra \citep[see, e.g.,][]{STELIB,Coelho:05,MILES}, so that -- from the pure point of view of information theory -- any synthetic model, no matter how complex, is always defined by a linear combination of a relatively reduced set of stellar spectra, of order $10^3$. This is the reason why information-based models can easily discriminate between synthetic and ``real'' spectra, as shown in \citet{IB:01} where the information content from observed 2dFGRS spectra was found to be much higher than the one derived from theoretical galaxy formation models that incorporated population synthesis\footnote{Also noting that noise -- inhererent to any observational data -- will affect the information content.} (see figure~3 in that paper). Fig.~\ref{fig:Ht} shows the entropy defined within a relatively wide spectral window, between 3,500 and 8,000\AA, normalized such that a totally uninformative scenario -- i.e. a constant $p(\lambda)$ in this interval -- corresponds to $H=1$, and the extreme case of a perfectly monochromatic signal, i.e. $p(\lambda)=\delta(\lambda-\lambda_0)$ has $H=0$. The simple stellar population models (SSPs) of \citet{BC03} have been used here to explore the variation with respect to stellar age (horizontal axis), with two choices of metallicity, as labelled. We adopt the \citet{Chabrier:03} stellar initial mass function in this analysis, but the differences are minimal for other reasonable choices. From an information theory point of view, the first salient feature of galaxy spectra is the closeness to a fully uninformative scenario: the deviation from the maximum entropy case is less than one part in $\sim$30. However, this is caused by the large spectral interval chosen and by the fact that, aside from absorption features (that usually amount to less than a few Angstrom in equivalent width), the continuum is a fairly smooth function of wavelength, with a relatively shallow gradient. Moreover, the trend found with respect to age stems from this: the younger populations are dominated by hotter stars with steeper, and blue, continua, whereas older populations have a much shallower wavelength dependence except for the prominent 4,000\AA\ break. In order to be able to discriminate better among the models, we measure the entropy within a narrower spectral window. This is the equivalent of defining individual probability distributions within prescribed wavelength intervals: $p(\lambda | [\lambda_1,\lambda_2])$, where only photons with wavelength between $\lambda_1$ and $\lambda_2$ are considered in the definition of the probability distribution function. We can also do this experiment adopting a running interval, of width $\Delta\lambda$, as we traverse the optical window, defining an ``entropy spectrum'', $H_{\Delta\lambda}(\lambda)$. The choice of $\Delta\lambda$ is important: too wide and we wash out all information, too narrow and the result becomes prohibitively dependent on the signal to noise ratio or velocity dispersion. At the resolution of SDSS galaxy spectra, around R$\sim$2,000, \citet{Rogers:10} found that $\Delta\lambda\sim$50-100\AA\ provides an optimal window, a result that is equally optimized for stellar spectra in SDSS \citep{Hawkins:14}. Fig.~\ref{fig:Hlam} shows the ``entropy spectrum'' of a few synthetic populations from the models of \citet{BC03}, using a running interval of width $\Delta\lambda$=100\AA. As comparison, we show the corresponding entropy spectrum for a sample of real galaxy spectra from SDSS in Fig.~\ref{fig:HlamSDSS} -- showing the median of subsamples classified according to their nebular emission properties (Q: quiescent, SF: star-forming, AGN: active nucleus). We will explore this sample in more detail in \S\ref{Sec:SDSS}. Hereafter, we quote estimates of entropy with the following definition: \begin{equation} \overline{\cal H}(\lambda)\equiv [1- H_{\Delta\lambda=100A}(\lambda)]\times 1,000, \label{eq:H2} \end{equation} which can be interpreted as negentropy, or information content. The maximally uninformative case corresponds to $\overline{\cal H}$=0. The factor 1,000 is chosen from the actual range of entropy values measured in galaxy spectra. This figure provides more insight into the information encoded in galaxy spectra. At redder wavelengths, the spectrum becomes less informative, as it is dominated by a continuum with relatively weak absorption features, except for some prominent regions, as labelled. The strongest contribution comes from the region around 4,000\AA, where the pile up of spectral lines produces a prominent break, with the entropy reaching its lowest value. Note how this blind approach to the information content of spectra produces the standard features targeted for the analysis of stellar populations: the Balmer regions at 4,100\AA\ (H$\delta$); 4,340\AA\ (H$\gamma$); 4,862\AA\ (H$\beta$) and 6,563\AA\ (H$\alpha$) are more prominent in younger populations around 1\,Gyr, a well-known result caused by the dominance of A-type stars to the net luminosity budget in populations with that age. Older populations produce conspicuous features around 4,300\AA\ (G band) and 5,100-5,400\AA, where a number of Mg and Fe absorption features are very prominent. The optical Na absorption at 5,890\AA\ is also present in the entropy spectra. These features have been intensively explored in galaxy spectra for decades, so their presence is not surprising. However, our analysis confirms that these regions are the ones that carry most of the information -- as in negentropy -- from a pure information theory approach. This result is especially relevant to Deep Learning algorithms, where the complex ``machines'' critically depend on the information content of the spectra to classify or learn about the underlying components. In essence, machine learning methods rely on a quantitative assessment, based on some figure of merit, whose maximisation leads either to a decision (in a classification algorithm), or to a set of best-fit parameters (in a regression approach). The success of any ML method -- either supervised or unsupervised -- will ultimately depend on how this figure of merit can discriminate among the basis of sources. These sources correspond here to the generic base set of simple stellar populations that describe any star formation history. Blind source separation methods also operate on the same basis \citep[see, e.g.,][]{ICA}, namely that the original sources that produce the final data carry enough information to be told apart from the observed mixtures. Whilst our approach does not optimally fit specific line strengths given a set of models, the analysis of entropy marks an intrinsic limit regarding the ability of any such method to unambiguously extract the details of the underlying stellar populations from spectroscopy. This paper is focused on how information is stored, and can be retrieved, from galaxy spectra in the most fundamental way based on entropy. Therefore, we simplify the methodology by selecting six targeted regions that correspond to the highest values of negentropy in this spectral window, avoiding H$\beta$ and H$\alpha$, that are strongly affected by emission from the gas component, and therefore bias any analysis focused on stellar populations. Entropy in each of these six intervals is measured within the $\Delta\lambda$=100\AA\ window. The vertical dotted lines mark those regions -- adopted below for the analysis of real galaxy spectra from SDSS (see Sec.~\ref{Sec:SDSS}). Fig.~\ref{fig:H2D} shows the (neg)entropy of a set of SSPs at fixed solar metallicity, for a range of ages, within the spectral window where these six regions are defined. $\overline{\mathcal{H}}$ is encoded as a grey scale, and appears at all ages strongest in the vicinity of the 4,000\AA\ break. The other features are significantly weaker, with some becoming more relevant at young ages ($\sim$0.5-1\,Gyr, Balmer absorption) and others more prominent at older ages ($\gtrsim$2\,Gyr, Mgb and Fe regions). This approach allows us to produce a simplified description of each galaxy spectrum with only six ``coordinates'', that represent the regions that carry the maximum amount of variation in their entropy. It is worth noting that the interpretation of a galaxy spectrum as a probability distribution of the energy of the incoming photons is affected by two observational pitfalls: the signal-to-noise ratio and the effective spectral resolution. The former depends on the efficiency of the observing apparatus and the exposure time, whereas the latter depends both on the spectrograph as well as the galaxy under scrutinity, as the distribution of stellar orbits impose an effective ``kinematic kernel'' that broadens -- via Doppler shifts -- all spectral features. Fig.~\ref{fig:Hvd} illustrates the effect on the entropy, with the noiseless case at the fiducial resolution of the stellar models shown as a red line. A reasonable S/N still produces manageable results, with the shaded region spanning the range of entropy produced by a bootstrap that adds Gaussian noise to the model spectra, corresponding to S/N=5 per \AA. A lower spectral resolution -- here shown by producing the equivalent observation of a (massive) galaxy with velocity dispersion of 300\,km/s (dark red) degrades the entropy quite dramatically. As expected, lower resolution (or higher velocity dispersion) produces smoother spectra, with less defined absorption features, approaching a homogeneous probability distribution, and therefore tending towards maximum entropy, i.e. minimum information. For those reasons, the analysis presented in the next section -- comprising actual galaxy spectra -- will constrain the data to high S/N and relatively low velocity dispersion. Fig.~\ref{fig:HRes} shows how entropy changes with respect to spectral resolution within the six features defined here. The results are shown for two population synthesis models at two different stellar ages (both with solar chemical abundance), as labelled. The \citet{BC03} models are shown as solid lines, and the MIUSCAT models \citep{MIUSCAT} are plotted with dashed lines. We emphasize that these models are independent, and even adopt different stellar libraries. Note the trends are equivalent in both sets of models, although there are some systematic variations caused by the different choice of model prescriptions and stellar libraries. For reference, we include an estimate of the 1\,$\sigma$ uncertainties if the spectra have a S/N of 20\,\AA$^{-1}$, as a shaded region for the 10\,Gyr BC03 model. Noting that R$\sim$2,000 is an optimal spectral resolution for galaxy spectra given the typical velocity dispersion of the stellar component, we find that the information content -- defined as the entropy -- gets halved at R$\sim$500-1000 (varying with the spectral index). The left part of the horizontal axis corresponds to the resolution expected in slitless grism spectroscopy or medium band photometry, where the discriminatory power of spectral indices is reduced -- in this case, the continuum is used instead, to constrain the population parameters \citep[see, e.g.][]{PEARS:09,SHARDS,LDG:15}. \section{The entropy content of SDSS galaxy spectra} \label{Sec:SDSS} The trends presented in the previous section have been obtained with synthetic models of stellar populations. In this section, we apply the same methodology to a set of high quality galaxy spectra from the Sloan Digital Sky Survey \citep[SDSS,][]{York:00}. We use the original, legacy, survey that consists of single fibre spectroscopy at resolution ${\cal R}\sim$2,000. From the original dataset (we retrieve the data from their Data Release 16, \citealt{DR16}), we select a sample of 76,570 spectra with a high S/N ($\gtrsim$10\,\AA$^{-1}$, measured in the $r$ band), constrained in velocity dispersion between $\sigma$=100 and 150\,km/s. It is a well-known fact that the stellar velocity dispersion -- a tracer of the gravitational potential -- correlates strongly with population properties \citep[e.g.,][]{Bernardi:03,SAMIGrad}. By choosing a relatively reduced range in $\sigma$, we aim at simplifying the working sample. Our goal in this paper is to explore how well entropy can disentangle subtle differences in stellar populations, so a focused data set is preferred. The median redshift is z=0.079 with a 95\% interval $\Delta$z=[0.054,0.099]. The spectra are corrected for foreground dust contamination using the dust extinction law of \citet{Fitz:99}, are brought to the rest-frame, and are resampled to a common wavelength grid with 1\AA\ spacing. In each spectrum, bad pixels flagged as problematic by the SDSS team -- that represent a very small fraction in these high S/N data -- were replaced by their best model fits, also provided within the data structure of the SDSS spectra. Fig.~\ref{fig:HlamSDSS} shows the negentropy spectrum of SDSS data, i.e. the measurement of $\overline{\mathcal{H}}(\lambda)$ (eq.~\ref{eq:H2}) with a running $\Delta\lambda$=100\AA\ interval, as in Fig.~\ref{fig:Hlam}, which was presented for a few simple stellar populations from the synthetic models of \citet{BC03}. The spectra are median-stacked into three classes, defined according to nebular emission as quiescent (Q, red), star-forming (SF, blue) and AGN (green). These three classes represent a fundamental transition during the evolution of galaxies, from an initial star-forming stage (in the Blue Cloud) to passive evolution in a quiescent phase (in the Red Sequence). AGN activity appears to dominate the transitioning stage (mostly populating the intermediate region, termed the Green Valley). These regions stem from the inherent bimodality of galaxy properties \citep{Strateva:01, Salim:14}. Therefore, by splitting the sample according to this classification scheme, we can assess how such a transition affects the entropy of galaxy spectra. The classification follows the standard BPT diagram \citep{BPT}, and is taken from the official galspecExtra SDSS catalogue \citep{MPA_JHU}, where we include an additional constraint on the equivalent width of H$\alpha$ emission to select quiescent galaxies \citep[e.g.][]{CF:11} -- as the bpt=$-$1 flag only refers to galaxies whose spectra cannot be mapped on the standard BPT diagram. The real spectra show a similar behaviour to the models, except at the location of prominent emission lines such as H$\beta$, [O{\sc III}], H$\alpha$+[N{\sc II}], and [S{\sc II}]. We focus on the most relevant features of the negentropy spectra, as found in the models, namely 4000\AA\ break strength (termed $D4K$), two Balmer absorption lines (H$\gamma$, H$\delta$) and the strongly metallicity dependent indices Mgb, $\langle$Fe$\rangle$ and NaD \citep[see, e.g.][]{WO:97,Trager:98,Balogh:99}. Fig.~\ref{fig:SDSSHist} compares the distribution of these features as measured with the traditional line strength approach (top, given as equivalent widths, following the methodology of \citealt{Rogers:10}), the measurement of negentropy ($\overline{\mathcal{H}}$), as defined in eqs.~\ref{eq:H1} and \ref{eq:H2}, for the same set of spectral windows (middle), and given by relative entropy, also termed Kullback-Leibler divergence, D$_{\rm KL}$, \citealt{Dkl} (bottom). The colour coding separates subsamples based on nebular emission (Q: red, SF: blue, AGN: green). The definition of relative entropy rests on the idea of the information content of a probability distribution, $p(\lambda)$, with respect to another one that acts as baseline, $q(\lambda)$. In this case, entropy refers to the amount of information relative to the baseline: \begin{equation} D_{\rm KL} \equiv -\sum_i p(\lambda_i)\log \left[p(\lambda_i)/q(\lambda_i)\right]. \label{eq:Dkl} \end{equation} Note that when $q(\lambda)$ is constant (i.e. a totally uninformative baseline), we recover Shannon entropy. Here we use as reference the median of all spectra, taking the galaxies from the whole sample, regardless of nebular emission (i.e. including Q, SF and AGN), to define the median $q(\lambda)$. Therefore, relative entropy in this paper is a measure of departure from the overall behaviour of the general sample. The most significant difference between SF, AGN and Q subsamples in Fig.~\ref{fig:SDSSHist} is obtained for D4K, and the SF subset is the one with the widest range of values in all cases. Comparing the standard line strengths with negentropy, we find some analogous results (D4K, Mgb, $\langle$Fe$\rangle$), whereas Balmer absorption behaves differently: while the line strength analysis finds a larger variation regarding nebular activity in H$\delta$, negentropy discriminates better with H$\gamma$. In contrast, relative entropy does not produce such clear segregation, with only a more extended tail in the distribution of SF galaxies. It would be difficult to look for differences between the distributions in this six dimensional parameter space. Therefore, once we quantify each individual spectrum by six numbers, we proceed to further reduce the dimensionality of parameter space via Principal Component Analysis (PCA). PCA provides a simple way to visualize the variance of the data by diagonalizing the corresponding covariance (represented here by a $6\times 6$ matrix). The eigenvectors, or principal components, are ranked in decreasing order of their associated variance, and the original data are projected onto these eigenvectors to produce an alternative representation that is ranked as a function of variance. \begin{table} \caption{Normalized PCA eigenvalues.\label{tab:scree}} \begin{center} \begin{tabular}{cccc} \hline Rank & \multicolumn{3}{c}{Eigenvalue ratio}\\ & Line Strength & Entropy & D$_{\rm KL}$\\ \hline 1 & 0.50022 & 0.87499 & 0.75479\\ 2 & 0.25449 & 0.06983 & 0.09225\\ 3 & 0.11622 & 0.02057 & 0.05972\\ 4 & 0.09691 & 0.01687 & 0.04688\\ 5 & 0.02616 & 0.01478 & 0.03986\\ 6 & 0.00600 & 0.00296 & 0.00650\\ \hline \end{tabular} \end{center} \end{table} We apply PCA independently to the data from the three methods described above, finding the eigenvalues shown in Table~\ref{tab:scree}. They are quoted as a ratio of total variance, and show that the standard line strength analysis produces the most mixed distribution, needing up to four components to account for 90\% of total variance, whereas the first principal component for standard entropy already carries $\sim$87\% of total variance, with the second one accounting for around 7\% and the subsequent components featuring a lower share. Relative entropy also shows a high weight of the first principal component, accounting for $\sim$75\% of total variance. Fig.~\ref{fig:PCA_weight} is a graphic representation of the weights of the first three principal components, where symbol size denotes the weight of a given spectral feature, as labelled in the horizontal axis, and the colour means positive (red) or negative (blue) sign for each weight. The principal components derived from the standard line strength data (LS) correspond to mixtures of Balmer absorption (mostly in PC1) and metallicity dependent strengths (PC2). In contrast, the components corresponding to Shannon (neg)entropy ($\overline{\mathcal{H}}$) produce a simpler scheme, where PC1 is mostly represented by D4K and PC2 is dominated by the information in the Balmer absorption lines (H$\delta$ and H$\gamma$). The third component -- whose associated variance is only 2\% -- depends mostly on H$\gamma$, with residual dependence on the other features, most notably H$\delta$ and NaD. The relative entropy representation (D$_{\rm KL}$) produces a mixed set of weights, with PC1 mostly having a strong dependence on D4K and Balmer absorption (both H$\delta$ and H$\gamma$ being represented). The second component is dominated by H$\delta$, with a substantial dependence on D4K (although with negative sign, in contrast to PC1). The third component mostly relates to H$\gamma$ with some dependence on D4K and NaD. Surprisingly, the metallicity dependent features (Mgb and $\langle$Fe$\rangle$) play a minor role in these components, which may be caused by the strong correlation among spectral indices (see Epilogue below). Overall, it is worth emphasizing that the bulk of the entropy variation found in both representations refer to the 4000\AA\ break strength and H$\delta$. This type of scheme has been extensively used in standard analyses of stellar populations in galaxies \citep[e.g.,][]{Kauff:03} as the 4000\AA\ break indicator is overall sensitive to the average age and metallicity, whereas H$\delta$, or H$\gamma$ trace recent episodes of star formation \citep[see, e.g.][]{Kauff:14, Wu:18}. In this paper, we retrieve this result purely from the information point of view, without any reference to the modelling of stellar populations. Fig.~\ref{fig:PCA_conts} shows the distribution of projections of the observed spectra on the first two principal components for line strength data and both definitions of entropy, as labelled. The contours map the density from 50\% to 80\% of total number of datapoints (in steps of 10\%), and are plotted separately with respect to nebular activity, as above, into star-forming (blue), quiescent (red), and AGN (green). We emphasize that the PCA output depends only on data in spectral regions where emission line activity is absent or weak, so that most of the variance cannot be directly ascribed to their classification into SF, Q and AGN. The standard line strength data (LS, top) separate the subsamples along a diagonal line in PC1-PC2 space, with AGN galaxies located between the SF and Q subsets. Similarly, Shannon entropy (middle) shows substantial differences between the SF and the Q subsets, with the AGN group populating an intermediate area, something that hints at nuclear activity representing a transition from star formation to quiescence, as already proposed in the literature \citep[see, e.g.,][]{Schawinski:07, Salim:14}. This diagram suggests that the ``horizontal direction'', i.e. PC1 (mainly dependent on D4K for $\overline{\mathcal{H}}$), is the main discriminator, supporting the definition of the green valley, i.e. transitioning galaxies from star formation to quiescence, using this spectral feature \citep{JA:19, JA:20}. In contrast, relative entropy (bottom) does not separate the spectra so well, and only produces an extended tail of star-forming galaxies towards high, positive values of PC1. Differences within this diagram (PC1 vs PC2 projections) are explored in more detail in the next figure for entropy and relative entropy. Fig.~\ref{fig:histogH} shows the distribution of some observables when taking subsets corresponding to the 5th (teal colour) and 95th (brown) percentiles of the distribution of PC1 projections (from negentropy, $\overline{\mathcal{H}}$) for the star-forming (top), quiescent (middle) and AGN (bottom) samples. Fig.~\ref{fig:histogDkl} is the equivalent for the PC1 projections of relative entropy. The segregation according to negentropy produces substantial differences in PC1 with respect to D4K, Mgb and $\langle$Fe$\rangle$, regardless of nebular emission properties. Relative entropy gives a more homogeneous picture of Q galaxies, that show a weaker dependence on the observables. Most of the diversity, in terms of information content, is found in the sample of star-forming galaxies. We should stress that the parent sample is restricted to a relatively narrow range of stellar velocity dispersion (100-150\.km/s), so that the distributions are as ``homogeneous'' as possible, concerning variations derived from the general mass-age and mass-metallicity trends \citep[see, e.g.,][]{Gallazzi:05}. The entropy analysis confirms the well-known view that the Q sample is more homogeneous, and that the AGN subsample sits between this one and the more diverse set of SF spectra. It is worth pointing out that this analysis also produces a segregation in the distribution of SNR -- a property only dependent on the technical aspects of the observations -- in many instances, except for SF galaxies. This is an important trend that all methods based on deep / shallow learning should take into account. Taken at face value, it would imply that an important part of the variance/information/signal one can extract from spectra could be due to technical details of the data acquisition process. Note that such segregation is not present in velocity dispersion, stellar mass, or redshift, for $\overline{\mathcal{H}}$, although relative entropy also shows a redshift and velocity dispersion dependence for Q and AGN galaxies. This result illustrates the need to select clean, well-defined samples in any statistical blind search of clues to the underlying star formation history of galaxies. In addition, these methods can be applied to separate out the observational effects from the physical ones -- as, in, e.g., the removal of night sky lines \citep{WH:05}. \section{Conclusions} \label{Sec:Conc} This paper focuses on the fundamental limitations of extracting information from galaxy spectra to derive star formation histories. This work is orthogonal to the standard approach based on comparisons of stellar population synthesis models with the observational data via full spectral fitting or targeted line strengths. There is a vast literature devoted to the traditional methodology that perform careful analyses to mitigate the inherent degeneracies among the population properties \citep[to name a few:][]{MOPED:03,SL:05,STECMAP,VESPA:07,FSPS:10,ULySS:11,pPXF:12,FLB:13,Firefly}. However, it is not the goal of this paper to assess these methods, but to explore the limitations at a more fundamental level, based on the information content (here defined as negentropy, alternatively defined as variance, as in PCA-based work, e.g., \citealt{Rogers:07}). The essence of the problem lies in the information content of the spectra, that we base in this paper on entropy, either taking the fundamental definition of \citet{SW:75} or relative entropy as defined by the Kullback-Leibler divergence \citep{Dkl}. While alternative methods are adopted in the literature to define information content, entropy represents the 'building block' as it directly relates to, e.g. how many bits of information are needed for a full representation of the data \citep{Shannon:53}. Entropy lies at the core of all methods aimed at blind classification of galaxies, such as Principal Component Analysis, Independent Component Analysis, Cross-entropy methods, factor analysis, convolutional neural networks, etc. We explore both synthetic models and real galaxy spectra from SDSS, to find that a reduced set of spectral regions encode most of the information, unsurprisingly tracing the traditional Lick system \citep[e.g.][]{Trager:98} commonly used in most studies. The definition of entropy is initially determined for the full spectral range of interest, say the optical window, and then narrower intervals can be explored. Fig.~\ref{fig:H2D} suggests that stellar populations feature a reduced set of well-defined regions where the information content is highest. Overall, the entropy of galaxy spectra is rather high -- as in uninformative -- reflecting the difficult task of constraining the details of the underlying populations, regardless of the apparently high number of data points in each spectra. Applying this method to SDSS galaxy spectra confirms this high entropy (Fig.~\ref{fig:HlamSDSS}), with a clear, and fully expected, dependence on nebular emission, with AGN systems representing an intermediate stage between star forming and quiescent galaxies. A detailed analysis based on PCA applied to the entropy estimates suggest that the 4000\AA\ break strength and Balmer absorption are the most important sources of entropy variation, with metal-dependent indicators being subdominant. This trend confirms previous work that focuses on, e.g. the D$_n$(4000) vs H$\delta$ bivariate diagram as a fundamental tool in the analysis of star formation histories \citep[see, e.g.][]{Kauff:03}. Intriguingly, Figs.~\ref{fig:histogH} and \ref{fig:histogDkl} show that S/N may also affect the analysis (even at S/N$\gtrsim$10\,\AA$^{-1}$), which implies that blind methods are highly susceptible to the quality of the data. Our results suggest that detailed estimates of star formation histories are hampered by the sizeable covariance of the spectral elements, which result in an effectively low number of ``degrees of freedom'', regardless of the large number of data units in a spectrum. Somehow, this addresses the long-known fact that a reduced set of high quality colours based on broadband photometry can produce constraints on the stellar populations that are comparable with the analogous study at high spectral resolution. For instance, the derivation of stellar masses from photometry is comparable with the equivalent analysis making use of spectra -- once the redshift is well known \citep[e.g.][]{Santini:15}. Moreover, non-solar abundance ratios produce variations over large spectral windows, especially at shorter wavelengths \citep[e.g.][]{AV:15}, so that broad- and medium-band photometry, in principle, carry such detailed information. This would make surveys such as J-PAS (\citealt{JPAS}, featuring 56 filters with 150\AA\ bandwidth), or slitless grism spectroscopy (with e.g. ACS or WFC3 at the Hubble Space Telescope, NIRISS at the JWST or NISP at Euclid) as informative, from the point of view of stellar populations, as the more expensive spectral surveys. Studies of entropy variations {\sl within} spectral data of the same galaxy -- from Integral Field Unit observations -- can also be exploited to understand radial variations of the underlying stellar populations, a concept that will be explored in future work. \section{Epilogue: A note about fitting stellar population data} \label{Sec:Cov} The entropy-based approach presented here illustrates the challenge of extracting a unique solution from this particular inverse problem: from a set of observed galaxy spectra, we need to derive the fundamental components, i.e. the base stellar populations that lead to the underlying star formation histories. While signal extraction from superpositions is a relatively easier task in, say, audio files or superposition of independent time series, galaxy spectra reveal a high level of entanglement regarding the information content as a function of wavelength, that prevents us from extracting details of the age distribution or chemical composition even in spectra with the highest quality. In this epilogue, we show a simple exercise that makes use of the optimal set of six spectral windows presented in this paper, defined by the entropy content (see Fig.~\ref{fig:H2D}). Fig.~\ref{fig:cov} shows the covariance matrix of a set of 16,386 simple stellar populations covering a wide range of ages (0.5$<$t$_{\rm SSP}$$<$14\,Gyr) and metallicities ($-$1.5$<$log\,Z/Z$_\odot$$<$$+$0.3), from the models of \citet{BC03}. Other sets of models produce very similar results. Each of the six indices is renormalized to unit variance, to be able to compare all indices in the same way. Note that all indices are correlated or anticorrelated at a rather high level, amounting to $\gtrsim$90\% of the intrinsic variance. Also note that these indices have been produced as having most of the information content, or variance, across the optical window, and are typically used in most analyses of stellar population in the literature. For instance, the 4000\AA\ break strength is often associated to an overall stellar age, and Balmer absorption reveals episodes of star formation within the past $\sim$1\,Gyr. However, the otherwise strong metallicity sensitive indices Mgb, $\langle$Fe$\rangle$ and NaD are strongly correlated with the former, as expected from the well-known age-metallicity degeneracy \citep[see, e,g,][]{Wo:94b,FCS:99}. Observational errors and spectral resolution will obviously affect the outcome, but this simple test focuses on a perfect observation, taking the models at face value and assuming zero velocity dispersion (only limited by the inherent resolution of the stellar library of the models). The off-diagonal terms of the 6$\times$6 covariance matrix are quite significant. This result should be taken as a note of caution in the interpretation of any statistic that produces a figure of merit based on the comparison of observations with population synthesis models. The most commonly used one, $\chi^2$, is expected to produce a minimum at the best fit, with a value roughly similar to the number of degrees of freedom \citep{Andrae:10}. If we use the more standard definition of the $\chi^2$ statistic for a generic covariance $\mathbb{C}$: \begin{equation} \chi^2(\pi) \equiv \Big ( {\bf x} - {\bf m(\pi)}\Big )^T\cdot\mathbb{C}^{-1}\cdot \Big ({\bf x} - {\bf m(\pi)}\Big ), \end{equation} where ${\bf x}$ represents the measurements and ${\bf m}$ the model output for a specific set of parameters ($\pi$). Perfectly uncorrelated data -- the assumption most often invoked -- implies a covariance $\mathbb{C}=\mathbb{I}$, and typically yields a minimum $\chi^2$ around 6, whereas the covariance estimated from the model grid produces a minimum $\chi^2$ of $\sim$30, clearly inconsistent with the assumption that the six line strengths are independent units of information. At face value, the substantially higher $\chi^2$ when accounting for covariance suggests that the fitting procedure effectively involves fewer degrees of freedom than the six (or more) spectral features. Therefore, even though we may have a large number of observable constraints from galaxy spectra, we can only produce a rough estimate of stellar age, metallicity and possibly some non-solar abundance ratios. This is not only applicable to line strength analysis, but to full spectral fitting, where each 'pixel', i.e. a flux unit within a relatively narrow spectral window ($\Delta\lambda$$\sim$1\,\AA), is highly correlated with most of the other pixels in the spectrum within the typical NUV-optical-NIR spectral interval used in such analyses. Highly targeted studies of line strengths are perhaps the only effective way to constrain the stellar population content of galaxies, whereas methods based on full spectral fitting should take into consideration this important caveat about covariance, and should not make the assumption that the large number of flux measurements are independent, a result that would falsely lead to a high constraining power. \section*{Acknowledgments} IF acknowledges support from the Spanish Research Agency of the Ministry of Science and Innovation (AEI-MICINN) under the grant with reference PID2019-104788GB-I00. OL acknowledges STFC Consolidated Grant ST/R000476/1 and a Visiting Fellowship at All Souls College, Oxford. This paper was meant to be produced during a visit to the Flatiron Institute (FI), but it was not possible due to the SARS-CoV-2 pandemic. Nevertheless, IF warmly thanks the FI for offering to host his visit. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. \section*{Data availability} This work has been fully based on publicly available data: galaxy spectra were retrieved from the SDSS DR16 archive (https://www.sdss.org/dr16/) and stellar population synthesis models can be obtained from the respective authors. \bsp % \label{lastpage}

Title: Delay-Weighted Calibration: Precision Calibration for 21 cm Cosmology with Resilience to Sky Model Error

Abstract: One of the principal challenges of 21 cm cosmology experiments is overcoming calibration error. Established calibration approaches in the field require an exquisitely accurate sky model, and low-level sky model errors introduce calibration errors that corrupt the cosmological signal. We present a novel calibration approach called Delay-Weighted Calibration, or DWCal, that enables precise calibration even in the presence of sky model error. Sky model error does not affect all power spectrum modes equally, and DWCal fits calibration solutions preferentially from error-free modes. We apply this technique to simulated data, showing that it substantially reduces calibration error in the presence of realistic levels of sky model error and can improve 21 cm power spectrum sensitivity by approximately 2 orders of magnitude.

https://export.arxiv.org/pdf/2208.04406

command. \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \newcommand\fitparams{\boldsymbol{\xi}} \newcommand\fitparam{\xi} \newcommand\freq{f} \newcommand\delay{\eta} \newcommand\visfunc{\boldsymbol{\zeta}} \newcommand\visfuncscalar{\zeta} \newcommand\freqparams{\boldsymbol{\gamma}} \newcommand\freqparam{\gamma} \newcommand\thermalcov{\boldsymbol{\mathsf{C}}_\text{T}} \newcommand\modelcov{\boldsymbol{\mathsf{C}}_\text{M}} \newcommand\modelcovscalar{C_\text{M}} \newcommand\data{\boldsymbol{v}} \newcommand\datascalar{v} \newcommand\fisherinfo{\boldsymbol{\mathsf{I}}} \newcommand\fisherinfoscalar{I} \newcommand\gains{\boldsymbol{g}} \newcommand\gain{g} \newcommand\gainsmat{\boldsymbol{\mathsf{G}}} \newcommand\fitparamsu{\boldsymbol{u}} \newcommand\fitparamu{u} \newcommand\modelvals{\boldsymbol{m}} \newcommand\modelval{m} \newcommand\thermalvar{\sigma_\text{T}^2} \newcommand\thermalvarvector{\boldsymbol{\sigma}_\text{T}^2} \newcommand\modelvar{\sigma_\text{M}^2} \newcommand\modelvarvector{\boldsymbol{\sigma}_\text{M}^2} \newcommand\matrixa{\boldsymbol{\mathsf{A}}} \newcommand\ascalar{A} \newcommand\uvcoord{\boldsymbol{u}} % \newcommand\beamscalar{B} \newcommand\beamvec{\boldsymbol{B}} \newcommand\beammat{\boldsymbol{\mathsf{B}}} \newcommand\jonesmat{\boldsymbol{\mathsf{J}}} \newcommand\jonesscalar{J} \newcommand\coherencyscalar{S} \newcommand\coherency{\boldsymbol{S}} \newcommand\electricfield{E} \newcommand\vismat{\boldsymbol{\mathsf{V}}} \newcommand\negloglikelihood{\chi^2} \newcommand\racoord{\text{R}} \newcommand\deccoord{\text{D}} \newcommand\muellermat{\boldsymbol{\mathsf{M}}} \newcommand\unitvec{\boldsymbol{e}} \newcommand\basistransform{\boldsymbol{\mathsf{K}}} \newcommand\muellerbasistransform{\boldsymbol{\mathsf{L}}} \newcommand\rotangle{\beta} \newcommand\rotmeas{R} \newcommand\wavelength{\lambda} \newcommand\skypos{\boldsymbol{\theta}} \newcommand\holographplane{\boldsymbol{P}_\text{holo}} \newcommand\uniformplane{\boldsymbol{P}_\text{uni}} \newcommand\weights{\boldsymbol{W}} \newcommand\numvis{\boldsymbol{N}} \newcommand\fillingfactor{C} \newcommand\obsindex{n} \newcommand\radweight{W} \newcommand\weightmat{\boldsymbol{\mathsf{W}}} \newcommand\gainamp{A} \newcommand\note[1]{\textcolor{red}{#1}} \newcommand\groundcoord{\boldsymbol{r}} \newcommand\antres{F} \newcommand\antresponse{\epsilon} \newcommand\antresmat{\boldsymbol{\mathsf{F}}} \newcommand\unnormgains{\boldsymbol{h}} \newcommand\unnormgain{h} \newcommand\modelcorr{\boldsymbol{\mathsf{C}}_\text{R}} \newcommand\modelcorrscalar{C_\text{R}} \newcommand\efield{E} \newcommand\transferfunc{T} \newcommand\transferfuncmat{\boldsymbol{\mathsf{T}}} \newcommand\ewinstpol{\text{p}} \newcommand\nsinstpol{\text{q}} \newcommand\parallacticangle{\phi} \newcommand\raval{\alpha} \newcommand\decval{\delta} \newcommand\noise{n} \newcommand\ft{\mathcal{FT}} \newcommand\re{\operatorname{Re}} \newcommand\im{\operatorname{Im}} \newcommand\overallphase{\Delta} \newcommand\argument{\operatorname{Arg}} \newcommand\overallamp{A} \newcommand\phasegradx{\Delta_x} \newcommand\phasegrady{\Delta_y} \newcommand\relcalgains{\boldsymbol{h}} \newcommand\relcalgain{h} \newcommand\crosspolphase{\Delta} \newcommand\costcomp{r} \usepackage{commath, graphicx, bbold, endnotes, graphicx, subfigure, multirow, afterpage} \begin{document} \title{Delay-Weighted Calibration: Precision Calibration for 21 cm Cosmology with Resilience to Sky Model Error} \author[0000-0003-4980-2736]{Ruby Byrne} \affiliation{Astronomy Department, California Institute of Technology, 1200 E California Blvd, Pasadena, CA, 91125, USA} \section{Introduction} \label{s:intro} Interferometric measurement of 21 cm emission from neutral hydrogen at high redshift has great potential for mapping large volumes of the universe, probing the Cosmic Dawn and Epoch of Reionization (EoR), and measuring large-scale structure evolution throughout cosmological history (see \citealt{Furlanetto2006}, \citealt{Pritchard2008}, \citealt{Morales2010}, and \citealt{Liu2020} for reviews). The success of 21 cm cosmology experiments is currently limited by the precision with which bright intervening emission, or foregrounds, can be separated from the faint cosmological signal. While the foreground emission is a staggering 4-5 orders of magnitude brighter than the 21 cm signal, it is very spectrally smooth and therefore can, in principle, be distinguished from the spectrally structured cosmological signal. However, the success of this approach requires extremely precise frequency-dependent, or bandpass, calibration. Low-level spectral errors in calibration introduce spectral structure into the foreground signal, preventing effective foreground removal. In recent years, this problem has inspired a proliferation of precision bandpass calibration methods \citep{Mitchell2008, Yatawatta2009, Kazemi2011, Sullivan2012, Kazemi2013, Salvini2014, Sievers2017, Dillon2020, Sob2020, Kern2020, Byrne2021b, Ewall-Wice2021, Sims2022, Sims2022b}. Nevertheless, precision calibration approaches generally require exquisite prior knowledge of the sky signal and instrumental response, and calibration precision remains a principal limitation of 21 cm cosmology analyses. We introduce a novel approach to bandpass calibration called Delay-Weighted Calibration, or DWCal, that combats spectral calibration error. DWCal imposes no prior assumptions about the instrument's bandpass response and can accurately calibrate even when the instrumental bandpass has substantial frequency structure. It fits the same number of free calibration parameters as traditional sky-based calibration, avoiding the computational challenges and over-fitting concerns associated with many-parameter calibration approaches. DWCal can be combined with other advanced calibration techniques such as redundant calibration or unified calibration. DWCal specifically combats a mechanism of calibration error by which errors in the calibration sky model introduce spurious spectral structure into the calibration solutions. Missing or mis-modeled sources and structures in the calibration sky model produce bandpass calibration errors, even when both the true sky signal and the sky model are intrinsically spectrally smooth. For low levels of sky model error, the resulting calibration error is nonetheless at a level that precludes a detection of the cosmological 21 cm signal \citep{Barry2016, Trott2016, Ewall-Wice2017, Byrne2019}. DWCal joins a suite of techniques designed to combat this error mechanism. First is simply the development of highly accurate and complete sky models \citep{Yatawatta2013a, Offringa2015, Carroll2016, Hurley-Walker2017, Patil2017, Eastwood2018, deGasperin2020, Byrne2022a}. These have substantially improved calibration performance, but some degree of sky model error is nonetheless unavoidable. Next, assuming stable and uniform antenna responses, calibration benefits from averaging across antennas and time \citep{Barry2016, Kern2020}. If the intrinsic antenna response is relatively spectrally smooth, the calibration solutions can be averaged across frequency or fit to a simple function such as a low-order polynomial \citep{Yatawatta2015, Barry2016, Dillon2018, Gehlot2018, Barry2019b, Li2019, Eastwood2019, Mertens2020}. Short baselines produce less spectral contamination in calibration, so calibration to short baselines only can improve calibration performance \citep{Ewall-Wice2017, Patil2017}. Taking this idea to its extreme, \citealt{Barry2019b} and \citealt{Li2019} calibrate with baselines of zero length, using the autocorrelation visibilities to fit the antennas' bandpass responses. Other calibration approaches address this error mechanism by incorporating additional calibration parameters to directly fit the sky signal. Redundant calibration, applicable to highly regular arrays, constrains parameters of the sky signal by matching repeated baseline measurements, thereby reducing reliance on the sky model \citep{Wieringa1992, Liu2010, Grobler2018, Dillon2018, Li2018, Dillon2020, Kern2020, Zhang2020}. However, regular arrays are particularly sensitive to sky model error, and they experience substantial bandpass calibration error from low-level sky model error even when redundantly calibrated \citep{Byrne2019}. \citealt{Ewall-Wice2021} directly fits the spectrally smooth components of the sky signal measured by each baseline, fitting out much of the sky model error at the cost of a large increase in calibration parameters. The DWCal framework takes a new approach to tackling the problem of sky model error in calibration. Interferometric measurements reconstruct spectrally smooth signals in a compact region of 2D power spectrum space (here the two dimensions refer to power spectrum modes parallel and perpendicular to the line-of-sight, respectively). This region is known as the ``foreground wedge'' and has been explored in depth in the literature \citep{Morales2012, Trott2012, Vedantham2012, Pober2013, Thyagarajan2013, Hazelton2013, Dillon2015, Morales2019}. Consequently, sky model error appears within this foreground wedge region. DWCal exploits this feature, incorporating knowledge of the foreground wedge to downweight contaminated power spectrum modes in calibration. Unlike short baseline calibration or autocorrelation calibration, DWCal calibrates with all baseline measurements, employing the full measurement set to constrain the calibration solutions. However, each baseline preferentially constrains the modes on which it is unaffected by sky model error. It assumes that the sky model error is compact in the 2D power spectrum space but makes no such assumption of the instrumental response, so it can fit any instrumental spectral features. It does not require averaging across antennas or time and supports arrays with heterogeneous antenna responses and time-dependence. DWCal does not introduce any additional calibration degeneracies and has the same number of degenerate parameters as sky-based calibration (see \S\ref{s:regularization} for a discussion of these degeneracies). While it is not immune to sky model error --- DWCal still requires a good sky model --- we show that it is significantly more robust against sky model error than established precision calibration techniques. DWCal introduces cross-frequency constraints on the gains. This means that---like calibration approaches such as those discussed in \citealt{Yatawatta2015}, \citealt{Mertens2020}, \citealt{Ewall-Wice2021}, and \citealt{Sims2022}---DWCal cannot be parallelized across frequency. Instead, the gains at all frequencies must be jointly calculated. This paper introduces a simple implementation of DWCal, demonstrating the principles of the technique. This implementation can be considered an extension to simple sky-based calibration. We apply DWCal to simulated data, demonstrating that it improves calibration performance in the presence of sky model error. Future work could optimize the DWCal algorithm to reduce its computational cost. It could also combine DWCal with other calibration techniques such as redundant calibration or unified calibration. \section{DWCal Formalism} In general, interferometric calibration consists of minimizing a cost function, often denoted $\chi^2$, to constrain parameters of the instrument response. The fundamental innovation underlying DWCal is the incorporation of delay-dependent expected error into the calibration $\chi^2$. In \S\ref{s:basic_formalism} we derive the DWCal $\chi^2$ as an extension to simple sky-based calibration. In \S\ref{s:regularization} we present a new approach to constraining calibration degeneracies, and in \S\ref{s:extensions} we discuss further extensions to DWCal. \subsection{The Basic DWCal Formalism} \label{s:basic_formalism} In its simplest form, sky-based calibration consists of calculating gain values that minimize the quantity \begin{equation} \negloglikelihood = \sum_f \sum_{jk} \left| \gain_j(f) \gain_k^*(f) \datascalar_{jk}(f) - \modelval_{jk}(f) \right|^2. \label{eq:sky_cal_simple} \end{equation} Here $f$ represents frequencies and $j$ and $k$ index antennas. $\sum_f$ indicates the sum over all frequency channels and $\sum_{jk}$ indicates the sum across all baselines. $\datascalar_{jk}(f)$ is the visibility derived by correlating antennas $j$ and $k$ at frequency $f$; $\modelval_{jk}(f)$ is a model of that visibility derived from a sky model propagated through an instrument simulator. $\gains(f)$ are the antenna gains, parameterized as a complex number for each antenna and frequency, and the asterisk $^*$ denotes the complex conjugate. We do not consider time dependence in this expression and assume a single time step. We further assume a single instrumental polarization mode. Note that here we define a calibration convention in which the gains multiply the data, not the model visibilities. This diverges from the convention typically used in the literature (see, for example, \citealt{Hamaker1996a}) for reasons delineated below. Under our convention, we calibrate data by multiplying the raw visibilities by the calculated gains. Minimizing Equation \ref{eq:sky_cal_simple} produces a maximum-likelihood estimate of the gains, assuming independent Gaussian-distributed thermal noise and model visibility error at each frequency and baseline (see \citealt{Byrne2021b} for a derivation of this expression based on the Bayesian likelihood function). Because Equation \ref{eq:sky_cal_simple} assumes uncorrelated frequency channels, it is separable in frequency. Each frequency can be calibrated independently. This simple sky-based calibration could be augmented with a frequency- and baseline-dependent weighting function. For example, we could replace Equation \ref{eq:sky_cal_simple} with an expression \begin{equation} \negloglikelihood = \sum_f \sum_{jk} \radweight_{jk}(f) \left| \gain_j(f) \gain_k^*(f) \datascalar_{jk}(f) - \modelval_{jk}(f) \right|^2, \label{eq:sky_cal_weighted} \end{equation} where $\radweight_{jk}(f)$ is a positive, scalar weighting function. $\radweight_{jk}(f)$ can be adjusted to downweight particular baselines and frequency channels with known contamination. For example, if we knew the model visibility $m_{jk}(f)$ was particularly inaccurate at frequency $f = f_0$, we could reduce the value of $\radweight_{jk}(f_0)$ to protect the calibration solutions from that error. Calibrating with Equation \ref{eq:sky_cal_weighted} is appropriate when the model visibility error appears in distinct frequency channels. However, sky model error produces error in the model visibilities at all frequencies. This model visibility error is not compact in the frequency domain but \textit{is} compact in the Fourier dual, or delay, domain \citep{Parsons2009, Parsons2012, Pober2013, Morales2019}. We therefore aim to remap the calibration problem into delay space. We return to Equation \ref{eq:sky_cal_simple}. To make our expressions more concise, we define a new quantity \begin{equation} \costcomp_{jk}(f) = \gain_j(f) \gain_k^*(f) \datascalar_{jk}(f) - \modelval_{jk}(f). \end{equation} Equation \ref{eq:sky_cal_simple} can then be written as \begin{equation} \negloglikelihood = \sum_{jk} \sum_f \left| \costcomp_{jk}(f) \right|^2. \end{equation} From the Plancherel theorem, this is equivalent to \begin{equation} \negloglikelihood = \frac{1}{N_\text{freq}} \sum_{jk} \sum_{\eta} \, \left| \widetilde{\costcomp}_{jk}(\eta) \right|^2, \label{eq:delay_space_chisquared} \end{equation} where $N_\text{freq}$ is the number of frequency channels and $\eta$ is delay, the Fourier dual of frequency $f$ with units of time. The tilde denotes the Fourier transformed quantity: \begin{equation} \widetilde{\costcomp}_{jk}(\eta) = \sum_f \costcomp_{jk}(f) e^{-2 \pi i \eta f}. \end{equation} Via the convolution theorem, \begin{equation} \widetilde{\costcomp}_{jk}(\eta) = \widetilde{\gain}_j(\eta) * \widetilde{\gain}_k^*(-\eta) * \widetilde{\datascalar}_{jk}(\eta) - \widetilde{\modelval}_{jk}(\eta), \label{eq:convolution_expression} \end{equation} where $*$ indicates the convolution. Equation \ref{eq:delay_space_chisquared} represents a delay-space reformulation of the calibration problem. We can now introduce a delay-dependent weighting function, such that \begin{equation} \negloglikelihood = \frac{1}{N_\text{freq}} \sum_{jk} \sum_{\eta} \widetilde{\radweight}_{jk}(\eta) \left| \widetilde{\costcomp}_{jk}(\eta) \right|^2. \label{eq:delay_weighting} \end{equation} $\widetilde{\radweight}_{jk}(\eta)$ can now be tuned to downweight particular delay modes for a given baseline $\{ j, k \}$. This allows us to capture the delay dependence of the model visibility error. At this point we leave the weighting function $\widetilde{\radweight}_{jk}(\eta)$ fully general. In the next section we describe development and implementation of a specific weighting function targeting model visibility error in the foreground wedge. This treatment motivates the calibration convention defined above, in which the gains multiply the data. If we were to use the convention established in \citealt{Hamaker1996a}, the model visibilities in Equation \ref{eq:convolution_expression} would be convolved with the gains. Nonzero gain values at high delay would spread model error beyond the foreground wedge modes, meaning that model error would no longer inhabit a compact delay-space region. Equation \ref{eq:delay_space_chisquared} achieves the delay-dependent weighting we desire. However, the double convolution in Equation \ref{eq:convolution_expression} makes this an unweildy expression to evaluate directly. We therefore transform back into the frequency domain. Expanding Equation \ref{eq:delay_space_chisquared} gives \begin{equation} \negloglikelihood = \frac{1}{N_\text{freq}} \sum_{jk} \sum_\eta \widetilde{\radweight}_{jk}(\eta) \bigg| \sum_f \costcomp_{jk}(f) e^{-2 \pi i \eta f} \bigg|^2, \end{equation} or \begin{equation} \negloglikelihood = \frac{1}{N_\text{freq}} \sum_{jk} \sum_\eta \widetilde{\radweight}_{jk}(\eta) \sum_f \sum_{f'} \costcomp_{jk}(f) \costcomp^*_{jk}(f') e^{2 \pi i \eta (f'-f)}. \end{equation} We can rewrite this expression as \begin{equation} \negloglikelihood = \sum_{jk} \sum_f \sum_{f'} \bigg[ \frac{1}{N_\text{freq}} \sum_\eta \widetilde{\radweight}_{jk}(\eta) e^{2 \pi i \eta (f'-f)} \bigg] \costcomp_{jk}(f) \costcomp^*_{jk}(f'), \end{equation} where the bracketed expression is simply the inverse Fourier transform of the weighting function. We then get that \begin{equation} \negloglikelihood = \sum_{jk} \sum_f \sum_{f'} \radweight_{jk}(f'-f) \costcomp_{jk}(f) \costcomp^*_{jk}(f'). \end{equation} Expanding the quantity $\costcomp_{jk}(f)$ gives \begin{equation} \begin{split} \negloglikelihood = & \sum_{jk} \sum_f \sum_{f'} \radweight_{jk}(f'-f) \\ & \times \left[\gain_j(f) \gain_k^*(f) \datascalar_{jk}(f) - \modelval_{jk}(f) \right] \\ & \times \left[\gain_j(f') \gain_k^*(f') \datascalar_{jk}(f') - \modelval_{jk}(f') \right]^*. \end{split} \label{eq:dwcal} \end{equation} Note that this can be written more concisely as a matrix multiplication operation: \begin{equation} \negloglikelihood = \sum_{jk} \left[\gains_j \gains_k^* \data_{jk} - \modelvals_{jk} \right]^\dag \weightmat_{jk} \left[\gains_j \gains_k^* \data_{jk} - \modelvals_{jk} \right], \label{eq:dwcal_matrix} \end{equation} where $\gains_j$, $\data_{jk}$, and $\modelvals_{jk}$ are each vectors of length $N_\text{freq}$. $\weightmat_{jk}$ is an $N_\text{freq} \times N_\text{freq}$ matrix, and $\dag$ denotes the conjugate transpose. Equations \ref{eq:dwcal} and \ref{eq:dwcal_matrix} take the form of a maximum-likelihood estimate with correlated frequency channels. We find that if model error is compact in delay it is necessarily not independent across frequencies. The function $\radweight_{jk}(f'-f)$ can be interpreted as encoding covariances between frequency channels. If the weighting function is appropriately set, it reduces calibration's dependence on error-prone delay modes such as the foreground wedge modes. We recover sky-based calibration in the limit that $\radweight_{jk}(f'-f) = 0$ when $f \ne f'$. \subsection{Constraining the Degenerate Phase} \label{s:regularization} Like sky-based calibration, DWCal is degenerate in the overall complex phase of the gains at each frequency. From Equation \ref{eq:dwcal}, note that for each frequency $f$ the transformation $\gains(f) \rightarrow \gains(f) e^{i \phi}$ leaves $\negloglikelihood$ unchanged, where $\phi$ is an arbitrary real constant. This degeneracy is typically constrained by setting the phase of a reference antenna's gain to zero or requiring that the average phase of the gains across all antennas be zero. This constraint can be imposed after minimizing the degenerate $\negloglikelihood$. However, this risks degrading the performance of an iterative optimization algorithm as it cannot fit to a unique minimum. We choose instead to introduce a regularization term to break the phase degeneracy. We use L2 regularization to constrain the mean phase at each frequency to be zero. Equation \ref{eq:dwcal} then becomes \begin{equation} \begin{split} \negloglikelihood = & \sum_{jk} \sum_f \sum_{f'} \radweight_{jk}(f'-f) \\ & \times \left[\gain_j(f) \gain_k^*(f) \datascalar_{jk}(f) - \modelval_{jk}(f) \right] \\ & \times \left[\gain_j(f') \gain_k^*(f') \datascalar_{jk}(f') - \modelval_{jk}(f') \right]^* \\ + & \lambda \sum_f \bigg( \sum_j \operatorname{Arg} \big[ \gain_j(f) \big] \bigg)^2, \end{split} \label{eq:dwcal_regularized} \end{equation} where the final term imposes the phase regularization. Here $\operatorname{Arg}$ denotes the complex phase and $\sum_j$ indicates the sum over all antennas. $\lambda$ is an arbitrary positive, real constant. In principle the value of $\lambda$ does not affect the global minimum of $\negloglikelihood$, as the global minimum occurs where the regularization term is identically zero. However, in practice we found that tuning $\lambda$ impacts the speed and accuracy of the optimization algorithm. Equation \ref{eq:dwcal_regularized} produces unique, non-degenerate solutions, and we use this expression to calculate the calibration solutions presented throughout this paper. Because it constrains the mean phase to be zero, we cannot calibrate a non-zero absolute phase at any frequency. \subsection{Further Extensions to DWCal} \label{s:extensions} The DWCal formalism described above and represented by Equations \ref{eq:dwcal} and \ref{eq:dwcal_regularized} can be considered an extension to traditional sky-based calibration (Equation \ref{eq:sky_cal_simple}). However, the principle behind DWCal can be applied to other, more advanced calibration techniques. DWCal can be applied to redundant calibration, specifically to the absolute calibration step that uses a sky model to constrain the bulk array response \citep{Liu2010, Dillon2018, Li2018, Kern2020}. Absolute calibration is susceptible to spectral error from low-level errors in the sky model \citep{Byrne2019}. Using the DWCal formalism to downweight the foreground wedge modes in absolute calibration could reduce the impact of sky model error on redundant calibration solutions. Unified calibration represents a middle ground between sky-based and redundant calibration \citep{Byrne2021b}. It directly fits the sky signal measured by each baseline or redundant baseline set, and it uses a sky model to impose a prior on that fit. DWCal can be applied to unified calibration by incorporated a delay-dependent weighting function into the prior. This would decrease the strength of the prior in foreground wedge modes and improve unified calibration's spectral performance. Throughout this section we have assumed calibration of a single polarization. However, DWCal can be extended to fully polarization calibration. For a dual-polarization interferometer, fully polarized calibration parameterizes the gains as a $2\times2$ polarization matrix for each antenna and frequency interval \citep{Sault1996}. If an instrument experiences negligible cross-polarization signal coupling, the gains can be more simply parameterized with two values per antenna and frequency, corresponding to the two instrumental polarizations \citep{Byrne2022b}. DWCal is fully applicable to polarized calibration, and the delay-dependent weighting function can be applied to each polarization mode. Gains need not be parameterized per-frequency. While it is common practice to fit parameters of a low-order polynomial or other simple function to the calibrated bandpass after calculating per-frequency gains \citep{Barry2016, Dillon2018, Gehlot2018, Barry2019b, Li2019, Eastwood2019, Mertens2020}, these parameters can instead be calculated directly during the $\chi^2$ minimization operation \citep{Byrne2021b}. This technique could be applied to DWCal, in which case the gains in Equation \ref{eq:dwcal} or \ref{eq:dwcal_regularized} would be considered functions of the tunable calibration parameters. However, it is critical that the gain parameterization appropriately represents the true instrumental response in order for calibration to accurately capture all spectral features. Inaccurate gain parameterizations risk introducing additional calibration error. For the remainder of this paper we focus on the simplest version of DWCal and calibrate by minimizing Equation \ref{eq:dwcal_regularized}. However, we emphasize that DWCal is not an alternative to precision calibration approaches such as redundant or unified calibration. Rather, the DWCal technique can supplement established calibration approaches to reduce spectral calibration error and build resilience to sky model error. \section{Methods} We demonstrate the DWCal technique with simulated data and a simple off-the-shelf optimization algorithm. The calibration code is written in Python and is publicly available on GitHub.\footnote{\texttt{https://github.com/rlbyrne/dwcal}} We use \textsc{pyuvdata}\footnote{\texttt{https://github.com/RadioAstronomySoftwareGroup/pyuvdata}} \citep{Hazelton2017} for interfacing with the visibility data and calibration solutions. \subsection{Visibility Simulation} \label{s:vis_sim} We simulate visibilities with \textsc{fhd}\footnote{\texttt{https://github.com/EoRImaging/FHD}} \citep{Sullivan2012, Barry2019a, Byrne2022b, Sullivan2022}. The simulation is based on the Murchison Widefield Array (MWA) Phase I configuration \citep{Tingay2013}. This array consists of 128 antennas in a non-regular imaging configuration. The \textsc{fhd} simulation corresponds to a zenith-pointed observation of the MWA's ``EoR-0'' field, centered at RA 0 h, Dec.\ $-27^\circ$. The simulation spans a frequency range of 167-198 MHz at a frequency resolution of 80 kHz. It corresponds to a 2 minute snapshot image with 2 second time resolution; for calibration, we average the visibilities across the full 2 minutes. We use the beam model developed by \citealt{Sutinjo2015} and average it across the full frequency range to represent a frequency-invariant beam. We simulate only a single polarization, corresponding to the east-west aligned dipoles. We represent the sky signal with the GLEAM source catalog \citep{Hurley-Walker2017}, and we model point sources only. The simulated ``true'' visibilities are derived from the full catalog. Following the approach used in \citealt{Barry2016} and \citealt{Byrne2019}, we simulate model visibilities $\modelvals(f)$ from an incomplete catalog that omits the faintest GLEAM sources. Our incomplete catalog includes 90\% of the full catalog's power and omits sources fainter than 91 mJy. Figure \ref{fig:model_error} plots model visibility error as a function of delay mode and baseline length. Due to the foreground wedge effect, the missing sources in the sky model produce model visibility error predominantly in a wedge-shaped region of delay space. We simulate the instrument response by defining randomized gains for each antenna and frequency. We then divide the true visibilities simulated from the full GLEAM catalog by the gains to derive uncalibrated data visiblities $\data(f)$. Calibration then attempts to recover the gains as accurately as possible, enabling reconstruction of the simulated true signal from the data visibilities. Our simulation does not include thermal noise, so any calibration error results from sky model error only. \subsection{Defining the Weighting Function} \label{s:weighting_func} DWCal uses a weighting function that represents the model visibility error for each baseline and delay mode. The form of this weighting function is fully arbitrary, but calibration performance improves with weightings that accurately estimate the true model visibility error. We choose to define the weighting function with a relatively simple parameterization of the model visibility error, presented in Figure \ref{fig:weighting_function}. We divide the foreground wedge into inner and outer wedge regions and empirically calculate the variance in each region from the model visibility error (plotted in Figure \ref{fig:model_error}). We further represent short baseline error near delays $\eta \approx 0$ by fitting a decaying exponential function in $|\eta|$. From Equation \ref{eq:delay_weighting}, $\widetilde{\radweight}_{jk}(\eta)$ can be interpreted as the reciprocal of the variance of the error on models of baseline $\{j, k\}$ at delay mode $\eta$. We therefore set $\widetilde{\radweight}_{jk}(\eta)$ equal to the reciprocal of the values plotted in Figure \ref{fig:weighting_function}. Figure \ref{fig:weighting_func_select_bls} plots the weighting function for three baseline lengths: 5, 500, and 2500 m. The left panel plots $\widetilde{\radweight}_{jk}(\eta)$, and the right panel presents the Fourier transformed quantities $\radweight_{jk}(\Delta f)$. In practice, we represent the weighting function as an $N_\text{freq} \times N_\text{freq}$ matrix $\weightmat_{jk}$ for each baseline $\{j,k\}$ (see Equation \ref{eq:dwcal_matrix}). This allows us to evaluate $\chi^2$ as a matrix multiplication operation. Figure \ref{fig:weighting_matrices} plots two such matrices, for baselines of lengths 5 m and 500 m. We generate these matrices for each baseline and pass them to the calibration optimization algorithm. In \S\ref{s:results} we compare calibration performance using this weighting function to that of sky-based calibration. We implement sky-based calibration by simply setting the weighting matrices equal to the identity matrix for each baseline, i.e.\ $\weightmat_{jk} = \mathbb{1}$. In order to perform a direct comparison between calibration methods, we desire that the $\chi^2$ do not differ by orders of magnitude. While the overall normalization of the weighting function is arbitrary, vastly different $\chi^2$ magnitudes risk introducing variations in optimization precision. We therefore normalize the weighting functions such that \begin{equation} \sum_{jk} \operatorname{Tr}\big( \weightmat_{jk} \big) = N_\text{bls} N_\text{freq}, \end{equation} where $N_\text{bls}$ is the number of baselines and $N_\text{freq}$ is the number of frequencies. $\operatorname{Tr}$ denotes the trace. \subsection{Calibration Optimization} \label{s:optimization} We choose to use an off-the-shelf Python-based optimizer for the DWCal explorations presented in this paper. We use \textsc{scipy}'s \texttt{optimize.minimize} function, with the ``method'' option set to ``Newton-CG.'' As this optimizer fits real-valued variables only, we represent the complex gains with their real and imaginary parts. The inputs to the \texttt{optimize.minimize} function are the data, model visibilities, an initial guess for the gains, and a function that evaluates $\chi^2$. The ``Newton-CG'' minimization method also accepts functions that explicitly calculate Jacobian and Hessian. We analytically calculate those quantities from Equation \ref{eq:dwcal_regularized} and supply them to the optimizer. See \texttt{https://github.com/rlbyrne/dwcal} for those calculations. We randomize the initial gains, independently selecting the value for each antenna and frequency channel from a complex circular Gaussian distribution with mean value 1 and standard deviation 0.01. We set $\lambda$, the coefficient of the regularization term (see \S\ref{s:regularization}), equal to 0.1. The primary limitation of this optimization implementation is its speed. Each calibration trial, corresponding to the full 128-antenna array with 384 frequency channels, took several hours to run. A DWCal trial took 7.6 hours, while setting the weighting matrices equal to the identity reduced runtime to 4.3 hours. While this performance is sufficient for demonstrating the DWCal concept, we require a more efficient optimizer for calibrating large data volumes. \subsection{Power Spectrum Estimation} \label{s:power_spectrum} After calculating calibration gains, we apply them to the simulated data using \textsc{pyuvdata}'s calibration utility. We then propagate the calibrated data through the \textsc{fhd}/$\epsilon$\textsc{ppsilon} power spectrum estimation pipeline \citep{Sullivan2012, Jacobs2016, Barry2019a, Sullivan2022} to explore the effect of calibration on cosmological power spectrum measurements. \textsc{fhd} grids the visibilities, and $\epsilon$\textsc{ppsilon}\footnote{\texttt{https://github.com/EoRImaging/eppsilon}} calculates error-propagated power spectra. In the next section we present 2D and 1D power spectra generated with $\epsilon$\textsc{ppsilon}. \section{Results} \label{s:results} We test calibration on simulated data with an instrumental response represented by randomized gains. The gains are selected from a circular complex Gaussian distribution with mean 1 and standard deviation 0.01. We select the values independently for each antenna and frequency channel. To account for the calibration degeneracy in the per-frequency phase (see \S\ref{s:regularization}), we impose a phase rotation at each frequency to ensure that the mean complex phase of the true gains is zero for each frequency channel. Figure \ref{fig:true_gains_hist} plots the distribution of the gains. We divide the simulated visibilities by these gains to derive the data $\data(f)$ used in calibration. Fitting randomized gains in this way ensures that our calibration tests are relevant for instruments with substantial bandpass structure. We show that DWCal performs well for arbitrary instrument responses. We calibrate using model visibilities $\modelvals(f)$ simulated from an incomplete sky model (see \S\ref{s:vis_sim}). We calibrate twice, once performing typical sky-based calibration (analogous to setting the weighting matrices equal to the identity matrix) and once implementing DWCal, using the weighting matrices described in \S\ref{s:weighting_func}. Figure \ref{fig:gains_error_hist} plots the distributions of gain error for sky-based calibration (left) and DWCal (right). We find that DWCal produces more accurate calibration solutions than sky-based calibration. DWCal is fundamentally designed to improve bandpass calibration performance, and it is therefore instructive to explore its impact on the fit gains' frequency dependence. The left panel of Figure \ref{fig:gains_error} plots the amplitude of the fit gain error as a function of frequency. In the right panel we have Fourier transformed the gain error across frequency, and we plot the error amplitude as a function of delay mode. The solid lines indicate the average value across all antennas and the shaded regions enclose the central 50\% of values. The left panel of Figure \ref{fig:gains_error} indicates that DWCal produces better results than sky-based calibration for most frequencies, with the notable exceptions of the lowest and highest frequency channels. DWCal leverages constraints from neighboring frequency channels to calibrate, and the frequencies on the edges of the band are therefore less constrained. Fortunately, these channels typically contribute minimally to the power spectrum estimate, as analyses generally apply a spectral window function to improve analysis dynamic range (the $\epsilon$\textsc{ppsilon} power spectrum pipeline, for example, applies a Blackman-Harris window function across frequency). From the right panel of Figure \ref{fig:gains_error} we note that gain error appears predominantly at low delay magnitude. Model visibility error occurs primarily at $\eta=0$. However, the foreground wedge effect extends this error into neighboring delay modes, and sky-based calibration therefore exhibits error in delay modes around $\eta=0$. DWCal does not completely eliminate this effect, but it reduces gain error at these low $|\eta|$. To explore the effect of this calibration error on cosmological power spectrum measurements we propagate the visibilities through the \textsc{fhd}/$\epsilon$\textsc{ppsilon} pipeline (see \S\ref{s:power_spectrum}). Figure \ref{fig:2d_ps} depicts the resulting 2D power spectra. Here the horizontal axis denotes modes perpendicular to the line-of-sight ($k_\perp$) and the vertical axis denotes modes parallel to the line-of-sight ($k_\parallel$). We present the ``dirty'' power spectra with no foreground subtraction. The leftmost panel (a) depicts the power spectrum with no calibration error. Here we have used the visibilities directly output from our GLEAM catalog simulation. Power appears predominantly in the foreground wedge region. At high $k_\perp$, poor baseline coverage produces substantial power leakage into high $k_\parallel$. However, at lower $k_\perp$, while we see some leakage into the ``EoR window'' modes (power spectrum modes outside the foreground wedge), the window power is suppressed by many orders of magnitude. In the second panel (b) of Figure \ref{fig:2d_ps}, we have applied the randomized gains plotted in Figure \ref{fig:true_gains_hist}. This added instrumental response structure couples power into the EoR window, and we see substantially higher power in the window compared to (a). Panels (c) and (d) of Figure \ref{fig:2d_ps} depict the results of calibrating with sky-based calibration and DWCal, respectively. Here calibration attempts to recover the gains applied in panel (b), reducing power leakage in the EoR window and returning the power spectrum to its state in panel (a). While both methods substantially improve upon the uncalibrated power spectrum in panel (b), DWCal produces less power leakage in the EoR window than sky-based calibration. To further highlight the differences between sky-based calibration and DWCal, Figure \ref{fig:2d_ps_diff} plots the difference between Figures \ref{fig:2d_ps}(c) and \ref{fig:2d_ps}(d). Here positive (red) values indicate higher power when calibrated with sky-based calibration, and negative (blue) values indicate higher power when calibrated with DWCal. We see that, throughout the EoR window, sky-based calibration reconstructs more power than DWCal. Finally, in Figure \ref{fig:1d_ps} we average the EoR window modes in annuli to compare 1D power spectra produced with sky-based calibration and DWCal. Here we have subtracted the ``perfect calibration'' power spectrum plotted in Figure \ref{fig:2d_ps}(a) to highlight the effect of calibration error only. The power spectrum of sky-based calibration error exceeds that of DWCal by about 2 orders of magnitude. The black line represents the magnitude of the predicted EoR signal \citep{Furlanetto2006}. Sky-based calibration error exceeds the predicted EoR signal on all modes. For the level of sky model error explored here, where we have omitted 10\% of the sky signal from the calibration sky model, sky-based calibration does not enable a detection of the cosmological signal, even in the absence of other systematic error and thermal noise. Even when calibrating to the same sky model and making no assumptions about the instrumental response, DWCal performs substantially better than sky-based calibration. By incorporating information about the foreground wedge in the form of baseline-dependent weighting matrices, DWCal reduces power leakage into the EoR window and mitigates the impact of calibration error on power spectrum measurements. \section{Discussion} DWCal, or Delay-Weighted Calibration, is a new tool for precision bandpass calibration that is resilient to sky model error. Broadly speaking, it is a calibration framework that allows for baseline- and delay-dependent weighting in calibration. This is valuable for 21 cm cosmology experiments, for which sky model error affects a compact region of the 2D power spectrum. DWCal is it highly adaptable to a variety of calibration approaches. In this paper we present DWCal as an extension to simple sky-based calibration, however more advanced calibration techniques such as redundant calibration or unified calibration could likewise benefit from the DWCal technique. DWCal has several distinct benefits over other leading precision calibration techniques. Namely, it makes no assumptions about the antenna gains' bandpass structure, uniformity, or stability. This enables DWCal to fit arbitrary bandpass features and accurately calibrate time-dependent, heterogeneous arrays. DWCal fits the same number of calibration parameters as sky-based calibration and introduces no further calibration degeneracies. Like sky-based calibration, it is degenerate only in the overall phase of the gains at each frequency, and in \S\ref{s:regularization} we introduce an innovative solution for resolving that degeneracy. The weighting function implemented in DWCal could take any form. Here we present results using the weighting function plotted in Figure \ref{fig:weighting_function}, which fits a simple foreground wedge feature and is designed to approximate the model visibility error calculated in simulation and plotted in Figure \ref{fig:model_error}. We expect DWCal will perform best with weighting functions that most accurately describe the underlying distribution of model visibility error. Future work could explore alternative weighting functions. Model visibility error is highly dependent on the array configuration, antenna beam, and sky models used, so the optimal DWCal weighting function will be unique to each experiment. This paper demonstrates that DWCal improves calibration's resilience to sky model error. From Figure \ref{fig:1d_ps}, we find that our simulated power spectrum measurements are improved by approximately 2 orders of magnitude on all modes when using DWCal compared to simple sky-based calibration. This moves us from a regime in which calibration error precludes a detection of the 21 cm signal into one in which such a detection would be possible, provided other sources of systematic error are properly controlled. Given that calibration error is one of the dominant barriers to the success of 21 cm experiments, DWCal provides an important tool for enabling these cosmological measurements. \section*{Acknowledgements} This work was inspired by conversations with Miguel Morales and Bryna Hazelton. Thank you to Ian Sullivan, Bryna Hazelton, and Nichole Barry for support with the \textsc{fhd}/$\epsilon$\textsc{ppsilon} software pipeline and to Yuping Huang for support with computing resources. Michael Wilensky and Bryna Hazelton provided valuable input during revision of this paper. \bibliography{sample631}{} \bibliographystyle{aasjournal}

Title: MAGAZ3NE: High Stellar Velocity Dispersions for Ultra-Massive Quiescent Galaxies at $z\gtrsim3$

Abstract: In this work we publish stellar velocity dispersions, sizes, and dynamical masses for 8 ultra-massive galaxies (UMGs; log($M$/M$_\odot>11$, $z\gtrsim3$) from the Massive Ancient Galaxies At $z>3$ NEar-infrared (MAGAZ3NE) Survey, more than doubling the number of such galaxies with velocity dispersion measurements at this epoch. Using the deep Keck/MOSFIRE and Keck/NIRES spectroscopy of these objects in the $H$- and $K$-bandpasses, we obtain large velocity dispersions of $\sim400$ km s$^{-1}$ for most of the objects, which are some of the highest stellar velocity dispersions measured, and $\sim40$\% larger than those measured for galaxies of similar mass at $z\sim1.7$. The sizes of these objects are also smaller by a factor of 1.5-3 compared to this same $z\sim1.7$ sample. We combine these large velocity dispersions and small sizes to obtain dynamical masses. The dynamical masses are similar to the stellar masses of these galaxies, consistent with a Chabrier initial mass function (IMF). Considered alongside previous studies of massive quiescent galaxies across $0.2<z<4.0$, there is evidence for an evolution in the relation between the dynamical mass - stellar mass ratio and velocity dispersion as a function of redshift. This implies an IMF with fewer low mass stars (e.g., Chabrier IMF) for massive quiescent galaxies at higher redshifts in conflict with the bottom-heavy IMF (e.g., Salpeter IMF) found in their likely $z\sim0$ descendants, though a number of alternative explanations such as a different dynamical structure or significant rotation are not ruled out. Similar to data at lower redshifts, we see evidence for an increase of IMF normalization with velocity dispersion, though the $z\gtrsim3$ trend is steeper than that for $z\sim0.2$ early-type galaxies and offset to lower dynamical-to-stellar mass ratios.

https://export.arxiv.org/pdf/2208.04329

\title{\sc \magazine: High Stellar Velocity Dispersions for Ultra-Massive Quiescent Galaxies at $z\gtrsim3$} \footnote{The spectra presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.\\} \shorttitle{Velocity Dispersions of UMGs} \shortauthors{B. Forrest, et al.} \correspondingauthor{Ben Forrest} \email{bforrest@ucdavis.edu} \author[0000-0001-6003-0541]{Ben Forrest} \affiliation{Department of Physics and Astronomy, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA} \affiliation{Department of Physics and Astronomy, University of California, Riverside, 900 University Avenue, Riverside, CA 92521, USA} \author[0000-0002-6572-7089]{Gillian Wilson} \affiliation{Department of Physics and Astronomy, University of California, Riverside, 900 University Avenue, Riverside, CA 92521, USA} \author[0000-0002-9330-9108]{Adam Muzzin} \affiliation{Department of Physics and Astronomy, York University, 4700, Keele Street, Toronto, ON MJ3 1P3, Canada} \author[0000-0001-9002-3502]{Danilo Marchesini} \affiliation{Department of Physics and Astronomy, Tufts University, 574 Boston Avenue, Medford, MA 02155, USA} \author[0000-0003-1371-6019]{M. C. Cooper} \affiliation{Center for Cosmology, Department of Physics and Astronomy, University of California, Irvine, 4129 Frederick Reines Hall, Irvine, CA, USA} \author[0000-0002-7248-1566]{Z. Cemile Marsan} \affiliation{Department of Physics and Astronomy, York University, 4700, Keele Street, Toronto, ON MJ3 1P3, Canada} \author{Marianna Annunziatella} \affiliation{Department of Physics and Astronomy, Tufts University, 574 Boston Avenue, Medford, MA 02155, USA} \affiliation{Centro de Astrobiolog\'ia (CSIC-INTA), Ctra de Torrej\'on a Ajalvir, km 4, E-28850 Torrej\'on de Ardoz, Madrid, Spain} \author[0000-0002-2446-8770]{Ian McConachie} \affiliation{Department of Physics and Astronomy, University of California, Riverside, 900 University Avenue, Riverside, CA 92521, USA} \author{Kumail Zaidi} \affiliation{Department of Physics and Astronomy, Tufts University, 574 Boston Avenue, Medford, MA 02155, USA} \author{Percy Gomez} \affiliation{W.M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743, USA} \author[0000-0001-8169-7249]{Stephanie M. Urbano Stawinski} \affiliation{Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA} \author{Wenjun Chang} \affiliation{Department of Physics and Astronomy, University of California, Riverside, 900 University Avenue, Riverside, CA 92521, USA} \author{Gabriella de Lucia} \affiliation{INAF - Astronomical Observatory of Trieste, via G.B. Tiepolo 11, I-34143 Trieste, Italy} \author[0000-0003-1181-6841]{Francesco La Barbera} \affiliation{INAF - Osservatorio Astronomico di Capodimonte, sal. Moiariello 16, 80131 Napoli, Italy} \author[0000-0003-2119-8151]{Lori Lubin} \affiliation{Department of Physics and Astronomy, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA} \author[0000-0002-7356-0629]{Julie Nantais} \affiliation{Departamento de Ciencias F\'isicas, Universidad Andres Bello, Fern\'andez Concha 700, Las Condes 7591538, Santiago, Regi\'on Metropolitana, Chile} \author[0000-0002-0033-5041]{Theodore Pe\~na} \affiliation{Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter Street, Madison, WI 53706-1507, USA} \author[0000-0003-3959-2595]{Paolo Saracco} \affiliation{INAF - Osservatorio Astronomico di Brera, via Brera 28, 20121 Milano, Italy} \author[0000-0001-7291-0087]{Jason Surace} \affiliation{IPAC, Mail Code 100-22 Caltech 1200 E. California Blvd. Pasadena, CA 91125, USA} \author[0000-0001-7768-5309]{Mauro Stefanon} \affiliation{Leiden Observatory, Leiden University, 2300 RA Leiden, The Netherlands} \keywords{Galaxy evolution (594)-- High-redshift galaxies (734) -- Quenched galaxies (2016)} \section{Introduction} Deep near-infrared photometric surveys of the last decade have suggested larger numbers of massive galaxies at high redshifts than predicted by cosmological galaxy simulations \citep[\eg][]{Muzzin2013, Straatman2014, Sherman2019, Marsan2022}. More recent simulations have better agreement with observations, but the discrepancy is still a factor of a few to ten at the highest masses \citep[though see ][]{Donnari2021,Lustig2022}. In the last several years, spectroscopic confirmation of a handful of galaxies with stellar masses of \logM$>11$ and at redshifts of $z>3$ have shown that such galaxies do indeed exist in non-negligible numbers \citep{Marsan2015,Marsan2017,Glazebrook2017, Schreiber2018b, Tanaka2019, Valentino2020, Forrest2020a, Forrest2020b}, but a robust measurement of the number density of such galaxies is still lacking. This is largely due to the fact that the determination of stellar mass, particularly from photometry alone requires a number of assumptions which introduce the possibility for significant error. There are numerous programs which determine galaxy parameters via spectral energy distribution (SED) fitting. Nearly all require some assumptions about the geometry of dust and dust extinction, the initial mass function (IMF) of star-formation, a parametric form of star-formation history, strength of emission lines, and choice of stellar population synthesis models, each of which play a role in the determined stellar mass of a galaxy. For large populations of galaxies, the median mass determination appears sensitive to these choices with scatter $\sim0.2$~dex \citep[\eg][]{Wuyts2009, Mobasher2015}, though the choice of code can lead to systematic offsets up to $0.3$~dex \citep{Muzzin2009, Leja2019}. While these differences are perhaps tolerable, the differences for individual galaxies can greatly exceed these numbers in cases with significant flux contributions from strong emission lines \citep{Stark2013, Salmon2015, Forrest2017}, and active galactic nuclei \citep[AGN;][]{Leja2018}, as well as in outlier cases where photometric redshifts are highly discrepant from true redshifts, though this seems less common in massive galaxies even at $z>3$ \citep{Schreiber2018b, Forrest2020b}. As a result, probing stellar masses independently of the above assumptions is valuable. While the stellar velocity dispersion formally probes the total mass of a galaxy, the massive, high redshift galaxies of interest here typically have small sizes, and have central masses dominated by stars \citep[\eg][]{vanderWel2014,Straatman2015,Saracco2019}. Locally, stellar velocity dispersion is well correlated with the luminosity and radius of elliptical galaxies \citep[\eg][]{Faber1976, Djorgovski1987, Dressler1987, Shu2012}, the mass of the central black hole \citep[\eg][]{Gebhardt2000, Kormendy2013}, mass-to-light ratio \citep[\eg][]{Cappellari2006}, and numerous other properties including galaxy color \citep{Wake2012} and stellar mass \citep[\eg][]{Zahid2016}. Velocity dispersions have been studied out to higher redshifts as well, and many such correlations appear to hold for these data, though they may be offset from the local relations \citep[\eg][]{vanDokkum2009, Newman2010, Bezanson2012, Bezanson2013b, Thomas2013, vandeSande2013, Gargiulo2016, Hill2016, Belli2017}. However, like the measurement of stellar masses, the measurement of stellar velocity dispersions holds the potential for systematic and statistical errors, the latter of which can of course be significant for low signal-to-noise (SNR) spectra. The interpretation also requires careful analysis, as effects such as galaxy rotation and inclination can either increase or decrease measured velocity dispersions \citep{Bezanson2018, Newman2018b, Mendel2020}. Still, stellar velocity dispersions can be used in concert with structural measurements to calculate dynamical masses, which are sensitive to the gravitational potential of a galaxy, and therefore to the contribution of dark matter as well as the contributions of dust, gas, and stars. This then provides an effective upper limit on the stellar mass of a galaxy, independent of the numerous assumptions intrinsic to the calculation of stellar masses via SED fitting, including the shape of the initial mass function (IMF). Variability in the IMF, which traces the number of stars formed as a function of their mass in a star-forming molecular cloud, can contribute to non-negligible differences in the determination of stellar mass as it sets the effective mass-to-light ratio. The IMF of many galaxies, particularly local massive early-type galaxies (ETGs), is inferred via spectral fitting or dynamical modeling to have a 'heavy' mass-to-light ratio (with respect to the MW distribution), such as that of the \citet{Salpeter1955} IMF, which assumes a functional power-law with index $x=-2.35$ (termed `heavy' due to the larger effective mass-to-light ratio). However, observations have suggested that the IMF is not universal \citep[see][for a review]{Hopkins2018a} and can vary over cosmic time, between galaxies, or as a function of galaxy radius, metallicity, stellar mass, or star formation density \citep[\eg][]{Cappellari2006, vanDokkum2008, Conroy2012, Cappellari2013a, Cappellari2013b, Kroupa2013, vanDokkum2017, Villaume2017, LaBarbera2019}. As such it is important to note that any measurement of the IMF in a galaxy is a measurement of the super-position of the IMF during any and all episodes of star-formation in that galaxy. Recently, \citet{Mendel2020} homogeneously analyzed 58 massive quiescent galaxies at $1.4<z<2.1$ and found that galaxies with higher stellar velocity dispersions at a given epoch prefer a heavier IMF such as that from \citet{Salpeter1955}, while galaxies with lower stellar velocity dispersions are better described by a lighter IMF such as the \citet{Chabrier2003} IMF. This result agrees with lower redshift analysis from \citet{Posacki2015}, though the higher redshift galaxies have systematically higher velocity dispersions than lower redshift galaxies with the same dynamical-to-stellar mass ratio. Measurements of velocity dispersion require spectra with reasonable signal-to-noise which are difficult to obtain for galaxies at earlier epochs. As such, only six massive galaxies with stellar masses \logM$\gtrsim11$ at $z>3$ have measured stellar velocity dispersions \citep{Tanaka2019, Saracco2020, Esdaile2021}. In this work we measure velocity dispersions for 8 additional massive galaxies at $z\gtrsim3$ using the MOSFIRE \citep{McLean2010, McLean2012} and NIRES \citep{JWilson2004} instruments on Keck, more than doubling the size of the current sample in the literature - 4/8 of these galaxies are more massive than any of the $z>3$ sample with velocity dispersions in the literature. Combined with size measurements for these galaxies and values from the literature, we perform the first statistical comparison of dynamical and stellar masses at this early epoch using 14 massive galaxies. We present the data in Section \ref{Sec:Data}, the velocity dispersion calculations and image analysis process in Section \ref{Sec:Analysis}, and then a discussion of the results in Section \ref{Sec:Res} and the main conclusions in Section \ref{Sec:Conc}. All analysis here uses a $\Lambda$CDM cosmology with $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_M=0.3$, and $\Omega_\Lambda=0.7$ as well as the AB magnitude system \citep{Oke1983}. A \citet{Chabrier2003} IMF is used for calculation of stellar mass. \section{Data} \label{Sec:Data} \subsection{Parent Photometric Catalogs} Targets selected for spectroscopic followup in the \magazine\ survey were drawn from parent photometric catalogs in the UltraVISTA DR1 \citep{Muzzin2013a}, UltraVISTA DR3 (Muzzin et al., in prep) and XMM-VIDEO (Annunziatella et al., in prep) fields. The UltraVISTA survey \cite{McCracken2012} imaged over 1.62~deg$^2$ in the COSMOS field with deep near-infrared \mbox{$Y$-,} \mbox{$J$-,} \mbox{$H$-,} and $K$-bandpasses. The first data release \citep[DR1 catalogs;][]{Muzzin2013a} combined additional photometry from $0.15-24$~$\mu$m yielding a total of 30 bandpasses with 90\% completeness $K_s=23.4$ mag. Subsequent deep imaging over 0.84~deg$^2$ in the NIR furthered the value of the dataset, with DR3 \citep[][Muzzin et al., in prep]{Marsan2022} reaching deeper than DR1 by 1.1 mag in the $K_s$-band and $\sim1.2$ mag deeper in the IRAC $3.6$ and $4.5~\mu$m bandpasses \citep{Ashby2018}. A total of up to 49 bandpasses in DR3 allowed for highly accurate galaxy spectral energy distributions (SEDs) and photometric redshift determinations, as well as detection of massive, quiescent galaxies at $z>3$ which are too faint for accurate characterization with optical photometry alone. The VISTA Deep Extragalactic Observations \citep[VIDEO;][]{Jarvis2013} survey similarly acquired deep NIR imaging over several fields, including IRAC data from SERVS \citep{Mauduit2012} and the DeepDrill survey \citep{Lacy2021}. Catalogs used in this work are built from VIDEO DR4 data over 5.1~deg$^2$ in the XMM-Newton Large Scale Structure (XMM) field with up to 22 bandpasses from $u$-band to IRAC 8.0 $\mu$m and a $5\sigma$ depth of $K_s=23.8$ mag (Annunziatella, et al., in prep). While this catalog is somewhat shallower in $K$-band depth, it covers a wider area which is important for detection of the rare massive, quiescent objects at these redshifts. \subsection{Near-Infrared Spectroscopy} For this work we analyze $H$- and $K$-band spectroscopic observations from Keck-MOSFIRE \citep{McLean2010, McLean2012} taken as part of the \magazine\ survey \citep{Forrest2020b} and details of the spectroscopic target selection are provided therein. The general survey observing strategy called for targeting ultra-massive galaxies (UMGs) in the $K$-band, where the strong emission features \OIIIfive\ and \Hbeta\ fall at the redshift of the sample. On-the-fly reduction was used, and once a redshift was confirmed, observation of a UMG was stopped. As such, UMGs with strong emission lines and only faint detection of the continuum have insufficient SNR to calculate a stellar velocity dispersion. However, 6 of the 16 confirmed UMGs from \citet{Forrest2020b} have MOSFIRE observations in $H$-band where a greater number of spectral features lie (\eg\ \Dfour, Ca H\&K, and higher order Balmer absorption features), enabling a more reliable velocity dispersion calculation. Since publication of \citet{Forrest2020b}, a redshift has also been obtained for an additional UMG, \mbox{COS-DR1-99209}, at z = 2.983 observed in both $H$- and $K$-band with MOSFIRE. For these seven galaxies, the MOSFIRE DRP was used to reduce the raw spectroscopy to 2D spectra. From there, a custom code written by one of us (B.F.) was used to optimally extract a 1D spectrum and perform telluric corrections using stars observed on the same masks, modeled with the PHOENIX stellar models \citep{Husser2013}. A more detailed description is provided in \citet{Forrest2020b}. Galaxy photometry are fit in conjunction with a single-band spectrum using FAST++ \citep{Schreiber2018a} to obtain relative scaling of different spectral bandpasses. In the case of XMM-VID1-2075, the only MAGAZ3NE UMG in this work without $H$-band MOSFIRE spectroscopy, $J$-, $H$- , and $K$-band NIRES spectroscopy was independently obtained (PI: Gomez; Gomez et al., in prep). A comparison of the two (very similar) $K$-band spectra is presented in Appendix~\ref{App:K2075}. The NIRES data were reduced using Pypeit \citep[version 1.0.4;][]{Prochaska2020}. Pypeit flat fields the science data, performs wavelength calibration, models and subtracts the sky background, and performs a flux calibration. A telluric correction was also calculated using Molecfit \citep{Smette2015, Kausch2015}. In total, we thus present new velocity dispersions for 8 \magazine\ UMGs in this work: \mbox{COS-DR1-99209}, \mbox{COS-DR3-84674}, \mbox{COS-DR3-111740}, \mbox{COS-DR3-201999}, \mbox{COS-DR3-202019}, \mbox{XMM-VID1-2075}, \mbox{XMM-VID3-1120}, and \mbox{XMM-DR3-2457}. We also include a ninth UMG from the MAGAZ3NE sample, \mbox{COS-DR3-160748}, which has a velocity dispersion from a high SNR spectrum taken with the LBT published in \citet{Saracco2020} as C1-23152. \section{Analysis}\label{Sec:Analysis} \subsection{Velocity Dispersions} Absorption feature stellar velocity dispersions were calculated using the Penalized Pixel-Fitting method (pPXF) \citep{Cappellari2004,Cappellari2017} in conjunction with the UMG spectra. This maintains consistency with the analysis of other $z>3$ massive galaxies \citep{Tanaka2019, Saracco2020, Esdaile2021}. \subsubsection{Inputs for pPXF Velocity Dispersion Calculations} When available, spectroscopy from both the $H$- and $K$-bands was used. Observed spectra were logarithmically rebinned and corrected for instrumental resolution, and templates were resampled to match this resolution. In cases where the resolution for a template was less than that of the observed spectra, the spectra were binned using inverse variance weighting to match the model resolution. Numerous runs were performed for each galaxy using a variety of spectral template libraries, wavelength masking strategies, and a range of additive Legendre polynomial orders to limit the effects of template mismatch and telluric correction inaccuracies. The variety of inputs also allows us to characterize the systematic error on the velocity dispersion, which exceeds the statistical error provided by pPXF. Extensive testing of pPXF on a sample of five massive quiescent galaxies $1.4<z<2.1$ was performed in \citet{vandeSande2013}. In their Appendix A, they test the dependence of velocity dispersions output by pPXF on various inputs, including template choice, polynomial degree, and stellar population models. We do similar testing when fitting the \magazine\ sample, which is described in more detail in Appendix~\ref{App:pPXF}. Briefly, we ran pPXF using the templates from \citet[][BC03]{Bruzual2003}, SSPs constructed from the MILES library \citep{Sanchez-Blazquez2006, Vazdekis2010}, and the Indo-US library \citep{Valdes2004}. These libraries provide sufficient variety of spectral templates to fit the observed spectra well. However, pPXF does not incorporate galaxy photometry into the fit, and failure to do this can result in underestimating the velocity dispersions \citep[][though results are often consistent within the errors]{Mendel2020}. As such, we also use FAST++ \citep{Schreiber2018a} with the \citet[][BC03]{Bruzual2003} templates to jointly fit the observed photometry and spectroscopy and obtain a best-fit template, subsequently using pPXF with that template choice fixed - we designate these runs as BC03++. Runs with each of these four template sets (BC03++, BC03, MILES, and Indo-US) were also done with an additive Legendre polynomial from order $0\leq d \leq 50$. Such a polynomial corrects for differences in template and observed spectral shape as can result from \eg\ telluric correction inaccuracies and helps avoid template mismatch. The effect of adding a polynomial of very high order is to perturb a template to fit all the noise features in an observed spectrum, and thus a somewhat low order polynomial is preferred. Choice of polynomial order varies in the literature: \citet{vandeSande2013} use $d\sim17$, with velocity dispersions only showing a small dependence on this choice from $0 \leq d \leq 50$, \citet{Mendel2020} use $d=9$, \citet{Saracco2020} use $d=4$, \citet{Tanaka2019} use $d=1$, and \citet{Esdaile2021} do not use an additive polynomial (effectively $d=0$). In general we find that the velocity dispersion varies the least over the range $10<d<20$ for the UMGs in this sample. Finally, we also choose various methods of masking the spectral wavelengths used in the fit. We test pPXF while masking all observed emission lines as well as: 1) all Balmer features, 2) the \Hbeta\ feature, 3) no other wavelengths, 4-6) wavelengths in 1-3 plus sky lines. Exclusion of the Balmer features can result in a more stable velocity dispersion \citep{vandeSande2013} and remove any degeneracy between small scale emission and template choice, but also remove a strong constraint on the velocity dispersion for spectra with low SNR as is typical for galaxies at these redshifts \citep{Tanaka2019, Esdaile2021}. Masking only the \Hbeta\ feature in these quiescent galaxies strongly mitigates the emission issue. \subsubsection{Measured Velocity Dispersions} Resultant best-fit templates from each run were visually inspected and also compared to the galaxy photometry, with results involving clearly incorrect templates discarded (these were uncommon, on the order of a few percent). Our galaxies have sufficient SNR such that the results of the many runs form a distribution with a clear mode for each galaxy, which we use as the velocity dispersion in subsequent analysis. The (asymmetrical) spread of the distribution of results is used to derive errors on the velocity dispersion, which can differ from the output error of pPXF by up to a factor of $\sim2$. Median values of the fitted velocity dispersion distributions and averages weighted by reduced $\chi^2$ and reported error are all statistically consistent with the mode of the distribution. Models with the best-fit velocity dispersions are shown in Figure~\ref{fig:fits1}. Plots showing the dependencies on choice of input parameters, as well as a more complete discussion are included in Appendix \ref{App:pPXF}. \subsubsection{Aperture Correction of the Measured Velocity Dispersions} For comparison with other measurements in the literature, we correct the measured velocity dispersions to velocity dispersions at the effective radius, $\sigma_{\rm e}$ (size calculations are described in Section~\ref{Sec:SizeCalc}). This removes instrumental dependence and accounts for the effects of seeing. Such a correction is dependent upon the size and shape of the spectral aperture, the observing conditions (\ie\ seeing) and the size of the target. The MOSFIRE aperture size of interest, $r_\textrm{aperture}$, is the distance along the slit over which the 1D spectrum was optimally extracted, and is thus a function of both intrinsic size and seeing conditions which varies for different masks on which the same object is located. In theory, this could also be affected by the length of a slit if it was insufficiently long to cover the entire object (minimum MOSFIRE slit length is 7.1"), though this would only be a concern for very large objects or extremely poor conditions, which does not affect this sample. Extensive modeling in Appendix B of \citet{vandeSande2013} shows that for a rectangular aperture with weighted extraction, this correction factor is quite flat as a function of $r_\textrm{aperture}/r_\textrm{eff}$, when the PSF is taken into account. Indeed, the correction factors for our velocity dispersions calculated following \citet{vandeSande2013} range from 1.048 to 1.058, though the small differences in this correction are far exceeded by the errors on the measured velocity dispersions. The corrected values are shown in Table \ref{tab:props} and used for the remainder of this analysis. \subsection{Sizes}\label{Sec:SizeCalc} GALFIT \citep{Peng2002, Peng2010} was used to model the $K_{\rm S}$-band images of all objects, and the $HST/WFC3/$F160W images of COS-DR3-201999, COS-DR3-202019, and COS-DR3-84674. For the sources in the COSMOS field, the UltraVISTA DR4 $K_{\rm S}$ mosaic with pixel scale 0.15$^{\prime \prime}$ per pixel and FWHM=0.78$^{\prime \prime}$ \citep{McCracken2012} was adopted. For the sources in the XMM field, the VIDEO DR4 $K_{\rm S}$ mosaic with pixel scale 0.2$^{\prime \prime}$ per pixel and FWHM=0.82$^{\prime \prime}$ \citep{Jarvis2013} was adopted. The fitting process was similar for all the galaxies. A small cutout centered on the relevant galaxy was created, making sure to include the central object and any nearby objects along with enough empty region for the sky background calculation. In most cases, the central galaxy was fitted simultaneously with the neighboring objects. In a few cases, the neighboring objects were not fitted if they were far enough from the UMG that their light was not contaminating the objects. In this case, the neighboring objects were only masked out in the GALFIT fitting. All objects were fitted with a single S\'ersic profile. The free parameters had the following fitting constraints: the centroid of the object was allowed to vary at most by 2 pixels in each direction from the initial coordinates; $0.05 \leqslant r_{\rm e} [^{\prime \prime}] \leqslant 1$; $0.2 \leqslant n \leqslant 7$; and $0.1 \leqslant q \leqslant 1$. We allowed GALFIT to fit a constant sky background as a free parameter. Previous studies have shown this to be the preferred choice, and that GALFIT performs significantly better when allowed to internally measure a sky background, as opposed to being provided a fixed background \citep{Haussler2007, Cutler2022}. Furthermore, the convolution box was allowed to span the whole cutout. For each $K_{\rm S}$-band object fit, two to three nearby, unsaturated, uncontaminated, and background-subtracted stars were used as point-spread functions (PSFs) for model convolution. We also adopted as the model PSF a high signal-to-noise PSF constructed using 10 different nearby stars, stacking the corresponding sky-subtracted stamps after masking any nearby objects, re-centering the stars, and normalizing the integrated flux. Utilizing different stars/PSFs allows for a more realistic estimate of the size measurement error, which is generally underestimated by GALFIT. For the $WFC3$/F160W images, a position-dependent PSF model was created using \texttt{grizli} (https://grizli.readthedocs.io) to shift and drizzle HST empirical PSFs \citep{Anderson2015} at the position of the UMGs. \citet{Cutler2022} showed that there is no significant difference in GALFIT structural measurements between galaxies fit with position-dependent PSFs and those with PSFs determined over a larger area of the mosaic, even at $z>2$. While most of the galaxies are not resolved in the ground-based imaging, GALFIT can still recover fits down to FWHM/2 \citep{Haussler2013, Nedkova2021}, although \citet{Ribeiro2016} suggests such measurements tend to be underestimates. Additionally, the sizes derived from the unresolved $K_{\rm S}$-band and the resolved $WFC3$/F160W GALFIT modelings are consistent with each other, as shown in Figure~\ref{fig:sizecorr}. The SNR of the images is not sufficiently high to obtain a reliable value of the S\'ersic index. As the size and S\'ersic index are covariant in the fitting process, we also use GALFIT to perform fits with the S\'ersic index fixed to $n=1,3,4,6$ and compare to the reported best fit, in which n is allowed to vary, to discern another source of possible error on the size measurement. In some cases, these fits do not converge, and in some the reported fit is clearly incorrect upon visual inspection. Ignoring these cases, we find that objects with best-fit S\'ersic index $2<n<4$ from ground-based $K$-band imaging show size variations on the order of 10\% in these tests. In the other cases, variation on the order of up to 20\% is seen. For all UMGs, these variations are smaller than the reported errors based on different characterizations of the PSF. For galaxies with imaging in both bandpasses, the measured sizes are consistent within the errors. However, the two bandpasses are probing different wavelengths, which can be on opposite sides of the \Dfour\ feature. To avoid any issues on this front, we convert all measured sizes to rest-frame 5000$\rm \AA$ sizes following \citet{vanderWel2014} as: \begin{eqnarray} r_{\rm eff, 5000\AA} = r_{\rm eff, \lambda_{\rm obs}} \bigg{(}\frac{1+z}{1+z_{\rm pivot}}\bigg{)}^\frac{\Delta {\rm log} (r_{\rm eff})}{\Delta {\rm log} \lambda} \end{eqnarray} where, \begin{eqnarray} z_{\rm pivot} &=& \lambda_{\rm obs} / 5000{\rm \AA} - 1\\ \frac{\Delta {\rm log} (r_{\rm eff})}{\Delta {\rm log} \lambda} &=& -0.35 + 0.12z -0.25{\rm log}\bigg{(}\frac{M_*}{10^{10}M_\odot}\bigg{)} \end{eqnarray} For rest-frame optical sizes (5000${\rm \AA}$), $z_{\rm pivot}(F160W) = 2.2$ and $z_{\rm pivot}(K) = 3.3$. While the $\frac{\Delta {\rm log} (r_{\rm eff})}{\Delta {\rm log} \lambda}$ relation from \citet{vanderWel2014} was derived using less massive galaxies at $z<2$, we note that the corrections here are considerably smaller than the errors on the size measurements. Given the consistency of all these half-light radii for a given galaxy, in what follows we use a weighted average of the corrected size measurements in all available bands for determination of dynamical mass. This size is listed in Table \ref{tab:props}. We also note that the morphology of COS-DR3-201999 was analyzed in \citet{Lustig2020}, with the id~252568, which returned $r_{\rm e}$(5000 ${\rm \AA}$)/kpc = $2.37^{+0.58}_{-0.37}$, a size fully consistent with the analysis herein. \subsection{Dynamical Masses} The velocity dispersion and effective radius measurements can be used to calculate dynamical masses for the UMGs in this sample, \begin{eqnarray} M_{\rm dyn}(<r_{\rm e})&=& \kappa_{\rm e}\frac{\sigma_{\rm e}^2 r_{\rm e}}{G}, \end{eqnarray} where $\kappa_{\rm e}$ is a virial coefficient which depends upon the (an-)isotropy of the stellar velocities and the intrinsic mass profile of the galaxy. This value has been calibrated using lower redshift ellipticals, as such determinations for high-redshift, compact quiescent galaxies have not been done due to their small sizes and faint magnitudes. The typical value used for $z\sim2$ quiescent galaxies is $\kappa_{\rm e}=2.5$ \citep{Newman2012,Barro2014}. The resultant value of $M_{\rm dyn}(<r_{\rm e})$ is then doubled to estimate the total $M_{\rm dyn}$, which is then compared to the total stellar mass. \citet{Cappellari2006} also published an analytical estimator which folds in both the virial coefficient and the correction to total mass \begin{eqnarray} M_{\rm dyn}&=&\beta (n) \frac{\sigma_{\rm e}^2 r_{\rm e}}{G}\\ \beta (n) &=& 8.87-0.831n+0.0241n^2 \end{eqnarray} For a sample of massive, quiescent galaxies at $z\sim2$, a typical value of $\beta (n) \sim5$ is found, which is equivalent to the choice of $\kappa_{\rm e}=2.5$ \citep{vandeSande2013, Belli2014a}. Previous samples of UMGs at \mbox{$z\gtrsim3$} \citep{Esdaile2021, Saracco2020} have used this estimator and returned values in the range of $5.4<\beta (n)<6.4$, while \citet{Tanaka2019} also adopt $\beta (n)=5$ due to a lack of a confident measure of S\'ersic index. In this work we also adopt the value of $\beta (n)=5$, as the SNR of the images used for size calculations is not sufficiently high to obtain a reliable value of the S\'ersic index. Results of these calculations are provided in Table~\ref{tab:props}. \begin{table*} \centering \caption{Properties of massive quiescent galaxies in the $z\gtrsim3$ sample discussed in this work.} \begin{tabular}{ llcclcc } UMG & \zspec & \logM & \logMdyn & $\sigma_e$ (km s$^{-1}$) & $r_\textrm{eff, 5000{\rm \AA}}$ (kpc) & Reference \\ \hline \hline COS-DR1-99209 & $2.9834^{+0.0023}_{-0.0028}$ & $11.22^{+0.05}_{-0.06}$ & $11.31^{+0.12}_{-0.20}$ & $401^{+63}_{-84}$ & $1.08\pm0.24$ & This work \\ COS-DR3-84674 & $3.0094^{+0.0015}_{-0.0011}$ & $11.25^{+0.01}_{-0.02}$ & $11.33^{+0.23}_{-0.14}$ & $442^{+206}_{-68}$ & $0.95\pm0.23$ & This work \\ COS-DR3-111740 & $2.7988^{+0.0013}_{-0.0011}$ & $10.98^{+0.01}_{-0.00}$ & $11.02^{+0.13}_{-0.24}$ & $467^{+102}_{-131}$ & $0.89\pm0.33$ & This work \\ COS-DR3-201999 & $3.1313^{+0.0014}_{-0.0012}$ & $11.40^{+0.03}_{-0.01}$ & $11.28^{+0.15}_{-0.24}$ & $271^{+55}_{-58}$ & $2.26\pm0.31$ & This work \\ COS-DR3-202019 & $3.1326^{+0.0021}_{-0.0011}$ & $11.67^{+0.04}_{-0.05}$ & $12.00^{+0.14}_{-0.27}$ & $345^{+92}_{-111}$ & $7.54\pm1.16$ & This work \\ XMM-VID1-2075 & $3.4520^{+0.0014}_{-0.0017}$ & $11.52^{+0.00}_{-0.05}$ & $11.49^{+0.12}_{-0.11}$ & $379^{+85}_{-53}$ & $1.85\pm0.16$ & This work \\ XMM-VID3-1120 & $3.4919^{+0.0018}_{-0.0029}$ & $11.47^{+0.02}_{-0.03}$ & $11.54^{+0.10}_{-0.31}$ & $419^{+74}_{-148}$ & $1.71\pm0.19$ & This work \\ XMM-VID3-2457 & $3.4892^{+0.0032}_{-0.0024}$ & $11.26^{+0.02}_{-0.03}$ & $11.49^{+0.07}_{-0.29}$ & $396^{+40}_{-132}$ & $1.71\pm0.22$ & This work \\ ZF-COS-20115 & $3.715$ & $11.06^{+0.06}_{-0.04}$ & $10.86^{+0.14}_{-0.20}$ & $283^{+52}_{-52}$ & $0.66\pm0.08$ & \citet{Esdaile2021} \\ 3D-EGS-40032 & $3.219$ & $11.31^{+0.03}_{-0.03}$ & $11.41^{+0.11}_{-0.16}$ & $275^{+56}_{-56}$ & $2.40\pm0.19$ & \citet{Esdaile2021} \\ 3D-EGS-18996 & $3.239$ & $10.99^{+0.02}_{-0.03}$ & $10.56^{+0.13}_{-0.19}$ & $196^{+48}_{-48}$ & $0.63\pm0.05$ & \citet{Esdaile2021} \\ 3D-EGS-31322 & $3.434$ & $10.99^{+0.05}_{-0.04}$ & $10.85^{+0.20}_{-0.39}$ & $201^{+119}_{-119}$ & $0.61\pm0.05$ & \citet{Esdaile2021} \\ C1-23152$^a$ & $3.352^{+0.002}_{-0.002}$ & $11.30^{+0.19}_{-0.13}$ & $11.34^{+0.07}_{-0.09}$ & $409^{+60}_{-60}$ & $1\pm0.1$ & \citet{Saracco2020} \\ SXDS-27434 & $4.0127^{+0.0005}_{-0.0005}$ & $11.06^{+0.04}_{-0.04}$ & $<11.32$ & $268^{+59}_{-59}$ & $<1.3$ & \citet{Tanaka2019} \\ \hline \end{tabular}\\ \footnotesize{$a$. This object is renamed COS-DR3-160748 in the MAGAZ3NE Survey.} \label{tab:props} \end{table*} \section{Results \& Discussion} \label{Sec:Res} We compare our results to massive, quiescent galaxies at a range of redshifts. The first sample, from \citet{Posacki2015}, is a reanalysis of 55 massive early-type galaxies at $z\sim0.2$ from SLACS \citep{Treu2010} and a subset of 223 \Hbeta\ massive absorption line galaxies in the local volume from ATLAS$^{\rm 3D}$ \citep{Cappellari2013b}. Galaxies selected from SDSS with velocity dispersions $\sigma>350$~km/s at similar redshifts were also compared in an attempt to mitigate progenitor bias \citep{Bernardi2006, Saracco2020}. \citet{Mendel2020} compiled and reanalyzed spectra from early-type galaxies at $1.4<z<2.1$, including spectra presented in \citet{Cappellari2009, Newman2010, Toft2012, Bezanson2013a, vandeSande2013, Belli2014a, Belli2014b, Barro2016, Belli2017}. In addition to our eight $z\gtrsim3$ UMGs, we fold in six previously published $z>3$ UMGs with velocity dispersion measurements: SXDS-27434 \citep{Tanaka2019}; C1-23152 (published by members of our group in \citet{Saracco2020} and subsequently renamed as COS-DR3-160748 in the context of the MAGAZ3NE Survey; \citet{Forrest2020b}); ZF-COS-20115, 3D-EGS-40032, 3D-EGS-18996, and 3D-EGS-31322 \citep{Esdaile2021}. Galaxies in these works have also been studied spectroscopically in \citet{Valentino2020}; \citet{Marsan2015} and \citet{Forrest2020b}; \citet{Glazebrook2017} and \citet{Schreiber2018b}, respectively. For the most part, the massive galaxies at $z>1.4$ were selected for spectroscopic follow-up via a combination of magnitude/stellar mass, color/SFR, and photometric redshift cuts. Nonetheless it is important to keep in mind that these cuts are not identical given the different survey depths, photometric wavelength coverage, and photometric redshift tools. Thus it is possible that studies are selecting different sub populations of massive quiescent galaxies. In Figure~\ref{fig:colors}, we show the rest-frame colors, stellar masses, and star-formation rates of the objects in the $z\gtrsim3$ sample. Most of the galaxies are consistent with recently quenched post-starburst galaxies, as they lie in the lower left of the $UVJ$ quiescent wedge or slightly blueward of it and show SFRs significantly below the main sequence for their mass at this redshift. The two notable exceptions to this are COS-DR3-202019 and SXDS-27434 \citep{Tanaka2019}. The former has \mbox{$SFR=82$~$M_\odot$/yr} and is the reddest and most massive of the sample, consistent with a dusty star-forming galaxy \citep{Forrest2020b}, while the latter has \mbox{$SFR=24$~$M_\odot$/yr} and has the bluest $(U-V)_{\rm REST}$ color of the sample \citep{Valentino2020}. \subsection{Large Velocity Dispersions} The best-fit velocity dispersions for the MAGAZ3NE sample are very large, at $\sim400$ km s$^{-1}$. Nonetheless, several galaxies at $z\sim2$ have previously been measured with similarly high velocity dispersions \citep{vanDokkum2009a, vandeSande2013, Belli2014b, Belli2017}. These velocity dispersions confirm the large stellar masses of these objects while being independent of the various problems intrinsic to SED fitting such as choice of IMF and contamination by emission lines (see Section~\ref{Sec:MassProbs}). A positive correlation between stellar velocity dispersion and stellar mass is expected as the mass within the small sizes over which we probe stellar velocity dispersion is dominated by stars. At $1.4<z<2.1$, the data compiled in \citet{Mendel2020} show a positive correlation between the two, though individual galaxies show significant scatter. A least-squares regression to the entire $z\gtrsim3$ sample shows a vertical offset towards larger velocity dispersions at a given stellar mass, but a similar slope to both the $1.4<z<2.1$ sample and a sample of massive quiescent galaxies from SDSS \citep{Zahid2016}, shown in Figure~\ref{fig:VDmass}. \subsection{The Size-Stellar Mass Relation} At a given epoch, the effective radius and stellar mass of a galaxy are also correlated, though quiescent and star-forming galaxies tend to follow different relations, and those relations evolve with time to smaller sizes for a given stellar mass at earlier times \citep[\eg][]{vanderWel2014, Straatman2015a, Mowla2019a, Marsan2019}. Indeed, the $1.4<z<2.1$ sample from \citet{Mendel2020} is in agreement with the $z\sim1.75$ relation for early-type galaxies presented in \citet{vanderWel2014} using data from 3D-HST: \begin{eqnarray*} r_{\rm e}/{\rm kpc} &=& 1.23 \times (M_*/5\times10^{10}M_\odot)^{0.76}, \end{eqnarray*} or equivalently, \begin{eqnarray*} \log(r_{\rm e}/{\rm kpc}) &=& -8.04+0.76\log(M_*/M_\odot) \end{eqnarray*} From Monte Carlo resampling of the $z\gtrsim3$ galaxies with $SFR<3$~M$_\odot$/yr we find \begin{eqnarray*} \log(r_{\rm e}/{\rm kpc}) &=& -9.73(\pm 1.50)+0.87 (\pm 0.15) \log(M_*/M_\odot), \end{eqnarray*} that is, smaller sizes for a given stellar mass showing a statistically consistent, but perhaps slightly steeper relation with stellar mass (see Figure~\ref{fig:sizemass}). Relative to the $z\sim1.75$ relation, this $z\gtrsim3$ fit shows smaller sizes by a factor greater than 3 at \logM$\sim11$ and a factor of $\gtrsim2$ at \logM$\sim11.5$, which also agrees with the redshift size evolution shown in \citep{Straatman2015a}. Limiting the $z\gtrsim3$ sample to quiescent galaxies with $HST/WFC3$ imaging does not significantly change the best-fit relation, though including the galaxies with $SFR>3$~M$_\odot$/yr does result in a steeper slope. \subsection{Comparison of Dynamical Mass and Stellar Mass} The dynamical and stellar masses for the $z\gtrsim3$ sample are listed in Table~\ref{tab:props} and shown in Figure~\ref{fig:sdmass}. For massive, quiescent galaxies with little gas or dust and small sizes, the dynamical and stellar masses are expected to be quite similar as the central regions are dominated by baryons with little dark matter contribution. The most obvious exception to this in the MAGAZ3NE sample is COS-DR3-202019 (the most massive galaxy in the sample), which has a radius $\sim3\times$ larger than any other galaxy in the sample, and is also the only one that shows evidence of ongoing star formation (see Figure~\ref{fig:colors}), but is still consistent with a 1-to-1 ratio between stellar and dynamical mass within $1\sigma$. The consistency of the $z\gtrsim3$ sample's ratios of dynamical to stellar mass, \MDS, with unity suggests that the Chabrier IMF used to derive the stellar masses for these objects is in general reasonable. While similarly massive galaxies at lower redshifts appear to prefer heavier IMFs \citep[\eg][]{Conroy2012, Cappellari2013a, Zahid2017}, at $z\sim1.7$ \citet{Mendel2020} also find that a lighter IMF such as the Chabrier IMF is required to prevent stellar masses from exceeding dynamical masses. Dynamical masses in significant excess of stellar masses would be expected if either the choice of IMF is incorrect or if there is an appreciable fraction of dark matter in the galaxy. We note that the contribution of dark matter for similar galaxies at lower redshift, $\sim 5-20\%$ \citep{Cappellari2013b, Mendel2020}, is too small to be quantified here given the observational errors involved. That said, a comparison of \MDS\ to stellar velocity dispersion can still yield important insights. For instance, high redshift quiescent galaxies have lower ratios of \MDS\ for a given velocity dispersion than galaxies at lower redshifts \citep{vandeSande2013, Hill2016, Belli2017, Mendel2020, Esdaile2021}, which is suggestive of a preference for a lighter IMF in such systems in early times. While our data do not allow for significant constraints on dark matter content or IMF form for individual galaxies, a combination of the eight new MAGAZ3NE galaxies presented here with the four UMGs from \citet{Esdaile2021}, one from \citet{Saracco2020}, and one from \citet{Tanaka2019} allow the first look at these properties using a statistical sample at $z\gtrsim3$, shown in Figure~\ref{fig:alpha}. We perform a linear regression between the logarithm of \MDS\ and the logarithm of the velocity dispersion at the effective radius for our sample, as well as those at $z\sim0.2$ and $z\sim1.7$. Additionally, we use Monte Carlo resampling (accounting for the correlated errors) to characterize the uncertainties on the resulting best-fits: \begin{eqnarray} \log(M_{\rm dyn}/M_{*})_{z\sim0.2} & = & (0.29 \pm 0.02) +\\\nonumber && (0.40 \pm 0.05) \times \log(\sigma_{\rm e}/350)\\ \log(M_{\rm dyn}/M_{*})_{z\sim1.7} & = & (0.30 \pm 0.06) + \\\nonumber && (1.25 \pm 0.20) \times \log(\sigma_{\rm e}/350) \\ \log(M_{\rm dyn}/M_{*})_{z\gtrsim3} & = & (0.03 \pm 0.04) + \\\nonumber &&(1.29 \pm 0.36) \times \log(\sigma_{\rm e}/350) \end{eqnarray} The best-fit slope at $z\gtrsim3$ ($1.29 \pm 0.36$) is consistent with that of the fit at $z\sim1.7$ ($1.25 \pm 0.20$) and significantly steeper than the low-redshift relation ($0.40 \pm 0.05$). Additionally, the $z\gtrsim3$ sample is offset to lower \MDS\ by $\sim0.3$ dex relative to the $z\sim 1.7$ sample and $\sim0.5$ dex relative to the low redshift sample for a given velocity dispersion. This means that while the $z\gtrsim3$ sample shows the same trend of preferring a heavier IMF at higher velocity dispersions relative to lower velocity dispersions, many of the highest velocity dispersion objects prefer a Chabrier IMF (or an IMF lighter than Chabrier) to a Salpeter IMF (see Figure~\ref{fig:alpha}). In order for high velocity dispersion galaxies to prefer a bottom-heavy IMF such as Salpeter or even heavier \citep[\eg][]{Conroy2012}, at least one of several parameters must be systematically incorrect and provide a 0.2 dex ($\sim60\%$) gain in \MDS, addressed below. \subsubsection{Are the high velocity dispersions too low?} The reported velocity dispersions herein are some of the largest measured \citep[see also][for other galaxies with $\sigma>400$ km s$^{-1}$]{vanDokkum2009a, vandeSande2013, Saracco2020}. To reach agreement with a Salpeter IMF, the velocity dispersions would have to be even higher by $\sim100$ km s$^{-1}$ for the highest velocity dispersion objects (and $\sim 500$ km s$^{-1}$ for those galaxies with lower velocity dispersions). This increase is perhaps not unrealistic for some galaxies here given the errors on the measured velocity dispersions. Intriguingly, this is in line with the large velocity dispersion of $510~\rm{km\ s}^{-1}$ measured for a massive, compact galaxy at $z=2.2$ in \citet{vanDokkum2009}. Of course, while we have performed a robust investigation into the possible systematics involved in the calculation of velocity dispersions for this sample (see \mbox{Appendix~\ref{App:pPXF}}), the fact remains that the systematics may contribute to the results. Another complicating factor here is the possibility of significant rotation in these systems, which would make the use of the measured velocity dispersion in the calculation of dynamical mass incorrect. Several massive, quiescent galaxies at $z\sim2$ are disk-dominated and have been confirmed to have significant rotation thanks to gravitational lensing \citep{Toft2017, Newman2018b}. Resolving rotation is not possible with our data. Measured velocity dispersions could be inflated by a rotational component if a spectral slit is oriented with the major axis of the disk or could be underestimated if the spectral slit is misaligned. Our sample is not large enough to claim that these effects cancel each other out on average. \subsubsection{Are the size measurements too small?} The GALFIT package used is widely used and appears to be generally accurate in calculating sizes. Several of the objects in the $z\gtrsim3$ sample are not resolved in ground-based imaging, which can lead to incorrect size estimates below FWHM/2, particularly if the PSF is not well determined \citep{Haussler2013, Nedkova2021}. Fortunately, the agreement between sizes calculated from $HST$ and ground-based imaging indicates that the sizes are reliable. To find agreement with the $z\sim1.75$ relations from \citet{vanderWel2014} and the \citet{Mendel2020} dataset, the sizes must be $2-4\times$ larger than measured. To improve consistency with a Salpeter IMF, the sizes must be underestimated by $\sim30\%$, which is considered unlikely as objects which are barely resolved are more likely to have their sizes overestimated. \subsubsection{Are the dynamical masses calculated appropriately?} The calculation of dynamical mass, in addition to relying upon accurate size and stellar velocity dispersion measurements, also contains a factor to account for the distribution of mass in the system. The standard transformation used is a function of S\'ersic index, $n$. While the imaging used to calculate sizes is not deep enough to reliably recover a S\'ersic index \citep[this usually requires SNR $\sim3\times$ deeper than that required for size measurements;][]{Haussler2007, Haussler2013}, the assumption of $\beta(n)=5$ corresponds to $n\sim5.5$. An increase of $\sim60\%$ in this factor would require $n\sim1.2$, typically seen in larger galaxies with well developed disks, which remains a possibility. The correction factor $\beta(n)$ was originally derived using low redshift elliptical galaxies, and while it shows great precision across $2<z<10$ \citep{Bertin2002, Cappellari2006}, it is possible such a transformation is not accurate for these galaxies for some reason. Deeper, higher resolution imaging, as may be obtained with \textit{JWST} may allow for insights into this possibility. \subsubsection{Are the stellar masses accurate?}\label{Sec:MassProbs} Mass inaccuracies can be caused by the presence of strong emission lines, which can cause an overestimate of up to $\sim0.5$ dex \citep[\eg][]{Stark2013, Salmon2015, Forrest2017}. However, \citet{Forrest2020b} model stellar masses for the MAGAZ3NE galaxies in this sample after correcting broadband photometry for any strong emission lines seen in the spectra (\OII, \OIII, \Hbeta), though of the sample here only COS-DR3-202019 shows significant emission. The only other strong line which could normally be an issue is H$\alpha$, though at the redshifts of the sample, this line falls in between the $K$-band and the IRAC 3.6$\mu$m bandpass, and so should not affect the photometry either. It is also known that the choice of modeling parameters and program can lead to differences in stellar mass calculations of around 0.2 dex \citep[\eg][]{Mobasher2015}. \citet{Leja2019} compare stellar masses for objects in the 3D-HST study derived using FAST \citep{Kriek2009} and Prospector-$\alpha$ \citep{Leja2017}, and while they find a systematic offset of up to 0.4 dex in stellar mass, these differences appear to be $<0.1$ dex for high mass galaxies at high redshifts, as is the case for our sample. Regardless, the Prospector-$\alpha$ code outputs higher stellar masses than FAST, and thus any such offset would only increase the tension with \eg\ a Salpeter IMF. The possibility of young stars outshining older populations in a spectrum and leading to a light-weighted stellar mass different from the true stellar mass would similarly result in an \textit{underestimate} of the stellar mass. Of course, the calculation of stellar masses rests upon the modeling of stellar populations, often based on the spectra of local stars. It is possible that these model populations are not applicable to stellar populations in the early Universe. A test of this possibility would require high-resolution stellar spectra at high redshifts and is not currently technologically feasible. \subsection{Evolutionary Insights} The high velocity dispersions presented here for the $z\gtrsim3$ sample support the large stellar masses calculated through SED modeling for massive, high redshift galaxies and suggest that SED modeling of large photometric samples can be trusted to first order, outliers notwithstanding. We also note however, that the velocity dispersions do not support much more mass than the stellar mass, implying that the contribution of dark matter at the centers of these compact galaxies is small. \subsubsection{Progenitor Bias} Galaxies with velocity dispersions such as those measured for some of our UMGs are exceedingly rare in the local Universe. While analyzing the apparent trends seen in previous figures, we must carefully consider factors such as progenitor bias as well as the possibility that descendants of the rare $z\gtrsim3$ UMGs do not exist in the limited local volume. In an attempt to mitigate these effects, we compare the UMGs in the $z\gtrsim3$ sample to an additional sample of massive low-redshift ETGs which are among the most massive galaxies in SDSS and which are not actively forming stars \citep{Bernardi2006, Saracco2020}. We correct the published velocity dispersions, originally corrected to $r_{\rm e}/8$, and transform them to $r_{\rm e}$ using the relation from \citet{Jorgensen1995}, \begin{eqnarray} \frac{\sigma_{\rm ap}}{\sigma_{\rm e}/8} &=& \bigg{(}\frac{r_{\rm aper}}{r_{\rm e}/8}\bigg{)}^{-0.04} \end{eqnarray} which was used for the original correction in \citet{Bernardi2006} \citep[though see discussion about issues with this method for ETGs in][]{LaBarbera2019}. This corresponds to a correction factor of $8^{-0.04}=0.92$. Galaxies in this sample have larger stellar masses, velocity dispersions, and dynamical-to-stellar mass ratios than most of the $z\sim0.2$ sample from \citet{Posacki2015} (Figure~\ref{fig:bernardi}). Velocity dispersion is known to correlate well with age for SDSS ETGs \citep[\eg][]{VanDerWel2009, Zahid2017}, and we thus assume that the stellar populations of these galaxies are also quite old. We note that making cuts to the galaxy samples herein by stellar mass or velocity dispersion do not result in qualitative changes to our conclusions. Spatially resolved studies of local massive ETGs have shown that stellar populations at their cores appear to be older than stars on the outskirts, as well as being regions with higher velocity dispersions \citep[\eg][]{vanDokkum2017, LaBarbera2019}, consistent with the bulk of star formation occurring at $z\gtrsim2$, followed by passive evolution via gas-poor (dry) mergers. Dry major mergers, having a mass ratio between the two galaxies close to unity, increase both stellar mass and radius at similar rates with minimal new star formation (\ie\ retaining an old stellar age) and without much change in velocity dispersion \citep[\eg][]{Hopkins2009}. Dry minor mergers on the other hand are expected to increase the effective radius approximately twice as fast as the stellar mass while also decreasing velocity dispersion slightly, though the cores of these galaxies could still retain high velocity dispersions \citep[\eg][]{Bezanson2009, Saracco2020}. As seen in the left panel of Figure~\ref{fig:bernardi}, passive evolution of the $z\gtrsim3$ UMGs via dry minor mergers could lead to galaxies with sizes, stellar masses, and velocity dispersions of some of the most massive galaxies in SDSS \citep{Bernardi2006}. While the $z\gtrsim3$ UMGs could evolve into galaxies at the massive end of the $z\sim1.7$ sample via dry minor mergers, those galaxies in the $z\sim1.7$ sample with lower stellar masses and velocity dispersions descend from galaxies with different properties than the $z\gtrsim3$ UMGs. In particular, these progenitors have lower stellar masses and are possibly still forming stars at $z\sim3$. Similarly, the $z\sim1.7$ sample could plausibly evolve into galaxies in the \citet{Bernardi2006} sample, but only those with larger velocity dispersions. Most of the $z\gtrsim3$ UMGs herein are compact, post-starburst galaxies. A gas-rich (wet) merger may have triggered such a burst of star formation \textit{in situ}, thus boosting the stellar mass significantly while keeping the effective radius small in contrast to the dry merger scenarios above \citep[\eg][]{Hopkins2009}. Major mergers, wet or dry, are expected to be few in number for massive galaxies, and it is perhaps the case that the $z\gtrsim3$ UMGs have simply undergone additional major mergers relative to the progenitors of the lower mass half of the $z\sim1.7$ sample. If so, the possibility exists that further major mergers would evolve these galaxies into systems more massive than any in the local volume. Regardless, while the evolution of sizes, stellar masses, and velocity dispersions can be explained with mergers, the dynamical-to-stellar mass ratio is less easily explained. If dynamical mass is calculated as $M_{\rm dyn} \propto r_{\rm e} \sigma_{e}^2$, then $\Delta \log(M_{\rm dyn}/M_{*})<0.05$ for all three merger cases described above. Instead of an offset in dynamical-to-stellar mass at a given velocity dispersion, it may be that the increase in velocity dispersion from wet mergers is causing an offset in velocity dispersion at fixed dynamical-to-stellar mass, and we simply do not have any galaxies with large dynamical-to-stellar mass ratios in our $z\gtrsim3$ sample. Major mergers can also introduce rotation into the system, making the dynamical mass calculations incorrect. \subsubsection{IMF} Detailed studies of absorption lines in local massive quiescent galaxies have suggested that the cores of these galaxies require a bottom-heavy ``super-Salpeter" IMF \citep[\eg][]{Lasker2013, Saulder2015, Conroy2017}. The higher inferred mass-to-light ratios associated with such a bottom-heavy IMF also correlate with velocity dispersion \citep[\eg][]{Conroy2013, Cappellari2013b}. This is not only seen in samples of galaxies, but also within individual nearby galaxies, where the central cores have larger velocity dispersions and heavier inferred IMFs than the outskirts \citep[\eg][]{LaBarbera2019}. While the sample of $z\gtrsim3$ galaxies in this study do show a similar trend towards a heavier IMF with higher velocity dispersion, none of the galaxies in our sample show evidence for a ``super-Salpeter" IMF and most require a Chabrier IMF in order for the stellar mass to remain below the dynamical mass, despite having very large velocity dispersions. This creates a conundrum, as massive compact systems at high redshift, such as the $z\gtrsim3$ sample are generally thought to be the progenitors of the low redshift, high-mass sample such as that from \citet{Bernardi2006}, growing largely through mergers as described above \citep[\eg][]{Bezanson2009, VanDerWel2009, Saracco2020, Mendel2020}. Such a picture does not offer a way to significantly change the observed IMF from high-redshift progenitors to the cores of local, massive ETGs. An alternative possibility is that the IMF is determined by metallicity \citep[\eg][]{Koppen2007}, which shows a close positive correlation with inferred IMF slope for local ETGs from IFU data in the CALIFA survey \citep{Martin-Navarro2015}. In this view, massive galaxies at early times undergo gas-rich mergers and form substantial fractions of their stars with gas containing a significant amount of metals from previous generations of stars. This causes new star formation at high metallicity in the $z\gtrsim3$ UMGs \citep{Saracco2020} which occurs with a bottom-heavier IMF. Meanwhile, less massive galaxies, being located in less massive halos, are more likely to build up their stellar mass not through merger-induced star formation, but by inflows of pristine gas. Further, due to their lower masses, these galaxies lose many of the metals they produce via galactic outflows. This then creates a lower-metallicity environment for star formation, which generates a bottom-lighter IMF. This picture is also consistent with the mass-metallicity relationship seen for star forming galaxies out to $z>3$ \citep[\eg][]{Tremonti2004, Lian2018, Sanders2021}. Subsequent growth of massive galaxies via minor mergers then deposits stars from the lower mass galaxies in the outskirts of the massive galaxy, producing the radial IMF trends seen in spatially resolved data \citep[\eg][]{vanDokkum2017, LaBarbera2019}. \section{Conclusions}\label{Sec:Conc} We have calculated stellar velocity dispersions and sizes for 8 UMGs at $z\gtrsim3$, more than doubling the sample at this epoch. The high dispersions, on the order of $\sim400~\rm{km\ s}^{-1}$, are some of the largest measured, about $1.5\times$ those of galaxies at $z\sim1.7$ of similar stellar mass. They also agree with the large stellar masses derived from SED fitting, supporting the conclusion that ultramassive quiescent galaxies at $z>3$ do exist in non-negligible numbers. Size measurements for these objects additionally show a continuation of the evolution to smaller sizes at higher redshifts, with galaxies of similar stellar mass being about 1/3 the size of their $z\sim1.7$ counterparts. We have used these size and stellar velocity dispersion measurements to calculate the dynamical mass. The ratio of dynamical-to-stellar mass for these objects shows a trend with velocity dispersion as seen at lower redshifts, though it is offset to higher velocity dispersions / lower mass-to-light ratios. This favors a Chabrier (or even bottom-lighter) IMF for most of the sample and is in tension with the ``super-Salpeter" IMFs seen in the cores of the most massive galaxies in the local Universe. \section{Acknowledgements} The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. This work also uses data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under ESO programme ID 179.A-2005 and on data products produced by CALET and the Cambridge Astronomy Survey Unit on behalf of the UltraVISTA consortium. This material is based upon work supported by the National Science Foundation under Cooperative Agreement No. AST-2009442. We gratefully acknowledge support from the NASA Astrophysics Data Analysis Program (ADAP) through grant numbers 80NSSC17K0019 and NNX16AN49G, the National Science Foundation through grants AST-1517863 and AST-2205189 and HST program numbers GO-15294 and GO-16300 provided by NASA through grants from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. Data presented herein were obtained using the UCI Remote Observing Facility, made possible by a generous gift from John and Ruth Ann Evans. Some of the material presented in this paper is based upon work supported by the National Science Foundation under Grant No. 1908422. P.S. acknowledges support by the grant PRIN-INAF-2019 1.05.01.85.11. B.F. also thanks B. Lemaux, A. Pillepich, and L. Lewis for helpful discussions and input, as well as the authers of the codes referenced below, upon which this work has relied heavily. Thanks to the anonymous referee as well, whose comments improved the manuscript. \software{ Astropy \citep{Astropy2013,Astropy2018}, FAST++ \citep{Schreiber2018a}, \texttt{grizli} (ascl.net/1905.001), IPython \citep{Perez2007}, Matplotlib \citep{Hunter2007}, Molecfit \citep{Smette2015, Kausch2015}, MosfireDRP (ascl.net/1908.007), NumPy \citep{Oliphant2006}, pPXF \citep{Cappellari2017}, PyPeit \citep{Prochaska2020} } \bibliography{/Users/ben/Documents/library} \clearpage \appendix \label{App} \renewcommand\thefigure{\thesection.\arabic{figure}} \renewcommand\thetable{\thesection.\arabic{table}} \section{Comparison of MOSFIRE and NIRES spectra for XMM-VID1-2075}\label{App:K2075} \setcounter{figure}{0} \setcounter{table}{0} One of the UMGs in this work, XMM-VID1-2075, has both a MOSFIRE $K$-band spectrum and a NIRES spectrum, which also includes both $K$-band and $H$-band data with some signal-to-noise. The spectra appear quite similar (Figure~\ref{fig:App_2075}). We fit the $K$-band spectrum from each instrument with the galaxy photometry using FAST++ and compare the results. The redshifts from the two fits are very similar, with $z_{\rm MOSFIRE}=3.4523$ and $z_{\rm NIRES}=3.4482$, a difference of $\sim 0.1\%$. In both cases, the best fit indicates a galaxy with \logM$\sim11.5$, $A_V\sim0.3$, and age $\sim 300-500$ Myr. However, including the NIRES $H$-band data while performing the fit results in a slightly older, less massive, less dusty galaxy (\logM$\sim11.3$, $A_V\sim0$, age $\sim800$ Myr). When each spectrum is fit with pPXF with a set of inputs and assuming the best-fit redshift to that spectrum, the results are statistically consistent. In this work we use the values from the fit to the entire $H$- and $K$-band NIRES spectrum, as this provides a greater number of features for determination of the velocity dispersion. \section{Dependence of Velocity Dispersions on pPXF Inputs}\label{App:pPXF} \setcounter{figure}{0} \setcounter{table}{0} Due to the low SNR of our spectra (order $\sim1$/pixel) compared to those pPXF was originally tested on (order $\sim100$/pixel), the resultant velocity dispersions can be sensitive to various parts of the fitting mechanism, including choice of templates, additive polynomial order, and wavelength range, among others. Extensive tests along these lines have been performed by \citet{vandeSande2013} for a sample of galaxies at $z\sim2$, some of which we reproduce for our sample. \subsection{Age and Metallicity Template dependence}\label{App:AgeMet} As the spectra herein have low SNR/pixel, slightly different templates can yield similar fits to the spectra alone. In particular, the degeneracy between age and metallicity can affect line widths and depths in ways that are difficult to disentangle using a low SNR spectrum alone. These difficulties can be somewhat alleviated by taking into account the broadband photometry of a galaxy. Similar to \citet{vandeSande2013} and \citet{Hill2016}, we test model dependency in this regime using the BC03 models due to their extended wavelength coverage. We use FAST++ to fit the spectra and photometry in combination with age and metallicity fixed over a range of values (each combination of $\log$(Age/yr)=[8.0, 8.1, 8.2, 8.3, 8.4, 8.5, 8.55, 8.6, 8.65, 8.7, 8.75, 8.8, 8.85, 8.9, 8.95, 9.0, 9.05, 9.1, 9.15, 9.2, 9.25] and Z=[0.004, 0.008, 0.02 (solar), 0.05]). We then use pPXF to fit the velocity dispersion of the galaxy using the best-fit template from each combination of age and metallicity, and compare to the reduced $\chi^2$ value from the FAST++ fit. An example of the results are shown for COS-DR3-84674 in Figure~\ref{fig:App_dAgeMet}. In all cases, the models show clear minima for each choice of metallicity, though in some cases a particular metallicity is not statistically favored. The model with the lowest reduced $\chi^2$ was used for this paper and in subsequent tests. Importantly, this choice is independent of pPXF and therefore also independent of additive polynomial order and spectral wavelength range (see following sections). \subsection{Dependence on Additive Polynomial Order}\label{App:Poly} The pPXF program allows for addition of a $d$-dimensional Legendre polynomial to a template in order to better match the observed spectrum. This provides better fits to lower SNR features in the observed spectral line profiles. A choice of polynomial order which is too low can fail to accurately match the template and observed spectrum, while excessively large order polynomials end up perturbing a template to match observational noise which often yields nonsensical results. We test the dependence of output velocity dispersion on polynomial order by forcing pPXF to fit the observed spectrum with the single best-fit BC03 template as determined above with order fixed to each $d=[1,2,3,..,50]$. Example results are shown in Figure~\ref{fig:App_dPoly}. For the most part, we see the greatest variability in output velocity dispersion at $d>20$, as well as some at $d<5$, while between these values the output velocity dispersion appears generally stable. \subsection{Dependence on Spectral Wavelength Range}\label{App:Wave} Velocity dispersion fits are also dependent upon the wavelengths available in the observed spectrum, where the inclusion or exclusion of specific spectral features can alter results. We refit truncated spectra using a range of starting wavelengths from $3200<\lambda_{\rm rest, blue}/{\rm \AA}<5000$ and ending wavelengths $4200<\lambda_{\rm rest, red}/{\rm \AA}<6000$ and analyze the results (Figure~\ref{fig:App_dWave}). The most apparent result is that when the spectrum includes the Ca H\&K lines, the velocity dispersion results are significantly more stable. In many cases, there also appears to be variability with the inclusion or exclusion of H$\delta$. Notably, we tend not to see much dependence on the H$\beta$ feature, which suggests that there is little line infilling. Further insights are difficult due to the small sample, low SNR of the spectra, and dependence of results on polynomial order. Given the strong dependence of the results on the inclusion of Ca H\&K, we also fit the spectra over the narrow range of $3900<\lambda_{\rm rest}/{\rm \AA}<4000$, so as to isolate these features. However, doing so precludes the use of the higher order polynomials discussed above, as the narrow wavelength range means each order has outsize effects on the result. Nevertheless, the results of this fit are statistically consistent within $1\sigma$ for 5 of the galaxies. The remaining 3 (COS-DR1-99209, COS-DR3-111740, and COS-DR3-202019) showed significant deviations at $d=0$ when testing polynomial order above and so this discrepancy is not surprising. \subsection{Dependence on Template Library}\label{App:Lib} In this work we use the BC03 template library due to its longer wavelength coverage, which allows joint fitting with photometry using FAST++. However, the velocity dispersions from pPXF can be highly dependent upon the availability of templates which are appropriate to the data. As such, we analyze results from using solely pPXF with three libraries: the BC03 library, the Indo-US stellar library templates \citep{Valdes2004}, and SSPs from the MILES stellar library \citep{Sanchez-Blazquez2006, Vazdekis2010}. For half of the galaxies all three libraries yield statistically similar results with other parameters fixed (Figure~\ref{fig:App_dLib}), while in the other half the MILES and BC03 outputs are similar and the Indo-US library produces discrepant results. \subsection{Overall Distribution of Velocity Dispersions}\label{App:Dist} As mentioned in the text, we perform a large number of fits with pPXF for each galaxy. Due to the variety of results and uncertainties associated with any particular fit, we instead use the distribution of results as a whole to determine stellar velocity dispersion for a particular galaxy. Each fit was convolved with a Gaussian kernel with a standard deviation equal to the reported uncertainty on the velocity dispersion and additionally weighted by the reduced $\chi^2$ of the fit. The normalized summation of these fits for the 8 MAGAZ3NE UMGs are shown in Figure~\ref{fig:VDdist}.

Title: RINGO3 polarimetry of very young ZTF supernovae

Abstract: The early phases of the observed evolution of the supernovae (SNe) are expected to be dominated by the shock breakout and ``flash" ionization of the surrounding circumstellar medium. This material arises from the last stages of the evolution of the progenitor, such that photometry and spectroscopy of SNe at early times can place vital constraints on the latest and fastest evolutionary phases leading up to stellar death. These signatures are erased by the expansion of the ejecta within ~5 days after explosion. Here we present the earliest constraints, to date, on the polarization of ten transients discovered by the Zwicky Transient Facility (ZTF), between June 2018 and August 2019. Rapid polarimetric followup was conducted using the Liverpool Telescope RINGO3 instrument, including 3 SNe observed within <1 day of detection by the ZTF. The limits on the polarization within the first 5 days of explosion, for all SN types, is generally <2%, implying early asymmetries are limited to axial ratios >0.65 (assuming an oblate spheroidal configuration). We also present polarimetric observations of the Type I Superluminous SN 2018bsz and Type II SN 2018hna, observed around and after maximum light.

https://export.arxiv.org/pdf/2208.12285

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} (stars:) supernovae: general -- techniques: polarimetric \end{keywords} \section{Introduction} \label{sec:intro} With the advent of deep, wide-field and high-cadence surveys, it has been possible to discover new supernovae (SNe) within hours of explosion. Rapid followup observations of these SNe has provided a new insight into the stellar origins of these explosion. Early photometric observations of the emergence of the explosion shock has, in a number of cases \citep[e.g.][]{2008natur.453..469s,2010apj...724.1396o, 2016apj...820...23g, 2017apj...848....8r, 2018natur.554..497b}, revealed a behaviour that cannot be explained by just the shock breaking out of the stellar surface \citep{2017hsn..book..967w}. Instead, the early, rapid rise in brightness can be greatly affected by the presence of circumstellar material (CSM). For a subset of Type Ia SNe, early photometric observations have revealed an ultraviolet excess, which has been interpreted as the shock interaction between the ejecta and a companion star \citep{2015natur.521..328c, 2017apj...845l..11h}. Early spectroscopy has revealed the presence of ``flash ionized" species, such as He {\sc ii} and N {\sc iv}, which correspond to the very quick ionization of the surrounding CSM \citep{2016apj...818....3k,2019mnras.483.3762k}. In the case of SN~2013cu, early observations by \citet{2014natur.509..471g} were able to reveal the presence of a Wolf-Rayet-like wind immediately around the progenitor \citep{2007ara&a..45..177c}. Using flash spectroscopy, \citet{2017natph..13..510y} showed that some SN progenitors may exhibit enhanced levels of mass loss as pre-supernova instabilities become important in the final years before explosion. \citet{2020arxiv200809986b} report that at least $30\%$ of hydrogen-rich SNe exhibit such features in very early observations, consistent with elevated mass-loss just prior to explosion. Despite the power of early time observations to place important constraints on the nature of the progenitor system, the observable signatures disappear by $\sim 5$ days, after the ejecta overrun the immediately surrounding dense CSM \citep{2014natur.509..471g}. Early-time observations are, therefore, very sensitive to the very last phases of stellar evolution. The early-time optical observations of very young supernovae have, predominantly, been in the form of photometry and spectroscopy. Polarimetry has been established as a sensitive probe of the presence of departures from spherical symmetry in SN explosions \citep{2008ara&a..46..433w}. Observations around maximum light and at later epochs, as the photosphere recedes into the ejecta with time, have revealed important clues to the physics of the explosions responsible through their imprint on the geometry of the ejecta. In general, core-collapse (CC) SNe show increasing degrees of polarization with time indicating the ejecta becoming more asymmetric the closer to the origin of the explosion. On the other hand, Type Ia SNe exhibit the opposite behaviour appearing to become progressively more spherical closer to origin of the explosion \citep{2008ara&a..46..433w}. The application of polarimetry to SNe at very early times, however, has been limited, with the earliest spectropolarimetric observation of a Type Ia SN published to date, occurring at only $\sim 5$ days after explosion \citep{2020apj...902...46y}. Given the power of polarimetry to probe the 3D structures of these events, at early times it has the potential to probe the shape of the progenitor system, including the nature of the mass loss prior to explosion. Indeed, as shown by \citet{2014mnras.442.1166m} and \citet{2017mnras.470.1491r}, in the context of SN 2009ip, polarimetry can provide constraints on both the 3D physical characteristics of the explosions and their interaction with the CSM, that are not accessible with ordinary photometric and spectroscopic observations. A major difficulty with conducting rapid polarimetric followup of young supernovae is establishing the connection between the discovery surveys and facilities with an appropriate polarimetric observing capability. Here, we report a pathfinder campaign, using the 2.0m Liverpool Telescope \citep{2004spie.5489..679s} and the RINGO3 polarimeter \citep{2012spie.8446e..2ja}, to acquire early-time observations of explosive transients; in particular those discovered by the Zwicky Transient Facility \citep[ZTF;][]{2019pasp..131a8002b, 2019pasp..131g8001g, 2019aas...23313106g}. The RINGO3 instrument and its predecessors \citep{2006spie.6269e..5ms, 2010spie.7735e..49s} were designed for the rapid followup of the Gamma Ray Burst afterglows \citep{2007sci...315.1822m,2009natur.462..767s,2013natur.504..119m}, exploiting the flexibility to rapidly observe new targets afforded by the robotic nature of the Liverpool Telescope. Given the location of ZTF at Palomar Observatory, California, USA, it is possible to trigger polarimetric followup observations with the Liverpool Telescope (La Palma, Canary Islands, Spain) within $<24$ hours of discovery. \section{Observations} \label{sec:obs} \subsection{Data Acquisition \& Reductions} \label{sec:dataacq} All observations of the target SNe were conducted with the RINGO3 instrument\footnote{\url{https://telescope.livjm.ac.uk/TelInst/Inst/RINGO3/}} mounted on the Liverpool Telescope. RINGO3 operates with three separate channels, each with its own camera, with the light split by wavelength using two dichroics. The three channels are: ``d'' covering $7700 - 10\,000\mathrm{\angs}$,``e'' covering $3500 - 6400\mathrm{\angs}$ and ``f'' covering $6500 - 7600\mathrm{\angs}$; following the nomenclature of \citet{jermakphd}, we will refer to these band passes as $r^{\ast}$, $b^{\ast}$ and $g^{\ast}$, respectively. Each camera has a slightly different plate scale: $0.43\arcsec$, $0.44\arcsec$ and $0.49\arcsec$ for the $b^{\ast}$, $g^{\ast}$ and $r^{\ast}$ bands, respectively. Each RINGO3 channel has its own $512 \times 512\,\mathrm{px}$ Electron Multiplying CCD which have, for the type of observations considered here, negligible noise associated with readout. RINGO3 uses a rotating polaroid ($\sim 0.4$Hz) to sample all of the components of the Stokes parameters at eight separate polaroid position angles. Each camera produces, therefore, 8 exposures in 2.3 seconds (i.e. 24 exposures total for the 3 cameras). The RINGO Data Reduction Pipeline\citep{arnoldphd}\footnote{\url{https://telescope.livjm.ac.uk/TelInst/Inst/RINGO3/\#pipeline}} creates a series of mean stacked images: one for the entire duration of the observation and a series of mean stacked images for each minute of the observation. For this study we only consider the mean stacked image constructed from all exposures acquired at a given epoch. The reduced images of the science targets and the zero-polarization and highly-polarized standard calibration stars were retrieved from the Liverpool Telescope Archive\footnote{\url{https://telescope.livjm.ac.uk/cgi-bin/lt\_search}}. \subsection{Data Analysis Workflow} \label{sec:dawf} In order to analyse the data, we created a bespoke package to process all the observations and, ultimately, derive the linear Stokes parameters for the science targets. The approach to the analysis follows those presented by \citet{jermakphd} and \citet{2019mnras.482.4057m}. The data were first sorted into discrete datasets, containing all 24 files corresponding to one individual observation. For each dataset and for each camera, source detection was conducted on the image at the first rotor position. These positions were then used to conduct photometry on the images at all 8 rotor positions. Due to extreme vignetting for all three cameras, source detection was not conducted in the four corners of the images (corresponding to areas of $128 \times 128\mathrm{px}$). Aperture photometry was conducted using the {\it python} {\tt photutils}\footnote{\url{https://pypi.org/project/photutils/}} package. For the bright standards, we used a fixed aperture of radius 8 pixels. For the science targets, aperture sizes were selected to match the full width at half maximum to balance possible contamination from nearby stars or enhanced background (e.g. host galaxy), but maximise the signal-to-noise. In the event that it was not possible for the Liverpool Telescope Data Reduction Pipeline to confidently establish the World Coordinate System for each image, the target (or targets) of interest in each image were selected by hand. For all targets, the intensity ($I$) and the normalized linear Stokes parameters ($q$ and $u$) were calculated from the measured fluxes $f_{i}$ at each rotor position $i$. Following the prescription of \citet{2002A&A...383..360C}, the intensities corresponding to each of the three Stokes parameters is given by: \begin{equation} \begin{aligned} S_{I} & = \sum^{8}_{i = 1}f_{i}\\ S_{q} & = f_{2} + f_{3} + f_{6} + f_{7}\\ S_{u} & = f_{1} + f_{2} + f_{5} + f_{6} \\ \end{aligned} \label{eq:rawstokes} \end{equation} for which the normalised, instrumental Stokes parameters are: \begin{equation} \begin{aligned} q_{inst} & = \pi \left(\frac{1}{2} - \frac{S_{q}}{S_{I}} \right)\\ u_{inst} & = \pi \left(\frac{S_{u}}{S_{I}} - \frac{1}{2} \right) \end{aligned} \label{eq:measstokes} \end{equation} In order to correctly propagate the photometric uncertainties, we used Monte-Carlo sampling to create N = 10000 samples from the distribution $N(f_{i}, (\Delta f_{i})^2)$ and carried these distributions through Equations \ref{eq:rawstokes} and \ref{eq:measstokes}, to derive the corresponding distributions for $q_{inst}$ and $u_{inst}$. Observations of zero and highly polarized standards were used to remove any instrumental polarization signature. The standard stars observed as part of the Liverpool Telescope RINGO3 standard calibration plan are derived from \citet{1992aj....104.1563s}. The instrumental polarization offset $(q_{0}, u_{0})$ was calculated using the observations of the zero-polarization standard stars, such that: \begin{equation} \begin{aligned} q^{\prime} & = 1.14 (q_{inst} - q_{0}) \\ u^{\prime} & = u_{inst} - u_{0} \end{aligned} \end{equation} where the factor 1.14 corrects for elliptical distortion of the polarization circle of a constant polarization source in the $qu$ plane \citep{arnoldphd}. The polarization properties of the highly polarized standards in the appropriate RINGO3 wavelength channels are given in Table \ref{tab:app:hpol}, as previously used by \citet{2019mnras.482.4057m}. We used reported polarization values for 7 standards from the ultraviolet to the infrared \citep[in the $UBVRIJHK$ passbands][]{1992aj....104.1563s}, employing a fourth order polynomial to calculate the brightness-weighted polarization over the wavelength ranges corresponding to the three RINGO3 channels. \begin{table} \centering \caption{Polarized standards from \citet{1992aj....104.1563s} in the RINGO3 channels.\label{tab:app:hpol}} \begin{tabular}{ccccccccc} \hline Standard & \multicolumn{2}{c}{$b^{\ast}$/``e"}& & \multicolumn{2}{c}{$g^{\ast}$/``f"}& &\multicolumn{2}{c}{$r^{\ast}$/``d"}\\ \cline{2-3}\cline{5-6}\cline{8-9} & $p(\%)$ & $\theta$ ($^{\circ}$)& &$p(\%)$ & $\theta$ ($^{\circ}$)& &$p(\%)$ & $\theta$ ($^{\circ}$)\\ \hline BD$\,+25\,727$ & 5.99 & 31.2 & &6.13 & 31.5 & & 5.22 & 31.7\\ BD$\,+59\,389$ &6.40 & 98.2 & &6.27 & 98.2 & & 5.45 & 98.2\\ BD$\,+64\,106$ &5.48 & 96.9 & &5.00 & 96.8 & & 4.89 & 96.7\\ HD$\,$155528 &4.80 & 91.9 & &4.80 & 91.9 & & 4.80 & 91.8\\ HD$\,$215806 &1.80 & 66.6 & &1.74 & 69.0 & & 1.40 & 70.8\\ Hiltner 960 &5.61 & 55.2 & &4.98 & 54.3 & & 4.19 & 53.5\\ VI Cyg 12 &8.42 & 115.6 & &8.42 & 115.6 & & 8.42 & 115.6 \\ \hline \end{tabular} \end{table} For highly polarized stars, the observed polarization angle, in the instrument coordinate frame, is given as: \begin{equation} \theta_{obs} = \frac{1}{2} \arctan \left(\frac{u^{\prime}}{q^{\prime}} \right) \end{equation} from which the rotation offset of the instrument can be determined by: \begin{equation} K = \theta_{0} - (ROTSKYPA - \theta_{obs}) \end{equation} where $\theta_{0}$ is the previously determined polarization angle for the standard star in the reference catalogue and $ROTSKYPA$ is the instrument rotation angle. $K$ is, therefore, the relative offset of the polaroid positions, with respect to the standard astronomical definition of the Stokes parameter coordinate system ($+q$ aligned with North and a polarizaton angle of $0^{\circ}$), without any rotation of the instrument. The total degree of observed polarization was calculated as: \begin{equation} p_{obs} = \sqrt{q^{\prime\,2} +u^{\prime\,2}} \end{equation} Through comparison with previously catalogued values of the polarization for the highly-polarized standard stars ($p_{0}$), the degree of instrumental ``depolarization” was derived as: \begin{equation} D = \frac{p_{0}}{p_{obs}} \end{equation} We note that this definition of the instrumental depolarization is the inverse to that used by \citet{2016mnras.458..759s}, however the two approaches are otherwise equivalent. For the science targets, $q^{\prime}$ and $u^{\prime}$ were used to calculate the intrinsic polarization angle: \begin{equation} \theta_{0} = K + ROTSKYPA - \frac{1}{2}\arctan\left(\frac{u^{\prime}}{q^{\prime}} \right) \end{equation} and the intrinsic degree of polarization: \begin{equation} p_{0} = D \times p_{obs} \end{equation} The degree of polarization was further corrected for bias using the Modified ASymptotic (MAS) estimator of true polarization $p_{MAS}$ \citep{2014mnras.439.4048p}. We follow \citet{2019mnras.482.5023h}, by characterising polarization measurements with $p_{MAS}/\sigma_{p} < 3$ as non-detections, and quote the 95\% upper limit. \subsection{The Stability of the RINGO3 Instrument} \label{sec:ringostab} To establish a baseline calibration for each science observation we utilised the zero- and high-polarization standards, observed as part of the standard RINGO3 calibration plan, from the night of and the nights before and after the science observation. As a single-beam polarimeter \citep{2012spie.8446e..2ja}, in which orthogonal polarization components are measured in series, instrumental and background effects may be additive and not completely removed through the calculations presented in Equations \ref{eq:rawstokes} and \ref{eq:measstokes}. The determination of the instrumental polarization calibration parameters $q_{0}$, $u_{0}$, $D$ and $K$ may also be limited by the level of the sky background, the seeing and the throughput of each individual channel \citep{2016mnras.458..759s}. RINGO3 also uses two dichroics (to separate the three separate wavelength channels) and a depolarizing Lyot prism, for which it is estimated the minimum total systematic uncertainty is $\sim 0.5\%$ \citep{jermakphd}. The measured instrumental polarization parameters, for each polarization standard star observed, are shown in Figure \ref{fig:ringostab}. The mean and standard deviation of the instrumental polarization parameters of RINGO3, for the survey period, are summarised in Table \ref{tab:ringostab}. We also calculated the intra-night standard deviation of the instrumental polarization parameters $\sigma_{N}$ (and in Table \ref{tab:ringostab} we report the average over all nights). In general, we find that the limiting systematic precision of RINGO3 is consistent with previous estimates \citep{jermakphd, 2016mnras.458..759s}. Although the calibration of RINGO3 is relatively stable over the period of the survey, there is some structure present in Figure \ref{fig:ringostab} (e.g. around $MJD58300$, which coincided with the cleaning of a mirror in the optical path) which requires applying calibrations derived over short timescales (rather than an average derived over the entire length of the survey). \begin{table*} \caption{Average properties of the RINGO3 instrumental polarization between June 2018 and August 2019. \label{tab:ringostab}} \begin{tabular}{cc|ccc|ccc|ccc|ccc} \hline\hline \multicolumn{2}{c}{Channel} & $\overline{q_{0}}$ & $\sigma (q_{0})$ & $\overline{\sigma_{N} (q_{0})}$ & $\overline{u_{0}}$ & $\sigma (u_{0})$ & $\overline{\sigma_{N} (u_{0})}$ & $\overline{D}$ & $\sigma (D)$ & $\overline{\sigma_{N} (D)}$ & $\overline{K}$ & $\sigma (K)$ & $\overline{\sigma_{N} (K)}$ \\ \hline $b^{\ast}$ & e & -0.58 & 0.24 & 0.10 & -2.02 & 0.39 & 0.15 & 0.97 & 0.11 & 0.07 & 125.92 & 3.55 & 2.19 \\ $g^{\ast}$ & f & -1.18 & 0.28 & 0.14 & -3.44 & 0.38 & 0.11 & 1.03 & 0.10 & 0.05 & 125.62 & 2.70 & 1.16 \\ $r^{\ast}$ & d & -1.22 & 0.40 & 0.15 & -3.28 & 0.44 & 0.19 & 1.06 & 0.20 & 0.11 & 126.38 & 2.14 & 1.17 \\ \hline\hline \multicolumn{14}{l}{$\sigma =$ scatter (standard deviation) of the observed parameter across all observations.}\\ \multicolumn{14}{l}{$\overline{\sigma_{N}} = $ average scatter of the observed parameter measured on individual nights.}\\ \end{tabular} \end{table*} \subsection{Science Targets and Observations} \label{sec:scip} Science targets were observed as part of programmes PQ18A02, PL18A10, PL18B01 and PL19A16. Targets were selected, primarily from ZTF, for their brightness and location in the sky to be suitable for RINGO3 observations. The sample is composed of 4 Type II SNe, 3 Type Ia SNe, 2 1991T-like Type Ia SNe, 1 Type Ic SN, 1 Type I Superluminous SN (SLSN) and a single transient (AT2019hgp) of unknown classification. Details of the formal discovery and classification of these transients are shown in Table \ref{tab:scip}. We note, however, that in a number of cases the objects had ZTF observations prior to the date and time given in the formal discovery announcement. During the ZTF observations, difference images are generated based on the image subtraction algorithm by \citet{2016apj...830...27z} implemented in the real-time reduction pipeline \citep{2019pasp..131a8003m}. Only ZTF alert streams that are above a $5-\sigma$ threshold will generate alerts. Using the IPAC ZTF difference imaging pipeline, we performed forced point-spread function (PSF) photometry at the location of SNe discovered by ZTF following the procedure described in \citet{2019apj...886..152y}. We applied a $4-\sigma$ threshold and inspected both the last non-detection limit and the first detection in both $g$ and $r$ bandpasses. To establish the phase of our observations, for each transient, we consider the explosion $t_{\rm exp}$ to be midway between the last ZTF non-detection and the first ZTF detection of the candidate (see Table~\ref{tab:detection}). The estimated explosion time of the Type II SNe\,2018cyg and 2018dfi are consistent with \citet{2020arxiv200809986b}. The discovery time of SN\,2018gep agrees with the first $r-$band detection reported by \citet{2019apj...887..169h}. A log of the science observations is presented in Table \ref{tab:obs} and the locations of the SNe, with respect to their host galaxies, are shown on Fig. \ref{fig:scip}. It was not possible to observe two of the science targets (SN~2018bsz and 2018hna) at early times, and these constitute outliers from the main targets of the early-time polarimetry survey. As these were observed alongside our other targets, and using the same Liverpool Telescope programmes, we include them here for completeness. \begin{table*} \centering \caption{Science targets for RINGO3 observations. \label{tab:scip}} \begin{tabular}{llccccll} \hline Target & Original & $\alpha_{J2000}$ & $\delta_{J2000}$ & Discovery & Redshift & Type & Discoverer \\ & Name & & & Date & & & \\ \hline SN~2018bsz & ASASSN-18km & 16:09:39.1 & -32:03:45.6 & 2018-05-17 & 0.027 & SLSN-I \citep{2018atel11674....1a} & ASASSN$^{1}$ \citep{2018TNSTR.655....1S} \\ SN~2018cnw & ZTF~18abauprj & 16:59:05.0 & +47:14:11.2 & 2018-06-15 & 0.028 & SN Ia-91T-like \citep{2018TNSCR.833....1M} & ZTF$^{2}$ \citep{2018TNSTR.832....1F} \\ SN~2018cyg & ZTF~18abdbysy & 15:34:08.5 & +56:41:48.7 & 2018-06-30 & 0.011 & SN II \citep{2018TNSCR.939....1F} & LOSS $^{3}$\citep{2018TNSTR.907....1J} \\ SN~2018dfi & ZTF~18abffyqp & 16:50:50.1 & +45:23:52.5 & 2018-07-10 & 0.031 & SN IIb \citep{2020arxiv200809986b} & POSS$^{4}$ \citep{2018TNSTR.957....1G}\\ SN~2018eay & ZTF~18abgmcmv & 18:16:13.1 & +55:35:27.2 & 2018-07-15 & 0.018 & SN Ia-91T-like \citep{2018TNSCR1012....1Y} & ZTF$^{2}$ \citep{2018TNSTR1005....1F} \\ SN~2018gep & ZTF~18abukavn & 16:43:48.2 & +41:02:43.4 & 2018-09-09 & 0.032 & SN Ic-BL \citep{2018TNSCR1442....1B} & ZTF$^{2}$\citep{2018TNSTR1357....1H} \\ SN~2018gvi & ZTF~18abyxwrf & 02:55:36.0 & +43:03:27.3 & 2018-09-24 & 0.021 & SN Ia \citep{2018TNSCR1487....1F} & ZTF$^{2}$ \citep{2018TNSTR1456....1F} \\ SN~2018hna & $\cdots$ & 12:26:12.1 & +58:18:50.8 & 2018-10-22 & 0.002 & SN II \citep{2018TNSCR1638....1L} & K. Itagaki \citep{2018TNSTR1614....1I} \\ SN~2019np & ZTF~19aacgslb & 10:29:22.0 & +29:30:38.3 & 2019-01-09 & 0.004 & SN Ia \citep{2019TNSCR..71....1B} & K. Itagaki \citep{2019TNSTR..53....1I} \\ SN~2019ein & ATLAS19ieo & 13:53:29.1 & +40:16:31.3 & 2019-05-01 & 0.008 & SN Ia \citep{2019TNSCR.701....1B} & ATLAS$^{5}$ \citep{2019TNSTR.678....1T} \\ AT~2019hgp & ZTF~19aayejww & 15:36:12.9 & +39:44:00.6 & 2019-06-08 & $\cdots$ & $\cdots$ & ZTF$^{2}$ \citep{2019TNSTR.973....1B} \\ SN~2019nvm & ZTF~19abqhobb & 17:25:38.7 & +59:26:48.3 & 2019-08-19 & 0.019 & SN II \citep{2019TNSCR1557....1H} & ZTF $^{2}$\citep{2019TNSTR1546....1N} \\ \hline \end{tabular} $^{1}$ All-Sky Automated Survey for Supernovae \citep[ASAS-SN][]{2014ApJ...788...48S}; $^{2}$ Zwicky Transient Facility \citep[ZTF][]{2019pasp..131a8002b}; $^{3}$ Lick Observatory Supernova Search \citep[LOSS][]{2000aipc..522..103l}; $^{4}$ Puckett Observatory Supernova Search (POSS); $^{5}$ Asteroid Terrestrial-impact Last Alert System \citep[ATLAS;][]{2018pasp..130f4505t} \end{table*} \begin{table*} \centering \caption{Non-detection limits and first detection epochs for the observed SNe. \label{tab:detection}} \begin{tabular}{lccc} \hline Target  & Filter & Last ZTF Non-detection  & First ZTF detection \\         &        &(MJD$^{a}$  [S/N]$^{b}$)   & (MJD$^{a}$  [S/N]$^{b}$) \\ \hline SN\,2018bsz / ASASSN-18km  & $\cdots$  & $\cdots$     & $\cdots$   \\ SN\,2018cnw / ZTF18abauprj &    $g$    & 58282.385 [$\textless$0] &  58283.283 [5.1] \\                            &    $r$    & 58283.329 [3.2]          &  58284.280 [19]  \\ SN\,2018cyg / ZTF18abdbysy &    $g$    & 58294.223 [$\textless$0] &  58295.205 [5.2] \\                            &    $r$    & 58294.242 [2.1]          &  58294.257 [4.0] \\ SN\,2018dfi / ZTF18abffyqp &    $g$    & 58306.307 [$\textless$0] &  58307.214 [63]  \\                            &    $r$    & 58306.201 [$\textless$0] &  58307.186 [44]  \\ SN\,2018eay / ZTF18abgmcmv &    $g$    & 58311.345 [3.7]$^{d}$    &  58312.350 [11]  \\                            &    $r$    & 58311.198 [2.5]          &  58311.222 [5.5] \\ SN\,2018gep / ZTF18abukavn &    $g$    & 58369.254 [3.9]$^{d}$    &  58370.186 [19]  \\                            &    $r$    & 58370.141 [4.0]$^{d}$    &  58370.163 [7.3] \\ SN\,2018gvi / ZTF18abyxwrf &    $g$    & 58384.319 [2.3]          &  58385.413 [6.3] \\                            &    $r$    & 58386.328 [$\textless$0] &  58388.485 [21]  \\ SN\,2018hna / ZTF18acbwaxk & $\cdots$  & $\cdots$     & $\cdots$   \\ SN\,2019np  / ZTF19aacgslb &    $g$    & 58491.454 [$\textless$0] &  58494.483 [179] \\                            &    $r$    & 58491.530 [1.2]          &  58492.445 [18]  \\ SN\,2019ein / ATLAS19ieo   &cyan-ATLAS & 58602.267  &  58604.474$^{c}$ \\ AT\,2019hgp / ZTF19aayejww &    $g$    & 58640.362 [$\textless$0] &  58641.201 [3.5]$^{d}$ \\                            &    $r$    & 58640.291 [$\textless$0] &  58641.289 [4.9]  \\ SN\,2019nvm / ZTF19abqhobb &    $g$    & 58713.218 [$\textless$0] &  58714.163 [71]  \\                            &    $r$    & 58713.242 [1.2]          &  58714.185 [55]  \\ \hline \end{tabular}\\ $^{a}$Modified Julian Date; $^{b}$Signal-to-noise ratio for the forced difference image PSF-fit flux measurement; \\ $^{c}$Discovered by ATLAS \citep{2019TNSTR.678....1T}; $^{d}$Target was marginally detected; \end{table*} \begin{table*} \centering \caption{RINGO3 observations of the target SNe. The exposure time is the total spent on the target across all 8 polaroid rotator positions. The same exposure time is used for each of the three ($b^{\ast}/g^{\ast}/r^{\ast}$) RINGO3 channels. \label{tab:obs}} \begin{tabular}{lrrllr} \hline Date (UT) & MJD & Phase & Dataset & Target & Exposure \\ & & (days)$^{\dagger}$& & & Time(s) \\ \hline 2018-06-02 23:02 & 58271.96 & 69.5 & e\_20180602\_2\_0 & SN 2018bsz & 598 \\ 2018-06-05 22:53 & 58274.95 & 72.4 & e\_20180605\_8\_0 & SN 2018bsz & 598 \\ 2018-06-10 22:37 & 58279.94 & 77.4 & e\_20180610\_4\_0 & SN 2018bsz & 895 \\ 2018-06-13 22:58 & 58282.96 & 80.5 & e\_20180613\_4\_0 & SN 2018bsz & 895 \\ 2018-06-17 22:05 & 58286.92 & 84.4 & e\_20180617\_3\_0 & SN 2018bsz & 1196 \\ 2018-06-20 22:01 & 58289.92 & 87.4 & e\_20180620\_4\_0 & SN 2018bsz & 1197 \\ 2018-06-24 21:35 & 58293.90 & 91.4 & e\_20180624\_1\_0 & SN 2018bsz & 1196 \\ 2018-06-28 21:33 & 58297.90 & 95.4 & e\_20180628\_3\_0 & SN 2018bsz & 1198 \\ \hline 2018-06-18 02:00 & 58287.08 & 4.2 & e\_20180617\_10\_0 & SN 2018cnw & 948 \\ 2018-06-18 02:16 & 58287.09 & 4.3 & e\_20180617\_11\_0 & SN 2018cnw & 948 \\ 2018-06-18 02:32 & 58287.11 & 4.3 & e\_20180617\_12\_0 & SN 2018cnw & 948 \\ 2018-06-18 02:48 & 58287.12 & 4.3 & e\_20180617\_13\_0 & SN 2018cnw & 945 \\ 2018-06-18 03:04 & 58287.13 & 4.3 & e\_20180617\_14\_0 & SN 2018cnw & 945 \\ \hline 2018-06-26 23:00 & 58295.96 & 1.7 & e\_20180626\_3\_0 & SN 2018cyg & 1799 \\ 2018-06-26 23:30 & 58295.98 & 1.7 & e\_20180626\_4\_0 & SN 2018cyg & 1796 \\ 2018-06-27 00:01 & 58296.00 & 1.8 & e\_20180626\_5\_0 & SN 2018cyg. & 1796 \\ \hline 2018-07-08 22:12 & 58307.93 & 1.2 & e\_20180708\_5\_0 & SN 2018dfi & 1196 \\ 2018-07-08 22:32 & 58307.94 & 1.2 & e\_20180708\_6\_0 & SN 2018dfi & 1196 \\ 2018-07-09 01:37 & 58308.07 & 1.3 & e\_20180708\_7\_0 & SN 2018dfi & 1196 \\ 2018-07-09 01:58 & 58308.08 & 1.3 & e\_20180708\_8\_0 & SN 2018dfi & 1196 \\ \hline 2018-07-19 22:12 & 58318.93 & 7.7 & e\_20180719\_5\_0 & SN 2018eay & 996 \\ 2018-07-19 22:29 & 58318.94 & 7.7 & e\_20180719\_6\_0 & SN 2018eay & 999 \\ \hline 2018-09-23 22:15 & 58384.93 & 14.8 & e\_20180923\_16\_0 & SN 2018gep & 446 \\ 2018-09-23 22:23 & 58384.93 & 14.8 & e\_20180923\_17\_0 & SN 2018gep & 446 \\ \hline 2018-10-02 02:39 & 58393.11 & 8.2 & e\_20181001\_10\_0 & SN 2018gvi & 1199 \\ \hline 2019-01-10 05:07 & 58493.21 & 82.4 & e\_20190109\_4\_0 & SN 2018hna & 476 \\ 2019-01-14 02:09 & 58497.09 & 86.3 & e\_20190113\_9\_0 & SN 2018hna & 478 \\ 2019-01-20 01:44 & 58503.07 & 92.3 & e\_20190119\_7\_0 & SN 2018hna & 537 \\ 2019-01-30 03:35 & 58513.15 & 102.3 & e\_20190129\_5\_0 & SN 2018hna & 715 \\ 2019-02-04 02:54 & 58518.12 & 107.3 & e\_20190203\_5\_0 & SN 2018hna & 957 \\ 2019-02-12 01:45 & 58526.07 & 115.3 & e\_20190211\_7\_0 & SN 2018hna & 997 \\ \hline 2019-01-11 03:46 & 58494.16 & 2.2 & e\_20190110\_4\_0 & SN 2019np & 898 \\ 2019-01-12 03:48 & 58495.16 & 3.2 & e\_20190111\_4\_0 & SN 2019np & 1197 \\ 2019-01-12 04:35 & 58495.19 & 3.2 & e\_20190111\_5\_0 & SN 2019np & 1197 \\ 2019-01-13 01:13 & 58496.05 & 4.1 & e\_20190112\_9\_0 & SN 2019np & 1198 \\ 2019-01-14 01:50 & 58497.08 & 5.1 & e\_20190113\_8\_0 & SN 2019np & 998 \\ 2019-01-15 02:34 & 58498.11 & 6.1 & e\_20190114\_20\_0 & SN 2019np & 717 \\ 2019-01-16 06:40 & 58499.28 & 7.3 & e\_20190115\_10\_0 & SN 2019np & 598 \\ 2019-01-20 01:35 & 58503.07 & 11.1 & e\_20190119\_6\_0 & SN 2019np & 417 \\ 2019-01-21 01:40 & 58504.07 & 12.1 & e\_20190120\_3\_0 & SN 2019np & 417 \\ 2019-01-23 01:33 & 58506.06 & 14.1 & e\_20190122\_9\_0 & SN 2019np & 717 \\ 2019-01-26 04:55 & 58509.21 & 17.2 & e\_20190125\_3\_0 & SN 2019np & 717 \\ 2019-02-07 01:15 & 58521.05 & 29.1 & e\_20190206\_9\_0 & SN 2019np & 996 \\ 2019-02-25 00:11 & 58539.01 & 47.0 & e\_20190224\_3\_0 & SN 2019np & 1196 \\ \hline \end{tabular} \\ $^{\dagger}$ Relative to calculated explosion epoch (see Section \ref{sec:scip} and Table \ref{tab:detection}). \end{table*} \begin{table*} \centering \contcaption{RINGO3 observations of the target SNe. The exposure time is the total spent on the target across all 8 polaroid rotator positions. The same exposure time is used for each of the three wavelength channels. \label{tab:obs:cont}} \begin{tabular}{lrrllr} \hline Date (UT) & MJD & Phase & Dataset & Target & Exposure \\ & & (days) & & & Time(s) \\ \hline 2019-05-03 21:27 & 58606.89 & 3.5 & e\_20190503\_3\_0 & SN 2019ein & 1499 \\ 2019-05-04 21:46 & 58607.91 & 4.5 & e\_20190504\_3\_0 & SN 2019ein & 1497 \\ 2019-05-04 22:13 & 58607.93 & 4.6 & e\_20190504\_4\_0 & SN 2019ein & 1497 \\ 2019-05-05 21:32 & 58608.90 & 5.5 & e\_20190505\_3\_0 & SN 2019ein & 1497 \\ 2019-05-06 21:56 & 58609.91 & 6.5 & e\_20190506\_3\_0 & SN 2019ein & 1496 \\ 2019-05-07 22:47 & 58610.95 & 7.6 & e\_20190507\_3\_0 & SN 2019ein & 896 \\ 2019-05-09 22:47 & 58612.95 & 9.6 & e\_20190509\_3\_0 & SN 2019ein & 599 \\ 2019-05-11 00:04 & 58614.00 & 10.6 & e\_20190510\_3\_0 & SN 2019ein & 598 \\ 2019-05-11 23:03 & 58614.96 & 11.6 & e\_20190511\_5\_0 & SN 2019ein & 598 \\ 2019-05-14 00:45 & 58617.03 & 13.7 & e\_20190513\_26\_0 & SN 2019ein & 598 \\ 2019-05-16 23:03 & 58619.96 & 16.6 & e\_20190516\_4\_0 & SN 2019ein & 599 \\ 2019-05-19 21:40 & 58622.90 & 19.5 & e\_20190519\_1\_0 & SN 2019ein & 598 \\ 2019-05-22 22:30 & 58625.94 & 22.6 & e\_20190522\_1\_0 & SN 2019ein & 598 \\ 2019-05-27 22:20 & 58630.93 & 27.6 & e\_20190527\_3\_0 & SN 2019ein & 596 \\ 2019-06-04 23:10 & 58638.97 & 35.6 & e\_20190604\_3\_0 & SN 2019ein & 598 \\ 2019-06-24 23:24 & 58658.98 & 55.6 & e\_20190624\_8\_0 & SN 2019ein & 1196 \\ 2019-07-05 22:23 & 58669.93 & 66.6 & e\_20190705\_3\_0 & SN 2019ein & 1797 \\ 2019-07-26 21:09 & 58690.88 & 87.5 & e\_20190726\_3\_0 & SN 2019ein & 1798 \\ \hline 2019-06-09 22:56 & 58643.96 & 3.2 & e\_20190609\_3\_0 & AT 2019hgp & 1797 \\ \hline 2019-08-19 21:32 & 58714.90 & 1.2 & e\_20190819\_4\_0 & SN 2019nvm & 1797 \\ \hline \end{tabular} \end{table*} \section{Results \& Analysis} \label{sec:res} \subsection{SN~2018bsz} \label{sec:res:18bsz} Discovered by ASAS-SN on 2018 May 17 \citep{2018TNSTR.655....1S}, it was temporarily classified as a young Type II SN \citep{2018TNSCR.679....1H}. It was later reclassified as a superluminous supernova \citep[for a review see ][]{2019ara&a..57..305g}, albeit a lower luminosity example \citep{2018a&a...620a..67a}. Our observations commenced 4 days after the photometric light curve maximum or 69 days after the explosion date proposed by \citet{2018a&a...620a..67a}. The polarization measurements for SN~2018bsz are presented in Table \ref{tab:res:18bsz} and the time evolution, with respect to the photometric light curve, is shown on Fig. \ref{fig:res:18bsz}. We derive limits on the polarization, strictest in the blue, at the general level of $<1 - 2\%$. We do, however, make one single detection of $p(b^{\ast}) = 2 \pm 0.5$\% at 11.4 days after maximum (or MJD58267.5). \begin{table} \centering \caption{RINGO3 polarization measurements of SN~2018bsz. \label{tab:res:18bsz}} \begin{tabular}{lrccc} \hline Epoch & Phase & $p(b^{\ast})$ & $p(g^{\ast})$ & $p(r^{\ast})$ \\ (MJD) & (days) & (\%) & (\%) & (\%) \\ \hline 58271.96 & 69.5 &$<1.56$ & $<2.56$ & $<5.07$\\ 58274.95 & 72.4 &$<1.08$ & $<1.60$ & $<4.59$\\ 58279.94 & 77.4 &$<2.46$ & $<1.86$ & $<4.54$\\ 58282.96 & 80.5 &$<2.49$ & $<1.08$ & $<2.23$\\ 58286.92 & 84.4 &$2.02\pm0.53$ & $<1.53$ & $<4.83$\\ 58289.92 & 87.4 &$<1.85$ & $<1.23$ & $<2.56$\\ \hline \end{tabular} \end{table} \subsection{SN~2018cnw} \label{sec:res:18cnw} \citet{2018TNSCR.833....1M} classified SN~2018cnw as being a ``91T-like" Type Ia SN. Five sets of observations were acquired in the course of a single night (around Modfied Julian Date [MJD] 58287.0). The observation of the SN was subject to poor seeing of $\sim 3.2\arcsec$, with the point spread function appearing obviously elongated. We note that it was not possible, under these conditions, to detect the SN in the $r^{\ast}$ observations. For the other filters, we derived upper limits on the polarization of $b^{\ast} < 2.0\%$ and $g^{\ast} < 0.9\%$. \subsection{SN~2018cyg} \label{sec:res:18cyg} SN~2018cyg was observed three times in a single night (MJD58296.0). The SN was marginally detected (S/N = 4.0) in ZTF $r-$band observation on MJD58294.257, followed by the $g-$ and $r-$band detections at MJD58295.205 (S/N = 5.2) and MJD 58295.246 (S/N = 16.0), respectively. We consider the RINGO3 observation was, therefore, conducted at $\sim 0.8$ to $1.8$ days after the first detection with ZTF, or $\approx$2 days after the explosion. (see Table~\ref{tab:detection}). \citet{2018TNSCR.939....1F} later classified it as a Type II SN, with a short plateau. Given the faintness of the SN at this epoch and poor observing conditions, we could not establish a photometric detection of SN~2018cyg in the $b^{\ast}$-band and were only able to place limits on the degree of polarization of $g^{\ast} < 15\%$ and $r^{\ast} < 22.0\%$. \subsection{SN~2018dfi} \label{sec:res:18dfi} Two sets of observations of SN~2018dfi, consisting of two separate exposures each, were conducted on a single night (MJD58308.0). The RINGO3 observations commenced 0.7 days after the first detection by ZTF, which we estimate to correspond to 1.2 days post-explosion. The SN was classified (at 4.5 days post-explosion) as a Type II SN \citep{2018TNSCR.974....1H}, which was further refined to being Type IIb \citep{2020arxiv200809986b}. Combining the Stokes parameters determined for all four exposures, we derive limits on the degree of polarization of SN~2018dfi of $p(b^{\ast}) < 2.3\%$, $p(g^{\ast}) < 6.8\%$ and and $p(r^{\ast}) < 4\%$. \subsection{SN~2018eay} \label{sec:res:18eay} Forced PSF photometry of ZTF images at the location of SN\,2018eay shows that the S/N measured in both $g$ (S/N=3.7 at MJD 58311.345) and $r$ (S/N=5.5 at MJD 58311.222) bands are obviously higher compared to the previous non-detections. The S/N derived based on an $r-$band image obtained earlier during the same night (MJD 58311.198) yields 2.5. Therefore, we adopt an explosion epoch at MJD 58311.2 for SN\,2018eay, indicating that the first RINGO3 imaging polarimetry was acquired 7.7 days post the SN explosion (the night of MJD 58319.0). Similarly to SN~2018cnw, SN~2018eay was classified as a ``91T-like" Type Ia SN \citep{2018TNSCR1012....1Y}. In both sets of observations, the SN was only weakly detected in all three channels and it was only possible to derive limits on the degree of polarization of $p(b^{\ast}) < 2.5\%$, $p(g^{\ast}) < 6.1\%$ and $p(r^{\ast}) < 7.5\%$. \subsection{SN~2018gep} \label{sec:res:18gep} Two sets of observations of SN~2018gep were conducted, in sequence, on the night of MJD58385.0, corresponding to 14.8 days post-explosion. We note that SN~2018gep was discovered very close to the moment of explosion, with the last ZTF non-detection of the transient occurring only 0.02 days before the first detection \citep{2018TNSTR1357....1H}. They conducted a second-order polynomial fit to the first three days of the $g$-band flux and defined $t_{0}$ at 58370.146 as the time at which the flux of SN\,2018gep is zero. In fact, the transient exhibits pre-explosion emission extended $\approx$ a week prior to the rapid rise in the light curve (see Figure 7 of \citealp{2019apj...887..169h}). An observation of SN~2018gep at 10.1 days yielded a classification for SN~2018gep as a broad-lined Type Ic supernova at around maximum light \citep{2018TNSCR1442....1B}; although \citet{2020arxiv200804321p} suggest the fast rise-time ($<6.2\,\mathrm{days}$) may imply that SN2018gep may be more closely related to the family of Fast Blue Optical Transients \citep{2014apj...794...23d}. From our RINGO3 observations we do not detect any significant polarization for SN~2018gep, instead deriving polarization limits of $p(b^{\ast}) < 1.6\%$ and $p(g^{\ast}) <7.0\%$ and $p(r^{\ast}) < 5.1\%$. \subsection{SN~2018gvi} \label{sec:res:18gvi} We acquired a single observation of SN~2018gvi at 8.2 days post-explosion, or 7.7 days after the first detection by ZTF. The SN had earlier been classified by \citet{2018TNSCR1487....1F} as a Type Ia SN. We derive upper limits on the polarization of SN~2018gvi in all three channels to levels of $p(b^{\ast}) < 1.6\%$, $p(g^{\ast}) < 2.0\%$ and $p(r^{\ast}) < 5.0\%$. \subsection{SN~2018hna} \label{sec:res:18hna} SN~2018hna was observed with RINGO3 at 6 separate epochs, beginning $\sim 5$ days before the $V$-band maximum. Given the long rise time to maximum light, the observations commenced $\sim 82$ days post-explosion as shown in Fig. \ref{fig:res:18hna}\footnote{\url{https://lasair.roe.ac.uk/object/ZTF18acbwaxk/}} \citep{2019apj...882l..15s}. The polarization measurements at each of the six epochs are listed in Table \ref{tab:res:18hna}. Overall, we constrain the polarization of SN~2018hna at around maximum light to be $<1.5\%$. In the $b^{\ast}$ and $g^{\ast}$ bands, we make four separate detections of significant polarization $p \sim 0.5 - 1.0\%$, whilst at a single epoch we find $p(r^{\ast}) = 1.3\pm0.3\%$. From Fig. \ref{fig:res:18hna}, it is clear that any polarization associated with SN~2018hna, despite its brightness ($m_{\mathrm{max}}(r) \sim 14.0\,\mathrm{mags}$), is close to the systematic floor of the RINGO3 instrument. \begin{table} \centering \caption{RINGO3 polarization measurements of SN~2018hna \label{tab:res:18hna}} \begin{tabular}{lrccc} \hline Epoch & Phase & $p(b^{\ast})$ & $p(g^{\ast})$ & $p(r^{\ast})$ \\ (MJD) & (days) & (\%) & (\%) & (\%) \\ \hline 58493.21 & 82.4 &$0.83^{+0.12}_{-0.13}$ & $<0.81$ & $<1.01$\\ 58497.09 & 86.3 &$<0.39$ & $0.68 \pm 0.18$ & $<1.50$\\ 58503.07 & 92.3 &$<0.89$ & $<0.91$ & $<1.09$\\ 58513.15 & 102.3&$<0.37$ & $0.69\pm0.18$ & $1.29^{+0.28}_{-0.29}$\\ 58518.12 & 107.3&$0.72 \pm0.17$ & $<0.50$ & $<0.74$\\ 58526.07 & 115.3&$<0.39$ & $<0.73$ & $<0.89$\\ \hline \end{tabular} \end{table} \subsection{SN~2019np} \label{sec:res:19np} Polarimetric followup of SN~2019np commenced 2.2 days after explosion, or 1.7 days after the first detection by ZTF. In total there were 13 separate observations of the SN up to 47 days post-explosion. The polarization measurements for SN~2019np are presented in Table \ref{tab:res:19np} and shown in Figure \ref{fig:res:19np} (alongside ZTF photometry\footnote{\url{https://lasair.roe.ac.uk/object/ZTF19aacgslb/}}). The SN was discovered on the rise up to maximum light and polarization constraints, in particular in the blue, limit the polarization across the optical wavelength range to $<2.0\%$. At a later epoch, 28 days after discovery and 10 days after maximum, we detect significant polarization at the level of $p(b^{\ast}) = 0.26\pm0.07\% $ and $p(g^{\ast})= 0.67\pm0.16\%$ consistent with the earlier limits on the polarization and the general level of polarization of this SN being low. \begin{table} \centering \caption{RINGO3 polarization measurements of SN~2019np \label{tab:res:19np}} \begin{tabular}{lrccc} \hline Epoch & Phase & $p(b^{\ast})$ & $p(g^{\ast})$ & $p(r^{\ast})$ \\ (MJD) & (days) & (\%) & (\%) & (\%) \\ \hline 58494.16 & 2.2 &$<1.40$ & $<3.01$ & $<4.70$ \\ 58495.16 & 3.2 &$<0.34$ & $<1.12$ & $<1.98$ \\ 58495.19 & 3.2 &$<0.60$ & $<0.61$ & $<1.41$ \\ 58496.05 & 4.1 &$<0.59$ & $<0.97$ & $<1.60$ \\ 58497.08 & 5.1 &$<0.19$ & $<0.57$ & $<1.41$ \\ 58498.11 & 6.1 &$<0.42$ & $<0.44$ & $<1.46$ \\ 58499.28 & 7.3 &$<0.65$ & $<0.84$ & $<3.87$ \\ 58503.07 & 11.1&$<0.63$ & $<0.54$ & $<1.19$ \\ 58504.07 & 12.1&$<2.40$ & $<0.63$ & $<1.62$ \\ 58506.06 & 14.1&$<0.87$ & $<1.68$ & $<2.40$ \\ 58509.21 & 17.2&$<0.23$ & $<0.27$ & $<1.01$ \\ 58521.05 & 29.1&$0.26\pm0.07$ & $0.67\pm0.16$ & $<0.48$ \\ 58539.01 & 47.0&$<0.49$ & $<0.60$ & $<0.46$ \\ \hline \end{tabular} \end{table} \subsection{SN~2019ein} \label{sec:res:19ein} RINGO3 observations of SN~2019ein started 3.5 days post-explosion or 2.4 days after the first ZTF detection. The Type Ia SN \citep{2019TNSCR.701....1B} was heavily observed in the rise to maximum, and in general we were only able to assess upper limits on the degree of polarization, however we did measure significant levels of polarization in the $b^{\ast}$-band (see Table \ref{tab:res:19ein}) that appear to increase with time around the period of the second light curve maximum that was observable at redder wavelengths (see Figure \ref{fig:res:19ein}, in conjunction with ZTF photometry \footnote{\url{https://lasair.roe.ac.uk/object/ZTF19aatlmbo/}}). \begin{table} \centering \caption{RINGO3 polarization measurements of SN~2019ein \label{tab:res:19ein}} \begin{tabular}{lrccc} \hline Epoch & Phase & $p(b^{\ast})$ & $p(g^{\ast})$ & $p(r^{\ast})$ \\ (MJD) & (days) & (\%) & (\%) & (\%) \\ \hline 58606.89 & 3.5 &$<1.68$ & $<2.61$ & $<8.55$ \\ 58607.91 & 4.5 &$<0.90$ & $<1.74$ & $<5.31$ \\ 58607.93 & 4.6 &$<0.68$ & $<1.63$ & $<3.99$ \\ 58608.90 & 5.5 &$<0.98$ & $<0.93$ & $<1.50$ \\ 58609.91 & 6.5 &$<0.67$ & $<0.62$ & $<1.72$ \\ 58610.95 & 7.6 &$<0.40$ & $<0.85$ & $<1.64$ \\ 58612.95 & 9.6 &$0.64 \pm 0.15$ & $<0.39$ & $<2.33$ \\ 58614.00 & 10.6 &$<0.21$ & $<0.84$ & $<2.54$ \\ 58614.96 & 11.6 &$<0.48$ & $<0.64$ & $<1.27$ \\ 58617.03 & 13.7 &$<0.46$ & $<0.85$ & $<1.92$ \\ 58619.96 & 16.6 &$0.70 \pm 0.15$ & $<0.58$ & $<1.77$ \\ 58622.90 & 19.5 &$0.42\pm0.12$ & $<1.23$ & $<0.94$ \\ 58625.94 & 22.6 &$0.93\pm 0.18$ & $<3.15$ & $<6.83$ \\ 58630.93 & 27.6 &$1.12 \pm 0.25$ & $<2.33$ & $<4.64$ \\ 58638.97 & 35.6 &$0.99 \pm 0.22$ & $<0.50$ & $<8.31$ \\ 58658.98 & 55.6 &$<1.72$ & $<1.20$ & $<1.80$ \\ 58669.93 & 66.6 &$<1.44$ & $<1.10$ & $<2.75$ \\ 58690.88 & 87.5 &$<2.29$ & $<2.74$ & $<11.17$ \\ \hline \end{tabular} \end{table} \subsection{AT~2019hgp} \label{sec:res:19hgp} AT2019hgp was discovered as a young transient at MJD 58642.242 \citep{2019tnsan..30....1b}. An $r-$band detection at MJD 58641.289 (S/N$=$4.9) and a $g-$band signal at MJD 58641.201 (S/N$=$3.5) were recovered by forced PSF photometry. An early spectrum obtained at MJD 58642.43 revealed emission lines of highly ionized species \citep{2019tnsan..30....1b}, suggesting that the candidate was a young and hot transient. We note, however, that no further specific classification of this transient has been recorded. Our RINGO3 observation was conducted 2.8 days after discovery or $\sim$3.2 days post-explosion. The transient was only photometrically detected at significant levels in the $b^{\ast}$ observation, for which we derive an upper limit on the polarization of $p(b^{\ast}) < 5.8\%$. \subsection{SN~2019nvm} \label{sec:res:19nvm} The single RINGO3 observation of SN~2019nvm commenced 0.7 days after the SN was first detected by ZTF, or $\sim 1.2$ days post-explosion. A spectrum, acquired $\sim 16$ hours after the RINGO3 observation revealed a young Type II SN showing ``flash features'' \citep{2017natph..13..510y} of He {\sc ii} $\lambda 4686$ and N {\sc iv} $\lambda 4537$ \citep{2019TNSCR1557....1H}. This suggests that SN~2019nvm was discovered very soon after explosion, potentially making this RINGO3 polarimetric observation the earliest ever acquired for a Type II SN. The observations were, however, conducted under poor seeing conditions ($\approx 2.7\arcsec$), with an elongated point spread function possibly indicating the effect of wind on the telescope. SN~2019nvm is located close to the nucleus in the edge-on galaxy UGC 10858 and, given the seeing conditions, it was not possible to accurately separate the SN and the host galaxy. For all three channels, we do not significantly detect a polarization signal, with upper limits on the degree of polarization of $p(b^{\ast}) < 1.5\% $, $p(g^{\ast}) < 2.7\%$ and $p(r^{\ast}) < 2.2\%$. \section{Discussion \& Conclusions} \label{sec:disc} For the SNe observed at early times we have, in general, only been able to place upper limits on the possible polarization. The evolution of the polarization constraints for all the early time observations, within 20 days of explosion, are shown in Figure \ref{fig:disc:all}. The degree of the constraint on the early time polarization is limited by two key factors: the brightness of the SN and the relatively high level of the systematic floor of the RINGO3 instrument \citep{2016mnras.458..759s}. In general, from our observations, the limits on the instrumental polarization means that the upper limits on the polarization are relatively high $\sim 1\%$. As time progresses, we note that the polarization limits do become better, and this is correlated with increased levels of signal-to-noise as the target SNe rise to maximum light. In addition, we find that the throughput in the $b^{\ast}$ channel is the best of the three RINGO3 channels and, from Fig. \ref{fig:disc:all}, the systematic floor for RINGO3 does increase towards the red. The $r^{\ast}$-band polarization limits are less constraining by a factor of $\sim 2$. The constraints on the level of polarization in the $b^{\ast}$ and $g^{\ast}$ bands are around $<1\%$ for Type Ia SNe. From \citet{1991a&a...246..481h}, we can place a constraint on the axial symmetry of the ejecta of Type Ia SNe at early times (assuming a spheroidal configuration) of $>0.9$ at $t \sim 3\,\mathrm{days}$. These limits are consistent with the earliest spectropolarimetric observation for a Type Ia SN (2018gv) which exhibited a low-level of continuum polarization ($\lesssim 0.2\%$) at $\sim 5$ days post-explosion \citep{2020apj...902...46y}. These limits imply that Type Ia SNe generally appear almost, if not completely, spherically symmetric at the earliest times. For the small number of CCSNe in this sample, the earliest constraints available could allow for the presence of significant asymmetries in the first few days; however, for these SNe the lack of subsequent follow-up observations at later epochs (see Fig. \ref{fig:disc:all}), as the SNe get brighter, means it is not possible to establish a baseline level of polarization (and some measure of the constant interstellar polarization [ISP] components) at maximum light as was done, for example, with SN~2019ein. For the Type II SNe, the early limits, in particular in the $b^{\ast}$ band, constrain the axial ratio to be $>0.65$ within $\sim 1$ day of explosion. We note that these figures do not include corrections for the ISP, a constant source of polarization, across all epochs, arising from intervening dust along the line of sight that is independent of the evolution of the polarization of the SNe themselves \citep{1973iaus...52..145s}. We would expect, more generally, that the constraints on the intrinsic polarization of the SNe would be lower, if a correction of the ISP could be applied. Given the limited number of significant detections of polarization across the whole sample, the Type Ia SN 2019ein stands out as having a series of detections in the $b^{\ast}$-band with a possible increasing degree of polarization (see Fig. \ref{fig:res:19ein}), reaching a maximum at around the second light curve peak observed in the ZTF $r^{\prime}$-band. SN 2019ein has the highest recorded expansion velocities for a Type Ia SN at early times \citep{2020apj...897..159p, 2020apj...893..143k}. SN~2019ein also exhibited blue-shifted line profiles in early spectra (at -14 days relative to B-band maximum light), in particular for the strong Si II feature. The expansion velocities decreased as the SN approached the B-band lightcurve maximum. \citet{2020apj...897..159p} interpreted the peculiar high expansion velocities as indicative of an asymmetric off-centre explosion, in which intermediate mass elements were mixed into the highest velocity portions of the ejecta. The largest measured polarization of $p(b^{\ast}) = 1.12\pm0.25\%$ occurred $\sim 11$ days after the B-band lightcurve maximum. This measurement supports the presence of significant asymmetries in SN~2019ein, as suspected from the earlier spectroscopic observations, however the time delay may indicate that the peculiar expansion velocities and the polarization are measuring the asymmetry in different ways. As we are conducting broad-band polarimetry, however, we are not sensitive to the key diagnostic feature of Type Ia SN asymmetries, namely the polarization profile of the strong Si II line \citep{2007sci...315..212w, 2010apj...725l.167m, 2019mnras.490..578c}. The lower polarization detections for SN~2019np ($<1\%$; see Fig. \ref{fig:res:19np}) are more generally consistent with the low levels of intrinsic continuum polarization usually seen for Type Ia SNe \citep{2008ara&a..46..433w}. SN2018bsz joins a small, but growing collection of SLSNe with multi-epoch polarimetric observations \citep{2015apj...815l..10l, 2016apj...831...79i, 2017apj...837l..14l, 2019mnras.482.4057m}. In general, Type I SLSNe are noted for having low levels of polarization, however SN~2015bn exhibited a significant rise in polarization after $\sim 20\,\mathrm{days}$ post-maximum \citep{2017apj...837l..14l}, with the evolution from the pre-maximum to post-maximum state clearly evident in spectropolarimetric observations \citep{2016apj...831...79i}. Our detection of significant polarization for SN~2018bsz also occurs $\sim 20\,\mathrm{days}$ (rest-frame) after the time of the V-band light curve maximum \citep{2018a&a...620a..67a}. \citet{2016apj...831...79i} and \citet{2017apj...837l..14l} both explained the behaviour of SN~2015bn as being due to a fundamental change in the asymmetry of the ejecta, with early time emission arising from an almost spherical outer layer, whilst at later times the emission arise from a more aspherical interior (giving rise to the increase in polarization with time). Due to the limitations of RINGO3, we were only able to obtain single detection of polarization at $\sim 2\%$. At a similar epoch in the evolution of SN~2015bn, \citeauthor{2017apj...837l..14l} measured $\sim 1\%$ (although the degree of polarization later rose to $\sim 1.54\%$ by $\sim 46\,\mathrm{days}$). It is interesting to note that \citeauthor{2016apj...831...79i} and \citeauthor{2017apj...837l..14l} showed that, although the interior of 2015bn was more aspherical than the outer layers, the orientation of the asymmetry (in the plane of the sky) was the same at both early (before the light curve maximum) and later times (after the light curve maximum). Future dense time series of polarimetric observations of Type I SLSNe will be able to confirm if the rise in polarization, coupled with the stability of the polarization angle, is a common feature for this class of SLSN. SN~2018hna appeared, photometrically and spectroscopically, similar to SN~1987A, with the lightcurve peaks occurring at $\sim 87.5$ and $86$ days post-explosion, respectively \citep{2019apj...882l..15s}. Our RINGO3 observations straddle the lightcurve peak and the level of polarization observed ($\sim 0.7\%$) is slightly higher than seen for SN~1987A ($\sim 0.4\%$) at similar epochs \citep{1991apj...375..264j}. Given the brightness of SN~2018hna, if we had been able to conduct earlier observations of this SN the RINGO3 observation would have been sufficiently sensitive to detect the presence of a similar peaks in the polarization that were seen for SN~1987A \citep{1991apj...375..264j}, if they occurred. The observation of SN~2019nvm is one of the earliest polarimetric observations of a SN, potentially just beating the first observation by \citet{1987iauc.4319....1w} of SN 1987A. \citet{2000apj...536..239l} reported early spectropolarimetry of the Type IIn SN 1998S 5 days after discovery (corresponding to $\sim 5 - 11$ days post-explosion), reporting levels of continuum polarization $\sim 2\%$. Unlike the early emission-line features of SN~2013cu \citep{2014natur.509..471g}, the spectrum of SN1998S persisted for upto 14 days post-discovery, before cooling and developing the classical P Cygni profiles of a Type II SN \citep{2001mnras.325..907f, 2001apj...550.1030w}. Although the spectral evolution of SN~1998S occurred over timescales of weeks, rather than days normally associated with ``flash" observations \citep{2014natur.509..471g, 2017natph..13..510y,2019apj...872..141s}, it does demonstrate that large levels of polarization, due to presence of an aspherical CSM, could be observed at very early epochs. This RINGO3 survey has crucially demonstrated the feasibility of employing polarimeters on robotic telescopes and, coupled with the appropriate feeder survey, the potential for ``flash polarimetry". While it is expected that the CSM signature at early times would be erased by the expansion of the ejecta, in 7 cases we have been able to successfully observe targets within this 5 day window. On 2 occasions, we have been able to trigger Liverpool Telescope observations within a day of discovery by ZTF. The quality of the data presented in this paper has been limited by instrumental errors caused by the single-beam design of RINGO3 making the cancellation of systematic errors difficult. RINGO3 was decommissioned on the Liverpool Telescope in January 2020 and replaced with a prototype of a new dual beam polarimeter (MOPTOP). This uses a dual sCMOS imaging system to record the ordinary and extra-ordinary rays from a polarizing beam-splitter \citep{2016spie.9908e..4ij, 2018spie10702e..4qj}. It therefore has higher throughput than the polaroid-based RINGO3 as well as allowing differential cancellation of polarization errors. Commissioning observations with MOPTOP \citep{2020mnras.494.4676s} show uncorrected systematic errors reduced to $<0.2$ per-cent and the sensitivity increased by a factor $\sim 4 \times$ compared to RINGO3. These combined improvements mean that MOPTOP has a polarization accuracy of $<0.3$\% for a source with $R=17$ in a 600 second exposure. From our sample, 5 targets would be observable to this polarization precision, given that exposure time, at the earliest epochs; including SNe 2018hna and 2019ein. MOPTOP will make tighter constraints or even detections of the polarization feasible for the types of targets we have observed so far with RINGO3. The improved sensitivity of the MOPTOP instrument and the capability to observe in four wavelength bands, from the optical to the near-infrared, will potentially provide a better handle on the ISP. As demonstrated for SN~2019ein, changes in the level of polarization are perceptible, and relative changes in intrinsic polarization can be directly measured. A complete correction for the ISP could be derived, including inferring the wavelength dependence, if observations continued into later phases when a SN might be considered intrinsically unpolarized \citep[see e.g.][]{1991apj...375..264j} when, despite its faintness, it would still be accessible to MOPTOP. Despite the constraints placed on the early-time explosion geometries with this RINGO3 programme, the limited sample size and only a limited number of detections of significant polarization means that ``flash polarimetry" is still {\it terra incognita}. This survey has demonstrated it is possible to conduct these types of observations with very fast turnaround times and, with the advent of MOPTOP on the Liverpool Telescope, it will soon be possible to directly and systematically measure the polarization of SNe at the earliest times. \section*{Acknowledgements} The Liverpool Telescope is operated on the island of La Palma by Liverpool John Moores University at the Spanish Obervatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias, with financial support from the UK Science and Technologies Facilities Council (STFC). Financial support for the development RINGO3 was also provided by the STFC Project Research and Development (PRD) scheme. Based on observations obtained with the Samuel Oschin Telescope 48-inch and the 60-inch Telescope at the Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under Grant No. AST-1440341 and a collaboration including Caltech, IPAC, the Weizmann Institute for Science, the Oskar Klein Center at Stockholm University, the University of Maryland, the University of Washington, Deutsches Elektronen-Synchrotron and Humboldt University, Los Alamos National Laboratories, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW. The research of JRM was supported through a Royal Society Research Fellowship. The research of YY is supported through a Benoziyo Prize Postdoctoral Fellowship. The research of AGY is supported by the EU via ERC grant No. 725161, the ISF GW excellence center, an IMOS space infrastructure grant and BSF/Transformative and GIF grants, as well as The Benoziyo Endowment Fund for the Advancement of Science, the Deloro Institute for Advanced Research in Space and Optics, The Veronika A. Rabl Physics Discretionary Fund, Paul and Tina Gardner, Yeda-Sela and the WIS-CIT joint research grant; AGY is the recipient of the Helen and Martin Kimmel Award for Innovative Investigation. The work of X. Wang is supported by National Natural Science Foundation of China (NSFC grants 11325313, 11633002, and 11761141001), and the National Program on Key Research and Development Project (grant no. 2016YFA0400803). The authors thank S. Van Dyk for useful comments on the manuscript. \section*{Data Availability} The observational data presented here is available in the public archive of the Liverpool Telescope (\url{https://telescope.livjm.ac.uk/cgi-bin/lt_search}) \bibliographystyle{mnras} \bsp % \label{lastpage}

Title: ALCHEMI finds a "shocking" carbon footprint in the starburst galaxy NGC~253

Abstract: Centers of starburst galaxies may be characterized by a specific gas and ice chemistry due to their gas dynamics and the presence of various ice desorption mechanisms. This may result in a peculiar observable composition. We analyze abundances of $CO_2$, a reliable tracer of ice chemistry, from data collected as part of the ALMA large program ALCHEMI, a wide-frequency spectral scan toward the starburst galaxy NGC~253 with an angular resolution of 1.6$''$. We constrain the $CO_2$ abundances in the gas phase using its protonated form $HOCO^+$. The distribution of $HOCO^+$ is similar to that of methanol, which suggests that $HOCO^+$ is indeed produced from the protonation of $CO_2$ sublimated from ice. The $HOCO^+$ fractional abundances are found to be $(1-2)\times10^{-9}$ at the outer part of the central molecular zone (CMZ), while they are lower ($\sim10^{-10}$) near the kinematic center. This peak fractional abundance at the outer CMZ is comparable to that in the Milky Way CMZ, and orders of magnitude higher than that in Galactic disk star-forming regions. From the range of $HOCO^+/CO_2$ ratios suggested from chemical models, the gas-phase $CO_2$ fractional abundance is estimated to be $(1-20)\times10^{-7}$ at the outer CMZ, and orders of magnitude lower near the center. We estimate the $CO_2$ ice fractional abundances at the outer CMZ to be $(2-5)\times10^{-6}$ from the literature. A comparison between the ice and gas $CO_2$ abundances suggests an efficient sublimation mechanism. This sublimation is attributed to large-scale shocks at the orbital intersections of the bar and CMZ.

https://export.arxiv.org/pdf/2208.13983

command. \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \newcommand{\hocop}{HOCO$^+$} \newcommand{\cotw}{CO$_2$} \newcommand{\ps}{s$^{-1}$} \newcommand{\methanol}{CH$_3$OH} \newcommand{\water}{H$_2$O} \newcommand{\httcop}{H$^{13}$CO$^+$} \newcommand{\htw}{H$_2$} \newcommand{\enum}[1]{$10^{#1}$} \newcommand{\fnum}[2]{$#1\times 10^{#2}$} \newcommand{\kms}{km\,s$^{-1}$} \shorttitle{Protonated \cotw\ in NGC~253} \shortauthors{Harada et al.} \begin{document} \title{ALCHEMI finds a ``shocking" carbon footprint in the starburst galaxy NGC~253} \input{ALCHEMICollabAuthList} \section{Introduction} \label{sec:intro} The abundance of interstellar molecules depends on the balance between their formation and destruction processes in the gas phase and on grain surfaces. Exchange processes can transform molecules in one phase to the other; gas-phase molecules can freeze onto dust grains as ice (adsorption), while molecules on grain surfaces can sublimate into the gas phase (desorption). Knowing both gas- and ice-phase abundances is, therefore, necessary for a comprehensive understanding of the chemical composition and their related physical conditions. Carbon dioxide (\cotw) is one of the most dominant forms of ice on interstellar dust \citep{2011ApJ...740..109O} together with H$_2$O and CO. \cotw\ is, in fact, one of the ice species detectable in extragalactic sources \citep[e.g., ][]{2007ApJ...659..296L, 2015ApJ...807...29Y}. It is obvious from the strong CO rotational emission that the gas-phase CO is abundant, but \cotw\ is thought to reside more preferentially on dust due to the inefficient gas-phase formation \citep{2011ApJ...735...15G}. Because \cotw\ has higher desorption energy than that of CO, the presence of abundant gas-phase \cotw\ requires stronger desorption mechanisms. While \cotw\ can be detected via rotational-vibrational transitions in warm gas ($\gtrsim$ several 100 K) \citep[e.g., ][]{2003Boonman} or in ice with broader line features \citep[e.g., ][]{2015ApJ...807...29Y}, it cannot be observed in cold gas due to the lack of a permanent dipole moment. Although the gas-phase abundances of such species without a dipole moment cannot be directly measured through commonly observed rotational transitions, it has been proposed that they can be estimated from their protonated forms \citep{1977ApJ...215..503H,2015A&A...579L..10A,2019MNRAS.483L.114R}. The protonated form of \cotw, \hocop, was first detected in Sgr B2 by \citet{1981ApJ...246L..41T}, but its line identification required spectroscopic confirmation by \citet{1982ApJ...254..405D}. Since then, \hocop\ has been detected in translucent clouds \citep{turner_physics_1999}, low-/high-mass star-forming regions \citep{sakai_detection_2008,vastel_abundance_2016,majumdar_detection_2018,fontani_protonated_2018}, the Galactic Center \citep[Sgr A and B2 clouds;][]{minh_observations_1988,minh_abundance_1991}, starburst galaxies \citep{2006ApJS..164..450M,2015AA...579A.101A} and a $z\sim0.9$ molecular absorber \citep{2013A&A...551A.109M}. Among them, \citet{minh_abundance_1991} and \citet{2015MNRAS.446.3842A} found about 2 orders of magnitude higher fractional abundances of \hocop\ in the Galactic Center than in spiral-arm molecular clouds. The main formation paths of \hocop\ are gas-phase reactions: protonation of \cotw\ such as \begin{equation} {\rm CO_2 + H_3^+ \longrightarrow HOCO^+ + H_2,} \nonumber \end{equation} or an ion-neutral reaction \begin{equation} {\rm HCO^+ + OH \longrightarrow HOCO^+ + H} \nonumber \end{equation} \citep{vastel_abundance_2016,2017AandA...602A..34B}. In the former route, the \hocop\ abundance can increase due to the evaporation of \cotw\ from grain surfaces because \cotw\ is one of the most abundant forms of carbon on grain surfaces. The ice can sublimate thermally (e.g., in the vicinity of protostars), or non-thermally (e.g., photodesorption, cosmic-ray-induced evaporation, or shock sputtering). While the latter route from HCO$^+$ is considered dominant in at least some parts of high- or low-mass star-forming regions \citep{majumdar_detection_2018,fontani_protonated_2018}, the protonation of \cotw\ can be the dominant route when there is a fast mechanism of \cotw\ sublimation. Shocks, one of drivers of the ice sublimation process, are ubiquitous in galactic centers. In many barred-spiral galaxies, galactic centers host intersections of $x_1$ and $x_2$ orbits\footnote{Bars lie on the $x_1$ orbits, while $x_2$ orbits form inner nuclear rings. The location of nuclear rings may correspond to that of inner Lindblad resonances, but this is not always the case \citep{2012ApJ...758...14K}.} \citep{1992MNRAS.259..345A,2013ApJ...769..100S,2020MNRAS.494.6030S}. At these orbital intersections, shocks are naturally expected. Abundances of typical shock molecular tracers such as CH$_3$OH and SiO have been found to be enhanced at locations of orbital intersections in IC342 and M83 \citep{2005ApJ...618..259M,2019ApJ...884..100H}. In this paper, we report an enhancement of \hocop\ at the orbital intersections near the center of the starburst galaxy NGC~253. NGC~253 is one of the nearest \citep[$d=3.5$\,Mpc;][]{2005MNRAS.361..330R} and most studied starburst galaxies. Its nuclear ring forms a central molecular zone (CMZ) within a few hundred parsec scale, with a mass of $\sim 2\times 10^8\,M_{\odot}$ within a radius of $r = 150$\,pc \citep{leroy_alma_2015}. This large reservoir of molecular gas enables active star formation \citep{sakamoto_molecular_2006,sakamoto_star-forming_2011,bolatto_suppression_2013,krieger_molecular_2019,2020MNRAS.491.4573R}, which affects the properties of molecular gas \citep[e.g., through heating ][]{mangum_fire_2019}. The gas in the bar orbit ($x_1$ orbits) is being fed to the center of NGC~253 ($x_2$ orbits), and shocks are expected when this gas flow collides with the nuclear ring as discussed above for other galaxies. Figure \ref{fig:orbit}(left) shows IRAC 8$\mu$m \citep{2009ApJ...703..517D,slvl} and CO(2-1) integrated-intensity (ALMA data 2018.1.01321.S; PI: Faesi) images covering most of NGC~253 to indicate the locations of these orbits. The association between shocks at orbital intersections and the chemistry has been proposed by \citet{2000A&A...355..499G} and \citet{2015ApJ...801...63M}. At the center of NGC~253, a large number of molecular species are detectable \citep{2006ApJS..164..450M,2015ApJ...801...63M}. To fully explore the chemical complexity in this galaxy, we conducted the ALMA large program AL{\footnotesize MA} Comprehensive High-resolution Extragalactic Molecular Inventory \citep[ALCHEMI;][]{2021A&A...656A..46M}. ALCHEMI is a wide-frequency, unbiased spectral scan mosaic toward the CMZ of NGC~253 at a common 1.6$\arcsec$ resolution. This survey has discovered high cosmic-ray ionization rates \citep{2021A&A...654A..55H,2021ApJ...923...24H,2022arXiv220403668H}, detected a phosphorus-bearing species for the first time in an extragalactic source \citep{2022A&A...659A.158H}, and detected methanol masers \citep{2022arXiv220503281H} in the CMZ of NGC~253. We utilize the ALCHEMI data to study multiple transitions of \hocop. This paper is organized as follows. Section \ref{sec:obs} describes our observational parameters and data analysis, and images of integrated intensities are presented in Section \ref{sec:integint}. Derived column densities of \hocop\ are shown in Section \ref{sec:coldens}, while the \hocop/\cotw\ ratios are discussed using chemical models in Section \ref{sec:chem}. In Section \ref{sec:discussion}, we discuss our results including a comparison with ice abundances. Our results are summarized in Section \ref{sec:summary}. \section{Observations and data analysis} \label{sec:obs} The ALCHEMI spectral survey mosaic of the NGC~253 CMZ was performed between 2017 and 2019, with ALMA 12m-antenna and 7m-antenna arrays. It covered a broad frequency range between 84 and 373~GHz (Bands 3 to 7, avoiding deep atmospheric lines), down to sensitivities $\sim 10$~mK per 10\,km\,s$^{-1}$ channel. The ALCHEMI data products have a uniform angular resolution of 1$\farcs$6 and cover a field of view of $50\arcsec\times20\arcsec$ centered on the CMZ of NGC~253 (phase center $\alpha=00^h47^m33.28^s$, $\delta=-25^\circ17'17.7''$ (ICRS)). The extent of the largest recoverable angular scale is greater than or equal to 15$\arcsec$. A detailed description of the ALCHEMI survey products can be found in \citet{2021A&A...656A..46M}. We extracted the line cubes around the transitions of \hocop\ with a velocity resolution binned to 10\,km\,s$^{-1}$. Within the $\sim 290$~GHz coverage of the ALCHEMI survey, we find 14 detectable \hocop\ transitions, occurring every $\sim 21.4$~GHz. However, some transitions are severely blended with transitions from other species and are not used in this analysis. The spectroscopic parameters and spectral channel RMS values for the \hocop\ transitions used in this paper are listed in Table~\ref{tab:spec}. \begin{deluxetable*}{ccccccc} \tablecolumns{7} \tablewidth{0pc} \tablecaption{\hocop\ Spectroscopic Properties and RMS Noise Values\label{tab:spec}} \tablehead{\colhead{Transition} &\colhead{$\nu_{rest}^{(a)}$} &\colhead{$E_{\rm up}$$^{(b)}$} &\colhead{$log(A_{\rm ul})$$^{(c)}$} &\colhead{RMS$^{(d)}$} &\colhead{RMS$^{(e)}$} &\colhead{Blending$^{(f)}$}\\ \colhead{} & \colhead{(GHz)} &\colhead{(K)} &\colhead{(s$^{-1}$)} &\colhead{(mJy beam$^{-1}$)} &\colhead{(mK)} &\colhead{}} \startdata 4$_{0,4}$-3$_{0,3}$ &85.531&10.3&-4.63&0.19&12.&Potential blending with CH$_3$CCH. Minor blending with U-lines\\ 5$_{0,5}$-4$_{0,4}$&106.914&15.4&-4.33&0.20&8.4 & N/A\\ 6$_{0,6}$-5$_{0,5}$&128.295&21.6&-4.08&0.33&9.5 &Minor blending with U-lines\\ 7$_{0,7}$-6$_{0,6}$&149.676&28.7&-3.88&0.42&9.0 &Potential blending with U-lines\\ 8$_{0,8}$-7$_{0,7}$&171.056&36.9&-3.70&0.75&12. &Potential blending with CH$_3$CCH.\\ 12$_{0,12}$-11$_{0,11}$&256.566&80.0&-3.16&0.99&7.2 &Potential blending with CH$_3$CCH, HC$_3$N $v7=2$\\ \hline \multicolumn{7}{c}{Transitions below were not used for analysis}\\ \hline 9$_{0,9}$-8$_{0,8}$ &192.435&46.2&-3.54& &&Blended with U-line\\ 10$_{0,10}$-9$_{0,9}$ &213.813&56.4&-3.40& &&Blended with C$_2$H$_5$OH\\ 11$_{0,11}$-10$_{0,10}$ &235.190&67.7&-3.28&&&Blended with SO$_2$\\ 13$_{0,13}$-12$_{0,12}$ &277.941&93.4&-3.06&&&Blended with U-line\\ 14$_{0,14}$-13$_{0,13}$ &299.314&107.7&-2.96&&&Non detection\\ 15$_{0,15}$-14$_{0,14}$ &320.686&123.1&-2.87&&&Non detection\\ 16$_{0,16}$-15$_{0,15}$ &342.056&139.6&-2.78&&&Non detection\\ 17$_{0,17}$-16$_{0,16}$ &363.424&157.0&-2.70&&&Non detection\\ \enddata \tablecomments{$(a)$ Rest frequency; $(b)$ Upper level energy of the transition; $(c)$ $A_{ul}$: Einstein coefficient of spontaneous emission. All values were taken from the Cologne Database for Molecular Spectroscopy \citep[CDMS; https://cdms.astro.uni-koeln.de;][]{2001AA370L49M,2005JMoSt.742..215M,2017AandA...602A..34B}; (d) and (e) RMS values of a single channel with $\Delta v=10$\,km\,\ps\ in mJy beam$^{-1}$ and mK units; (f) presence of blending; Transitions are shown with quantum numbers $J_{K_a, K_c}$. Only $K_a = 0$ transitions are shown because transitions with $K_a \neq 0$ are not detected due to their higher energy state and lower Einstein coefficients. The upper part of this table shows transitions used for the analyses of this paper, whose line shapes are separable from neighboring lines. The lower part lists transitions with severe blending or without reliable detection. ``Potential blending" means the case where the line centers are separated by more than 200\,km\,s$^{-1}$, but the line wing can contaminate the moment maps of \hocop\ transitions.} \end{deluxetable*} \section{Integrated intensities}\label{sec:integint} Figure \ref{fig:mom0} (a-d) shows the velocity-integrated intensity (moment 0) images of \hocop\ in multiple transitions (see Table \ref{tab:spec} for their properties). Many transitions used in this work have neighboring lines, and it is not possible to make the moment 0 images simply by collapsing neighboring channels. To exclude this contamination, we applied the 3-D mask made from the position-position-velocity space only including pixels with CO $J=1-0$ detections above the $20\sigma$ level. The 20$\sigma$ cutoff may sound unnecessarily high, but the signal-to-noise ratios of \hocop\ transitions are more than 100 times lower than that of CO(1-0). Therefore, this mask does not exclude any notable \hocop\ emission but helps to exclude the contamination from nitrogen sulfide transitions neighboring with CO(1-0) to be included in the mask. Despite the elimination of contamination with this mask, the only transition that is free from contamination is $5_{0,5}-4_{0,4}$ (panel a) of Figure \ref{fig:mom0}. Other images that are relatively less affected from blending are also shown in Figure \ref{fig:mom0} (panels b-d). The level of contamination is usually very low ($<10\%$) except for GMC 5, where there is little \hocop\ emission and stronger emission from neighboring lines. Transitions $4_{0,4}-3_{0,3}$ and $8_{0,8}-7_{0,7}$ can still be used to obtain column densities as we use spectral fitting, but their images are not shown. A low-excitation line of \hocop\ ($5_{0,5}-4_{0,4}$; $E_{\rm up}=15.4\,$K) shows peaks near the outer CMZ, in giant molecular clouds (GMCs) 1, 7, and 9 \citep[GMC numbering is shown in panel f;][see also Appendix \ref{sec:app_gmc} for coordinates]{leroy_alma_2015}. On the other hand, the higher-excitation transitions ($E_{\rm up}\gtrsim 30\,$K) peak closer to the center (GMCs 3 and 6). Panels (e) and (f) of Figure \ref{fig:mom0} show integrated-intensity images of \methanol($2_k-1_k$) (group of transitions at $\nu_{rest} \sim 96.74$ GHz with the strongest transition $2_0^+ - 1_0^+$)\footnote{These transitions may not be in local thermodynamic equilibrium (LTE), but are ``quasi thermal" and they are not identified as masing \citep{2022arXiv220503281H}.} and H$^{13}$CO$^+$(1-0) ($\nu_{rest} = 86.75$ GHz) for comparison. The emission distribution of \methanol($2_k - 1_k$) is similar to that of the low-$J$ transitions of \hocop, as well as low-$J$ transitions of HNCO, and SiO \citep[][Huang et al., in preparation]{2015ApJ...801...63M}. On the other hand, the distribution of H$^{13}$CO$^+$(1-0), which is rather similar to that of molecules with strongest emission \citep[e.g., CO, HCO$^+$, HCN, CS, etc.][]{2015ApJ...801...63M, 2021A&A...656A..46M}, is clearly different from that of \hocop. The H$^{13}$CO$^+$(1-0) emission is concentrated near the center of NGC 253 (GMCs 3-7) instead of the outer CMZ. The similarity between integrated intensities of \hocop\ ($5_{0,5}-4_{0,4}$) and CH$_3$OH ($2_k-1_k$) and the difference between those of \hocop\ ($5_{0,5}-4_{0,4}$) and H$^{13}$CO$^+$(1-0) are highlighted in Figure \ref{fig:ratio}. While the \hocop($5_{0,5}-4_{0,4}$)/CH$_3$OH ($2_k-1_k$) ratios are relatively constant, the \hocop($5_{0,5}-4_{0,4}$)/H$^{13}$CO$^+$(1-0) ratio varies significantly. All these transitions have relatively low upper state energies (\hocop\ ($5_{0,5}-4_{0,4}$): $E_u=15.4\,$K, CH$_3$OH ($2_0^+-1_0^+$):$E_u=7.0\,$K, H$^{13}$CO$^+$(1-0): $E_u=4.2\,$K), and these ratio maps should be good proxies for variations in column densities of these molecules. \section{Column densities and fractional abundances}\label{sec:coldens} Figure \ref{fig:hocop_col}(a) shows the column density map of \hocop. These column densities were derived using the public software CASSIS \citep[http://cassis.irap.omp.eu/][]{2015sf2a.conf..313V} supplied with spectroscopic constants from the spectroscopic database from the CDMS \citep{2001AA370L49M,2005JMoSt.742..215M}. CASSIS calculates molecular column densities based on input spectral line brightness temperatures with consideration of optical depths, either with an LTE assumption or, if collisional rates are available, non-LTE assumption. We used a Markov Chain Monte Carlo (MCMC) algorithm assuming LTE to fit column densities, excitation temperatures, line velocities, and line widths (see Appendix \ref{sec:app_cassis}). Column densities were calculated only for the pixels with $>3\sigma$ detection of both \hocop($4_{0,4}-3_{0,3}$) and \hocop($5_{0,5}-4_{0,4}$) at velocities within 30\,\kms\ from the line center. Line-center velocities used for this 3-$\sigma$ detection criterion are determined from the image cube of CO(1-0) from ALCHEMI data. These line-center velocities from CO may be different from \hocop\ velocities fitted from CASSIS. Instead of deriving the column densities and excitation temperatures on a pixel-by-pixel basis, we bin the intensities within hexagonal pixels with a horizontal length of 0\farcs8, half of the image spatial resolution of 1.6\arcsec, to reduce the computational time running CASSIS. Examples of spectral fitting are shown for hexagonal pixels located at ($x_{\rm offset}$,$y_{\rm offset}$)=($-17.3\arcsec$,$-10.4\arcsec$) and ($2.0\arcsec$,$2.1\arcsec$) in Figure \ref{fig:spectra} where the offset is taken from the phase center. In general, the observed spectra fit well with the LTE spectra. The resulting column-density distribution appears similar to that of the moment~0 images of low-excitation transitions (e.g., Figure~\ref{fig:mom0}a). Excitation temperatures derived from the above spectral fitting are shown in Figure~\ref{fig:hocop_col}(b). While regions far from the kinematic center\footnote{The kinematic center of NGC~253 is located near GMC~5 \citep{turner_1_1985,2010ApJ...716.1166M}. Although there is a debate on the exact location of the kinematic center, the difference of 0\farcs7 appearing in the literature does not affect our discussion.} of NGC~253 show low excitation temperatures of $\lesssim 10\,$K (GMCs 1, 2, 8, and 9), the excitation temperatures become higher, up to 40 K, near the center of NGC~253 (GMCs 3 and 6). We also obtain the total hydrogen column densities (Figure \ref{fig:hocop_col}c) to calculate fractional abundances of \hocop\ (the \hocop\ column densities divided by total hydrogen column densities N(H$_2$)). The total H$_2$ column densities were derived from the dust continuum image at 361.5 GHz shown by \citet{2021ApJ...923...24H} with the derivation method based on \citet{1983QJRAS..24..267H} for pixels above $3\sigma$ detection. A simplified formula is given as Equation (3) in \citet{mangum_fire_2019}: \begin{equation} N(H_2) ({\rm cm^{-2}}) \sim 7.0\times 10^{22} R_{dg} \left (\frac{\lambda ({\rm mm})}{0.4}\right)^\beta \frac{T_R({\rm K})}{T_d ({\rm K})}, \end{equation} where $N(H_2)$ is the molecular hydrogen column density, $R_{dg}$ is the dust-to-gas mass ratio, $\lambda$ is the wavelength, $T_R$ is the radiation temperature, and $T_d$ is the dust temperature. This formula is valid for $h\nu\ll kT_d$ (the Rayleigh-Jeans approximation). We use the emissivity $\beta =1.5$, a dust temperature $T_{\rm dust}=30\,$K, and a dust-to-gas mass ratio of 150 following \citet{mangum_fire_2019}. This estimate of the dust temperature is close to the observed value, but some dust components may be warmer. \citet{2018ApJ...860...23P} derived dust temperature components of 37, 70, and 188\,K in the central region of NGC 253 using their assumed source size of $17.3\arcsec \times 9.2\arcsec$ from their Herschel and SOFIA observations. These components contain mass fractions of 65, 26, and 9\,\%, respectively. If the dust is warmer, the actual column density should be smaller by a similar factor; for instance, a factor of 5 smaller if $T_{\rm dust} = 150\,$K. The column-density dependence on the dust temperature becomes larger than $\propto \frac{1}{T_d}$ when the dust is cold ($T_d \lesssim 20\,$K) and one cannot use the Rayleigh-Jeans approximation, but we expect that there is a very small amount of cold dust in the center of NGC 253. The fractional abundance of HOCO$^+$ is higher at larger distances from the center of NGC~253, and it decreases by more than an order of magnitude at the center (Figure~\ref{fig:hocop_col}d). At peaks of \hocop (GMCs 1, 8, and 9), the fractional abundance is $\sim$\fnum{(1-2)}{-9}, similar to those observed in Galactic center clouds: $(2-8)\times10^{-9}$ \citep{minh_abundance_1991,2015MNRAS.446.3842A}. On the other hand, it is orders of magnitude higher than those observed in Galactic disk clouds, which range from \enum{-13} to \fnum{5}{-11} \citep{vastel_abundance_2016,fontani_protonated_2018,majumdar_detection_2018}. \section{\hocop/\cotw\ ratios}\label{sec:chem} To estimate the gas-phase abundances of \cotw\ from \hocop, we ran chemical abundance models based on Nautilus \citep{2016MNRAS.459.3756R}, accounting for gas, ice surface, and ice mantle phases. In addition to the thermal evaporation, desorption from dust heating due to cosmic rays \citep{1993MNRAS.261...83H} is included in the model. Desorption through cosmic-ray heating of dust is where the dust grain is temporarily heated to a certain maximum temperature for a very short time scale ($\sim 10^{-5}\,$s), then cools down. The model also includes photodesorption \citep{2009A&A...496..281O,2009ApJ...693.1209O} both from direct UV photons and cosmic-ray-induced UV photons with a default yield of \enum{-4} for all the grain species. We also ran a model with a desorption yield of \enum{-3} for \cotw. Our models do not include shocks. We calculated grid models with varying densities ($n=10^3 - 10^6$\,cm$^{-3}$) and cosmic-ray ionization rates ($\zeta = 10^{-17} - 10^{-12}$\,\ps) following a similar approach as \citet{2021ApJ...923...24H}. Temperatures were calculated in the Meudon Photodissociation Region (PDR) code (ver. 1.5.4) \citep{2006ApJS..164..506L} (Figure \ref{fig:chem_model}a), and were fed to Nautilus to run chemical abundance models with a larger chemical network. Despite the high gas temperature with the high cosmic-ray ionization rate ($T>1000\,$K when $\zeta=10^{-12}$\,\ps\ and $n=10^3$\,cm$^{-1}$), the dust temperature calculated from the Meudon code remains cold, around 11\,K. We adopted a maximum visual extinction $A_{\rm V}=20\,$mag with a turbulent velocity of $1\,$km~s$^{-1}$, and used the temperature in the model at $A_{\rm V}=10\,$mag (Figure \ref{fig:chem_model}d), where the effects of the PDRs are negligible. We note that, unlike in the description of observational results, fractional abundances are expressed as abundances of certain species over total hydrogen abundances ($N_{\rm Htotal} = N_{\rm Hatom} + 2N_{H2}$), instead of molecular hydrogen abundances. Figure~\ref{fig:chem_model} (b) shows \hocop/\cotw\ abundance ratios in the gas phase, that vary between $10^{-5} - 10^{-2}$ for the most part. Previous ALCHEMI studies have suggested that the cosmic-ray ionization rates in NGC~253 are at least a few orders of magnitude higher than that in the Galactic spiral arm clouds \citep{2021A&A...654A..55H,2021ApJ...923...24H,2022arXiv220403668H}, which increases the \hocop/\cotw\ abundance ratios due to an increased H$_3^+$ abundance. Although the cosmic-ray ionization rates are expected to be high, extremely high rates ($\zeta \gtrsim 10^{-13}$\,\ps\ for $n=10^5\,$cm$^{-3}$) would destroy \hocop\ (Figure \ref{fig:chem_model}c), especially in lower density regions. Therefore, we consider \hocop/\cotw\ ratios of $\sim 10^{-3} - 10^{-2}$ in later sections, which are taken from the parameter space where the \hocop\ abundance is moderately high\footnote{We note that the presence of \hocop\ is still consistent with the cosmic-ray ionization rates derived by \citet{2021A&A...654A..55H} and \citet{2021ApJ...923...24H} ($\zeta \gtrsim 10^{-14}$\,\ps\ for $n=10^5\,$cm$^{-3}$), but the value obtained by \citet{2022arXiv220403668H} ($\zeta \sim 10^{-13}$\,\ps\ for $n=10^5\,$cm$^{-3}$) would not allow high fractional abundances of \hocop.}. It is worth noting that cosmic-ray-induced desorption mechanisms do not change the abundances significantly. This is because cosmic-ray-induced photodissociation is a more efficient form of destruction than desorption if the desorption yield is on the order of \enum{-3} or lower. According to \citet{2009A&A...496..281O}, the photodesorption yield of \cotw\ cannot go higher than a few times \enum{-3}, which means photodesorption is not significant. Desorption of \cotw\ due to cosmic-ray heating of dust has an even lower effect than the photodesorption for models run with the commonly-used maximum dust temperature of 70\,K. In our model, the dominant formation reactions of \hocop\ vary with time. In general, protonation of \cotw\ is more dominant in early time ($<10^5$\,yr), and the gas-phase production with HCO$^+$ and OH becomes more efficient in later times (see Section \ref{sec:intro}). However, the dominant formation routes also vary with physical conditions and it is difficult to conclude which one is more dominant. Although our models do not include shocks, but we argue that this approach should be sufficient to estimate the \hocop/\cotw\ ratios, especially their upper limit. The \hocop/\cotw\ ratios are determined by the balance among the protonation of \cotw, electron recombination of \hocop, proton transfer from \hocop\ to species with higher proton affinity than \cotw, and the ion-neutral production of \hocop (HCO$^+$ + OH). These reactions occur regardless of shocks. If shocks evaporate \cotw\ significantly, there should be less contribution from the \hocop\ formation through HCO$^+$ and OH compared with the protonation of \cotw, and the \hocop/\cotw\ ratios should be lower, while shocks should increase the fractional abundances of both \hocop\ and \cotw. Therefore, our models without shocks are likely sufficient to obtain upper limits of the \hocop/\cotw\ ratios, but more realistic modeling with shocks will be conducted as future work. It should also be noted that the models without shocks severely underproduce \hocop\ fractional abundances compared with observed peak values, which implies the need for shocks to explain the observed abundances (see Section \ref{sec:disc_origin}). We include the gas-neutral reaction of HCO$^+$ + OH and many other related reactions in the model. The fact that our model could not reproduce the observed fractional abundances suggests that the formation route through this reaction is not enough to explain our observations. \section{Discussion}\label{sec:discussion} \subsection{Origins of \hocop\ emission}\label{sec:disc_origin} As discussed earlier, the formation routes of \hocop\ do not have to involve the protonation of \cotw. The gas-phase reaction between HCO$^+$ and OH may also contribute to \hocop. \citet{fontani_protonated_2018} argued that \hocop\ must be formed via the reaction above (HCO$^+$ + OH; see Sect. \ref{sec:intro}) in high-mass star-forming cores because \hocop\ fractional abundances derived from the $4_{0,4}-3_{0,3}$ transition are correlated with the fractional abundances of H$^{13}$CO$^+$, while there is no correlation with that of methanol. If \hocop\ is formed via protonation, a large amount of evaporated \cotw\ must be present, which also implies a large amount of methanol in the gas phase. This is because methanol is formed on the ice, and its gas-phase production is extremely inefficient \citep{2007A&A...467.1103G}. The CMZ of NGC~253 shows a different trend from the case of these high-mass star-forming cores. We do see a positive spatial correlation between \hocop\ and \methanol\ with low-excitation transitions of both molecules enhanced at the outer CMZ of NGC 253 (Figure \ref{fig:ratio}). Meanwhile, the correlation between \hocop\ and H$^{13}$CO$^+$ is weak because H$^{13}$CO$^+$ is more abundant near the center of the CMZ \citep[Figure \ref{fig:ratio}; see also ][ for abundances of H$^{13}$CO$^+$]{2021ApJ...923...24H}. There is a caveat that the ice composition may be different in high-mass star-forming regions observed by \citet{fontani_protonated_2018} and NGC 253 CMZ, and the presence or lack of correlation may not necessarily imply a difference in formation routes. On the other hand, the presence of a correlation between \methanol\ and \hocop\ and the lack of correlation between \hocop\ and most other species (e.g., CO, HCN, HCO$^+$, etc.) strongly suggest a similar mechanism enhancing abundances of both \methanol\ and \hocop. This mechanism must involve desorption, as methanol is only efficiently formed on ice. Therefore, we argue that \hocop\ in our observations is likely formed from \cotw\ through protonation. \subsection{Inferred gas-phase \cotw\ fractional abundances}\label{sec:co2frac} If \hocop\ is produced through the protonation of \cotw\ as we discussed above, \cotw\ must be evaporated from ice into the gas phase because \cotw\ is much more abundant in the ice than in the gas phase (Figure \ref{fig:chem_model}d). Obtaining the gas-phase fractional abundances of \cotw\ could provide essential hints helping us to evaluate the origin of the gas-phase \cotw. From the chemical model, we find that the range of the \hocop/\cotw\ ratio is $\sim 0.001-0.01$. Because the maximum fractional abundance of \hocop\ is $\sim 2\times10^{-9}$ in GMC 1 and $\sim 1\times10^{-9}$ in GMCs 2, 8, and 9, the gas-phase \cotw\ fractional abundances can be \fnum{(1-20)}{-7} at the outer CMZ, where the \hocop\ intensity peaks. \subsection{Comparison with ice observations of \cotw}\label{sec:ice} Here we compare the fractional abundance of \cotw\ gas estimated above with that of \cotw\ ice observations. We utilize the \cotw\ column densities of selected regions of AKARI observations by \citet{2015ApJ...807...29Y}. Although this reference describes these observations in detail, we include the summary of them in Appendix \ref{sec:app_ice}. These observations used a rectangle slit with a size of $5\arcsec \times 5.8 \arcsec$ (Fig \ref{fig:co2ice} left). Subsequently, we extracted the values of the continuum flux from the same regions to estimate the total \htw\ column density using the method described in Section \ref{sec:coldens}. We then derived the fractional abundances of \cotw\ in the ice phase in these regions shown in Figure \ref{fig:co2ice} (right). We note that the components traced with \cotw\ ice likely come from relatively lower-column-density regions than the ones traced by the dust continuum. Therefore, we have to be aware of the caveat that our estimation of the \cotw\ fractional abundances is rather crude, only accurate for an order of magnitude approximation. The \cotw\ ice fractional abundances near the center of NGC 253 (rectangular regions 2 and 3) are lower than in regions 4 and 5 by about an order of magnitude or more. Away from the center, the \cotw\ ice fractional abundances are similar to those of the Milky Way ISM of $\sim 10^{-6}-10^{-5}$ \citep{2003Boonman}, which is also consistent with fractional abundances of \cotw\ ice in our chemical model. Therefore, the \cotw\ ice abundance decreases towards the center of NGC 253, deviating from the Milky Way value. We note that, if we use a higher dust temperature, we would derive lower \htw\ column densities than those we show in Figure \ref{fig:hocop_col}. Subsequently, the fractional abundance estimate would increase. Although the warm dust does not fully explain the large difference in the derived fractional abundances between the outer part (rectangular regions 4 and 5) and center (rectangular regions 2 and 3) of the CMZ, this uncertainty should be considered in the interpretation of the data. If the \cotw\ ice fractional abundance is indeed lower in the center than at the outer CMZ, one of the possible factors that may attribute to this suppression is photodissociation, either directly by UV photons or cosmic-ray-induced UV photons. Because of the high star formation rate in the rectangular regions 2 and 3, strong photodissociation is expected \citep{2015ApJ...801...63M}. Warm dust can also lower the ice abundance of \cotw\ through desorbing \cotw\ ice into the gas phase, or desorbing the precursors of \cotw\ ice. Desorption of many \cotw\ precursors can take place with lower dust temperatures than the desorption of \cotw\ itself. \cotw\ ice is thought to be formed through \begin{eqnarray} {\rm CO + OH \longrightarrow CO_2 + H}\\ {\rm CO + O \longrightarrow CO_2} \end{eqnarray} \citep{2013A&A...559A..49M} and possibly \begin{equation} {\rm H_2CO + O \longrightarrow CO_2 + H_2} \end{equation} \citep{2015A&A...577A...2M}. Some of these reactants have lower binding energies than \cotw\ \citep[$E_b{\rm (CO_2)}\sim 3000\,$K, $E_b{\rm (CO)}\sim 1300\,$K, $E_b{\rm (O)}\sim 1600\,$K;][]{2017MolAs...6...22W,2022arXiv220107512M}\footnote{Note that the binding energy is not the temperature where desorption takes place. Effects of desorption appear when the desorption rate becomes significant enough compared with the accretion rate. There is also an additional complication caused by the ice composition. For example, even if a species has a low binding energy, it may not desorb if it is buried in the ice of another species with a higher binding energy.}. Higher dust temperatures increase the desorption rates of ice species, and species with lower binding energies can desorb with lower dust temperatures \citep[35-50\,K for CO, $\sim 50\,$K for O, $\sim 80\,$K for CO$_2$][]{2022arXiv220107512M}. If a desorption rate of any of these reactants is faster than the reaction rate to form \cotw\, there would be less abundances of \cotw\ ice. Although the mass of warm or hot dust is smaller than that of cold dust, these warm/hot components are likely concentrated in rectangular regions 2 and 3 in Figure \ref{fig:co2ice} where the star formation is active. This can explain the suppression of ice in these regions as well as low \hocop\ fractional abundances in GMCs 4 and 5 (see Figure \ref{fig:mom0}f). Cosmic rays could also contribute to ice desorption. We argued in Section \ref{sec:chem} that cosmic-ray-induced desorption is likely negligible, but its effect could possibly be enhanced in NGC~253. \citet{1993MNRAS.261...83H} estimated that the maximum dust temperature due to the cosmic ray heating of the dust is 70\,K, but this temperature can be different for the case of NGC~253, where some dust is already warm. With the higher maximum dust temperature, the evaporation rate of ice species can be enhanced \citep[e.g., ][]{2020A&A...633A..97K}. This dependence of desorption due to cosmic-ray heating on the dust temperature should be further explored with theoretical studies. Variation of the \cotw\ ice abundance can be also caused by the initial ice composition. For example, the ice may be rich in atomic or molecular hydrogen. If the ice is abundant in atomic H, frequent hydrogenation reactions can occur, and species such as water, CH$_4$, NH$_3$, and \methanol\ may be abundant. On the other hand, if atomic H is deficient, \cotw\ formation may be a more dominant route of destroying CO ice than \methanol\ formation. Although it is difficult to assess how much the ice compositions differ between Galactic star-forming regions and NGC 253 CMZ, we note that this is another factor that could affect the overall chemistry. \subsection{Ice and gas-phase chemistry in NGC~253} We find that the gas-phase fractional abundance of \cotw\ can be $\sim (1-20)\times 10^{-7}$ in GMCs 1, 2, 8, and 9 (Section \ref{sec:co2frac}), and the ice fractional abundance is $\sim 2\times 10^{-6}$ around GMC 1 (rectangular region 5 in Figure \ref{fig:co2ice}). This means that there is a process sublimating a large fraction of \cotw\ ice into the gas phase in these GMCs. One possible \cotw\ sublimation mechanism is a hydrodynamical shock. Chemical models have shown that shocks can sputter off the ice because some gas particles have enough kinetic energy to desorb ice \citep{2008A&A...482..549J,2011ApJ...740L...3V,2012MNRAS.421.2786F}. We also note that shock sublimation occurs even when the dust temperature is low, because the energy is provided by gas. When the shock velocity is high $>20\,$km\,s$^{-1}$, ice sputtering is efficient enough to desorb a large fraction of ice \citep{2015A&A...584A.102H}. Ubiquitous methanol emission is likely attributed to shock sublimation in the Milky Way Galactic Center \citep{2009ApJ...692...47M}, and this type of ice sublimation likely occurs also in NGC 253 with frequent shocks \citep{2015ApJ...801...63M}. The locations with enhanced \hocop\ abundances are considered as intersections of different orbits. As shown in Figure \ref{fig:orbit}, bar orbits ($x_1$ orbits) and inner orbits ($x_2$ orbits) intersect at the northeast and southwest parts of the CMZ, near GMCs 1, 8, 9, and 10. Shocks at GMCs 1, 2, 7, 8, 9, and 10 have been suggested by the detection of Class~I methanol masers \citep{2022arXiv220503281H}, and some other molecular tracers of shocks (Huang et al. in preparation). Because these regions are located at intersections of different orbits \citep[bar and nuclear ring; see ][ for the dynamical modeling]{2000PASJ...52..785S,2001ApJ...549..896D,2022arXiv220604700L}, shocks due to cloud collisions are not surprising. If the shock scenario is correct, ice sputtering of other species in addition to \cotw\ should be taking place. Methanol enhancement shown in Section \ref{sec:integint} is one example supporting this scenario. Water is another species abundant in ice, and ice sputtering should increase its gas-phase abundance. From the \water/\cotw\ ratio of $\sim 7$ in ice \citep{2015ApJ...807...29Y} and our estimated gas-phase fractional abundance of \cotw\ of $(1-20)\times 10^{-7}$, the gas-phase water abundances in shocked regions must be $\sim (0.7-14)\times 10^{-6}$. This estimate assumes that the same fraction of \cotw\ and \water\ ice is sublimated, and does not consider the higher desorption energy of water compared with \cotw. It also assumes that gas-phase reactions subsequent to sputtering do not change the \cotw\/\water\ ratio. \citet{2017ApJ...846....5L} derived fractional abundances of gas-phase water to be $\sim 10^{-7}$ from Herschel HIFI/PACS/SPIRE data using multiple transitions of water in analysis with all data convolved to $40\arcsec$ at the center of NGC 253. This value may be locally higher in shocked regions. However, it is impossible to confirm it without spatially resolving shocked regions and nuclear starburst regions with a higher-angular-resolution ($<10\arcsec$) observation. Another possible \cotw\ sublimation mechanism is thermal desorption. We do not know the distribution of the dust temperature due to the lack of high angular resolution data at the wavelengths of the peak black body radiation intensity ($\sim 10-100\,\mu$m for 30-300\,K). Yet, we do know that the most active star formation takes place around GMCs 3-5 from radio recombination line or 3mm continuum data \citep{2015MNRAS.450L..80B, 2021ApJ...923...24H}. If the hot/warm dust components are concentrated in GMCs 3-5, it is unlikely that the thermal desorption already takes place at GMCs 1 and 9. Therefore, thermal desorption unlikely contributes to the sublimation of \cotw\ in GMCs 1, 2, 8, and 9. However, it is quite possible that thermal desorption occurs around GMC 3-5 due to the high dust temperature. Another mechanism which can sublimate \cotw\ is cosmic-ray-induced desorption as described in Section \ref{sec:ice}. We argued that this desorption mechanism is unlikely unless the dust is warm so that the maximum dust temperature achieved from cosmic-ray heating of dust becomes significantly larger than 70\,K. Without high star formation rates in GMCs 1 and 9, it is unlikely that the dust is already warm. For the reasons above, we conclude that a shock is the most likely scenario driving the \cotw\ evaporation. Yet, to better constrain the ice fractional abundances, high angular resolution observations at infrared wavelengths, e.g., from JWST, are crucial. \section{Summary}\label{sec:summary} In this paper, we analyzed the abundances of \hocop, the protonated form of \cotw, in the central molecular zone of the starburst galaxy NGC~253, and discussed its relationship with the gas-and ice-phase \cotw. Below is the summary of our findings. \begin{itemize} \item The distribution of \hocop\ shows clear enhancements at locations of $x_1$ and $x_2$ orbital intersections where shocks are expected. This distribution is similar to that of methanol but is different from that of \httcop. There are two formation routes of \hocop; one is the ion-neutral reaction HCO$^+$ + OH and the other is the protonation of \cotw. If the former route is dominant, the \hocop\ distribution should be similar to that of HCO$^+$, while the latter route should cause similarity with the \methanol\ distribution. Therefore, \hocop\ is likely produced through the protonation of \cotw. \item We derive \hocop\ column densities across the CMZ using CASSIS, from which we also obtain its fractional abundances using the total H$_2$ column densities estimated from the dust emission. We find \hocop\ fractional abundances as high as $\sim 2\times 10^{-9}$, which is similar to those observed in the Galactic center, but orders of magnitude higher than those reported in Galactic spiral-arm molecular clouds. \item From the results of chemical modeling and values of cosmic-ray ionization rates derived from previous ALCHEMI works, we estimate that the gaseous \hocop/\cotw\ ratio is likely $10^{-3} - 10^{-2}$. This ratio suggests that the gas-phase \cotw\ fractional abundances are \fnum{(1-20)}{-7} at peaks of \hocop\ emission. \item We also estimate fractional abundances of \cotw\ ice from their column densities in the literature. The ice fractional abundance at the \hocop\ peak is similar to the value in the Galactic interstellar medium ($10^{-6}-10^{-5}$), but is lower ($\sim (1-3)\times 10^{-7}$) near the NGC~253 galactic center. \item The increased gaseous and ice fractional abundances of \cotw\ at the outer CMZ of NGC~253 imply that a large fraction of ice is sublimated. Because of the association of these locations with evidence of shocks, we propose that this efficient sublimation is attributed to shock-induced sputtering. \end{itemize} High spatial resolution observations of molecular emission in external galaxies, such as those performed by the ALCHEMI survey toward the central regions of the starburst galaxy NGC~253 with ALMA, have greatly improved our understanding of gas-phase abundances and will continue to do so. Now, complementary observations of ice at high angular resolutions with the JWST are required to obtain a complete picture of the chemical processes in starburst galaxies. \begin{acknowledgements} \sloppypar{We thank the anonymous referee for constructive comments. N.H. thanks Hideko Nomura for the helpful discussion on \cotw\ transitions in the infrared wavelength, and Kotomi Taniguchi for the initial help using CASSIS. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2017.1.00161.L, ADS/JAO.ALMA\#2018.1.00162.S, ADS/JAO.ALMA\#2018.1.01321.S.} ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. Data analysis was in part carried out on the Multi-wavelength Data Analysis System operated by the Astronomy Data Center (ADC), National Astronomical Observatory of Japan. N.H. acknowledges support from JSPS KAKENHI Grant Number JP21K03634. V.M.R. has received support from the Comunidad de Madrid through the Atracci\'on de Talento Investigador Modalidad 1 (Doctores con experiencia) Grant (COOL:Cosmic Origins of Life; 2019-T1/TIC-5379), and the Ayuda RYC2020-029387-I funded by MCIN/AEI /10.13039/501100011033. L.C. has received partial support from the Spanish State Research Agency (AEI; project number PID2019-105552RB-C41). P.H. is a member of and received financial support for this research from the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne. K.S. is supported by the grant MOST 109-2112-M-001-020 from the Ministry of Science and Technology, Taiwan. This research has made use of the NASA/IPAC Extragalactic Database (NED), which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology. \end{acknowledgements} \facilities{ALMA} \software{Astropy \citep{astropy:2013, astropy:2018}, CASA \citep{casa}, CASSIS \citep{2015sf2a.conf..313V}, Nautilus \citep{2016MNRAS.459.3756R}} \appendix \section{CASSIS fitting parameters}\label{sec:app_cassis} CASSIS constrains parameters such as column density, line velocity, line width (full width at half maximum; FWHM), excitation temperatures, and source sizes by fitting the spectra. We used the MCMC algorithm provided by CASSIS to derive these parameters. We use spectra from transitions shown in the upper half of Table \ref{tab:spec}. When using the MCMC method, users provide acceptable ranges of these parameters as well as the initial guesses. We used the column density range of [$10^{12}$,$10^{16}$] cm$^{-2}$ with the initial guess of \enum{14} cm$^{-2}$, the excitation temperature range [5,50] K with the initial guess of 6\,K, the velocity range [$v_{\rm CO} -30$,$v_{\rm CO}+30.$]\,\kms\ where $v_{\rm CO}$ is the velocity obtained from the moment 1 image of CO(1-0), which is also used for the initial guess. The range of FWHM we used is [$\sigma_V$ -20.,90.]\,\kms\ with the initial guess of $\sigma_V$ where $\sigma_V$ is the velocity dispersion of the CO(1-0) image\footnote{The relationship between the FWHM and the standard deviation of a Gaussian distribution is usually described as $FWHM \sim 2.355 \sigma_V$. In our case, the line width of \hocop\ is much smaller than that of CO(1-0), and the initial guess of $\sigma_V$ for the FWHM of \hocop\ is still a reasonable one.}. We note that the line width of \hocop\ is significantly different from that of CO(1-0), and this range of FWHM is simply determined by running CASSIS multiple times and checking the range to produce a reasonable fit. Uncertainties from the CASSIS fit are reasonably small for most cases. Figure \ref{fig:error_cassis}(left) shows that the errors of the column densities are around 10\,\% for most cases, and $\sim 20-30\,$\% for a small fraction of pixels with low signal-to-noise ratios. Uncertainties in excitation temperatures are within 0.5\,K for most pixels, with a maximum of 3\,K. There are also other sources of uncertainties, in addition to the spectral fitting. For example, \hocop\ column densities derived with spectroscopic constants from Jet Propulsion Laboratory (JPL; https://spec.jpl.nasa.gov/) would yield up to a factor of 2 larger values compared with ones from CDMS. Although this is a large factor, a use of different spectroscopic constants changes the results uniformly within the field of view. There are also observational uncertainties of up to 15\% \citep{2021A&A...656A..46M}. These uncertainties will not change our main conclusions. The \hocop\ transitions are optically thin even where the column densities are high. CASSIS does not provide the optical depths, but we also ran MADCUBA \citep{2019A&A...631A.159M}, a similar spectral fitting software to obtain optical depths. The optical depths are $<0.1$ for positions that we checked, which have high intensities of \hocop. The modeled intensities with CASSIS for pixels that are closest to each GMC position are shown in Table \ref{tab:model_int} if \hocop\ transition for a GMC is detected. \begin{deluxetable*}{ccccccccc} \tablecolumns{9} \tablewidth{0pc} \tablecaption{Modeled integrated intensities\label{tab:model_int}} \tablehead{\colhead{GMC} &\colhead{$x_{\rm offset}$} &\colhead{$y_{\rm offset}$} &\multicolumn{6}{c}{$\int I dv$}\\ \colhead{} &\colhead{($\arcsec$)} &\colhead{($\arcsec$)} &\multicolumn{6}{c}{(K\,km\,\ps)}} \startdata &&&($4_{0,4}-3_{0,3}$)&($5_{0,5}-4_{0,4}$)&($6_{0,6}-5_{0,5}$)&($7_{0,7}-6_{0,6}$)&($8_{0,8}-7_{0,7}$) &($12_{0,12}-11_{0,11}$)\\ \cline{4-9}\\ 1 &-17.3, &-10.4 &12.14 &9.00 &5.25 &2.47 &0.94 &0.00 \\ 2 &-13.6, &-2.8 &6.40 &4.93 &2.92 &1.39 &0.55 &0.00 \\ 3 &-6.5,& -4.2 &6.39 &7.99 &8.96 &9.06 &8.36 &3.02 \\ 6 &0.4, &2.1 &4.41 &6.05 &7.42 &8.37 &8.81 &6.28 \\ 7 &4.8,& 4.2 &11.12 &12.40 &11.84 &9.95 &7.48 &0.88 \\ 8 &10.0, &6.2 &7.30 &6.05 &4.03 &2.22 &1.01 &0.01 \\ 9 &12.0,& 5.6 &13.08 &10.73 &7.11 &3.85 &1.73 &0.01 \\ \enddata \tablecomments{Integrated intensities of \hocop\ transitions produced by CASSIS fitting for hexagonal pixels closest to GMCs. Results are shown only for GMCs with \hocop\ detection. \label{tab:gmc_intens}} \end{deluxetable*} \section{Ice data from AKARI}\label{sec:app_ice} Here we summarize analyses by \citet{2015ApJ...807...29Y}, and present spectra for regions used in our analysis. The observed wavelength range is about $2.5-5.0\,\mu$m. Within this range, CO$_2$ ice, H$_2$O ice, Br$\alpha$, and PAH $3.3\,\mu$m features were detected in addition to the continuum. \cotw\ ice features at $4.27\,\mu$m are fit using the data range of $4.1-4.4\,\mu$m as there is only this narrow ice feature in this wavelength range. Because the ice composition changes the spectral shape, multiple ice compositions were tested to best fit the spectra. Consequently, the ice composition of \water:\methanol:\cotw=$9:1:2$ was used in the final analysis. Figure \ref{fig:ice_spec} shows spectra used to derive \cotw\ ice column densities shown in Figure \ref{fig:co2ice}. Note that our Regions 1-5 correspond to ID 47-51 in Table 3 of \citet{2015ApJ...807...29Y}. \section{GMC positions}\label{sec:app_gmc} As already noted in \citet{2022arXiv220503281H} and Behrens et al. (submitted to ApJ), the GMC nomenclature was adopted from \citet{leroy_alma_2015}, but with modified positions. How these modifications are made is explained in Behrens et al. (submitted to ApJ). These positions are shown in Table \ref{tab:gmc}. \begin{deluxetable*}{ccc} \tablecolumns{3} \tablewidth{0pc} \tablecaption{GMC positions\label{tab:gmc}} \tablehead{\colhead{ID} &\colhead{RA (ICRS)} &\colhead{DEC (ICRS)}\\ \colhead{} &\colhead{$00^h47^m-^s$} &\colhead{$-25^\circ17'-''$}} \startdata 1 & 32.02 & 28.2 \\ 2 & 32.28 & 20.2 \\ 3 & 32.81 & 21.6 \\ 4 & 32.97 & 20.0 \\ 5 & 33.21 & 17.4 \\ 6 & 33.33 & 15.8 \\ 7 & 33.64 & 13.3 \\ 8 & 34.02 & 11.4 \\ 9 & 34.17 & 12.3 \\ 10 & 34.24 & 7.8 \\ \enddata \tablecomments{Modified coordinates of GMC positions in \citet{leroy_alma_2015} provided by A. K. Leroy (private communication).} \end{deluxetable*} \section{The archival CO(2-1) image} The large-scale CO(2-1) image of NGC 253 shown in Figure \ref{fig:orbit} was taken from the ALMA archive (project code \#2018.1.01321.S). These data use the configuration consisting on the 7-m array complemented by the total power antenna. Pipeline-reduced image cubes (QA2 products) for the 7-m array and total power single-dish data were combined with the CASA command {\tt feather}. This image was shown to indicate the rough positions of $x_1$ and $x_2$ orbits only. We expect that the PI team will present the data with better imaging quality and scientific analysis. \input{ALCHEMI_HOCOp.bbl}

Title: What Are Those Tiny Things? A First Study of Compact Star Clusters in the SMACS0723 Field with JWST

Abstract: We use the unprecedented resolution and depth of the JWST NIRCam Early Release Observations (ERO) at 1-5$\mu m$ to study the stellar mass, age, and metallicity of compact star clusters in the neighborhood of the host galaxies in the SMACS J0723.3-7327 galaxy cluster field at z = 0.39. The measured colors of these star clusters show a similar distribution as quiescent galaxies at the same redshift, but are >3 magnitudes fainter than the current depths of wide-field galaxy survey. The star clusters are unresolved in the NIRCam/F150W data suggesting sizes smaller than 50pc. This is significantly smaller than star forming clumps or dwarf galaxies in local galaxies. From fitting their photometry with simple stellar population (SSP) models, we find stellar metallicities consistent with 0.2-0.3Z$_{\odot}$ and ages of $5.0^{+0.5}_{-1.1}$ Gyrs. We rule out ages of <3 Gyrs at a 2$\sigma$ confidence, and metallicities <0.2 Z$_{\odot}$ and solar/super-solar at 4$\sigma$ significance. Assuming the mass-to-light ratio (1.22 M$_{\odot}$/L$_{\odot}$) obtained from the best-fit SSP, we estimate stellar masses of $4.1^{+3.5}_{-1.8}\times10^6\,{\rm M_{\odot}}$. These are between average masses of local globular clusters and dwarf galaxies. Our analysis suggests middle-aged globulars with relatively recent formation times at z=1-2, which could be subsequently stripped away from their host galaxies due to interactions in the cluster environment. However, we cannot rule out these objects being compact cores of stripped dwarf galaxies.

https://export.arxiv.org/pdf/2208.05502

command. \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \newcommand{\A}{\text{\normalfont\AA}} \newcommand{\red}[1]{{\bf\textcolor{red}{[#1]}}} \newcommand{\blue}[1]{{\bf\textcolor{blue}{[#1]}}} \newcommand\textlcsc[1]{\textsc{\MakeLowercase{#1}}} \newcommand{\comment}[1]{\textcolor{magenta}{ \textbf{[#1]}}} \def \hbeta{H$\beta$} \def \ebmv{E(B-V)} \def \ebmvs{ E_{s}{\rm (B-V)} } \def \ebmvn{ E_{n}{\rm (B-V)} } \def \halpha{H$\alpha$} \def \Msol{{M}_{\odot}} \def \Mstell{{M}_{*}} \def \Lsol{{\rm L}_{\odot}} \def \Zsol{{\rm Z}_{\odot}} \def \Msolyr{{\rm M_{\odot}\,yr^{-1}}} \def \logm{\log(M/\Msol)} \def \lya{Ly$\alpha$} \def \ewha{{\rm EW}({\rm H}\alpha)} \def \ewlya{{\rm EW}({\rm Ly}\alpha)} \def \h2{{\rm H_{2}}} \def \oabund{12+\log({\rm O/H})} \def \kms{{\rm km\,s}^{-1}} \def \myr{{\rm Myr}} \def \dex{{\rm dex}} \def \gyr{{\rm Gyr}} \def \hbeta{H$\beta$} \def \halpha{H$\alpha$} \def \siii{\ion{Si}{2}} \def \siiii{\ion{Si}{3}} \def \siiv{\ion{Si}{4}} \def \cii{\ion{C}{2}} \def \Cii{[\ion{C}{2}]} \def \niiha{[N{\scriptsize ~II}]/H$\alpha$} \def \lniiha{$\log($[N{\scriptsize ~II}]/H$\alpha$)} \def \niiniiha{[N{\scriptsize ~II}]/([N{\scriptsize ~II}]+H$\alpha$)} \def \niiniihatot{[N{\scriptsize ~II}]/([N{\scriptsize ~II}]+H$\alpha$)$_{\rm tot}$} \def \niipha{([N{\scriptsize ~II}]+H$\alpha$)} \def \niiphatot{([N{\scriptsize ~II}]+H$\alpha$)$_{\rm tot}$} \def \oiiihb{[O{\scriptsize ~III}]/H$\beta$} \def \oiiiha{[O{\scriptsize ~III}]/H$\alpha$} \def \loiiihb{$\log($[O{\scriptsize ~III}]/H$\beta$)} \def \civ{\ion{C}{4}} \def \oi{[\ion{O}{1}]} \def \oii{[\ion{O}{2}]} \def \oiii{[\ion{O}{3}]} \def \heii{\ion{He}{2}} \def \feii{\ion{Fe}{2}} \def \aliii{\ion{Al}{3}} \def \nii{[\ion{N}{2}]} \def \hii{\ion{H}{2}} \def \aco{\alpha_{{\rm CO}}} \def \Ga{\textit{GALEX0959+0151}} \def \Gb{\textit{GALEX1000+0157}} \def \Gc{\textit{GALEX1000+0201}} \def \Gaa{GALEX0959+0151} \def \Gbb{GALEX1000+0157} \def \Gcc{GALEX1000+0201} \def \CO{CO$(1-0)$} \def \fesc{f_{{\rm esc}}} \def \mgas{M_{{\rm gas}}} \def \fgas{f_{{\rm gas}}} \def \LIR{L_{{\rm IR}}} \def \LFIR{L_{{\rm FIR}}} \def \LUV{L_{{\rm UV}}} \def \LHA{L_{{\rm H\alpha}}} \def \DLHA{\Delta\log(L_{{\rm H\alpha}})} \def \LCO{L_{{\rm CO}}} \def \LCII{L_{{\rm CII}}} \def \logLha{\log(L_{\rm H\alpha})} \def \logL{\log(L)} \def \IRXB{IRX$-\beta$} \def \dn4000{D_{{\rm n}}(4000) } \def \flya{$f_{\rm Ly\alpha}$} \def \iracA{[$3.6\,{\rm \mu m}$]$-$[$4.5\,{\rm \mu m}$]} \def \iracB{[$4.5\,{\rm \mu m}$]$-$[$5.8\,{\rm \mu m}$]} \def \fion{$f^{\rm ion}_{\rm esc}$} \def \xiion{$\xi_{\rm ion}$} \def \wone{\textit{W1}} \def \WISE{\textit{WISE}} \def \NEOWISE{\textit{NEOWISE}} \def \wtwo{\textit{W2}} \newcommand{\ergPerSecondPerCM}{{\rm erg}\,{\rm s}^{-1}{\rm cm}^{-2}} \newcommand{\ergPerSecond}{{\rm erg}\,{\rm s}^{-1}} \def \thegalaxy{\textit{DC\_881725}} \newcommand{\Ranga}[1]{\textbf{\textcolor[rgb]{0.00,0.00,1.00}{RangaComments: [#1]}}} \begin{document} \title{What Are Those Tiny Things?\\ A First Study of Compact Star Clusters in the SMACS0723 Field with JWST} \correspondingauthor{Andreas L. Faisst} \email{afaisst@ipac.caltech.edu} \author[0000-0002-9382-9832]{Andreas L. Faisst} \affiliation{Caltech/IPAC, MS314-6, 1200 E. California Blvd. Pasadena, CA 91125, USA} \author[0000-0001-7583-0621]{Ranga Ram Chary} \affiliation{Caltech/IPAC, MS314-6, 1200 E. California Blvd. Pasadena, CA 91125, USA} \author[0000-0003-2680-005X]{Gabriel Brammer} \affil{Cosmic Dawn Center (DAWN), Copenhagen, Denmark} \affil{Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark} \author[0000-0003-3631-7176]{Sune Toft} \affil{Cosmic Dawn Center (DAWN), Copenhagen, Denmark} \affil{Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark} \keywords{Globular star clusters (656), Galaxy clusters (584), Stellar populations (1622), Metallicity (1031), Galaxy formation (595)} \section{Introduction} \label{sec:intro} The Early Release Observations \citep[ERO; ][]{PONTOPPIDAN22} have already showcased the capabilities of the new {\it James Webb Space Telescope} \citep[JWST;][]{RIGBY22} in terms of its sensitivity and resolution at near-infrared wavelengths. Here, we use the unprecedented sensitivity of the JWST ERO data to study for the first time, the faintest and most compact structures around the galaxies associated with the SMACS J0723.3-7327 galaxy cluster field (hereafter SMACS0723) at $z=0.39$. Although these objects appear to be consistent with compact star clusters (or globular clusters), they could just as well be the cores of stripped dwarf galaxies. Globulars around galaxies are abundant and have been observed in our Milky Way as well as many local galaxies \citep[e.g.,][]{HARRIS79,LARSEN01,BRODIE14,HARRIS14,MASSARI19}. The number of globulars hosted by galaxies range from a few to tens for dwarf galaxies such as the Magellanic clouds \citep[e.g.,][]{GEORGIEV10}, to a few hundreds for intermediate-mass galaxies such as our Milky Way or M31 \citep[e.g.,][]{HARRIS91,MASSARI19}, to thousands and more for massive brightest cluster galaxies (BCGs) in galaxy clusters \citep[e.g.,][]{PENG08,HARRIS17,ALAMOMARTINEZ13}, which early on were also associated with the intercluster medium \citep[ICM; e.g., ][]{JORDAN03,BASSINO03}. There are different mechanisms suggested for the formation of globulars. Old globulars, among the oldest stellar structures in our universe \citep[e.g.,][]{RICOTTI16}, could be formed through rapid processes \citep[such as the collapse of gas clouds, e.g.,][]{PEEBLES68,FORBES18} in the early stages of the universe (such as the Epoch of Reionization at $z\sim6$). On the other hand, the peak formation of globulars (i.e., their ages) may coincide with the peak of cosmic star formation density in galaxies around $z\sim2$, suggesting they are formed via continuous star formation in gas-rich environments \citep[][]{HARRIS91,BRODIE06,FORBES18}. Indeed, some observations have suggested such a scenario by finding young globulars in dense, gas-rich merging systems \citep[][]{SCHWEIZER98,DEGRIJS01,TRUJILLOGOMEZ21} and current theoretical models mostly assume such a co-evolutionary scenario \citep[e.g.,][]{PFEFFER18,CHOKSI19,LI19,ELBADRY19}. There are also observational hints of cluster formation from cooled gas from the ICM to the galaxy cluster core \citep[e.g.,][]{HOLTZMAN92}. For an in-depth discussion of the different globular cluster formation mechanisms, we refer the readers to \citet[][]{KRUIJSSEN15}. Age determination based on main-sequence turnoff from colors of single stars and cooling curves from long-lived white dwarfs suggest a dominant population of very old globulars with a range of metallicity as well as a ``young'' branch for which metallicity is anti-correlated to age \citep[][]{HARRIS79,KRAUSS03,MARINFRANCH09,FORBES10,VANDENBERG13,LEAMAN13,FORBES15,FORBES18,KRUIJSSEN19,MASSARI19}. The latter may be associated with disruption events of dwarf galaxies and accretion ({\it in-situ} vs. {\it ex-situ} formation, see also \citealt[][]{KRUIJSSEN19}). Among the young globulars, the metal-poor population is slightly older than the metal-rich population ($12.5$ vs. $11.5\,{\rm Gyrs}$; \citealt[][]{FORBES18}), however, these differences are of low significance due to uncertainties in the age measurements. Similarly, the absolute ages of the oldest and metal poor globulars have a significant range; Alternative age measurements by \citet[][]{FORBES15} (assuming that the metallicity of globulars follows the galaxy mass$-$metallicity relation) suggest that even the oldest, metal-poor globulars form at $z<5.9$, {\it i.e.} after the reionization of the early universe. The interacting galaxies in the SMACS0723 galaxy cluster environment may be the place to look for evidence for the formation of compact star clusters at $z~=~0.39$ $-$ approximately $4.3\,{\rm Gyrs}$ before the observation of Milky Way or local globulars. The galaxies are clearly interacting as shown by the evident diffuse intracluster light \citep[e.g.,][]{PASCALE22}, suggestive of stars being stripped from the galaxies. The new JWST/NIRCam observations at $1-5\,{\rm \mu m}$ show a considerable amount of relatively blue and faint ($>28\,{\rm AB\,mag}$) point sources around the cluster members \citep[see also ][]{LEE22}. As shown in Section~\ref{sec:stars}, this number of faint point sources cannot be explained by Milky Way stars, nor globulars in the Milky Way, which, at sizes of $\sim10\,{\rm pc}$ would be resolved and significantly brighter. These objects are good candidates for globular clusters and their study may inform us about the formation of globulars at higher redshifts. The large angular coverage of the two NIRCam fields of view (FoV), being offset from each other by $\sim1\,{\rm Mpc}$ at the cluster's redshift, allow also a spatial study of globulars around the brightest cluster galaxy (BCG). If recently formed star clusters are stripped from their host galaxies due to the interactions in the galaxy cluster, we would expect these to be relatively young and metal abundant compared to those originating from an early-formation scenario. Furthermore, the size and masses of these compact star clusters may differentiate them from stripped cores of dwarf galaxies \citep[e.g.,][]{IDETA04}. This paper is organized as follows: In Section~\ref{sec:dataandmeasurements}, we detail the data used in this work, the selection of the compact star clusters, and various measurements. In Section~\ref{sec:results}, we present the results from our analysis. We conclude in Section~\ref{sec:end}. Throughout this work, we assume a $\Lambda$CDM cosmology with $H_0 = 70\,{\rm km\,s^{-1}\,Mpc^{-1}}$, $\Omega_\Lambda = 0.7$, and $\Omega_{\rm m} = 0.3$. All magnitudes are given in the AB system \citep{OKE74} and stellar masses and star formation rates (SFRs) are normalized to a \citet[][]{CHABRIER03} initial mass function (IMF) unless noted differently. \section{Data and Measurements}\label{sec:dataandmeasurements} \subsection{Data} \label{sec:data} In this work, we focus on compact stellar clusters in the galaxy cluster environment SMACS J0723.3-7327 at $z = 0.39$ (07$^{\rm h}$23$^{\rm m}$13.3$^{\rm s}$, -73$^{\rm d}$27$^{\rm m}$25$^{\rm s}$). SMACS0723 was initially discovered as part of the southern extension to the ROSAT All-Sky Survey \citep[][]{VOGES99} based {\it Massive Cluster Survey} \citep[MACS; ][]{EBELING01} and also detected by Planck \citep[][]{PLANCK11} through the Sunyaev-Zel'dovich effect. It was then subsequently studied by the {\it Reionization Lensing Cluster Survey} using the Hubble space telescope \citep[RELICS,][]{COE19}. A new study using JWST by \citet[][]{MAHLER22} places the total mass of the cluster at $M_{\rm <400\,kpc} = 3\times10^{14}\,{\rm M_{\odot}}$ (compared to the measurement by Planck of $M_{\rm 500} = 8\times10^{14}\,{\rm M_{\odot}}$). The BCG has a stellar mass of $\sim2 \times 10^{11}\,{\rm M_\odot}$ (close to the ``knee'' of the galaxy stellar mass function, e.g., \citealt[][]{DAVIDZON17}) and resembles a relatively featureless elliptical galaxy with little to no ongoing star formation. Here, we use new JWST observations of the cluster \citep[ERO program ID 2736;][]{PONTOPPIDAN22} taken with the Near Infrared Camera \citep[NIRCam;][]{RIEKE05} broad-band filters F090W ($0.90\rm \mu m$), F150W ($1.50\rm \mu m$), F200W ($1.99\rm \mu m$), F277W ($2.76\rm \mu m$), F356W ($3.57\rm \mu m$), and F444W ($4.41\rm \mu m$). These filters correspond to wavelengths in the cluster's rest-frame of $0.65\rm \mu m$, $1.08\rm \mu m$, $1.43\rm \mu m$, $1.98\rm \mu m$, $2.57\rm \mu m$, and $3.17\rm \mu m$, respectively. NIRCam consists of two modules, each observing a $2.2\arcmin\times2.2\arcmin$ FoV, one centered on the cluster's BCG and the other one centered $\sim3\arcmin$ ($\sim1\,{\rm Mpc}$ at the cluster's redshift) to the south-west. The data were downloaded from the \textit{Mikulski Archive of Space Telescopes} (MAST)\footnote{\url{https://mast.stsci.edu/}} and subsequently reduced with the JWST data reduction pipeline. The reduced image data were then combined using the grism redshift and line analysis software for space-based spectroscopy \citep[\textsc{Grizli};][]{BRAMMER21} package to a final pixel scale of $0.02\arcsec/{\rm px}$ for F090W, F150W, and F200W and $0.04\arcsec/{\rm px}$ for F277W, F356W, and F444W, respectively \citep[][]{BRAMMER22_SMACSreduct}. Note that the recent \texttt{jwst\_0942.pmap} photometric calibration reference file was used with modifications as of September 5 2022 as described in the \textsc{Grizli} Github repository\footnote{\url{https://github.com/gbrammer/grizli/pull/107}}. The details of the data reduction will be presented in detail in an upcoming paper (Brammer et al., in prep). The field was also imaged by JWST/MIRI at mid-infrared wavelengths. However, the MIRI data is too shallow for this study and is therefore not being used. Figure~\ref{fig:overview} shows an overview of the data used. \subsection{Source Selection} \label{sec:selection} In this work, we study the faintest, compact stellar clusters in the SMACS0723 field, such as potential globular clusters around the cluster host galaxies. These sources are selected manually mainly on the F200W image, however, we require them also to be detected in at least two other bands in order to reject potential spurious sources. In addition, the sources are selected to be unresolved point sources as well as faint (to exclude Milky Way stars, see Section~\ref{sec:stars}). For example, a good upper limit in size of globulars is $\sim10\,{\rm pc}$ \citep[e.g.,][]{LARSEN01}, which, at $z=0.39$, corresponds to an angular size of $\sim1.4$ milliarcseconds (note that there are some larger globulars, for example NGC 2419 in the Milky Way and Lindsay 1 in the Small Magellanic Cloud). This is significantly smaller than the PSF size (between $20$ and $120$ milliarcseconds at $1-5\,{\rm \mu m}$), thus motivating the point-source assumption. Milky Way stars are the most likely contaminants in this selection, as discussed in Section~\ref{sec:stars}. We tried an automated selection of globulars, however, we found that this approach results in a severely incomplete selection due to blending with the BCG and other cluster galaxies, as well as the variation of the intracluster and scattered light across the field. In total, we identified $178$ compact star clusters in the full FoV of NIRCam around the cluster host galaxies (see Figure~\ref{fig:overview}), which we characterize and study in the following. The coordinates and NIRCam fluxes of the extracted star clusters are listed in Table~\ref{tab:photo}. We note that our selection is not complete, but serves as a starting point to understand the properties and origin of these faint sources. We refer to the work by \citet[][]{LEE22} for a detailed study of the distribution and number density of these star clusters in the ICM of SMACS0723. If these sources are globular clusters, we would expect to only see the tip of the iceberg ({\it i.e.} the brightest ones) even at the unprecedented depths of these observations, assuming the commonly used Gaussian globular cluster luminosity function \citep[e.g.,][]{ALAMOMARTINEZ13}. \subsection{Measurement of the PSF}\label{sec:psf} To measure the photometry of unresolved point sources, a robust determination of the point spread function (PSF) is crucial. Here, we measure the PSF in all $6$ NIRCam filters by stacking stars on the full FoV of the available observations. For the selection of stars, we use an identical approach as in \citet[][]{FAISST22}. Summarizing, the stars are selected based on the $R_{e}$ vs. magnitude diagram (produced by the \texttt{flux\_radius} and \texttt{flux\_auto} quantities derived by \textsc{SExtractor} \citep{Bertin1996} run the NIRCam images). We note that, based on the upturn in size measurements on that diagram, point sources at $\sim25\,{\rm mag}$ already start to enter the non-linear regime towards saturation at the depth of these NIRCam observations. We therefore require stars between $25\,{\rm mag}$ and $27\,{\rm mag}$, to avoid the aforementioned saturation effect as well as to exclude potential faint, spurious sources. In addition, a half-light radius of less than $3\,{\rm pixels}$ is required to select unresolved sources. For each of the $58$ stars, we create a $2\arcsec \times 2\arcsec$ cutout and subsequently center the stars before stacking them to obtain the final PSF. The final PSF for each filter (normalized and of the same pixel scale as the images, see Section~\ref{sec:data}) are available for download\footnote{\url{https://github.com/afaisst/JWST_SMACS_PSFs}}. The relatively low number density of stars does not allow us to quantify variations of the PSF. However, the recent study by \citet[][]{NARDIELLO22} measured PSF variations across the NIRCam FoV based on stars in the globular cluster M 92, finding variations in the FWHM of up to $15-20\%$. Such variations have a negligible effect on the measured photometry in our case, but we will include this effect later when we constrain the sizes of our compact star clusters. \vspace{-0.1cm} \subsection{Measurement of Photometry}\label{sec:photomeasure} The photometry of the selected globulars is measured using the software \textsc{Tractor}\footnote{\url{http://thetractor.org/}} \citep{LANG16b,LANG16a,WEAVER22b}, which performs a prior-based forced photometry including the PSF of the observations. We first compared the 1-dimensional projected light profiles from the stack of all compact star clusters to the one of the PSF (left panel of Figure~\ref{fig:psfs}). This test shows that the star clusters are indeed consistent with unresolved point sources, hence we use the point-source fitting option of \textsc{Tractor}. In the following, we fit the photometry of individual sources to study the variations in the stellar population across our sample. We also derive the photometry from the stacks of the sources in each of the $6$ NIRCam filters to study the average properties. In both cases, we create cutouts of $0.8\arcsec$ size of all the sources. The stacks were created by first performing a local background subtraction (measured in a sky annulus with inner radius of $5\,\rm px$ and outer radius of $10\,\rm px$) on the individual cutouts and then combining them via median stacking. The following steps are then identical for the fitting of individual sources and the stacks. \textsc{Tractor} is run on the $0.8\arcsec$ cutouts by applying a point source model, using the F200W position of the sources as prior (determined from the 0-order moment of light on the source). During the fitting, the position is free to vary within $\pm2\,{\rm pixels}$ of the prior position. This helps to mitigate potential misalignments between the different image products. In addition, we let \textsc{Tractor} fit and subtract the (slowly varying on $0.8\arcsec$ scales) local background at the position of the source to remove the contribution of the halo from nearby cluster galaxies and the intracluster light. Note that for that reason, simple aperture photometry will result in overestimated flux measurements. The inverse variance image (used for error estimation) is created as $1/\sigma^2$ with $\sigma$ derived from $\sigma-$clipping statistics on a large region of the image not contaminated by the intracluster light and other galaxies. The variance output by \textsc{Tractor} is used to obtain the uncertainties on the flux measurements. Figure~\ref{fig:fitexample} shows the fits to the stacked cutouts of the compact star clusters in the six different bands. The point source assumption is also validated by the residuals, which are consistent with zero. The NIRCam $6$-band photometry derived for the stack is listed in Table~\ref{tab:photostack}.\vspace{+0.7cm} \subsection{SED Fitting}\label{sec:sedfitting} To study the properties of the compact star clusters, we fit simple stellar population (SSP) models to the photometry of the median stacks obtained in Section~\ref{sec:photomeasure}. We assume that the star clusters have been formed in a single burst, thus are well described with an SSP model. Given the low sampling in wavelength of these data, we think that this is a good approximation as multiple stellar populations would be difficult to tell apart if they exist. Before fitting, we need to correct the extracted flux densities for Milky Way dust extinction, which we obtain from the \citet[][]{SCHLEGEL98} dust maps\footnote{Specifically, we use the \texttt{sfdmap} Python package (\url{https://github.com/kbarbary/sfdmap})} recalibrated by \citet[][]{SCHLAFLY11}. The dust extinction towards the SMACS0723 is significant at an $E(B-V)\sim0.2\,{\rm mag}$ level, which results in an wavelength-dependent extinction of $\sim 0.25\,{\rm mag}$ at $0.9\,{\rm \mu m}$, $\sim 0.07\,{\rm mag}$ at $2.0\,{\rm \mu m}$, and less than $0.03\,{\rm mag}$ at $4.4\,{\rm \mu m}$ assuming the \citet[][]{FITZPATRICK99} dust extinction curve. We use these values to correct the measured flux densities of the star clusters. For our fiducial SSP models, we chose Padova 1994 stellar evolutionary tracks \citep[][]{BERTELLI94} together with the Basel Stellar Library \citep[BaSeL, version 3.1;][]{LEJEUNE97,LEJEUNE98,WESTERA03} complemented by the empirical STELIB library \citep[][]{LEBORGNE03}\footnote{\url{http://svocats.cab.inta-csic.es/stelib/index.php}} at wavelengths blueward of $9500\,{\rm \AA}$ \citep[see][]{BRUZUALCHARLOT03}. In the following, we assume a \citet{CHABRIER03} IMF. These fiducial SSP models were created using the software \textsc{GALAXEV}\footnote{\url{http://www.bruzual.org/bc03/}} in a grid of different ages and stellar metallicities. Specifically, the age grid ranges from $0.5\,{\rm Gyrs}$ to $11\,{\rm Gyrs}$ in steps of $0.5\,{\rm Gyrs}$ and we use $6$ different stellar metallicities of $0.005$, $0.02$, $0.2$, $0.4$, $1$, and $2.5\,{\rm Z_{\odot}}$. We compared the results using these fiducial models to other SSP models as well. Specifically, we compared to the MIST stellar tracks \citep[][]{DOTTER16,CHOI16}, based on the MESA isochrones \citep{PAXTON11,PAXTON13,PAXTON15,PAXTON18}\footnote{\url{https://waps.cfa.harvard.edu/MIST/index.html}}, assuming a BaSeL stellar library. These additional SSP models are created using the {\it Flexible Stellar Population Synthesis} code \citep[FSPS;][]{CONROY09,CONROY10}\footnote{\url{https://github.com/cconroy20/fsps}. A Python wrapper \citep[][]{JOHNSONFSPSPython21} exists at this link: \url{https://dfm.io/python-fsps/current/}.} for a Chabrier IMF and using the same age and metallicity grid. Generally, we find that wavelengths blueward of $\sim1\,{\rm \mu m}$ rest-frame are most affected by the choice of these different models. We will comment on the differences in the results in Section~\ref{sec:results}. The fit to the various models was performed by $\chi^2$ minimization using the \texttt{SciPy} {\it least-squares} \textsc{Python} package. We note that we fixed the redshift of the compact star clusters to the one of the galaxies in the SMACS0723 cluster ($z=0.39$). In Section~\ref{sec:photoandcolors} we will show that this is indeed a good assumption based on the similarity in colors of the star clusters and galaxies at $z=0.39$. Also, small changes in redshift (e.g., due to the kinematics in the cluster environment) do not have an impact on the fitted properties. \section{Results}\label{sec:results} \subsection{Properties of Globular Clusters}\label{sec:properties} \subsubsection{Photometry and Colors}\label{sec:photoandcolors} Figure~\ref{fig:properties} show the observed photometric properties of the extracted compact star clusters in SMACS0723. As shown on the left panel, the brightness of the star clusters ranges from $28$ to $30\,{\rm mag}$ in F090W and $27.5$ to $29.5\,{\rm mag}$ in F200W (left panel). The NIRCam filters cover the (redshifted) $2\,{\rm \mu m}$ peak, hence the F090W and F444W brightness is similar. The middle left panel shows the color distribution of the compact star clusters (purple circles) compared to star forming (gray) and quiescent (red) galaxies selected in the COSMOS2020 catalog \citep[][]{WEAVER22a} at a redshift $z=0.39$ with $\Delta z~=~0.2$\footnote{We interpolate the Suprime-Cam/$z$, UltraVISTA/$H$, UltraVISTA/$K_s$, and \textit{Spitzer}/IRAC $3.6\,{\rm \mu m}$ fluxes from COSMOS2020 to obtain the corresponding fluxes in the JWST/NIRCam filters F090W, F150W, F200W, and F277W.}. Due to their SED shape, which is due to the redshifting of the 1.6$\mu$m bump through the bandpasses, the globulars reside mostly at red [F090W]$-$[F150W]$\,\gtrsim0$ colors and blue [F200W]$-$[F277W]$\,\lesssim0$ colors. This parameter space is similar to quiescent galaxies at the same redshift, suggesting similar stellar populations. However, the star clusters are significantly fainter, $2-4\,{\rm mag}$ below the detection limits of the COSMOS surveys (right panel). \subsubsection{Population-Averaged Ages and Metallicities}\label{sec:agesandmetal} We note that these star clusters are faint even for JWST and most of the individual star clusters are detected at very low S/N in blue and red bands where their SED flux drops. We therefore first fit SSP models (see Section~\ref{sec:sedfitting}) to the photometry derived from the stacked cutouts to obtain a population-averaged median age and stellar metallicity measurement. The following values for metallicity and age are therefore an average of the studied relatively massive cluster population (see also Section~\ref{sec:localgcs}). Later in Section~\ref{sec:variations} we will fit individual star clusters to study the population variations of these properties. Figure~\ref{fig:fitresults} shows the results from SED fitting using our fiducial SSP models (Padova stellar tracks) to the stacked photometry. The left panel shows a $\chi^2$ map for the adopted age and metallicity grid with $3\sigma$, $4\sigma$, and $5\sigma$ contours indicated. We find a best-fit age of $1.5^{+0.5}_{-0.5}\,{\rm Gyr}$ and a best-fit metallicity of $0.2-0.3\,{\rm Z_{\odot}}$ ($3\sigma$). Interestingly, the $\chi^2$ map also indicates the presence of a younger ($<1\,{\rm Gyr}$) and more metal rich ($0.4-1.0\,{\rm Z_{\odot}}$) solution. The right panel of Figure~\ref{fig:fitresults} shows the data (black symbols) with the best-fit SED (black line) as well as other models (see legend in figure) for comparison with the same age or metallicity as the best fit. The data excludes with high significance metallicities below $0.2\,{\rm Z_{\odot}}$ as well as solar/super-solar. This is because changes in metallicity affect the colors significantly at a fixed age. For example, compare the $1.5\,{\rm Gyr}$ model at $0.02\,{\rm Z_{\odot}}$ (gray dotted line) and at $2.5\,{\rm Z_{\odot}}$ (gray dashed line) with the best fit ($0.2\,{\rm Z_{\odot}}$) in the right panel of Figure~\ref{fig:fitresults}. On the other hand, ages are not constrained well by our current data, especially with the JWST filters available. For example, ages up to $5\,{\rm Gyrs}$ cannot be excluded at better than $4\sigma$ level and maximal ages of the Universe at $z=0.39$ ($9.3\,{\rm Gyrs}$) are possible at $5\sigma$ level. This is expected by the lack of blue coverage of the JWST filters, and in addition the relation between color and age only changes weakly for SSPs with ages older than $\sim3\,{\rm Gyr}$. For example, the [F090W]$-$[F200W] color (corresponding to rest-frame $(r-H)$ color) changes by $0.4-0.6\,{\rm mag}$ (depending on metallicity) for ages $<3\,{\rm Gyr}$ (caused by hot massive stars), while it only changes by $\sim0.2\,{\rm mag}$ thereafter. We note that a better sampling of the observed $0.5-1.5\,{\rm \mu m}$ regime could constrain the older ages better (see Figure~\ref{fig:fitresults}). In addition, we quantify the reliability of these results by comparing to fits using the MIST isochrones as discussed in Section~\ref{sec:sedfitting}. We find that the models differ significantly at wavelengths blueward of $\sim1\,{\rm \mu m}$ rest-frame for low metallicities and low ages. However, we find overall that our results are robust against the choice of different model parameterizations. Specifically, using the {\it MIST} isochrones, we find consistent ages of $1.5^{+0.5}_{-0.5}\,{\rm Gyr}$ and a best-fit metallicity of $0.4\,{\rm Z_{\odot}}$ with a slightly larger $3\sigma$ range of $0.2-0.4\,{\rm Z_{\odot}}$. Similarly to the other models, metallicities of $<0.2\,{\rm Z_{\odot}}$ as well as solar metallicities are robustly excluded.\vspace{10mm} \subsubsection{Stellar Masses}\label{sec:masses} From the best-fit SSP models obtained for each individual star cluster, we derive a $V-$band mass-to-light (M/L) ratio from which we can estimate the total stellar masses of the compact star clusters. For the best-fit SSP model fit to the stacked photometry, we derive a M/L ratio of $0.45\,{\rm M_{\odot}\,L_{\odot}^{-1}}$. The M/L ratios of the individual star clusters range from $0.2-4.8\,{\rm M_{\odot}\,L_{\odot}^{-1}}$. With the $V$-band luminosity derived from the NIRCam/F090W filter (which is close to the $\sim5500\,{\rm \AA}$ $V$-band in rest-frame), we derive stellar masses for the individual star clusters and we find a median of $2.4^{+3.0}_{-1.5}\times 10^6\,{\rm M_{\odot}}$ (using the M/L ratio derived from the stacked SED would result in a stacked median stellar mass of $3.9^{+3.2}_{-1.8}\times 10^6\,{\rm M_{\odot}}$). As shown on the right panel in Figure~\ref{fig:properties}, the star clusters line up well with quiescent galaxies at $z=0.39$ on the $K-$band magnitude vs. stellar mass diagram. This suggest that these compact star clusters have a similar M/L ratio as quiescent galaxies at the same redshift. Assuming the quiescent galaxy M/L relation, we would obtain an average stellar mass of $3.3^{+1.6}_{-1.1}\times 10^6\,{\rm M_{\odot}}$. We can compare this obtained M/L ratio to what is measured for globular clusters in the Milky Way and local galaxies. The M/L ratios depend on the age (hence mass itself) and metallicity of the stellar population. Given the estimated age and metallicity of our star clusters, we find a possible range in M/L of $1-4\,{\rm M_{\odot}\,L_{\odot}^{-1}}$ in rest-frame $V$-band according to the observations and models in \citet[][]{KRUIJSSEN08}. Note that the M/L ratio increases with age, hence the upper end of this range would be applicable to old globulars, while the lower end is more common in young star clusters. Our measured M/L is consistent with younger ages. Summarizing, the masses derived for our compact star clusters are on the high-end of what is expected for globular star clusters in local galaxies (see also Figure~\ref{fig:sizes}). Furthermore, the high-mass end of the derived mass range may overlap with stellar masses estimated for Milky Way dwarf galaxies. To assess this, we compare the size constraints on our compact star clusters with the sizes of local dwarf galaxies. \subsubsection{Sizes}\label{sec:sizes} As established already in Section~\ref{sec:photomeasure} and Figure~\ref{fig:psfs}, the selected compact star clusters are unresolved, hence we are only able to place an upper limit on their sizes. A conservative upper limit would be the NIRCam PSF size at $\sim1.5\,{\rm \mu m}$, which is $\sim180\,{\rm pc}$ at $z=0.39$. To measure a more accurate upper size limit, we carry out a simple simulation as shown in the right panel of Figure~\ref{fig:psfs}. Specifically, we simulate point sources of different sizes (from $20$ to $200\,{\rm pc}$), which we convolve with the NIRCam/F150W PSF (Section~\ref{sec:psf}). We add realistic background noise measured from the real NIRCam/F150W image and assume a point source peak flux of $20\,{\rm nJy}$, which is consistent with the one measured for the stack in the same filter (see left panel of Figure~\ref{fig:psfs}). This simple simulation shows that we would be able to resolve sources larger than $50\,{\rm pc}$. We therefore place an upper limit to the sizes of our compact star clusters of $50\,{\rm pc}$. Figure~\ref{fig:sizes} compares this size limit to other Galactic and extra-galactic sources at similar stellar masses as derived for our compact star clusters (Section~\ref{sec:masses}). The sizes of the galaxies at $z=0.39$ are taken from the HST COSMOS-{\it DASH} morphology catalog \citep[][]{MOWLA19,CUTLER22} in HST WFC3/F160W. At lower masses, we complement that sample with galaxies at the same redshift directly extracted from the NIRSpec/F150W image\footnote{The masses and redshift are computed using the SED-fitting code \textsc{EAZY} \citep[][]{BRAMMER08,BRAMMER22_SMACSreduct}. Note, that we only extract sources from the NIRCam field offset from the cluster's BCG to avoid the effects of lensing on the size measurements.}. The sizes of these lower-mass galaxies are consistent with extrapolating the $M-R_{e}$ relation at $>10^9\,{\rm M_{\odot}}$ (dashed line) to these lower masses. The galaxies are clearly more extended than the compact star clusters selected here. We also show average sizes of star-forming clumps (blue hatched area showing $1\sigma$) in local galaxies taken from \citet[][]{DRAZINOS13} and \citet[][]{LARSON20}. At the resolution of the NIRCam observations, we would be able to marginally resolve such structures if they were present in the (quiescent) SMACS0723 host galaxies. Similarly, we would resolve objects similar in mass and size as the Milky Way dwarf galaxies, here taken from \citet[][]{STRIGARI08}\footnote{The masses were derived from total luminosities given in \citet[][]{STRIGARI08} and assuming the mass-to-light ratios of the galaxies measured by \citet[][]{REVAZ09}.}. We caution however, if the dwarf galaxies have been tidally stripped in the dense cluster environment such that only the bright central core remains, we would likely not be able to distinguish those from compact star clusters. For example, a dwarf galaxy could lose more than $90\%$ of its mass during the first few pericenter passages around its host galaxy \citep[][]{IDETA04}. This would bring them to a mass consistent with our compact star clusters. \subsection{Variations in the Properties of Star Clusters}\label{sec:variations} As shown in the middle panel of Figure~\ref{fig:overview}, the selected compact star clusters span about $1.5-2\,{\rm mags}$ in color space, which could be indicative of variations in age and metal properties. In the following, we try to characterize these color variations in terms of variations in these physical parameters. For this, we fit the same SSP models as described in Section~\ref{sec:sedfitting} to each individual compact star cluster. We note that while the uncertainties of these measurements for individual compact star clusters are rather large due to their faintness, we still are able to investigate statistically trends of age and metallicity in our sample. Figure~\ref{fig:variations} shows the individual star clusters on the [F090W]$-$[F150W] vs. [F200W]$-$[F277W] color$-$color diagram color-coded by the best-fit age (blue scale) and metallicity (red scale). Note that each symbol has two colors assigned (metallicity as edge-color and age as face-color). Light symbols correspond to star clusters with lower ages and metallicities, while darker symbols correspond to star clusters with older ages and higher metallicities. The background shows a Gaussian kernel density estimation (including the uncertainties of the data points) to visualize the most probably density of points. The lines indicate colors derived from the SSP models for different metallicities and ages (running from left to right, dots on lines indicated ages of 0.5, 1, 4, and $9\,{\rm Gyrs}$. Older models result in redder [F090W]$-$[F150W] colors, while more metal rich models result in a reddening of both colors. The range in colors of the individual compact star clusters are coupled with variations in ages and metallicity. This suggests different formation times and possibly different formation processes (causing different metal contents) of the star clusters in the SMACS0723 environment. We also note that the observed color (hence metallicity) distribution may be expected to be affected by the ``blue tilt'', caused by a lack of the most massive, metal poor star clusters \citep[see][]{FORBES10,USHER18}. \subsection{Comparison to Local Globular Clusters} \label{sec:localgcs} In the following, we compare the BCG and the identified compact star clusters to systems in the local universe. The BCG of the SMACS0723 cluster has a stellar mass of $\sim2\times 10^{11}\,{\rm M_\odot}$ and a $V$-band absolute magnitude of $\rm M_V^{Vega} \sim -23.2$\footnote{The $V$-band magnitude was derived from HST photometry and subsequently converted to the Landolt $V$ in Vega system to be consistent with the literature.}, which is at the faint end of the brightness distribution of the BCGs studied in \citet[][]{HARRIS14} ($-24.1 \lesssim \rm M_V^{Vega} \lesssim -23.0$). The $V$-band luminosities of our compact star clusters range between $2.5\times 10^6\,{\rm L_\odot}$ and $3.1\times 10^7\,{\rm L_\odot}$, which is about $1-2$ orders of magnitudes brighter than the common turnover luminosity of the globular cluster luminosity function at $\sim 10^5\,{\rm L_\odot}$ \citep[][]{LARSEN01,HARRIS14}. (Note that the lower limit is due to the sensitivity limit of the JWST observations.) In this luminosity range, the luminosity functions of globulars around local BCGs with similar luminosity find about $100$ globular clusters \citep[][]{HARRIS14}, which is $60\%$ less than what we find in the SMACS0723 cluster field. Several reasons could contribute to this differences. First, the SMACS0723 galaxy cluster is more massive ($8\times 10^{14}\,{\rm M_\odot}$ as measured by Planck) compared to the Abell clusters ($1-5\times 10^{14}\,{\rm M_\odot}$), which affects the number of globulars. For example, as shown in \citet[][]{HARRIS14}, the luminosity function of globulars around the BCG in Abell 3558 ($4.7\times 10^{14}\,{\rm M_\odot}$) peaks at two times higher number densities as compared to the one of the less massive cluster Abell 1736 ($2.8\times 10^{14}\,{\rm M_\odot}$). Second, our selection includes the full ICM (over several $100\,{\rm kpc}$, see Figure~\ref{fig:overview}), which may lead to the inclusion of globular clusters around other galaxies in the vicinity of the BCG. Finally, as pointed out in Section~\ref{sec:sizes}, we might also include the tidally stripped compact cores of dwarf galaxies in our sample. Given the above points, the number of compact star clusters is close to what is expected based on local observations. Note that $>50\%$ of our compact star clusters have $\log(L/L_\odot) > 6.7$ and thus would be considered as superluminous globulars according to \citet[][]{HARRIS14}. Such superluminous objects could be related to ultra compact dwarf galaxies (or the stripped cores thereof) or bridge the gap between globulars and dwarf galaxies. These object have been found in several rich local galaxy clusters \citep[e.g.,][]{MISGELD11,MIESKE12}. SMACS0723 is a massive galaxy cluster at these redshifts (approximately a factor of two more massive than local galaxy clusters studied in Harris et al.), hence the large occurrence of such superluminous globulars would not surprising. Finally, we compared the color distribution of our compact star clusters to the globulars around local BCGs from \citet[][]{HARRIS14}. Figure~\ref{fig:colorlocal} shows the $(B-I)_{\rm Vega}$ vs. M$_{I}^{\rm Vega}$ color-magnitude diagram for our extracted compact star clusters at $z=0.39$ (purple circles)\footnote{These quantities have been measured on the best-fit de-reddened rest-frame SSP models to the individual star clusters in the Landolt filter system (in Vega magnitudes) to be consistent with other works. Note that the JWST data does not cover the rest-frame $B$ and $I-$bands. Their fluxes are extrapolated from the best-fit models.} as well as globulars around the local galaxies {\it NGC 7720} (gray) and {\it ESO444-G046} (brown), which are the BCGs of the clusters Abell 2634 and 3558, respectively. The former BCG is brighter ($-23.8$ vs. $-23.4\,{\rm Vega\,mag}$) and resides in a more massive cluster ($4.7\times10^{14}$ vs. $1.5\times10^{14}\,{\rm M_\odot}$), which, however, is still $1.7$ times less massive than SMACS0723. The compact star clusters studied in this work at $z=0.39$ reside at the bright end of the distribution of local globulars. Their colors are consistent with the colors of globulars in Abell 3558, the more massive one of the two local galaxy clusters shown in Figure~\ref{fig:colorlocal}. This is consistent with the idea that the brightness of the globular clusters scales with the mass of the galaxy clusters. \subsection{Comparison to Higher Redshift Globulars}\label{sec:highz} Recent studies by \citet[][]{MOWLA22} and \citet[][]{CLAEYSSENS22} examined the properties of individual globular star clusters in SMACS0723 field of lensed galaxies at higher redshifts. The significant magnification (up to $100\times$) due to lensing allows the authors of these studies to probe fainter star clusters in the mass range of $10^5-10^7\,{\rm M_\odot}$. However, the wavelength range probed by the JWST NIRCam filters is bluer due to the higher redshift of the sources, which may affect how well different properties of the clusters can be constrained with the given photometry. The \citet[][]{CLAEYSSENS22} study finds relatively young ages of 100s of Myrs up to $\sim1\,{\rm Gyr}$ for the studied star clusters around $18$ galaxies in the redshift range $1.3 < z < 7.7$. The star clusters are consistent local globulars in terms of their size and stellar mass. These ages are $\sim 1-2\,{\rm Gyr}$ younger than what is measured for our BCG star clusters at $3\sigma$, which is in line with the formation of star clusters over cosmic time. On the other hand, \citet[][]{MOWLA22} study in detail a system dubbed as the ``Sparkler'' at $z=1.378$ and find ages of $3-5\,{\rm Gyrs}$ for $6$ out of the $12$ star clusters. This would indicate that these star clusters have been formed during the early reionization phase of the intergalactic medium at $z=8-11$, which would be an interesting but also extreme scenario for cluster formation as understood by current models and other observations of lensed galaxies at high redshifts (see discussion in \citealt[][]{CLAEYSSENS22} and references therein as well as, for example, \citealt[][]{VANZELLA17}). We note that Claeyssens et al. find much younger ages ($\sim 100\,{\rm Myrs}$) for the same star clusters. Understanding this disagreement is beyond the scope of our paper, however, there could be multiple reasons such as aperture vs. PSF fitting photometer methods, the inclusion of galactic extinction, differences in SSP models and SED fitting codes, as well as differences in the JWST photometric calibration references. \subsection{A Final Word on Milky Way Stars}\label{sec:stars} Due to their point-like nature and faintness, cool dwarf stars in our Milky Way could be mistaken as globular clusters in the $z=0.39$ host galaxies. A significant contamination of our sample by such stars is, however, ruled out by two reasons. First, as shown in Figure~\ref{fig:fitresults}, black body models (representing cool Milky Way dwarf stars from $3000-3400\,{\rm K}$) are not consistent with the extracted photometry of our sources. Specifically, the mid-infrared peak is redshifted consistent with the redshift of the cluster galaxies ($z=0.39$). A combination of black body models ({\it i.e.} stars of different temperatures) would be necessary to explain the observed photometry. We note that due to the stacking analysis, stars of different temperatures can be combined, leading to a similar SED as the stacked photometry of the star clusters in our sample. As an additional test, we therefore compared the photometry of randomly picked star clusters to point sources ({\it i.e.} stars) at comparable faint magnitudes selected from the parallel NIRCam module off the center of SMACS0723. We find that the mid-infrared peak emission between the two samples is significantly offset (as a results of the redshift), suggesting negligible contribution of stars to the stack. Second, the predicted number of stars in the NIRCam FoV is low compared to the selected compact star clusters. Specifically, we predicted the number of stars and brown dwarfs from the \citet[][]{WAINSCOAT92} as well as the \citet[][]{KIRKPATRICK21} models. The former model is in good agreement with the number of stars in the COSMOS field \citep[][]{SCOVILLE07} as shown in \citet[][]{FAJARDOACOSTA22}, however, it does not include consistently M, L, and T spectral types, which start to contribute significantly at fainter magnitudes. The expected number of stars from the Wainscoat and Kirkpatrick models are shown on the left panel of Figure~\ref{fig:properties} as blue histogram and dark blue horizontal lines, respectively. We would expect on the order of $\sim20$ stars and dwarfs over the full FoV of NIRCam, and approximately $5$ over the area where the compact star clusters are found (see Figure~\ref{fig:overview}). The negligible contribution of Milky Way stars to our sample is also suggested by comparing the number of selected star clusters in the FoVs of the two NIRCam modules. Specifically, the parallel module off-center of the SMACS0723 cluster should be dominated by Milky Way stars. Repeating the sample selection on that field results in only $\sim10$ point sources at similar magnitudes as the selected star clusters. Conservatively assuming that these are all stars and that the stellar density is the same in both FoVs, we would expect $<10$ out of the $178$ selected compact star clusters to be Milky Way stars. Vice versa, we note that globulars and dwarf galaxies in and around our Milky Way would be easily resolved by JWST and can therefore be ruled out. \section{Summary and Conclusions}\label{sec:end} JWST's depth and resolution provides the exciting opportunity to study the faintest and most compact star clusters in our universe. Here, we use the NIRCam early release observations of the SMACS0723 galaxy cluster at $z=0.39$ to study a population of $178$ compact star clusters residing in the intracluster region. We extract the photometry of these point sources with the \textsc{Tractor} software using robust PSF measurements. We note that there is considerable Galactic dust extinction towards the SMACS0723 field, which needs to be corrected for. We find that the compact star clusters occupy a similar location in [F090W]$-$[F150W] vs. [F200W]$-$[F277W] color space as quiescent galaxies at the same redshifts (Figure~\ref{fig:properties}). However, the star clusters are more than $2-4$ magnitudes fainter than the detection limits of current wide-field surveys and have sizes that are smaller than $50\,{\rm pc}$ (Figures~\ref{fig:psfs} and~\ref{fig:sizes}). The photometry of the star clusters is fit with SSP models to obtain constraints on their masses, age, and stellar metallicity (Figure~\ref{fig:fitresults}). Different stellar models are tested and the results are found to be robust under the different choices. We derive metallicities between $0.2-0.3\,{\rm Z_{\odot}}$ and robustly exclude $<0.2\,{\rm Z_{\odot}}$ and solar/super-solar metallicities at $4\sigma$ level. For the ages, we find $1.5^{+0.5}_{-0.5}\,{\rm Gyrs}$, ruling out ages $>5\,{\rm Gyrs}$ at $4\sigma$ confidence. The photometry is, however, consistent with older ages up to $9\,{\rm Gyrs}$ at $5\sigma$ level, mainly due to the relatively flat relation between color and age at ages $>3\,{\rm Gyrs}$. In addition, we find indications of variations of age and metallicity in this cluster population (Figure~\ref{fig:variations}). Assuming the M/L ratios of the best fit SSP models, we derive stellar masses for the star clusters of $2.4^{+3.0}_{-1.5}\times 10^6\,{\rm M_{\odot}}$ (Figure~\ref{fig:properties}). We note that this M/L ratio is similar to the one of quiescent galaxies at the same redshift and consistent with models and observed ratios of local globulars. The stellar masses of our compact star clusters lie at the high-mass end of average masses of local globulars and there could be overlap with masses of local dwarf galaxies (Figure~\ref{fig:sizes}). All in all, this suggests that these objects are likely young to middle-aged, compact star clusters (sizes smaller than $50\,{\rm pc}$) with formation times at $z=0.5-0.7$. Such a young age and modest metallicity support a scenario where the star clusters have been formed recently and then were stripped away several $100\,{\rm kpc}$ (projected) from their host galaxies due to the interactions in the cluster field. On the other hand, these clusters could have been formed recently in cold flows onto the cluster core \citep[e.g.,][]{HOLTZMAN92}. Related to this, the recent work by \citet[][]{LEE22} studies the spatial distribution of these star clusters and indicates that they follow closely the intercluster light and trace well the dark matter structure of the galaxy cluster. This would be more in support of a scenario in which the star clusters are stripped from their host galaxies in this highly interactive environment. A narrower sampling of the $1-5\,{\rm \mu m}$ wavelength with photometry or spectroscopy could help in determining more robust ages to place more stringent constraints on the formation scenario. The star clusters probed here are $1-2$ orders of magnitudes brighter than the turnover of the globular cluster luminosity function measured around BCGs in local galaxy clusters. The lower end of the luminosity distribution is limited by JWST's sensitivity. At these luminosities, the clusters are in the regime of superluminous globulars as defined in \citet[][]{HARRIS14}, however, it is to note that SMACS0723 is relatively massive compared to local galaxy clusters (although its BCG is at the lower end of the luminosity distribution of local BCG probed in Harris et al.). Interestingly, the $(B-I)$ color distribution of compact star clusters in SMACS0723 is similar to the one of globulars in the $\sim1.7\times$ less massive Abell 3558 cluster, one of the most massive clusters studied in that work (Figure~\ref{fig:colorlocal}). We note that the star cluster here are, at {\it maximal} sizes of $50\,{\rm pc}$ and masses at the high end of local globulars, among the most compact stellar agglomerations (Figure~\ref{fig:sizes}). Compact star clusters or globulars are a likely explanation of these objects, however, we note that the stripped compact cores of dwarf galaxies could be an alternative explanation (see for example \citet[][]{IDETA04} for a discussion of the origin of $\omega$ Centauri). For example, number estimates based on the Illustris simulation \citep[][]{GENEL14,VOGELSBERGER14a,VOGELSBERGER14b} suggest several hundreds of dwarf galaxies at masses of $10^7\,{\rm M_{\odot}}$ in clusters with $M_{\rm 200} > 5\times 10^{13}\,{\rm M_{\odot}}$, including SMACS0723 at a mass of $8.4\times10^{14}\,{\rm M_{\odot}}$ \citep[][]{MISTANI16,COE19}. Simulations suggest that a dwarf galaxy could lose $\sim90\%$ of its mass during the first few pericenter passages around its host galaxy \citep[e.g.,][]{IDETA04}, making their masses consistent with the ones measured for the clusters (Figure~\ref{fig:sizes}). If a fraction of the expected number of dwarf galaxies gets stripped in the cluster environment, compact cores of these dwarfs could become comparable in number with the compact star clusters seen here. \begin{acknowledgments} {\it Acknowledgments:} We thank Sergio Fajardo-Acosta, Alastair Edge, and Jessica Krick for fruitful discussions on the topic of stars and globular clusters. We also thank the anonymous referee for the helpful input, which significantly improved the quality of this manuscript. This research is partially funded by the Joint Survey Processing effort at IPAC/Caltech through NASA grant NNN12AA01C. This research made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant No. 140. This work has been based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with JWST programs 2736. The authors acknowledge the ERO teams led by Klaus M. Pontoppidan for developing their observing programs with a zero-exclusive-access period. \end{acknowledgments} \vspace{5mm} \facilities{JWST (NIRCam), VLT (MUSE)} \software{ \texttt{astropy} \citep{ASTROPY13,ASTROPY18}, \texttt{EAZY} \citep[][]{BRAMMER08}, \texttt{FSPS} \citep[][]{CONROY09,CONROY10}, \texttt{GALAXEV} \citep[][]{BRUZUALCHARLOT03}, \texttt{Grizli} \citep[][]{BRAMMER21}, \texttt{Source Extractor} \citep{Bertin1996} } \newpage \begin{center} \begin{longtable*}{l l l l l l l l l l} \caption{Coordinates, NIRCam fluxes, and stellar masses for all extracted compact star clusters. All fluxes are corrected for Galactic extinction. The errors in masses are derived from the photometry only and do not include uncertainties in the M/L ratio.} \label{tab:photo} \\ \hline\hline \colhead{ID} & \colhead{R.A.} & \colhead{Decl.} & \colhead{F090W} & \colhead{F150W} & \colhead{F200W} & \colhead{F277W} & \colhead{F356W} & \colhead{F444W} & \colhead{log(M/M$_{\odot}$)} \\[-0.2cm] \colhead{} & \colhead{(J2000)} & \colhead{(J2000)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{}\\ \hline \endfirsthead \multicolumn{10}{c}{\tablename\ \thetable{} -- continued} \\ \hline\hline \colhead{ID} & \colhead{R.A.} & \colhead{Decl.} & \colhead{F090W} & \colhead{F150W} & \colhead{F200W} & \colhead{F277W} & \colhead{F356W} & \colhead{F444W} & \colhead{log(M/M$_{\odot}$)} \\[-0.2cm] \colhead{} & \colhead{(J2000)} & \colhead{(J2000)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{}\\ \hline \endhead \hline \multicolumn{10}{r}{continued on next page}\\ \endfoot \hline \hline \endlastfoot SC\_000 & 110.83478 & -73.45518 & $9.4 \pm 1.2$ & $22.7 \pm 0.3$ & $20.3 \pm 0.3$ & $21.6 \pm 1.2$ & $12.0 \pm 1.1$ & $11.2 \pm 1.5$ & $6.73 \pm 0.05$\\ SC\_001 & 110.83487 & -73.45463 & $20.5 \pm 1.2$ & $29.2 \pm 0.5$ & $33.8 \pm 0.7$ & $30.7 \pm 0.6$ & $15.4 \pm 1.1$ & $13.4 \pm 1.5$ & $7.02 \pm 0.02$\\ SC\_002 & 110.83238 & -73.45431 & $34.9 \pm 1.2$ & $57.5 \pm 0.8$ & $70.3 \pm 0.7$ & $67.7 \pm 0.5$ & $39.6 \pm 0.5$ & $33.7 \pm 0.5$ & $7.25 \pm 0.02$\\ SC\_003 & 110.83262 & -73.45475 & $29.8 \pm 1.2$ & $29.8 \pm 0.7$ & $33.3 \pm 0.7$ & $31.5 \pm 1.1$ & $17.3 \pm 0.7$ & $12.8 \pm 1.5$ & $6.52 \pm 0.02$\\ SC\_004 & 110.83263 & -73.45367 & $12.8 \pm 1.2$ & $20.8 \pm 0.7$ & $14.7 \pm 0.5$ & $10.6 \pm 1.2$ & $11.1 \pm 0.7$ & $4.5 \pm 1.5$ & $6.68 \pm 0.04$\\ SC\_005 & 110.82962 & -73.45353 & $16.4 \pm 1.2$ & $16.7 \pm 0.5$ & $19.2 \pm 0.5$ & $12.2 \pm 0.8$ & $9.4 \pm 0.6$ & $7.6 \pm 1.5$ & $6.59 \pm 0.03$\\ SC\_006 & 110.83044 & -73.45322 & $15.5 \pm 1.2$ & $20.0 \pm 0.5$ & $22.5 \pm 0.7$ & $16.3 \pm 1.2$ & $11.1 \pm 0.7$ & $6.4 \pm 1.5$ & $6.94 \pm 0.04$\\ SC\_007 & 110.82800 & -73.45331 & $3.2 \pm 1.2$ & $12.2 \pm 0.7$ & $14.0 \pm 0.7$ & $18.9 \pm 0.5$ & $16.6 \pm 1.1$ & $15.7 \pm 1.5$ & $5.75 \pm 0.14$\\ SC\_008 & 110.82647 & -73.45389 & $23.8 \pm 1.2$ & $20.6 \pm 0.5$ & $19.7 \pm 0.6$ & $25.2 \pm 0.8$ & $9.5 \pm 1.1$ & $8.6 \pm 1.5$ & $6.27 \pm 0.02$\\ SC\_009 & 110.82579 & -73.45360 & $21.7 \pm 1.2$ & $26.3 \pm 0.5$ & $27.0 \pm 0.4$ & $24.6 \pm 0.7$ & $15.0 \pm 1.1$ & $11.7 \pm 0.7$ & $6.42 \pm 0.02$\\ SC\_010 & 110.82474 & -73.45357 & $5.3 \pm 1.2$ & $14.9 \pm 0.4$ & $18.9 \pm 0.4$ & $13.0 \pm 1.2$ & $11.4 \pm 1.1$ & $7.8 \pm 1.5$ & $6.62 \pm 0.09$\\ SC\_011 & 110.82139 & -73.45397 & $17.0 \pm 1.2$ & $18.6 \pm 0.6$ & $18.0 \pm 0.3$ & $14.9 \pm 0.5$ & $9.6 \pm 0.7$ & $5.9 \pm 1.5$ & $6.46 \pm 0.03$\\ SC\_012 & 110.82147 & -73.45443 & $6.3 \pm 1.2$ & $16.8 \pm 0.8$ & $14.9 \pm 0.7$ & $12.8 \pm 1.2$ & $11.5 \pm 1.1$ & $1.8 \pm 1.5$ & $6.65 \pm 0.10$\\ SC\_013 & 110.82276 & -73.45429 & $7.5 \pm 1.2$ & $16.6 \pm 0.7$ & $20.8 \pm 0.4$ & $15.2 \pm 1.2$ & $2.0 \pm 1.1$ & $5.4 \pm 1.5$ & $6.55 \pm 0.07$\\ SC\_014 & 110.82239 & -73.45520 & $14.7 \pm 1.2$ & $17.8 \pm 0.8$ & $22.2 \pm 0.6$ & $20.7 \pm 1.2$ & $12.0 \pm 1.1$ & $13.3 \pm 1.5$ & $6.50 \pm 0.04$\\ SC\_015 & 110.82412 & -73.45540 & $26.1 \pm 1.2$ & $42.1 \pm 0.2$ & $42.7 \pm 0.3$ & $47.2 \pm 1.2$ & $27.2 \pm 1.0$ & $21.7 \pm 1.5$ & $6.36 \pm 0.02$\\ SC\_016 & 110.82460 & -73.45579 & $15.2 \pm 1.2$ & $16.6 \pm 0.5$ & $18.5 \pm 0.5$ & $19.5 \pm 1.1$ & $7.3 \pm 1.1$ & $7.4 \pm 1.1$ & $6.13 \pm 0.03$\\ SC\_017 & 110.82441 & -73.45585 & $8.1 \pm 1.2$ & $14.6 \pm 0.6$ & $13.0 \pm 0.5$ & $16.9 \pm 0.5$ & $15.3 \pm 0.9$ & $13.5 \pm 1.5$ & $5.92 \pm 0.06$\\ SC\_018 & 110.82575 & -73.45549 & $20.8 \pm 1.2$ & $28.0 \pm 0.6$ & $21.5 \pm 0.5$ & $21.9 \pm 1.2$ & $16.8 \pm 1.1$ & $16.7 \pm 1.5$ & $6.16 \pm 0.02$\\ SC\_019 & 110.82953 & -73.45540 & $18.3 \pm 1.2$ & $21.3 \pm 0.8$ & $28.3 \pm 0.2$ & $33.8 \pm 1.2$ & $20.3 \pm 1.1$ & $24.1 \pm 1.5$ & $6.52 \pm 0.02$\\ SC\_020 & 110.83033 & -73.45513 & $26.0 \pm 1.2$ & $34.4 \pm 0.4$ & $38.6 \pm 0.4$ & $42.2 \pm 1.2$ & $26.4 \pm 1.1$ & $13.8 \pm 1.5$ & $6.43 \pm 0.02$\\ SC\_021 & 110.83040 & -73.45523 & $12.0 \pm 1.2$ & $21.5 \pm 0.7$ & $27.5 \pm 0.6$ & $39.3 \pm 1.2$ & $29.3 \pm 1.1$ & $30.0 \pm 1.5$ & $6.34 \pm 0.04$\\ SC\_022 & 110.83355 & -73.45528 & $11.5 \pm 1.2$ & $12.9 \pm 0.5$ & $13.0 \pm 0.6$ & $12.9 \pm 1.0$ & $5.3 \pm 1.1$ & $6.3 \pm 1.5$ & $5.95 \pm 0.05$\\ SC\_023 & 110.83300 & -73.45532 & $6.8 \pm 1.2$ & $15.5 \pm 0.5$ & $14.4 \pm 0.5$ & $13.7 \pm 1.0$ & $7.1 \pm 0.7$ & $6.2 \pm 1.5$ & $6.63 \pm 0.08$\\ SC\_024 & 110.83395 & -73.45505 & $15.7 \pm 0.1$ & $15.8 \pm 0.6$ & $11.1 \pm 0.6$ & $10.1 \pm 1.2$ & $3.8 \pm 1.1$ & $1.2 \pm 1.5$ & $6.22 \pm 0.00$\\ SC\_025 & 110.83219 & -73.45462 & $19.3 \pm 1.2$ & $22.8 \pm 0.6$ & $24.3 \pm 0.7$ & $20.5 \pm 0.5$ & $17.1 \pm 1.0$ & $12.7 \pm 1.5$ & $6.37 \pm 0.03$\\ SC\_026 & 110.83015 & -73.45389 & $19.9 \pm 0.0$ & $30.4 \pm 0.8$ & $34.9 \pm 0.5$ & $36.0 \pm 0.9$ & $18.9 \pm 0.7$ & $18.3 \pm 1.5$ & $6.18 \pm 0.00$\\ SC\_027 & 110.82938 & -73.45402 & $12.7 \pm 1.2$ & $17.0 \pm 0.8$ & $21.8 \pm 0.5$ & $17.4 \pm 1.2$ & $12.8 \pm 1.1$ & $9.3 \pm 1.5$ & $6.49 \pm 0.04$\\ SC\_028 & 110.83031 & -73.45406 & $18.4 \pm 1.2$ & $14.4 \pm 0.7$ & $14.7 \pm 0.6$ & $19.7 \pm 1.2$ & $9.2 \pm 1.1$ & $10.7 \pm 1.5$ & $6.11 \pm 0.03$\\ SC\_029 & 110.83097 & -73.45398 & $4.4 \pm 1.2$ & $13.7 \pm 0.8$ & $12.0 \pm 0.5$ & $9.8 \pm 0.4$ & $2.3 \pm 1.1$ & $2.5 \pm 1.5$ & $6.17 \pm 0.12$\\ SC\_030 & 110.83071 & -73.45574 & $46.2 \pm 1.2$ & $55.4 \pm 0.4$ & $59.1 \pm 0.7$ & $52.6 \pm 1.2$ & $29.8 \pm 0.5$ & $20.5 \pm 1.5$ & $7.08 \pm 0.01$\\ SC\_031 & 110.83011 & -73.45603 & $10.3 \pm 1.2$ & $21.3 \pm 0.8$ & $18.8 \pm 0.6$ & $15.2 \pm 0.3$ & $9.5 \pm 1.1$ & $4.6 \pm 1.5$ & $6.76 \pm 0.04$\\ SC\_032 & 110.83222 & -73.45585 & $10.8 \pm 1.2$ & $13.2 \pm 0.4$ & $14.6 \pm 0.5$ & $13.4 \pm 1.2$ & $7.1 \pm 0.1$ & $2.9 \pm 1.5$ & $6.76 \pm 0.04$\\ SC\_033 & 110.82036 & -73.45593 & $5.2 \pm 1.2$ & $12.7 \pm 0.8$ & $13.1 \pm 0.5$ & $15.8 \pm 0.6$ & $7.3 \pm 1.0$ & $6.0 \pm 1.5$ & $6.53 \pm 0.09$\\ SC\_034 & 110.82258 & -73.45596 & $10.0 \pm 1.2$ & $12.7 \pm 0.3$ & $12.6 \pm 0.4$ & $8.3 \pm 1.2$ & $3.7 \pm 1.1$ & $3.3 \pm 1.5$ & $6.68 \pm 0.05$\\ SC\_035 & 110.80869 & -73.45551 & $15.6 \pm 1.2$ & $18.8 \pm 0.5$ & $20.8 \pm 0.5$ & $19.6 \pm 0.6$ & $10.4 \pm 1.1$ & $6.6 \pm 1.5$ & $6.37 \pm 0.04$\\ SC\_036 & 110.81668 & -73.45396 & $15.0 \pm 1.2$ & $10.1 \pm 0.6$ & $13.3 \pm 0.5$ & $16.7 \pm 1.2$ & $7.7 \pm 1.1$ & $8.0 \pm 1.5$ & $6.12 \pm 0.04$\\ SC\_037 & 110.81672 & -73.45406 & $3.9 \pm 1.2$ & $21.2 \pm 0.8$ & $19.9 \pm 0.4$ & $16.8 \pm 0.9$ & $7.1 \pm 0.6$ & $3.5 \pm 1.5$ & $6.45 \pm 0.14$\\ SC\_038 & 110.81860 & -73.45359 & $6.8 \pm 1.2$ & $20.8 \pm 0.7$ & $23.1 \pm 0.6$ & $24.1 \pm 1.2$ & $14.1 \pm 1.1$ & $7.6 \pm 1.5$ & $6.89 \pm 0.07$\\ SC\_039 & 110.82047 & -73.45328 & $31.2 \pm 1.2$ & $32.3 \pm 0.4$ & $32.5 \pm 0.4$ & $31.2 \pm 0.5$ & $20.6 \pm 0.5$ & $8.4 \pm 1.5$ & $6.38 \pm 0.01$\\ SC\_040 & 110.82181 & -73.45330 & $12.8 \pm 1.2$ & $19.8 \pm 0.1$ & $15.7 \pm 0.2$ & $16.9 \pm 1.2$ & $15.1 \pm 1.1$ & $15.6 \pm 1.5$ & $6.12 \pm 0.04$\\ SC\_041 & 110.82086 & -73.45378 & $2.8 \pm 1.2$ & $11.5 \pm 0.7$ & $11.6 \pm 0.4$ & $8.6 \pm 1.2$ & $7.5 \pm 0.7$ & $2.9 \pm 1.5$ & $6.32 \pm 0.18$\\ SC\_042 & 110.82852 & -73.45620 & $12.0 \pm 0.1$ & $17.6 \pm 0.4$ & $17.1 \pm 0.6$ & $16.0 \pm 0.4$ & $8.8 \pm 1.1$ & $5.9 \pm 1.5$ & $6.37 \pm 0.00$\\ SC\_043 & 110.82804 & -73.45646 & $17.9 \pm 1.2$ & $20.7 \pm 0.8$ & $15.5 \pm 0.3$ & $15.2 \pm 1.2$ & $8.4 \pm 1.1$ & $6.9 \pm 1.5$ & $6.26 \pm 0.03$\\ SC\_044 & 110.82936 & -73.45614 & $18.6 \pm 1.2$ & $17.1 \pm 0.5$ & $18.0 \pm 0.5$ & $20.0 \pm 1.2$ & $16.6 \pm 1.1$ & $21.3 \pm 1.5$ & $6.28 \pm 0.03$\\ SC\_045 & 110.82436 & -73.45603 & $8.0 \pm 1.2$ & $14.9 \pm 0.7$ & $16.0 \pm 0.6$ & $13.3 \pm 0.8$ & $6.9 \pm 1.1$ & $1.9 \pm 1.5$ & $6.39 \pm 0.06$\\ SC\_046 & 110.82245 & -73.45393 & $11.0 \pm 1.2$ & $13.9 \pm 0.3$ & $11.9 \pm 0.5$ & $11.0 \pm 0.3$ & $4.6 \pm 0.7$ & $4.4 \pm 1.5$ & $6.59 \pm 0.04$\\ SC\_047 & 110.82245 & -73.45386 & $9.5 \pm 1.2$ & $14.6 \pm 0.7$ & $14.3 \pm 0.6$ & $14.5 \pm 0.6$ & $18.8 \pm 0.4$ & $10.9 \pm 1.5$ & $5.99 \pm 0.06$\\ SC\_048 & 110.82226 & -73.45405 & $6.0 \pm 1.2$ & $7.1 \pm 0.6$ & $8.2 \pm 0.5$ & $7.4 \pm 1.2$ & $8.3 \pm 1.1$ & $1.9 \pm 1.5$ & $5.95 \pm 0.09$\\ SC\_049 & 110.81272 & -73.45459 & $19.8 \pm 1.2$ & $25.4 \pm 0.8$ & $25.6 \pm 0.6$ & $21.9 \pm 1.2$ & $12.9 \pm 1.1$ & $9.5 \pm 1.5$ & $7.05 \pm 0.02$\\ SC\_050 & 110.82056 & -73.45527 & $10.7 \pm 1.2$ & $20.9 \pm 0.8$ & $20.4 \pm 0.6$ & $24.2 \pm 1.2$ & $10.7 \pm 0.7$ & $12.8 \pm 1.5$ & $6.74 \pm 0.04$\\ SC\_051 & 110.83121 & -73.45448 & $10.5 \pm 1.2$ & $18.1 \pm 0.8$ & $18.0 \pm 0.6$ & $17.6 \pm 1.2$ & $10.5 \pm 1.1$ & $9.6 \pm 1.5$ & $6.80 \pm 0.05$\\ SC\_052 & 110.83858 & -73.45495 & $12.0 \pm 1.2$ & $19.7 \pm 0.8$ & $18.9 \pm 0.4$ & $17.0 \pm 1.2$ & $10.8 \pm 1.1$ & $6.6 \pm 1.5$ & $6.36 \pm 0.05$\\ SC\_053 & 110.83849 & -73.45514 & $19.3 \pm 1.2$ & $24.8 \pm 0.6$ & $22.9 \pm 0.5$ & $22.8 \pm 0.7$ & $13.3 \pm 0.7$ & $7.7 \pm 1.5$ & $6.17 \pm 0.02$\\ SC\_054 & 110.83820 & -73.45532 & $9.5 \pm 1.2$ & $8.4 \pm 0.6$ & $10.7 \pm 0.7$ & $11.2 \pm 1.2$ & $5.0 \pm 1.1$ & $4.8 \pm 1.5$ & $5.93 \pm 0.06$\\ SC\_055 & 110.83400 & -73.45490 & $6.1 \pm 1.2$ & $9.3 \pm 0.8$ & $15.3 \pm 0.6$ & $15.0 \pm 1.1$ & $6.4 \pm 1.1$ & $7.5 \pm 1.5$ & $6.75 \pm 0.10$\\ SC\_056 & 110.83486 & -73.45413 & $7.2 \pm 1.2$ & $11.4 \pm 0.6$ & $12.0 \pm 0.3$ & $9.5 \pm 1.2$ & $7.8 \pm 1.1$ & $8.4 \pm 1.5$ & $6.63 \pm 0.07$\\ SC\_057 & 110.82368 & -73.45312 & $7.7 \pm 1.2$ & $11.5 \pm 0.5$ & $16.3 \pm 0.2$ & $13.4 \pm 0.6$ & $8.0 \pm 1.1$ & $2.2 \pm 1.5$ & $6.78 \pm 0.07$\\ SC\_058 & 110.82875 & -73.45347 & $5.8 \pm 1.2$ & $8.2 \pm 0.6$ & $10.7 \pm 0.5$ & $8.6 \pm 1.2$ & $5.3 \pm 1.1$ & $4.1 \pm 1.5$ & $6.20 \pm 0.07$\\ SC\_059 & 110.84541 & -73.45423 & $9.5 \pm 1.2$ & $17.9 \pm 0.5$ & $15.5 \pm 0.7$ & $11.1 \pm 1.2$ & $5.6 \pm 1.1$ & $7.5 \pm 0.3$ & $6.72 \pm 0.06$\\ SC\_060 & 110.84582 & -73.45393 & $6.3 \pm 1.2$ & $12.8 \pm 0.6$ & $11.3 \pm 0.7$ & $10.0 \pm 1.2$ & $7.9 \pm 1.0$ & $6.0 \pm 0.5$ & $6.58 \pm 0.08$\\ SC\_061 & 110.84403 & -73.45389 & $3.9 \pm 1.1$ & $10.4 \pm 0.8$ & $11.2 \pm 0.2$ & $11.9 \pm 1.2$ & $6.3 \pm 1.1$ & $4.7 \pm 1.5$ & $6.68 \pm 0.14$\\ SC\_062 & 110.84351 & -73.45305 & $9.3 \pm 1.2$ & $14.2 \pm 0.8$ & $12.2 \pm 0.7$ & $11.7 \pm 1.2$ & $6.9 \pm 1.1$ & $3.0 \pm 1.5$ & $6.72 \pm 0.06$\\ SC\_063 & 110.84426 & -73.45268 & $7.1 \pm 0.1$ & $19.4 \pm 0.5$ & $19.6 \pm 0.5$ & $19.7 \pm 1.2$ & $9.7 \pm 0.8$ & $6.6 \pm 1.5$ & $6.56 \pm 0.01$\\ SC\_064 & 110.84474 & -73.45277 & $10.8 \pm 1.2$ & $18.6 \pm 0.8$ & $19.3 \pm 0.6$ & $21.3 \pm 1.2$ & $12.0 \pm 1.1$ & $8.3 \pm 1.5$ & $6.83 \pm 0.05$\\ SC\_065 & 110.84351 & -73.45347 & $2.0 \pm 1.2$ & $13.4 \pm 0.7$ & $14.9 \pm 0.7$ & $11.8 \pm 1.2$ & $6.6 \pm 0.0$ & $3.1 \pm 1.5$ & $5.23 \pm 0.22$\\ SC\_066 & 110.83753 & -73.45430 & $10.7 \pm 1.2$ & $16.5 \pm 0.5$ & $12.7 \pm 0.7$ & $18.2 \pm 1.2$ & $12.1 \pm 0.5$ & $7.0 \pm 1.5$ & $6.04 \pm 0.05$\\ SC\_067 & 110.82613 & -73.45409 & $4.1 \pm 1.2$ & $24.4 \pm 0.7$ & $21.5 \pm 0.6$ & $28.9 \pm 1.2$ & $17.8 \pm 1.1$ & $28.6 \pm 1.5$ & $6.69 \pm 0.13$\\ SC\_068 & 110.83132 & -73.45512 & $4.6 \pm 1.2$ & $6.0 \pm 0.8$ & $10.7 \pm 0.6$ & $5.3 \pm 1.1$ & $1.9 \pm 1.1$ & $0.8 \pm 1.5$ & $6.06 \pm 0.11$\\ SC\_069 & 110.82314 & -73.45583 & $42.1 \pm 1.2$ & $53.0 \pm 0.0$ & $37.3 \pm 0.1$ & $32.8 \pm 1.1$ & $35.2 \pm 1.1$ & $29.7 \pm 1.5$ & $6.50 \pm 0.02$\\ SC\_070 & 110.82385 & -73.45596 & $17.6 \pm 1.2$ & $26.8 \pm 0.8$ & $29.5 \pm 0.7$ & $28.9 \pm 1.2$ & $18.3 \pm 1.0$ & $13.9 \pm 1.5$ & $6.57 \pm 0.03$\\ SC\_071 & 110.83472 & -73.45482 & $6.5 \pm 1.2$ & $12.4 \pm 0.7$ & $11.3 \pm 0.4$ & $9.5 \pm 1.2$ & $5.9 \pm 1.1$ & $8.6 \pm 1.5$ & $6.57 \pm 0.08$\\ SC\_072 & 110.84418 & -73.45358 & $7.7 \pm 1.2$ & $7.2 \pm 0.2$ & $5.2 \pm 0.7$ & $7.2 \pm 1.2$ & $4.4 \pm 1.1$ & $5.7 \pm 1.5$ & $5.74 \pm 0.05$\\ SC\_073 & 110.84534 & -73.45344 & $8.4 \pm 1.2$ & $13.2 \pm 0.7$ & $15.0 \pm 0.7$ & $13.5 \pm 1.0$ & $6.5 \pm 1.1$ & $5.3 \pm 1.5$ & $6.59 \pm 0.06$\\ SC\_074 & 110.80991 & -73.45428 & $7.1 \pm 1.2$ & $5.8 \pm 0.8$ & $7.3 \pm 0.6$ & $6.1 \pm 1.2$ & $5.8 \pm 1.1$ & $-0.1 \pm 1.5$ & $6.47 \pm 0.07$\\ SC\_075 & 110.81175 & -73.45398 & $4.1 \pm 1.2$ & $11.9 \pm 0.4$ & $13.9 \pm 0.7$ & $10.8 \pm 1.2$ & $7.9 \pm 1.0$ & $3.8 \pm 1.5$ & $6.48 \pm 0.15$\\ SC\_076 & 110.81713 & -73.45409 & $12.9 \pm 1.2$ & $22.3 \pm 0.0$ & $17.0 \pm 0.6$ & $17.3 \pm 1.2$ & $12.1 \pm 0.7$ & $6.3 \pm 1.5$ & $5.99 \pm 0.04$\\ SC\_077 & 110.82002 & -73.45379 & $20.2 \pm 1.2$ & $21.5 \pm 0.5$ & $20.4 \pm 0.7$ & $-0.7 \pm 1.2$ & $-1.2 \pm 1.1$ & $-2.9 \pm 1.5$ & $6.92 \pm 0.03$\\ SC\_078 & 110.83525 & -73.45489 & $4.9 \pm 1.2$ & $8.4 \pm 0.6$ & $12.0 \pm 0.7$ & $12.5 \pm 1.2$ & $6.9 \pm 1.1$ & $4.3 \pm 1.4$ & $6.24 \pm 0.11$\\ SC\_079 & 110.83585 & -73.45481 & $6.6 \pm 1.2$ & $11.6 \pm 0.8$ & $11.6 \pm 0.7$ & $11.3 \pm 1.2$ & $6.2 \pm 0.8$ & $1.7 \pm 1.5$ & $6.30 \pm 0.09$\\ SC\_080 & 110.84533 & -73.45445 & $7.7 \pm 1.2$ & $14.5 \pm 0.7$ & $14.1 \pm 0.5$ & $14.0 \pm 1.1$ & $7.5 \pm 1.1$ & $3.9 \pm 1.5$ & $6.75 \pm 0.06$\\ SC\_081 & 110.84509 & -73.45449 & $4.9 \pm 1.2$ & $10.0 \pm 0.7$ & $9.8 \pm 0.4$ & $7.9 \pm 1.2$ & $4.6 \pm 0.6$ & $-2.5 \pm 1.5$ & $6.28 \pm 0.11$\\ SC\_082 & 110.82796 & -73.45581 & $4.7 \pm 1.2$ & $8.8 \pm 0.8$ & $12.6 \pm 0.6$ & $8.9 \pm 1.2$ & $8.7 \pm 1.1$ & $4.8 \pm 1.5$ & $6.55 \pm 0.12$\\ SC\_083 & 110.82717 & -73.45708 & $7.1 \pm 1.2$ & $18.8 \pm 0.6$ & $20.3 \pm 0.5$ & $19.4 \pm 0.7$ & $11.0 \pm 1.0$ & $8.0 \pm 1.5$ & $6.63 \pm 0.07$\\ SC\_084 & 110.78770 & -73.45499 & $6.6 \pm 1.2$ & $10.7 \pm 0.6$ & $9.7 \pm 0.6$ & $10.5 \pm 1.2$ & $3.8 \pm 1.1$ & $6.5 \pm 0.5$ & $6.15 \pm 0.07$\\ SC\_085 & 110.81329 & -73.45452 & $23.5 \pm 1.2$ & $13.6 \pm 0.8$ & $14.3 \pm 0.5$ & $2.1 \pm 1.2$ & $1.8 \pm 1.1$ & $-0.4 \pm 1.5$ & $6.98 \pm 0.02$\\ SC\_086 & 110.82160 & -73.45648 & $7.5 \pm 1.2$ & $8.3 \pm 0.8$ & $13.7 \pm 0.7$ & $10.9 \pm 0.8$ & $5.8 \pm 1.0$ & $7.8 \pm 1.5$ & $6.37 \pm 0.07$\\ SC\_087 & 110.81876 & -73.45635 & $10.8 \pm 0.1$ & $11.7 \pm 0.8$ & $15.3 \pm 0.7$ & $10.6 \pm 1.2$ & $5.0 \pm 1.0$ & $5.3 \pm 1.5$ & $6.75 \pm 0.00$\\ SC\_088 & 110.81799 & -73.45675 & $2.8 \pm 1.2$ & $8.5 \pm 0.8$ & $7.8 \pm 0.5$ & $9.5 \pm 0.4$ & $5.3 \pm 1.1$ & $2.3 \pm 1.5$ & $5.73 \pm 0.19$\\ SC\_089 & 110.81732 & -73.45681 & $11.4 \pm 1.2$ & $12.8 \pm 0.7$ & $13.3 \pm 0.4$ & $11.0 \pm 0.7$ & $4.5 \pm 1.1$ & $4.0 \pm 1.5$ & $6.45 \pm 0.04$\\ SC\_090 & 110.81038 & -73.45548 & $14.1 \pm 1.2$ & $17.6 \pm 0.7$ & $18.7 \pm 0.7$ & $21.8 \pm 1.2$ & $12.9 \pm 1.1$ & $12.1 \pm 1.5$ & $6.16 \pm 0.04$\\ SC\_091 & 110.78922 & -73.45409 & $14.0 \pm 0.7$ & $8.6 \pm 0.6$ & $9.6 \pm 0.4$ & $10.2 \pm 0.9$ & $4.7 \pm 1.1$ & $3.1 \pm 1.5$ & $6.02 \pm 0.02$\\ SC\_092 & 110.78970 & -73.45433 & $7.7 \pm 1.2$ & $10.8 \pm 0.5$ & $11.2 \pm 0.4$ & $8.8 \pm 1.2$ & $4.7 \pm 0.7$ & $5.5 \pm 1.5$ & $6.64 \pm 0.07$\\ SC\_093 & 110.79125 & -73.45385 & $9.8 \pm 1.2$ & $10.2 \pm 0.5$ & $10.7 \pm 0.5$ & $9.8 \pm 1.2$ & $3.5 \pm 1.1$ & $2.4 \pm 1.5$ & $6.33 \pm 0.05$\\ SC\_094 & 110.78340 & -73.45303 & $6.7 \pm 0.1$ & $8.2 \pm 0.5$ & $9.5 \pm 0.4$ & $7.4 \pm 1.2$ & $2.8 \pm 1.1$ & $2.2 \pm 1.5$ & $6.57 \pm 0.01$\\ SC\_095 & 110.78423 & -73.45257 & $1.9 \pm 1.2$ & $6.7 \pm 0.4$ & $7.7 \pm 0.4$ & $6.8 \pm 1.2$ & $2.8 \pm 1.1$ & $1.2 \pm 1.5$ & $6.41 \pm 0.24$\\ SC\_096 & 110.78713 & -73.45280 & $22.1 \pm 1.2$ & $22.9 \pm 0.6$ & $23.7 \pm 0.7$ & $21.4 \pm 1.2$ & $12.1 \pm 0.8$ & $9.5 \pm 1.5$ & $6.64 \pm 0.03$\\ SC\_097 & 110.78789 & -73.45194 & $13.2 \pm 1.2$ & $17.7 \pm 0.4$ & $18.5 \pm 0.5$ & $14.6 \pm 1.2$ & $9.5 \pm 1.1$ & $5.8 \pm 1.5$ & $6.86 \pm 0.03$\\ SC\_098 & 110.78635 & -73.45047 & $7.2 \pm 1.2$ & $14.7 \pm 0.6$ & $14.0 \pm 0.4$ & $18.4 \pm 1.2$ & $10.8 \pm 0.6$ & $9.2 \pm 1.5$ & $5.88 \pm 0.08$\\ SC\_099 & 110.78172 & -73.45070 & $10.3 \pm 1.2$ & $9.8 \pm 0.8$ & $10.8 \pm 0.6$ & $9.1 \pm 0.6$ & $4.5 \pm 1.1$ & $1.8 \pm 1.5$ & $6.35 \pm 0.04$\\ SC\_100 & 110.79601 & -73.44790 & $6.0 \pm 1.2$ & $16.9 \pm 0.7$ & $19.2 \pm 0.5$ & $18.7 \pm 1.2$ & $9.8 \pm 0.6$ & $6.0 \pm 1.5$ & $6.83 \pm 0.08$\\ SC\_101 & 110.79508 & -73.44825 & $14.7 \pm 1.2$ & $13.2 \pm 0.7$ & $14.7 \pm 0.7$ & $13.0 \pm 1.2$ & $6.2 \pm 0.8$ & $10.0 \pm 1.5$ & $6.30 \pm 0.04$\\ SC\_102 & 110.79584 & -73.44829 & $9.4 \pm 1.2$ & $16.7 \pm 0.7$ & $15.7 \pm 0.4$ & $14.3 \pm 1.2$ & $8.9 \pm 1.1$ & $7.0 \pm 1.5$ & $6.32 \pm 0.06$\\ SC\_103 & 110.80174 & -73.44977 & $13.2 \pm 1.2$ & $17.9 \pm 0.6$ & $17.8 \pm 0.5$ & $19.1 \pm 0.7$ & $9.5 \pm 1.1$ & $8.6 \pm 1.5$ & $6.07 \pm 0.05$\\ SC\_104 & 110.80198 & -73.44992 & $9.3 \pm 1.2$ & $13.8 \pm 0.4$ & $16.7 \pm 0.1$ & $14.3 \pm 1.1$ & $8.0 \pm 1.0$ & $5.7 \pm 1.5$ & $6.75 \pm 0.06$\\ SC\_105 & 110.80116 & -73.44958 & $9.9 \pm 1.2$ & $16.3 \pm 0.7$ & $16.0 \pm 0.5$ & $18.8 \pm 0.6$ & $13.8 \pm 1.1$ & $14.2 \pm 1.5$ & $6.00 \pm 0.04$\\ SC\_106 & 110.80534 & -73.44852 & $8.6 \pm 1.2$ & $11.5 \pm 0.7$ & $16.0 \pm 0.5$ & $9.4 \pm 0.5$ & $8.0 \pm 0.5$ & $6.0 \pm 1.5$ & $6.29 \pm 0.05$\\ SC\_107 & 110.80524 & -73.44870 & $8.0 \pm 1.2$ & $10.6 \pm 0.3$ & $8.2 \pm 0.7$ & $11.9 \pm 1.2$ & $5.5 \pm 0.9$ & $2.2 \pm 1.5$ & $5.75 \pm 0.06$\\ SC\_108 & 110.78716 & -73.45552 & $8.9 \pm 1.2$ & $14.5 \pm 0.5$ & $15.0 \pm 0.5$ & $15.0 \pm 1.2$ & $9.2 \pm 1.1$ & $7.6 \pm 1.5$ & $6.28 \pm 0.06$\\ SC\_109 & 110.78741 & -73.45586 & $9.9 \pm 1.2$ & $11.8 \pm 0.7$ & $15.5 \pm 0.6$ & $12.2 \pm 1.2$ & $6.4 \pm 0.5$ & $4.2 \pm 0.4$ & $6.72 \pm 0.06$\\ SC\_110 & 110.78030 & -73.45425 & $7.6 \pm 1.2$ & $13.3 \pm 0.5$ & $14.7 \pm 0.6$ & $12.7 \pm 0.4$ & $6.1 \pm 0.7$ & $3.1 \pm 1.5$ & $6.55 \pm 0.07$\\ SC\_111 & 110.78126 & -73.45340 & $5.6 \pm 1.2$ & $11.9 \pm 0.4$ & $15.3 \pm 0.7$ & $15.7 \pm 0.4$ & $9.0 \pm 1.1$ & $8.3 \pm 1.5$ & $6.76 \pm 0.08$\\ SC\_112 & 110.78087 & -73.45291 & $10.7 \pm 1.2$ & $17.9 \pm 0.5$ & $20.0 \pm 0.5$ & $20.6 \pm 1.2$ & $13.2 \pm 0.9$ & $8.3 \pm 1.5$ & $6.75 \pm 0.04$\\ SC\_113 & 110.79191 & -73.45265 & $9.3 \pm 1.2$ & $15.2 \pm 0.7$ & $14.4 \pm 0.4$ & $13.2 \pm 1.2$ & $8.5 \pm 0.2$ & $4.0 \pm 1.5$ & $6.27 \pm 0.06$\\ SC\_114 & 110.79176 & -73.45274 & $11.4 \pm 1.2$ & $13.2 \pm 0.8$ & $13.5 \pm 0.6$ & $10.8 \pm 1.2$ & $5.9 \pm 1.1$ & $5.0 \pm 1.5$ & $6.66 \pm 0.04$\\ SC\_115 & 110.79931 & -73.45132 & $7.9 \pm 1.2$ & $15.4 \pm 0.7$ & $13.7 \pm 0.4$ & $11.7 \pm 0.8$ & $6.4 \pm 1.1$ & $7.9 \pm 1.5$ & $6.65 \pm 0.07$\\ SC\_116 & 110.79995 & -73.45124 & $12.0 \pm 1.2$ & $9.8 \pm 0.8$ & $11.8 \pm 0.6$ & $10.0 \pm 1.2$ & $4.6 \pm 0.3$ & $6.4 \pm 1.5$ & $6.09 \pm 0.04$\\ SC\_117 & 110.79854 & -73.45139 & $4.4 \pm 1.2$ & $12.8 \pm 0.5$ & $10.2 \pm 0.5$ & $10.9 \pm 1.2$ & $5.4 \pm 0.6$ & $3.4 \pm 1.5$ & $6.41 \pm 0.12$\\ SC\_118 & 110.79830 & -73.45146 & $7.1 \pm 1.2$ & $8.5 \pm 0.6$ & $9.3 \pm 0.4$ & $10.9 \pm 0.3$ & $5.2 \pm 0.6$ & $0.2 \pm 1.5$ & $5.87 \pm 0.07$\\ SC\_119 & 110.79798 & -73.45131 & $7.7 \pm 1.2$ & $9.6 \pm 0.5$ & $11.5 \pm 0.5$ & $9.5 \pm 1.2$ & $5.3 \pm 1.1$ & $3.6 \pm 1.5$ & $6.25 \pm 0.07$\\ SC\_120 & 110.79658 & -73.45191 & $13.1 \pm 1.2$ & $15.0 \pm 0.7$ & $13.5 \pm 0.6$ & $11.4 \pm 1.2$ & $7.4 \pm 0.7$ & $2.6 \pm 1.5$ & $6.25 \pm 0.04$\\ SC\_121 & 110.78631 & -73.45810 & $23.0 \pm 1.2$ & $25.2 \pm 0.8$ & $24.1 \pm 0.7$ & $25.6 \pm 1.2$ & $19.6 \pm 1.1$ & $11.2 \pm 1.5$ & $6.26 \pm 0.02$\\ SC\_122 & 110.78559 & -73.45643 & $10.1 \pm 1.2$ & $14.8 \pm 0.7$ & $14.9 \pm 0.3$ & $13.9 \pm 1.2$ & $6.0 \pm 1.1$ & $5.0 \pm 1.5$ & $6.75 \pm 0.05$\\ SC\_123 & 110.78135 & -73.45553 & $8.2 \pm 1.2$ & $21.7 \pm 0.7$ & $19.7 \pm 0.7$ & $22.2 \pm 1.2$ & $14.3 \pm 0.9$ & $9.3 \pm 1.5$ & $6.86 \pm 0.07$\\ SC\_124 & 110.67039 & -73.48220 & $17.8 \pm 1.2$ & $14.8 \pm 0.7$ & $19.1 \pm 0.7$ & $26.5 \pm 1.2$ & $17.0 \pm 1.1$ & $14.8 \pm 1.5$ & $6.85 \pm 0.03$\\ SC\_125 & 110.68257 & -73.47759 & $15.3 \pm 1.2$ & $12.4 \pm 0.6$ & $16.0 \pm 0.6$ & $19.9 \pm 1.2$ & $8.7 \pm 1.1$ & $8.9 \pm 1.5$ & $6.79 \pm 0.03$\\ SC\_126 & 110.73865 & -73.47510 & $5.6 \pm 1.2$ & $14.7 \pm 0.6$ & $14.0 \pm 0.1$ & $14.0 \pm 1.2$ & $29.0 \pm 1.1$ & $20.4 \pm 1.5$ & $6.35 \pm 0.09$\\ SC\_127 & 110.70747 & -73.49255 & $21.9 \pm 1.2$ & $32.1 \pm 0.8$ & $25.4 \pm 0.5$ & $34.1 \pm 1.2$ & $32.7 \pm 1.1$ & $33.2 \pm 1.5$ & $6.95 \pm 0.02$\\ SC\_128 & 110.68775 & -73.48465 & $15.1 \pm 1.2$ & $18.6 \pm 0.8$ & $13.4 \pm 0.7$ & $18.7 \pm 0.5$ & $21.6 \pm 0.8$ & $19.9 \pm 1.5$ & $6.79 \pm 0.03$\\ SC\_129 & 110.69481 & -73.48426 & $9.5 \pm 1.2$ & $13.6 \pm 0.7$ & $20.6 \pm 0.7$ & $37.8 \pm 1.2$ & $34.1 \pm 1.0$ & $35.6 \pm 1.5$ & $6.58 \pm 0.06$\\ SC\_130 & 110.70230 & -73.48263 & $8.9 \pm 1.2$ & $12.4 \pm 0.8$ & $12.8 \pm 0.7$ & $13.5 \pm 0.8$ & $14.8 \pm 1.1$ & $17.8 \pm 1.5$ & $6.56 \pm 0.06$\\ SC\_131 & 110.69502 & -73.48136 & $15.5 \pm 1.2$ & $18.3 \pm 0.7$ & $19.4 \pm 0.7$ & $29.6 \pm 1.2$ & $28.7 \pm 1.1$ & $24.5 \pm 1.5$ & $6.80 \pm 0.03$\\ SC\_132 & 110.67261 & -73.47396 & $15.6 \pm 1.2$ & $21.2 \pm 0.8$ & $19.9 \pm 0.7$ & $29.5 \pm 1.2$ & $21.2 \pm 1.1$ & $15.9 \pm 1.5$ & $6.81 \pm 0.03$\\ SC\_133 & 110.85435 & -73.45440 & $8.1 \pm 1.2$ & $10.3 \pm 0.6$ & $10.5 \pm 0.7$ & $7.0 \pm 1.2$ & $5.1 \pm 1.1$ & $3.9 \pm 1.5$ & $6.66 \pm 0.06$\\ SC\_134 & 110.85244 & -73.45419 & $3.8 \pm 1.2$ & $13.3 \pm 0.6$ & $13.6 \pm 0.7$ & $13.0 \pm 1.2$ & $8.3 \pm 0.6$ & $3.6 \pm 1.5$ & $6.61 \pm 0.15$\\ SC\_135 & 110.85889 & -73.45459 & $4.2 \pm 1.2$ & $12.8 \pm 0.8$ & $9.7 \pm 0.6$ & $6.7 \pm 1.2$ & $3.9 \pm 0.8$ & $0.6 \pm 1.5$ & $6.37 \pm 0.10$\\ SC\_136 & 110.85791 & -73.45462 & $6.6 \pm 1.2$ & $10.5 \pm 0.7$ & $10.9 \pm 0.6$ & $8.3 \pm 0.7$ & $5.7 \pm 1.1$ & $-1.6 \pm 1.5$ & $6.43 \pm 0.08$\\ SC\_137 & 110.85896 & -73.45420 & $6.3 \pm 1.2$ & $11.6 \pm 0.6$ & $12.0 \pm 0.6$ & $10.6 \pm 0.9$ & $4.7 \pm 1.1$ & $5.1 \pm 1.5$ & $6.46 \pm 0.10$\\ SC\_138 & 110.86045 & -73.45428 & $10.7 \pm 1.2$ & $12.3 \pm 0.8$ & $13.8 \pm 0.7$ & $12.1 \pm 1.2$ & $7.3 \pm 1.1$ & $6.5 \pm 1.5$ & $6.12 \pm 0.06$\\ SC\_139 & 110.86098 & -73.45427 & $6.2 \pm 1.2$ & $5.4 \pm 0.6$ & $6.0 \pm 0.7$ & $6.1 \pm 1.2$ & $4.0 \pm 1.1$ & $-0.0 \pm 1.5$ & $6.43 \pm 0.07$\\ SC\_140 & 110.85891 & -73.45394 & $10.7 \pm 1.2$ & $9.3 \pm 0.7$ & $10.6 \pm 0.6$ & $7.4 \pm 0.6$ & $4.3 \pm 1.1$ & $1.0 \pm 1.5$ & $6.15 \pm 0.05$\\ SC\_141 & 110.82008 & -73.46628 & $21.7 \pm 1.2$ & $27.3 \pm 0.3$ & $18.4 \pm 0.7$ & $15.3 \pm 0.3$ & $9.1 \pm 0.8$ & $5.6 \pm 1.5$ & $6.17 \pm 0.03$\\ SC\_142 & 110.82292 & -73.46619 & $0.9 \pm 1.2$ & $8.3 \pm 0.8$ & $9.5 \pm 0.7$ & $6.9 \pm 1.2$ & $5.4 \pm 1.1$ & $0.4 \pm 1.5$ & $6.29 \pm 0.35$\\ SC\_143 & 110.79685 & -73.45404 & $2.9 \pm 1.2$ & $7.2 \pm 0.7$ & $8.4 \pm 0.4$ & $7.2 \pm 1.2$ & $3.7 \pm 0.3$ & $3.5 \pm 1.5$ & $6.08 \pm 0.22$\\ SC\_144 & 110.66959 & -73.47230 & $4.1 \pm 1.2$ & $15.2 \pm 0.5$ & $14.1 \pm 0.5$ & $20.5 \pm 0.5$ & $61.5 \pm 1.0$ & $76.3 \pm 1.5$ & $6.21 \pm 0.13$\\ SC\_145 & 110.81957 & -73.45875 & $11.0 \pm 1.2$ & $13.2 \pm 0.8$ & $13.6 \pm 0.4$ & $9.7 \pm 1.2$ & $6.1 \pm 0.6$ & $5.3 \pm 1.5$ & $6.64 \pm 0.05$\\ SC\_146 & 110.82110 & -73.45871 & $5.1 \pm 1.2$ & $12.6 \pm 0.5$ & $12.4 \pm 0.6$ & $9.3 \pm 1.2$ & $4.7 \pm 1.1$ & $4.1 \pm 1.5$ & $6.37 \pm 0.10$\\ SC\_147 & 110.82095 & -73.45843 & $3.9 \pm 1.2$ & $8.1 \pm 0.7$ & $7.4 \pm 0.7$ & $6.5 \pm 1.2$ & $3.2 \pm 1.1$ & $2.5 \pm 1.5$ & $6.33 \pm 0.14$\\ SC\_148 & 110.81494 & -73.45791 & $9.8 \pm 1.2$ & $6.5 \pm 0.7$ & $8.5 \pm 0.5$ & $8.7 \pm 0.5$ & $6.0 \pm 1.1$ & $4.2 \pm 1.5$ & $5.93 \pm 0.05$\\ SC\_149 & 110.81525 & -73.45789 & $4.7 \pm 1.2$ & $9.2 \pm 0.3$ & $11.1 \pm 0.5$ & $11.4 \pm 0.7$ & $6.3 \pm 1.1$ & $5.7 \pm 1.5$ & $6.22 \pm 0.09$\\ SC\_150 & 110.81370 & -73.45776 & $10.4 \pm 1.2$ & $11.2 \pm 0.6$ & $11.8 \pm 0.5$ & $11.3 \pm 0.7$ & $6.9 \pm 1.0$ & $8.0 \pm 1.5$ & $5.96 \pm 0.06$\\ SC\_151 & 110.81586 & -73.45765 & $15.9 \pm 1.2$ & $25.2 \pm 0.6$ & $24.4 \pm 0.6$ & $22.4 \pm 1.2$ & $12.1 \pm 0.6$ & $9.4 \pm 0.3$ & $6.95 \pm 0.03$\\ SC\_152 & 110.81782 & -73.45708 & $21.2 \pm 1.2$ & $19.1 \pm 0.7$ & $20.1 \pm 0.6$ & $16.1 \pm 1.2$ & $9.0 \pm 1.1$ & $5.7 \pm 1.5$ & $6.46 \pm 0.02$\\ SC\_153 & 110.82499 & -73.45730 & $9.6 \pm 1.2$ & $27.0 \pm 0.7$ & $31.1 \pm 0.5$ & $26.5 \pm 0.6$ & $14.6 \pm 1.1$ & $10.0 \pm 0.5$ & $6.85 \pm 0.05$\\ SC\_154 & 110.82503 & -73.45740 & $6.7 \pm 1.2$ & $17.6 \pm 0.3$ & $16.3 \pm 0.5$ & $16.4 \pm 1.2$ & $15.2 \pm 1.1$ & $13.1 \pm 1.5$ & $6.78 \pm 0.07$\\ SC\_155 & 110.82519 & -73.45764 & $4.6 \pm 1.2$ & $9.7 \pm 0.7$ & $10.7 \pm 0.7$ & $8.3 \pm 1.2$ & $5.0 \pm 0.7$ & $5.9 \pm 1.5$ & $6.50 \pm 0.12$\\ SC\_156 & 110.82888 & -73.45774 & $10.3 \pm 1.2$ & $16.0 \pm 0.7$ & $17.5 \pm 0.7$ & $13.6 \pm 1.2$ & $8.8 \pm 1.1$ & $3.5 \pm 1.5$ & $6.36 \pm 0.04$\\ SC\_157 & 110.88178 & -73.45305 & $5.6 \pm 1.2$ & $13.1 \pm 0.8$ & $16.2 \pm 0.5$ & $12.4 \pm 1.2$ & $8.2 \pm 1.1$ & $4.5 \pm 1.5$ & $6.63 \pm 0.09$\\ SC\_158 & 110.87357 & -73.45284 & $9.0 \pm 1.2$ & $12.3 \pm 0.8$ & $10.6 \pm 0.7$ & $10.0 \pm 1.2$ & $4.2 \pm 1.1$ & $4.9 \pm 1.5$ & $6.52 \pm 0.05$\\ SC\_159 & 110.87397 & -73.45293 & $7.5 \pm 0.1$ & $7.0 \pm 0.5$ & $7.6 \pm 0.7$ & $9.3 \pm 1.2$ & $5.0 \pm 1.0$ & $1.6 \pm 1.5$ & $5.72 \pm 0.01$\\ SC\_160 & 110.87266 & -73.45103 & $9.4 \pm 1.2$ & $7.5 \pm 0.6$ & $7.1 \pm 0.3$ & $11.3 \pm 1.2$ & $5.6 \pm 1.1$ & $2.6 \pm 1.5$ & $5.82 \pm 0.06$\\ SC\_161 & 110.87221 & -73.45115 & $5.2 \pm 1.2$ & $5.5 \pm 0.4$ & $10.9 \pm 0.5$ & $9.1 \pm 1.2$ & $9.1 \pm 1.1$ & $8.8 \pm 0.3$ & $5.97 \pm 0.09$\\ SC\_162 & 110.75752 & -73.45268 & $7.9 \pm 1.2$ & $11.5 \pm 0.8$ & $12.1 \pm 0.4$ & $10.6 \pm 1.2$ & $5.3 \pm 1.1$ & $6.3 \pm 1.5$ & $6.26 \pm 0.06$\\ SC\_163 & 110.75739 & -73.45370 & $8.8 \pm 1.2$ & $14.7 \pm 0.8$ & $11.3 \pm 0.5$ & $6.9 \pm 1.2$ & $6.8 \pm 1.1$ & $5.5 \pm 1.5$ & $6.52 \pm 0.05$\\ SC\_164 & 110.74766 & -73.45285 & $18.4 \pm 1.2$ & $22.9 \pm 0.2$ & $20.9 \pm 0.5$ & $20.5 \pm 1.2$ & $10.0 \pm 1.1$ & $8.1 \pm 1.5$ & $6.83 \pm 0.03$\\ SC\_165 & 110.74990 & -73.44874 & $21.2 \pm 1.2$ & $29.8 \pm 0.4$ & $32.3 \pm 0.5$ & $27.7 \pm 0.5$ & $17.5 \pm 0.6$ & $14.4 \pm 1.5$ & $6.68 \pm 0.02$\\ SC\_166 & 110.77654 & -73.44677 & $13.1 \pm 1.2$ & $12.5 \pm 0.7$ & $10.7 \pm 0.7$ & $8.0 \pm 0.5$ & $5.9 \pm 0.7$ & $3.1 \pm 1.5$ & $6.12 \pm 0.03$\\ SC\_167 & 110.77515 & -73.44679 & $10.0 \pm 1.2$ & $10.5 \pm 0.3$ & $7.4 \pm 0.3$ & $16.1 \pm 1.2$ & $13.3 \pm 1.1$ & $10.1 \pm 1.5$ & $5.84 \pm 0.04$\\ SC\_168 & 110.78108 & -73.44683 & $9.3 \pm 1.2$ & $8.7 \pm 0.7$ & $10.2 \pm 0.3$ & $11.1 \pm 1.2$ & $6.4 \pm 1.1$ & $7.3 \pm 0.4$ & $5.98 \pm 0.05$\\ SC\_169 & 110.80459 & -73.44832 & $10.5 \pm 1.2$ & $10.6 \pm 0.7$ & $7.2 \pm 0.6$ & $6.7 \pm 1.2$ & $5.2 \pm 0.9$ & $1.5 \pm 1.5$ & $5.90 \pm 0.05$\\ SC\_170 & 110.79816 & -73.44769 & $5.8 \pm 1.2$ & $11.9 \pm 0.5$ & $10.0 \pm 0.3$ & $10.9 \pm 1.0$ & $7.9 \pm 0.1$ & $2.7 \pm 1.5$ & $5.79 \pm 0.09$\\ SC\_171 & 110.79823 & -73.44778 & $11.9 \pm 1.2$ & $15.5 \pm 0.6$ & $12.8 \pm 0.4$ & $10.3 \pm 1.2$ & $6.8 \pm 1.1$ & $5.0 \pm 1.5$ & $6.21 \pm 0.04$\\ SC\_172 & 110.86482 & -73.45783 & $3.7 \pm 1.2$ & $13.8 \pm 0.6$ & $16.7 \pm 0.6$ & $13.5 \pm 1.2$ & $6.4 \pm 1.1$ & $3.0 \pm 1.5$ & $6.65 \pm 0.14$\\ SC\_173 & 110.86402 & -73.45812 & $6.7 \pm 1.2$ & $10.8 \pm 0.7$ & $10.2 \pm 0.5$ & $10.1 \pm 1.2$ & $6.6 \pm 0.6$ & $6.9 \pm 1.5$ & $5.77 \pm 0.09$\\ SC\_174 & 110.90066 & -73.45376 & $7.9 \pm 1.2$ & $13.4 \pm 0.7$ & $12.1 \pm 0.6$ & $10.2 \pm 1.0$ & $5.4 \pm 1.1$ & $3.4 \pm 0.4$ & $6.62 \pm 0.07$\\ SC\_175 & 110.70823 & -73.47715 & $32.5 \pm 1.2$ & $23.8 \pm 0.5$ & $17.1 \pm 0.5$ & $19.8 \pm 1.2$ & $25.9 \pm 1.1$ & $18.0 \pm 1.5$ & $7.12 \pm 0.01$\\ SC\_176 & 110.68084 & -73.47226 & $18.9 \pm 1.2$ & $24.8 \pm 0.6$ & $18.8 \pm 0.5$ & $12.3 \pm 0.6$ & $10.2 \pm 1.1$ & $7.9 \pm 1.5$ & $6.89 \pm 0.03$\\ SC\_177 & 110.64783 & -73.47067 & $5.3 \pm 1.2$ & $10.4 \pm 0.8$ & $10.6 \pm 0.6$ & $14.5 \pm 1.2$ & $12.2 \pm 1.1$ & $8.8 \pm 1.5$ & $6.33 \pm 0.09$\\ \end{longtable*} \end{center} \newpage \begin{deluxetable*}{l l l l l l} \tabletypesize{\scriptsize} \tablecaption{Stacked photometry for all of the compact star clusters. All flux densities have been corrected for Galactic extinction.\label{tab:photostack}} \tablewidth{0pt} \tablehead{ \colhead{F090W} & \colhead{F150W} & \colhead{F200W} & \colhead{F277W} & \colhead{F356W} & \colhead{F444W}\\[-0.2cm] \colhead{(nJy)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{(nJy)} & \colhead{(nJy)} } \startdata $9.79 \pm 0.12$ & $13.11 \pm 0.07$ & $13.98 \pm 0.06$ & $13.02 \pm 0.12$ & $8.16 \pm 0.11$ & $6.64 \pm 0.15$ \enddata \end{deluxetable*} \bibliography{bibli}{} \bibliographystyle{aasjournal}

Title: Detailed Evolutionary Models for Twins in Sight of New Spectral Data: AN Cam, RS Ari, and V455 Aur

Abstract: We present the evolutionary scenarios for three eclipsing twin ($q(M_2/M_1)\sim$1) binary systems using their combined spectroscopic and photometric data. Using accurate \textit{TESS} photometric data, RV measurements, and spectroscopic data enabled us to calculate fundamental parameters, such as mass and radius, better than 2 percent. The temperature of each component and metallicity of the systems have been obtained via high-resolution spectra. According to our spectral analysis, the metallicity values of AN Cam, RS Ari, and V455 Aur are \text{[M/H]}=\,0.00$\pm$0.12, 0.05$\pm$0.08, and -0.07$\pm$0.07, respectively. Using the derived metallicity for each system, initial orbital parameters and detailed evolutionary status of these three systems are calculated with high precision by using \textsc{mesa}. According to our analysis, both components of AN Cam have passed the terminal age main-sequence, the primary component of RS Ari is in the giant phase while the secondary component has passed the terminal age main-sequence, finally, both components of V455 Aur are still on the main-sequence. The current ages of the three systems AN Cam, RS Ari, and V455 Aur are 3.0, 3.3, and 1.4 Gyrs, respectively, and they will approximately start to transfer mass between components in 400, 250, and 2700 Myrs, respectively.

https://export.arxiv.org/pdf/2208.06196

\defcitealias{imbert1987}{I87} \defcitealias{imbert2002}{I02} \defcitealias{Griffin2001}{G01} \defcitealias{Griffin2013}{G13} \defcitealias{Southworth2021}{S21a} \defcitealias{Southworth2021b}{S21b} \label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} binaries: eclipsing -- techniques: spectroscopic -- stars: fundamental parameters -- stars: evolution \end{keywords} \section{Introduction} The evolution of a star mainly depends on its mass and chemical composition, where mass plays a major role in evolution. There are several ways to measure the mass of a star, which is explained substantially by \cite{Serenelli}. Masses of stars, which are in binary systems, can be calculated directly via radial velocities of components if the angular inclination of its orbit is known. For visual binaries, calculation of inclination of the orbit is relatively easy but for eclipsing binaries, one needs the light curve (LC) of the system, which also helps not only masses but also to calculate fundamental physical parameters of the components of the system such as the radius, therefore log g, temperature ratios, light contributions, etc. By combining radial velocity (RV) and LC solutions, fundamental parameters of an eclipsing binary can be deduced within 1\% uncertainties \citep[][]{Andersen,Torres}. This feature of eclipsing binaries and its results have made them the focus point in areas such as stellar formation \citep[e.g.][]{Shu}, stellar evolution \citep[e.g.][]{Baraffe,Hurley,Pietrinferni,Bressan}, stellar populations \citep[e.g.][]{Renzini,Duquennoy,Chabrier,Moe}, empirical MLR relation studies \citep[e.g.][]{Demircan,Benedict,Eker} and many more. On observational time bases, RV measurements can be obtained relatively easily compared to the LC of an eclipsing binary. Along with technological developments, ground-based robotic/automated small telescopes had took the job of observation of variables stars, which relaxed the pressure on relatively big telescopes for observation time requests \citetext{\citealp[ASAS,][]{Pojmanski}; \citealp[OGLE,][]{Udalski}}. Over the last decade, with a focus on exoplanets and needing more precise observations to find more exoplanets, ground-based photometric telescopes have been superseded by space-based photometric telescopes \citetext{\citealp[Kepler]{Borucki}; \citealp[TESS]{TESS}}. Although the main goal of space-based photometric telescopes is to find exoplanets, the way they work is basically the same for building a LC, which is detecting and measuring the brightness change of a star in a certain time interval. Also, since these telescopes are not affected by the day-night cycle of Earth, they can make uninterrupted observations, which provides valuable LC data for eclipsing binaries with long periods \citep[e.g.][]{Prsa2011,Prsa2022}. Binaries with a mass ratio bigger than 0.95 are described as twin binaries \citep[][]{Tokovinin2000}. Statistical studies on twin binaries have shown that spectral types of twin binaries are predominantly F-G-K \citep[][]{Halb2003,Simon2009}. Although there are studies to explain the reasons for forming equal components, there is still no exact solution for this twin-phenomenon \citep[][]{Bate2000,Bate2002,Lucy2006,Moe,Kounkel2019,Elbadry2019}. Lately, there have been studies that focus on discovering eclipsing twin binary systems based on photometric surveys and space-based photometric telescopes, which have increased the number of known twin-binaries extensively \citep[][]{Zhang2017,Bakis2020,Yucel2022}. \cite{Bate2019} shows that low metallicity produces low mass-ratio binaries with low-mass stellar primaries ($M_1 = 0.1 - 1.5\, \mathrm{M}_\odot$) but noted that this result has been obtained for a limited number of systems. Clearly, there is still in need for more analyzed twin binary systems to understand this phenomenon. Especially, well-determined metallicities and initial orbital conditions of these systems would greatly help scientists to understand the mystery of how twin binaries have been formed. In this respect, we have analyzed three eclipsing twin binary systems, AN Cam, RS Ari, and V455 Aur, using photometric and RV data in the literature as well as new high resolution spectral data we have obtained. Fundamental parameters of these three systems together with metallicity values have been obtained, and their evolutionary models have been calculated. The paper is structured as follows. In \S2, we present our target selection criteria and properties of the observational data used. In \S3, calculated fundamental parameters of these systems are presented. In \S4, we present the detailed evolutionary analysis for three systems. Finally, in \S5, we have discussed our overall results. \section{Targets and Data} \subsection{The Target Selection} The systems that we have analyzed were selected specifically from The 9th Catalogue of Spectroscopic Binary Orbits \citep[SB9,][]{Pourbaix2004} under four conditions, which are 1- they must be a twin binary system, 2- they must have similar total mass to see the different evolutionary stages due to their ages, 3- they must have available LC data, and 4- they must be bright enough to obtain its spectrum with our 24-inch telescope and $R=12000$ resolution spectrograph. AN Camelopardalis is a twin-binary system with a period of 20.9986 days. Its twin feature was discovered by \citet[hereafter I87]{imbert1987} from RV data. Just recently, \citet[hereafter S21a]{Southworth2021} has calculated the radius of its components by using the TESS data. With help of Gaia EDR3 optical/infrared apparent magnitude values, they calculated the effective temperature of the components. They noted that the primary star is less massive than the secondary one. There is no spectroscopic study of the system. RS Arietis is a twin binary system with a period of 8.80318 days. Its twin feature was discovered by \citet[hereafter I02]{imbert2002} from RV data. There is no prior study on this system. V455 Aurigae is a triple system. Its eclipsing nature has been revealed by \cite{Hipparcos} but its nature as a twin binary system within a three-body system has been revealed by analyzing RVs of the system, which were obtained by \cite{Griffin2001}. In that analysis, unusual changes in systemic velocity ($V_\gamma$) values were noticed, and the cause of this was expected from a third body in the system. Afterward with additional observation, \citet[hereafter G13]{Griffin2013} has obtained the third body's orbital parameters and indicates that the mass of the third body could be no less than 0.5 $\mathrm{M}_\odot$. Recently, \citet[hereafter S21b]{Southworth2021b} has obtained the radius of the components in the twin-binary system and radius of the third body by using TESS data, and temperatures of bodies in the system by using theoretical spectra with empirical calibrations. There is no spectral study of this system. \subsection{RV Data} RV data for AN Cam, RS Ari, and V455 Aur have been taken from \citetalias{imbert1987}, \citetalias{imbert2002}, and \citetalias{Griffin2013}, respectively. Details can be found in those papers, but we briefly give a summary here. \citetalias{imbert1987} observed AN Cam at Observatoire de Haute-Provence\footnote{\url{http://www.obs-hp.fr/}} with 1-meter Swiss telescope, equipped with Coravel \citep[][]{Baranne1979} instrument and managed to measure 40 and 34 RVs for the primary and secondary components, respectively. They have noted that exposure times had required 10 to 15 minutes due to the presence of two neighboring components causing the cross-correlation peaks to be shallower. \citetalias{imbert2002} observed RS Ari in years between 1978-1992, and have measured 62 RVs for both components. Besides two measurements, which had been obtained via 1.5-meters Danish telescope of ESO\footnote{\url{https://eso.org/public/teles-instr/lasilla/danish154/coravel/}}, all measurements were obtained using a 1-meter Swiss telescope, which is located at the Observatoire de Haute-Provence. \citetalias{Griffin2013} observed V455 Aur in two different observation schedules using the 36-inch telescope equipped with a photoelectric RV spectrometer \citep[][]{Griffin1967}, which is located at the Cambridge Observatory, and managed to measure a total of 200 RVs (100 for primary and 100 for secondary) for the system. \subsection{Photometric Data} The photometric data used to study three systems were produced by the Transiting Exoplanet Survey Satellite (\textit{TESS}) of NASA. Although, the main object of this satellite is finding the exoplanets, it is also a good source for eclipsing binaries \citep[][]{Prsa2022}. Details of the spacecraft and camera of TESS have been explained extensively by \cite{TESS}. Therefore, we will give a brief information about the photometric data that we have used to analyze these systems. AN Cam had been observed by the TESS telescope in three sectors, 19, 25, and 26, with long cadence (1800s) observations. We have used Sector 19, which covers observation dates between 2019/11/27 and 2019/12/24 for analysis since both eclipses are well resolved. AN Cam will also be observed in sectors 52, 53, and 59. RS Ari had been observed by the TESS telescope in four sectors, 18, 42, 43, and 44, with short cadence (120s) and long cadence (1800s) observations. We have used Sector 18, which covers observation dates between 2019/11/02 and 2019/11/27. RS Ari will also be observed in Sector 58. V455 Aur had been observed by the TESS telescope in only one sector (20), with short cadence (120s) observations. Sector 20 covers observation dates between 2019/12/24 and 2020/01/21. V455 Aur will also be observed in Sector 60. \subsection{Spectral Data} Spectroscopic observations for determining the temperature of components and metallicity of the system were carried out with a 24-inch telescope (UBT60), which is equipped with a Shelyak Instruments eShel Spectrograph (SIeS). It provides spectra between $\lambda$ 4050 and 8160 in 27 \'{e}chelle orders with a resolving power of $R \sim 12000$. More details on CCD and telescope can be found in \citet[]{Bakis2020}. Since RV data are enough to calculate spectroscopic orbits of the systems we analyzed, we observed each system in one observation night in an orbital phase which the spectral lines of both components are clearly visible. Since two of the systems in our list, AN Cam and RS Ari, are relatively faint for our instruments, 6 spectra with exposure times of 1200s have been collected and combined to reach a sufficient S/N ratio. 3 spectra with exposure times of 600s were sufficient for V455 Aur since it is a relatively bright system. The S/N ratio together with other observing information for the spectral data are given in Table\,\ref{tab:SpecObs}. The spectra were reduced, wavelength calibrated, and continuum normalized using the Image Reduction and Analysis Facility, {\textsc{iraf}}\footnote{{\sc iraf} is provided by National Optical Astronomy Observatories (NOAO) in Tucson, Arizona, USA \citep[][]{Tody}.}. \begin{table} \caption{Log for spectroscopic observations. S/N refers to 6500 \AA.} \resizebox{0.48\textwidth}{!}{\begin{tabular}{cccccc} \toprule Date & HJD & System & Mid-Phase & Exposure & S/N \\ & +2459000 & & & (s) & \\ \hline 2021/02/24 & 270.36145 & V455 Aur & 0.903 & 1800 & 107\\ 2021/03/05 & 279.23978 & RS Ari & 0.733 & 7200 & 62\\ 2021/03/29 & 303.27457 & AN Cam & 0.129 & 7200 & 68\\ \bottomrule \label{tab:SpecObs} \end{tabular}} \end{table} \section{FUNDAMENTAL PARAMETERS} \subsection{Spectroscopic Orbit and LC Modelling} Even though spectroscopic orbits of the three systems have already been resolved using RV data, simultaneous solutions of both RV and photometric data were carried out by using the Wilson-Devinney code \citep[WD,][]{1971ApJ...166..605W,1979ApJ...234.1054W,1990ApJ...356..613W,2008ApJ...672..575W,2010ApJ...723.1469W,2014ApJ...780..151W} to obtain fundamental parameters such as mass, radius, temperature ratio and light contributions of components for all three systems. Since these systems are detached, mode 2 of the code has been used. During the analysis, we fixed the temperature of the primary component, which is selected as the one that has higher temperature. Orbital parameters that have been obtained via spectroscopic orbits such as orbital period ($P$), mass ratio ($q$), eccentricity ($e$), and longitude of periastron ($w$) are also fixed during analysis. The adjusted parameters are conjunction time $T_0$, semi-major axis ($a$), systemic velocity ($V_\gamma$), the orbital inclination ($^{\circ}$), the temperature of secondary component ($T_2$), dimensionless surface potentials of both components ($\Omega_{1,2}$), and monochromatic luminosity ($L_1$). It should be noted that for V455 Aur, we have adjusted the light contribution of the third body and found it as $l_3$=0.032$\pm$0.004, which is a close value to 0.028$\pm$0.002 given by \citetalias{Southworth2021b}. Photometric and spectroscopic orbit models for the three systems are given in Figs.\ref{fig:rvlcancam}, \ref{fig:rvlcrsari}, and \ref{fig:rvlcv455aur} for AN Cam, RS Ari, and V455 Aur, respectively. In Table \ref{tab:funpar}, we present the parameters obtained as a result of the analysis. Since stellar spots are detected on the components of RS Ari, the spot features are also given in Table \ref{tab:funpar}. The systematics in the middle of eclipses for all systems are noteworthy. For AN~Cam, the TESS magnitude is $m_{TESS}=$9.01 mag \citep[][]{Stassun2019} and the RMS of the systematics is 2 millimag (mmag) which is about 10 times bigger than the precision of the TESS data ($\sigma\sim200$ ppm) for such a brightness \citep[][]{Oelkers2018}. Nevertheless, the out of eclipse data show O--C RMS of 0.5 mmag corresponding to 500 ppm. Noticing the same systematic in AN Cam LC solution, \citetalias{Southworth2021} stated that it may be due to inhomogeneities on the components' surfaces such as star spots. In order to see the magnitude of the effect of systematic on the fitting parameters of AN Cam, we added artificial noise to the TESS data with a similar amplitude of the residuals and re-analyzed the TESS data. The resulting stellar radius parameter changed as follows: $\delta R_1=$ 0.4 percent and $\delta R_2=$ 0.3 percent. For RS~Ari, the TESS magnitude is $m_{TESS}=$9.305 mag \citep[][]{Stassun2019} and the RMS of the systematics is 3 mmag which is about 15 times bigger than the precision of the TESS data ($\sigma\sim200$ ppm) for such a brightness \citep[][]{Oelkers2018}. A similar approach is applied to see the effect of systematics on the model parameters, especially the radius of components. The artificially added noise on the TESS data yielded a change in the stellar radius as follows: $\delta R_1=$ 0.3 percent and $\delta R_2=$ 0.6 percent. For V455~Aur, the TESS magnitude is $m_{TESS}=$6.814 mag \citep[][]{Stassun2019} and the RMS of the systematics in the eclipse region is 1 mmag which is close to RMS of out of eclipse residuals (0.6 mmag) and about 13 times bigger than the precision of the TESS data ($\sigma\sim80$ ppm) for such a brightness \citep[][]{Oelkers2018}. Applying similar approach yielded a change in the stellar radius as follows: $\delta R_1=$ 0.07 percent and $\delta R_2=$ 0.4 percent. We see that the changes in radius of components for three systems are within the uncertainty limits of parameters listed in Table~\ref{tab:funpar}. The logarithmic limb-darkening (LD) law was adopted during the LC solutions. In order to see if the selection of LD law has an effect on the improvement of LC solutions, especially to reduce systematics, linear and square root laws have also been tested. Since the LD coefficients for TESS passband are not available in \cite{vanhamme1993}, we adopted the closest passband of Cousins\,I. It has been seen that the selection of LD law has no significant contribution to reduce the systematics seen near the eclipses of program stars. We suspect that using LD coefficients of Cousins-I band instead of TESS passband may cause low order systematics. \subsection{Determination of Temperature and Metallicity} Although modern photometric calibrations for temperature and metallicity determination are introduced, the spectral analysis still gives direct and the most reliable results. Hence, the temperature of each component and metallicity of each system were acquired from the observed spectra fitted by synthetic spectra which are calculated using {\sc atlas9} \citep[][]{Kurucz1979, Castelli}, and {\sc spectrum} \citep[][]{Gray} codes. By using the component light contributions, the synthetic spectrum of each component has been combined to have a composite spectrum of the system. H$_{\beta}$ region was selected for temperature determination. We had built several synthetic spectra with an interval of 100 K for AN Cam and RS Ari, and an interval of 50 K for V455 Aur since its S/N is better than others, which provides smaller uncertainties. After the temperatures of components for each system have been obtained, several synthetic spectra with different metallicities, in an interval of 0.1 dex, with calculated temperatures have been produced. We have selected 5 regions for metallicity determination, including H$_{\beta}$ region. For each spectral region, metallicity values corresponding to minimum ${\chi}^{2}$ were determined. Then, we have calculated the weighted average of the metallicity ($\Bar{Z}_w$) in Eq.~\eqref{average} by adopting weights as $1/{\chi}^{2}$ for each model. In Eq.~\eqref{average}, $Z$ refers to the metallicity value for each region, while $w$ and $n$ refer to the weight and number of regions, respectively. \begin{equation} \Bar{Z}_w=\frac{\displaystyle\sum_{i=1} ^{n} Z_i w_i}{\displaystyle\sum_{i=1} ^{n} w_i} \label{average} \end{equation} The weighted standard deviation was calculated using Eq.~\eqref{deviation} where $\sigma_w$ is the weighted standard deviation, $Z_i$ is the determined metallicity for each region, $\Bar{Z}_w$ is the determined weighted metallicity, $w$ is the weight for each region, and $n$ is the number of regions. As an example, H$_\beta$ regions of the three systems are presented in Fig.\ref{fig:hbeta}. \begin{equation} \sigma_w=\sqrt{\frac{\displaystyle\sum_{i=1} ^{n} w_i \left(Z_i-\Bar{Z}_w\right)^2}{\frac{\left(n-1\right)\displaystyle\sum_{i=1} ^{n} w_i}{n}}} \label{deviation} \end{equation} \subsection{Astrophysical parameters and the distances of three systems} As explained in the previous section, combining precise RV data with accurate photometric data made us reach the fundamental parameters of the analyzed systems with a precision better than $\sim$2 percent. Calculating the temperatures of components of each system enables us to determine the spectral types and intrinsic colours of the components. By using Solar absolute visual magnitude (4.75 mag) and bolometric corrections from \cite{2020eker}, bolometric magnitudes and absolute magnitudes of the components of each system have been derived. Thus, photometric distances of each system have been carried out using the distance modulus. The calculated photometric distances of three systems are in excellent agreement with \textit{Gaia} parallax (see Table \ref{tab:funpar}), which implies that our solutions are on the mark. As the photometric data of three systems are from the TESS mission, LC model parameters such as relative radii of the components have been obtained with better than 0.1 per cent. However, this is not the case when the solutions have been made with RV data. In the case of simultaneous solutions the uncertainty of parameters are on the order of 1-3 per cent. Therefore, our error estimations of parameters (except temperature) are dominated by the uncertainty of spectroscopic orbital parameters, mainly semi-major axis. The code {\sc wd} estimates the errors based on dependence of $\chi^2$ on the model parameters in the region of the model fit. This is an underestimated approach in terms of calculating errors. However, according to Cramer-Rao theorem \citep[][]{aitken1942xv,frechet1943,albrecht2009findings}, any unbiased estimator for the parameters will deliver a covariance matrix on the parameters that is no better than this \citep[][]{Maxted2020}. Therefore, the error propagation is done as the following; minimum and maximum values of each parameter are calculated by considering the minimum and maximum values of its dependent parameters. Thus, the plus and minus uncertainties of the respective parameter are the differences of the parameter from the maximum and minimum values, respectively. Only, surface temperatures of the components and their uncertainties are determined directly from the spectroscopy by using grid of synthetic spectroscopic models. \begin{table*} \centering \caption{Results of analysis, and fundamental parameters of AN Cam, RS Ari, and V455 Aur, respectively. } \resizebox{\textwidth}{!}{\begin{tabular}{lcccccc} \toprule \multirow{2}*{Parameter} & \multicolumn{2}{c}{AN~Cam} & \multicolumn{2}{c}{RS~Ari} & \multicolumn{2}{c}{V455~Aur}\\ & Primary & Secondary & Primary & Secondary & Primary & Secondary \\ \hline \textbf{\textsc{wilson-devinney} Analysis} \\ $P$(days) & \multicolumn{2}{c}{20.99846} & \multicolumn{2}{c}{8.803172} & \multicolumn{2}{c}{3.145799} \\ $T_{0}$(HJD-2400000) & \multicolumn{2}{c}{58826.6769 $\pm$ 0.0003} & \multicolumn{2}{c}{58802.2524 $\pm$ 0.0008} & \multicolumn{2}{c}{58858.5671 $\pm$ 0.0002}\\ $K_{1,2}$(km s$^{-1}$) & 62.03 $\pm$ 0.31 & 61.01 $\pm$ 0.28 & 73.10 $\pm$ 0.47 & 72.28 $\pm$ 0.43 & 96.2066 $\pm$ 0.06 & 100.6080 $\pm$ 0.06\\ $e$ & \multicolumn{2}{c}{0.47001 $\pm$ 0.00028} & \multicolumn{2}{c}{0 (fixed)} & \multicolumn{2}{c}{0.00930 $\pm$ 0.00029} \\ $w$ ($^o$) & \multicolumn{2}{c}{194.4 $\pm$ 0.5} & \multicolumn{2}{c}{0 (fixed)} & \multicolumn{2}{c}{132.5 $\pm$ 1.9} \\ $V_{\gamma}$(km s$^{-1}$) & \multicolumn{2}{c}{-38.00 $\pm$ 0.14} & \multicolumn{2}{c}{20.81 $\pm$ 0.24} & \multicolumn{2}{c}{0 (fixed)}\\ $q (M_2/M_1)$ & \multicolumn{2}{c}{1.02 $\pm$ 0.01} & \multicolumn{2}{c}{1.01 $\pm$ 0.01} & \multicolumn{2}{c}{0.96 $\pm$ 0.01} \\ $M_{1,2}{\mathrm sin}^{3}i (M_{\odot})$ & 1.382 $\pm$ 0.023 & 1.405 $\pm$ 0.022 & 1.397 $\pm$ 0.026 & 1.412 $\pm$ 0.027 & 1.270 $\pm$ 0.002 & 1.214 $\pm$ 0.002 \\ $a {\mathrm sin}i (R_{\odot})$ & \multicolumn{2}{c}{45.058 $\pm$ 0.215} & \multicolumn{2}{c}{25.286 $\pm$ 0.161} & \multicolumn{2}{c}{12.162 $\pm$ 0.004}\\ $T_{\mathrm{eff}1}/T_{\mathrm{eff}2}$ & \multicolumn{2}{c}{1.025 $\pm$ 0.003} & \multicolumn{2}{c}{0.833 $\pm$ 0.009} & \multicolumn{2}{c}{1.012 $\pm$ 0.001} \\ $i$ ($^o$) & \multicolumn{2}{c}{88.972 $\pm$ 0.006} & \multicolumn{2}{c}{88.725 $\pm$ 0.031} & \multicolumn{2}{c}{84.971 $\pm$ 0.001} \\ $l_1/l_{\mathrm{total}}$ & \multicolumn{2}{c}{0.449 $\pm$ 0.005} & \multicolumn{2}{c}{0.461 $\pm$ 0.002} & \multicolumn{2}{c}{0.508 $\pm$ 0.001} \\ $l_3/l_{\mathrm{total}}$ & & & & & \multicolumn{2}{c}{0.032 $\pm$ 0.004} \\ Limb-darkening $X_{1,2}$ & 0.543 & 0.563 & 0.556 & 0.633 & 0.532 & 0.539\\ Limb-darkening $Y_{1,2}$ & 0.267 & 0.258 & 0.262 & 0.169 & 0.268 & 0.266\\ $\Omega_{1,2}$ & 22.370 $\pm$ 0.007 & 19.041 $\pm$ 0.009 & 11.246 $\pm$ 0.015 & 8.125 $\pm$ 0.006 & 9.711 $\pm$ 0.001 & 9.839 $\pm$ 0.001\\ $r_{1,2}$ & 0.049 $\pm$ 0.001 & 0.059 $\pm$ 0.001 & 0.095 $\pm$ 0.006 & 0.139 $\pm$ 0.003 & 0.115 $\pm$ 0.001 & 0.109 $\pm$ 0.001\\ \textbf{Spot Parameters} \\ Co-latitude ($^o$) & & & 90 $\pm$ 1 & 88 $\pm$ 1 \\ Longitude ($^o$) & & & 270 $\pm$ 1 & 260 $\pm$ 1 \\ Radius ($^o$) & & & 10 $\pm$ 1 & 19 $\pm$ 1 \\ Temperature Ratio & & & 0.74 $\pm$ 0.05 & 1.02 $\pm$ 0.02 \\ \textbf{Spectral Analysis} \\ $T_{\mathrm{eff}1,2}$ (K) & 6050 $\pm$ 100 & 5900 $\pm$ 100 & 6000 $\pm$ 100 & 5000 $\pm$ 100 & 6500 $\pm$ 50 & 6424 $\pm$ 50 \\ \text{[M/H]} & \multicolumn{2}{c}{0.00 $\pm$ 0.12} &\multicolumn{2}{c}{0.05 $\pm$ 0.08} & \multicolumn{2}{c}{-0.07 $\pm$ 0.07} \\ $v {\mathrm {sin}} i_{1,2}$ (km s$^{-1}$) & 25 $\pm$ 5 & 25 $\pm$ 5 & 20 $\pm$ 5 & 20 $\pm$ 5 & 25 $\pm$ 5 & 25 $\pm$ 5\\ \textbf{Fundamental Parameters} \\ Spectral Type (Sp) & F9.5 IV & G1.5 IV & G0 IV & K1.5 III & F4 V & F4.5 V \\ M$_{1,2}$ ($\mathrm{M}_\odot$) & 1.383 $\pm$ 0.025 & 1.406 $\pm$ 0.024 & 1.398 $\pm$ 0.028 & 1.413 $\pm$ 0.029 & 1.287 $\pm$ 0.003 & 1.231 $\pm$ 0.003\\ R$_{1,2}$ ($\mathrm{R}_\odot$) & 2.206 $\pm$ 0.059 & 2.667 $\pm$ 0.058 & 2.403 $\pm$ 0.041 & 3.531 $\pm$ 0.048 & 1.409 $\pm$ 0.013 & 1.339 $\pm$ 0.013\\ a ($\mathrm{R}_\odot$) & \multicolumn{2}{c}{45.065 $\pm$ 0.252} & \multicolumn{2}{c}{25.312$\pm$0.160} & \multicolumn{2}{c}{12.286 $\pm$ 0.007} \\ log $g_{1,2}$ (cm s$^{-2}$) & 3.892 $\pm$ 0.059 & 3.734 $\pm$ 0.055 & 3.822 $\pm$ 0.044 & 3.492 $\pm$ 0.040 & 4.250 $\pm$ 0.009 & 4.275 $\pm$ 0.010\\ log L$_{1,2}$/L$_\odot$ & 0.787 $\pm$ 0.036 & 0.891 $\pm$ 0.034 & 0.829 $\pm$ 0.044 & 0.847 $\pm$ 0.047 & 0.505 $\pm$ 0.021 & 0.440 $\pm$ 0.022\\ Combined visual magnitude (mag)\footnote{taken from SIMBAD} & \multicolumn{2}{c}{9.69 $\pm$ 0.02} &\multicolumn{2}{c}{10.02 $\pm$ 0.04} & \multicolumn{2}{c}{7.28 $\pm$ 0.01} \\ Individual visual magnitudes (mag) & 10.360 $\pm$ 0.011 & 10.137 $\pm$ 0.011 & 10.391 $\pm$ 0.040 & 10.561 $\pm$ 0.047 & 7.894 $\pm$ 0.010 & 8.191 $\pm$ 0.013\\ Combined colour index $(B-V)$(mag)\footnote{taken from SIMBAD} & \multicolumn{2}{c}{0.61 $\pm$ 0.05} & \multicolumn{2}{c}{0.82 $\pm$ 0.10} & \multicolumn{2}{c}{0.43 $\pm$ 0.02} \\ Bolometric magnitude (mag) & 2.773 $\pm$ 0.078 & 2.513 $\pm$ 0.083 & 2.678 $\pm$ 0.110 & 2.633 $\pm$ 0.118 & 3.488 $\pm$ 0.053 & 3.650 $\pm$ 0.055\\ Absolute visual magnitude (mag) & 2.766 $\pm$ 0.071 & 2.483 $\pm$ 0.083 & 2.655 $\pm$ 0.110 & 2.890 $\pm$ 0.075 & 3.417 $\pm$ 0.050 & 3.584 $\pm$ 0.055\\ Bolometric correction (mag) & 0.029 $\pm$ 0.007 & 0.007 $\pm$ 0.006 & 0.023 $\pm$ 0.016 & -0.257 $\pm$ 0.043 & 0.071 $\pm$ 0.003 & 0.066 $\pm$ 0.003\\ Computed synchronization velocities (km s$^{-1}$) & 6.2 $\pm$ 0.1 & 5.6 $\pm$ 0.1 & 13.8 $\pm$ 0.2 & 20.3 $\pm$ 0.3 & 22.7 $\pm$ 0.2 & 21.5 $\pm$ 0.2\\ Distance (pc) & \multicolumn{2}{c}{326 $\pm$ 12} & \multicolumn{2}{c}{338 $\pm$ 19} & \multicolumn{2}{c}{75 $\pm$ 4}\\ \textit{Gaia} Distance (pc) & \multicolumn{2}{c}{311 $\pm$ 1} & \multicolumn{2}{c}{337 $\pm$ 2} & \multicolumn{2}{c}{77 $\pm$ 1} \\ \bottomrule $T_{\mathrm{0}}$ refers to time of periastron passage.\\ \footnotesize{${^4}{^,}{^5}$ Taken from \citet[]{Hog}}\\ \label{tab:funpar} \end{tabular}} \end{table*} \section{Evolutionary Analysis} One of the main goals of this paper is to test our results with up-to-date theoretical evolutionary tracks. In that regard, we have used version 12115 of Modules for Experiments in Stellar Astrophysics \citep[MESA,][]{Paxton2011, Paxton2013, Paxton2015, Paxton2018, Paxton2019} to calculate the evolution of these systems. We have done that procedure in two sections, single star evolution and binary star evolution. \subsection{Single Star Evolution} Components of the systems we analyzed are still in their Roche lobes, which means that there is no mass transfer in the systems. Thus, components of the systems can be analyzed as a single star. We have calculated evolutionary tracks by using mass and metallicity, in the range of uncertainties, with a single rotating star as the component of each system. We have also calculated evolutionary tracks of several different masses with measured metallicities by using spectral data to build Zero Age Main Sequence (ZAMS) lines and Terminal Age Main Sequence (TAMS) lines with each metallicity value with uncertainty, which are represented by dashed lines and black dots, respectively. Our results are given in Fig.\ref{fig:single} as log $T_{\mathrm{eff}}$-log $L$, log $T_{\mathrm{eff}}$-log $R$/$R_\odot$, and log $T_{\mathrm{eff}}$-log$g$ planes. Continuous lines represent the primary components, and dashed lines represent the secondary components in each system, respectively. Metallicities are represented by associated colour. According to our analysis, and as can be seen in Fig.\ref{fig:single}, the primary component of AN Cam is in the phase of thin hydrogen shell burning around its helium core while the secondary one is almost finishing thick hydrogen shell burning around its helium core and will start thin hydrogen shell burning around its helium core. The primary component of RS Ari is in the phase of becoming a red giant while the secondary component of RS Ari has just started to burn thin shell hydrogen around its helium core. Finally, both components of V455 Aur are on the main-sequence and still burning hydrogen in their cores. \subsection{Binary Stars Evolution} Since the three systems are detached, a single star evolution approximation is suitable. But in need of understanding the processes during the evolution of each system, one must calculate the evolution from the start, which requires the inclusion of initial orbital parameters. \textsc{mesa} provides this opportunity with its binary module, which enables the evolution of both components in a system with relevant orbital changes with respect to given initial period, or semi-major axis, and eccentricity values. Since these systems are detached, there has been no change in masses due to mass transfer to this day. Therefore, initial masses in the systems have the same value as they had started their evolution, with only two parameters that have changed over time due to orbital evolution, which are period and eccentricity. For finding initial period and eccentricity values for each system, a method similar to \cite{Rosales}, has been used. Several model calculations were made for each system with different initial periods and initial eccentricity values with a stopping condition of the current eccentricity value of that system. When the model has reached to current eccentricity value for each system, the evolution is stopped and a $\chi^2$ calculation has been made, which includes calculated radius and temperature of the components, and period with the current value of corresponding parameters of each system. In our model calculations, for the angular momentum change, we activated the setting of magnetic braking \citep[][]{Rappaport}, for tidal synchronization, we had used "Orb$\_$period" option, which synchronizes the orbit relevant to the timescale of the orbital period. We also applied tidal circularisation, given by \cite{Hurley}, on both components of AN Cam and V455 Aur since these systems have eccentric orbits. Binary evolution calculations were made by using calculated metallicity values. \subsubsection{AN Cam} As explained in the previous section, we built evolution models that initial parameters change for the period and eccentricity between 21.50 and 21.80 days with an interval of 0.005 days and between 0.4820 and 0.4900 with an interval of 0.0002, respectively, and kept the evolution continued until eccentricity drops to 0.47001, which is the up-to-date eccentricity value of AN Cam. Finally, we selected the best model, which gives the smallest $\chi^2$ value of 0.0044, with initial orbital parameters for period and eccentricity as, 21.61 and 0.4860, respectively (given in Fig. \ref{fig:ancamchi}). After that, we made an evolution model with those initial orbital parameters and stopped the model until the mass transfer started through the Roche lobe. According to our model, the age of AN Cam is 3.00$\pm$0.15 Gyrs, and the primary component of AN Cam will start transferring mass through its Roche lobe when the primary component is in the phase of thermal pulses while the secondary component is in the phase of a red giant when the age of the system is 3.511 Gyrs. Changes in orbital parameters and radii of the components during the evolution are presented in Fig. \ref{fig:mesaancam}. Detailed evolution of both components of AN Cam with timetables are given in Table \ref{tab:ancamtime} and shown in Fig.\ref{fig:ancamhr}. It is also interesting to note what happens after the mass transfer begins in the system. Considering the fact that the more massive component of AN Cam starts mass transferring to the less massive component just after the thermal pulses begins (AGB-stage) (see Table \ref{tab:ancamtime}), there will also be a non-conservative mass loss from the donor on rates changing between $10^{-8}-10^{-5} M_\odot year^{-1}$ \citep[][]{Hofner2018}. This means that the donor will have lost about 50-70 percent of its mass during this stage not only because of the mass transfer but also due to mass loss causing it to be a white dwarf (WD) in the end of its evolution. The mass transfer and loss rates play an important role in the evolution of the system. It is, therefore, difficult to predict the fate of the system without exact knowledge of these parameters. However, we adopted average rates (i.e. \cite{Paxton2015,Rosales,Soydugan}) while calculating the binary star evolution to have an idea about the end of the system. These average rates refer to $\alpha=$0.4, $\beta=$0.1 and $\gamma=$0.1 for fraction of lost from the vicinity of the donor as fast wind, fraction of lost from the vicinity of the accretor as fast wind, and fraction of lost from the circumstellar disk, respectively. These assumptions lead to a WD-giant binary with stellar masses of $M_{\mathrm{WD}}$=0.32 $M_\odot$ and $M_{\mathrm{Giant}}$=1.82 $M_\odot$. We need to have more evolution studies of semi-detached or contact binary systems in the literature \citep[among others;][]{Budding2005, HBakis2008,HBakis2016,HBakis2021} to have a more reliable knowledge about the mass transfer and loss rates in these systems for a better estimation about the end of the evolution of detached binary systems. \begin{table*}\centering \caption{Detailed evolution of AN Cam with time stamps} \resizebox{\textwidth}{!}{\begin{tabular}{lccccccccccc} \toprule & \multirow{2}*{Mark} & \multirow{2}*{Evolutionary Status} & Age & Period & \multirow{2}*{Eccentricity} & \multicolumn{3}{c}{Primary} & \multicolumn{3}{c}{Secondary} \\ & & & (Myears) & (days) & & log $T_\mathrm{eff}$ (K) & log $L$ (L$_\odot$) & Radius (R$_\odot$) & log $T_\mathrm{eff}$ (K) & log $L$ (L$_\odot$) & Radius (R$_\odot$)\\ \hline \parbox[t]{2mm}{\multirow{7}{*}{\rotatebox[origin=c]{90}{Primary}}} & A & Zero age main sequence & 0 & 21.610 & 0.4860 & 3.824 & 0.531 & 1.380 & 3.829 & 0.563 & 1.403 \\ & B & Core contraction & 2566 & 21.354 & 0.4798 & 3.791 & 0.663 & 1.877 & 3.806 & 0.814 & 2.085 \\ & C & Terminal age main sequence & 2710 & 21.282 & 0.4780 & 3.803 & 0.786 & 2.041 & 3.797 & 0.823 & 2.188 \\ & D & Thin H shell burning & 3105 & 20.818 & 0.4659 & 3.777 & 0.833 & 2.429 & 3.797 & 0.837 & 2.230 \\ & E & Entering red giant phase & 3293 & 19.766 & 0.4355 & 3.716 & 0.674 & 2.682 & 3.692 & 1.008 & 4.400\\ & F & Circularisation of Orbit & 3359 & 14.415 & 0 & 3.703 & 0.734 & 3.050 & 3.684 & 1.222 & 5.839 \\ & G & Starting of mass transfer & 3454 & 14.338 & 0 & 3.695 & 0.900 & 3.826 & 3.646 & 1.766 & 13.099 \\ \midrule \parbox[t]{2mm}{\multirow{8}{*}{\rotatebox[origin=c]{90}{Secondary}}} & a & Zero age main sequence & 0 & 21.610 & 0.4860 & 3.824 & 0.531 & 1.380 & 3.829 & 0.563 & 1.403 \\ & b & Core contraction & 2450 & 21.389 & 0.4806 & 3.793 & 0.653 & 1.841 & 3.792 & 0.692 & 1.932 \\ & c & Terminal age main sequence & 2571 & 21.352 & 0.4797 & 3.791 & 0.664 & 1.879 & 3.805 & 0.815 & 2.090 \\ & d & Thin H shell burning & 2922 & 21.084 & 0.4729 & 3.792 & 0.814 & 2.220 & 3.778 & 0.859 & 2.489\\ & e & Entering red giant phase & 3097 & 20.829 & 0.4662& 3.778 & 0.832 & 2.418 & 3.716 & 0.690 & 2.731 \\ & f & Circularisation of Orbit & 3359 & 14.415 & 0 & 3.703 & 0.734 & 3.050 & 3.684 & 1.222 & 5.839 \\ & g & Thermal Pulses of Secondary & 3441 & 14.362 & 0 & 3.696 & 0.874 & 3.695 & 3.651 & 1.720 & 12.060 \\ & h & Starting of mass transfer & 3454 & 14.338 & 0 & 3.695 & 0.900 & 3.826 & 3.646 & 1.766 & 13.099\\ \bottomrule \end{tabular}} \label{tab:ancamtime} \end{table*} \subsubsection{RS Ari} We had used the same procedure on RS Ari, except for the fact that since there is no eccentricity in the system, there is no way to know whether the system has started its evolution with an eccentricity, if it has, when the orbit of the system has circularized, or system has started its evolution with no eccentricity. Hence, we assumed that RS Ari had started its evolution with a circular orbit; therefore, the only initial orbital parameter that would change in time is the period of the system. Therefore, we started evolution with periods between 8.87 days and 8.88 days with an interval of 0.0005 days. We have used the current period of the system as a stopping condition, which is 8.803172 days. When the models have stopped, we calculated a $\chi^2$, which was obtained by using the temperature and radius of components’ current values and values models give. Lastly, we fitted the $\chi^2$ values against the period and found the best-fitted period of the system, 8.8759386 days. In Fig.\ref{fig:rsariperiod}, fitting procedure is presented. Hence, we have used that period value as the initial period of the system and calculated the evolution model until the start of the mass transfer in the system. According to our model, the age of RS Ari is 3.23 $\pm$ 0.11 Gyrs, and the primary component of RS Ari will start transferring mass through its Roche lobe when the primary component is in the phase of a red giant, while the secondary component becomes giant when the age of the system is 3.54 Gyrs. The changes in orbital parameters and radii of components during evolution are presented in Fig.\ref{fig:mesarsari}. Detailed evolution of both components of RS Ari with timetables are given in Table \ref{tab:rsaritime} and Fig.\ref{fig:rsarihr}. Similar to AN Cam, RS Ari will become a WD-Giant binary in the late stages of its evolution. The stellar masses will be $M_{\mathrm{WD}}$=0.30 $M_\odot$ and $M_{\mathrm{Giant}}$=1.84 $M_\odot$ under the mass loss and transfer rate assumptions as taken for AN Cam. \begin{table*}\centering \caption{Detailed evolution of RS Ari with time stamps} \resizebox{\textwidth}{!}{\begin{tabular}{lcccccccccc} \toprule & \multirow{2}*{Mark} & \multirow{2}*{Evolutionary Status} & Age & Period & \multicolumn{3}{c}{Primary} & \multicolumn{3}{c}{Secondary} \\ & & & (Myears) & (days) & log $T_\mathrm{eff}$ (K) & log $L$ (L$_\odot$) & Radius (R$_\odot$) & log $T_\mathrm{eff}$ (K) & log $L$ (L$_\odot$) & Radius (R$_\odot$)\\ \hline \parbox[t]{2mm}{\multirow{6}{*}{\rotatebox[origin=c]{90}{Primary}}} & A & Zero age main sequence & 0 & 8.876 & 3.821 & 0.525 & 1.392 & 3.824 & 0.546 & 1.408\\ & B & Core contraction & 2651 & 8.838 & 3.787 & 0.653 & 1.888 & 3.806 & 0.779 & 2.000\\ & C & Terminal age main sequence & 2823 & 8.832 & 3.798 & 0.785 & 2.092 & 3.792 & 0.817 & 2.226\\ & D & Thin H shell burning & 3152 & 8.817 & 3.775 & 0.819 & 2.410 & 3.741 & 0.745 & 2.591\\ & E & Entering red giant phase & 3351 & 8.800 & 3.713 & 0.661 & 2.665 & 3.695 & 0.857 & 3.648 \\ & F & Starting of mass transfer & 3541 & 8.678 & 3.691 & 0.940 & 4.079 & 3.657 & 1.526 & 9.382 \\ \midrule \parbox[t]{2mm}{\multirow{6}{*}{\rotatebox[origin=c]{90}{Secondary}}} & a & Zero age main sequence & 0 & 8.876 & 3.821 & 0.525 & 1.392 & 3.824 & 0.546 & 1.408\\ & b & Core contraction & 2508 & 8.841 & 3.789 & 0.644 & 1.846 & 3.789 & 0.670 & 1.905 \\ & c & Terminal age main sequence & 2675 & 8.837 & 3.787 & 0.658 & 1.897 & 3.800 & 0.800 & 2.104 \\ & d & Thin H shell burning & 3028 & 8.824 & 3.786 & 0.810 & 2.271 & 3.776 & 0.838 & 2.452 \\ & e & Entering red giant phase & 3213 & 8.814 & 3.766 & 0.810 & 2.493 & 3.714 & 0.672 & 2.702 \\ & f & Starting of mass transfer & 3541 & 8.678 & 3.691 & 0.940 & 4.079 & 3.657 & 1.526 & 9.382 \\ \bottomrule \end{tabular}} \label{tab:rsaritime} \end{table*} \subsubsection{V455 Aur} Orbit of V455 Aur is almost circular, $e$=0.00930, but regarding evolutionary status, it indicates that the system has started its evolution with an eccentricity. Hence, we used the same procedure, which we used for AN Cam. We built evolution models that initial parameters change for the period and eccentricity between 4.050 and 4.350 days with an interval of 0.005 days and between 0.370 and 0.420 with an interval of 0.002, respectively, and kept the evolution continued until eccentricity drops to 0.00930, which is the up-to-date eccentricity value of V455 Aur. Finally, we selected the best model, which gives the smallest $\chi^2$ value, 0.0003678, with initial orbital parameters for period and eccentricity as, 4.195 and 0.396, respectively (given in Fig. \ref{fig:v455aurchi}). After that, we made an evolution model with those initial orbital parameters and stopped the model until the mass transfer started through the Roche lobe. According to our model, the age of V455 Aur is 1.37$\pm$0.25 Gyrs, and the primary component of V455 Aur will start transferring mass through its Roche lobe when the primary component is becoming a red giant while the secondary component is in the phase of burning thick hydrogen shell around its helium core when the age of the system is roughly at 4 Gyrs. The changes in orbital parameters and radii of the components during evolution are presented in Fig. \ref{fig:mesav455aur}. Detailed evolution of both components of V455 Aur with timetables are given in Table \ref{tab:v455time} and Fig.\ref{fig:v455aurhr}. Similar to AN Cam and RS Ari, V455 Aur will become a WD-Giant binary in the late stages of its evolution. The stellar masses will be $M_{\mathrm{WD}}$=0.26 $M_\odot$ and $M_{\mathrm{Giant}}$=1.64 $M_\odot$ under the mass loss and transfer rate assumptions as taken for AN Cam and RS Ari. \begin{table*} \centering \caption{Detailed evolution of V455 Aur with time stamps} \resizebox{\textwidth}{!}{\begin{tabular}{lccccccccccc} \toprule & \multirow{2}*{Mark} & \multirow{2}*{Evolutionary Status} & Age & Period & \multirow{2}*{Eccentricity} & \multicolumn{3}{c}{Primary} & \multicolumn{3}{c}{Secondary} \\ & & & (Myears) & (days) & & log $T_\mathrm{eff}$ (K) & log $L$ (L$_\odot$) & Radius (R$_\odot$) & log $T_\mathrm{eff}$ (K) & log $L$ (L$_\odot$) & Radius (R$_\odot$)\\ \hline \parbox[t]{2mm}{\multirow{8}{*}{\rotatebox[origin=c]{90}{Primary}}} & A & Zero age main sequence & 0 & 4.195 & 0.396 & 3.816 & 0.434 & 1.283 & 3.805 & 0.343 & 1.213 \\ & B & Circularisation of Orbit & 2479 & 2.982 & 0 & 3.800 & 0.569 & 1.613 & 3.800 & 0.479 & 1.450\\ & C & Core contraction & 2859 & 2.899 & 0 & 3.793 & 0.587 & 1.704 & 3.795 & 0.492 & 1.507\\ & D & Terminal age main sequence & 3055 & 2.849 & 0 & 3.803 & 0.696 & 1.841 & 3.792 & 0.497 & 1.539\\ & E & Thin H shell burning & 3684 & 2.543 & 0 & 3.772 & 0.770 & 2.313 & 3.794 & 0.621 & 1.762 \\ & F & Entering red giant phase & 3897 & 2.315 & 0 & 3.711 & 0.646 & 2.650 & 3.789 & 0.643 & 1.848\\ & G & Starting of mass transfer & 4002 & 2.064 & 0 & 3.694 & 0.764 & 3.284 & 3.786 & 0.657 & 1.903\\ \midrule \parbox[t]{2mm}{\multirow{5}{*}{\rotatebox[origin=c]{90}{Secondary}}} & a & Zero age main sequence & 0 & 4.195 & 0.396 & 3.816 & 0.434 & 1.283 & 3.805 & 0.343 & 1.213 \\ & b & Circularisation of Orbit & 2479 & 2.982 & 0 & 3.800 & 0.569 & 1.613 & 3.800 & 0.479 & 1.450\\ & c & Core contraction & 3287 & 2.766 & 0 & 3.795 & 0.720 & 1.961 & 3.789 & 0.510 & 1.582\\ & d & Terminal age main sequence & 3533 & 2.647 & 0 & 3.785 & 0.755 & 2.141 & 3.797 & 0.609 & 1.711 \\ & e & Starting of mass transfer & 4002 & 2.064 & 0 & 3.694 & 0.764 & 3.284 & 3.786 & 0.657 & 1.903\\ \bottomrule \end{tabular}} \label{tab:v455time} \end{table*} \subsubsection{Uncertainty of Fundamental Stellar Parameters Prior to Mass-Transfer} The orbital period affects the size of the orbit prior to mass-transfer which determines the fate of a binary system. Due to the uncertainty of each orbital parameter (orbital period, eccentricity), a deviation is expected in the initial orbital parameters used in the {\sc mesa}-code which would affect the final evolutionary parameters. A grid of evolutionary models is calculated with orbital parameters with changing values within their individual uncertainty box to see how initial orbital parameters deviate. It is seen that the deviations in the initial orbital period and orbital eccentricity are within 1 percent and 0.1 percent, respectively. These deviations show themselves in the final fundamental parameters with similar amounts. Therefore, the timestamps given in Tables~\ref{tab:ancamtime},\ref{tab:rsaritime} and \ref{tab:v455time} are stable within the uncertainty box of the orbital parameters obtained in this study. \begin{table*} \centering \caption{Comparison of results of AN Cam and V455 Aur with \citetalias{Southworth2021} and \citetalias{Southworth2021b}.} \resizebox{\textwidth}{!}{\begin{tabular}{ccccccccc} \toprule \multirow{3}*{Parameters} & \multicolumn{4}{c}{AN Cam} & \multicolumn{4}{c}{V455 Aur} \\ & \multicolumn{2}{c}{This Study} & \multicolumn{2}{c}{\citetalias{Southworth2021}} & \multicolumn{2}{c}{This study} & \multicolumn{2}{c}{\citetalias{Southworth2021b}} \\ & Primary & Secondary & Primary & Secondary & Primary & Secondary & Primary & Secondary \\ \hline Mass (M$_\odot$) & 1.383$\pm$0.025 & 1.406$\pm$0.024 & 1.380$\pm$0.021 & 1.402$\pm$0.025 & 1.287$\pm$0.003 & 1.231$\pm$0.003 & 1.2887$\pm$0.0063 & 1.2316$\pm$0.0050\\ Radius ($R_\odot$) & 2.206$\pm$0.059 & 2.667$\pm$0.058 & 2.159$\pm$0.012 & 2.646$\pm$0.014 & 1.409$\pm$0.013 & 1.339$\pm$0.013 & 1.389$\pm$0.011 & 1.318$\pm$0.014\\ a (R$_\odot$) & \multicolumn{2}{c}{45.065$\pm$0.252} & \multicolumn{2}{c}{45.05$\pm$0.24} & \multicolumn{2}{c}{12.286$\pm$0.007} & \multicolumn{2}{c}{12.295$\pm$0.017}\\ log g (cgs) & 3.892$\pm$0.059 & 3.734$\pm$0.055 & 3.9095$\pm$0.0030 & 3.7400$\pm$0.0037 & 4.250$\pm$0.009 & 4.275$\pm$0.010 & 4.2626$\pm$0.0070 & 4.2887$\pm$0.0089\\ T$_\mathrm{eff}$ (K) & 6050$\pm$100 & 5900$\pm$100 & 6050$\pm$150 & 5750$\pm$150 & 6500$\pm$50 & 6424$\pm$50 & 6500$\pm$200 & 6400$\pm$200\\ log$L$ (L$_\odot$) & 0.787$\pm$0.036 & 0.891$\pm$0.034 & 0.750$\pm$0.043 & 0.839$\pm$0.046 & 0.505$\pm$0.021 & 0.440$\pm$0.022 & 0.492$\pm$0.054 & 0.419 $\pm$ 0.055\\ \text{[M/H]} & \multicolumn{2}{c}{0.00$\pm$0.12} & \multicolumn{2}{c}{0.00$\pm$0.07} & \multicolumn{2}{c}{-0.07$\pm$0.07} & \multicolumn{2}{c}{-0.07$\pm$0.07} \\ Age (Gyr) & \multicolumn{2}{c}{3.00$\pm$0.15} & \multicolumn{2}{c}{3.3} & \multicolumn{2}{c}{1.37$\pm$0.25} & \multicolumn{2}{c}{1.8$\pm$0.2}\\ \bottomrule \label{tab:comparison} \end{tabular}} \end{table*} \section{Conclusions} Twin binary systems (q $>$ 0.95), generally, are interesting objects with their characteristic features. In this study, we have selected three twin binary systems and derived their fundamental parameters, including temperature, metallicity, and age, of three eclipsing binaries, by combining their TESS data, precise RV measurements, and new high resolution spectra. Two of the systems (AN Cam and RS Ari) are found to be evolved, which means they are post terminal age main sequence (T.A.M.S.) systems, while the other one (V455 Aur) is still a main-sequence system. More accurate spectroscopy related parameters of the systems such as metallicity and effective temperature are determined then previous studies on the same objects \citep[][]{Southworth2021, Southworth2021b}. All three systems are found to have nearly Solar metallicity within the uncertainty of measurements. Besides, the distances of these three systems we analyzed are in excellent agreement with \textit{Gaia} eDR3\citep[][]{2020gaia} distances, which emphasized our measurements are accurate. The third body in V455 Aur has not been studied in this paper except its light contribution. We obtained a similar light contribution to the one obtained by \cite{Southworth2021}, which will yield similar physical parameters of the third star. Finally, comparison of our results with \citetalias{Southworth2021} and \citetalias{Southworth2021b} for AN Cam and V455 Aur, respectively, is given in Table \ref{tab:comparison}. The primary and secondary components of AN Cam are rotating much faster than their synchronisation velocity, which is expected as the orbit of the system is still not circularized (see Table \ref{tab:ancamtime}). In other two systems, component stars of V455 Aur have synchronised rotation velocities with their orbits as expected from their evolutionary times stamps (see Table \ref{tab:v455time}) while the secondary component of RS Ari is rotating faster than its synchronisation velocity. This is interesting because RS Ari completed the orbital circularization time scale and the primary component has a synchronised rotational velocity. Calculating the synchronisation time scale using \cite{Zahn} for RS Ari yields $\sim$ 61 Myr which is much smaller than RS Ari's present age. With help of modern evolution code, \textsc{MESA}, we have calculated the evolutionary phases of these three systems including initial orbital parameters, which could shed light on, generally, understanding the properties of current semi-detached binaries. There are still very few systems, in which their evolutionary phases have been revealed. We believe that finding the initial orbital properties of twin binaries would help to understand the formation mechanism of twin binaries in detail. \section*{Acknowledgements} We are grateful to the anonymous referee for her/his valuable suggestions. This work is a part of the PhD thesis of GY. We thank to Akdeniz University Space Sciences and Technologies Department for granting us observing time with UBT60 telescope. We would like to thank Dr. A. J. Rosales for helping us in {\sc mesa} calculations. This paper includes data collected with the TESS mission, obtained from the MAST data archive at the Space Telescope Science Institute (STScI). Funding for the {\it TESS} mission is provided by the NASA Explorer Program. STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement. This research has made use of NASA’s Astrophysics Data System. \section*{Data Availability} All data are incorporated into the article. \bibliographystyle{mnras} \bibliography{reference} % \appendix \section{Selected Spectral Regions for Metallicity Analysis} \bsp % \label{lastpage}

Title: CO(J = 1-0) Observations toward the Filamentary Cloud in the Galactic Region of $153.60^{\circ} \leqslant l \leqslant 156.50^{\circ}$ and $1.85^{\circ} \leqslant b \leqslant 3.50^{\circ}$

Abstract: We present observations of $J$=1-0 transition lines of ${ }^{12} \mathrm{CO}$, ${ }^{13} \mathrm{CO}$, and $\mathrm{C}^{18} \mathrm{O}$ towards the Galactic region of $153.60^{\circ} \leqslant l \leqslant 156.50^{\circ}$ and $1.85^{\circ} \leqslant b \leqslant 3.50^{\circ}$, using the Purple Mountain Observatory (PMO) 13.7 m millimeter telescope. Based on the \tht data, one main filament and five sub-filaments are found together as a network structure in the velocity interval of $[-42.5, -30.0] \,\mathrm{km} \mathrm{\,s}^{-1}$. The kinematic distance of this molecular cloud (MC) is estimated to be $\sim4.5 \mathrm{\,kpc}$. The median length, width, excitation temperature, line mass of these filaments are $\sim49 \mathrm{\,pc}$, $\sim2.9 \mathrm{\,pc}$, $\sim8.9 \mathrm{\,K}$, and $\sim39 \,M_{\odot} \mathrm{pc}^{-1}$, respectively. The velocity structures along these filaments exhibit oscillatory patterns, which are likely caused by the fragmentation or accretion process along these filaments. The maximum accretion rate is estimated to be as high as $\sim700 \,M_{\odot} \mathrm{pc}^{-1}$. A total of $\sim162$ \tht clumps and $\sim 103$ young stellar objects (YSOs) are identified in this region. Most of the clumps are in gravitationally bound states. Three \hii regions (G154.359+2.606, SH2-211, SH2-212) are found to be located in the apexes of the filaments. Intense star forming activities are found along the entire filamentary cloud. The observed results may help us to better understand the link between filaments and massive star formation.

https://export.arxiv.org/pdf/2208.07506

\title{CO(J = 1-0) Observations toward the Filamentary Cloud in the Galactic Region of 153\fdg60 $\leqslant l \leqslant$ 156\fdg50 and 1\fdg85 $\leqslant b \leqslant$ 3\fdg50} \correspondingauthor{Weihua Guo, Xuepeng Chen} \email{whguo@pmo.ac.cn, xpchen@pmo.ac.cn} \author{Weihua Guo} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \affiliation{School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China} \author{Xuepeng Chen} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \affiliation{School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China} \author{Jiancheng Feng} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \affiliation{School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China} \author{Li Sun} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \affiliation{School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China} \author{Shiyu Zhang} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \affiliation{School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China} \author{Chen Wang} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \author{Yang Su} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \affiliation{School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China} \author{Yan Sun} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \affiliation{School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China} \author{Qingzeng Yan} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \author{Shaobo Zhang} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \author{Xin Zhou} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \author{MiaoMiao Zhang} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \author{Min Fang} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \author{Ji Yang} \affiliation{Purple Mountain Observatory and Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, 10 Yuanhua Road, Nanjing 210034, China} \affiliation{School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, China} \keywords{Molecular clouds(1072) --- Interstellar filaments(842) --- Star forming regions(1565) --- Young stellar objects(1834)} \section{Introduction} \label{sec:intro} Surveys at multi-wavelengths, e.g., infrared surveys \citep{2010A&A...518L.102A,2014prpl.conf...27A}, high-resolution HI surveys \citep{2020A&A...642A.163S}, and CO surveys \citep[][]{2021A&A...655L...2C,2021ApJS..257...51Y}, have revealed that filamentary structures are ubiquitous in cold Galactic interstellar medium (ISM). The filaments are found to have lengths ranging from $0.1\,\mathrm{pc}$ to $1\,\mathrm{kpc}$ , containing gas mass of $\sim 10^{3} - 10^{5} M_{\odot}$ \citep{2014A&A...568A..73R,2018A&A...610A..77H,2021arXiv211101057S}. As part of the ISM cycle, dense molecular filament plays a key role as it connects several important processes, such as the compression of diffuse gas, the fragmentation of cloud structures, and the formation of dense cores. For example, about $75\%$ pre- and proto-stellar cores are found to be located in filaments in the $Herschel$ Gould Belt Survey \citep[e.g.,][]{2014prpl.conf...27A,2015A&A...584A..91K}. Large-scale filaments containing dense molecular gas appear to form massive stars and clusters, e.g., Nessie \citep{2010ApJ...719L.185J,2019AAS...23312706J,2018A&A...619A.166M}, Orion \citep{1981PhDT.........3B,1999ApJ...510L..49J,2008PASJ...60..407T}, California \citep{2009ApJ...703...52L,2017A&A...606A.100L,2021ApJ...921...23G}, and NGC 6334 \citep{2013A&A...554A..42R,2018PASJ...70S..41F}, though the relevant mechanisms are still in study. \cite{2018ARA&A..56...41M} suggested an evolutionary picture for the formation of high-mass stars, which involves scenarios of a global hierarchical collapse and clump-fed accretion. A typical example is the DR21 ridge, the most massive cloud structure in the Cygnus-X region \citep[e.g.,][]{2010A&A...520A..49S}. Compared to the Taurus filament B211/B213 \citep{2013A&A...550A..38P}, it has a more spherical/elliptical geometry with a few sub-filaments connected to the hub ridge. The mass-flows transported along sub-filaments toward the hub ridge are considered to account for the mass growth of the massive clumps and cores therein \citep{2012A&A...543L...3H}. The observations toward other filamentary clouds, e.g., Monoceros R2 \citep{2019A&A...629A..81T}, G22 \citep{2018ApJ...852...12Y}, and W33 \citep{2021A&A...646A.137L}, support this scenario. In the hub ridges, with the material convergence, the mass accumulates and density increases, mini-starburst activities, with $\Sigma_{\mathrm{SFR}}> 1 \,M_{\odot} \mathrm{yr}^{-1} \mathrm{kpc}^{-2}$ and $\Sigma_{M_{\mathrm{gas}}}> 100 \,M_{\odot} \mathrm{pc}^{-2}$ \citep{2016ApJ...833...23N}, can even take place. The filaments IRDC G035.39–00.33 \citep{2011A&A...535A..76N} and W43-MM1 ridge \citep{2014A&A...570A..15L} were found to have a SFE of $\sim 15 \%$ within an area of $\sim 8 \mathrm{\,pc}^{2}$ and a SFE of $\sim 6 \%$ within an area of $\sim 8 \mathrm{\,pc}^{3}$, respectively. Thus the two regions were suggested to form starburst clusters. Investigating the multiscale kinematics will provide crucial clues to understand the relation between filamentary structures and high-mass star formation. It is necessary to search for more filamentary clouds to reveal their gas dynamics in terms of filaments fragmentation, cores accretion, high-mass star formation. The Milky Way Imaging Scroll Painting (MWISP) project\footnote{\url{http://www.radioast.nsdc.cn/mwisp.php}} is an unbiased, high sensitivity CO multi-line survey towards the northern Galactic plane \citep{2019ApJS..240....9S,2021ApJS..256...32S}, which provides us a good opportunity to study large-scale filamentary clouds \citep[e.g.,][]{2021ApJ...921...23G,2021ApJS..257...51Y}. In this work, we present the observations of the three CO isotope lines towards the Galactic region of 153\fdg6 $\leqslant l \leqslant$ 156\fdg5 and 1\fdg85 $\leqslant b \leqslant$ 3\fdg5. The observations are described in Section 2. Section 3 presents the main results including the basic physical parameters of the MC, the process of the discrimination of the filaments, the physical properties and kinematic structures of the filaments, and the star formation activities therein. Section 4 discuss these results and a summary is made in Section 5. \section{Observations and Data Reduction} \label{sec-data} As part of the MWISP, the observations were performed with the PMO 13.7 m millimeter telescope from April 2012 to February 2017. The three isotope emission lines of \tw at $115.271 \mathrm{\,GHz}$, \tht at $110.201 \mathrm{\,GHz}$, and \ei at $109.782 \mathrm{\,GHz}$ were simultaneously observed with the 3$\times$3 beams Superconducting Spectroscopic Array Receiver (SSAR) system in the sideband separation mode\citep{2012ITTST...2..593S}. The ${ }^{12} \mathrm{CO}$ was at the upper sideband $(\mathrm{USB}),$ and ${ }^{13} \mathrm{CO}$ and $\mathrm{C}^{18} \mathrm{O}$ were at the lower sideband $(\mathrm{LSB})$. Typical system temperatures were around $\sim 270\mathrm{\,K}$ for the USB and around $175\mathrm{\,K}$ for the LSB, respectively. In the position-switch On-The-Fly (OTF) mode, the telescope scanned the sky along both the longitude and latitude directions at a constant rate of $50^{\prime \prime} \mathrm{\,s}^{-1}$. The half-power beamwidth (HPBW) is approximately 52$\arcsec$ at $115.2\mathrm{\,GHz}$ and 55$\arcsec$ at $110.2\mathrm{\,GHz}$. The beam efficiencies ($\eta_{\rm MB}$) are 44$\%$ for $115.2 \mathrm{\,GHz}$ and 48$\%$ for $110.2 \mathrm{\,GHz}$, respectively. The pointing uncertainty is approximately within $5^{\prime \prime}$. The total 1GHz bandwidth is separated into 16384 channels, resulting in a resolution of 61 kHz in frequency, which is corresponding to $\sim 0.16 \mathrm{\,km}\mathrm{\,s}^{-1}$ for $^{12}$CO, $\sim 0.17 \mathrm{\,km}\mathrm{\,s}^{-1}$ for \tht and \ei in velocity, respectively. Velocities were all given with respect to the local standard of rest (LSR). The rms noises levels are $\sim 0.5 \mathrm{\,K}$ for \tw, and $\sim 0.3 \mathrm{\,K}$ for \tht and $\mathrm{C}^{18} \mathrm{O}$, respectively. More information about the telescope can be found in the telescope status report\footnote{\url{http://www.radioast.csdb.cn/zhuangtaibaogao.php}}. All data were reduced by the dedicated pipelines based on C, IDL, and Python by MWISP working group\footnote{\url{http://www.iram.fr/IRAMFR/GILDAS}} and the GILDAS/CLASS package by IRAM \citep{2005sf2a.conf..721P}. After removing bad channels and correcting the first order (linear) base line fitting, the data were regridded into the standard FITS files with a pixel size of $30^{\prime \prime} \times 30^{\prime \prime}$ (approximately half of the beam size). \section{Results}\label{sec-Result} \subsection{Filamentary Network Cloud}\label{sec-cloud} As shown in Figure \ref{fig-lv}, we construct the Longitude-Velocity (LV) diagram by summing the emission along the latitude interval of $[1\fdg85, 3\fdg5]$. The CO emission mainly appears in the velocity interval of $[-45, 5] \mathrm{\,km}\mathrm{\,s}^{-1}$. According to the analysis in \cite{2001ApJ...547..792D}, the emission in $[-15,5] \mathrm{\,km}\mathrm{\,s}^{-1}$ is generally attributed to the Local arm. \cite{2017ApJ...838...49X} partially presented the MWISP CO data analysis in this interval (towards the region of $147\fdg75 \leqslant l \leqslant 152\fdg0 , 1\fdg5 \leqslant b \leqslant 5\fdg25$). In this work, we focus on the CO emission in the velocity interval of [-42.5, -30.0]$\mathrm{\,km}\mathrm{\,s}^{-1}$. The zoom-in LV diagram shows that the emission could be further divided into eastern\footnote{All the directions mentioned in this work are in the Galactic coordinate system.} ($l \sim$[155\fdg0, 156\fdg5]) and western ($l \sim$[154\fdg0, 155\fdg0]) regions. Figure \ref{fig-thrcol} shows the velocity-integrated intensity image for the three CO lines. For $^{12}$CO and $^{13}$CO, to obtain clear maps, we use pixels with emission above $3 \,\sigma$ and at least $3$ continuous channels for integration. For C$^{18}$O, in order to retain as many valid signals as possible, we apply for the DBSCAN algorithm \citep[connectivity 1 and MinPts 4, see][for details]{2020ApJ...898...80Y} to make a mask. The parameters of cutoff threshold and the peak brightness temperature are selected as $1.5 \,\sigma$ and $>3 \,\sigma$ to filter noise in this process. Then pixels within the mask greater than $2 \,\sigma$ are used for integration. As shown in Figure \ref{fig-thrcol}, the three-color intensity image clearly shows the relative spatial distribution traced by the three isotope CO lines, respectively. In the western region, the emission of \tw and \tht outlines a network structure. Compared with the relatively diffuse emission traced by \tw, the \tht emission shows the distinct skeletons of the network structure, while the \ei emission is mainly concentrated at the main node structure. The peak intensity of the \tw emission appears at the position of (l, b) $\sim$ (154\fdg358, 2\fdg600). In the eastern region, two isolated filamentary structures are separately distributed up and down. The upper one extends about 1$\degr$ along the direction from southwest to northeast with respect to the Galactic plane by an inclination angle of about 45$\degr$. The emission in the southwestern part seems brighter than the northeastern part. The lower one is roughly parallel to the Galactic plane. Table\ref{tb-IRAS} lists \hii regions and IRAS sources associated with the MC. The association between the \hii regions and the MC is based on the consistency of the velocities of the ionized gas and the CO gas. For IRAS sources, four of them are selected from the catalog of \cite{2002ApJS..141..157Y}. The rest three are selected from the analysis of previous studies \citep{2001ApJ...555..613I,2005ApJS..161..361K}, e.g., according to correlation between the luminosities of IR and $\mathrm{H} \alpha$, the IRAS04368+5021 and IRAS04329+5045 are identified to be associated with the \hii regions of SH2-211 and SH2-212 respectively. \begin{deluxetable*}{lcccc} \tabletypesize{\small} \setlength{\tabcolsep}{0.25in} \tablecaption{associated\hii regions and IRAS sources \label{tb-IRAS}} \tablewidth{0pt} \tablehead{ \colhead{Name} & \colhead{$l$} & \colhead{$b$} & \colhead{Velocity} & \colhead{references for association} \\ & \colhead{($\degr$)} & \colhead{($\degr$)} & \colhead{($\mathrm{\,km}\mathrm{\,s}^{-1}$)} & } \startdata G154.346+2.60 & 154.346 &2.606 & $-43.60 \pm 24.00$ \tablenotemark{a} & \cite{2015ApJS..221...26A} \\ SH2-211 & 154.661 & 2.452 & $-35.22 \pm 28.50$ \tablenotemark{a} & \cite{2011ApJ...738...27B} \\ & & & $-33.00$ \tablenotemark{b} & \cite{1991PASP..103..843P} \\ SH2-212 & 155.357 & 2.609 & $-43.95 \pm 25.57 $ \tablenotemark{a}& \cite{2011ApJ...738...27B} \\ & & & $-38.4$ \tablenotemark{b}& \cite{1991PASP..103..843P} \\ & & & $-40.1 \pm 1.0$ \tablenotemark{c}& \cite{1989ApJS...71..469L} \\ IRAS04324+5102 & 154.397 & 2.548 & $-37.55$ & \cite{2002ApJS..141..157Y} \\ IRAS04324+5106 & 154.347 & 2.606 & $-36.02$ & \cite{2002ApJS..141..157Y} \\ & & & $-37.3$ \tablenotemark{d}& \cite{Pirogov1999} \\ IRAS04335+5110 & 154.414 & 2.774 & $-34.89$ & \cite{2002ApJS..141..157Y} \\ IRAS04366+5022 & 155.331 & 2.596 & $-34.00$ & \cite{2002ApJS..141..157Y} \\ IRAS04368+5021 & 155.367 & 2.607 & $\sim$ & \cite{2001ApJ...555..613I} \\ IRAS04329+5045 & 154.650 & 2.435 & $\sim$ & \cite{2001ApJ...555..613I} \\ % IRAS04329+5047 & 154.637 & 2.445 & $-37.1$ \tablenotemark{e}& \cite{1990ApJ...352..139S} \\%$-38.30$ & & & $-38.3$ \tablenotemark{d}& \cite{Pirogov1999} \\%$-38.30$ \enddata \tablenotetext{}{a: velocity of hydrogen radio recombination lines ($v_{\rm LSR,\mathrm{HRRLs}}$) $\pm$ Full width at half height (FWHM). b: $v_{\rm LSR,\mathrm{ H} \alpha}$. c: $v_{\rm LSR,\mathrm{ H109} \alpha}$. d: $v_{\rm {LSR,\mathrm{ HCN}}(J=1-0)}$. e: $v_{\rm LSR,\mathrm{ CO}}$} \end{deluxetable*} \subsection{Distance of the cloud}\label{Distance} In the second quadrant of the Milky Way, the systemic velocity ($v_{\rm LSR}$) of molecular gas increases negatively from $\sim 0 \mathrm{\,km}\mathrm{\,s}^{-1}$ to trace subsequently the Local arm, the Perseus Arm, the Outer Arm, and the Scutum-Centaurus Arm \citep{2008AJ....135.1301V, 2015ApJ...798L..27S,2019ApJ...885..131R}. \cite{2016ApJS..224....7D, 2017ApJS..229...24D} analyzed the MC structures in the region of l $\sim$ [100\fdg0, 150\fdg0], b $\sim$ [-3\fdg0, 5\fdg0], using the CO data from the MWISP survey. According to the tendency of LV map \cite[see, e.g., Figure 5 in][]{2016ApJS..224....7D}, the filamentary MC in this work is located in the Outer arm. According to previous optical observations of stars associated with the \hii regions, the distance of IRAS$04324+5106$ was suggested to be $\sim 6 \mathrm{\,kpc}$ \citep{1993ApJ...407..657C}. The distance of SH2-212 was suggested to range from $\sim 4.81 \mathrm{\,kpc}$ to $\sim 6.8 \mathrm{\,kpc}$ \citep{1979A&AS...38..197M,2011MNRAS.411.2530J,2015AJ....149..127L,2020A&A...633A..99C}. For SH2-211, the spectroscopic distance was relatively uncertain, ranging from $\sim 3 \mathrm{\,kpc}$ to $\sim 7.8 \mathrm{\,kpc}$ \citep{1984A&A...139L...5C,1990AJ.....99..846H,2015AJ....150..147F}. We try to estimate the distances using the Bayesian distance calculator provided by \cite{2019ApJ...885..131R}. We select nine pixels with strong radiation and apply single Gaussian fits to the spectral lines to derive the peak velocities. The rotation model from \cite{2019ApJ...885..131R} is then applied to each coordinate and peak velocity to get the distance. The results are listed in Table \ref{tb-distance}. As shown in Figure \ref{fig-3d-dis}, the position with strongest emission (named as ``NO.1'') is found to have a higher combined probability ($70\%$) to be associated with portion of the Outer arm at a distance of $4.51 \pm 0.49 \mathrm{\,kpc}$ than to be associated with the Perseus arm at $2.03 \pm 0.34 \mathrm{\,kpc}$. The other eight positions go through the similar situation. The average distance is $\sim 4.50 \pm 0.51 \mathrm{\,kpc}$. We therefor adopt $4.5 \mathrm{\,kpc}$ as the distance of the filamentary cloud, which is compariable with the distance estimated in previous studies (see above). \begin{deluxetable}{lllcccc} \tabletypesize{\small} \setlength{\tabcolsep}{0.30in} \tablecaption{Measured distances of the six selected positions. \label{tb-distance}} \tablewidth{0pt} \tablehead{ \colhead{NO.} & \colhead{$l$} & \colhead{$b$} & \colhead{$v_{\rm LSR}$} & \colhead{distance} & \colhead{probability} & \colhead{Arm}\\ \colhead{} & \colhead{($\degr$)} & \colhead{($\degr$)} & \colhead{(km s$^{-1}$)} & \colhead{(kpc)} & \colhead{} & \colhead{} } \startdata 1 & 154.35 & 2.58 & -35.1 & 4.50$\pm$0.51 & 0.63 & Out \\ 2 & 154.48 & 2.49 & -35.9 & 4.52$\pm$0.51 & 0.68 & Out \\ 3 & 154.43 & 2.76 & -33.8 & 4.48$\pm$0.51 & 0.55 & Out \\ 4 & 154.44 & 2.87 & -35.1 & 4.49$\pm$0.51 & 0.60 & Out \\ 5 & 154.59 & 2.64 & -35.2 & 4.50$\pm$0.51 & 0.66 & Out \\ 6 & 154.84 & 2.43 & -37.2 & 4.53$\pm$0.51 & 0.74 & Out \\ 7 & 154.78 & 2.48 & -35.5 & 4.51$\pm$0.51 & 0.69 & Out \\ 8 & 154.77 & 2.81 & -36.2 & 4.51$\pm$0.51 & 0.68 & Out \\ 9 & 155.35 & 2.59 & -36.6 & 4.50$\pm$0.53 & 0.70 & Out \\ \enddata \end{deluxetable} \subsection{Properties of the cloud}\label{Properties} Assuming \tw is optically thick and the background temperature (${T}_{\rm bg})$ is $2.73 \mathrm{\,K}$, the excitation temperature ($T_{\mathrm{ex}}$) could be calculated from the peak value of the main beam temperature of \tw ($T_{\rm mb, peak,^{12}CO}$) following the method of \citet{1991ApJ...374..540G}. The formula is simplified as \begin{equation} T_{\rm ex}=5.532 \left[{\rm log}(1+\frac{5.53}{T_{\rm mb,peak,^{12}CO}+0.819})\right]^{-1}\,({\rm K}). \label{eq:tex} \end{equation} The distribution of excitation temperature is shown in Figure \ref{fig-Tex}. The value ranges from $3.1$ to $26.4 \mathrm{\,K}$. The position of the peak value is spatially coincident with the \hii region G154.346+02.606. In the assumption of local thermodynamic equilibrium (LET) and equal $T_{\mathrm{ex}}$ of the isotopologue pair, the $^{13}$CO column density ($N_{\rm ^{13}CO}$) can be estimated by \begin{equation} N_{\rm ^{13}CO} = 2.42 \times 10^{14} \cdot \frac{\int T_{\rm mb, ^{13}CO} dv}{1-{\rm exp}(-5.29/T_{\rm ex})}\cdot \frac{\tau_{13}}{1-e^{-\tau_{13}}},\mathrm{where} \label{N13} \end{equation} \begin{equation} \tau_{13}=-\ln \left[1-\frac{T_{\mathrm{mb}, \text { peak },{ }^{13} \mathrm{CO}}}{5.29}\left(\left[e^{5.29 / T_{\mathrm{ex}}}-1\right]^{-1}-0.164\right)^{-1}\right] \end{equation} As emission of the molecular cloud primarily occurs in regions with excitation temperatures greater than 5K, we set the excitation temperature that below 5K to 5K when estimating optical depth of \tht($\tau_{13}$). We get that the range of $\tau_{13}$ is 0.08 - 1.98. The H$_2$ column density could be derived by multiplying the $^{13}$CO column density with the ratio of $N_{\rm H_2}/N_{\rm ^{13}CO}$ $\sim$ 7$\times$10$^{5}$ \citep{1982ApJ...262..590F}. For \ei is always thin, we use a similar but simpler method under the same assumption to derive the \ei column density: \begin{equation} N_{\rm C^{18}O} = 2.24 \times 10^{14} \cdot \frac{(1+0.88/T_{\rm ex})\int T_{\rm mb, C^{18}O} dv}{1-{\rm exp}(-5.27/T_{\rm ex})}. \label{N18} \end{equation} The H$_2$ column density could be derived by a conversion ratio of $N_{\rm H_2}/N_{\rm C^{18}O}$ $\sim$ 7$\times$10$^{6}$ \citep{1995A&A...294..835C}. The H$_2$ column density can also be estimated by multiplying the \tw integrated intensity with the CO-to-H$_2$ conversion factor $X \sim$ 1.8$\times$10$^{20}$\,cm$^{-2}$\,K$^{-1}$\,km$^{-1}$\,s \citep{2001ApJ...547..792D}: \begin{equation} N_{\rm H_2} = X \int T_{\rm mb, ^{12}CO} dv. \label{N12} \end{equation} Figure \ref{fig-columndensity} shows the distributions of the derived H$_2$ column density. The volume density is obtained by dividing the column density by an equivalent radius, where the equivalent radius can be obtained if the emission area is assumed to be circle in shape. In the estimation process of the mass and volume density, a mean molecular weight with the value of 2.83 is adopted \citep{2008A&A...487..993K}. The projected emission area, column density, volume density and mass are listed in Table\ref{tb-cloud}. The emission area of \tw ($2347$ arcmin$^{2}$) is much larger than that of \tht ($506$ arcmin$^{2}$) and \ei ($49$ arcmin$^{2}$). The low-density molecular gas traced by \tw occupies most of the total mass. In the following section \ref{filaments}, we focus on the \tht emission as it better traces the filamentary skeletons. \begin{deluxetable}{ccccccc} \tabletypesize{\small} \setlength{\tabcolsep}{0.15in} \tablecaption{Properties of the cloud \label{tb-cloud}} \tablewidth{0pt} \tablehead{ \colhead{Velocity interval}&\colhead{Distance}&\colhead{Molecule Tracer} & \colhead{Area} & \colhead{$N(\rm H_{2})$\tablenotemark{a}} & \colhead{$n(\rm H_{2})$\tablenotemark{a}} & \colhead{$M(\rm H_{2})$\tablenotemark{b}}\\ \colhead{(km s$^{-1}$)}&\colhead{(kpc)}&\colhead{}& \colhead{(arcmin$^{2}$)} & \colhead{($10^{21}$cm$^{-2}$)} & \colhead{(cm$^{-3}$)} & \colhead{($10^{4}M_{\sun}$)} } \colnumbers \startdata {} &{ }&\tw & 2347 & 2.85 & 25 & 11.7 \\ {[-42.5, -30.0]} &4.5 &\tht & 506 & 3.53 & 68 & 2.96 \\ { }&{ }&\ei & 49 & 1.87 & 110 & 0.42 \\ \enddata \tablecomments{a, mean value with weight of integrated intensity. b, the estimated values at the distance of 4.5 kpc.} \end{deluxetable} \subsection{Filaments}\label{filaments} Generally, a filament is characterized by an elongated structure with an aspect ratio greater than $\sim 3-5$ \citep{2014prpl.conf...27A}, and is significantly overdense with respect to its surroundings. The identification of filaments in this work is based on two aspects: the characteristic of morphology and the velocity coherence. As shown in Figure \ref{fig-cubemoment}, through visual inspecting the channel maps, six skeletons (F1 $\sim$ F6) are carefully depicted and emphasized by different colored dashed lines. To avoid arbitrary judgment, we apply for the Discrete Persistent Structures Extractor (DisPerSE) \citep{2011MNRAS.414..350S} to the H$_{2}$ column density map derived from \tht to further identify the persistent filamentary structures. In order to not only guarantee a clear skeleton construction, but also retain the information of persistence in morphology as far as possible, the persistence threshold is set to be $3.8\times 10^{20}$\,cm$^{-2}$ ($\sim 4 \,\sigma$), and the trimBelow threshold is set to be $9.9\times 10^{20}$\,cm$^{-2}$ ($\sim 10 \,\sigma$). The results of a set of 1-pixel wide curves are marked out by black solid lines in Figure \ref{fig-disperse}. As shown in Figure \ref{fig-disperse}, the skeletons obtained by the two methods are in good agreement with each other, especially for F1, F5, and F6. The DisPerSE identification only relies on column density and is not sensitive enough to the velocity coherence. On the other hand, visual inspection has considered the information of velocity gradients instead of connecting the filaments randomly in spatial distribution. The mean column density profile of each filament is constructed from radial cuts using a similar procedure as described in \cite{2011A&A...529L...6A} and \cite{2013A&A...550A..38P}. The width (deconvolved FWHM) of each filament is derived by applying for Gaussian fitting to the \tht column density profile (see Figure \ref{fig-profile}). The average spectra of the filaments are extracted to get average $v_{\mathrm{LSR}}$ by applying Gaussian fits to \tht (see Figure \ref{fig-fil-spectrum}). Adopting a distance of $4.5 \mathrm{\,kpc}$, the lengths of these filaments range from $32$ to $90 \mathrm{\,pc}$. The widths range from $1.3$ to $3.9 \mathrm{\,pc}$. We extract each filament along the visual inspection skeleton within the corresponding width and estimate the mean $T_{\mathrm{ex}}$, the mean column density, and the total mass using the same method as in the calculations of the cloud physical parameters. The average line mass ($M_{\text {line}}$) is measured by dividing the mass by length. All the results are listed in Table \ref{tb-filaments}. \begin{deluxetable}{cccccccc} \tabletypesize{\small} \setlength{\tabcolsep}{0.15in} \tablecaption{Properties of filaments \label{tb-filaments}} \tablewidth{0pt} \tablehead{ \colhead{Name} & \colhead{Length} & \colhead{Width} & \colhead{$v_{\mathrm{LSR}}$} & \colhead{Mass} & \colhead{$ T_\mathrm{ex} $} & \colhead{$ N_\mathrm{H_2} $} & \colhead{$M_\mathrm{line}$} \\ & \colhead{(pc)} & \colhead{(pc)} & \colhead{kms$^{-1}$}& \colhead{(\msun)} & \colhead{(K)} & \colhead{($ 10^{21} $ cm$^{-2}$)} & \colhead{(\msun\,pc$^{-1}$)}} \colnumbers \startdata F1 & 51 & 3.2 & -35.7 & 13650 & 10.0 & 5.74 & 248 \\ F2 & 32 & 3.3 & -36.4 & 1725 & 9.1 & 1.47 & 49 \\ F3 & 75 & 2.6 & -36.7 & 2245 & 8.6 & 1.32 & 28 \\ F4 & 38 & 3.9 & -38.6 & 4045 & 9.6 & 3.17 & 98 \\ F5 & 90 & 1.3 & -36.5 & 2343 & 6.7 & 1.18 & 24 \\ F6 & 47 & 2.5 & -35.6 & 227 & 6.6 & 0.41 & 4 \\ \enddata \end{deluxetable} Figure \ref{fig-m11} shows the velocity field of the observed region. In morphology, F1 $\sim$ F4 mesh together in the \tw image, while they are isolated relatively in the \tht image. F1 displays a V-shape opening to east. F2, F3 and F4 are located to the east of F1 and all the four filaments arrange in the order from Northwest to Southeast. F1 connects with F4 at the cross-A and connects with F2 at the cross-B. F3 connects with F2 at the cross-C. F5 and F6 are separated from the complex network and continue to extend eastward. Compared to F1, F2 $\sim$ F6 have lower integrated intensities and similar aspect ratios. The three \hii regions are located at the apexes of the filaments, e.g., G154.346+2.606 located within F1, SH2-212 located within F5. The seven IRAS sources are located within the filaments. Five of them are associated with \hii regions. For F1$-$F4, the average $v_{\rm LSR}$ changes from redshifted to blueshifted in a direction from northwest to southeast. The velocity difference is up to $\sim 10 \mathrm{\,km}\mathrm{\,s}^{-1}$. Correspondingly, as shown in Figure \ref{fig-fil-spectrum} and Table\ref{tb-filaments}, the line center velocities of the four filament change form from $-35.7 \mathrm{\,km}\mathrm{\,s}^{-1}$ to $-38.6 \mathrm{\,km}\mathrm{\,s}^{-1}$ by a step of $\sim 0.5 \mathrm{\,km}\mathrm{\,s}^{-1}$. In order to better analysis kinematic properties of the six filaments, we extend their paths appropriately based on velocity consistency of \tw, e.g., connecting F4 with F1 and extending the ends of F2, F3, F5 and F6 northward. As shown in Figure \ref{fig-pvfit}, we extract five PV slices (widths $\sim 2$ pixels) along these extended paths from south end to north end. The filaments show large-scale kinematic oscillation pattern, e.g., F4+F1,F3, F5. We perform a linear fitting to velocity difference to obtain velocity gradient ($\bigtriangledown v_{\shortparallel}$) when the velocity difference is greater than $0.5 \mathrm{\,km}\mathrm{\,s}^{-1}$ (3 times velocity resolution) over a length of about 4 beam widths ($\sim 4.5 \mathrm{\,pc}$). For example, around the G154.346+2.606, the velocity difference is $\sim 1.34 \mathrm{\,km}\mathrm{\,s}^{-1}$ along a length of $\sim$ 0\fdg15 in the south side and $\sim 1.27 \mathrm{\,km}\mathrm{\,s}^{-1}$ along a length of $\sim$ 0\fdg13 length in the north side. The fitted velocity gradients are $\sim 1.27 \mathrm{\,km}\mathrm{\,s}^{-1} \mathrm{\,pc}^{-1}$ and $\sim 0.39 \mathrm{\,km}\mathrm{\,s}^{-1} \mathrm{\,pc}^{-1}$, respectively. The feature that velocity gradients with opposite directions existing between the IRAS sources, or \hii regions or intersection points is common along these filaments, which implies ongoing mass-flow process around the star forming zones (see discussion below). The results including the center coordinates and half-length of the segments, the length of the segments, the velocity differences, and the $\bigtriangledown v_{\shortparallel}$ are listed in Table \ref{tb-segment}. In order to investigate the periodicity of the six filaments, we construct the 1-D second-order structure functions (SF) of velocity and column density. More specifically, we extract slices \footnote{The width of slice is 8 pixels for F1 and 5 pixels for F2 - F6.} along the splines and derive the average values as well as the standard deviations from the intensity-weighted mean velocities of ${ }^{12} \mathrm{CO}$ and the column densities estimated from ${ }^{13} \mathrm{CO}$. The SF is defined as $S_{p}(\ell)=\left\langle|x(r)-x(r+\ell)|^{p}\right\rangle$, where $x(r)$ is the velocity or density at position, $r$, $\ell$ is the spatial displacement from $r$, $p$ is the order\citep{2015ARA&A..53..583H,2020NatAs...4.1064H}. The maximum lag to compute the SF is set to be the half of the total length for each filament, as the method is only valid in this range \citep{2020NatAs...4.1064H}. We present the results in Figure\ref{fig-structurefun}. We do not observe significant dip in the SF of F2 and F4, indicating that no evidence for periodicity is found in the data of these two filament; we visually identify dips in F1, F3, F5 and F6 for the column densities profile (their minimum locations are [20,30,11,8]pc), and in F3, F5 and F6 for the velocity profile (with minimum locations of [25,20,13]pc).The minimum potential characteristic lengths as revealed by these dips are approximately 8-30 pc, which are larger but still comparable with the average length of the segments($\lesssim 10$ pc). The flat slopes of the density structure functions imply that the \tht's signal-to-noise ratio is not high enough, which makes the periodicity less evident. \begin{deluxetable}{cccccccc} \tabletypesize{\small} \setlength{\tabcolsep}{0.11in} \tablecaption{Properties of the segments along the major filament \label{tb-segment}} \tablewidth{0pt} \tablehead{\colhead{Name}& \colhead{($l,b, \mathrm{half-length}$)} & \colhead{segment}& \colhead{Length} & \colhead{Velocity difference} & \colhead{$\bigtriangledown v_{\shortparallel}$} & \colhead{Mass} & \colhead{$\dot{M}$\tablenotemark{a}}\\ \colhead{}&\colhead{(\degr, \degr,\degr)}&\colhead{(\degr, \degr)}&\colhead{($\mathrm{pc}$)} &\colhead{($\mathrm{\,km}\mathrm{\,s}^{-1}$)} & \colhead{$\mathrm{\,km} \mathrm{\,s}^{-1} \mathrm{\,pc}^{-1}$} & \colhead{($M_{\odot}$)} & \colhead{($M_{\odot} \mathrm{Myr}^{-1}$)} } \startdata F4+F1 & (154.76, 2.43, 0.090 )& (0.15 , 0.33 ) & 14.1 & 0.64 & 0.05 & 2261 & 115 \\ & (154.53, 2.48, 0.060 )& (0.43 , 0.55 ) & 9.4 & 2.56 & 0.31 & 842 & 262 \\ & (154.44, 2.54, 0.075 )& (0.53 , 0.68 ) & 11.7 & 0.39 & 0.04 & 2920 & 114 \\ & (154.37, 2.67, 0.065 )& (0.75 , 0.88 ) & 10.2 & 1.27 & 0.14 & 5149 & 720 \\ & (154.43, 2.86, 0.075 )& (0.95 , 1.10 ) & 11.7 & 1.33 & 0.13 & 1499 & 189 \\ & (154.30, 3.31, 0.050 )& (1.48 , 1.58 ) & 7.8 & 0.77 & 0.11 & 337 & 39 \\ \hline F2 & (154.47, 2.72, 0.070 )& (0.10 , 0.24 ) & 11.0 & 2.13 & 0.22 & 2105 & 462 \\ & (154.52, 2.66, 0.035 )& (0.22 , 0.29 ) & 5.5 & 1.39 & 0.29 & 322 & 95 \\ \hline F3 & (154.54, 2.68, 0.040 )& (0.04 , 0.12 ) & 6.3 & 0.71 & 0.13 & 226 & 30 \\ & (154.57, 2.58, 0.055 )& (0.13 , 0.24 ) & 8.6 & 1.34 & 0.18 & 355 & 65 \\ & (154.75, 2.84, 0.040 )& (0.65 , 0.73 ) & 6.3 & 1.71 & 0.31 & 802 & 246 \\ & (154.72, 2.91, 0.035 )& (0.74 , 0.81 ) & 5.5 & 1.28 & 0.27 & 295 & 81 \\ & (154.79, 3.07, 0.075 )& (0.90 , 1.05 ) & 11.7 & 2.53 & 0.24 & 384 & 92 \\ \hline F5 & (155.28, 2.61, 0.050 )& (0.35 , 0.45 ) & 7.8 & 2.22 & 0.32 & 390 & 123 \\ & (155.40, 2.66, 0.050 )& (0.50 , 0.60 ) & 7.8 & 1.42 & 0.20 & 756 & 152 \\ & (155.83, 2.95, 0.050 )& (1.10 , 1.20 ) & 7.8 & 1.41 & 0.20 & 284 & 57 \\ & (155.99, 2.96, 0.050 )& (1.30 , 1.40 ) & 7.8 & 2.90 & 0.41 & 303 & 125 \\ \hline F6 & (155.35, 2.36, 0.035 )& (0.18 , 0.25 ) & 5.5 & 3.12 & 0.66 & 233 & 155 \\ \enddata \tablenotetext{a}{The mass-flow rates ($\dot{M}$) is estimated using a simple cylindrical model from \cite{2013ApJ...766..115K}, $\dot{M} = \frac{\bigtriangledown v_{\shortparallel}M_{\rm gas}}{tan(\alpha)}$, where $\alpha$ is the angle of the inclination to the plane of the sky and $M_{\rm gas}$ is the mass of the filament segment, which is estimated from ${ }^{13} \mathrm{CO}$ in this work.} \end{deluxetable} \subsection{Clumps}\label{clumps} The general models of infinite self-gravitating fluid cylinder \citep{1953ApJ...118..116C,1964ApJ...140.1056O,2010ApJ...719L.185J} suggest that the entire filament will collapse toward the short axis and then fragment into clumps or cores when the $M_{\text {line }}$ of an isothermal filament exceeds the critical value for equilibrium. To better understand the evolutionary phases of the filaments and star formation therein, we apply for the Gaussclumps algorithm in GILDAS \citep{1990ApJ...356..513S} to the \tht datacube to search for molecular clumps in the velocity interval of [-42.5, -30.0] $\mathrm{\,km} \mathrm{s}^{-1}$. The threshold is set to be $5\,\sigma$ to avoid false clumps. The rest control parameters $s_{0}, s_{a}, s_{c}, w$ are set to be 1, 1, 1, 2, respectively, which are suggested by \cite{1998A&A...329..249K}. After removing the clumps with a short axis less than the spatial resolution ($50^{\prime \prime}$) and those located on the edge of the observed region, a total of $162$ clumps are identified. The Gaussclumps fitting procedure gives the information of the clumps including the position in the Galactic coordinate system, angular sizes of the major ($\Theta_{\mathrm{maj}}$) and minor axis ($\Theta_{\mathrm{min}}$), $v_{\mathrm{LSR}}$, peak temperature ($T_{\rm peak}$) and spectral FWHM ($\Delta V$). Using the calculation methods in \cite{2016A&A...588A.104G}, the physical parameters including the radii after the beam deconvolution ($R_{\mathrm{eff}}$), excitation temperature ($T_{\rm ex}$), optical depth ($\tau$), column density ($N_{\mathrm{H}_{2}}$), volume density ($n_{\mathrm{H}_{2}}$), surface density ($\Sigma_{c}$), LTE mass ($M_{\text {LTE }}$), virial mass ($M_{\mathrm{vir}}$), and virial parameter ($\alpha_{\mathrm{vir}}=M_{\mathrm{vir}} / M_{\mathrm{LTE}}$) of these clumps are derived. The results are tabulated in Table\ref{tb-clumps}. The effective radius after deconvolution of the clumps range from $0.3 \mathrm{\,pc}$ to $1.9 \mathrm{\,pc}$ with a median value of $0.8 \mathrm{\,pc}$. The masses under local equilibrium range from $14 \,M_{\odot}$ to $2416 \,M_{\odot}$ with a median value of $78 \,M_{\odot}$. The excitation temperatures range from $4.6 \mathrm{\,K}$ to $22.8 \mathrm{\,K}$ with a median value of $9.5 \mathrm{\,K}$. The volume densities range from $70 \mathrm{\,cm}^{-3}$ to $6554 \mathrm{\,cm}^{-3}$ with a median value of $590 \mathrm{\,cm}^{-3}$. The virial parameters span a range of $0.33 \sim 2.96$ with a median value of $1.15$. As shown in Figure \ref{fig-dv}, we calculate the thermal and non-thermal velocity dispersions of the $162$ clumps following the methods of \cite{2017ApJ...838...49X}. The thermal velocity dispersion of each species is given by $\sigma_{\rm T}(\mu_{\rm obs})=\sqrt{\frac{k_{\rm B}T_\mathrm{kin}}{\mu_{\rm obs}m_{\rm H}}}$, where $k_{\mathrm{B}}$ is the Boltzmann constant, $\mu_{\rm obs}$ is atomic weight of the observed molecule ($\mu_{\rm obs}=29$ for \tht), $T_{\text {kin }}$ is the kinetic temperature that equals the excitation temperature and $m_{\mathrm{H}}$ is the mass of hydrogen atom. The non-thermal velocity dispersion is estimated by subtracting the thermal velocity dispersion from the measured linewidth, $\sigma_{\rm NT}=\sqrt{{\sigma_{\rm obs}}^{2}-{\sigma_{\rm T}}^{2}(\mu_{\rm obs})}$, where $\sigma_{\rm obs}=\Delta V /\sqrt{8\ln2}$, and $\Delta V$ is the measured FWHM from the Gaussclumps procedure. The three dimensional velocity dispersion ($\sigma_{\mathrm{3 D}}$) is estimated as $\sigma_{\mathrm{3 D}}=\sqrt{3} \sigma_{\mathrm{obs}}$. \subsection{Young Stellar Objects}\label{YSO} Using data from 2MASS \citep[Two Micron All Sky Survey,][]{2006AJ....131.1163S} and WISE \citep[Wide-field Infrared Survey Explorer,][]{2010AJ....140.1868W} and a categorization scheme supplied by \cite{2014ApJ...791..131K}, we investigate the disk-bearing candidate YSOs in this region. To complement our catalogue, we use the YSO catalogues from \cite{2016MNRAS.458.3479M,2019MNRAS.487.2522M} who used machine learning classifiers to search for YSOs through WISE photometry and Gaia DR2 \citep[the second $Gaia$ Data Release,][]{2018A&A...616A...1G}. The disk-bearing YSOs can be categorized into three classes based on the \cite{1994ApJ...434..614G} concept: Class I ($\mathrm{ALPHA} \geqslant 0.3$), flat-spectrum ($0.3 \geqslant \mathrm{ALPHA} \geqslant-0.3$), and Class II ($-0.3 \geqslant \mathrm{ALPHA} \geqslant-1.6$). The infrared (IR) spectral index ALPHA is defined as $\frac{dlog(\lambda F_{\lambda})}{d log(\lambda)}$, where $F\rm _{\lambda}$ is the flux as a function of wavelength $\lambda$. A total of $279$ disk-bearing candidate YSOs are identified in this region, of which 221 sources have matched Gaia objects and 203 sources have stellar parallax ($Plx$). To eliminate the foreground objects, we adopt a $30\%$ distance uncertainty of the cloud \citep{2019A&A...622A..52Z} and compare it with the parallax distance from Gaia ($1/Plx$). Finally, a total of 103 sources are reserved as the YSO sample, including 26 (77) sources with (without) Gaia parallax distance information. If $3 \,\sigma$ of the \tw emission is used to define the boundary of the cloud, $79$ sources are located within the cloud. The fraction of forground/backgroud contamination is estimated to be $\sim 6.6\%$ by the density of sources outside the cloud and the emission area of the cloud. In the YSO sample, $39\%$ are from \cite{2016MNRAS.458.3479M} and $29\%$ are from \cite{2019MNRAS.487.2522M}. The contamination level may be underestimated as the YSOs identified from \cite{2019MNRAS.487.2522M} are only retained in the region where the dust opacity value is higher than $1.3\times10^{-5}$. Of the $79$ sources, $32$, $24$, and $23$ are classified as Class I, flat-spectrum, and Class II, respectively. The IR photometric magnitudes, the ALPHA values and the origin of the YSOs sample are listed in Table\ref{tb-YSOs}. As shown in Figure \ref{fig-yso}, most of the candidate YSOs are distributed along the ridgelines of the filaments. In addition, the Class I objects (magenta plus) appear to be assembled nearby the three \hii regions. \section{Discussion} \subsection{Large-scale Filaments}\label{Large-Scale Filaments} Giant molecular filaments (GMF) can reach up to Galactic scale and play an important role as part of the spiral arms \citep{2014A&A...568A..73R, 2015MNRAS.450.4043W, 2015ApJ...815...23Z,2017ApJS..229...24D}. However, there is no clear definition of the GMF. According to the description of \cite{2019A&A...622A..52Z}, in position-position-velocity space, an elongated molecular cloud structure with length greater than 10\,pc and mass greater than $10^{3}M_{\odot}$ could be considered as GMF. In this MC, except F6, these filaments can be considered as GMF as their length range from $\sim$ 32\,pc to $\sim$ 90\,pc and mass range from $\sim 1725\,M_{\odot}$ to $\sim 13650 \,M_{\odot}$. The fitted widths of these filaments are $1.3-3.9 \mathrm{\,pc}$, which are much larger than the typical value with of `$Herschel$ filaments' \citep[0.1$\mathrm{\,pc}$,][]{2011A&A...529L...6A}. The low-J CO only trace relatively diffuse gas and the angular resolution ($\sim 1.1 \mathrm{\,pc}$ at a distance of $\sim 4.5 \mathrm{\,kpc}$) is far from enough to resolve the widths of the filaments. According to the criteria of hub-filament that a central body of low aspect ratio and high column density surrounded by branches with features of greater aspect ratio and lower column density \citep{2009ApJ...700.1609M}, and the criteria of ridge-nest structure that a single dominating denser filament with a disorganised network of lower-density filaments \citep{2011A&A...533A..94H}, we suggest that the construction of the identified filaments is similar to the `ridge-nest' rather than `hub-filament'. We consider F1 as the main filament and F2 to F6 as sub-filaments. As shown in Table\ref{tb-filaments}, F1 $\sim$ F5 have larger $M_{\text {line }}$ than the critical mass per unit length ($M_{\text {line, crit}}$) in the isothermal cylinder model \citep{1964ApJ...140.1056O, 1997ApJ...480..681I}, here $M_{\rm line, crit} = 2{c_{s}}^{2}/G \sim$ 16.4 ($T/10 \mathrm{K}$) $M_{\sun} \rm pc^{-1}$ ($c_{\rm s}$ is the isothermal sound speed $\sim 0.2 \mathrm{\,km}\mathrm{\,s}^{-1}$, and $G$ is the gravitational constant). The $M_{\text {line, crit}}$ is the critical value required for a filament to be gravitationally unstable to radial contraction and fragmentation along its length \citep{1997ApJ...480..681I}. The observed $M_{\text {line }}$ of F1 $\sim$ F5 exceed $M_{\text {line, crit}}$ except F6. It indicates that filaments F1 $\sim$ F5 are gravitationally unstable and should fragment into dense cores, while F6 may expand eventually. \subsection{Kinematics of the Filaments}\label{Velocity Structures} \begin{deluxetable*}{lccccccc} \tabletypesize{\small} \setlength{\tabcolsep}{0.10in} \tablecaption{Characteristic fragmentation lengths of the filaments \label{tb-lengthscale}} \tablewidth{0pt} \tablehead{ \colhead{Name} & \colhead{$l_{v}$\tablenotemark{a}} & \colhead{$l_{\mathrm{N_{H_{2}}}}$\tablenotemark{b}} & \colhead{$c_{s}$} & \colhead{$\sigma_{\mathrm{obs}}$} & \colhead{$n_{c}$\tablenotemark{c}} & \colhead{$\lambda_{\mathrm{crit,th}}$}& \colhead{$\lambda_{\mathrm{crit,turb}}$}\\ & \colhead{pc} & \colhead{pc} & \colhead{(km s$^{-1}$)} & \colhead{(km s$^{-1}$)}& \colhead{$\mathrm{~cm}^{-3}$} & \colhead{pc}& \colhead{pc} } \startdata F1 & - & 20 & 0.17 & 0.89 & 366 &1.7 &9.2 \\ F3 & 25 & 30 & 0.16 & 0.54 & 67 &3.8 &13.0 \\ F5 & 20 & 11 & 0.15 & 0.56 & 157 &2.3 &8.8 \\ F6 & 13 & 8 & 0.14 & 0.93 & 12 &7.9 & 52.8\\ \enddata \tablenotetext{a}{Potential characteristic length scales estimated from \tw velocity structure function.} \tablenotetext{b}{Potential characteristic length scales estimated from \tht column density function.} \tablenotetext{c}{$n_{c}= N_{\mathrm{c}}/\mathrm{width}$.} \end{deluxetable*} According to the theoretical model of an infinite self-gravitating fluid cylinder \citep{1953ApJ...118..116C,1964ApJ...140.1056O,2010ApJ...719L.185J}, a hydrostatic filament can fragment if perturbations are larger than the critical wavelength of $\lambda_{\mathrm{crit}} =11 H$, and its gravitational instability would peak at $\lambda_{\max }=2\lambda_{\mathrm{crit}}$. The isothermal scale height $H$ is expressed as $c_{s}\left(4 \pi G \rho_{c}\right)^{-1 / 2}$, where $c_{\mathrm{s}}$ is the isothermal sound speed ($\sqrt{\frac{k_{\mathrm{B}} T_{\mathrm{kin}}}{\mu_{\mathrm{obs}} m_{\mathrm{H}}}}$), $G$ is the gravitational constant, $\rho_{c}$ is an initial gas mass density at the center of the filament. If turbulent pressure dominates over thermal pressure, $c_{s}$ should be replaced by velocity dispersion $\sigma_{\mathrm{obs}}$. We adopt the initial central density $\rho_{\mathrm{c}}=\mu m_{\mathrm{p}} n_{\mathrm{c}}= \mu m_{\mathrm{p}}N_{\mathrm{c}}/\mathrm{width}$, where $N_{\mathrm{c}}$ is the central column density along the filament, ``width" is the deconvolved FWHM from Gaussian fitting of the radial column density profiles (see, Figure \ref{fig-profile}), $m_{\mathrm{p}}$ is the proton mass, and $\mu=2.33$ is the molecular weight of molecular gas. The typical observed velocity dispersion $\sigma_{\mathrm{obs}}$ is estimated by $\Delta V / \sqrt{8 \ln 2}$, where $\Delta V$ is the average value of FWHM (Gaussian fitting) derived from each \tht spectral line along the filaments. The theoretical predicted fragmentation characteristic lengths for the filaments with dips in their SF are listed in Table \ref{tb-lengthscale}. The observed fragmentation characteristic lengths of F1, F3, and F5 are comparable with the predicted critical wavelength ($\lambda_{\text {crit }}$) or maximum wavelength ($\lambda_{\max }$) of fragmentation of the self gravitating fluid cylinder. As described in \cite{2011A&A...533A..34H}, based on the self-gravitating cylinder model, if the oscillatory pattern along filament is caused by core-forming motions, there will be a phase-shift of $\lambda/4$ between density and velocity. However, using the method of cross-correlation as demonstrated by \cite{2020NatAs...4.1064H}, we have not found significant evidence of the co-oscillation at such phase-shift. The oscillation pattern along PV diagrams commonly appears in spectral line observations towards filamentary clouds \citep[see, e.g.,][]{2010ApJ...719L.185J,2011A&A...533A..34H,2013A&A...554A..55H,2021ApJ...921...23G}, which is generally interpreted as mass transport along the segment due to core accretion\citep[][]{2013ApJ...766..115K,2014MNRAS.440.2860H,2013ApJ...769..115H,2018ApJ...855....9L}. We estimate mass-flow rates ($\dot{M}$) using a simple cylindrical model from \cite{2013ApJ...766..115K}, $\dot{M} = \frac{\bigtriangledown v_{\shortparallel}M_{\rm gas}}{tan(\alpha)}$, where $\alpha$ is the angle of the inclination to the plane of the sky, which is assumed to be $45^{\circ}$ \citep{2018ApJ...852...12Y,2021A&A...646A.137L}. $M_{\rm gas}$ is the mass of the filament segment, which is estimated from ${ }^{13} \mathrm{CO}$ in this work. We apply for this method to the regions where mass-flow may exist along the filaments (see, Figure\ref{fig-pvfit}). The results of the mass-flow rates ($\dot{M}$) along the filaments are listed in Table \ref{tb-segment}. Around the \hii region G154.346+02.606, the estimated $\dot{M}$ are $\sim 720 \,M_{\odot} \mathrm{Myr}^{-1}$ and $\sim 114 \,M_{\odot} \mathrm{Myr}^{-1}$, respectively. This result is consistent with the infall rate of the high-mass star formation regions from high resolution observations, e.g. the hub-filamentary cloud G22 \cite[$\sim 400\,M_{\odot}\mathrm{Myr}^{-1}$,][]{2018ApJ...852...12Y}, the infrared dark cloud G14.225-0.506 \cite[$\sim 100\,M_{\odot}\mathrm{Myr}^{-1}$,][]{2019ApJ...875...24C}. There may be deviations in mass estimation using different probes, e.g., the mass of G22 is derived from multiple mid-infrared extinction data, while the mass of G14.225-0.506 is estimated from N$_{2}$H$^{+}$(1-0) which has a higher critical density. Due to the assumption of LTE conditions throughout the cloud, optical thin of ${ }^{13} \mathrm{CO}$ and a constant \tht to H$_{2}$ abundance, the mass maybe underestimated by 2-3 times from \tht in this work \citep[see, e.g.,][]{2009ApJ...699.1092H,2008ApJ...680..428G}. Even so, the estimated mass-flow rate is still higher than $100\,M_{\odot}\mathrm{Myr}^{-1}$ which is thought to be high enough to allow the formation of even O-type stars \citep{2018PASJ...70S..53I}. For the region with a high mass flow rate of $\sim 720 M_{\odot} \mathrm{Myr}^{-1}$, we obtain a mean volume density ($\rho$) of $272 \mathrm{\,cm}^{-3}$ from \tht. The timescale of the free-fall time $t_{\mathrm{ff}}=\left(\frac{3 \pi}{32 G\rho}\right)^{0.5}$ is estimated to be about $3.1 \mathrm{\,Myr}$. Within one $t_{\mathrm{ff}}$, the mass would have accumulated to $2232\,M_{\odot}$, roughly 7.5\% of the total mass of the molecular cloud estimated from \tht. There are about nine regions with mass-flow rate around them greater than $100\,M_{\odot}\mathrm{Myr}^{-1}$ (marked by bold numbers in Figure \ref{fig-pvfit}). These regions include the three \hii regions, the intersections of the filaments (cross-A and cross-B), and a dense region along F3. The \ei emission also appears in most of these areas correspondingly. All the systematic velocities at the intersections are relatively redshifted compared to that of the ambient gas, which implies that the intersection could play a role similar to a hub. The IRAS04335+5110 is overlaid with the cross-B and the velocity dispersion there seems relatively larger than its periphery, implying star formation activity is ongoing there. In addition to the star forming zones and intersections, the most eye-catching region is the dense area along F3, which has a material flow rate of $246\,M_{\odot}\mathrm{Myr}^{-1}$. It is of interest to study star formation activity therein in the further observations. \subsection{Distribution of the Clumps }\label{distribution of clumps} As shown in Figure \ref{fig-coredistr}, most identified clumps are distributed along the filaments. About $43 \%(66 / 153)$ clumps have a separation less than $2 \mathrm{\,pc}$ ($\sim$ half of the filaments width) from the nearby backbones of the filaments. The spatial association between filaments and clumps is remarkable, which suggests that clumps form primarily along filaments \citep[see, e.g.,][]{2013ASPC..476...95A}. The ratios of clumps with $\alpha_{\mathrm{vir}}<1$ and $\alpha_{\mathrm{vir}}<2$ are approximately $32 \%(52 / 162)$ and $90 \%(147 / 162)$, respectively. Most of the clumps are in gravitationally bound states, implying that the global contraction along filaments is efficient. As shown in Figure \ref{fig-dv}, similar distributions between the observed velocity dispersion and the calculated non-thermal velocity dispersion suggest that the clumps are dominated by non-thermal motions, e.g., hierarchical, global gravitational collapse, stellar feedbacks, or accretion-driven turbulence, etc. As shown in Figure \ref{fig-Lar2}, the clump C1 (MWISP G154.349+2.609) is located above the demarcation line ($M_{\rm c}=870 \,M_{\odot}(\mathrm{R} / \mathrm{pc})^{1.33}$) between high- and low-mass star formation that suggested by \cite{2010ApJ...716..433K}. The surface densities of the clump C1 and C29 (MWISP G154.350+2.610) are higher than the lower limit ($0.05 \mathrm{\,g}\mathrm{\,cm}^{-2}$) of high-mass star formation suggested by \cite{2013MNRAS.431.1752U}. It implies that the two clumps (C1 and C29) have potential to form high-mass stars. These two clumps are located in the apex of filament F1, which is similar with the scene revealed by $Herschel$ maps that massive stars tend to form at the junctions of supercritical filaments \citep[see,e.g.,][]{2010A&A...520A..49S,2012A&A...543L...3H}. The most massive clump in this sample is Clump C1. It has a mass of $\sim 2416 \,M_{\odot}$ with a diameter of $\sim 1.1 \mathrm{\,pc}$ and a viral parameter of $\sim 0.33$. In addition, we note that clumps with $\alpha_\mathrm{vir}\leqslant 1$ mainly distribute in the apexes of filaments, e.g., the apex of F1, F4, the north of F1, F3, the south of F5. Clumps in these regions may go through rapid collapses to form stars. Generally, a sample of star-forming regions with roughly constant column density should have a mass distribution $M \propto R^{\sigma}$ with $\sigma \simeq 2$. The linear fitted slope of this sample is $2.57$ with a correlation coefficient $r \simeq 0.66$, which agrees with the results of \cite{2015ApJ...805..157E} and \cite{2018MNRAS.477.2220T}, greater than the results $(\simeq 1.6-1.7)$ of \cite{2010ApJ...724..687L}, \cite{2010ApJ...716..433K}, and \cite{2014MNRAS.443.1555U}, and less than the results ($\geq 2.7$) from \cite{2009ApJ...698..324R}. The slope greater than 2 may be caused by the superposition of mass in the line of sight direction \citep{2020SSRv..216...76B}. \subsection{Star Formation Rate and Efficiency} \begin{deluxetable}{lcccc} \tabletypesize{\scriptsize} \setlength{\tabcolsep}{0.1in} \tablecaption{The 10 massive candidate YSOs in the observed field. \label{tb-massiveyso}} \tablewidth{0pt} \tablehead{ \colhead{allwise} & \colhead{$l$} & \colhead{$b$} & \colhead{from} & \colhead{Infrared luminosity} \\ & \colhead{($\degr$)} & \colhead{($\degr$)} & & \colhead{($L_{\odot}$)} } \startdata \input{table_massiveyso.txt} \enddata \tablenotetext{a}{Integral luminosity estimated from WISE.} \end{deluxetable} The star formation efficiency is defined as the ratio of the mass of the YSOs to the total mass of YSOs and gas: $\mathrm{SFE}=\frac{M_{\mathrm{YSO}}}{M_{\mathrm{YSO}}+M_{\mathrm{gas}}}$. We use the trapezoid rule \citep{2008ApJS..179..249D,2015ApJS..220...11D} to integrate over the SEDs of the YSOs sample and the sources that located within the cloud, respectively. As shown in Figure\ref{fig-Lbol}, the peak position of histogram is $\sim 10 L_{\odot}$, indicating that the YSOs sample is complete down to $\sim 10 L_{\odot}$. There are 9 out of 10 sources with luminosities above $100 L_{\odot}$ located within the $3\,\sigma$ boundary of \tw emission (see Figure\ref{fig-yso} and Table\ref{tb-massiveyso}), three of which are distributed around \hii region G154.346+02.606, one in \hii region SH2-211, four in \hii region SH2-212, and one to the south of the filamentary molecular cloud. The contamination from foreground/background on the candidate massive YSOs is estimated to be $\sim 2.3\%$, which is neglected in our following calculation. A total of $8$ sources overlap with the filamentary cloud. According to \cite{2012AJ....144...31K}, the luminosity function (LF) of high-mass star forming clouds (HLF) have a characteristic tail extending toward luminosities above $100 L_{\odot}$. Thus, the candidate YSOs with infrared luminosities above $100 L_{\odot}$ could be candidate massive YSOs. However, the number of candidate massive YSOs is incomplete due to the saturation problem of WISE survey. We refer to the massive young stellar catalog form the Red MSX Source Survey\citep{2013ApJS..208...11L} for supplement. Only two \hii regions (G155.3319+02.5989 and G154.3472+02.6099) with bolometric luminosity up to $10^{4} L_{\odot}$ are found and they are coincides with the candidate massive YSOs in our sample (J044027.19+502828.8 and J043620.99+511254.1, see Table\ref{tb-massiveyso}). Based on the Kroupa IMF, the total number of YSO within the filamentary cloud is estimated to be $1255$ assuming that the $8$ sources would evolve into high-mass protostellars with mass greater than $8 M_{\odot}$. The total mass of YSOs is estimated to be $\sim 627\,M_{\odot}$ provided that the mean mass of a single candidate YSO is $0.5 \,M_{\odot}$. Given that the mass of molecular cloud ($2.96 \times10^{4} M_{\odot}$, from \tht), the SFE will be $\sim 2.1\%$ in this case. The \tht emission area is $\sim 506$ arcmin$^{2}$ which is corresponded to $\sim 867 \mathrm{\,pc}^{2}$ at a distance of $4.5 \mathrm{\,kpc}$. Taking 2 Myr \citep{2009ApJS..181..321E} and $0.54 \mathrm{\,Myr}$ \citep{2015ApJ...806..231H} as the Class $\mathrm{I}$ + $\mathrm{II}$ and Class $\mathrm{I}$ lifetimes, the estimated star formation rate ($\mathrm{SFR}=\frac{M\rm_{YSOs}}{\tau}$, $\tau$ is the average lifetime of YSOs), the gas mass surface density ($\Sigma_{\mathrm{^{13}CO}}$) and the SFR surface density $\left(\Sigma_{\mathrm{SFR}}\right)$ are $\sim 313 \,M_{\odot} \mathrm{Myr}^{-1}$, $\sim 34 \,M_{\odot} \mathrm{pc}^{-2}$, and $\sim 0.36 \,M_{\odot} \mathrm{Myr}^{-1} \mathrm{pc}^{-2}$, respectively. These results are comparable with that of \cite{2019A&A...622A..52Z}, who investigated systematically the star-forming content of a sample of $57$ GMFs and suggested that the SFRs of the GMFs scale similarly with dense gas as those of nearby molecular clouds. However, limited by the resolution, the seemingly individual object could potentially be an unresolved cluster at such a great distance. Considering the influence of YSO clustering, the estimated SFR can only be regarded as a lower limit which may be underestimated by a factor of $1.5-7$\citep{2019A&A...622A..52Z}. \subsection{WISE Three-Color Diagram}\label{wise} Figure \ref{fig-wise} shows the multi-color WISE image ($22 \,\mu \mathrm{m}$ in red, $4.6 \,\mu \mathrm{m}$ in green, and $3.4 \,\mu \mathrm{m}$ in blue) in the observed region. The $22 \,\mu \mathrm{m}$ emission is used to trace the small warm dust grains heated by ionized gases. The $4.6 \,\mu \mathrm{m}$ emission mostly originates from stars associated with the \hii regions and is considered as a promising diagnostic in the search for massive candidate YSOs \citep{2008AJ....136.2391C,2011ApJ...743...56C}. The $3.4 \,\mu \mathrm{m}$ band contains polycyclic aromatic hydrocarbon (PAH) emission at $3.3 \,\mu \mathrm{m}$ as well as a prominent molecular hydrogen feature at $3.234 \,\mu \mathrm{m}$. As shown in Figure \ref{fig-wise}, the infrared emission in the observed region is dominated by the three \hii regions and we describe them below. \subsubsection{\hii Regions in the Filaments}\label{HII regions} \paragraph{G154.346+02.606}\label{G154.346+02.606} As shown in Figure \ref{fig-IR-region1}, the infrared emission is enclosed by the \tht emission. In addition, the velocity of hydrogen radio recombination lines (HRRLs) is consistent with that of the CO emission (see Table\ref{tb-IRAS}). Thus the \hii region G154.346+02.606 is confirmed to be associated with filament F1. IRAS04324+5106 was identified as high-mass protostellar object due to its low FIR luminosity to virial mass ratio \citep{2018AJ....156..210A}, which is spatially coincident with the \tht peak emission. One H$_{2}$O maser towards IRAS04324+5106 was detected by \cite{1988A&A...191..323W} using the $100 \mathrm{\,m}$ Effelsberg telescope. The peak velocity of the maser is $-41 \mathrm{\,km}\mathrm{\,s}^{-1}$. \cite{2007PASJ...59.1185S} confirmed this maser in the $\mathrm{H}_{2} \mathrm{O}$ survey with the Nobeyama $45 \mathrm{\,m}$ telescope. The $\mathrm{H}_{2} \mathrm{O}$ maser is suggested to be one of the best tracers to the early phases of high-mass star formation \citep{1989ApJ...346..983E,2001ApJ...559L.143F} as it is often associated with outflow or accretion processes \citep{2007PASJ...59.1185S}. A chain of $350 \,\mu \mathrm{m}$ clumps (orange ellipse) were identified by Bolocam Galactic Plane Survey \citep[BGPS,][]{2015ApJS..218....1M}. In our observations, about 10 \tht clumps are located around the \hii region. Among them, C1 has potential to form high-mass star (see above) and undergoes rapid collapse as it has the largest mass and the smallest virial parameter (0.39). In the sub-region (154\fdg30$\leqslant$ l $\leqslant$ 154\fdg40, 2\fdg55$\leqslant$ b $\leqslant$ 2\fdg65, $\sim 61 \mathrm{\,pc}^{2}$), the gas mass estimated from \tht is $\sim 7803 \,M_{\odot}$. Three Class $\mathrm{I}$ sources are found with luminosity greater than 100 $L_{\odot}$. Assuming that the three Class $\mathrm{I}$ sources would evolve into high-mass protostellars ($>8 M_{\odot}$), the total number of YSOs is estimated to be $\sim 470$ based on the Kroupa IMF. Thus, the SFE, SFR, $\Sigma_{\mathrm{^{13}CO}}$, and $\Sigma_{\mathrm{SFR}}$ are estimated to be about $2.9\%$, 117 $M_{\odot}$My$^{-1}$, 124 $M_{\odot}\mathrm{pc}^{-2}$, and 1.88 $M_{\odot}$My$^{-1}\mathrm{pc}^{-2}$, respectively. Within the green contour ($\sim 8.1\,\mathrm{pc}^{-2}$, the \tht integrated intensity greater $\gtrsim 12.65 \mathrm{\,K} \mathrm{\,kms}^{-1}$ , equivalent to $N_{\mathrm{H}_{2}} \gtrsim 6.1 \times 10^{21} \mathrm{\,cm}^{-2}$ ), the gas mass is estimated to be $\sim 2832 \,M_{\odot}$. The SFE, SFR, $\Sigma_{\mathrm{^{13}CO}}$, and $\Sigma_{\mathrm{SFR}}$ in this area is estimated to be $\sim 7.6 \%$, $\sim 117 \,M_{\odot} \mathrm{My}^{-1}$, $\sim 388 \,M_{\odot} \mathrm{pc}^{-2}$, $\sim 16 \,M_{\odot} \mathrm{My}^{-1} \mathrm{pc}^{-2}$. According to the starburst certification standard of $\Sigma_{\mathrm{SFR}}>1 M_{\odot} \mathrm{yr}^{-1} \mathrm{kpc}^{-2}$ \citep{2016ApJ...833...23N}, G154.346+02.606 would be a promising candidate for mini-starburst. \paragraph{SH2-211}\label{SH2-211} As shown in Figure \ref{fig-IR-region2}, in the zoom-in region, the \tht morphology displays a ringlike structure and is spatially consistent with the infrared bubble in the northwestward. The rough consistency of the $v_{\rm LSR}$s ranges between HRRLs and CO suggests that the CO emission is associated with SH2-211 (see Table\ref{tb-IRAS}). \cite{1986BAAS...18Q.921M} revealed three gas components of HII, HI and $\mathrm{CO}$ associated with SH2-211 and SH2-212, and suggested that SH2-211 may be located in a more complex molecular environment according to the estimated parameters, e.g, sizes, masses. Two IRAS sources \citep[IRAS04329+5047 and IRAS04329+5045, ][]{2005ApJS..161..361K} are located on the opposite sides of the IR bubble separately. Peak emission of \tht appears around IRAS04329+5047. Three early-type stars (BOV, V=15.78; O9V, V=15.23; and O9Ib, V=13.54) are suggested as the ionization sources of SH2-211 \citep{1984A&A...139L...5C}. Three clumps (C118, C101, C46) are located around the bubble. The velocities of C118 ($-39.2 \mathrm{\,km}\mathrm{\,s}^{-1}$) and C46 ($-36.7 \mathrm{\,km}\mathrm{\,s}^{-1}$) are consistent with F4 and F3 respectively (see Figure \ref{fig-fil-spectrum}), while C101 has the moderate velocity among the three. The gas mass calculated from \tht within the green box ($0\fdg07 \times 0\fdg07$) is $\sim 275 \,M_{\odot}$. The \tht emission area is $\sim 14 \mathrm{\,pc}^{2}$. Compared with G154.346+02.606 and SH2-212, the mass-flow rate along the filament around SH2-211 is the lowest. There are 1 candidate massive YSOs (J043649.54+505242.8) at the center of the IR bubble. Based on the Kroupa IMF, the SFE, SFR, $\Sigma_{\mathrm{^{13}CO}}$, and $\Sigma_{\mathrm{SFR}}$ within the green box are estimated to be about $22\%$, $39 M_{\odot}$My$^{-1}$, $19 M_{\odot}\mathrm{pc}^{-2}$, and $2.8 M_{\odot}$My$^{-1}\mathrm{pc}^{-2}$, respectively. The \hii region SH2-211 may host ongoing starburst event\citep{2016ApJ...833...23N}. As mentioned before, F3 and F4 has a distinct $v_{\rm LSR}$ (F3 at $-36 \mathrm{\,km}\mathrm{\,s}^{-1}$, F4 at $-38 \mathrm{\,km}\mathrm{\,s}^{-1}$). Based on the velocity continuity of \tw, the northwest part is likely to be associated with F3 and the southeast part is likely to be associated with F4. The bubble appears in the intersection of F3 and F4. It is likely that the collision of the two filaments accounts for the star formation and then the \hii region. The heated bubble has a relatively circular morphology. As shown in Figure \ref{fig-IR-region2_pv}, the CO PV diagram along the bubble also shows a circle morphology, indicating the expansion of the bubble. Assuming the bubble is produced by stellar wind, the mechanical luminosity of the stellar wind can be estimated by $L_{\text {wind }} \approx \frac{1}{3}\left(\frac{n_{\mathrm{gas}}}{\mathrm{cm}^{-3}}\right)\left(\frac{R_{\mathrm{c}}}{\mathrm{pc}}\right)^{2}\left(\frac{V_{\mathrm{c}}}{\mathrm{km} \mathrm{s}^{-1}}\right)^{3} \times 10^{30} \text { erg } \mathrm{s}^{-1}$ \citep{1977ApJ...218..377W}, where $R_{\mathrm{c}}$ is the radius of the shell, $V_{\mathrm{c}}$ is the shell expansion velocity, $n_{\mathrm{gas}}$ is the density of the cavity within the MC. The $R_{\mathrm{c}}$ ($\sim 2.1 \mathrm{pc}$) and $V_{\mathrm{c}}$ ($\sim 1.5 \,\mathrm{km\,s}^{-1}$) of the shell are obtained from Figure \ref{fig-IR-region2_pv}. The $n_{\mathrm{gas}}$ ($n_{\mathrm{gas}}=\frac{3 N_{\text {shell }}}{R_{\mathrm{c}}}$) estimated from the \tw emission within the shell is $\sim 1.7\times 10^{3} \mathrm{\,cm}^{-3}$. Then the $L_{\text {wind }}$ is estimated to be $\sim 8.4 \times 10^{33} \mathrm{\,erg} \mathrm{\,s}^{-1}$. The kinetic time-scale ($t_{\text {kin}} \approx \frac{16}{27}\frac{R_{\rm c}}{\rm pc}\frac{\rm kms^{-1}}{v_{\rm c}}$) for opening such a shell is estimated to be about $\sim 8.2 \times 10^{5} \mathrm{\,yr}$. \paragraph{SH2-212}\label{SH2-212} As shown in Figure \ref{fig-IR-region3}, the \tht morphology displays an arc-like structure encircling the infrared bubble. The velocities of the HRRLs (see Table\ref{tb-IRAS}) differ from the CO molecular gas with an offset of $\sim 5 \mathrm{\,km}\mathrm{\,s}^{-1}$. \cite{2008A&A...482..585D} attributed this velocity difference to a scenario of ``champagne flow'' \citep{1979A&A....71...59T}, the ionized gas flows away from the cloud. About six \tht clumps are semicircling the ionized gas. C37 and C128 have a velocity at $-36 \mathrm{\,km}\mathrm{\,s}^{-1}$, which is roughly same as that of F5 (see, Figure \ref{fig-fil-spectrum}). C11, C24, and C75 have velocities at $-35 \mathrm{\,km}\mathrm{\,s}^{-1}$, differ from F5 with $\sim 1 \mathrm{\,km}\mathrm{\,s}^{-1}$. C115 is the closest clumps to the infrared bubble and has velocity at $ \sim -34 \mathrm{\,km}\mathrm{\,s}^{-1}$, differ from F5 with $\sim 2 \mathrm{\,km}\mathrm{\,s}^{-1}$. \cite{2008A&A...482..585D} suggested the origin of the fragmented molecular gas around the infrared bubble through the collect and collapse process. Near C11 and C37, where the infrared cluster IRAS 04366+5022 is located, a possible second-generation UC \hii region is forming \citep{2008A&A...482..585D}. The velocity difference between C11 and C37 may be caused by the formation and ionization of the secondary UC \hii region. The mass of the exciting star \citep[No. 228 in][]{2008A&A...482..585D} of the second-generation UC \hii region estimated from SED between $1.25 \,\mu \mathrm{m}$ and $21.3 \,\mu \mathrm{m}$ is $\sim 14 \,M_{\odot}$, and the spectral class is $\sim$ B0.5V \citep{2011MNRAS.411.2530J}. The gas mass calculated from \tht within the green box ($0\fdg1 \times 0\fdg1$) is $\sim 905 \,M_{\odot}$. The \tht emission area is $\sim 24 \mathrm{\,pc}^{2}$. Four candidate massive YSOs are found at the boundary of IR bubble, and the locations of two candidates (J044027.19+502828.8 and J044037.27+502740.7) are consistent with the region of 22 $\mu \mathrm{m}$ emission. Based on the Kroupa IMF, the lower limits of SFE, SFR, $\Sigma_{\mathrm{^{13}CO}}$, and $\Sigma_{\mathrm{SFR}}$ within the green box are estimated to be about $15\%$, $78 M_{\odot}$My$^{-1}$, $36 M_{\odot}\mathrm{pc}^{-2}$, and $3.1 M_{\odot}$My$^{-1}\mathrm{pc}^{-2}$, respectively. Therefor, the \hii region SH2-212 may also host ongoing starburst event. \subsubsection{Other Regions}\label{HII} As shown in Figure \ref{fig-wise-LMC}, based on the analysis of CO molecular dynamics structure (see Figure \ref{fig-pvfit}), we zoom in to nine regions where mass-flow activities exist, including cross-A, cross-B, and cross-C (corresponding to the zoom-in areas 1, 2, and 4). The WISE point sources appear in clusters in these regions and the emission at $4.6 \,\mu \mathrm{m}$ and $22 \,\mu \mathrm{m}$ bands is obvious. The $4.6 \mu \mathrm{m}$ emission is consistent with the strong \tht emission, which implies that YSOs are embedded in the CO clouds. The estimated accretion rates around these regions are $\sim 100 \,M_{\odot} \mathrm{Myr}^{-1}$, several times smaller than those around the \hii regions. Among the nine regions, the area 2 (cross-B) is probably the most active one with star formation as it has the largest accretion rate. Actually, when examining the entire WISE image, we consider that the entire MC is undergoing active star formation. \section{Conclusion} We present a larg-field survey of the $J=1-0$ transition lines of $^{12}$CO, $^{13}$CO, and C$^{18}$O towards the Galactic region of l=[153\fdg6, 156\fdg5] and b=[1\fdg85, 3\fdg5], using the PMO 13.7 m millimeter telescope. The main results are summarized as follows: 1, A network-shaped cloud is found in the velocity interval of $[-42.5,-30.0] \mathrm{\,km}\mathrm{\,s}^{-1}$. The distance is estimated to be $\sim$ 4.5 kpc. The basic physical parameters including excitation temperatures, H$_2$ column densities, and masses are estimated from ${ }^{12} \mathrm{CO}$ and ${ }^{13} \mathrm{CO}$, respectively. 2. Six large-scale filaments are identified from the \tht emission. By comparing the observed average $M_{\text {line}}$ and the $M_{\text {line, crit}}$, filaments F1-F5 are suggested to be under the gravitationally unstable conditions and will fragment into dense cores. 3. The PV diagrams of the filaments (e.g. F1, F5) show oscillation patterns, which is explained by the mass-flow caused by accretion activity. % The material seems to transport along the filaments to feed the \hii regions G154.346+2.606, SH2-212 and cross-B, with high mass-flow rates larger than $100 \,M_{\odot} \mathrm{Myr}^{-1}$. The mass-flow rates in the rest parts of filaments are relatively lower, at a level of a few tens of $M_{\odot} \mathrm{Myr}^{-1}$. 4. A total of $162$ CO clumps are extracted from the \tht datacube, of which $32 \%$ have $\alpha_\mathrm{vir}$ less than 1 and $90 \%$ have $\alpha_\mathrm{vir}$ less than 2. Rapid collapse seems to take place in the apexs of the filaments as most clumps with $\alpha_\mathrm{vir}$ less than 1 are located therein. Global contraction along filaments is efficient. We find $163$ candidate YSOs within the cloud traced by \tw. The SFE along the filaments is $\sim 2.1 \%$, which is comparable with that of other GMFs. 5. Three \hii regions, including G154.346+02.606, SH2-211 and SH2-212, are associated with the filaments. According to the estimated SFR densities ($> 1 M_{\odot}$My$^{-1}\mathrm{pc}^{-2}$), the three \hii regions are considered to host ongoing mini starburst events. In addition, other 9 regions with material flow around show signature of star forming activity in the WISE three-color diagram. The results from both the CO and infrared emission suggest that the entire filamentary cloud is undergoing intense star formation. \begin{acknowledgments} This work was supported by the National key R\&D Program of China (grant No.2017YFA0402702), the National Natural Science Foundation of China (grant No. 12041305), and CAS International Cooperation Program (grant No. 114332KYSB20190009). M.Zhang was supported by the National Natural Science Foundation of China (grants No. 12073079). This research made use of the data from the Milky Way Imaging Scroll Painting (MWISP) project, which is a multi-line survey in ${ }^{12} \mathrm{CO}$/${ }^{13} \mathrm{CO}$/$\mathrm{C}{ }^{12} \mathrm{O}$ along the northern galactic plane with PMO-13.7m telescope. We are grateful to all the members of the MWISP working group, particularly the staff members at PMO-13.7m telescope, for their long-term support. MWISP was sponsored by National Key R\&D Program of China with grant 2017YFA0402701 and CAS Key Research Program of Frontier Sciences with grant QYZDJ-SSW-SLH047. \end{acknowledgments} \vspace{5mm} \facilities{PMO-13.7m} \software{GILDAS/CLASS \citep{2005sf2a.conf..721P}} \bibliography{export-bibtex}{} \begin{longrotatetable} \begin{deluxetable*}{cccccccccccccccccc} \setlength{\tabcolsep}{0.020in} \tablecaption{Properties of the identified dense clumps \label{tb-clumps}} \tablewidth{0pt} \tablehead{ \colhead{NO.} & \colhead{Clump} & \colhead{$ \Theta_\mathrm{maj} $} & \colhead{$ \Theta_\mathrm{min} $} & \colhead{$PA$} & \colhead{$v$} & \colhead{$T_{\rm peak}$} & \colhead{$\Delta V$} & \colhead{$R_{\rm eff}$\tablenotemark{a}} & \colhead{$T_{\rm ex}$} & \colhead{$\tau$} & \colhead{$N_{\mathrm{H}_{2}}$} & \colhead{$ \Sigma_{c} $} & \colhead{$n_{\mathrm{H}_{2}}$} & \colhead{$ M_\mathrm{LTE} $} & \colhead{$M_\mathrm{vir}$} & \colhead{$\alpha_{\text {vir}}$}\\ & & \colhead{($ \arcmin $)} & \colhead{($ \arcmin $)} & \colhead{($\degr$)} & \colhead{($\mathrm{km} \mathrm{s}^{-1}$)} & \colhead{(K)} &\colhead{($\mathrm{km} \mathrm{s}^{-1}$)} & \colhead{( pc)} & \colhead{(K)} & & \colhead{($10^{21} \mathrm{\,cm}^{-2}$)} & \colhead{($\mathrm{M}_{\odot} \mathrm{pc}^{-2}$)} & \colhead{($ 10^{3}\mathrm{cm}^{-3}$)} & \colhead{($\mathrm{M}_{\odot}$)} & \colhead{($\mathrm{M}_{\odot}$)} & \\ \colhead{$(1)$} & \colhead{$(2)$} & \colhead{$(3)$} & \colhead{$(4)$} & \colhead{$(5)$} & \colhead{$(6)$} & \colhead{$(7)$} & \colhead{$(8)$} & \colhead{$(9)$} & \colhead{$(10)$} & \colhead{$(11)$} & \colhead{$(12)$} & \colhead{$(13)$} & \colhead{$(14)$} & \colhead{$(15)$} & \colhead{$(16)$} & \colhead{$(17)$} } \startdata \input{table_2.txt} \enddata \tablenotetext{a}{Deconvolution radius: $R_{eff} = \frac{1}{2}\times d \times \frac{\pi}{2} \sqrt{\Theta_\mathrm{maj}\Theta_\mathrm{min}-\theta_\mathrm{beam}^2}$.} \end{deluxetable*} \end{longrotatetable} \begin{longrotatetable} \begin{deluxetable*}{lcccccccccccccc} \tabletypesize{\tiny} % \setlength{\tabcolsep}{0.015in} \tablecaption{Infrared Photometric Magnitudes of Candidate YSOs \label{tb-YSOs}} \tablewidth{0pt} \tablehead{\colhead{allwise} & \colhead{$l$} & \colhead{$b$} & \colhead{$W1$ (3.4 $\mu m)$} & \colhead{$W2$ (4.6 $\mu m)$} &\colhead{$W3$ (12 $\mu m)$} & \colhead{$W4$ (22 $\mu m)$} & \colhead{$J$ (1.25 $\mu m)$} & \colhead{$H$ (1.65 $\mu m)$} & \colhead{$K_{\rm s}$ (2.16 $\mu m)$} &\colhead{FROM\tablenotemark{a}} & \colhead{ GAIA$\_$SOURCE$\_$ID } &\colhead{ALPHA} &\colhead{$plx$}& \colhead{distance} \\ & \colhead{($\degr$)} & \colhead{($\degr$)} & \colhead{(mag)} &\colhead{(mag)} & \colhead{(mag)} & \colhead{(mag)} &\colhead{(mag)} &\colhead{(mag)} & \colhead{(mag)} & \colhead{} &\colhead{} & \colhead{} &\colhead{mas} & \colhead{kpc} \\ \colhead{$(1)$} & \colhead{$(2)$} & \colhead{$(3)$} &\colhead{$(4)$} & \colhead{$(5)$} & \colhead{$(6)$} &\colhead{$(7)$} & \colhead{$(8)$} & \colhead{$(9)$} &\colhead{$(10)$} & \colhead{$(11)$} & \colhead{$(12)$} &\colhead{$(13)$} & \colhead{$(12)$} &\colhead{$(13)$} } \startdata \input{table_103.txt} \enddata \tablenotetext{a}{The marks of Class$\ast$, svmgaia, and SVMClass$\ast$ indicate the candidate YSOs identified through the scheme of \cite{2014ApJ...791..131K} and obtained from \cite{2016MNRAS.458.3479M}, and \cite{2019MNRAS.487.2522M}, respectively. } \end{deluxetable*} \end{longrotatetable}

Title: Unifying High- and Low-resolution Observations to Constrain the Dayside Atmosphere of KELT-20b/MASCARA-2b

Abstract: We present high-resolution dayside thermal emission observations of the exoplanet KELT-20b/MASCARA-2b using the MAROON-X spectrograph. Applying the cross-correlation method with both empirical and theoretical masks and a retrieval analysis, we confirm previous detections of Fe\,\textsc{i} emission lines and we detect Ni\,\textsc{i} for the first time in the planet (at 4.7$\sigma$ confidence). We do not see evidence for additional species in the MAROON-X data, including notably predicted thermal inversion agents TiO and VO, their atomic constituents Ti\,\textsc{i} and V\,\textsc{i}, and previously claimed species Fe\,\textsc{ii} and Cr\,\textsc{i}. We also perform a joint retrieval with existing \textit{Hubble Space Telescope}/WFC3 spectroscopy and \textit{Spitzer}/IRAC photometry. This allows us to place bounded constraints on the abundances of Fe\,\textsc{i}, H$_2$O, and CO, and to place a stringent upper limit on the TiO abundance. The results are consistent with KELT-20b having a solar to slightly super-solar composition atmosphere in terms of the bulk metal enrichment, and the carbon-to-oxygen and iron-to-oxygen ratios. However, the TiO volume mixing ratio upper limit (10$^{-7.6}$ at 99\% confidence) is inconsistent with this picture, which, along with the non-detection of Ti\,\textsc{i}, points to sequestration of Ti species, possibly due to nightside condensation. The lack of TiO but the presence of a large H$_2$O emission feature in the WFC3 data is challenging to reconcile within the context of 1D self-consistent, radiative-convective models.

https://export.arxiv.org/pdf/2208.04759

\title{Unifying High- and Low-resolution Observations to Constrain the Dayside Atmosphere of KELT-20b/MASCARA-2b} \correspondingauthor{David Kasper} \email{kasperd@uchicago.edu} \author[0000-0003-0534-6388]{David Kasper} \affil{Department of Astronomy \& Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA} \author[0000-0003-4733-6532]{Jacob L.\ Bean} \affil{Department of Astronomy \& Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA} \author[0000-0002-2338-476X]{Michael R.\ Line} \affil{School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85281, USA} \author[0000-0003-4526-3747]{Andreas Seifahrt} \affil{Department of Astronomy \& Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA} \author[0000-0003-2404-2427]{Madison T. Brady} \affil{Department of Astronomy \& Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA} \author[0000-0003-3667-8633]{Joshua Lothringer} \affil{Department of Physics, Utah Valley University, Orem, UT 84058, USA} \author[0000-0002-1321-8856]{Lorenzo Pino} \affil{INAF-Osservatorio Astrofisico di Arcetri Largo Enrico Fermi 5I-50125 Firenze, Italy} \author[0000-0002-3263-2251]{Guangwei Fu} \affil{Department of Astronomy, University of Maryland, College Park, MD 20742, USA} \author[0000-0002-8573-805X]{Stefan Pelletier} \affil{Institut de Recherche sur les Exoplan\`etes, D\'epartement de Physique, Universit\'e de Montr\'eal, 1375 Avenue Th\'er\`ese-Lavoie-Roux, Montreal, H2V 0B3, Canada} \author[0000-0002-4410-4712]{Julian St\"urmer} \affil{Landessternwarte, Zentrum f\"ur Astronomie der Universit\"at Heidelberg, K\"onigstuhl 12, 69117 Heidelberg, Germany} \author[0000-0001-5578-1498]{Bj\"orn Benneke} \affil{Institut de Recherche sur les Exoplan\`etes, D\'epartement de Physique, Universit\'e de Montr\'eal, 1375 Avenue Th\'er\`ese-Lavoie-Roux, Montreal, H2V 0B3, Canada} \author[0000-0002-7704-0153]{Matteo Brogi} \affil{Department of Physics, University of Warwick, Coventry CV4 7AL, UK} \affil{INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025 Pino Torinese, Italy} \affil{Centre for Exoplanets and Habitability, University of Warwick, Coventry, CV4 7AL, UK} \author{Jean-Michel D\'esert} \affil{Anton Pannekoek Institute for Astronomy, University of Amsterdam, 1098 XH Amsterdam, The Netherlands} \keywords{Hot Jupiters (753), Exoplanet atmospheres (487)} \section{Introduction} \label{sec:intro} \begin{deluxetable*}{lccccccccclcc} \tabletypesize{\scriptsize} \tablecolumns{13} \tablewidth{0pc} \tablecaption{\label{tab:obs_log} Log of the MAROON-X observations} \tablehead{ \colhead{UT Date} & \colhead{Exposures} & \colhead{Planet Phase Range} & \colhead{Airmass} & \colhead{Conditions} & \colhead{Seeing} & \colhead{Average Blue Arm SNR} } \startdata 2021 May 29 10:47 $\rightarrow$ 14:03 & 34 & 0.63 $\rightarrow$ 0.67 & 1.28 $\rightarrow$ 1.03 & stable, clear & 0.6\arcsec & 176\\ 2021 June 04 11:37 $\rightarrow$ 14:34 & 31 & 0.37 $\rightarrow$ 0.40 & 1.09 $\rightarrow$ 1.02 $\rightarrow$ 1.08 & stable, clear & 0.5\arcsec & 196\\ \enddata \end{deluxetable*} KELT-20b/MASCARA-2b (hereafter KELT-20b for brevity) is a 1.56\,$R_{\text{Jup}}$ radius exoplanet orbiting a bright ($m_V \sim 7.6$) A star with a 3.47\,day orbital period \citep{lund17,Talens18}. These parameters place it in the population of the $\sim30$\footnote{\url{https://exoplanetarchive.ipac.caltech.edu}} known transiting hot Jupiters orbiting early-type stars. KELT-20b is very highly for dayside emission atmospheric follow-up due to the apparent magnitude of the host star, the size of the planet relative to the host star, and the planet's high equilibrium temperature of $\sim$2250\,K. Furthermore, the unambiguous detection of both iron \citep{borsa22,Yan22,Johnson22} and water \citep{Fu22} in the emission spectrum of this exoplanet, which is one of the coolest exoplanets with a detected thermal inversion, gives a special opportunity to unify ground-based, high-resolution and space-based, low-resolution observations of exoplanet atmospheres. The thermal emission spectrum of KELT-20b has been previously observed with ground-based, high-resolution spectrographs by \citet{borsa22}, \citet{Yan22}, and \citet{Johnson22}. Using data from HARPS-N, \citet{borsa22} found a 6.8$\sigma$ detection of Fe\,\textsc{i}, as well as 3$\sigma$ detections of Fe\,\textsc{ii} and Cr\,\textsc{i}. The \citet{borsa22} detections of Fe\,\textsc{ii} and Cr\,\textsc{i} were only obtained for their post-secondary eclipse dataset and did not appear in their pre-eclipse dataset, thus suggesting the presence of chemical inhomogeneities. \citet{Yan22} used data from CARMENES and found Fe\,\textsc{i} at 7.5$\sigma$. Using PEPSI, \citet{Johnson22} found Fe\,\textsc{i} in emission at 15.1$\sigma$ with non-detections of Fe\,\textsc{ii} and Cr\,\textsc{i}, among others. All the detected species in the previous works were observed in emission, which indicates a thermal inversion in KELT-20b's atmosphere. \citet{Johnson22} inferred upper limits on the abundances of their TiO, VO, FeH, and CaH non-detections with volume mixing ratios in the $\sim 10^{-9}-10^{-10}$ range, with the exception of FeH, for which they found at an upper limit of $3\times10^{-7}$. With space-based data, \citet{Fu22} found H$_{2}$O and CO by analyzing the emission spectrum of KELT-20b derived from \textit{Hubble Space Telescope}/WFC3 and \textit{Spitzer}/IRAC eclipse observations. The water feature is the strongest of those in emission observed to-date with WFC3 \citep{Mansfield2021}. \citet{Fu22} claimed that TiO is necessary to cause the deep inversion that leads to the large water emission feature. However, this requirement is in tension with the previous ground-based measurements that all failed to see TiO in KELT-20b's thermal emission spectrum. In this paper we present ground-based, high-resolution observations of the dayside emission from KELT-20b that were obtained using the MAROON-X instrument on the Gemini North telescope \citep{seifahrt16, seifahrt18, seifahrt20}. We aim to determine the composition of the planet's dayside and perform a joint analysis including existing \textit{HST} and \textit{Spitzer} data to unify ground- and space-based results for this planet. We present our observations in \S\ref{sec:obs}. We describe two analyses of the data in \S\ref{sec:template} and \ref{sec:ret} that yield the detection of the planet's atmosphere and constraints on its properties. We contextualize our atmospheric retrieval with a comparison to atmospheric forward models in \S\ref{sec:RCEmodels}. We conclude with a discussion of the results in \S\ref{sec:discussion}. \section{Observations} \label{sec:obs} We used MAROON-X to obtain 4.51\,hours of data (6.2\,hours total observing time including overheads) of KELT-20b during a post-secondary eclipse phase on May 29 2021 and a pre-secondary eclipse phase on June 04 2021 UTC (Program ID: GN-2021A-Q119). We obtained data with both the MAROON-X ``blue'' (500 -- 663\,nm) and ``red'' (654 -- 920\,nm) channels. The two channels were observed simultaneously with 250\,s exposures. Table~\ref{tab:obs_log} gives a log of the observations. The data were reduced with the standard MAROON-X pipeline \citep{seifahrt20}. The pipeline includes detector calibration, one-dimensional spectral extraction, barycentric corrections calculated for the flux-weighted midpoint of each observation, and wavelength solutions and instrumental drift corrections based on the simultaneous Fabry-Perot etalon calibration data. \section{Detection of the Planetary Atmosphere Signal} \label{sec:template} We performed two analyses of our data to characterize KELT-20b's dayside atmosphere. The first analysis used a cross-correlation function (CCF) technique with line-weighted binary templates following \citet{pino20} and \citet{kasper2021}. We used our established approach to create a time (i.e., phase) evolving spectrum containing planet lines normalized to the planet plus star continuum. We used eleven stellar templates built from observed stellar spectra \citep[e.g.,][]{suarez20} as masks for the CCF analysis. These masks correspond to F9, G2, G8, G9, K2, K6, M0, M2, M3, M4, and M5 stellar types and come from the ESPRESSO data reduction pipeline (``ESPRESSO DRS''\footnote{\url{http://eso.org/sci/software/pipelines/}}). The mask ensemble spans the transition from ionized and neutral atoms to molecules as absorption features in stellar spectra. We used the masks in emission as analogs for the dayside of the hydrogen-dominated atmosphere of KELT-20b. Figure \ref{fig:ccf_comp} shows the peak signal-to-noise ratio (SNR) from the CCF analysis when combining the pre- and post-eclipse datasets and utilizing each of the masks. The SNR normalization was computed via a 3$\sigma$ background clipping method \citep[as in][]{kasper2021}. The high significance correlation with the earlier spectral type masks implicates neutral atomic lines, and Fe\,\textsc{i} in particular, as the dominant opacity source in KELT-20b's atmosphere at the wavelengths probed. In comparison with the earlier-type masks applied to the blue channel, M-type masks give lower detection confidences, down to a non-detection at M5. Additionally, M-type masks do not yield a significant detection in the red channel. This indicates, in agreement with previous ground-based results, that molecules like TiO are likely not present in the planet's atmosphere. Complementary to the above analysis, we also explored the initial detectability of a variety of specific gases via data-model template cross-correlation \citep{kasper2021,line2021}. As in past works we removed stationary telluric and stellar features via the singular value decomposition method (we removed between 2-8 singular values and found little difference. We ultimately settled on 2 to prevent accidental removal of the planetary signal). The high-resolution template models are derived from the converged output of a KELT-20b-specific 1D radiative-convective-thermochemical equilibrium (1D-RC) model calculated using the {\tt ScCHIMERA} code \citep{pis18,Arcangeli2018,Mansfield2021, kasper2021} assuming solar composition. The high-resolution model spectra are then convolved with a planetary rotation kernel and a Gaussian spectral response function at an R\,=\,85,000, as appropriate for MAROON-X. We searched for CaH, Ca\,\textsc{i} and \textsc{ii}, CrH, Cr\,\textsc{i}, FeH, Fe\,\textsc{i} and \textsc{ii}, MgH, Mg\,\textsc{i}, Na\,\textsc{i}, Ni\,\textsc{i}, Si\,\textsc{i}, Ti\,\textsc{i} and \textsc{ii}, TiO, V\,\textsc{i} and \textsc{ii}, and VO. Of all the species tested only Fe\,\textsc{i} and Ni \textsc{i} were found with a significant response (9.8$\sigma$ and 4.7$\sigma$, respectively). We also explored a grid of constant-with-altitude abundances for each gas \citep[using the same converged temperature-pressure profile found above; e.g.,][]{Giacobbe2021} to ensure we did not miss any species detections due to the assumption of thermochemical equilibrium. No additional species were found in this analysis. Figure~\ref{fig:FeI} shows an SNR map for Fe\,\textsc{i} in the planet velocity (K$_{p}$) vs.\ system velocity (V$_{\text{sys}}$) plane (a similar plot for Ni\,\textsc{i} can be found in the Appendix). The SNR in the map was computed via a 3-sigma background clipping method \citep[as in][]{kasper2021}. With the combination of pre- and post-eclipse phases we find agreement with the system velocity and planetary orbital speed. This $K_{p}$ vs.\ $V_{\text{sys}}$ mapping was also performed on the stellar template CCFs to ensure that the signal originated from the planet. \begin{deluxetable}{lcl} \tablecaption{\label{tab:priors} Description of retrieved parameters and their prior ranges. All priors are assumed uniform between the bounds given. Variables correspond to the labeling in the corner plot shown in the Appendix.} \tablehead{ \colhead{Parameter} & \colhead{Description} & \colhead{Prior} } \startdata $K_{p\text{,pre/post}}$ & Keplerian velocity for & 150 -- 210\,km\,s$^{-1}$\\ & pre/post eclipse nights & \\ $V_{\text{sys,pre/post}}$ & systemic velocity & -50 -- 10\,km\,s$^{-1}$ \\ log($a_{\text{pre/post}}$) & model multiplicative & -1 -- 1\\ & scale factor & \\ log($\gamma_1$) & vis-to-IR opacity & -3 -- 4 \\ log($\kappa_{\text{IR}}$) & IR opacity & -3 -- 0 (cgs) \\ T$_{\text{irr}}$ & irradiation temperature & 1000 -- 5000\,K\\ $\Delta$log\,$g$ & differential log-gravity & -1 -- 0 (cgs)\\ & from max (log\,$g$=3.28) & \\ H$^-$, Fe, TiO, & log gas volume & -12 -- 0 \\ H$_2$O, CO & mixing ratios & \\ H*e$^-$ & log mixing ratio& -18 -- 0 \\ & hydrogen $\times$ electron & \\ & for free-free cont. & \\ \enddata \end{deluxetable} \section{Retrieval Analysis} \label{sec:ret} Following \citet{line2021} and \citet{kasper2021}, we applied the \citet{Brogi2019} cross-correlation-to-log-likelihood retrieval framework to derive abundances and the vertical temperature structure in KELT-20b's dayside atmosphere. The \texttt{CHIMERA} forward model underlying the retrieval assumed constant-with-altitude abundances and used the \citet{Guillot2010} parameterization of the temperature-pressure (T-P) profile. The retrieval parameters and their prior ranges are given in Table \ref{tab:priors}. A more detailed description of the radiative transfer method, including opacity sources, and log-likelihood implementation is given in \cite{line2021} and \cite{kasper2021} (with opacity sources there-in including \cite{mckemmish19,John1988,Grimm2015,Grimm2021}) as well as new cross-sections utilized in this work from the EXOPLINES data base \citep{GharibNezhad21} as sourced from the line lists for VO \citep{Mckemmish16}, H$_2$O \citep{Polyansky2018}, MgH \citep{GharibNezhad2013}, CrH and CaH \citep{Bernath2020MOLLIST}. For our ``MAROON-X only'' retrieval we combined the blue and red arm MAROON-X data for both the pre- and post-eclipse observations. We also performed a joint retrieval on the combination of the MAROON-X observations and the \textit{HST}/WFC3 and \textit{Spitzer}/IRAC 4.5\,$\mu$m observations presented in \citet{Fu22}. The joint retrieval was performed by combining the high-resolution and low-resolution likelihoods as described in \cite{Brogi2019}. In all the retrievals we included H$_2$O, CO, TiO, Fe\,\textsc{i}, and abundance proxies for the H$^-$ bound-free and free-free continua. The detection of Ni\,\textsc{i} occurred after we performed the retrieval analysis. Because the mass of the planet is not well constrained \citep[upper limit of 3.5\,M$_{\text{Jup}}$,][]{lund17}, we also included a log\,$g$ free parameter that spanned a range between maximum log\,$g$ (3.28, cgs) and one dex below. The results are summarized in Figure \ref{fig:retrieval} and the full corner plot is given in the Appendix (Figure~\ref{fig:staircase}). We note that in comparing the maximum log-likelihood difference between the two retrievals we found the difference as entirely due to the additional $red\chi^2 \sim\,1$ data-points and that the MAROON-X contribution did not change in any meaningful way. This is unsurprising as there is little overlap in common parameters between the two datasets, with the exception of the T-P profile. E.g., water and CO do not present themselves in the MAROON-X data and the atomic species don’t show up in the WFC3. As relevant below, this suggests that any T-P profile information arising from the low-res contribution is fully consistent with what the MAROON-X data prefers. The joint retrieval to all the datasets provides an excellent fit to the \textit{HST} and \textit{Spitzer} observations (median $\chi^2$ per number of data points is $\sim 0.9$, p-value=0.02). As expected from the results in the literature and the analysis in \S\ref{sec:template}, the retrieval prefers a strong thermal inversion. The temperature gradient proxy parameter, log($\gamma_1$), is +21$\sigma$ away from zero, where a value of zero indicates an isothermal T-P profile, negative is a monotonically decreasing T-P profile, and positive is an inverted T-P profile (increasing temperature with decreasing pressure above the inversion level). The combined MAROON-X + \textit{HST} and \textit{Spitzer} retrieval provides a much more stringent constraint on this temperature gradient proxy than with MAROON-X alone (only 1.8$\sigma$ above zero, see blue vs. red histograms as well as additional caption details in Figure~\ref{fig:staircase}). We note that the {\it pressure level} at which the primary line forming temperature gradient resides is not well constrained [controlled by the log($\kappa_{IR}$) parameter] due to the lack of prior constraint on gravity and the degeneracy between metallicity and the ``$\tau$=2/3" pressure level. In the top left inset on the left panel of Figure~\ref{fig:retrieval}, this lack of constraint is apparent as the reconstructed T-P profiles invert at a continuum of pressure levels, with pressures below the inverted portion naturally becoming isothermal due to the \citet{Guillot2010} profile parameterization employed, as it assumes a double gray formalism. The retrieved T-P profile {\it gradient} (as fully apparent in some reconstructions within the bounds of the figure) matches quite well the expectation from 1D-RC. However, this forward model includes TiO, which is seemingly absent given the data. We explore this issue more in the next section (\S\ref{sec:RCEmodels}). The joint retrieval between the optical and near-to-mid IR enables constraints on both refractory (Fe) and volatile (C, O) elements. We derived bounded constraints on the abundances of Fe\,\textsc{i}, H$_2$O, and CO (Figure~\ref{fig:staircase}). The retrieved abundances are found to be in qualitative agreement with the expectations for a $\sim$3$\times$solar composition gas in thermochemical equilibrium (see the right panel of Figure~\ref{fig:retrieval}). Inferring elemental abundance ratios from retrieved molecular abundances, especially with simplistic retrieval forward modeling assumptions, (e.g., constant-with-altitude volume mixing ratios) is not always straightforward [e.g., \cite{Sheppard2017} vs. \cite{Arcangeli2018}]. This is especially true in ultra-hot Jupiters where thermal dissociation can deplete the abundances of measurable species (e.g., Fe\,\textsc{i}, H$_2$O) into non-measured species (Fe\,\textsc{ii}, OH, O\,\textsc{i}). In order to place the retrieved abundances into context, we compare a series of elemental abundance proxies to a battery of, again, 1D-RC models, summarized in Figure~\ref{fig:abund_summary}. We use the total sum of the bounded retrieved gases (Fe\,\textsc{i}+H$_2$O+CO) as a proxy for total metal enrichment (a.k.a, ``metallicity"), the ratio, CO/(H$_2$O+CO) as a tracer of the carbon-to-oxygen ratio (C/O), and Fe\,\textsc{i}/(H$_2$O+CO) as a proxy for the iron-to-oxygen ratio (Fe/O). Figure~\ref{fig:abund_summary} shows these secondary retrieval data-products as histograms compared to those same quantities (and their dependencies with altitude) from different composition 1D-RC models. We find that the overall metallicity (left most panel) is consistent with enrichment values $\lessapprox 30\times$solar, with a relatively loose lower bound of $\sim$0.1$\times$solar. Depending on the exact pressure level probed, the C/O (middle panel) can range anywhere between solar (C/O=0.55) to super-solar (C/O=0.85). Finally, we find the most stringent constraint on the Fe/O (right most panel). The retrieved values are largely consistent with solar, if not up to a few times solar (but below 10$\times$). We can be more quantitative about the elemental abundance constraints if we make further assumptions about the pressure levels probed to correct for potential biases due to the constant-with-altitude gas mixing ratio assumption. As discussed above, we are unable to constrain the absolute pressure-level location of the base of the inversion. However, these degeneracies work out in such away that the $\tau$=2/3 level probed by the different wavelengths should always see the same temperature gradient, and largely the same abundance along a mixing ratio profile. An example of the pressure levels probed within the 3$\times$solar 1D-RC is shown in the Appendix ($\sim$0.1\,bar to 0.1\,mbar, see Figure~\ref{fig:tau_sfc}). Additionally, the constant-with-altitude gas mixing ratios assumed in the retrieval will be more heavily weighted towards the deeper, greater abundance layers, before molecular dissociation and atomic ionization occurs. Conveniently, our chosen elemental abundance proxies are fairly constant with pressure/altitude (within a 1D-RC) at layers deeper than the dissociation/ionization level (as also can be seen by the pressure levels probed in Figure~\ref{fig:tau_sfc}). It is also true that in these deeper layers the C/O and Fe/O proxies are {\it exact} measures for those elemental ratios as Fe\,\textsc{i}, H$_2$O, and CO are the sole carriers of elemental Fe, O, and C (at altitudes where dissociation begins, O becomes predominately locked into OH and O\,\textsc{i}, in addition to CO, and Fe into Fe\,\textsc{ii}). Given the above caveats and assumptions, we provide log of the abundances relative to the uniform regions of the solar composition 1D-RC model (thick black curves in each Figure \ref{fig:abund_summary} panel): [Fe\,\textsc{i}+H$_2$O+CO] = -1.28 -- 1.49, [CO/(H$_2$O+CO)]=-0.47 -- 0.19, and [Fe\,\textsc{i}/(H$_2$O+CO)] = -0.20 -- 0.86 at 95\% confidence (here ``[X/Y]'' is the usual bracket notation used for stellar abundances where zero is equal to solar). Taken together, the elemental abundance ratios are relatively unremarkable and largely consistent with solar-composition (perhaps a modest enhancement in the Fe/O and overall metallicity), and generally in-line with current trends \citep[e.g.][]{kreidberg14,Benneke2015} -- any interpretation beyond that would not be supported by these data-sets. The retrieval also provides an upper limit on the TiO volume mixing ratio of 10$^{-7.6}$ at 99\% confidence. This is about one order of magnitude lower than the expectation from 3$\times$solar 1D-RC model. It has long been recognized that TiO could potentially play a major role in causing thermal inversions in hot Jupiter atmospheres \citep[e.g.,][]{Hubeny2003, Fortney2008}. However, detections of this species have been rare and not without controversy \citep[see][and references therein]{prinoth22}. KELT-20b has a strong thermal inversion, yet we and others have failed to find TiO in the emission spectrum \citep{Yan22, borsa22, Johnson22}. \section{Radiative-Convective Equilibrium Atmosphere Models} \label{sec:RCEmodels} The lack of TiO in the atmosphere of KELT-20b raises the question of what causes the strong thermal inversion. To explore this issue we modeled a number of scenarios with the \texttt{PHOENIX} 1D self-consistent radiative-convective equilibrium atmosphere model \citep{hauschildt:1999,Barman2001}. Our ultra-hot Jupiter model set-up was similar to those presented in \cite{lothringer18}. We first modeled the atmosphere of KELT-20b with all available opacities, including atomic opacity from hydrogen up to uranium, continuous opacity sources like H$^{-}$, and molecular opacity, including TiO and VO. The resulting spectrum shows a qualitative match to the low-resolution \textit{HST} and \textit{Spitzer} observations of \citet{Fu22}, in agreement with the \texttt{ScCHIMERA} self-consistent models. However, we also computed a model without TiO and VO, since we do not find evidence of these species with our high-resolution observations. The results are summarized in Figure~\ref{fig:phoenix}. While the \texttt{PHOENIX} models with TiO and VO opacity artificially removed are a better match to the results of our (and others') high-resolution observations, importantly, they do not match the low-resolution, space-based observations. This is due to the fact that without TiO and VO, the heating from the absorption of the short-wavelength stellar irradiation by species like the atomic metals is balanced by cooling by molecules like H$_{2}$O and CO. An inversion only forms at lower pressures once these latter molecules have thermally dissociated and can no longer radiatively cool the atmosphere. This means that the region that H$_{2}$O probes in the thermal emission spectrum is below the large temperature inversion and we would not expect to see H$_{2}$O in emission (see Figure \ref{fig:retrieval}). On the other hand, when TiO and VO are present, they can heat up the atmosphere in the region that H$_2$O probes, resulting in a strong H$_2$O emission feature. Thus, the non-detection of TiO and VO at high-resolution is in tension with the strong H$_2$O emission feature found by \cite{Fu22}. \citet{lothringer19} pointed out the importance of the host star spectral energy distribution in setting the thermal structure of highly irradiated planets. \citet{Fu22} suggested that KELT-20b's A-type host star, with its higher proportion of short-wavelength flux than the typical hot Jupiter host star, is likely responsible for the large water emission feature seen in the planet's spectrum. Therefore, if TiO is not present to cause the thermal inversion at the deep pressures where H$_2$O is still intact then it could be that the models are missing opacity at the short wavelengths where the host star is particularly bright. \section{Discussion} \label{sec:discussion} Our detection of atomic emission lines agrees with the broad conclusions of recent publications analyzing the dayside of the ultra-Hot Jupiter KELT-20b: there is a thermal inversion in the atmosphere and neutral iron (as opposed to TiO or VO) is the dominant optical opacity source observed in the spectrum \citep{borsa22,Yan22,Johnson22}. Additionally, for the first time with KELT-20b we search for and find Ni\,\textsc{i} (at 4.7$\sigma$ confidence). KELT-20b is one of the coolest exoplanets known to have a thermal inversion and it has the strongest water emission feature seen in over two dozen hot Jupiter emission spectra obtained with \textit{HST}/WFC3 \citep{Fu22}. However, there have been no detections of TiO or atomic Ti in either its emission spectrum or in the many observations of its transmission spectrum \citep{belloarufe22, Langeveld22, Rainer21, nugroho20, Stangret20, hoeijmakers20, Kesseli20, Casasayas19}. Given the presence of the other refractory species that have been detected for this planet (in addition to those described already, Na, Mg, Ca, and Cr species have also been detected in the transmission spectrum), sequestration of Ti in the unobservable parts of the atmosphere due to cold trapping seems a likely explanation \citep{Spiegel2009, Parmentier2013}. Intriguingly, models with TiO cold trapping can also fit the ensemble of WFC3 spectra \citep[see Figure 4 in][]{Mansfield2021}. This fact combined with the rarity of TiO detections suggests that cold-trapping of Ti might be a common phenomenon in hot and ultra-hot Jupiter atmospheres. More work is needed to explore this emerging population-level trend. \citet{lothringer21} have proposed that the ratio of refractory and volatile species abundances is an important tracer of planet formation. KELT-20b is one of the few planets where Fe, H$_2$O, and CO spectral features are present in the thermal emission spectrum. The thermal emission spectrum typically arises from deeper atmospheric layers than the transmission spectrum, which can probe such high altitudes that mass fractionation of different chemical species can be an issue, thus complicating the interpretation of abundance measurements. By exhibiting Fe, H$_2$O, and CO features that arise from the bulk atmosphere, KELT-20b in principle presents an important opportunity to connect Fe/O and Fe/C abundance ratios to models of giant planet formation. We have constrained the abundances of Fe\,\textsc{i}, H$_2$O, and CO for KELT-20b by performing a unified retrieval on ground-based, high-resolution optical spectroscopy and space-based low-resolution infrared spectroscopy. This extends our previous work by \citet{bro17}, which only conditioned an analysis of high-resolution data on a low-resolution retrieval, and the work of \citet{Brogi2019}, which developed our joint retrieval framework on simulated data. To our knowledge only one other such retrieval has been performed on real data before, by \citet{Gandhi19}. Our measured Fe\,\textsc{i}, H$_2$O, and CO abundances and their ratios are consistent with a solar elemental Fe, C, and O under the assumption of 1D radiative-convective-thermochemical equilibrium. Further observations of KELT-20b could build on this work to determine more precise abundances of these elements as a constraint on the planet's formation. Of particular value would be a measurement of the mass of the planet, broader wavelength space-based spectroscopy from the \textit{James Webb Space Telescope}, and ground-based observations to probe the H$_2$O and CO lines at higher spectral resolution \citep[e.g.,][]{line2021, Pelletier2021, vansluijs22, yan22b, holmberg22}. In addition, advances in retrieval techniques are needed to ensure that the derived abundances are also accurate. A key limitation of the current state-of-the-art retrievals applied to high-resolution spectroscopy is the predominant assumption of 1D atmospheres and constant-with-altitude abundances. Future work that incorporates variations of temperature and abundance with longitude and altitude \citep[e.g.][]{gandhi22} will be critical to ensuring valid composition inferences as we continue to push towards higher fidelity datasets. \begin{acknowledgements} The University of Chicago group acknowledges funding for the MAROON-X project from the David and Lucile Packard Foundation, the Heising-Simons Foundation, the Gordon and Betty Moore Foundation, the Gemini Observatory, the NSF (award number 2108465), and NASA (grant number 80NSSC22K0117). We thank the staff of the Gemini Observatory for their assistance with the commissioning and operation of the instrument. J.L.B.\ and M.R.L.\ acknowledge support from NASA XRP grant 80NSSC19K0293. M.B.\ acknowledges support from the STFC research grant ST/T000406/1. This work was enabled by observations made from the Gemini North telescope, located within the Maunakea Science Reserve and adjacent to the summit of Maunakea. We are grateful for the privilege of observing the Universe from a place that is unique in both its astronomical quality and its cultural significance. \end{acknowledgements} \facilities{Gemini-North (MAROON-X), Hubble Space Telescope(WFC3), and Spitzer(IRAC)} \software{\texttt{astropy} \citep{astropy:2018}, \texttt{barycorrpy} \citep{barycorrpy}, \texttt{corner} \citep{corner}, \texttt{matplotlib} \citep{matplotlib:2007}, \texttt{numpy} \citep{numpy:2020}, \texttt{python} \citep{python3:2009}, \texttt{scipy} \citep{2020SciPy-NMeth}} \newpage \appendix \bibliography{kelt20_dayside_2022}{} \bibliographystyle{aasjournal}

Title: Effect of impact velocity and angle on deformational heating and post-impact temperature

Abstract: The record of impact induced shock-heating in meteorites is an important key for understanding the collisional history of the solar system. Material strength is important for impact heating, but the effect of impact angle and impact velocity on shear heating remains poorly understood. Here, we report three-dimensional oblique impact simulations, which confirm the enhanced heating due to material strength and explore the effects of impact angle and impact velocity. We find that oblique impacts with an impact angle that is steeper than 45 degree produce a similar amount of heated mass as vertical impacts. On the other hand, grazing impacts produce less heated mass and smaller heated regions compared to impacts at steeper angles. We derive an empirical formula of the heated mass, as a function of the impact angle and velocity. This formula can be used to estimate the impact conditions (velocity and angle) that had occurred and caused Ar loss in the meteoritic parent bodies. Furthermore, our results indicate that grazing impacts at higher impact velocities could generate a similar amount of heated material as vertical impacts at lower velocities. As the heated material produced by grazing impacts has experienced lower pressure than the material heated by vertical impacts, our results imply that grazing impacts may produce weakly shock-heated meteorites.

https://export.arxiv.org/pdf/2208.07630

\title{Effect of impact velocity and angle on deformational heating and post-impact temperature} \authors{S. Wakita\affil{1,2}, H. Genda\affil{3}, K. Kurosawa\affil{4}, T. M. Davison \affil{5}, and B. C. Johnson \affil{1,6}} \affiliation{1}{Department of Earth, Atmospheric, and Planetary Sciences, Purdue University, West Lafayette, IN, USA} \affiliation{2}{Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA} \affiliation{3}{Earth-Life Science Institute, Tokyo Institute of Technology, Meguro, Japan} \affiliation{4}{Planetary Exploration Research Center, Chiba Institute of Technology, Narashino, Japan} \affiliation{5}{Department of Earth Science and Engineering, Imperial College London, London, UK} \affiliation{6}{Department of Physics and Astronomy, Purdue University, West Lafayette, IN, USA} \correspondingauthor{Shigeru Wakita}{shigeru@mit.edu} \begin{keypoints} \item We examined the dependence of impact heating on impact angle and velocity using a shock physics code. \item The amount of heated mass is similar among $>45^\circ$ impacts, while it is smaller for shallower impacts. \item We derived an empirical formula for the cumulative heated mass over 1000 K during oblique impacts. \end{keypoints} \section*{Plain Language Summary} Meteorites are extraterrestrial materials that have been delivered to the Earth from asteroids. The materials in meteorites can record information about their formation and subsequent evolution. Thus, they are excellent sources of information used to explore the history of the solar system. One such feature recorded is evidence of shock: high pressures and temperatures caused by collisions between asteroids. Previous work investigating impacts found that material strength is a key factor in determining the amount of impact heating, especially in low-speed collisions, like those expected to occur in the main asteroid belt. In this work, we explore various oblique incidence impacts to study the effects of material strength by using a shock physics model. We confirm that material strength plays a key role in oblique impacts, just as in head-on impacts. Our results show that head-on and 45 degree impacts can generate nearly the same amount of heated mass in total. However, more oblique impacts with a shallower angle produce less heated mass than other steeper-angle impacts (i.e., head-on and 45 degree impacts). We also find that low-speed vertical impacts and high-speed grazing impacts can produce the same amount of material in asteroids that have experienced a given temperature. \section{Introduction} Asteroids may impact other terrestrial bodies at various impact velocities and angles. Current asteroids collide with each other at a mean impact velocity of 5 km/s \cite{Bottke:1994,Farinella:1992}, while asteroids hit the Moon and Mars at over 10 km/s \cite{Ivanov:2001,Yue:2013}. We observe the evidence of impacts as craters on Moon, Mars, and asteroids, even on small-sized bodies, like Ryugu and Bennu \cite{Walsh:2019,Morota:2020}. As asteroids have experienced such collisions over the history of the solar system, exploring ancient evidence of impacts can reveal the early collisional history of asteroids \cite<e.g.,>{Sugita:2019}. Ejecta produced by impacts on asteroids sometimes travel to the Earth and are found as meteorites. Shock metamorphism in meteorites provides evidence of past impacts on their parent bodies \cite<e.g.,>{Keil:1994,Scott:2002}. Using shock-induced textures in meteorites, we categorize them by the degree of shock metamorphism \cite{Stoffler:1991, Scott:1992, Rubin:1997, Stoffler:2018}. When a strong shock propagates in the parent body, it might lead to the melting of materials. Weak shocks, however, are unable to produce melted textures; weakly shocked meteorites have experienced shock pressure less than 40 GPa. Nevertheless, weakly shocked meteorites also show evidence of moderate impact heating ($\sim700-800$ $^\circ$C), such as the dehydration of phyllosilicate minerals \cite{Nakamura:2005,Nakato:2008,Abreu:2013} and the reset of $^{40}$Ar-$^{39}$Ar ages \cite{Bogard:1995,Bogard:2011,Weirich:2012,Cohen:2013}. Thus, studying impact heating is essential for a better understanding of the solar system. Oblique impacts are more likely to occur than head-on impacts in the solar system \cite{Shoemaker:1962}. The most probable impact angle ($\theta_{\rm imp}$) is $45^\circ$. The probability of impact occuring with an impact angle between $\theta_{\rm imp}$ and $\theta_{\rm imp} + d\theta_{\rm imp}$ is given as $\sin(2\theta_{\rm imp}) d\theta_{\rm imp}$ \cite{Shoemaker:1962}. To understand the nature of oblique impacts, numerical simulations have been performed using smoothed particle hydrodynamics \cite<SPH; e.g.,>{Genda:2012,Monaghan:1992,Benz:1994,Jutzi:2015,Sugiura:2021,Okamoto:2020,Citron:2022} and grid-based hydrodynamic codes \cite<CTH, SOVA, iSALE-3D; e.g.,>{Pierazzo:2000, Pierazzo:2000a, Elbeshausen:2009,Elbeshausen:2011,Elbeshausen:2013,Davison:2014,Artemieva:2017,Wakita:2019a}. Previous work has examined the crater volume and heated mass \cite{Pierazzo:2000, Pierazzo:2000a, Elbeshausen:2009, Elbeshausen:2013, Davison:2011, Davison:2014}. The volume heated to any given temperature depends on the impactor diameter, mass, velocity, and angle \cite{Pierazzo:2000a}. Recently, the material strength in the target has been recognized as an additional important parameter for heating \cite{Quintana:2015, Kurosawa:2018, Kurosawa:2021a, Wakita:2019, Wakita:2019a}. As extraterrestrial materials have experienced impacts on their parent asteroids, impact heating is crucial to understand their record. When we consider the material strength in rocks, the temperature increase is much higher than previously expected \cite<e.g.,>{Kurosawa:2018}. Dissipation of the kinetic energy due to plastic deformation in pressure-strengthened rocks is equivalent to an increase in internal energy, leading to higher temperatures during, and after, decompression. \citeA{Kurosawa:2018} confirmed the post-shock heating due to plastic deformation in head-on impacts using a shock physics code. Although the following work explored an oblique impact of 45$^\circ$ at 5 km/s \cite{Wakita:2019a}, impacts at a variety of impact angles and velocities must be explored to better understand the effects of deformational heating. \citeA{Davison:2014} showed the dependence of the heated mass on impact angle, however, the estimated post-shock temperature is based on the peak shock pressure, and ignores enhanced deformational heating \cite{Kurosawa:2018}. Here we perform oblique impact simulations with material strength to examine the dependence of impact heating on impact velocities and angles. While we vary impact velocity and angle systematically with the same method as in previous work \cite{Wakita:2019a}, we also conduct a companion series of the impacts without material strength. Simulations without material strength have no deformational heating. Thus, comparison of simulations with and without strength allows us to quantitatively determine the effect of deformational heating \cite{Kurosawa:2018, Wakita:2019a}. Considering 1000 K as a reference temperature, which is the temperature for Ar age resetting \cite<following previous work, e.g.,>{Marchi:2013}, we provide an empirical relationship of the heated mass which exceeds that temperature. \section{Methods} We use the iSALE-3D shock physics code \cite{Elbeshausen:2009,Elbeshausen:2011,Collins:2016}, to simulate a spherical impactor striking a flat-surface target at oblique angles. iSALE-3D uses a solver as described in \citeA{Hirt:1974}. iSALE-2D is limited to simulating vertical-incidence impacts due to its use of axial symmetry. Thus, to simulate a range of impact angles, we use the fully three-dimensional version, iSALE-3D. The results of iSALE-2D and vertical impacts of iSALE-3D have previously been shown to agree well \cite<e.g.,>{Elbeshausen:2009, Davison:2011, Davison:2014, Raducan:2022}. Note that some examined about crater formation and impact ejecta, but \citeA{Davison:2014} compared the impact heating (similar to ours but without the shear heating). Since material strength is important for studying the effect of impact heating, we employ a strength model appropriate for rocky materials \cite<see below,>{Collins:2004,Melosh:1992,Ivanov:1997}. It is well known that porous targets produce more melt than non-porous targets in iSALE-2D \cite{Wunnemann:2006,Davison:2010}, and a porosity compaction model is implemented into the iSALE-3D \cite{Wunnemann:2006,Collins:2011}. However, to reduce the parameter space and compare it with previous work \cite<e.g.,>{Kurosawa:2018}, we only consider non-porous materials in this study. Nevertheless, we can apply our results in this study to well-consolidated rocky materials, such as ordinary chondrites having a porosity less than 10\% \cite<e.g.,>{Flynn:2018,Ostrowski:2019}. We assume the impactor and the target have the same composition of dunite, using the ANEOS equation of state \cite{Benz:1989}. In this work, we model dunite as non-porous with material strength. Since the dunite has a well-defined equation of state \cite{Benz:1989}, it is widely used to simulate the bodies in the inner solar system \cite<e.g.,>{Davison:2010, Johnson:2015}. It also represents meteoritic material well \cite<ordinary chondrite,>{Svetsov:2015}. Thus, material parameters for dunite without porosity in this work are representative of compact bodies in the inner solar system. As previously noted, simulations without strength are only used to quantify the effects of material strength and are not meant to represent specific solar system bodies. For the input parameters, we take the same values shown in \citeA<Table S1 of>{Kurosawa:2018}. When we simulate impacts with material strength, we use two models to describe the yield strength of intact and damaged rock; the Lundborg strength model \cite{Lundborg:1968} and the Drucker-Prager model \cite{Drucker:1952}, respectively. We combine these two models using a damage parameter $D$ which depends on total plastic strain \cite<e.g.,>{Collins:2004}. $D$ ranges from 0 (intact rocks) to 1 (thoroughly fractured rock). A damage parameter $D$ is initially set as 0 and damage is accumulated as material deforms and accumulates plastic strain according to the damage model of \citeA{Ivanov:1997}. When the shock pressure exceeds the Hugoniot elastic limit, materials become thoroughly damaged. We find material is completely damaged out to approximately 4 impactor radii from the point of impact (see also Fig. 5 of \citeA{Wakita:2019a}). The yield strength $Y$, that of intact rock $Y_i$, and that of damaged rock $Y_d$ are written as follows, \begin{equation} Y = (1-D)Y_i + DY_d, \label{eq:D} \end{equation} \begin{equation} Y_i = Y_{\rm coh,i} + \cfrac{\mu_{\rm int}P}{1+\cfrac{\mu_{\rm int}P}{Y_{\rm limit}-Y_{\rm coh, i}}}, \label{eq:Lund} \end{equation} \begin{equation} Y_d = \rm min (Y_{\rm coh}+\mu P, Y_{\rm i}), \label{eq:DrPr} \end{equation} where $Y_{\rm coh,i}$ is the cohesion for intact rock at zero pressure, $\mu_{\rm int}$ is the coefficient of internal friction for intact rock, $P$ is the temporal mean pressure, $Y_{\rm limit}$ is the limiting strength at high pressure, and $\mu$ is the damaged friction coefficient, respectively. To simultaneously handle both intact and thoroughly fractured rocks (Equation 1), we use the Lundborg model for intact rock (Equation \ref{eq:Lund}) and the Druker-Prager model for the thoroughly fractured rock (Equation \ref{eq:DrPr}), respectively. The damaged friction coefficient $\mu$ is one of the important parameters for impact heating \cite{Kurosawa:2018,Wakita:2019, Wakita:2019a}. The dependence of the damaged friction coefficient has been explored and the smaller value represents the case without material strength \cite{Kurosawa:2018}. Following the fiducial case in their work, we also adopt $\mu$ = 0.6 which is a typical value pertaining to granular materials made of rocky materials \cite<e.g.,>{Collins:2004}. Note that the limiting strength $Y_{\rm limit}$ is another influential parameter \cite<see also Figure S1 of>{Kurosawa:2018}. The limiting strength, also known as the von-Mises strength, is the maximum strength material can have regardless of confining pressure. Following \citeA{Kurosawa:2018}, we use $Y_{\rm limit}$ of 3.5 GPa in this work. To examine the dependence of impact heating on impact properties, we vary the impact velocity ($v_{\rm imp}$ = 2, 3, 5, 7, 10 km/s) and the impact angles ($\theta_{\rm imp}$ = 15$^\circ$, 30$^\circ$, 45$^\circ$, 60$^\circ$, 75$^\circ$, 90$^\circ$). Note that we measure the impact angles from the target surface, i.e., 90$^\circ$ is a vertical impact. We fix a radius of the impactor ($R_{\rm imp}$) as 2 km with a resolution of 0.1 km, which corresponds to 20 cells per projectile radius (CPPR). In this study, we assume that the size ratio of the impactor to the target is small enough to neglect the curvature of the target body. A resolution of 20 CPPR is sufficient to resolve cumulative heated mass as demonstrated in previous work \cite<see Text S3 and Figure S2 in>{Wakita:2019a}. As the numerical diffusion exaggerates the temperature of material near the contact boundary between the impactor and the target \cite{Kurosawa:2018}, we need to omit these artificial overheated regions from further analysis. In our case, that region corresponds to 0.3 times of impactor mass ($M_{\rm imp})$. Though we count that region in the results, we do not discuss the results that are less than 0.3 $M_{\rm imp}$ (see following sections). We place Lagrangian tracer particles in each cell of a high-resolution zone at the initial state of simulations. Note that the computational region in iSALE-3D consists of a high-resolution zone and an extension zone. The cell size in the extension zone is larger than that in the high-resolution zone and increases according to the distance from the boundary between the high-resolution zone \cite<see also Figure 1 of>{Davison:2011}. To track the highly heated region (i.e., $>$ 1000 K), we take the number of high-resolution cells as 220 in horizontal direction (x), 100 in vertical direction (z), and 150 in depth direction (y), respectively. We also note that we horizontally shift the origin of the impact point according to the impact angle; the origin is located in the middle at $\theta_{\rm imp}$ = 90$^\circ$ and it is shifted 20 cells in the downrange direction at $\theta_{\rm imp}$ = 45$^\circ$. The total number of tracer particles is as large as $\sim 2 \times 10^6$. The Lagrangian tracers move through the Eulerian grid tracking the temperature and movement of a parcel of material. Note that we neglect radiative and conductive cooling during impact events. These are less effective than the cooling due to expansion over the timescales considered here \cite<e.g.,>{Sugita:2002}. \section{Results} The heated region depends on the impact obliquity. Figure \ref{fig:material} represents the distribution of peak temperature ($T_{\rm peak}$) at a time of 5 $t_s$ after impact, in a suite of simulations with material strength and an impact velocity of 5~km/s, where $t_s$ is a characteristic time for impactor penetration, $t_s = 2R_{\rm imp}/v_{\rm imp}$. As shown in the provenance plots (right panels), the heated area becomes shallower for more oblique impacts (see dashed lines, which show the $T_{\rm peak}=1000$~K isotherm). Comparing impacts with and without material strength (Figure \ref{fig:hydro}), we can confirm material strength enhances the temperature increase. This is consistent with previous work \cite{Kurosawa:2018,Wakita:2019a}. As confirmed in \citeA{Kurosawa:2018}, more kinetic energy is converted into internal energy in the target with material strength. Since deformational heating cannot occur in simulations without material strength, we observe such additional heating only in case with material strength. Most material has reached its peak temperature at 2.5 $t_s$, as the isothermal line of 1000 K indicates (see white dotted line and black dashed line in Figure \ref{fig:a45} (b)). To ensure material has reached its peak temperature we focus our analysis of peak temperatures reached before 5 $t_s$. Also, we have confirmed that the cumulative heated mass is similar at 2.5 -- 10 $t_s$ and does not change significantly after 5 $t_s$ \cite<see Figure S5 in>{Wakita:2019a}. To examine the effect of material strength, we compare the cumulative heated mass in the target with/without material strength (see Figure \ref{fig:cummass}). Note that we consider the number of the tracer particles for which the peak temperature is beyond a given $T_{\rm peak}$, and regard their total mass as the cumulative heated mass. As shown in Figure \ref{fig:cummass} (c) ($v_{\rm imp}$ = 5 km/s), the heated mass with material strength (right-hand panel) is always larger than that without material strength (pure hydrodynamic case, left-hand panel) at a given impact angle. The difference between the case with/without material strength reaches a factor of ten at the most. Material strength enhances the impact heating regardless of $\theta_{\rm imp}$. On the other hand, the effect of shear heating for the cumulative heated mass depends on $v_{\rm imp}$. For lower impact velocity scenarios (Figure \ref{fig:cummass} (a) and (b)), our results show the cumulative heated mass in the target with material strength is $\sim$10 times larger than that without material strength. Previous work indicated a combination of the material strength and movement of the impactor results in the enhanced heating in the oblique impacts \cite{Wakita:2019a}. Since the heated mass without material strength at lower velocity has a larger difference between vertical and oblique impacts, it implies that material strength is more effective than a movement of the impactor. The heated mass without material strength approaches that with material strength as $v_{\rm imp}$ increases (see Figure \ref{fig:cummass} (d)). This is also consistent with previous findings \cite{Quintana:2015,Kurosawa:2018}. As a result, the difference between the heated mass of 90$^\circ$ and 45$^\circ$ in the case without material strength approaches to that with material strength (Figure \ref{fig:cummass} (d)). Thus, the material strength more effectively increases for the cumulative heated mass in lower $v_{\rm imp}$ scenarios than higher $v_{\rm imp}$. We now consider the results with material strength and discuss the effect of $\theta_{\rm imp}$ on impact heating. Previous work focused on oblique impacts at $v_{\rm imp}$ = 5 km/s showed that the cumulative heated mass of 90$^\circ$ and 45$^\circ$ are almost the same \cite{Wakita:2019a}. We find the heated mass of 75$^\circ$ and 60$^\circ$ are between that of 90$^\circ$ and 45$^\circ$, and their difference is within a factor of 1.5 (Figure \ref{fig:cummass} (c)). On the contrary, we find that shallower impacts ($\theta_{\rm imp} \le 30^\circ$) produce much less heated mass than 45$^\circ$ impacts. Grazing impactors are decapitated before they penetrate into the target and heating is limited mainly to the lower hemisphere \cite{Schultz:1990,Davison:2011}. While the heated area of $\theta_{\rm imp} \le 30^\circ$ impacts has a similar width as the 45$^\circ$ impacts, the shallower penetration of grazing impactors results in the heated area extending to smaller depths (see dashed lines in Figure \ref{fig:material} (c) and (d)). As a result, the heated mass of $\theta_{\rm imp} \le 30^\circ$ becomes smaller than other higher angle impacts. Note that it is beyond the focus of our paper to find the threshold impact angle where cumulative heated mass begins decreasing. While the cumulative heated mass takes a similar value for $>$ 45$^\circ$ impacts, the cumulative heated mass by $<$ 30$^\circ$ impacts is always less than that regardless of the impact velocities ($v_{\rm imp}$). Our results show impacts of $\theta_{\rm imp} \ge 45^\circ$ produce a similar heated mass within less than a factor of 1.5 (right panels in Figure \ref{fig:cummass}). Note that the results saturates around $M_{\rm target}^{\rm MS}/M_{\rm imp} \sim 100$ (see Figure \ref {fig:cummass} (d)), because our total computational region in the target that we track with tracer particles is as large as $M_{\rm target}/M_{\rm imp} \simeq 10^2 $ (see Methods). On the contrary, the ratio of the cumulative heated mass with shallower angles ($\theta_{\rm imp} \le 30^\circ$) to that with $\theta_{\rm imp}=45^\circ$ ($M_{\rm target}^{\rm MS} (\theta_{\rm imp})/M_{\rm target}^{\rm MS} (\theta_{\rm imp} = 45^\circ))$ is $\sim$ 0.6 ($\theta_{\rm imp}=30^\circ$) and $\sim$ 0.2 ($\theta_{\rm imp}=15^\circ$). Nevertheless, the heated mass from shallower angle impacts ($\theta_{\rm imp} \le 30^\circ$) is less than steeper angle impacts, the effect of material strength on the degree of impact heating is still significant (see Figure \ref{fig:cummass}). \section{Discussion} Here we discuss the impact induced heated materials using our results. If the impact heats the target enough beyond a threshold temperature, it could trigger Ar loss and reset the target's $^{40}$Ar-$^{39}$Ar age. \citeA{Kurosawa:2018} showed that the threshold of impact velocity for Ar loss would be 2 km/s in the target with the material strength, which is lower than 8 km/s in the case without material strength. We investigate the cumulative heated mass of the impact-induced Ar age resetting during oblique impacts, by assuming 1000 K as an index temperature, which is the closure temperature of typical Ar carrier mineral \cite<e.g., feldspar,>{Cohen:2013}. Figure \ref{fig:tpeak} summarizes the cumulative heated mass of $T_{\rm peak} > 1000$ K in case of the material strength. The dependence of the cumulative heated mass on $\theta_{\rm imp}$ are similar regardless of $v_{\rm imp}$; the heated mass of $\theta_{\rm imp} \ge 45^\circ$ are always larger than $\theta_{\rm imp} \le 30^\circ$ at given $v_{\rm imp}$. The high-velocity impacts produce a larger amount of the cumulative heated mass than the low-velocity case. While $^{40}$Ar-$^{39}$Ar age has been used to estimate the latest impact event on the parent body of meteorites \cite{Bogard:2011}, it is also possible to estimate their original depth. When $M^{\rm MS}_{\rm target}/M_{\rm imp}$ is sufficiently large (e.g., $\ge 1$), we can regard such an impact condition as resetting the Ar age. Thus, our results showed that grazing impacts (e.g., $\theta_{\rm imp} = 30^\circ$ with 5 few km/s) can contribute to the Ar age resetting (Figure \ref{fig:tpeak}). Since grazing impacts excavate relatively shallower material (see Figure \ref{fig:material} (d)), it may imply that meteorites might originate from shallower depth than previously thought. We need to mention that the usage of $T_{\rm peak}$ can exaggerate the cumulative heated mass. Because the $T_{\rm peak}$ records the temperature during the shock, it may be overestimated due to the numerical diffusion. Even if the $T_{\rm peak}$ is the same, the difference in pressure may indicate the different status (i.e., shock state or post-shock state). In such a case, especially for the phenomena that would take a time to occur (e.g., dehydration), the post-shock temperature ($T_{\rm post}$, the temperature after the pressure becomes less than $10^5$ Pa) can be useful. When we compare the peak temperature with the post-shock temperature, we find that the former is overestimated by about 100 K in comparison with the latter at $T_{\rm peak}=$ 1000 K in the case of $v_{\rm imp}=5$ km/s (Figure \ref{fig:diff}). Although $T_{\rm post}$ is a more accurate way, it is computationally expensive to examine all tracer particles and incapable in our current work. Because some particles, that are at the contact between the impactor and the target, take a longer time ($> 5 t_s$) to decrease pressure to $10^5$ Pa (see white region in Figure \ref{fig:tdiff_plot}). However, the difference between $T_{\rm peak}$ and $T_{\rm post}$ are acceptable and our estimate on $T_{\rm peak}$ is appropriate in accounting the cumulative heated mass. \citeA{Kurosawa:2018} calculated the cumulative heated mass for Ar loss using the entropy corresponding to 1000 K at $10^5$ Pa. Our result of $M^{\rm MS}_{\rm target}/M_{\rm imp} = 7.04$ with a vertical impacts at $v_{\rm imp}=$ 10 km/s are within $11\%$ difference from their result of 6.32 \cite<see Figure 3 in>{Kurosawa:2018}. This agreement, suggests that the use of peak temperature to estimate the cumulative heated mass after the impact-induced shock heating is reasonable. It is worth exploring the dependence of the cumulative heated mass on impact angle and velocity. To estimate the cumulative heated mass at various impact velocities and angles, we derive an empirical formula. Note that an artificial increase in temperature around the contact boundary between impactor and target in numerical simulations is reported \cite{Kurosawa:2018}. We use the numerical results over 0.3 $M_{\rm imp}$ which are free from the artificial overheating and thus more conservative. To minimize the number of coefficients in the empirical formula, we adopt the following equation, \begin{linenomath*} \begin{equation} M^{\rm formula}_{\rm target} (T_{\rm peak})/M_{\rm imp} = \left\{ C_1 \sin(\theta_{\rm imp}) + C_2 \sin^2(\theta_{\rm imp}) + (1-C_1-C_2)\sin^3(\theta_{\rm imp}) \right\} C_3(0.5 v^2_{\rm imp}/E_{T_{\rm peak}})^{C_4}, \label{eq:MS} \end{equation} \end{linenomath*} where $E_{T_{\rm peak}}$ is the specific internal energy at $T_{\rm peak}$. The procedure to obtain the empirical formula is described as follows. We prepare five datasets at a given impact velocity (2, 3, 5, 7, and 10 km/s), which are the normalized heated mass as a function of impact angle. We confirm that the five datasets almost converge into a single trend against the change in impact angle. Then, we fit the all data with a third order polynomial function with two physical constraints, which are (1) the normalized heated mass equals zero at $\theta_{\rm imp} = 0^\circ$, and (2) the normalized heated mass equals unity at $\theta_{\rm imp} = 90^\circ$. Thus, the dependence of impact angle on the heated mass at a given impact velocity can be described with two fitting coefficients ($C_1$ and $C_2$ within bracket in Equation \ref{eq:MS}). Next, we divide the heated masses by the effects of the impact angle, resulting in the corrected heated mass depending only on impact velocity, which corresponds to the specific internal energy. The corrected data points can be fitted well by a power-law with two fitting constants ($C_3$ and $C_4$ in Equation \ref{eq:MS}). By combining impact angle and velocity dependence, we obtain the empirical formula shown as Equation (\ref{eq:MS}) with four coefficients. For the case of $T_{\rm peak}=$ 1000 K (Figure \ref{fig:tpeak}), we find that the coefficients are $C_1=-0.249$, $C_2=3.40$, $C_3=1.07$, and $C_4=0.749$, respectively. Note that $E_{T_{\rm peak}} = 4.16$ MJ/kg at $T_{\rm peak}=$ 1000 K. Figure \ref{fig:tpeak_comp} represents that the empirical formula works properly within the error of $\pm 21\% (2 \sigma)$ to our numerical results with material strength. This formula would be useful to estimate the cumulative heated mass resulting from various impact conditions. We also discuss the occurrence of dehydration reactions of phyllosilicate (e.g., serpentine). As the dehydration of hydrous materials is an endothermic reaction, additional calculations are required to assess the dehydrated mass accurately \cite<e.g.,>{Kurosawa:2021a}. Additionally, \citeA{Kurosawa:2021a} experimentally showed that the efficiency of impact devolatilization on carbonaceous asteroid-like materials (e.g., calcite) is considerably low ($<10\%$ of theoretical prediction). Thus, the dehydrated mass estimated by numerical simulations will overestimate regardless of considering the reaction heat. Although we may overestimate the heated mass by using peak temperature but ignoring the reaction heat, it is worth considering the impact conditions that may induce the dehydration. We here take a temperature threshold for the dehydration as 873 K, at which the dehydration of phyllosilicate starts to occur \cite{Lange:1982,Nozaki:2006,Nakato:2008}. Note that the Hugoniot curve for dunite and serpentine differs \cite{Benz:1989,Brookshaw:1998}. Based on shock heating alone, the serpentine will reach a similar temperature of the dunite at a given impact condition. Thus, the amount of dehydrated mass may not change even if the equation state of serpentine is used. Please also note that our numerical setup is different from previous work of investigating the fate of hydrous minerals during the impacts which suggested that the serpentine in the core can avoid the dehydration \cite{Wakita:2019}. Since they considered a dunite layer over the serpentine core, the dunite layer might insulate the serpentine core from the direct impact-induced heating that we explore in this work. The vertical impact with material strength at $v_{\rm imp}$ = 2 km/s produces dehydrated materials of $M^{\rm MS}_{\rm target}/M_{\rm imp} \sim 0.4$ (Figures \ref{fig:cummass} and \ref{fig:tpeak_873K}). Grazing impacts ($\theta_{\rm imp} \le 30^\circ$) into the target with material strength require a higher impact velocity to produce a similar amount of impact heated material as vertical impacts (see also Figure \ref{fig:tpeak_873K}); $\theta_{\rm imp} = 30^\circ$ with $v_{\rm imp}$ = 3 km/s ($M^{\rm MS}_{\rm target}/M_{\rm imp} \sim 0.5$) and $\theta_{\rm imp} = 15^\circ$ with $v_{\rm imp}$ = 5 km/s ($M^{\rm MS}_{\rm target}/M_{\rm imp} \sim 0.4$). Nevertheless, those grazing impacts with material strength require lower velocities than vertical impacts without material strength to produce a similar amount of impact heated material (at an impact velocity of $v_{\rm imp}$ = 7 km/s, $M^{\rm Hydro}_{\rm target}/M_{\rm imp} \sim 0.5$, Figure \ref{fig:tpeak_873Khydro}). Oblique impacts generate heated materials at lower peak pressure than vertical impacts at a given impact velocity \cite{Wakita:2019a}. We also find grazing impacts at higher impact velocities could produce dehydrated material at lower peak pressures than vertical impacts with lower impact velocity (see Figure \ref{fig:ppeak_tpeak}). As the high temperature region of oblique impacts is widely distributed (see Figure \ref{fig:material}), the weakly shocked region is also close to the surface, potentially to be ejected. If this ejected material eventually lands on the Earth as meteorites, this implies that shock heated meteorites could have experienced a wide range of peak pressures. Thus, oblique impacts of $\theta_{\rm imp} < 45^\circ $ may have produced dehydrated minerals in weakly shock-metamorphosed meteorites. The time takes for hydrous minerals to dehydrate depends on the reaction temperature. While dehydrated materials in carbonaceous chondrites indicate shock heating \cite<e.g.,>{Nakamura:2005,Nakato:2008,Abreu:2013}, carbonaceous chondrites have generally experienced weak shocks. The dehydration starts to occur at about 873 K \cite<600 $^\circ$C,>{Lange:1982,Nozaki:2006,Nakato:2008}, but some work examines the duration time at higher temperature. \citeA{Nozaki:2006} conducted heating experiments on carbonaceous chondrites: Both short (10 s) and long (120 s) heating at a temperature of 1173 K (900 $^\circ$C) decomposed the hydrous minerals. They indicated that the temperature is more important for dehydration than duration. Other experimental work implies that a weakly-shocked and dehydrated carbonaceous chondrite would have experienced 1--100 hour heating at 1173 K or a 10-1000 days heating at 973 K (700 $^\circ$C) \cite{Nakato:2008}. While our impact simulations are unable to consider the duration materials spend at elevated temperatures, we can estimate the cooling time of the hydrous materials. Assuming that hydrous material is heated to a depth of 10 m ($d$, smaller than our cell size), its cooling time scale would be over 10,000 days. Note that we take the thermal diffusivity ($\kappa$) of $10^{-7} {\rm m}^2/{\rm s}$, which is a typical value of the carbonaceous chondrite \cite{Opeil:2020}, and calculate $t_{\rm cool} \sim d^2/\kappa$. The material in much deeper locations would take longer time. Although we are unaware of the time to dehydrate at 873 K, the cumulative heated mass above 873 K may represent the amount of dehydrated minerals. \section{Conclusions} The mass heated by oblique impacts is a key to the understanding of the history of asteroids and meteorites. While \citeA{Kurosawa:2018} clarified the importance of the material strength of the target in the vertical impact, the following work confirmed this in the oblique impacts at a given impact velocity and angles \cite{Wakita:2019a}. We advanced their work by examining the dependence of the impact velocities and angles. Considering material strength in the target, we have performed a series of oblique impact simulations over a range of impact velocities and angles. The cumulative heated mass in the target with material strength is always larger than that without material strength, which indicates the material strength enhances impact heating. Our oblique impacts simulations with material strength showed that vertical impacts and impacts with steeper angles ($\geq$ 45$^\circ$) generate a similar cumulative heated mass, within a factor of 1.5. Grazing angle impacts ($\leq$ 30$^\circ$) produce less heated mass than other oblique impacts regardless of impact velocity. From our impact simulations of a wide parameter space, we derived an empirical formula for material with peak temperature over 1000 K, which can be used to understand $^{40}$Ar-$^{39}$Ar age resetting. Vertical impacts at low impact velocity and grazing impacts at high impact velocity produce a similar heated mass, but have differences in their peak pressure, indicating that grazing impacts are more likely to be responsible for impact heating in weakly shocked meteorites. \section*{Open Research} All our data are given by using iSALE-3D and our input files are available \cite{Wakita:2022a}. We also provide the cumulative heated mass of various impacts with/without material strength for $T_{\rm peak}$ of every 10 K as a Data Set. Please note that usage of the iSALE-3D code is restricted to those who have contributed to the development of iSALE-2D, and iSALE-2D is distributed on a case-by-case basis to academic users in the impact community. It requires a registration from the iSALE webpage (https://isale-code.github.io/) and usage of iSALE-2D and computational requirements are also shown in there. We directly plot figures from our binary data using pySALEPlot which is included in iSALE-3D and developed by TMD. Please also note that pySALEPlot in the current stable release of iSALE-2D (Dellen) would not work for the data from iSALE-3D. \acknowledgments We gratefully acknowledge the developers of iSALE-3D, including Dirk Elbeshausen, Kai W{\"u}nnemann, and Gareth Collins. This work has been supported in part by JSPS, Japan KAKENHI Grant Number JP17H06457 and JP17H02990. HG and KK are supported by JSPS KAKENHI Grant JP19H00726. KK is supported by JSPS KAKENHI Grants JP18H04464 and JP21K18660. TMD is funded by STFC grant ST/S000615/1.

Title: Visible extreme adaptive optics for GMagAO-X with the triple-stage AO architecture (TSAO)

Abstract: The Extremely Large Telescopes will require hundreds of actuators across the pupil for high Strehl in the visible. We envision a triple-stage AO (TSAO) system for GMT/GMagAO-X to achieve this. The first stage is a 4K DM controlled by an IR pyramid wavefront sensor that provides the first order correction. The second stage contains the high-order parallel DM of GMagAO-X that has 21000 actuators and contains an interferometric delay line for phasing of each mirror segment. This stage uses a Zernike wavefront sensor for high-order modes and a Holographic Dispersed Fringe Sensor for segment piston control. Finally, the third stage uses a dedicated 3K dm for non-common path aberration control and the coronagraphic wavefront control by using focal plane wavefront sensing and control. The triple stage architecture has been chosen to create simpler decoupled control loops. This work describes the performance of the proposed triple-stage AO architecture for ExAO with GMagAO-X.

https://export.arxiv.org/pdf/2208.07330

\keywords{Adaptive optics, Exoplanets, Wavefront sensing, Wavefront control, coronagraphy, Giant Magellan Telescope} \section{INTRODUCTION} \label{sec:intro} Direct imaging of exoplanets is a challenging endeavor. Especially if we are using ground-based telescopes. However, detecting Earth-like planets, and hopefully even habitable planets, is the primary science driver for the next generation of extremely large telescopes (ELTs). The ELTs will be build on Earth, and will have to image through the Earth's atmosphere. The turbulence in the atmosphere degrades the spatial resolution of the telescopes and limits their resolution to an arcsecond. A second challenge is the large contrast between the exoplanet and its host star. Earth-like planets are easily tens of billions times fainter at separation of a couple times the diffraction limit of the telescopes. Imaging such exoplanets will require extreme high performance adaptive optics systems. Most of the current direct imaging instruments have been designed to operate at near-infrared wavelengths (Y to L band). This was mainly a technical limitation. Longer wavelengths need less precise correction to achieve similar performance as compared to short wavelengths. Therefore, it was easier to start at longer wavelengths for the first generation of exoplanet imagers such as VLT/NACO\cite{lenzen2003naos, rousset2003naos}. This made such instruments suited to hunt for young hot giant gas planets. These planets still retain a large amount of the heat that built up during their formation phase. The heat makes the planets self-luminous in the infrared and therefore bright at infrared wavelengths. Such planets usually are at contrast levels of only $10^{-4}$ to $10^{-6}$. The first generation of HCI instruments lead to the development of instruments that were finely tuned for exoplanet imaging. The second generation of HCI instruments such as SPHERE, GPI and scexao focused on improving the contrast within the control radius. This was done by adding more advanced coronagraphs and wavefront control. The success of these instruments lead to deeper sensitivities and the discovery of more planets and brown dwarfs. The Magellan Adaptive Optics system eXtreme (MagAO-X) has been specifically designed to push extreme AO towards optical wavelengths\cite{males2022magaox}. There is exciting exoplanets science at such short wavelengths: planets in reflected light\cite{serindag2019testing}, detection of bio-signatures in atmospheres \cite{schwieterman2018exoplanet} and accreting proto-planets\cite{haffert2019two}. Recent commissioning observations have shown that MagAO-X is close to its designed specification\cite{males2022magaox}. MagAO-X will start to tackle the optical science cases. However, only a handful targets are available at the angular resolution and sensitivity of MagAO-X. We need a system with similar specifications to MagAO-X on the ELTs to survey many targets to get a broad understanding of exoplanets. The Giant Magellan Telescope Adaptive Optics Extreme\cite{males2022gmagaox} (GMagAO-X) instrument is a direct imaging instrument under development at the UofA for the 24.5-meter Giant Magellan Telescope (GMT). The University of Arizona Space Institute (UASI) funded a conceptual design study for GMagAO-X beginning in February 2021. A positive Conceptual Design Review (CoDR) on September 14th, 2021, determined that the project could move into a UASI-funded preliminary design phase. GMagAO-X is currently the only direct imaging instrument under development for any of the large ground-based telescopes (GMT, Thirty Meter Telescope (TMT) and the European Extremely Large Telescope (E-ELT)) that can observe exoplanets in the visible spectral range. We will be able to image and characterize several hundred of planets with GMagAO-X. GMagAO-X will use a 21000 actuator deformable mirror (DM) to achieve high sensitivity at visible wavelengths. The 21000-actuator DM will use novel pupil slicing and recombining techniques to implement a parallel DM where each GMT segment will get its own 3K Boston Micromachine DM\cite{close2022gmagaoxDM}. A major challenge will be the wavefront sensing and control architecture for such a large actuator DM. Current cameras put strong limitations on the number of available pixels and read-out speeds. A strong driver in the design of GMagAO-X is to use hardware components that are available today to make sure we can build the instrument whenever the instrument gets greenlighted. This proceeding will describe an approach that is being investigated for GMagAO-X that works with current hardware. In Section 2 we will describe the overall architecture. Each of the sub-components of the full loop are described in Sections 3, 4 and 5. \section{OVERVIEW OF THE ARCHITECTURE} The main challenge for the AO system of GMagAO-X is the large amount of pixels that are required to sample the pupil. The 21000 actuator DM has a projected pitch of 14 cm onto the primary mirrors of the GMT. An equivalent monolithic DM would have 175x175 actuators across the pupil. We follow the same approach as MagAO-X which increased its pupil sampling by 25\% to lower aliasing errors. This means we have to sample the GMT pupil with $175 \times 1.25\approx220$ pixels across the full aperture (over all segments). Most of the new AO system use either the OCAM 2K (240x240 pixels) for wavefront sensing in the visible or the CRED ONE (320 x 256) for wavefront sensing in the near-infrared. These sensors are cameras that have Electron Multiplying (EM) gain capability which reduces the read noise to below 1 electron rms. Low read noise detectors are necessary as this is a major limitation for the low-flux performance of AO systems. While larger sensors are in development, GMagAO-X follows the design philosophy that we will use currently available hardware to make us ready to build the system as soon as possible. This does not mean that new technology will not be considered for GMagAO-X, but that we do not want to be dependent on the development future technology. If we limit our available pixels to 240 pixels (the lowest denominator of the current high-speed EMCCDs) then that is already close to the required sampling of a single pupil. Any wavefront sensor that needs multiple pupil images can not be used with a single detector. One of the first designs for GMagAO-X was to use either a four-sided or three-sided pyramid wavefront sensor\cite{ragazzoni1996pupil,schatz2021three} where each pupil is send to its own camera. While this might solve the pixel issue, it comes with a lot of synchronization and software challenges. Single pupil wavefront sensors are interferometric wavefronts sensors. Starlight is generally temporally incoherent because we use broadband light. This precludes the use of heterodyne-like interferometry measurements. The starlight itself has to create its own reference beam. Going even further, the interferometer needs to be a common path interferometer. If the reference beam is created in a separate optical path any differential aberration (most likely vibrations) will lower the performance. From all common path interferometers it is the Zernike wavefront sensor (ZWFS) that is the most efficient in its use of light \cite{zernike1942phase,n2013calibration}. The major downside of any interferometer is the dynamic range. The dynamic range of a ZWFS is significantly smaller than what is required to sense open-loop atmospheric turbulence. This is also the reason why the ZWFS has only been used as a second stage sensor \cite{n2013calibration}. We propose to use a multi-stage system where a first stage AO system is used to take care of the large low-order aberrations. The HODM has a stroke requirement of 3.5 um which is not enough to capture the full range of turbulence. GMagAO-X was always envisioned to operate with a Woofer-Tweeter architecture similar to that of MagAO-X. Here we propose to add a separate low-order wavefront sensor to directly control the woofer (e.g. the low-order DM (LODM)) to create the first stage AO loop. This cleans up the wavefront aberrations enough that we can use a ZWFS for the high-order loop that controls the 21-kilo DM. A second part of the high-order loop contains a differential piston sensor. Sensing differential piston is possible with the ZWFS, but because the ZWFS is an interferometric wavefront sensor it is sensitive to phase wrapping effects. Therefore, we include the Holographic dispersed fringe sensor (HDFS) as a piston sensor for the GMT \cite{haffert2022phasing}. This sensor has been demonstrated to reach <30 nm piston rms for the GMT in a lab environment \cite{haffert2022phasing,hedglen2022lab}. We assume that the HDFS is tracking and controlling the differential piston for the simulations in this work. This is implemented by removing all differential piston from the input disturbances. Later work will include the HDFS in the end-to-end simulations. The low-order AO loop will control the monolithic 64x64 actuator woofer DM. We are currently baselining an ALPAO 3228 DM beacuse of its 15 um stroke. The low-order loop will only need $62\times 1.25\approx80$ pixels across the full pupil. Such a pupil easily fits multiple times on either the OCAM2K or the CRED ONE. We are considering either a three or four-sided pyramid wavefront sensor to control the LODM. In this proceeding we explore the four-sided PWFS. However, the three-sided PWFS has been shown to have equivalent performance for bright objects\cite{schatz2021three} and superior performance for faint objects\cite{codona2018comparative}. The LO loop and HO loop together will deliver PSFs with high Strehl which are send to the coronagraph. Any non-common path aberration (NCPA) between the AO loops and the coronagraph optics will create stellar leakage around the coronagraph. These NCPA will have to be controlled and removed. GMagAO-X will contain several focal-plane wavefront sensors (FPWFS) and coronagraphic wavefront sensors (CWS) to measure the NCPA and create dark holes around the PSF. The main problem then comes to control of the aberrations. The NCPA could in principle be off-loaded to the HO loop. However, operating the ZWFS (or even the PWFS for that matter) around a non-zero point will lead to non-linear interactions that are difficult to track and calibrate. It is easier if the responsibilities of the AO system and the coronagraph are separated. This can be achieved by adding a dedicated DM in the coronagraph that is purely used for NCPA correction and dark hole digging. This is the approach that is implemented on MagAO-X \cite{males2022magaox}. Separating the responsibilities also makes the control loop design much easier. We do not have to take into account any dynamic interaction between the coronagraph and the AO system. All together, GMagAO-X will have three stages in its wavefront control architecture. An overview of the proposed multi-stage AO system can be seen in Figure \ref{fig:overview}. \section{THE LOW-ORDER AO LOOP} The LO loop consists of the 3000 actuator DM and a variant on the pyramid wavefront sensor. One of the major design choices for the low-order loop is to include a dedicated LODM inside GMagAO-X. GMagAO-X will not use the Adaptive Secondary Mirrors (ASM) of the GMT. This is mainly driven by ease of calibration. It is tricky to calibrate the ASM without complicated illumination optics, while it is much easier for an internal DM. One of the main choices we have to consider is the wavelength that will be used to drive the LO loop. MagAO-X uses I-band photons for wavefront sensing with its PWFS. That wavelength band provides the best sensitivity for M stars \cite{guyon2018wavefront}, which are the prime targets of GMagAO-X. However, the short wavelength of 0.8 $\mu$m means that we have to modulate the PWFS to increase its capture ranges. The PWFS at MagAO-X is often modulated by 3 $\lambda/D$ as this provides a stable system that is robust in most seeing conditions. The modulation lowers the sensitivity of the wavefront sensor significantly\cite{fauvarque2017general}. While visible wavefront sensing like I-band is the most common band, more and more systems are considering near-infrared wavefront sensors \cite{bond2018adaptive,boccaletti2020sphere}. This is driven by both the amount of targets and the non-linearity effects of the PWFS. The amount of non-linearity is set by the wavelength. Sensing the wavefront at longer wavelengths makes the PWFS more linear\cite{bond2018adaptive}. Most importantly, the amount of targets increases substantially. Figure \ref{fig:guidestars} shows the amount of targets that are available for GMagAO-X if we use an I-band wavefront sensor or a K-band wavefront sensor. There are two important science cases: \begin{itemize} \item Exoplanets in reflected light. These companions need to orbit their host star at relatively close orbits to capture and reflect enough light to make them detectable\cite{guyon2018wavefront}. The sample of targets are all the stars that are in the Solar Neighborhood. We made a selection by looking for all objects within 10 pc within in the GAIA catalogue and all objects within 30 pc in the SIMBAD database. \item Accreting proto-planets. These planets are still young and embedded in a (gapped) circum-stellar disk. Such stars are usually classified as T-Tauri stars. Our second sample contains all T-Tauri stars that are visible from the southern hemisphere. \end{itemize} The amount of available targets increases by a factor 3 if we use a K-band wavefront sensor instead of an I-band wavefront sensor. The loss in sensitivity in K-band is compensated by the additional amount of targets that are observable. All most all T-Tauri stars are significantly brighter in K-band, which will partially compensate for the loss in sensitivity. Another driver for the choice for K-band is that GMagAO-X will support science observations from R-band to H-band. All of the K-band photons can be used for wavefront sensing. This will increase the throughput to the science channels because no photons have to be shared. The K-band LO-loop performance is predicted by end-to-end simulations. The parameters of the LO system can be found in Table \ref{tab:loe2esims}. The LO loop uses a simple leaky integrator for the controller, where the gain is optimized for each guide star magnitude. We do not expect significant optical gain effects because the sensing happens at K-band. Therefore, we have not applied any optical gain corrections for these simulations. We ran 5 trials of 500 iterations for each combination of guidestar magnitude and gain. The median Strehl across the last 250 iterations of all 5 trials is shown for each magnitude in Figure \ref{fig:firststage}. The band around the curve are the 16\% and 84\% quantiles of the Strehl. \begin{table}[] \centering \caption{The parameters of the end-to-end simulations of the LO loop.} \begin{tabular}{l|l|l} \hline Parameter & Value & Unit \\ \hline\hline $\lambda_0$ & 2.2 & $\mu$m \\ \hline $\Delta \lambda / \lambda$ & 0.2 & \\ \hline read-noise & 0.3 & $e^{-}$/pixel \\ \hline background & 12.5 & mag arcsecond$^{-2}$ \\ \hline loop speed & 1.0 & kHz \\ \hline number of modes & 3200 & kHz \\ \hline modulation & 1 & $\lambda/D$ \\ \hline pupil sampling & 80 & pixels \\ \hline pupil distance & 100 & pixels \\ \hline \end{tabular} \label{tab:loe2esims} \end{table} There are not enough photons to control all modes for the faintest targets. We also ran the same gain optimization procedure for magnitudes 10 to 13 with 800 modes and with 1200 modes. We saw a significant increase in performance at 11, 12 and 13th magnitude. The system still delivers 10\% Strehl at 12th magnitude in I-band. \section{THE HIGH-ORDER AO LOOP} The high-order loop uses a ZWFS to control the high-order DM. The phase reconstruction is still a challenge even with the reduced number of pixels that ZWFS uses. The ZWFS can in principle use a linear reconstructor in the small phase regime\cite{n2013calibration}. Most AO systems would use a straightforward Matrix-Vector Multiplication (MVM) for that. This matrix has a size of 21000 by 35000 elements. Calibrating this matrix with a push-pull approach would take a significant amount of time. Therefore, we are reconstructing the input phase directly. Mathematically the ZWFS can be described as, \begin{equation} E_{o} = E_{i} + m*(e^{i\delta}-1)E_{i}. \end{equation} Here, $E_{o}$ is the output electric field and $E_{i}$ the input electric field. The ZWFS applies a low-pass filter through the phase dot (with step size $\delta$) on the input electric field. This is described by the convolution of $E_{i}$ with the pupil response of the ZWFS mask $m$. For a well corrected beam the spatially filtered electric field is very close to the diffraction-limited electric field \cite{n2013calibration,steeves2020picometer}. The filtered part can be replaced by its diffraction-limited counterpart with a scalar correction for the Strehl loss. This effectively means we are interfering the aberrated electric field with a spatially filtered version of the diffraction-limited beam. This is a simple two beam interference problem, \begin{equation} E_o = E_i + E_r. \end{equation} In this equation, $m*(e^{i\delta}-1)E_{i}$ has been replaced by the modelled reference electric field $E_r$. The intensity of the measurement is, \begin{equation} I_o = I_i + I_r + 2|E_i||E_r|\cos{\left(\phi_i - \phi_r\right)}. \end{equation} The input intensity, $I_i$, is just the measured pupil intensity and does not depend on the phase. This can be measured and subtracted. Similarly, the reference intensity can also be subtracted. The phase reconstruction is, \begin{equation} \phi_i = \arccos{\left[\frac{I_o - I_i - I_r}{2\sqrt{I_iI_r}}\right]} + \phi_r. \end{equation} This reconstructor takes all non-linearities of the ZWFS into account, if the reference field has been corrected for the Strehl loss. The direct phase reconstruction is much quicker than a MVM multiplication. The actuator voltages signals are calculated by projecting the reconstructed phase onto the voltages. This is done through a sparse MVM because the actuators have a very localized influence function. Therefore, it is faster to project the phase onto the actuators with a sparse MVM. The HO loop is closed after 15 iterations of the LO loop in the end-to-end simulations. In practice this will most likely take longer if human interaction is necessary to close both loops. We use a 0th magnitude guidestar for our first end-to-end simulation of the multi-stage AO system (MSAO). For this guidestar magnitude both the LO and HO loop run at 2 kHz with optimized gains. The ZWFS uses J-band light for these simulations. The PSFs at various stages during the loop are shown in Figure \ref{fig:expectedpsf}. The seeing halo is completely removed up to 90 $\lambda/D$ after the second stage. The Strehl reaches nearly 95\% at $1\mu$m which can be seen in Figure \ref{fig:msstrehl}. This shows that this system can reach the required performance for high-contrast imaging at short wavelengths. \section{CONCLUSION AND OUTLOOK} This proceeding describes the current wavefront sensing and control architecture for GMagAO-X. A system with multiple stages is considered because of the design philosophy. The system will use a first stage K-band pyramid wavefront sensor running at 1 or 2 kHz. Followed by a Zernike wavefront sensor at bluer wavelengths to control the high-order loop. This combination of wavefront sensors allows us to control our 21000 actuator DM and achieve a Strehl of 95\% at 1$\mu$m on the GMT. The current simulations are the first end-to-end simulations of GMagAO-X and do not include all sources of noise and all input disturbances. Therefore, our next step will be to expand the simulations to make them more representative: \begin{itemize} \item The input disturbance will need to include vibrations caused by the telescope and windshake. \item Include differential piston, which we have now neglected. \item More representative detector noise. \item Advanced control algorithms such a predictive control\cite{haffert2021data} and optical gain compensation. \end{itemize} \acknowledgments Support for this work was provided by NASA through the NASA Hubble Fellowship grant \#HST-HF2-51436.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. \bibliography{report} % \bibliographystyle{spiebib} %

Title: PICASO 3.0: A One-Dimensional Climate Model for Giant Planets and Brown Dwarfs

Abstract: Upcoming James Webb Space Telescope (JWST) observations will allow us to study exoplanet and brown dwarf atmospheres in great detail. The physical interpretation of these upcoming high signal-to-noise observations requires precise atmospheric models of exoplanets and brown dwarfs. While several one-dimensional and three-dimensional atmospheric models have been developed in the past three decades, these models have often relied on simplified assumptions like chemical equilibrium and are also often not open-source, which limits their usage and development by the wider community. We present a python-based one-dimensional atmospheric radiative-convective equilibrium model. This model has heritage from the Fortran-based code (Marley et al.,1996} which has been widely used to model the atmospheres of Solar System objects, brown dwarfs, and exoplanets. In short, the basic capability of the original model is to compute the atmospheric state of the object under radiative-convective equilibrium given its effective or internal temperature, gravity, and host--star properties (if relevant). In the new model, which has been included within the well-utilized code-base PICASO, we have added these original features as well as the new capability of self-consistently treating disequilibrium chemistry. This code is widely applicable to Hydrogen-dominated atmospheres (e.g., brown dwarfs and giant planets).

https://export.arxiv.org/pdf/2208.07836

command. \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \usepackage{amsmath} \usepackage{tabularx} \usepackage{natbib} \usepackage{float} \usepackage[colorlinks=true,linkcolor=blue,citecolor=blue]{hyperref}% \usepackage[all]{hypcap} \usepackage{lineno} \usepackage{footnote} \DeclareUnicodeCharacter{2212}{-} \newcommand{\mj}{$M_{\mathrm{J}}$} \newcommand{\rj}{$R_{\mathrm{J}}$} \newcommand{\me}{$M_{\oplus}$} \newcommand{\re}{$R_{\oplus}$} \newcommand{\te}{$T_{\rm eff}$} \newcommand{\teff}{$T_{\rm eff}$} \newcommand{\tint}{$T_{\rm int}$} \newcommand{\teq}{$T_{\rm eq}$} \newcommand{\co}{CO} \newcommand{\meth}{CH$_4$} \newcommand{\amon}{NH$_3$} \newcommand{\cotwo}{CO$_2$} \newcommand{\water}{H$_2$O} \newcommand{\phos}{PH$_3$} \newcommand{\pq}{P$_Q$} \newcommand{\cp}{\citep} \newcommand{\ct}{\citet} \newcommand{\MNep}{M_{\rm Nep}} \newcommand{\RNep}{R_{\rm Nep}} \newcommand{\tb}{$T_{\rm B}$} \newcommand{\qp}{$Q_{\rm p}$} \newcommand{\tchem}{$t_{\rm chem}$} \newcommand{\tmix}{$t_{\rm mix}$} \newcommand{\kzz}{$K_{zz}$} \newcommand{\gjf}{GJ 436b} \newcommand{\gjt}{GJ 3470b} \newcommand{\wasp}{WASP-107b} \newcommand{\orcid}[1]{\href{https://orcid.org/#1}{\includegraphics[width=10pt]{Orcid-ID.png}}} \shorttitle{\texttt{PICASO 3.0}: 1D Climates} \shortauthors{Mukherjee et al.} \graphicspath{{./}{figures/}} \begin{document} \title{\texttt{PICASO 3.0}: A One-Dimensional Climate Model for Giant Planets and Brown Dwarfs } \email{samukher@ucsc.edu} \author{Sagnick Mukherjee$^{1}$ \orcid{0000-0003-1622-1302}, Natasha E. Batalha$^{2}$ \orcid{0000-0003-1240-6844}, Jonathan J. Fortney$^{1}$ \orcid{0000-0002-9843-4354}, Mark S. Marley$^{3}$ \orcid{0000-0002-5251-2943}} \affiliation{{$^1$}Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA \\ {$^2$} NASA Ames Research Center, MS 245-3, Moffett Field, CA 94035, USA \\ {$^3$} Lunar and Planetary Laboratory, The University of Arizona, Tucson, AZ 85721, USA\\} \keywords{ Brown Dwarfs, exoplanets} \section{Introduction}\label{sec:intro} There are three broad categories of substellar atmosphere modeling frameworks: one-dimensional (1D) physically self-consistent models, three-dimensional (3D) General Circulation Models (GCMs) and atmospheric retrieval. Atmospheric models enable the understanding of physical and chemical processes driving the behaviour of substellar atmospheres. When combined with observational data, they can be used to infer the physical state of atmospheres. Additionally, the adiabat calculated by the atmospheric models provides the upper boundary conditions of the interior evolutionary models of substellar objects like giant planets and brown dwarfs \citep{hubbard77,burrows97,chabrier00,saumonmarley08}. Although substellar atmospheres generally constitute a small fraction ($\sim$ 1\%) of the total mass of their bodies, their radiative properties control the overall cooling of objects throughout their evolution. Therefore, atmospheric models are crucial for understanding both physical (e.g. climate and chemistry) and global properties of substellar objects (e.g. radius/luminosity evolution over time). A variety of 1D radiative--convective--thermochemical equilibrium models of exoplanetary and brown dwarf atmospheres have been developed \citep[e.g.][]{Marley96,barman01,fortney2005comp,fortney08,marley21,Philips20,Piskorz_2018,goyal2020}. These models originate both from stellar atmosphere modeling codes \citep[e.g.][]{barman01,sudarsky03,seager1998} and planetary atmospheric modeling codes \citep[e.g.][]{Marley96,baudino15}. These models rely on iterating to a self-consistent radiative-convective-thermochemical equilibrium (RCTE) solution. Self-consistency here means that the model iterates on all components of the atmospheric structure simultaneously such that they are physically consistent with each other (e.g., chemical abundances are consistent with the current temperature profile). The models do not capture 3D dynamics, but instead are able to incorporate the effects of radiative/convective energy transport, chemistry, and clouds. These 1D models are also computationally efficient, which enhances their exploratory power significantly over 3D circulation models. For example, \citep[e.g.][]{zhang21} used the \texttt{SONORA BOBCAT} grid of models, generated using the 1D RCTE model described in \citet{marley21}, to infer the physical properties of 55 T-dwarfs in an uniform analysis. The computational speed of these 1D self-consistent models along with their ability to treat atmospheric physics and chemistry self-consistently makes them a powerful tool for interpreting atmospheres of planets and brown dwarfs. Besides the model used in \citet{marley21}, several other independent 1D RCTE models have been widely used in the literature \citep[e.g.][]{tremblin15,Philips20,gandhi17,burrows08,marley21,malik17}. For example, the \texttt{GENESIS} code from \citet{gandhi17} was used for the high significance detection of various Carbon and Nitrogen bearing species in the atmosphere of HD 209458b by \citet{giacobbe21}. The \texttt{ATMO} model \citep{tremblin15,Philips20} was used by \citet{goyal18} to perform a uniform fitting analysis of the transmission spectra of 10 hot Jupiters which led to important conclusions about their atmospheric composition. Despite the many insights gained from these models, their simplicity often makes them insufficient to explain a variety of atmospheric spectra \citep[e.g.][]{noll97,oppenheimer98,saumon03,golimowski04,geballe09,sorahana2012,leggett12,Miles20}. For example, \citet{Miles20} found the presence of disequilibrium chemistry in a series of late T and early Y-dwarfs by measuring their significantly enhanced photospheric CO abundances. In another example, \citet{zhang21} concluded that presence of clouds can lead to much better overall fit to the available infrared spectra of 55 brown dwarfs with a uniform analysis using the \texttt{SONORA} cloudless grid of models. This motivates adding physical complexity to 1D RCTE models in order to enable better interpretation of upcoming data, for example, from \textit{JWST}. Models for exoplanetary and brown dwarf atmospheres have been developed in three-dimensions (3D) \citep[e.g.][]{showman08,showman20,tan21,menou09,roman19,Wolf_2022,Lee_2021} as well. The 3D models work to capture the detailed global dynamics like horizontal transport/winds in the atmospheres along with other atmospheric components like radiation and clouds. However, these models are computationally intensive which limits their power of exploration of the vast parameter space in question. Moreover, in order to make the models computationally feasible, they rely on approximations such as gray atmospheres in the radiative transfer calculations \citep[e.g.][]{tan21,tanandshowman19,menou09,tan22}. As a result, 3D models are unable to resolve some atmospheric properties predicted by 1D models -- like detached convective zones in brown dwarfs and giant planets. Atmospheric retrievals are also widely used to interpret observational data. These studies aim to recover the atmospheric state of any planet/brown dwarf based on the observed spectral data in a Bayesian framework \citep[e.g.][]{madhu18,line17,burningham17}. The free parameters of interest typically include the atmospheric temperature-pressure ($T(P)$) profile, chemical abundances and cloud properties \citep{line17,burningham17,burningham21,mukherjee20,taylor21}. The multi-dimensional parameter space of these free parameters is sampled by repeatedly calculating model observables (e.g., spectra) with simple and computationally fast forward models. Comparison of these modeled observables with the observed data helps in retrieving likelihood distributions for all the retrieved free parameters. However, these retrieved properties are often not required to have physical constraints. For example, the retrieval analysis of hot Jupiter, WASP-18 b, was predicted to have a thermal inversion and relatively high metallicity (283$\times$Solar), corresponding to a CO abundance of $>$10\% \citep{sheppard17}. Later using a self-consistent grid of models, \citet{arcangeli18} showed that this planet is more likely to have a solar metallicity and hot dayside, which is more consistent with what is expected from a hot Jupiter with little heat redistribution. This means that even though retrieval studies are effective in inferring the various atmospheric properties like abundances and temperature structures, they are not equipped to provide the physical interpretations of why the atmosphere is in the retrieved state, and they often result in unphysical solutions. Therefore, physically motivated self-consistent models are required to understand the physical and chemical processes that drive the structure of planetary and substellar atmospheres. In this work, we focus on 1D physically motivated radiative--convective--thermochemical equilibrium models for substellar atmospheres. In addition to include complexities like disequilibrium chemistry within these models, another important modification we must make to these codes, is to transform them to open sourced, ``FAIR'' codes (findable, accessible, interpretable, and reproducible). Doing so will enable the community to interpret the upcoming influx of data from missions like JWST \citep{JWSTERO}. Currently, there are a handful of 1D climate models that are open source (e.g., \texttt{HELIOS} \citep{malik17,malik19}, and \texttt{TLUSTY} \citep{hubeny88,hubeny03,sudarsky03}). \texttt{HELIOS} is a Python-based GPU dependent model and it has been used for modeling atmospheres of a variety of substellar objects \citep[e.g.][]{fossati21,rockymalik19,yan22,deline22}. \texttt{TLUSTY} is a stellar atmosphere model written in FORTRAN77. This code has been modified to be applied to substellar atmospheres as well and is called \texttt{COOLTLUSTY} \citep{hubeny03}. \texttt{COOLTLUSTY} also has been widely used for brown dwarfs and exoplanets \citep[e.g.][]{burrows06,spiegel10,spiegel12,lacy19,lacy20}. In this work, we aim to add to develop a new open-source Python based one-dimensional radiative-convective equilibrium model for H-dominated substellar atmospheres -- \texttt{PICASO 3.0} that is both: 1) open source and 2) capable of including disequilibrium chemistry induced by vertical mixing self-consistently within the 1D radiative--convective equilibrium framework. \texttt{PICASO 3.0} has been released publicly as an extension of the already open-source 1D and 3D radiative transfer tool \texttt{PICASO} \citep{batalha19}. It is also accompanied by detailed tutorials exploring all of its uses and limitations. Here, we focus on the description of the numerical techniques used in the code along with benchmarking to previous studies to demonstrate its functionality. In \S\ref{sec:model}, we discuss the methodology of \texttt{PICASO 3.0}, the new Python atmospheric model. We present the benchmarking of our model with other models in \S\ref{sec:benchmark} followed by recommendations on how to use this model in \S\ref{sec:recommendations}. We briefly discuss ongoing and future improvements in \S\ref{sec:improve} and conclude in \S\ref{sec:summary}. \section{Model Setup of \texttt{PICASO 3.0}}\label{sec:model} The heritage of our code is the Fortran based \texttt{EGP} sub-stellar atmospheric model. The \texttt{EGP} model has been used for substellar atmospheres including solar system planets -- Titan and Uranus \citep{mckay1989thermal,marley1999thermal}, the L--T--Y brown dwarf sequence \citep[e.g.][]{Marley96,morley14water,morley2012neglected,marley21,karilidi21,zhang21}, cloudy atmospheres \citep[e.g.][]{cushing08,stephens09} and a wide variety of extrasolar planets \citep[e.g.][]{fortney2005comp,fortney2007planetary,fortney08, Marley_2012,fortney20, Morley2015super,Morley2017gj436} for over two decades. It has also been used as the primary radiative-transfer scheme in the SPARC GCM \citep{Showman2009}. Briefly, this 1D model solves for the self-consistent temperature, chemical and cloud structure of H-dominated atmospheres under the assumption of radiative-convective equilibrium. The code structure of \texttt{PICASO 3.0} is composed of Python classes \footnote{https://docs.python.org/3/tutorial/classes.html} which generally include multiple Python functions. We refer to these classes and functions as modules here. In general, Python is significantly slower than Fortran. However, with the use of `numba`'s `just-in-time` framework \citep{numba} to create on-the-fly compiled machine code, \texttt{PICASO 3.0} has comparable run times to the original Fortran. Figure \ref{fig:figschematic} shows a simplified schematic of \texttt{PICASO 3.0}'s workflow. First, the user provides an initial set of physical properties for the object to be modeled, shown within the red boxes at the top of Figure \ref{fig:figschematic}. The model supports both non-irradiated (e.g., brown dwarf) and irradiated (e.g., planets) calculations. Therefore, we specify optional inputs with dashed outlines. For example, for modeling a field brown dwarf atmosphere, the user only must specify the {\teff} ({\tint} in case of a planet), gravity, atmospheric metallicity and C/O ratio of the brown dwarf. For modeling an irradiated planet, the user must specify the internal temperature of the planet {\tint}, planet gravity, atmospheric metallicity, C/O ratio, semi-major axis, and host star properties (e.g., stellar temperature, metallicity, and gravity). {\tint} represents the temperature obtained by converting the internal heat flow of the planet via the Stefan-Boltzmann law. Along with these inputs, the user must also specify an initial pressure-temperature profile ($T(P)$) guess, which is divided into plane-parallel logarithmically spaced pressure layers ($\sim$ 60-90). Then, through an iterative process, which we describe in \S \ref{sec:physics}, the model iterates through computing chemistry, opacities, and net upwelling and downwelling fluxes until a radiative-convective equilibrium threshold has been met. During this iteration, the model solves for both radiative and convective parts of the atmosphere and takes into account the possibility of multiple radiative and convective zones. Ultimately, the model produces the final atmospheric state of the object ($T(P)$ and associated chemistry). These can then be used to compute transmission, emission, and/or reflected light spectra used to compare with observations (see Figure \ref{fig:figschematic}). We discuss these physical and chemical aspects one by one in the following \S \ref{sec:physics}. \subsection{Physics and Chemistry of Substellar Atmospheres}\label{sec:physics} \subsubsection{Radiative-Convective Equilibrium}\label{sec:RCE} The key physical basis of this model is the assumption of radiative-convective equilibrium in substellar atmospheres. The radiative equilibrium represents the physical scenario where within each atmospheric layer the energy emitted must be balanced by the energy absorbed. This means that each atmospheric layer must allow the transfer of the same amount of radiative energy through it. \citet{hubeny17} provides a detailed derivation of the radiative--convective equilibrium criteria. For completeness, we present some of the important steps here. Radiative equilibrium scenario is represented by: \begin{equation}\label{eq:eqRCE} \int_0^\infty (\chi_{\nu}J_{\nu}-\eta_{\nu})d{\nu} =0 \end{equation} where $\chi_{\nu}$ and ${\eta_{\nu}}$ are absorption and emission coefficients, respectively and $J_{\nu}$ is the first moment of the intensity field. Assumption of local thermodynamic equilibrium allows Equation \ref{eq:eqRCE} to be rewritten as: \begin{equation}\label{eq:eqRCE1} \int_{\nu} \kappa_{\nu}(J_{\nu}-B_{\nu})d{\nu} =0 \end{equation} where $\kappa_{\nu}$ is the wavelength dependent total absorption opacity of the layer and $B_{\nu}$ is the local Planck function of the layer. This is the integral form of the radiative equilibrium condition for each atmospheric layer. The integral form of the radiative--equilibrium condition (Equation \ref{eq:eqRCE1}) is applicable and numerically stable throughout the atmosphere. Another method of computing radiative-equilibrium condition, which is used by our model, is referred to as the differential form. The differential form is numerically stable at optically thicker parts of the atmosphere and might become numerically unstable at parts of the atmosphere which are optically thin ($\tau <<$ 1). In practice, this instability only appears at parts of the atmosphere with low temperatures and very small optical depths. In our first release of the climate code, we stick to our original methodology of using the differential form, despite the instabilities pointed out by \citet{hubeny17}. In a later update we will explore and implement the integral form as well within our model to make our solutions more accurate at the optically thinner parts of the atmosphere. The differential form can be derived by using Equation \ref{eq:eqRCE1} along with the second moment of the radiative transfer equation, \begin{equation} \int_{\mu} \mu\dfrac{dI_{\nu}}{d\tau_{\nu}}d\mu = \int_{\mu}\mu(I_{\nu} - S_{\nu})d{\mu} \end{equation} which under local thermodynamic equilibrium leads to the form, \begin{equation}\label{eq:eqRT} \dfrac{dH_{\nu}}{dz}= \kappa_{\nu}(J_{\nu} - B_{\nu}) \end{equation} where $H_{\nu}$ is the wavelength dependent second moment of the specific intensity field. Comparing Equation \ref{eq:eqRCE1} and \ref{eq:eqRT} leads to the condition, \begin{equation}\label{eq:eqRT1} \int_{\nu}\dfrac{dH_{\nu}}{dz}d{\nu}= 0 \end{equation} which means that the integral of the second moment of the intensity, {$H_{\nu}$}, must be constant at all layers. Mathematically, this can be written as: \begin{equation}\label{eq:RCEfin} \int_{\nu} H_{\nu}d{\nu} - \dfrac{\sigma_{sb}T_{\rm eff}^4}{4\pi} = 0 \end{equation} where $\sigma_{sb}$ is the Stefan--Boltzmann constant and \teff\ is the effective temperature of the object. This calculation can be found in the \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/climate.py#L448}{\texttt{t\char`_start}} module in the code. The model tries to achieve this radiative equilibrium in all the atmospheric layers which are stable against convection. Given Equation \ref{eq:RCEfin}, which defines the radiative-equilibrium condition, the general procedure of the model is to start with an initial guess of the $T(P)$ profile of the atmosphere. This can be seen in the required inputs box at the top of Figure \ref{fig:figschematic}. Radiative fluxes through all the atmospheric layers are then calculated given a chemical state of the atmosphere. These radiative fluxes are used to compute the wavelength integrated flux carried by each layer $H(z)$. If the atmospheric layers are not in radiative equilibrium, this quantity $H(z)$ would not be equal to the target flux given by $\dfrac{\sigma_{sb}T_{\rm eff}^4}{4\pi}$, and the $T(P)$ solution would be perturbed until the convergence criteria set by Equation \ref{eq:RCEfin} is met. The methodology of the convergence is discussed further in \S\ref{sec:model_convergence}. \subsubsection{Radiative Transfer}\label{sec:RT} In order to compute radiative-convective equilibrium with Equation \ref{eq:RCEfin}, radiative fluxes must be calculated for all atmospheric layers. The fundamental equation describing atmospheric radiative transfer is, \begin{equation}\label{eq:RT1} \begin{aligned} \mu\dfrac{dI_{\nu}(\mu,\tau_{\nu},\phi)}{d\tau_{\nu}}= &I_{\nu}(\mu,\tau_{\nu},\phi)-S_{\nu}(\mu,\tau_{\nu},\phi) \\ & - \dfrac{\omega_0}{4\pi}\int_0^{2\pi}\int_{-1}^{1}P_{\nu}(\mu,\mu^{'},\phi,\phi^{'}) \\ & \cdot I_{\nu}(\mu^{'},\tau_{\nu},\phi^{'})d\mu^{'}d\phi^{'} \end{aligned} \end{equation} where $\mu = \cos(i)$ where $i$ is the incident angle of the ray, $\phi$ is the azimuthal angle, $I_{\nu}$ is the specific intensity, $\tau_{\nu}$ is the frequency dependent optical depth, $S_{\nu}$ is the frequency dependent source function and $P_{\nu}$ is the scattering phase--function. The first term on the right--hand--side of Equation \ref{eq:RT1} describes the attenuation of the intensity with increasing optical depth. The source function $S_{\nu}$ is used to describe the emission of the atmosphere itself (e.g. thermal emission) or any external radiation source (e.g. incident stellar radiation). The third term in Equation \ref{eq:RT1} represents scattering of the intensity within the atmosphere where the phase--function $P_{\nu}(\mu,\mu^{'},\phi,\phi^{'})$ is the probability that intensity $I_{\nu}(\mu^{'},\tau_{\nu},\phi^{'})$ will be scattered from the direction ($\mu^{'},\phi^{'}$) to the direction ($\mu,\phi$). In order to solve Equation \ref{eq:RT1}, we follow the two-stream radiative transfer methodology described in \citet{toon1989rapid} for the calculation of radiative fluxes. The specific intensity $I_{\nu}(\mu,\tau_{\nu},\phi)$ can be integrated over the azimuthal angle $\phi$ to calculate the azimuthally integrated intensity $I_{\nu}(\mu,\tau_{\nu})$. The upward and downward fluxes can also be defined by, \begin{equation}\label{eq:RT2} \begin{aligned} F_{\nu}^{+}= & \int_{0}^{1} {\mu}I^{+}_{\nu}(\mu,\tau_{\nu})d\mu \\ F_{\nu}^{-}= & \int_{0}^{1} {\mu}I^{-}_{\nu}(\mu,\tau_{\nu})d\mu \end{aligned} \end{equation} where $I^{+}_{\nu}(\mu,\tau_{\nu})$ is the azimuthally integrated intensity for upward values of $\mu$ and $I^{-}_{\nu}(\mu,\tau_{\nu})$ is the azimuthally integrated intensity for downward values of $\mu$. Equation \ref{eq:RT1} can be integrated to produce two separate coupled equations in terms of the upward and downward fluxes instead of the $\mu$ and $\phi$ dependent intensities. These equations for the upward and downward fluxes are\footnote{there is a typo in Eqn. 12 of \citet{toon1989rapid} which incorrectly swaps in $S^{+}_{\nu}$ for $S^{-}_{\nu}$ in the negative partial flux}, \begin{equation}\label{eq:RT3} \begin{aligned} \dfrac{{\partial}F_{\nu}^{+}}{\partial\tau_{\nu}}= & \gamma_1F_{\nu}^{+} - \gamma_2F_{\nu}^{-} -S^{+}_{\nu} \\ \dfrac{{\partial}F_{\nu}^{-}}{\partial\tau_{\nu}}= & \gamma_2F_{\nu}^{+} - \gamma_1F_{\nu}^{-} +S^{-}_{\nu} \end{aligned} \end{equation} where $\gamma_1$ and $\gamma_2$ are functions of the scattering properties of the medium and $S^{+}_\nu$ and $S^{-}_\nu$ are modified versions of the source function. The radiative transfer calculation within our model is divided into two distinct parts: 1) the transfer of thermal emission through the atmosphere and 2) the transfer of the reflected external (e.g. stellar) radiation throughout the atmosphere. Equation \ref{eq:RT3} is applicable to both of these components but the functions $\gamma_1$, $\gamma_2$, $S^{+}_\nu$, and $S^{-}_\nu$ are defined differently for each. A brief description of the radiative transfer of thermal radiation in our model is provided below followed by a discussion of the radiative transfer of external stellar radiation. We use the two-stream source function technique described in \citet{toon1989rapid} to calculate the thermal upward and downward fluxes in each layer. The hemispheric mean approximation approach is used in this method where $\gamma_1$ is given by 2-$\omega_0(1+g)$ and $\gamma_2$ is $\omega_0(1+g)$. $\omega_0$ and $g$ are the single scattering albedo and the scattering asymmetry parameter of the atmospheric layer, respectively. This hemispheric mean approximation was used to obtain functional forms for the source functions $S^{+}_\nu$ and $S^{-}_\nu$ in \citet{toon1989rapid} (see Table 3 and Equations 53 \& 54 in \citet{toon1989rapid}). With these source functions, the upward azimuthally averaged intensities at the top ($I^{+}_{n}(0,\mu)$) and bottom ($I^{+}_{n}(\tau,\mu)$) of an atmospheric layer with optical depth $\tau$ can be written as, \begin{equation}\label{eq:RT4} \begin{aligned} I^{+}_{n}(0,\mu) = & I^{+}_{n}(\tau,\mu)e^{-\tau/\mu} + \dfrac{G}{\lambda\mu-1}(e^{\lambda\tau}e^{-\tau/\mu}-1) \\ & + \dfrac{H}{\lambda\mu+1}(1-e^{-\lambda\tau}e^{-\tau/\mu}) +\alpha_1(1-e^{-\tau/\mu}) \\ & + \alpha_2(\mu-(\tau+\mu)e^{-\tau/\mu}) \end{aligned} \end{equation} This calculation can be found in the \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/fluxes.py#L1750}{\texttt{get\char`_thermal\char`_1d\char`_gfluxi}} module. Similarly the downward azimuthally averaged intensities at the top ($I^{-}_{n}(0,-\mu)$) and bottom ($I^{-}_{n}(\tau,-\mu)$) of the same atmospheric layer can be written as, \begin{equation}\label{eq:RT5} \begin{aligned} I^{-}_{n}(\tau,-\mu) = & I^{-}_{n}(0,-\mu)e^{-\tau/\mu} + \dfrac{K}{\lambda\mu-1}(e^{-\tau/\mu}-e^{-\lambda\tau}) \\ & + \dfrac{J}{\lambda\mu+1}(e^{\lambda\tau}-e^{-\tau/\mu}) +\sigma_1(1-e^{-\tau/\mu}) \\ & + \sigma_2({\mu}e^{-\tau/\mu}+\tau-\mu) \end{aligned} \end{equation} and can also be found in the \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/fluxes.py#L1736}{\texttt{get\char`_thermal\char`_1d\char`_gfluxi}} module. The functions $G$, $H$, $K$, $I$, $\alpha_1$, $\alpha_2$, $\sigma_1$, and $\sigma_2$ have been computed for the hemispheric-mean approximation in \citet{toon1989rapid} and are used in our calculation as well. Solving equations \ref{eq:RT4} and \ref{eq:RT5} also require boundary conditions for the diffuse flux at the top and bottom of the atmosphere. These boundary conditions are set using the thermal blackbody intensities at the top and bottom of the atmosphere using, \begin{equation}\label{eq:RT6} \begin{aligned} B_{\rm top}= & (1.-e^{-\tau^{'}/\mu_1})B({ T_{\rm top}}) \\ B_{\rm bot} = & B({ T_{\rm bot}})+\mu_1\dfrac{B({ T_{\rm bot}})-B({ T_{\rm bot-1}})}{\tau}\\ \end{aligned} \end{equation} where $B(T)$ represents the blackbody function, $T_{\rm top}$ is the temperature of the top most atmospheric layer, $T_{\rm bot}$ is the temperature of the bottom most atmospheric layer, and $\tau$ is the optical depth of the bottom most atmospheric layer. $\mu_1$ is assumed to be 0.5 due to the hemispheric-mean approximation. Note that the bottom boundary condition is valid only for gas giant atmospheres where the highest pressure grid point corresponds to the end of the user-defined grid and does not correspond to a ``surface''. \texttt{PICASO 3.0} does have the option to swap boundary conditions that are pertinent to the hard surfaces needed for terrestrial atmospheres, but it is not relevant for this work on gas giants and as such we do not discuss it here. $\tau^{'}$ is given by, \begin{equation}\label{eq:RT7} \tau^{'}= \tau_{top}\dfrac{{ P_{\rm top}}}{({ P_{\rm top+1}}-{ P_{\rm top}})} \end{equation} This expression along with the top boundary condition in Equation \ref{eq:RT6} captures the downward thermal flux arising from the part of the atmosphere which has pressure less than the minimum pressure in the used atmospheric pressure grid. This formulation prevents arbitrary artificial cooling of the top-most atmospheric layer. Equations \ref{eq:RT4} and \ref{eq:RT5} can be used to calculate the incidence angle ($\mu$) dependent upward and downward -- $I_n^{+}(\mu)$ and $I_n^{-}(\mu)$ intensity field in each atmospheric layer. But in order to use these radiative intensities ultimately for the convergence criteria in Equation \ref{eq:RCEfin}, they need to be integrated with Equation \ref{eq:RT2} to calculate the direction independent upward and downward fluxes -- $F^{+}_{\nu}$ and $F^{-}_{\nu}$. In order to compute these disk-averaged, layer fluxes, the upward and downward thermal intensity for each atmospheric layer is calculated using Equation \ref{eq:RT4} and \ref{eq:RT5} at five incident angles. The cosine of these incident angles ($\mu$) are determined using the Gaussian quadrature method with 5 points\citep{abramowitz}. The choice of 5 Gauss points is the default setting in the code but other choices are also available in the \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/disco.py#L46}{\texttt{get\char`_angles\char`_1d}} module. Table \ref{table:tab1} shows the five default values of $\mu$ and the corresponding Gauss weights used for this Gaussian quadrature integration. The intensities at different incident angles are used to compute the integral in Equation \ref{eq:RT2} with the Gaussian-quadrature integration technique using the weights in Table \ref{table:tab1}. The module \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/fluxes.py#L1551}{\texttt{get\char`_thermal\char`_1d\char`_gfluxi}} uses the formulation described above to calculate the wavelength dependent upward and downward thermal fluxes at the edges of each atmospheric layer. \begin{table}[h!] \centering \begin{tabular}{||c c c||} \hline $\theta$(deg) & $\mu=cos(\theta)$ & Gauss Weight \\ [0.5ex] \hline\hline 84.345 & 0.09853 & 0.015747 \\ 72.270 & 0.30453 & 0.073908 \\ 55.804 & 0.56202 & 0.146386 \\ 36.680 & 0.80198 & 0.167174 \\ 16.221 & 0.96019 & 0.096781 \\ [1ex] \hline \end{tabular} \caption{Gauss points and weights used for Gaussian quadrature integration of thermal flux over different angles.} \label{table:tab1} \end{table} For the radiative transfer of the reflected stellar light, we use the quadrature approximation. The calculation is performed for a single incidence angle of the stellar radiation beam (60$^{\circ},\mu_0$=0.5) unlike the 5 Gauss point method of the thermal counterpart. Under the quadrature approximation, $\gamma_1$ is $0.5\sqrt{3}(2-\omega_0(1+g))$, $\gamma_2$ is $\omega_0\sqrt{3}(1-g)/2$, $\gamma_3$ is ($1-\sqrt{3}g\mu_0$)/2, and $\gamma_4$ is 1-$\gamma_3$. These functions can be found in the \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/fluxes.py#L1219}{\texttt{get\char`_reflected\char`_1d\char`_gfluxv}} module. The source functions $S_{\nu}^{+}$ and $S_{\nu}^{-}$ for this component are, \begin{equation}\label{eq:RT8} \begin{aligned} S_{\nu}^{+}= & \gamma_3{\pi}F_s\omega_0e^{-(\tau_c+\tau)/\mu_0} \\ S_{\nu}^{-}= & \gamma_4{\pi}F_s\omega_0e^{-(\tau_c+\tau)/\mu_0}\\ \end{aligned} \end{equation} where $\tau_c$ is the cumulative optical depth of all the atmospheric layers above the layer of calculation and $\tau$ is the optical depth of the layer itself. $F_s$ here represents the stellar flux incident on the top of the atmosphere. $F_s$ is interpolated from the \texttt{PHOENIX} grid of models \citep{allard12} available as a part of the \texttt{PySynPhot} \citep{pysynphot13} package. Using these functions, Equation \ref{eq:RT3} can be solved for the upward and downward fluxes for the $n$'th atmospheric layer as has been shown in \citet{toon1989rapid}, \begin{equation}\label{eq:RT9} \begin{aligned} F_{\nu,n}^{+}= & k_{1n}e^{\lambda_n\tau_n} + \Gamma_n{k_{2n}}e^{-\lambda_n\tau_n} + C_n^{+} \\ F_{\nu,n}^{-}= & \Gamma_{n}k_{1n}e^{\lambda_n\tau_n} + {k_{2n}}e^{-\lambda_n\tau_n} + C_n^{-}\\ \end{aligned} \end{equation} where the quantities $\Gamma_n$, $k_{1n}$, $k_{2n}$, $\lambda_n$, $C_n^{+}$, and $C_n^{-}$ are defined in \citet{toon1989rapid} (Equation 21, 22, 23, and 24). Equation \ref{eq:RT9} is solved in the \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/fluxes.py#L1288}{\texttt{get\char`_reflected\char`_1d\char`_gfluxv}} module. However, an additional term needs to be added to the downward fluxes solved from Equation \ref{eq:RT9} in this formulation. This term is, \begin{equation}\label{eq:RT10} \begin{aligned} F_{\nu,n}^{-}= F_{\nu,n}^{-} + \mu_{0}{F_s}e^{-\tau_c/\mu_0} \\ \end{aligned} \end{equation} Like the thermal component, complete solutions of Equation \ref{eq:RT9} also require boundary conditions on the diffuse flux at the top and bottom of the atmosphere. We use the boundary conditions, \begin{equation}\label{eq:RT11} \begin{aligned} F_{\rm top,st}= & 0 \\ F_{\rm bot,st} = & {R_s}\mu_{0}F_{s}e^{-\tau_{\rm c,bot}/\mu_0}\\ \end{aligned} \end{equation} where $F_{\rm top,st}$ and $F_{\rm bot,st}$ are the diffuse flux at the top and bottom of the atmosphere, $R_s$ denotes the reflectivity of the bottom surface and $\tau_{c,bot}$ is the cumulative optical depth of the deepest layer. The reflectivity of the bottom surface is typically assumed to be 1\% in the model, but users should check that the bottom surface is not optically thin. This is further discussed in the \S\ref{sec:recommendations}. This formulation is used by the \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/fluxes.py#L1123}{\texttt{get\char`_reflected\char`_1d\char`_gfluxv}} module to calculate the wavelength dependent upward and downward reflected stellar light fluxes at the edges of each atmospheric layer. But the calculation of both the thermal and reflected stellar light fluxes following the procedure described above requires another important parameter -- the optical depth $\tau$. The implemented $\tau$ calculation procedure in this model is described below. Generally, two approaches are used for calculating the optical depth $\tau$ -- 1) the line--by--line approach (e.g. \citet{gandhi17},\citet{burrows08}) or 2) the correlated-k approach (e.g. \citet{marley1999thermal}, \citet{malik17}, \citet{fortney08}). This model uses a correlated-k opacity approach for computationally efficient inclusion of gaseous opacities. In the correlated-k approach, each wavelength bin is represented by 8 distinct Gaussian quadrature points ($g_i$), each with its associated weight ($w_i$). Table \ref{table:tab2} shows the Gauss points and the Gauss weights used for the 8 point correlated-k approach in the model. Each of these Gauss-points also have an optical depth $\tau_i$ associated with them. These ``Gauss" points should not be confused with the ``Gauss" points used to calculate the disk-integrated fluxes. This means that the radiative transfer equations for each wavelength bin must be computed 8 times, once for each value of optical depth $\tau_i$ corresponding to the $i$'th Gauss point. This calculation is done by calling \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/fluxes.py#L1551}{\texttt{get\char`_thermal\char`_1d\char`_gfluxi}} and \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/fluxes.py#L1123}{\texttt{get\char`_reflected\char`_1d\char`_gfluxv}} modules 8 times by the \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/climate.py#L1122}{\texttt{climate}} module. The 8 thermal and radiative upward and downward fluxes are then added together following, \begin{equation}\label{eq:RT12} F^{\pm}_n(\lambda) = \sum_{i=1}^{i=8} w_iF^{\pm}_{i,n}(\lambda,\tau_{i,n}) \end{equation} where $F^{\pm}_{i,n}$ is the thermal/reflected light flux at the $n$'th atmospheric layer calculated with the optical depth $\tau_{i,n}$ corresponding to the $i$'th Gauss point. Our calculation of correlated-k opacities is described in more details in \S\ref{sec:opa}. This summation with Equation \ref{eq:RT12} within the \texttt{climate} module ultimately produces thermal/reflected wavelength dependent upward and downward fluxes in each atmospheric layer. These fluxes can now be used to calculate the convergence criteria in Equation \ref{eq:RCEfin}. The net thermal flux for the $n$'th layer can be simply obtained with, \begin{equation}\label{eq:RT13} \begin{aligned} F^n_{\rm thermal,net} = & \int_{\lambda}( F^+_n(\lambda) -F^-_n(\lambda))d\lambda \\ F^n_{\rm stellar,net} = & \int_{\lambda}( F^+_{n,s}(\lambda) -F^-_{n,s}(\lambda))d\lambda\\ \end{aligned} \end{equation} where $F^+_n(\lambda)$ represents the upward thermal flux from the $n$'th layer within the wavelength bin between $\lambda$ and $\lambda$+d$\lambda$ and $F^-_n(\lambda)$ is the downward thermal flux from the same layer within the same wavelength bin. $F^{\pm}_{n,s}$ also represent the same quantity for the reflected light component. The total net radiative flux in the $n$'th atmospheric layer can be calculated using, \begin{equation}\label{eq:net} F^n_{\rm net} = r_{\rm th}F^n_{\rm thermal,net} + r_{\rm st}F^n_{\rm stellar,net} \end{equation} where r$_{\rm th}$ is the contribution factor of the thermal radiation to the net flux and is generally fixed at 1 and r$_{\rm st}$ is the contribution factor of the stellar radiation to the net flux. Non-zero values of r$_{\rm st}$ is only relevant when the external irradiation on the atmosphere is non-zero. In the scenario when a user is computing a planet-wide average $T(P)$ profile, the stellar irradiation is contributing to 50\% (one hemisphere) of the planet and as a result $r_{\rm st}=0.5$. If instead the goal is to compute a night-side average atmospheric state, $r_{\rm st}$ is set to be 0. On the other extreme, to compute the day-side atmospheric state of a tidally locked planet $r_{\rm st}$ should be set at 1. This full net flux, $F^n_{\rm net}$, is same as $4\pi\int_{\nu}H_{\nu}d\nu$ in Equation \ref{eq:RCEfin}. Therefore, $F^n_{\rm net}$ can be used to check if the convergence criteria (Equation \ref{eq:RCEfin}) is satisfied at all the radiative layers of the atmosphere. This check is done in the ``Converged?" box of the flow chart shown in Figure \ref{fig:figschematic}. Now that we have described the radiative transfer within our model, we will discuss other blocks of the model flow chart shown in Figure \ref{fig:figschematic} starting with atmospheric chemistry. \begin{table}[h!] \centering \begin{tabular}{||c | c||} \hline Gauss Points ($g_i$) & Gauss Weights ($w_i$)\\ [0.5ex] \hline\hline 0.065960251992824 & 0.165231051440291 \\ \hline 0.313509004297193 & 0.309768948559709\\ \hline 0.636490995702807 & 0.309768948559709\\ \hline 0.884039748007175 & 0.165231051440291\\ \hline 0.953471592210149 & 0.008696371128436358\\ \hline 0.966500473910379 & 0.01630362887156367\\ \hline 0.983499526089621 & 0.01630362887156367\\ \hline 0.996528407789851 & 0.008696371128436358\\[1ex] \hline \end{tabular} \caption{Gauss points and weights used for calculation of the correlated-k opacities} \label{table:tab2} \end{table} \subsubsection{Equilibrium Chemistry}\label{sec:chem} As shown in Figure \ref{fig:figschematic}, in order to compute the radiative fluxes for checking the convergence criteria, as outlined in the previous \S \ref{sec:RT}, we first need a method to provide the chemical state of the atmosphere. This is required because the chemistry dictates the optical depth ($\tau$) required for the radiative transfer calculations. In the simplest case, we determine the chemical state of the atmosphere using the $T(P)$ profile of the atmosphere, atmospheric metallicity, and C/O ratio assuming chemical equilibrium. The atmospheric metallicity (M/H) is defined as the ratio of the abundances of all heavy elements to Hydrogen abundance in the atmosphere. Our climate model uses a pre-calculated grid of molecular abundances on a pressure vs. temperature vs. [M/H] vs. C/O grid. This chemistry grid is computed using the thermochemical equilibrium models presented in \citet{gordon1994computer,lodders99,lodders02,visscher06,channon10} and using protosolar elemental abundances from \citet{Lodders10}. The grid includes 73 temperature points between 75 K and 4000 K and 20 pressure points logarithmically spaced between 10$^{-6}$ and 3000 bars. This corresponds to a pre-computed grid with 1460 grid points. The molecular abundances included in this grid are H$_{2}$, H, H$^{+}$, H$^{−}$, H$_2{^{-}}$, H$_2^{+}$, H$_3^{+}$, He, H$_2$O, CH$_4$, CO, CO$_2$, OCS, HCN, C$_2$H$_2$, C$_2$H$_4$, C$_2$H$_6$, NH$_3$, N$_2$, PH$_3$, H$_2$S, SiO, TiO, VO, Fe, FeH, MgH, CrH, Na, K, Rb, Cs, Li, LiOH, LiH, LiCl, and e-. The metallicities included in the grid are: [M/H] (relative to Solar)= -1, -0.75, -0.5, -0.3, -0.25, 0.0, 0.5, 0.7, 1.0, 1.5, 1.7 and 2. The C/O ratios included in the grid are: C/O (relative to Solar) = 0.25, 0.5, 1.0, 1.5, 2, and 2.5. In this way, Solar values are [M/H]=0 and C/O=1. In order to change the C/O ratio for a given atmospheric metallicity, both the elemental abundances ratios -- C/H and O/H are increased/decreased slightly while maintaining a constant (C+O)/H. The version of \texttt{PICASO 3.0}, described in this work, does not allow to change other elemental ratios ( e.g. S/H or N/H ). Future releases will include this flexibility. Figure \ref{fig:figabun} shows the pre-computed volume mixing ratios of four major atmospheric gases -- {\meth}, {\co}, {\water}, and {\amon} for a solar mixture of chemical elements. Converged brown dwarf $T(P)$ profiles with log(g)=5 and multiple {\teff} values between 300 K and 2300 K are also overplotted in the pressure-temperature space to depict parts of the parameter space relevant for objects with different {\teff}. There are a few important features that strongly influence the climates of sub-stellar atmospheres that are worth noting. First, the sharp drop in {\water} vapor abundance at $T\lessapprox$200~K which can be seen in Figure \ref{fig:figabun} lower left panel. This drop in {\water} vapor abundance is caused by the condensation of {\water} into cloud particles. A similar condensation effect can also be seen in the case of {\amon} at $T\le$100 K. Condensation induced changes in the gaseous abundances are included in the pre-computed grid of gaseous abundances. Second, the most abundant carbon bearing gas in the atmosphere also changes from \co\ to \meth\ between temperatures of $\sim$ 1000 and 1200 K. Figure \ref{fig:figabun} top right panel shows that \co\ is the major carbon bearing gas above $\sim$ 1200 K but at temperatures cooler than 1200 K, \co\ abundance decreases rapidly and \meth\ abundance rises. This change can be understood by the net chemical reaction responsible for inter-conversion between \co\ and \meth\ under chemical equilibrium, \begin{equation} {\rm CH_4 + H_2O \leftrightarrow CO + 3H_2} \\ \end{equation} This reaction is favored in the forward direction for higher temperatures than $\sim$ 1200 K and as result \co\ is more dominant for such temperatures. The reverse reaction dominates for temperatures lower than $\sim$ 1200 K and \meth\ becomes the dominant C-bearing gas there. Many other similar interesting trends has been explored in the literature for several gases \citep[e.g.][]{lodders02,visscher06,visscher10,moses05,moses11,moses13}. These trends can be directly explored from the available chemical grid within the model but we move on to another challenge of using a pre-computed chemistry grid -- limited number of grid points. Due to the finite number of grid points (1460 points), chemical abundances cannot be calculated exactly at any pressure--temperature point of interest. Therefore, each time the chemistry routine \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/justdoit.py#L1527}{\texttt{premix\char`_atmosphere}} is called, the climate model uses an interpolation scheme to compute abundances at each pressure-temperature layer point. This interpolation is performed by the module \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/justdoit.py#L2247}{\texttt{chem\char`_interp}}. The 2D interpolation scheme used relies on finding the four surrounding grid points namely $T_{\rm low}(P_{\rm low}$), $T_{\rm low}(P_{\rm high}$), $T_{\rm high}(P_{\rm low}$), and $T_{\rm high}(P_{\rm high}$). Then the abundances in these surrounding points represented by $\xi^i_{low,low}$, $\xi^i_{low,high}$, $\xi^i_{high,low}$, and $\xi^i_{high,high}$ are used to interpolate the abundance of each species in the $T(P)$ point. This interpolation is done using, \begin{align} \ln\xi^i(T,P) = (1-t_{\rm int})(1-p_{\rm int})\ln\xi{^i_{low,low}} + & \\ t_{\rm int}(1-p_{\rm int})\ln\xi{^i_{high,low}} +t_{\rm int}p_{\rm int}\ln\xi{^i_{high,high}} + \nonumber &\\ (1-t_{\rm int})p_{\rm int}\ln\xi{^i_{low,high}} \nonumber \end{align} where $t_{\rm int}$ and $p_{\rm int}$ are given by, \begin{align} t_{\rm int} = \dfrac{1/T-{1}/{T_{low}}}{{1}/{T_{high}}-{1}/{T_{low}}} \\ p_{\rm int} = \dfrac{lnP-lnP_{low}}{lnP_{high}-lnP_{low}} \end{align} In the case where the $T(P)$ moves outside the edges of this chemistry grid, a linear interpolation scheme is instead adopted within the range 50-5200~K. This allows some flexibility in the iteration scheme. However, beyond these values, the model is not valid. Note, choices in interpolation routines cause discrepancies in resulting abundances. For example, only using two nearest-neighbors causes instabilities in chemical profiles. The interpolation chosen here was tested in multiple chemical regimes to ensure stability. After computing the atmospheric chemistry, the next step in Figure \ref{fig:figschematic} is the computation of the atmospheric opacities, which we describe next. \subsubsection{Pre-mixed Opacities}\label{sec:opa} The chemical structure of the atmosphere dictates the optical depth ($\tau(\lambda)$) of each atmospheric layer which is necessary for solving the radiative transfer equations described in \S\ref{sec:RT}. This calculation of atmospheric optical depths occurs in the ``Opacity" block of the model flow chart in Figure \ref{fig:figschematic}. As previously stated, we use the correlated-k approach to handle the molecular opacities in this model. An alternative approach is the line--by--line method which is more precise but comparatively speaking, more computationally expensive than the correlated-k approach at low resolution, and over large wavelength ranges. For completeness, we describe the general methodology of our approach here. However, the data were originally computed by \citet{lupu_roxana_2021_cks}, available on Zenodo, and detailed in \citet{marley21}. We include a table of the main reference data used to compute the pre-mixed opacities in Table \ref{tab:my_label}. \begin{table*} \centering \begin{tabular}{c|c} C$_2$H$_2$ & \citet{hitran2012}\\ C$_2$H$_4$ & \citet{hitran2012}\\ C$_2$H$_6$ & \citet{hitran2012} \\ CH$_4$ & \citet{yurchenko13vibrational, yurchenko_2014}\\ CO & \citet{HITEMP2010,HITRAN2016,li15rovibrational}\\ CO$_2$ & \citet{HUANG2014reliable}\\ CrH & \citet{Burrows02_CrH}\\ Fe & \citet{Ryabchikova2015,oBrian1991Fe,Fuhr1988Fe, Bard1991Fe,Bard1994Fe} \\ FeH & \citet{Dulick2003FeH, Hargreaves2010FeH} \\ H$_2$ & \citet{HITRAN2016} \\ H$_3^+$ & \citet{Mizus2017H3p}\\ H$_2$--H$_2$ & \citet{Saumon12} with added overtone from \citet{Lenzuni1991h2h2} Table 8\\ H$_2$--He & \citet{Saumon12} \\ H$_2$--N$_2$ & \citet{Saumon12} \\ H$_2$--CH$_4$ & \citet{Saumon12} \\ H$_2^-$ & \citet{bell1980free}\\ H$^-$ bf & \citet{John1988H}\\ H$^-$ ff & \citet{Bell1987Hff}\\ H$_2$O & \citet{Polyansky2018H2O}\\ H$_2$S & \citet{azzam16exomol}\\ HCN & \citet{Harris2006hcn,Barber2014HCN,hitran2020}\\ LiCl & \citet{Bittner2018Lis}\\ LiF & \citet{Bittner2018Lis}\\ LiH & \citet{Coppola2011LiH} \\ MgH & \citet{Yadin2012MgH,GharibNezhad2013MgH} assembled in \citet{GharibNezhad2021}\\ N$_2$ & \citet{hitran2012}\\ NH$_3$ & \citet{yurchenko11vibrationally,Wilzewski16} \\ OCS & \citet{HITRAN2016}\\ PH$_3$ & \citet{sousa14exomol} \\ SiO & \citet{Barton2013SiO} \\ TiO & \citet{McKemmish2019TiO} assembled in \citet{GharibNezhad2021}\\ VO & \citet{McKemmish16} assembled in \citet{GharibNezhad2021}\\ Li,Na,K & \citet{Ryabchikova2015,Allard2007AA, Allard2007EPJD,Allard2016, Allard2019}\\ Rb,Cs & \\ \end{tabular} \caption{Data used to compute correlated-K opacities. Correlated-K opacities are available at \citet{lupu_roxana_2021_cks} and detailed in \citet{marley21}.} \label{tab:my_label} \end{table*} In the correlated-k approach, the relevant wavelength range for the radiative transfer is first divided to 196 wavelength bins. These carefully chosen wavelength bins are shown in Figure \ref{fig:figwbin} and chosen to reflect the approximate spectral energy distributions of planetary atmospheres. The molecular opacity $\kappa$ in any one of these wavelength bins has numerous individual molecular lines, as shown in Figure \ref{fig:figkcoeff} left panel. The cumulative distribution function (CDF) of the opacity can be defined as $G(\kappa_0) = N(\kappa \le \kappa_0)$ where $N$ denotes the total number of instances when the condition $\kappa \le \kappa_0$ is satisfied within the wavelength bin. The CDF $G(\kappa)$ of the opacity within each wavelength bin is computed. This function, $G(\kappa)$, is then inverted to obtain the ``k-distribution''. An example k-distribution for the the opacity window shown in the left panel in Figure \ref{fig:figkcoeff}, is shown in Figure \ref{fig:figkcoeff} right panel. After computing a k-distribution for each of the 196 wavelength bins, the 8 Gauss points ($g_i$) shown in Table \ref{table:tab2} are used to represent each distribution. Generally, the slope of k-distributions are shallow and slowly changing at lower Gauss point values ($G$) between 0 and 0.9. Then, the slope steepens rapidly between $G$= 0.9--1, as can be seen in Figure \ref{fig:figkcoeff}. To capture the complex shape of the k-distribution with just 8 Gauss points, a double-Gauss method is used for the integration. The first set of four Gauss points are shown in Figure \ref{fig:figkcoeff} right panel with black points. These four Gauss points sparsely sample most of the k-distribution between $G=$0--0.95 whereas the last four Gauss points, shown with red points in Figure \ref{fig:figkcoeff}, sample the small, but rapidly changing, part of the k-distribution between $G=$0.95--1. The first and second set Gauss points and weights for this double-Gauss point method can be generated from sample points of the generally used Gauss-Legendre quadrature \citep{abramowitz} using, \begin{equation} \begin{aligned} g_{i,1} = & f(g_i^{'} + 1)/2 \\ w_{i,1} = & fw_i^{'}/2 \\ g_{i,2} = & f+ (1-f)(g_i^{'} + 1)/2 \\ w_{i,2} = & (1-f)w_i^{'}/2 \\ \end{aligned} \end{equation} where $g_{i,1}$ and $w_{i,1}$ represent the first set of our Gauss points (shown with black points in Figure \ref{fig:figkcoeff}) and $g_{i,2}$ and $w_{i,2}$ represents the second set (shown with red points in Figure \ref{fig:figkcoeff}), $g_i^{'}$ and $w_i^{'}$ are the sampling points for Gauss-Legendre quadrature of some order (4 in our case) defined within the interval [-1,1], and $f$ is the adjustable parameter which sets the division in the values of $G$ which will be sampled by the first and second set of Gauss points. For example, if $f=0.8$ then $g_{i,1}$ will sample values between 0 and 0.8 while the rest will be sampled by $g_{i,2}$. For our purpose, this $f$ is set at 0.95 which is a reasonable choice based on the typical shape of k-distributions as shown in Figure \ref{fig:figkcoeff}. This double-Gauss method for calculating correlated k-coefficients has a large impact on the time and computational efficiency of the code as it reduces the number of Gauss points required for radiative transfer calculations but still maintains sufficient accuracy required for the model. It has been shown that the radiative transfer with double-Gauss method with 8 Gauss points is as accurate as a normal set of 20 Gauss points \citep[Michael Line, priv. comm. as detailed in][]{marley21}. The choice of the wavelength-bins for the correlated-k opacities are also crucial for the model and are described next. The 196 wavelength bins used span 0.2 $\mu$m -- 227 $\mu$m to capture the general spectral energy distribution of planetary atmospheres. However, these wavelength bins are not all of equal wavelength width. These wavelength bins are shown in Figure \ref{fig:figwbin} where three blackbody curves corresponding to temperatures of 300 K, 1000 K and 2000 K are also shown for comparison. The bins are narrow between 0.5-10 $\mu$m to capture the large number of molecular rovibrational bands in this range. At the tail ends of the blackbody distributions, larger bins help boost computational speed and maintain sufficient precision in radiative transfer calculations required for the application cases of this model. As our model uses a pre-computed chemistry grid during iterations on the $T(P)$ profile, the k-coefficients are also pre-computed on the same pressure--temperature--metallicity--C/O ratio grid. For intermediately $T-P$ values, the molecular opacity is interpolated using the same formalism as has been described for the gas abundances in \S\ref{sec:chem}. Figure \ref{fig:figpremixopa} shows heat maps of pre-computed Planck function and abundance weighted molecular cross-sections in the same pressure-temperature grid as Figure \ref{fig:figabun}. For each $P-T$ point in the grid, the abundance weighted molecular cross-sections are integrated over all wavelengths using the Planck function (corresponding to the temperature $T$) as the integrating kernel (see Equation 2 in \citet{freedman14}). Each panel corresponds to cross-sections at different Gauss points which we use in our models. Planck mean cross sections at only the first four Gauss points are shown here in the four panels. Dashed black lines marked on the cross-section maps depict different converged $T(P)$ profiles of brown dwarfs from \citep{marley21} with {\teff} between 300 K and 2300 K and log(g)=5. As the order of the Gauss points increases in Figure \ref{fig:figpremixopa}, the cross-sections become higher as higher order Gauss points trace higher opacity parts of the k-distribution. Achieving atmospheric convergence in regions of the $P-T$ space where the cross-section change rapidly with small changes in temperature or pressure can be difficult. Such an ``opacity cliff" can be seen at $T$ values between$\sim$ 900-1700 K in Figure \ref{fig:figpremixopa}. This cliff appears in the Planck mean cross-sections due to the overlap of the peak of the Planck function at 900-1700 K with large {\water} opacity bands between 1.5-3 $\mu$m. These ``opacity cliffs" were also seen in \citet{freedman2008opacities,freedman14} with Rosseland-mean opacities. We discuss the effect of these cliffs on model convergence in more detail in \S\ref{sec:convergence_rec}. In addition to molecular opacities, collision induced absorption (CIA) of H$_2$-H$_2$, H$_2$-H, H$_2$-He, H$_2$-N$_2$, H$_2$-CH$_4$, and continuum opacities such as H-bf, and H-ff are also accounted for. These are included separately from our correlated-k table. The CIA opacities are pre-calculated between temperatures of 75 K and 7000 K with 1000 wavelength bins and then interpolated to the correct $P-T$ combination with a spline. Note that even though the CIA temperatures are computed up to 7000~K, the temperature valid range of our model is still limited by the 1460 opacity grid. We note that the pre-computed opacity grid used with this model is known to be incomplete with regards to atomic and ionic opacities, which are particularly important at the high temperature and low pressure parts of the atmosphere \citep[e.g.][]{hoeijmakers2018atomic, hoeijmakers2020hot}. % This limits, to some extent, the current code's ability to treat the ``ultra-hot Jupiters" ($T_\mathrm{eq}\gtrapprox$2200~K). In a future update we will add these opacities to the correlated-K tables. \subsubsection{Disequilibrium Chemistry}\label{sec:deq_chem} A significant addition to this model is the capability to treat vertical mixing induced disequilibrium chemistry self-consistently within the radiative-convective equilibrium framework. This is an optional part of the model, as shown in Figure \ref{fig:figschematic}. In 1D models, vertical mixing is often parametrized as a diffusive process which is described by the eddy diffusion coefficient, {\kzz} \citep{allen81}. The mixing timescales of all relevant gases in each atmospheric layer is given by, \begin{equation}\label{eq:tmix} t_{\rm mix} = \dfrac{H^2}{K_{\rm zz}} \end{equation} where $H$ is the local scale height of that atmospheric layer. All thermochemical reactions within the atmosphere like CO$\leftrightarrow$CH$_4$ and NH$_3${$\leftrightarrow$}N$_2$ also proceed with a characteristic timescale -- {\tchem}. \citet{Zahnle14} parametrized the {\tchem} of several such gases using 1D chemical kinetics models. The parametrized {\tchem} from \citet{Zahnle14} and \citet{visscher06} are given by, \begin{align*} t_{\rm CO,CH_4,H_2O} &= \dfrac{3\times10^{-6}}{P_{\rm bar}}exp\left(\dfrac{42000 K}{T}\right)\\ t_{\rm NH_3,N_2} &= \dfrac{10^{-7}}{P_{\rm bar}}exp\left(\dfrac{52000 K}{T}\right) \\ t_{\rm CO_2} &= \dfrac{10^{-10}}{\sqrt{P_{\rm bar}}}exp\left(\dfrac{38000 K}{T}\right)\\ t_{\rm HCN} &= \dfrac{1.5\times10^{-4}}{P_{\rm bar}m^{0.7}}exp\left(\dfrac{36000 K}{T}\right)\\ t_{\rm PH_3} &= \dfrac{1.9\times10^{12}}{\rm [OH]}exp\left(\dfrac{6013.6 K}{T}\right) \end{align*} \label{eqn:tchem} where $P_{\rm bar}$ is the atmospheric pressure in bars, $T$ is the temperature and $m$ is the atmospheric metallicity relative to solar metallicity. {\tchem} is generally short at high pressure-high temperature regions of the atmosphere. As the atmosphere gets colder at lower pressures the {\tchem} rises exponentially and becomes large. The volume mixing ratios of {\co}, {\meth}, {\amon}, {\cotwo}, {\water} and HCN are expected to follow equilibrium chemistry at pressures where {\tmix} $>>$ {\tchem} which happens in the deeper atmosphere. But as the atmosphere becomes colder with lowering pressure, {\tchem} may exceed {\tmix}. The pressure at which this occurs is called the ``quench pressure". At pressures less than the ``quench pressure", gases are expected to depart from chemical equilibrium and their mixing ratios become constant. In this model, we include these disequilibrium chemistry effects for the 8 gas species shown in Eqn. \ref{eqn:tchem}: {\water}, {\meth}, {\co}, {\cotwo}, {\amon}, N$_2$, HCN, and PH$_3$. The quench pressure is determined using the parametrized {\tchem} in Eqn. \ref{eqn:tchem}. The abundance then follows the equilibrium chemistry at pressures more than the quench level, and is held constant at pressures less than the quench level. The uncertain parameter in the disequilibrium modeling framework is the eddy diffusion parameter, {\kzz} \citep{Philips20,fortney20,karilidi21}. Therefore, this model is flexible in assumptions regarding {\kzz}. Currently, there are two user-options for defining \kzz\ within a model run: 1) a fully user-defined {\kzz} value which is either constant or variable throughout the height of the atmosphere, but which does not change during the iterative climate solution \citep[as was done in][]{karilidi21,Philips20}, and 2) a model-predicted {\kzz}, which is calculated from the $T(P)$ profiles in the convective zones, using mixing length theory \citep{gierasch85energy}, and in the radiative zones, using parametrizations \citep[e.g.][]{moses21}. In the second case, along with the $T(P)$ profile, the \kzz\ will also change simultaneously with the iterations in the model. The {\kzz} in the convective zone can be calculated with mixing length theory and is given by \citep{gierasch85energy}, \begin{equation}\label{eq:kzz_con} K_{zz}= \dfrac{H}{3}{\left(\dfrac{L}{H}\right)}^{{4}/{3}}{\left(\dfrac{RF}{\mu{\rho_a}c_p}\right)}^{{1}/{3}} \end{equation} where $H$ is the local scale height of the atmosphere, $L$ is the turbulent mixing length, $R$ is the universal gas constant, $\mu$ is the mean molecular weight of the atmosphere, $\rho_a$ is the atmospheric density, $c_p$ is the atmospheric specific heat at constant pressure and $F$ is the convective heat flux. The convective heat flux can be calculated by the difference between the net thermal radiative flux and $\sigma{T_{\rm eff}}^4$ within the convective zones of the atmosphere. Therefore, $\sigma{T_{\rm eff}}^4$ is the maximum allowed value of $F$ in this framework if the energy transport within the convective atmosphere is assumed to be completely convective. While various parametrizations of {\kzz} in the radiative zones of substellar atmospheres have been discussed in the literature \citep{wang15,zhang18,parmentier13,tan22}, we have included the parametrization from \citet{moses21} as a starting point. Further parametrizations of {\kzz} can easily be swapped in, in the future. The \citet{moses21} radiative zone {\kzz} is given by, \begin{equation}\label{eq:kzz_rad} K_{zz} = \dfrac{5\times10^{8}}{\sqrt{P_{\mathrm{bar}}}}\left(\dfrac{H}{{620} {\rm km}}\right)\left(\dfrac{T_{\rm eff}}{1450 {\rm K}}\right)^{4} \end{equation} where $P_{\mathrm{bar}}$ is the pressure of the radiative level in bars and $H$ is the atmospheric scale height. Both Equation \ref{eq:kzz_con} and \ref{eq:kzz_rad} can be found in the \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/fluxes.py#L3455}{\texttt{get\char`_kzz}} function. As the $T(P)$ profile iterates towards the converged solution, the quench levels of various gases change as well. This means that the abundances of quenched species will depart from chemical equilibrium, and the pre-computed k-coefficient tables described in \S\ref{sec:opa} are no longer valid. The methodology to remix the k-coefficients with updated abundances and recompute resulting optical depth calculation during the convergence process, is referred to ``on--the--fly" mixing, and is described in the following \S\ref{sec:onthefly}. \subsubsection{Mixing Opacities ``on--the--fly"}\label{sec:onthefly} With disequilibrium chemistry, the atmospheric chemistry depends on the quench pressures, which again depend on the $T(P)$ profile of the atmosphere. Therefore, because the chemistry of the atmosphere cannot be pre-determined, the atmospheric opacities also need to be calculated ``on--the--fly". We mix the correlated-k opacities of individual gases using the methodology of \citet{amundsen17} called the resort-rebin technique. Currently, in this model we focus on the quenching of {\co}, {\meth}, {\water}, {\amon}, {\cotwo}, N$_2$, HCN and PH$_3$. However, the major opacity sources among these gases are mainly {\co}, {\meth}, {\water}, and \amon. Meaning, the contribution of N$_2$, HCN and PH$_3$ on the total gas opacity is negligible for the relatively small departures from the chemical equillibrium with $\log$(M/H)$\le$2.5 explored in this analysis. % Therefore, we mix the correlated-k opacities of {\co}, {\meth}, {\water}, and {\amon} with the correlated-k opacities of all the other sets of background gases which follow equilibrium chemistry as the volume mixing ratio of these gases evolve due to quenching with the $T(P)$ profile of the atmosphere. Should the motivation to include more gases in the ``on--the--fly'' methodology present themselves in future observations, they can be included in a future code release. One of the drawbacks of ``on--the--fly mixing" outlined in \citet{amundsen17} is the dependence of the accuracy of the technique on the spectral resolution of the correlated-k opacities. This effect has been well explored in \citet{karilidi21}. Specifically, it was found that the 196 wavelength bins, traditionally used in the chemical equilibrium version of \texttt{EGP} and this new python version, is not sufficiently high to use with the resort-rebin technique. \citet{karilidi21} found that 661 wavelength bins in the correlated-k opacities are required to counteract the inaccuracies in the resort-rebin technique. Therefore, when disequilibrium calculations are requested by the user, \texttt{PICASO} automatically switches to 661 wavelength bins. With the opacities and mixing routines described, the final module, shown in Figure \ref{fig:figschematic} is the computation of the convective zones, which we outline in the following \S\ref{sec:convec}. \subsubsection{Convective Zones}\label{sec:convec} As the $T(P)$ profile iterates towards radiative equilibrium, parts of the atmosphere will become unstable against convection, and energy transport will be expected to occur via convection instead of radiation. These parts of the atmosphere are forced to follow the local adiabat. Figure \ref{fig:figconvec} shows a heat map of the adiabatic lapse-rate, ${\rm dlnT}/{\rm dlnP}$, used in \texttt{PICASO}. This grid is pre-computed assuming a solar H--He mixture with He mass fraction of Y=0.28. This includes H$_2\leftrightarrow$ 2H dissociation and a detailed accounting of the molecular vibrational and rotational levels\footnote{Raw grid can be found in GitHub file \href{https://github.com/natashabatalha/picaso/blob/climate/reference/climate_INPUTS/specific_heat_p_adiabat_grad.json}{specific\_heat\_p\_adiabat\_grad.json}}. The calculations for this grid was done by Didier Saumon as described in \citet{marley21}. The pre-computed grid has 53 temperature points between 10 K and 3981 K and 26 pressure points between 10$^{-2}$ and 10$^3$ bars. The lapse-rate of parts of the profile which become unstable against convection are interpolated from this grid. For every atmospheric layer, the lapse-rate, ${\rm dlnT}/{\rm dlnP}$, is first calculated from the $T(P)$ profile. The grid of adiabatic lapse-rates shown in Figure \ref{fig:figconvec} is then used to interpolate the local adiabatic lapse rate for each atmospheric layer. This interpolation is done using 2D interpolation similar to the technique described in \S\ref{sec:chem}. If the ratio between the lapse-rate obtained from the $T(P)$ profile and the interpolated, lapse rate, $\nabla$ is greater than $\sim$1 (numerically set to 0.98) in any of the layers, then these layers are considered convective and are forced to follow the interpolated local adiabatic lapse rate according to the following equation, \begin{equation} T^{i+1} = \exp\left({T^{i} +{\nabla}ln\left(\dfrac{P^{i+1}}{P^{i}}\right)}\right) \end{equation} where $T^{i}$ and $P^{i}$ are the temperature and pressure of the i'th convective layer and $T^{i+1}$ and $P^{i+1}$ are the temperature and pressure of the $i$+1'th convective layer starting from below the radiative-convective boundary layer. We should note that this convective adjustment is done for one layer at a time followed by a iteration of the entire $T(P)$ as this adjustment for each layer, in principle, would lead to temperature changes in all the other layers. This approach used in this model is different from forcing all the layers which are unstable against convection to be convective at once. The \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/climate.py#L603}{\texttt{t\char`_start}} module implements this convective adjustment of the $T(P)$ profile. This methodology is used to develop and grow convective zones in the atmosphere with the \href{https://github.com/natashabatalha/picaso/blob/caf63752563215e76ec713f65182f7efc367f3fc/picaso/justdoit.py#L3589}{\texttt{find\char`_strat}} module during the iterations of the model. Note that this module is designed to always form or grow convective zones and not shrink or remove them. We discuss the important implications of this for the user in \S\ref{sec:recommendations}. \subsection{Iteration Scheme And Model Convergence}\label{sec:model_convergence} Each required and optional physical component of the model shown in Figure \ref{fig:figschematic} has been described separately in \S\ref{sec:physics}. But as our model is iterative in nature, a description of the iterative scheme of the model along with its convergence criteria is a necessary component. We provide a brief outline of the model iterative scheme for both the chemical equilibrium and disequilibrium runs separately below. \subsubsection{Iterative Scheme for Chemical Equilibrium}\label{sec:chem_eq_convergence} This model aims to achieve radiative-convective equilibrium using the Newton-Rhapson iterative scheme. As a first step, the chemistry and the correlated-k opacities of the atmosphere for the first given guess $T(P)$ profile are calculated as detailed in \S\ref{sec:chem} and \ref{sec:opa}. Then, the methodology described in \S\ref{sec:RT} is used to compute the net radiative fluxes in all atmospheric layers. The temperature of each layer is then perturbed by a small arbitrary $dT$, historically set at 0.01\% of the current layer temperature, while keeping the other layer temperatures fixed. The radiative fluxes in all layers are then recomputed after this perturbation. These two sets of radiative fluxes are used to compute the Jacobian A$_{ij}$ which quantifies the change in radiative flux in layer $i$ due to perturbed temperature in layer $j$. The Jacobian is given by \citep[e.g.][]{hubeny17}, \begin{equation} A_{ij} = \dfrac{F^{j}_{i,p}-F_{i}}{dT_{j}} \end{equation} where $F_{i}$ is the net radiative flux in layer $i$ before the perturbation and $F^{j}_{i,p}$ is the perturbed net radiative flux in layer $i$ due to a change in temperature $dT_{j}$ in layer $j$. The ultimate temperature correction, $\delta{T}$, needed to ensure radiative equilibrium (in the radiative zones) is then solved using the equation, \begin{equation}\label{eq:correction} A\delta{T} = \sigma{T_{\rm eff}}^4 - F(T(P)) \end{equation} where $A$ is the Jacobian and $F(T(P))$ are the net radiative fluxes in each layer with the current atmospheric state. % This process continues iteratively until a tolerance of maximum allowed radiative flux difference from $\sigma$\teff$^4$ is reached in all the radiative layers of the atmosphere. This convergence criteria can be expressed as, \begin{equation}\label{eq:tolerance} \left|\dfrac{F^n_{\rm net} - {\sigma}T_{\rm eff}^4}{{\sigma}T_{\rm eff}^4}\right| \le \epsilon \end{equation} where $\epsilon$ denotes the tolerance parameter, set at $5\times10^{-3}$ in our model. This equation is a numerical version of Equation \ref{eq:RCEfin}. The model is also considered to be converged if the maximum fractional temperature correction among all the radiative layers $|\delta{T/T}|$, calculated from Equation \ref{eq:correction}, during any iteration is smaller than $\epsilon$. It is important to note that the model satisfies the convergence criteria in Equation \ref{eq:tolerance} in the mid-point of each atmospheric layer while the temperature at the edges of each atmospheric layer (levels) is iterated. For example, if the optical depths at the bottom and top (edges) of the $i$'th atmospheric layer are $\tau_i$ and $\tau_{i-1}$, then the model aims to satisfy Equation \ref{eq:tolerance} in this layer for an optical depth of $(\tau_i+\tau_{i-1})/2$. But, the iteration of the temperature profile is done to the temperature at the bottom and top (or edges) of this atmospheric layer. This is important for the stability of the iterative process. As the iteration progresses, the stability of each atmospheric layer against convective mixing is also checked using the technique described in \S\ref{sec:convec}. If layers unstable against convection are found, the temperature of these convectively unstable layers are forced to a H$_2$-He gas mixture adiabat from the grid shown in Figure \ref{fig:figconvec}. Once the convergence criteria of Equation \ref{eq:tolerance} is met, the model run stops and it produces the outputs outlined by the green boxes in Figure \ref{fig:figschematic}. Outputs such as the converged $T(P)$ profile, chemical abundances and outgoing radiative fluxes in the 196 wavelength grid are always returned to the user after model converges. These outputs can further be optionally used for calculating various observables such as transmission, emission or reflected light spectra with \texttt{PICASO}. For a deeper understanding of the iteration scheme of this model, three atmospheric iteration steps for a brown dwarf with {\teff} of 1000 K and log(g)=5 are shown in Figure \ref{fig:figiter_bd} and iterations for an irradiated planet at a distance of 0.1 AU from a sun like star with {\tint} of 300 K and log(g)=3.4 are shown in Figure \ref{fig:figiter_exo}. In Figure \ref{fig:figiter_bd} and \ref{fig:figiter_exo}, each row corresponds to the atmospheric state in a certain iteration of the model. The first column shows the $T(P)$ profile of the atmosphere in each iterative step of this model with the red solid line whereas \texttt{SONORA} $T(P)$ profile for this case is shown as a black dashed line. The second column shows the lapse-rate of the $T(P)$ profile as a function of pressure in each iteration with the solid blue line and the adiabatic lapse-rate is shown with a black solid line. The third column shows the volume mixing ratio profiles of four gases -- {\water}, {\meth}, {\amon}, and {\co} as a function of pressure and the fourth column shows the emergent thermal spectrum from the top of the atmosphere. In Figure \ref{fig:figiter_bd}, the first row shows a simple initial isothermal guess profile of 500 K and the last row shows the final converged solution. For our initial guess, the bottom four layers of the atmosphere have been assumed to be convective. The initial thermal emission spectrum resembles a blackbody because the atmosphere is isothermal. As the model iterates, the isothermal $T(P)$ profile is perturbed to reach an atmospheric state such that Equation \ref{eq:tolerance} is satisfied for all radiative layers of the atmosphere. In the iteration shown in the middle row, the bottom of the atmosphere becomes unstable against convection and therefore in the final solution the lapse-rate in that part of the atmosphere is forced to follow the local adiabatic lapse-rate. As the model iterates towards the converged solution, the $T(P)$ profile becomes more complex and leads to a redistribution of flux in each wavelength bin. This leads to molecular absorption lines becoming more evident in the thermal spectrum with each iteration. The final converged solution matches well with the \texttt{SONORA BOBCAT} grid of models (shown as black dashed line). For the irradiated planet convergence shown in Figure \ref{fig:figiter_exo}, the initial guess is an isothermal profile with $T= 700$ K and the iterations continue until Equation \ref{eq:tolerance} is satisfied. A second helpful method of visualizing a converged run is to look at the contribution of radiative and convective fluxes as compared to the target flux in brown dwarfs given by $\sigma$\teff$^{4}$ or the internal heat flux for planets given by $\sigma$\tint$^{4}$. Figure \ref{fig:fig_bd_exo1_exo2} shows these layer--by--layer radiative and convective fluxes for converged models of a representative brown dwarf (left panel), warm Jupiter (middle panel), and hot Jupiter (right panel). The net radiative flux in each layer is shown with the orange shaded region. The shaded blue region depicts the convective flux at each layer. The red dashed line shows the target flux for the brown dwarf -- $\sigma$\teff$^{4}$. This converged brown dwarf model has two convective and radiative zones as can be seen from its $T(P)$ profile shown in black. As a result, the convective flux (blue) peaks at the location of these convective zones. The net radiative flux decays rapidly at the deeper convective zone and convection carries the majority of the energy in these deep convective layers. The proof that this is a converged model and Equation \ref{eq:tolerance} is satisfied lies in the fact that the sum of the net radiative and convective fluxes is equal to the target flux at all atmospheric layers. For the planet case where there is an additional energy flux from the host star, we show the additional energy with black hatched shading (2nd and 3rd column of Figure \ref{fig:fig_bd_exo1_exo2}). This incident flux is downward, compared to the thermal radiative and convective fluxes, which is upward. Therefore, the upper atmosphere needs larger quantities of upward thermal radiative fluxes to balance this downward stellar flux. Ultimately, the goal is to maintain a summed flux of $\sigma$\tint${^4}$ (red dashed line) at all the atmospheric layers of the planet. Therefore, the upper atmosphere of irradiated planets, with a given \tint\ , is pushed toward hotter temperatures when compared to a brown dwarf with a comparable {\teff}. \subsubsection{Iteration Scheme for Chemical Disequilibrium}\label{sec:EGP_tk} The primary difference in the iterative scheme of the disequilibrium model is that it uses a two-step convergence method. With the user specified input parameters, the code first uses the basic chemical equilibrium model (\S\ref{sec:chem_eq_convergence}) to converge to a chemical equilibrium atmospheric solution. This atmospheric solution is then used as the initial guess to the disequilibrium chemistry model in the second step of the iterative process. Of particular importance, is the need to recalculate the convective zones when disequilibrium chemistry is turned on. This is important because recent work has shown that there are major differences in the location, extent and number of convective zones between the equilibrium chemistry and disequilibrium chemistry atmospheric solutions (Mukherjee et al. (2022 submitted)). The general procedure is to partially reset the convective zones before moving to disequilibrium solver. First, any upper convective zones are removed. Next, the upper boundary of the deepest convective zone is reset to slightly higher pressures (usually $\sim$5 levels in a 91-level atmosphere from 10$^{-4}$ to 200 bars). This enables the convective zones in the presence of vertical mixing to be re-calculated because we cannot assume there will be minor perturbations from the equilibrium chemistry solutions. In order to highlight the convergence of a brown dwarf disequilibrium chemistry model, a few atmospheric iterations are shown in Figure \ref{fig:figiter_bd_deq} for a brown dwarf with {\teff} of 700 K and log(g)=5.25. The figure is similar to Figure \ref{fig:figiter_bd} except here the third column now shows the mixing (black dashed line) and chemical timescales (solid colored lines) as a function of pressure and the fourth column shows the volume mixing ratio profiles of various gases. The first row shows the atmospheric state reached after the first step of the convergence process, where a converged chemical equilibrium solution is achieved. The last row shows the final converged solution with disequilibrium chemistry. The chemical equilibrium solution from the \texttt{SONORA} grid is shown with the black dashed line in the first column. As the model iterates, chemical equilibrium $T(P)$ profile is perturbed to reach an atmospheric state where the $T(P)$ profile is visibly colder than the chemical equilibrium solution. This is a result of different atmospheric optical depths as a result of quenching of several gases in the deeper atmosphere. The chemical equilibrium solution has a single deep convective zone for this brown dwarf. However, in the iteration shown in the middle row and last row, a second convective zone also develops when disequilibrium chemistry is treated self-consistently. This shows that chemical disequilibrium also impacts the location and number of convective zones in the atmosphere. As a second convective zone develops in the middle and the third row, a decrease in mixing timescale can also be seen in that pressure range of the convective zone because convective zones are assumed to be much more efficient in mixing and thus have higher {\kzz} and lower {\tmix} than radiative zones. Comparing the fourth column in the first and the last row shows that mixing causes several orders of magnitude change in the abundance of {\co} in the upper atmosphere when compared to chemical equilibrium solutions. With these examples, we conclude our detailed discussion on the methodology of the model and move on to the model benchmarking analysis. \section{Benchmarking Analysis}\label{sec:benchmark} We have benchmarked this model in four ways in order to ensure all the described methodology is functioning as expected: 1) \texttt{PICASO 3.0} chemical equilibrium non-irradiated functionality vs. \texttt{SONORA BOBCAT} grid from \citet{marley21}, 2) \texttt{PICASO 3.0} chemical equilibrium non-irradiated functionality vs. \texttt{ATMO 2020} code from \citet{Philips20}, 3) \texttt{PICASO 3.0} chemical equilibrium irradiated functionality vs. \texttt{ATMO} code from \citep{goyal2020}, and 4) \texttt{PICASO 3.0} chemical disequilibrium non-irradiated functionality vs. \texttt{ATMO 2020} code from \citet{Philips20}. \subsection{Benchmarking chemical equilibrium non-irradiated atmospheres} Figure \ref{fig:figbenchmark1} shows the comparison between the three chemical equilibrium non-irradiated models. The models described in this paper are shown in red, the \texttt{BOBCAT} models are shown with a black dashed line, and the blue dot-dashed line show the models from \citet{Philips20}. The left columns show a comparison between the $T(P)$ profiles of the three models. The right column shows a comparison between the lapse-rate of the profiles. Included in the benchmarking is four different log(g) = 5 brown dwarf models with {\teff} values of 600 K, 700 K, 800 K, and 1000 K, which covers significant chemical transitions of carbon species. \texttt{PICASO 3.0} and the models from \citep{marley21} match well with $T(P)$ profiles differences smaller than 0.1\%. The lapse-rates from this model also match excellently with those from \citet{marley21}. Given \texttt{PIASO}'s heritage in the EGP model used to compute the grid in \citet{marley21}, this is validation that the update to Python from Fortran did not introduce numerical issues. The disagreement between \texttt{PICASO 3.0} and the chemical equilibrium models from the \texttt{ATMO 2020} grid are typically smaller than 1\% in the deeper atmosphere below 10 mbars. At pressures less than 10 mbars, the differences between the two models are between 1\% to 5\%. These differences are considered minor given the independent code setup, opacity calculations, and chemistry routines. For example, models presented in \citet{Philips20} use the ``on--the--fly" opacity mixing method described in \S\ref{sec:onthefly} even when modeling atmospheres with chemical equilibrium whereas \texttt{PICASO 3.0} uses interpolations of pre-mixed opacities for iterations of chemical equilibrium models. Differences such as these along with differences in opacity line lists used by the two models could account for the 1\% to 5\% differences in $T(P)$ profiles in the upper atmosphere. \subsection{Benchmarking chemical equilibrium irradiated atmospheres} We also benchmark the equilibrium chemistry version of the code by comparing models of irradiated exoplanets calculated using \texttt{PICASO 3.0} with the published \texttt{ATMO} models presented in \citet{goyal2020}. We use the RCTE models for WASP-25 b \citep{enoch11,Southworth_2014} presented in \citep{goyal2020} to perform this benchmarking. WASP-25 b is a hot Saturn with an estimated equilibrium Temperature of $\sim$ 1210 K (assuming 0 albedo) \citep{enoch11,Southworth_2014}. We use the same system parameters for WASP-25 b as are used in the models presented in \citet{goyal2020} and detailed in the Appendix section of their article. Figure \ref{fig:figbenchmark4} shows the comparison between the two models at sub-solar and solar metallicities with a super-solar C/O ratio. The models generally agree for both the metallicities. For the sub-solar metallicity models, the agreement is better than 1\% across all the pressures. For the solar metallicity models, the maximum disagreement between the models is $\sim$ 3\%. These minor differences were also found between the \texttt{PICASO 3.0} and \texttt{ATMO 2020} models for brown dwarfs. As described above, these minor differences can be attributed to different opacities and numerical methodologies used. \subsection{Benchmarking non-irradiated atmospheres with diseequilibrium chemistry} In order to benchmark the version of the model with disequilibrium chemistry we have done two types of comparisons. First, we set {\kzz} = 0 in our disequilibrium chemistry model and compare to the results of our equilibrium chemistry model from the \texttt{SONORA} grid. Second, we benchmark our disequilibrium chemistry model with results from the \texttt{ATMO 2020} grid. In the first test, if {\kzz} is assumed to be zero, {\tmix} becomes infinitely large according to Equation \ref{eq:tmix}. Therefore, in principle, none of the gases will be quenched and their volume mixing ratios will follow equilibrium chemistry throughout the atmosphere. Therefore, a disequilibrium chemistry model run with {\kzz} = 0 must produce the same result as the equilibrium chemistry models. Of particular importance, this test checks if our ``on--the--fly" mixing routines, and double-iterative routine are performing as expected. Figure \ref{fig:figbenchmark2} shows the benchmarking between our disequilibrium chemistry model with {\kzz} = 0 and the equilibrium chemistry models from the \texttt{SONORA BOBCAT} grid. The left column shows the comparison of the $T(P)$ profiles between the two models and the right column shows the comparison of the lapse-rates between the two models. Each row from the top to bottom shows log(g) = 5.0 brown dwarf models with {\teff} of 600, 700, 800, and 1000 K. The two models match with differences $\le$ 1\% level. The small $\le$ 1\% deviations are due to the slight inaccuracies in the ``on--the--fly" mixing technique \citep{amundsen17}, which is sensitive to number of wavenumber bins, versus the pre-weighted technique (see \S\ref{sec:onthefly}). In the second test, we directly benchmark our disequilibrium chemistry model with disequilibrium chemistry models from \citet{Philips20}. Disequilibrium chemistry models in \citet{Philips20} assume a constant-with-pressure {\kzz} for the atmospheres. Therefore, we use a constant {\kzz} in our benchmarking test and show the results in Figure \ref{fig:figbenchmark3}. For a brown dwarf with {\teff} of 700 K and log(g) of 5.5, the top left panel in Figure \ref{fig:figbenchmark3} shows: 1) the differences that arise in the $T(P)$ profiles when disequilibrium is turned on for both the models from \citet{Philips20} and \texttt{PICASO 3.0}, and 2) the differences that arise between \citet{Philips20} and \texttt{PICASO 3.0}. Both the equilibrium chemistry and disequilibrium chemistry models show excellent matches with among themselves which is an excellent benchmarking demonstration of our methods and codes. Additionally, this also shows the impact of disequilibrium chemistry on the $T(P)$ profiles of this representative brown dwarfs, which results in $\sim$50~K colder $T(P)$ profiles compared to equilibrium chemistry models. The rest of the panels in Figure \ref{fig:figbenchmark3} show comparisons between various disequilibrium chemistry $T(P)$ profiles with different {\kzz} and gravity values produced with our model with the grid of models from \citet{Philips20}. Our disequilibrium models generally agree well within 5\% levels in the deeper atmosphere with models from \citet{Philips20}. However, the disagreements are higher in the much lower pressure upper atmosphere which was also the case for the equilibrium chemistry model comparisons. The small disagreements between the models are mainly due to two reasons -- 1) \citet{Philips20} use chemical kinetics models for calculation of quenched abundances whereas we use the ``quench-time" approximation (\S\ref{sec:deq_chem}) to do so and 2) differences between the accuracy of the ``on--the--fly" mixing method used between the two models due to differences in number of wavelength bins used. These disagreements show the uncertainty in state--of--the--art models which perhaps can be explained in future with comparisons with high signal-to-noise data. \section{Modeling Recommendations}\label{sec:recommendations} The code is generally well-behaved across the parameter space of interest for brown dwarfs and exoplanets. However, iterative schemes are sometimes notoriously tricky to converge. Therefore, here we outline recommendations regarding the use of code such that users can get meaningful and accurate results. These recommendations are implemented in the publicly available code tutorials available via the \texttt{PICASO} documentation page \footnote{https://natashabatalha.github.io/picaso/tutorials.html released with publication acceptance}. \subsection{Choosing Model Pressure Grid} We recommend using typically 50-90 atmospheric pressure layers, corresponding to 51-91 pressure levels. In general, higher number of pressure layers increases the computational time required for convergence substantially and lower number of layers makes the atmospheric grid too coarse for an accurate calculation. The maximum and minimum values of the pressure grid are also ultimately chosen by the user. While the choice of the minimum pressure can be made somewhat arbitrarily, we recommend that the model not be run at pressures lower than 10$^{-6}$. This is the lowest pressure for which chemistry and opacities are computed (shown in Figure \ref{fig:figabun} and \ref{fig:figpremixopa}). Another uncertainty from the double-Gauss correlated-k approach can arise from running the model at very low pressures ($\le$ 10$^{-4}$ bars). As the molecular lines become very narrow at such low pressures, most of the opacities resides at very high values of $G(\kappa)$ (e.g. between 0.995-1.0) in the k-distribution. Therefore, these opacities are not taken into account even with the double-Gauss method. This can make the radiative transfer at such low pressures inaccurate. The maximum pressure of the atmospheric model needs to chosen carefully in the case of giant planet atmospheres and brown dwarfs. If the maximum pressure of the atmospheric model is too low, then the model can become transparent to the deepest atmospheric layer, especially in wavelengths with little gaseous opacities (e.g. optical wavelengths $\lessapprox1\mu$m). This would result in an inaccurate $T(P)$ profile. A good way to check if the profile was run with low maximum pressure is to plot the wavelength dependent brightness temperature of the converged model along with the converged temperature of the deepest atmospheric layer. If the brightness temperature at any wavelength exceeds the brightness temperature associated with the bottom temperature, then the pressure grid is not well-suited for the calculation. However, if the brightness temperature is smaller than this bottom temperature at all wavelength, then the pressure grid is well-suited for the calculation. In the first scenario, the pressure grid needs to be extended to higher pressures. The problem with this approach to choose a pressure-grid is that it can only be done after running the model once. A more practical alternative to this approach is to find a comparable model from the \texttt{SONORA BOBCAT} \footnote{\href{https://zenodo.org/record/5063476\#.YkX8tjfMJhE}{Link to published SONORA BOBCAT models}} to the temperature, gravity parameter space wanted and use the maximum pressure corresponding to that model. Also, the highest pressure for which chemistry and opacities are computed (shown in Figure \ref{fig:figabun} and \ref{fig:figpremixopa}) is 3000 bars. Therefore, runs with pressure grids extending to higher pressures than this will be inaccurate as they would require linear extrapolation of both gas abundances and opacities. When computing disequilibrium chemistry runs, the user must take another aspect into consideration while choosing the maximum pressure of the atmospheric pressure grid: the quench levels of various gases. If the maximum pressure of the atmospheric grid is too low such that gases are expected to quench at deeper parts of the atmosphere than the pressure grid, then the model run will be incorrect. Therefore, users must check the quench levels of various gases after performing a disequilibrium chemistry calculation with the model. If the quench levels of any of the gases are at the last pressure level, then the user needs to increase the maximum pressure of the run to get a correct converged solution. \subsection{Choosing the Initial Guess T(P) Profile} Climate models that leverage an iterative scheme require a first guess of the $T(P)$ profile. This guess $T(P)$ profile is then iterated to reach the converged solution. Even though we used isothermal $T(P)$ profiles as our first guess in both Figure \ref{fig:figiter_bd} and \ref{fig:figiter_exo}, it is usually preferable to start with an initial guess that is close to the expected solution. Using simple profiles like isothermal profiles can lead to significant increase in run times and in the worst cases also lead to solutions which do not converge. A far better alternative to isothermal profiles, are publicly available model grids like \texttt{SONORA BOBCAT} or the \citet{Philips20} models. For exoplanets, we recommend using parametrized $T(P)$ profiles from, for example \citet{guillot10}, as a first guess. Functionalities to browse these profiles directly are already available in \texttt{PICASO}, via the \href{https://natashabatalha.github.io/picaso/picaso.html?highlight=guillot_pt#picaso.justdoit.inputs.guillot_pt}{\texttt{guillot\_pt()}}, and \href{https://natashabatalha.github.io/picaso/picaso.html?highlight=sonora#picaso.justdoit.inputs.sonora}{\texttt{sonora()}} functions \footnote{also see various tutorials e.g. \href{https://natashabatalha.github.io/picaso/notebooks/6_BrownDwarfs.html\#Download-and-Query-from-Sonora-Profile-Grid}{Download and Query from Sonora Profile Grid}, and \href{https://natashabatalha.github.io/picaso/notebooks/FAQs.html\#How-do-I-access-the-pressure-temperature-profile-parameterizations?}{Accessing the pressure-temperature profile parameterizations}}. Another first guess required to run the model is the extent of the deepest convective zone. Specifically, the user needs to specify a guess of the pressure level of the radiative-convective boundary in the atmosphere which the code will then modify using methods described in \S\ref{sec:convec}. Even though this guess is modified within the iteration, we recommend always setting the bottom 3-5 levels to be convective first. If the user sets too many layers to be convective, the converged solutions may be inaccurate because the model is only equipped to grow and merge convective zones, not shrink them. Therefore, the best practice is to start with 3-5 bottom convective levels. If the code runs into convergence issues after doing so, we recommended increasing this number slowly to check if it helps converge the solution. \subsection{Convergence}\label{sec:convergence_rec} In most cloud-free cases, the code will converge. However, there are a few known cases where extra steps are needed to reach a converged solution. For equilibrium chemistry brown dwarf models with {\teff} between 1500 K--1700 K, an oscillating behaviour is seen in the iterations of the $T(P)$ profile. In these cases, the $T(P)$ profile oscillates between two profiles with a constant temperature offset. This behaviour is not unexpected and is caused by the sharp ``cliff" in gaseous opacity at these effective temperatures. This can be seen in the cross-section maps in Figure \ref{fig:figpremixopa}. All the four panels in Figure \ref{fig:figpremixopa} show a sharp change in gaseous opacity around the 900 K--1700 K temperature range. The overlap of this cliff with the converged brown dwarf $T(P)$ profile at {\teff}=1700 K can also be seen in Figure \ref{fig:figpremixopa}. As already mentioned in \S\ref{sec:opa}, this sharp opacity ``cliff" makes the convergence difficult around this temperature range. In order to overcome this, we recommend performing multiple runs for the same object by recycling the unconverged final $T(P)$ profile of each run as an initial guess profile for the next run until a converged solution is reached. A similar convergence issue also occurs for objects with {\teff} $<$ 300 K. Figure \ref{fig:figabun} bottom left panel shows how {\water} abundance in the vapor phase falls drastically due to {\water} condensation at such low temperatures. This sharp drop in the {\water} abundance causes the $T(P)$ profiles of these cold objects at the low pressure regions to oscillate between multiple possible solutions. This behaviour can be seen in the upper parts of the {\teff}=300 K $T(P)$ profile in Figure \ref{fig:figpremixopa}. We recommend two solutions to this problem: 1) either set the minimum pressure of the model pressure grid to no less than 1 mbar, or 2) use a \texttt{SONORA} profile as an initial guess and recycle the unconverged profile multiple times ($\sim$ 4 times) to reach a converged state. The first recommendation helps the user exclude the part of the $T(P)$ profile which causes the instability in the convergence. As these objects are cold, the excluded pressure region is extremely cold ($<$ 120 K), which means that there is practically no contribution to the observables (e.g. total emergent flux). \section{Calculating Observables} After using the radiative-convective equilibrium model to calculate the atmospheric structure of a brown dwarf or an exoplanet, the user may want to compute various observables. \texttt{PICASO} has been updated to include the capability of calculating the 1D and 3D thermal emission spectrum, the 1D transmission spectrum, and the 1D and 3D reflection spectroscopy, and phase curves. With a 1D $T(P)$ profile, users will likely want to produce 1D higher resolution thermal emission spectra, reflection spectra, and/or transmission spectra of the planet or brown dwarf. These calculated observables can be directly compared with observational data. Figure \ref{fig:figobservable} shows an example of this with an exoplanet. The exoplanet has been assumed to be a Jupiter mass and size planet with {\tint}=300 K at a distance of 0.1 AU around a Sun like star. The radiative-convective equilibrium model described in this work has been used to calculate the planet-wide average atmospheric structure of this hypothetical planet once with chemical equilibrium and then again with chemical disequilibrium. The top left panel in Figure \ref{fig:figobservable} shows the $T(P)$ profile of this planet in each case. It is clear that self-consistent treatment of disequilibrium chemistry also impact the $T(P)$ profile of irradiated planets compared to equilibrium chemistry models. We have then used the 1D radiative transfer routines of \texttt{PICASO} to compute the reflection albedo spectrum of this exoplanet. This albedo spectrum has been shown in Figure \ref{fig:figobservable} top right panel. The planet is not very reflective after 0.5 $\mu$m, after which brightness from Rayleigh scattering tapers off. The transmission spectra of the planet is also modeled with \texttt{PICASO} and is shown in Figure \ref{fig:figobservable} bottom left panel. This panel shows that the presence of disequilibrium chemistry can be clearly detected for a hot Jupiter like this with its transmission spectra between 3-5 $\mu$m. The thermal spectra are shown in the bottom right panel and the presence of disequilibrium chemistry in hot Jupiters also can be easily detected with their thermal spectra in the M-band, or in various NIRSpec and/or NIRCam JWST modes. Note that for this illustrative example, we have used the planet-wide average computed $T(P)$ profile for all these viewing geometries. However, a more self-consistent method to compute these viewing geometries would be to compute a day-side averaged profile for use in the zero-phase reflection and thermal spectrum, and a planet-wide average for the transmission spectrum. \texttt{PICASO} has also been equipped with simple modules to couple the atmospheric models produced from this code with the evolutionary models from \citet{marley21} to calculate absolute Vega magnitudes of the modeled object in any filter of choice. The atmospheric model presented in this paper used with these radiative transfer tool are already available in \texttt{PICASO} and can be immensely helpful in modeling or planning observations using models. \section{Future Improvements}\label{sec:improve} Currently, our publicly released model has the capability to model non-cloudy brown dwarfs and exoplanets with equilibrium chemistry and disequilibrium chemistry. However, several future improvements of the model are needed to enhance its capability to capture a larger parameter space. Here we briefly discuss these needed future improvements. We note that all \texttt{PICASO} development is public, and open to community involvement. \subsection{Clouds} Our iterative climate model currently does not include the capability to treat atmospheric clouds self-consistently. The \texttt{EGP} code already includes condensation clouds with the \citet{ackerman2001cloud} model and has been used to model cloudy L-dwarf atmospheres \citep[e.g.][]{cushing08,stephens09}. Clouds can become important opacity sources in brown dwarf and exoplanetary atmospheres due to their strong scattering tendencies. This also has large effects on the $T(P)$ profile of these atmospheres \citep{morley14water,morley2012neglected}. We plan to couple this model with the Python-based cloud model \texttt{VIRGA} \citep{virga}. Recently, \texttt{VIRGA} has been updated to include variable sedimentation efficiency (f$_{\rm sed}$) with height \citep{rooney21}. We will couple this updated \texttt{VIRGA} model with our model so that our models can be used to model hotter L-dwarfs which are generally assumed to be cloudy and also much colder Y-dwarfs with {\water} clouds. This improvement will also enable us to apply our model to cloudy exoplanets and brown dwarfs. \subsection{1D Chemical Kinetics Model} We use the quench time approximation to model the effects of vertical mixing on the chemical abundance profiles in this model. This approximation is based on the parametrized mixing timescales from \citet{Zahnle14} outlined in \S\ref{sec:deq_chem}. However, more robust 1D chemical kinetics models of treating disequilibrium chemistry caused by vertical mixing are now open-sourced like the \texttt{VULCAN} model \citep{tsai17,tsai21}. We plan to couple our Python model with the \texttt{VULCAN} chemical kinetics model to increase the flexibility of our model. % This will make our disequilibrium chemistry models more robust. Moreover, \texttt{VULCAN} also includes the capability to treat stellar irradiation induced photochemistry. This will lead to a significant improvement for applicability of our model to irradiated exoplanets where photochemistry can largely impact atmospheric chemistry especially at lower pressures. This update will allow us to explore the impact of photochemistry on the atmospheric structure of exoplanets. We have already made progress for this coupling but still need to test and benchmark several aspects of the coupled code before this update is publicly available for use. \subsection{Time Evolution Version} Recently, \citet{mayorga21,robinson14} enhanced the \texttt{EGP} model to \texttt{EGP+} by including a time-stepping version of the code which can model the dynamic temporal response of the atmosphere to time-varying physical conditions like stellar irradiation changing with time. This is especially relevant to planets with highly eccentric orbits as the stellar irradiation can change by a large amount within one orbital time-period for these planets. We plan to adapt this improvement within our code in the near future as well. Studying the atmospheres of eccentric exoplanets is a promising research area and a subject of future JWST observing campaigns. \section{Conclusions and Summary}\label{sec:summary} We presented a new open-source Python based 1D radiative--convective equilibrium model as part of the \texttt{PICASO} package. This code derives its heritage from the \texttt{EGP} code developed by \citet{marley1999thermal} based upon a Titan atmosphere model developed by \citet{mckay1989thermal}. The \texttt{EGP} code has been used to model brown dwarf and exoplanetary atmospheres for almost three decades now. The model is applicable to H-dominated atmospheres of both brown dwarfs and irradiated exoplanets. The model includes the capability to do calculations with both equilibrium chemistry and disequilibrium chemistry due to vertical mixing. The model includes options to use {\kzz} values constant with height while performing disequilibrium chemistry runs. We have also included the capability to use self-consistent prescriptions of {\kzz} within the model where the {\kzz} will also iterate along with the $T(P)$ profile and atmospheric chemistry to ultimately reach a converged atmospheric state. We have benchmarked this model with publicly available models from the \texttt{SONORA BOBCAT} grid and also with results from an independent model used by \citet{Philips20} to produce the \texttt{ATMO 2020} atmospheric grid. For irradiated planets, the \texttt{PICASO 3.0} models were benchmarked against the \texttt{ATMO} grid presented in \citet{goyal2020}, using the hot Saturn WASP 25-b as a test case. The chemical equilibrium version of the model was benchmarked both with models from the \texttt{SONORA} grid, \texttt{ATMO 2020}, and the \texttt{ATMO} models for WASP 25-b. The chemical disequilibrium models were benchmarked against models from the disequilibrium chemistry atmospheric models from the \texttt{ATMO 2020} grid of models. This benchmarking analysis showed excellent agreement with \texttt{PICASO}. Our model is open-source and publicly available for the community (we include several in text links to code throughout this manuscript). Additionally, we outlined many recommendations in this work for proper usage of the model. This includes recommendations on choosing the atmospheric pressure grid for a science object, choosing the initial guess $T(P)$ profile for a particular run and also the various parts of the parameter space where the model is known to face convergence issues. We also included ways of resolving these convergence issues. We will also release tutorials to apply this model for various science cases with the code, upon paper acceptance. We plan to actively develop this model further to include clouds, couple it with 1D chemical kinetics codes for better robustness and also include time dependent effects like variable stellar irradiation within our model. \section{Acknowledgments} SM thanks the UC Regents Fellowship award for supporting him for this work. NEB acknowledges support from NASA Astrophysics Division. NEB and JJF acknowledge support from NASA’S Interdisciplinary Consortia for Astrobiology Research (NNH19ZDA001N-ICAR) under award number 19-ICAR19\_2-0041. JJF and MSM acknowledge the support of NASA XRP grant 80NSSC19K0446. We thank the anonymous referee for helping us to improve the manuscript significantly. {\it Software:} PICASO \citep{batalha19}\footnote{upon acceptance we will formally release PICASO 3.0 and update this with the Zenodo link}, pandas \citep{mckinney2010data}, NumPy \citep{walt2011numpy}, IPython \citep{perez2007ipython}, Jupyter \citep{kluyver2016jupyter}, matplotlib \citep{Hunter:2007} \bibliography{sample_arxiv}{} \bibliographystyle{apj}

Title: MHD Simulation of Homologous Eruptions from Solar Active Region 10930 Caused by Sunspot Rotation

Abstract: The relationship between solar eruption and sunspot rotation has been widely reported, and the underlying mechanism requires to be studied. Here we performed a full 3D MHD simulation of data-constrained approach to study the mechanism of flare eruptions in active region (AR) NOAA 10930, which is characterized by continuous sunspot rotation and homologous eruptions. We reconstructed the potential magnetic field from the magnetogram of Hinode/SOT as the initial condition and drove the MHD system by applying continuous sunspot rotation at the bottom boundary. The key magnetic structure before the major eruptions and the pre-formed current sheet were derived, which is responsible for the complex MHD evolution with multiple stages. The major eruptions were triggered directly by fast reconnection in the pre-formed current sheet above the main polarity inversion line between the two major magnetic polarities of the AR. Furthermore, our simulation shows the homologous eruption successfully. It has reasonable consistence with observations in relative strength, energy release, X-ray and H{\alpha} features and time interval of eruptions. In addition, the rotation angle of the sunspot before the first eruption in the simulation is also close to the observed value. Our simulation offers a scenario different from many previous studies based on ideal instabilities of twisted magnetic flux rope, and shows the importance of sunspot rotation and magnetic reconnection in efficiently producing homologous eruptions by continuous energy injection and impulsive energy release in a recurrent way.

https://export.arxiv.org/pdf/2208.08957

\title{MHD Simulation of Homologous Eruptions from Solar Active Region 10930 Caused by Sunspot Rotation} \author{Xinyi Wang} \affiliation{SIGMA Weather Group, State Key Laboratory for Space Weather, National Space Science Center, Chinese Academy of Sciences,Beijing 100190, PR China} \affiliation{College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing 100049, PR China} \author[0000-0002-7018-6862]{Chaowei Jiang} \affiliation{Institute of Space Science and Applied Technology, Harbin Institute of Technology, Shenzhen 518055, PR China, \url{chaowei@hit.edu.cn}} \author[0000-0001-8605-2159]{Xueshang Feng} \affiliation{College of Earth and Planetary Sciences, University of Chinese Academy of Sciences, Beijing 100049, PR China} \affiliation{Institute of Space Science and Applied Technology, Harbin Institute of Technology, Shenzhen 518055, PR China, \url{chaowei@hit.edu.cn}} \author[0000-0002-1916-1053]{Aiying Duan} \affiliation{Planetary Environmental and Astrobiological Research Laboratory (PEARL), School of Atmospheric Sciences, Sun Yat-sen University, Zhuhai 519000, PR China} \author{Xinkai Bian} \affiliation{Institute of Space Science and Applied Technology, Harbin Institute of Technology, Shenzhen 518055, PR China, \url{chaowei@hit.edu.cn}} \keywords{Magnetohydrodynamic (MHD) --- Sun: corona --- Methods: numerical --- Sun: magnetic fields} \section{Introduction} Solar eruption is considered as the most magnificent phenomenon in the solar system. It is manifested as flares and coronal mass ejections along with high energetic particle events, and these solar transients from solar corona can affect heavily the solar-terrestrial environment. By estimating the typical parameters of the eruptive source regions, it has been well recognized that only the magnetic free energy stored in the coronal current exceeds the required energy density as released in a typical eruption \citep{2000JGR10523153F}. Based on this, many theories of solar eruption have been proposed, which converged into the standard CSHKP model \citep{1964NASSP..50..451C, 1966Natur.211..695S, 1974SoPh...34..323H, 1976SoPh...50...85K} by grasping the key structure of magnetic field and can be applied to incorporate many observations (such as flare, particle acceleration, shock wave and radio burst). Although the basic scenario is well established, the initiation mechanism of solar eruption remains not fully understood. Currently two kinds of initiation mechanisms are frequently invoked, one is based on the ideal plasma macro-instabilities and the other on non-ideal, micro process, i.e., magnetic reconnection \citep{2011LRSP....8....1C}. The coronal magnetic field in the non-eruptive evolution is nearly force-free, and a particular force-free structure, magnetic flux rope (MFR), holds the central position in models based on ideal instabilities. In the earliest model of such kind, the MFR is simply taken as an electric wire as in \cite{2000JGR...105.2375L} (later an MFR has a twisted 3D structure), which is confined to be in equilibrium by the overlying magnetic arcade that is anchored at the photosphere. The ideal instabilities of such a pre-existing MFR, mainly the torus and kink instability, provide an effective way for driving eruptions as developed by many theoretical and simulation researches \citep{1978mit..book.....B,2004A&A...413L..27T,2005ApJ...630..543F,2006PhRvL..96y5002K}. When the MFR reaches the critical point of instability, the eruption is triggered with a quick lifting up of the MFR. Meanwhile a current sheet (CS) forms under the erupting MFR in a dynamic way. A flare is resulted when magnetic reconnection sets in at the CS, which converts magnetic energy to thermal and nonthermal energies that power the flare. Usually the torus instability is considered to be more efficient, while the kink one can only bring MFRs to be torus unstable or, otherwise, confined flares \citep{2005ApJ...630L..97T,2013AdSpR..51.1967S}. The second type of mechanisms as built upon magnetic reconnection needs a CS forms before eruption, such as the bipolar tether-cutting model \citep{1980IAUS...91..207M,1992LNP...399...69M,2001ApJ...552..833M} and the quadrupolar breakout model \citep{1999ApJ...510..485A,2012ApJ...760...81K}. In these models, the CS first forms, and reconnection then triggers the eruption, while MFR forms during the eruption, which is distinct from the first type in which the CS is built up at the wake of the erupting MFR. In the breakout model, a magnetic null point needs to be pre-existing above a sheared core. The expansion of the sheared core will compress the null point to form the breakout CS. Slow reconnection in the CS progressively weakens the overlying field, which in turn allows more expansion of the core field, and it is proposed that a positive feedback is established, which finally leads to the formation of the flare CS (i.e., the vertical CS within the sheared core) and an eruption \citep{2012ApJ...760...81K}. The tether-cutting model is simpler in the requirement of magnetic topology, since it is based on only a bipolar arcade. Shearing motion in bipolar field near the polarity inversion line (PIL) forms a CS. The reconnection in the CS cuts gradually the tethering field lines, which allows the expansion of the core field until a global disruption of the system is triggered, similar to that of the breakout model. The reconnection in a newly-formed large-scale CS, as resulted from stretching of the large-scale overlying field by the rising core field (as an MFR), further plays an important role in supporting the eruption. However, previous numerical simulations (e.g., \citealt{2003ApJ...585.1073A,2010ApJ...708..314A}) show that shearing motion alone can only help to form an MFR (along with flux cancellation), while its eruption is triggered by some ideal instabilities. Recently a ultra-high accuracy MHD simulation has established a new point: the sling-shot effect of reconnection can accelerate impulsively the plasma to fast eruption without ideal instabilities taking place \citep{2021NatAs...5.1126J}, thus emphasizing the key role of reconnection in both triggering and driving a eruption. In the above models, magnetic energy ought to be released mainly by fast reconnection \citep{1964NASSP..50..425P} even in the presence of an erupting MFR. Ideal instabilities alone can only release a small amount of magnetic energy \citep{1991ApJ...373..294F}, which is inadequate for accelerating the coronal plasma. Even though so many theoretical models exist, it is not easy to determine which one operates in the realistic events, since the coronal magnetic fields and their evolutions associated with eruptions are often much more complex than as described in the models. In the recent years, numerical models that are constrained or directly driven by the observed data have been developed and proved to be a powerful tool in probing the mechanisms of realistic solar eruptions (e.g., \citealt{2014ApJ...788..182I,2017ApJ...840...37P,2018ApJ...866...96J,2018shin.confE.208J,2021ApJ...919...39G}), and the progresses have been reviewed by \cite{JIANG2022100236}. In this paper we performed an MHD simulation of a data-constrained approach to study the mechanism of the flare eruptions in active region (AR) NOAA 10930. This AR is very eruption-productive and has been studied extensively in many previous papers (e.g., \citealt{2007PASJ...59S.785S,2008ApJ...676L..81J,2010AGUFMSH31A1785I,2011ApJ...740...19R,2011ApJ...740...68F,2014Natur.514..465A}). It appeared on the solar disk on December 2006 and produced a number of flares including four X-class ones in a few days (e.g., an X3.4 flare on December 13th and an X1.9 flare on December 14th). The most prominent dynamics of the AR is that a sunspot newly emerging into the AR showed continual rotation for days in the period with the flares. For example, previous studies found that the sunspot had rotated about $240^{\circ}$ in 2 days \citep{2007ApJ...662L..35Z} or $540^{\circ}$ in 5 days \citep{2009SoPh..258..203M} as measured by different methods. Such rotation resulted in a strong shearing flow near the main PIL of the AR and the magnitude of the shearing speed has also been estimated \citep{2008PASJ...60.1181M,2009ApJ...690.1820T}. There are some static modeling of the coronal magnetic field for this AR \citep{2008ApJ...675.1637S,2008ApJ...679.1629G}, i.e., by using nonlinear force-free field (NLFFF) extrapolation, verifying that the magnetic structure has a highly sheared core or MFR. A few works have been done also using dynamic MHD simulations with focus on the mechanism of eruptions, but the conclusions are at odds with each other. For instance, based on the observed vector magnetograms from Hinode/SOT \citep{2008SoPh..249..167T}, \cite{2014Natur.514..465A} first reconstructed a series of NLFFF solutions to follow the pre-flare evolution of the AR from December 9th to December 12th, and found that a sigmoidal MFR was progressively built up. Then with the pre-flare NLFFF solutions as the initial condition, they managed to simulate the eruption of the X3.4 flare by using 3 different types of ad-hoc boundary conditions at the bottom surface and concluded that the main trigger of the flare is torus instability of the MFR. \cite{2011ApJ...740...68F} constructed a background potential field by the line-of-sight magnetogram from SOHO/MDI for this AR and then introduced into the core field an artificial MFR through rigid emergence from the bottom boundary to simulate how the emerging MFR leads to the eruption in the background field. In such a study, they suggested a different trigger, i.e., kink instability rather than the torus instability because the decay index near the erupting MFR is found to be smaller than the critical value ($n<1.5$). Another study \citep{2017ApJ...842...86M} was performed by triggering the eruption with a small bipole emerging at the main PIL in a large-scale stable NLFFF field. By adjusting the orientations of the small bipole relative to the main PIL, they concluded that the so-called opposite polarity and reversed shear types' emergence can effectively trigger the eruption, as the same mechanism originally developed in \cite{2012ApJ...760...31K}. We note that none of the aforementioned dynamic simulations have taken into consideration the effect the continual and significant rotation of the AR's sunspot in leading to the eruptions, which, however, is strongly suggested by observations \citep{1909MNRAS..69..454E,2003SoPh..216...79B,2008MNRAS.391.1887Y,2018ApJ...856...79Y,2012ApJ...761...60V}. Although the preflare sheared magnetic structure is undoubtedly resulted by the sunspot rotation, there is no self-consistent model of such a process, and this is also the motivation of this paper. Here we employed an MHD model as driven by the sunspot rotation to follow the coronal magnetic evolution of AR 10930 from its energy slow accumulation to fast releasing process. We started the simulation with a potential magnetic field reconstructed from the observed magnetogram and then applied a rotational motion to the positive sunspot of the AR to mimic the observed rotation. With a continual rotational driving, our model displayed a full evolution from the initial potential field to two homologous eruptions (which may correspond to 2 X-class flares). We found that reconnection in a quasi-statically pre-formed CS triggered the homologous eruptions, which is consistent with a fundamental mechanism of solar eruption initiation as recently established \citep{2021NatAs...5.1126J, 2021arXiv211104984B, 2022ApJ...925L...7B}. Furthermore, our results of the coronal magnetic configuration have reasonable consistency with the observed soft X-ray and $\rm H\alpha$ features. Also the time interval and relative strength of the simulated eruptions are on the same scale of the quantities of eruptions as derived from observations. The mechanism in our research is different from other works that requires the pre-formed MFR and is initiated by ideal MHD instabilities. We also suggest that many homologous eruptions of rotational sunspots may be triggered by the same mechanism in this paper. The paper is organized as follows. We first show the observation and data in Section $\ref{Data and Observation}$, then describe the model and method in Section $\ref{Model and Method}$. Simulation results are displayed in Section $\ref{Results}$ and finally we give discussion and conclusion in Section $\ref{Conclusion}$. \section{Data and Observation}\label{Data and Observation} The AR NOAA 10930 is highly dynamic in which 4 X$-$class flares were produced and 3 of them occurred on December, 2006; X6.5 on December 6th, X3.4 on December 13th and X1.5 on December 14th \citep{2007PASJ...59S.779K}. In this research we focus on the X3.4 flare located at S07W22 on December 13th and the X1.5 flare located at S06W46 on December 14th \citep{2013ApJ...778...48B}. Figure~\ref{sunspot} taken from Stoke V images of HINODE/SOT \citep{2007SoPh..243....3K,2008SoPh..249..167T} shows the complex magnetic flux distribution of this AR. As denoted in the 4th panel of the figure, we define 4 areas of the photospheric magnetic field, which are, respectively, the strong positive (SP, which is the rotating sunspot), the weak positive (WP, the weak field region with positive polarities at the west side of the rotating sunspot), the strong negative (SN, i.e., the large sunspot) and the weak negative (WN, the weak field region of negative polarity at the west side of the large sunspot). There are mainly two sunspots with opposite magnetic polarities. The leading sunspot (SN) shows nearly no change during two flare events, while the smaller one in the south (SP) shows evident growth, and this indicates that the main sunspots are not a pair. The positive sunspot emerged later than the main negative sunspot, translating from west to east (right to left) \citep{2008ApJ...687..658W} with an obvious counter-clockwise rotation from December 10th to December 14th. It is connected not only to the main negative sunspot in the north but also to the west (right) dispersed polarities (WN) \citep{2009SoPh..258..203M}. The positive sunspot became diffused and rotated more slowly on December 14th. The evolution of an inverse-S sigmoid during two flares is shown in Figure~\ref{XRAY} taken from Hinode/XRT \citep{2005AdSpR..36.1489D,2007SoPh..243...63G}. The sigmoid formed near the main PIL and had a big tail around December 12th, which is likely formed by the rotation of the positive sunspot. The post-flare arcades spread from left to right during the first flare. In the middle of December 13th, the first flare ended, and the sigmoid reformed at the same position was involved in the second flare. The post-flare arcades during the second eruption didn't spread too much to the right as the flare ribbons. Both flares exhibited two ribbons signature as shown in the 1st and 3rd columns of Figure~\ref{QSLflare}, which are taken from Broadband Filter Imager (BFI) of SOT. The negative ribbon of both flares was located between the two main spots, spreading to right and was longer in the first eruption. While the positive ribbon was initially on the left side of the positive polarity (since the footpoints at the positive polarity have been rotated counter-clockwise to left) and it shrank into a circle to the right (Figure~\ref{QSLflare}). Except the length of the negative ribbon, the corresponding ribbons in both events resemble each other in both position and shape, which reflects the similarity in their underlying magnetic configurations and thus their trigger mechanisms. There was an Earth$-$directed CME with a projected speed of $\rm 1780~km~s^{-1}$ \citep{2010BASI...38..147R} on December 13th and another CME with a speed of $\rm 1042~km~s^{-1}$ on December 14th. A major geomagnetic storm was observed on December 15th. \section{Model and Method}\label{Model and Method} We used the DARE--MHD model \citep{2016NatCo...711522J} to study the dynamic evolution of the solar corona. The model was developed based on the CESE method \citep{2002JCoPh.175..168Z,1993LNP...414..396C} in Cartesian coordinate system combined with adaptive mesh refinement (AMR) technique by utilizing the PARAMESH \citep{2000CoPhC.126..330M} to solve the full MHD equations: \begin{equation}\label{MHD equation} \begin{array}{r} \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho \mathbf{v})=-\nu_{\rho}\left(\rho-\rho_{0}\right) \\ \rho \frac{D \mathbf{v}}{D t}=-\nabla p+\mathbf{J} \times \mathbf{B}+\rho \mathbf{g}+\nabla \cdot(\nu \rho \nabla \mathbf{v}) \\ \frac{\partial \mathbf{B}}{\partial t}=\nabla \times(\mathbf{v} \times \mathbf{B}-\eta \mu_{0} \mathbf{J}) \\ \frac{\partial T}{\partial t}+\nabla \cdot(T \mathbf{v})=(2-\gamma) T \nabla \cdot \mathbf{v} \end{array} \end{equation} where $\mathbf{J}=\nabla \times \mathbf{B}/\mu_{0}$, $\mathbf{g}$ is the solar gravity, $\mu_{0}$ is the magnetic permeability in vacuum, $\nu$ is the kinetic viscosity and $\gamma=1$ is the adiabatic index. We choose $\nu_{\rho}= 0.05~V_{A}$ (the $\rm Alfv \acute{e} n$ speed) to avoid the very low density in the strong magnetic field region, which may lead to a very small time step. By setting this, the plasma density will be relaxed to its initial value $\rho_{0}$ in a time scale of 20 $\rm Alfv \acute{e} n$ time $\tau_{\rm A}$. This time scale is sufficient large such that the fast dynamics of $\rm Alfv \acute{e} nic$ speed is not influenced. The viscosity $\nu$ is given as $\nu=0.05\Delta x^{2}/\Delta t$, where $\Delta x$ (varies from $1^{\prime \prime}$ to $4^{\prime \prime}$ in our simulation) and $\Delta t$ are the grid resolution and time step respectively. No explicit resistivity was applied in our simulation. Since to mimic the real corona environment, any explicit value of $\eta$ will give a larger value of resistivity than only a numerical method has, which will affect the reconnection process. The computational domain is sufficiently large of $\rm [-553,553]~Mm$ in $x$,$y$-direction and $\rm [0,1106]~Mm$ in $z$-direction to prevent the influence of the side and top boundary conditions on the computation of the eruption initiation. The Powell source terms and the diffusion control term are added to maintain the divergence-free condition of magnetic field as described in \cite{2010SoPh..267..463J}. \subsection{Initial Conditions} We smoothed the magnetogram at December 12th 2006, 20:30 UT which is taken from \cite{2008ApJ...675.1637S} using Gaussian smoothing with FWHM of 20 pixels. This makes the maximum value of $B_{z}$ decreases from 2619 G to 1595 G and then we constructed a potential field as the initial condition. The background plasma density satisfies a hydrostatic isothermal model with a value of $\rm 2.3\times 10^{-15}~g~cm^{-3}$ at bottom. To save the computation time, the strength of magnetic field from the smoothed magnetogram is reduced by a factor of 25. But this will make the plasma pressure and density decay slower than the background magnetic field, causing a higher plasma $\beta$, if we use the real value of solar gravity ($g_{\odot}=\rm 274~m~s^{-2}$). To avoid such a situation, we modified the gravity in the same way as \cite{2021NatAs...5.1126J}: \begin{equation}\label{modified gravity} g=\frac{k}{(1+z / L)^{2}} g_{\odot} \end{equation} where $k=5.7$ and $L=76.8$ Mm. In this way, the plasma $\beta$ around the active region is less than 0.1 under 340 Mm. The miminum value of plasma $\beta$ is $\beta = 2.5\times 10^{-3}$. The $\rm Alfv \acute{e} n$ speed $V_{\rm A}>1000$~km~s$^{-1}$ below 190 Mm. These mimic the real corona environment better. \subsection{Boundary Conditions} We energized the system by applying a photospheric rotational motion to the positive polarity at the bottom boundary, as shown in Figure~$\ref{velocityfield}$. To ensure that such a flow will not modify the magnetic flux distribution $B_{z}$ at the photosphere, the velocity can be specified by employing a potential function $\psi(B_{z})$ with $\mathbf v = \nabla \times (\psi \mathbf e_z) $. While the specific forms of the potential function in many previous researches (e.g., \citealt{2003ApJ...585.1073A,2010ApJ...708..314A,2013SoPh..286..453T,2021ApJ...922..108J}) made the line speed of rotation $|\mathbf v|$ vanish ($|\mathbf v|=|\nabla \psi|=0$) at PIL where $B_{z}=0$. As a result the shear flow near PIL is relative weak. To make the shear flow stronger, the velocity potential was modified as: \begin{equation}\label{velocity potential} \psi = v_{0}B_{z} \end{equation} and \begin{equation}\label{velocity field} v_{x}=\frac{\partial \psi\left(B_{z}\right)}{\partial y}, \quad v_{y}=-\frac{\partial \psi\left(B_{z}\right)}{\partial x} \end{equation} This velocity profile can reproduce the strongest shear flow $v = |\nabla \psi| = v_{0} |\nabla B_{z}|$ at PIL and the faster line speed of rotation in the north than south as observation shows \citep{2009SoPh..258..203M}. To save the computation time, $v_{0}$ is scaled such that the maximum speed is $\rm 34.1~km~s^{-1}$, which is about 60 times of the real value as $\rm 0.5~km~s^{-1}$ \citep{2009ApJ...690.1820T,2009SoPh..258..203M} but still smaller than the typical $\rm Alfv \acute{e} n$ speed by around two orders of magnitude. With such a large driving speed, the time scale of quasi-static evolution is shortened by the same times. The photospheric motion is coupled with the magnetic field evolution by the frozen-in theorem of ideal MHD, manifested as the line-tied condition, which is important to the success of simulation. To self-consistently update the bottom magnetic field, we solve the induction equation: \begin{equation}\label{induction equation} \frac{\partial \mathbf{B}}{\partial t}=\nabla \times(\mathbf{v} \times \mathbf{B})+\eta_{\rm stable} \nabla_{\perp}^{2}\mathbf{B} \end{equation} at the photosphere. The last term $\eta_{\rm stable} \nabla_{\perp}^{2}\mathbf{B}$ is used to maintain the numerical stability near the PIL (see also \citealt{2021FrP.....9..224J}). Here we set: \begin{equation}\label{etastable} \eta_{\rm stable}=1\times 10^{-2}e^{-B^{2}_{z}} \end{equation} On the side/top boundary, if we fix the plasma variables ($\rho,\mathbf{v},T$), there will be reflection. Instead, all the variables are extrapolated from the neighboring inner points using a zero gradient along the normal direction of the boundary surface. The normal component of magnetic field at side and top boundary is updated by divergence$-$free condition to avoid the accumulation of numerical error. This mimics the open boundary. \subsection{Topology}\label{Topology} To analyse the magnetic structure, we calculated the $Q$ factor to identify the quasi separatrix layers (QSLs) \citep{2002JGRA..107.1164T,2016ApJ...818..148L} as follows: \begin{equation} Q=\frac{a^{2}+b^{2}+c^{2}+d^{2}}{|a d-b c|} \end{equation} where \begin{equation} a=\frac{\partial X}{\partial x}, \quad b=\frac{\partial X}{\partial y}, \quad c=\frac{\partial Y}{\partial x}, \quad d=\frac{\partial Y}{\partial y} \end{equation} and $(X,Y)$ and $(x,y)$ are a pair of footpoints of the same magnetic field line. The region with large value of $Q$ (e.g., $\geq 10^5$) denotes the most possible location where reconnection will take place and is often used to be compared with the position and shape of the flare ribbons. \section{Results}\label{Results} \subsection{Overall Process}\label{Overall Process} Figure~\ref{energy} shows the evolution curves of the total magnetic and kinetic energies in the computational volume as well as their changing rates. The magnetic energy injection by the surface motion is also shown (the dashed line in Figure~\ref{energy}A), which is computed by time integration of the total Poynting flux at the bottom surface. As driven by the continual rotation of the sunspot for a time duration of 190~min (in which SP has rotated about 3 turns), the AR in the MHD model experiences firstly an overall increase of magnetic energy and then two eruption events with rapid release of part of the magnetic energy. The two eruptions can be identified clearly from the energy evolution, with onset time $t_{\rm E1}=119$~min for the first eruption (will be referred to as E1) and $t_{\rm E2} = 161$~min for the second eruption (E2), respectively. From the beginning to time of around $t=28$~min, the kinetic energy keeps a very low value of below $10^{-3}~E_{\rm p}$, and the magnetic energy injection curve matches well with increase of the total magnetic energy, owing to the line-tied boundary condition and the low numerical dissipation. This ideal process is followed by two small episodes of magnetic energy release that occur before E1. The first one (P1) starts at $t_{\rm P1} = 28$~min, after which the kinetic energy rises to $10^{-3}~E_{\rm p}$, and it results in a small deviation of the bottom surface energy input and the total magnetic energy accumulation. The second one (P2) occurs at $t_{\rm P2}= 80$~min, after which the kinetic energy first rises to a peak value of $3.4\times 10^{-3}~E_{\rm p}$ and then decreases slightly. The reason for these small energy release will be analyzed in the next sections. The first major eruption (E1) begins when the magnetic energy reaches about $1.44 E_{\rm p}$ (and the sunspot has been rotated about 1.5 turns). Through this eruption, the magnetic energy decreases to about $1.36 E_{\rm p}$ ($8\times 10^{-2}~ E_{\rm p}$ free energy loss) and the kinetic energy increases impulsively to $3.6 \times 10^{-2} E_{\rm p}$. That is, about a half of the magnetic energy loss is converted to kinetic energy in 10~min. The amounts of magnetic energy released and total kinetic energy obtained, on the order of magnitude of $10^{32} $~erg, are consistent with the estimations from previous studies that used NLFFF extrapolations for the pre-flare and post-flare magnetic fields~\citep{2008ApJ...675.1637S,2010BASI...38..147R}. After the first eruption, the magnetic energy increases again while the kinetic energy drops to a low value close to that of the pre-eruption state. At $t\sim 161 $~min starts the second eruption (E2), a weaker one than the first eruption. The magnetic energy decreases from $1.41 E_{\rm p}$ to $1.36 E_{\rm p}$ ($5\times 10^{-2}~E_{\rm p} $ free energy loss). The kinetic energy increases to $2.8\times 10^{-2}E_{\rm p}$ and $56\%$ magnetic energy has been converted to kinetic energy in around 10 min. The maximum erupting speed reaches $1500$~km~s$^{-1}$ and $1100$~km~s$^{-1}$ in E1 and E2, respectively. Both eruptions drive a fast shock wave with speeds of about $500$~km~s$^{-1}$. The complex distribution of magnetic flux in this AR renders the eruptions highly asymmetrical in both north-south and west-east directions. Since the magnetic field is multiplied by a factor of 0.04, the kinetic energy should be underestimated in our simulation because if we strengthen the magnetic field used to calculate, the ratio of kinetic energy to magnetic energy will be increased. Although it is not likely to reproduce realistically the observed flares with such simple setting of sunspot rotation, these two eruptions can still mimic approximately the observed two X-class flares on December 13th and 14th, respectively, as the first one, X3.4, is stronger than the second one of X1.5. The positive sunspot has rotated over 1.5 turns before E1, which is comparable with \cite{2009SoPh..258..203M}. Furthermore, if multiplied by a factor of 60 determined by the speeding up in our velocity-driven simulation, the quasi-static evolution time before E1 is about 5 days and time interval between E1 and E2 is about 40 hours. Both time scales are comparable with observations: 3 days of sunspot rotation before E1 and another 44-hour interval between E1 and E2. Interestingly, there will be more eruptions, produced in a homologous way, if the simulation is continued with further rotation of the sunspot, confirming that the sunspot rotation is an efficient mechanism in producing eruptions. Finally it is worthy noting that the total magnetic energy is always below the open field energy $E_{\rm open}\sim 1.51 E_{\rm p}$~\citep{1984ApJ...283..349A,1991ApJ...375L..61A,1991ApJ...380..655S} during the whole process, suggesting that eruption is efficient at keeping the magnetic energy below its upper limit, i.e., the open field energy. \subsection{Evolution of Magnetic Field and Electric Current}\label{Magnetic Evolution} To understand why the energies evolve in the manner as described in the last subsection, here we give a detailed study on the evolution of magnetic field, topology, and current density. First we consider the magnetic field evolution before the major eruption. The initial magnetic topology is shown in the first column of Figure~\ref{P1slice}C and F. It shows complex X-points around, QSLs above and at the west (right) side of SP. These initial QSLs play an important role in magnetic evolution. Before $t_{\rm P1}$, P1-QSL, i.e., the QSL above SP, was strengthened by stress between the sheared core field (which expands outward as driven by the rotation) and the surroundings. This contributed to the formation of a current layer, referred to as P1-CS at the location of P1-QSL. At the same time a current layer was also developed above the main PIL between SP and SN, for which we called the PIL-CS. It developed with rotated SP between sunspots and didn't show any sudden changes. These current layers before $t_{\rm P1}$ were not strong enough, i.e., not sufficiently thin to trigger reconnection, so the kinetic energy remained to be a very small value. The magnetic energy injection from the bottom boundary and its increase in the coronal volume matched each other well, showing the signature of quasi-static evolution in this period. P1-CS took effect when it became strong enough after $t_{\rm P1}$. The core expansion let the P1-CS (the gray iso-surface in Figure~\ref{P1slice}B) form at the top of SP and translate to west side subsequently, with exactly the same location of P1-QSL (Figure~\ref{P1slice}C, D, F and G). Reconnection in P1-CS let WN connect to SP continuously, leading to the exchanges of SP-SN and WP-WN (as in Figure~\ref{P1slice}A). The current layer PIL-CS was still too weak before $t_{\rm P2}$ while kept developing (Figure~\ref{P1slice}B). Weak outflow ($\rm 500~km~s^{-1} $) were produced by slow reconnection in P1-CS (Figure~\ref{P1slice}E and H) which accounts for the deviation between magnetic energy injection from the bottom surface and energy accumulation between $t_{\rm P1}$ and $t_{\rm P2}$. The third stage began after $t_{\rm P2}$. The magnetic structure is very similar with an eruption: a rising MFR, the reconnected arcades and a PIL-CS can be seen (Figure~\ref{P2slice}A, B and C). Though with these similarities, the distribution of outflow and speed show the difference (Figure~\ref{P2slice}D). The MFR is located at the PIL-CS and two parts of outflow has the same position with the intersection of the slice and MFR (Figure~\ref{P2flow}A). After we move the slice to east side, the two parts become a single one (Figure~\ref{P2flow}A). This suggests the location of reconnection was at the east side of SP. Checking the topology of magnetic structure we found that, due to the complexity of magnetic flux distribution, the initial field has a QSL (referred to as P2-QSL) at east (left) (Figure~\ref{P2flow}B). The existence of the initial P2-QSL made reconnection can take place at the location of P2-QSL before PIL-CS's width reached grid resolution. When enough magnetic field and current were transported to SP east, the second weak energy release process began with outflow speed about $\rm 500~km~s^{-1} $. Since we only rotated SP and other parts stayed nearly potential (Figure~\ref{velocityfield}), the slow outflow will be restricted by the overlying field to be the `horizontal flow' (Figure~\ref{P2slice}D), which has merely velocity in $x$ and $y$ directions. The same as in P1, the mechanism here is different from eruption: the energy conversion is resulted by the slow reconnection near the initial QSL but not fast reconnection in PIL-CS. This is the key reason why P2 is also a weak energy release episode. At the end of P2, the reconnected arcades connected to SP and SN, keeping rotating and preparing for the next eruption. Reconnection in side P1-CS existed all the time and transformed WP-WN to SP-WN (Supplementary Video 1), which let more field lines participate in the formation of PIL-CS next time. These are ready for the first major eruption (E1). During the periods as described above, converging motion towards PIL induced by rotation kept thinning the CS between SP and SN (PIL-CS) with a speed as the same order of rotation, i.e., 2 orders of magnitude lower than local $V_{\rm A}$ ($\rm Alfv \acute{e} n$ speed), thus representing the `quasi-static evolution'. Owing to the very low magnetic diffusion in our code, we can get a very thin CS even with such a low speed. Otherwise, a larger magnetic diffusion will widen the CS against the converge motion, as pointed out in \cite{2021NatAs...5.1126J}. The fourth bunch of field lines (labeled by the red arrow in Figure~\ref{E1slice}A) became SP-WN (Figure~\ref{E1slice}A) and took effect to form a stronger PIL-CS by rotational post-P2 arcades. The trigger PIL-CS grow up from the bottom of simulation box by continuous rotation (Figure~\ref{E1slice}C) until the thickness of PIL-CS reached 2-3 grid resolution. Then numerical diffusion became non-negligible and triggered the fast reconnection and the fourth stage, namely, the major eruption (E1) began. The PIL-CS also extended to WN (Figure~\ref{E1slice}F), which corresponds to the longer flare ribbon on December 13th 2006. Reconnection in PIL-CS formed an MFR during the eruption (Figure~\ref{E1slice}B). The plasma outflow originated from the PIL-CS with a speed reaching up to $\rm 1500~km~s^{-1} $ (Figure~\ref{E1slice}D) and impulsively drove MFR to erupt (as shown in Supplementary Video 2A). Meanwhile, the PIL-CS became longer in the vertical direction, thinner and stronger (Figure~\ref{E1slice}C and Supplementary Video 2B), with more flux involved into reconnection, which provides the energy required for this eruption. With such a high speed, this eruption was strong enough to remove the restriction of overlying field (which also occurred in E2) and no `horizontal flow' can be seen in Figure~\ref{E1slice}D and \ref{E2slice}D. At the end of E1, the reconnected arcades SP-SN restored and kept rotating as before (Figure~\ref{E1slice}A). While SP-WN returned to its origin WP-WN (labeled by the red arrow in Figure~\ref{E1slice}A) and was out of control of rotation. When the arcades SP-SN formed after E1 was sheared enough again, the fifth stage began (after $t_{\rm E2}$). The same as in E1, the PIL-QSL along with PIL-CS grew up again from bottom near PIL. Reconnection in PIL-CS formed an MFR (Figure~\ref{E2slice}B), which was lift up by the outflow (with the speed of $\rm 1100~km~s^{-1}$) initiated from PIL-CS (Figure~\ref{E2slice}D and Supplementary Video 3A). The side P1-CS always existed and transformed the field line connection of WP-WN to that of SP-WN, while the time duration of side reconnection before $t_{\rm E2}$ was not so long as that before $t_{\rm E1}$. As a result, less magnetic flux was involved in the formation of PIL-CS (the field lines labeled by red arrow in Figure~\ref{E2slice}A remain WP-WN), which made the eruption CS weaker than E1 (Figure~\ref{E2slice}C) and shorter in the vertical direction (Supplementary Video 3B). The flare ribbon and PIL-CS (Figure~\ref{E2slice}F) were shorter also in the horizontal direction. This naturally leads to the fact that the magnetic energy release in E2 is less than that in E1. We note that when E2 began, the current sheet of E1 didn't disappear (Figure~\ref{E2slice}C), and in a short interval, the latter eruption (E2) caught up with the former one (E1), making the shock in E2 clearer. After the eruption, the post-flare arcades should restore to the pre-flare configuration again, and if with further rotation of the sunspot, it will lead to the third eruption which is beyond the scope of this event research. We stopped the simulation at $t\sim190$ min, showing the whole process of magnetic evolution of 2 eruptions and the reasons for such changes. \subsection{Comparison with Observations}\label{observations} To show the credibility of our simulation, our results are compared with the observed X-ray and $\rm H\alpha$ features, time scale, rotation angle, magnetic energy release and relative strength of the eruptions. In general, QSLs denotes the location where reconnection is most likely to take place and their footpoints at the bottom surface represent the position of flare ribbons \citep{2002JGRA..107.1164T}. Figure~\ref{QSLflare} shows the comparison of the bottom QSLs for the two simulated eruptions with the flare ribbons as observed for the two flares. During both eruptions, the QSLs are overall consistent in shape and position with observed flare ribbons: QSL-N (corresponding to the negative flare ribbon) was located near the main PIL between two sunspots. QSL-P (corresponding to the positive ribbon) was initially at the east (left) of the positive sunspot and then shrank into a quasi-circular shape to the west (right). Both are comparable with the evolution of flare ribbons, especially the QSL-N, which extended longer in E1 than E2 (Figure~\ref{QSLflare}). This was formed by the side reconnection in P1-CS, which transformed field connection of WP-WN to SP-WN as described in Section \ref{Magnetic Evolution} and as a result the west field lines were involved in the eruption. Sigmoids in soft X-ray images (Figure~\ref{obslice}) before both flares were located at the main PIL and bent towards the positive sunspot by sunspot rotation. These observed features are comparable with the synthetic images of coronal emission from current density \citep{2016ApJ...828...62J} and simulated magnetic structure (Figure~\ref{obslice}). Quantitatively, the simulation can also yield consistence in timing and magnetic energy release as mentioned in Section~\ref{Overall Process}. There were 5 days rotation before E1 and another 40-hour interval between E1 and E2. Both time scales are comparable with actual evolution time: 3 days rotation before E1 and 44 hours time interval between eruptions on December 13th and December 14th. The positive sunspot has rotated over 1.5 turns before the first eruption in our simulation, which is consistent with the total rotational angle of $540^{\circ}$ as derived in \cite{2009SoPh..258..203M}. The magnetic energy release in E1 in our simulation is $\Delta E_{\rm mag}=3.6\times 10^{32}~\rm erg$, which is very close to the values derived with other methods in previous researches of $\Delta E_{\rm mag}\sim 3\times 10^{32}~\rm erg$ \citep{2008ApJ...675.1637S,2010BASI...38..147R}. From observation, the CME on December 13th (1780 km~s$^{-1}$) is faster than on December 14th (1042 km~s$^{-1}$), which is consistent with our simulation. The H$\alpha$ figures also show some observational evidence corresponding to P1 and P2 episodes as labeled by the white arrow in Figure~\ref{ob_p1p2}. The H$\alpha$ brightening has the similar location of P1-CS in Figure~\ref{ob_p1p2} A and P2-QSL in Figure~\ref{ob_p1p2}B and C respectively. This indicates the slow reconnection there before the major eruption as described in Section \ref{Magnetic Evolution}. These results enhance the credibility of our simulation. It should be noted that our simulation simplified the photospheric motions in many aspects, which could affect the results. We did not include the flux emergence process of the rotating sunspot, its shearing motion (from west to east) with respect to the leading sunspot SP, and the colliding motion between the two main sunspots \citep{2008ApJ...687..658W}. For example, if we move the positive sunspot from west to east, the QSL-N in E1 may be longer since when the positive sunspot is located further east, it will connect to WN with a stronger sheared configuration. Larger computational domain is also helpful to obtain a longer QSL: once the MFR reaches the top or lateral boundaries, the closed field lines will be taken as the open field and can't be shown by Q factor calculation. These adjustment has potential to get a higher degree of consistence between simulated QSLs and observed flare ribbons. Also the converge motions (i.e., the collision of the two sunspots) will shorten the evolution time since it will enhance the building up of the PIL-CS and enhance the amount of the magnetic energy release by strengthening the magnetic gradient near PIL \citep{2022A&A...658A.174B}. Though more complex motions and settings may reproduce the flares more realistically, our result shows the key role played by sunspot rotation in leading to the eruptions and can shed light on the onset mechanism of this homologous event. \subsection{Eruption Initiation Mechanism}\label{trigger} There are two types of CS in our simulation: the formation of CSs in P1 and P2 which are responsible for slow reconnection depends on the initial topology while PIL-CS formed by continuous shear near PIL which accounts for the main energy release in E1 and E2. As the sunspot rotation brought field lines together, the magnetic field expanded slowly in P1 and was translated to P2-QSL in P2. This leaded to the squeeze between core field and the surroundings. Then a squeezed QSL formed at top (Figure~\ref{P1slice}G) and the east (left) side of the positive sunspot (Figure~\ref{P2flow}). Slow reconnection here changed the magnetic topology without eruptions. During the same period, converging motion induced by rotational flow made PIL-CS stronger and thinner. When the CS's thickness reached down to the grids width, magnetic gradient near CS will be strong enough to let the diffusion kick in. This mimicked essentially the non-uniform magnetic diffusivity as required in the Pescheck-type reconnection \citep{1994ApJ...436L.197Y}: the resistivity depends sensitively on the local current density, and finally leaded to fast reconnection and eruption. Figure~\ref{MFRv}C shows the temporal evolution of velocity at approximately the middle point of the field lines as shown in Figure~\ref{MFRv}A and B, respectively. These field lines are used to illustrate the dynamics of the field that experienced reconnection and became part of the MFR subsequently in the two eruptions, E1 and E2. Once the reconnection took place, the coronal plasma as frozen with the field lines was accelerated impulsively from a few 10 $\rm km~s^{-1}$ to over 1000 $\rm km~s^{-1}$. This acceleration was accomplished by the strong slingshot effect of the upward concave magnetic field lines as labeled by the white arrow in the middle panel of Figure~\ref{MFRv}A and B. Shortly after the impulsive acceleration, the upward tension force changed sign to a downward one since the magnetic field lines relaxed quickly from upward to downward concave shape. As a result, the field lines experienced deceleration from above 1000 $\rm km~s^{-1}$ to around 600 $\rm km~s^{-1}$, which is consistent with the MFR acceleration process described in \cite{2021NatAs...5.1126J}, suggesting the magnetic reconnection played the key role in initiating the two eruptions. We also estimated the possible role played by torus instability in driving the eruptions of E1 and E2. To do this, we need to calculate the decay index $n$ of the strapping field (often approximated by the potential field model) overlying the erupting MFRs. Since the potential field is not always a good approximation of the strapping field (especially when the overlying field is substantially sheared), we also calculated the decay index of our simulated field for comparison in Figure~\ref{MFRv}D and E. The decay index was derived along the white dashed line in Figure~\ref{MFRv}A and B, which denotes the eruption direction following the method proposed by \cite{2019ApJ...884...73D}. The critical height of simulated field (above which $n>1.5$) is located at 50 Mm in E1 and above 60 Mm in E2. The reconnection point (labeled by the white arrow in the middle panel of Figure~\ref{MFRv}A) is located at the height of 50-60 Mm in E1, which indicates the MFR axis entered the unstable region. Therefore, when the MFRs in E1 was formed, the torus instability was possible to be triggered to drive the eruption in addition to the reconnection. While the MFR axis in E2 is located below 60 Mm and the torus instability had little chance to take effect. This may be an additional reason why E1 is stronger than E2, as the overlying field of E1 decays faster with height than that in E2. It is also worthy noting that, PIL-CSs were formed before the onset of both eruptions, or in other words, they were all formed in a quasi-static way before MFR exists. Then an MFR was formed synchronously with the reconnection and acceleration in PIL-CSs (Supplementary Video 2A and 3A). The acceleration process of the erupting MFR was accomplished under the critical height of torus instability in E2 while above the critical height in E1 as shown in Figure~\ref{MFRv}C. Furthermore, the MFRs experienced a deceleration process after the impulsive acceleration phase, and this deceleration occurs even in the torus unstable region of two eruptions, which clearly indicated the torus instability was not the main factor controlling the dynamics of the MFRs. Therefore, though torus instability had the potential to be triggered and helped the acceleration in E1, magnetic reconnection was the main initiation mechanism of both eruptions. The P1-CS formed at the top of the positive sunspot initially and the four polarities: SP, SN, WP and WN constituted a quadrupolar topology. One may compare this situation with the breakout model: P1-CS corresponds to the breakout CS which opens the overlying field of the eruptive core in the quadrupolar configuration. However, our case is unlike the breakout model in which the reconnection at the breakout CS plays the key role in triggering eruption. In our simulation, the main consequence of slow reconnection in P1-CS is changing the magnetic connectivity, which can make E1 stronger, but it is not required to trigger the eruption. Continuous sunspot rotation can initiate the eruption alone from sheared PIL-CS. This clearly suggests that the mechanism as demonstrated here is a fundamental one, which is consistent with that shown in \cite{2021NatAs...5.1126J} and \cite{2022ApJ...925L...7B}. \section{Conclusions and Discussions}\label{Conclusion} In this paper, using our velocity-driven DARE-MHD model, we have simulated the two eruptions of NOAA AR 10930 on December 13th 2006 and December 14th 2006 continuously. Our simulation started from a potential field obtained by the observed magnetogram and with a simple rotation flow applied to one of the main magnetic polarities at the bottom surface to mimic the sunspot rotation. Owing to the complex distribution of the magnetic flux, there were two slow reconnection processes in P1 and P2 before the first major eruption, which helped building the special magnetic topology. When sunspot rotated over 1.5 turns, the most strong CS formed near the main PIL. Fast reconnection in PIL-CS formed an MFR and the reconnection outflow ejected the coronal plasma violently. The PIL-CS was stretched to be longer, stronger and thinner, and continuous reconnection released the energy required by E1. After this eruption, the post-flare arcades of E1 were further stressed by the rotating sunspot with about another half turn, during which the PIL-CS forms again and then the second major eruption began, which is very similar to the homologous eruption mechanism as shown in \cite{2022ApJ...925L...7B}. The PIL-CS between sunspots was developed from bottom and reconnection sets in to trigger the eruption when the width was comparable with grid resolution like in E1. Though E1 and E2 had the same mechanism, there were less magnetic flux participated in E2, which made the CS and also the eruption in E2 weaker than in E1. Two eruptions have reasonable consistency with observations in relative strength, magnetic energy loss, sunspot rotation angle in the pre-flare duration, observed X-ray and $\rm H\alpha$ features, as well as eruption time interval. Our simulation offers a scenario different from many previous studies. For example, different from \cite{2014Natur.514..465A}, in which they drove an NLFFF extrapolated for about 6 hours before the eruption to erupt by using three different types of photospheric boundary conditions, we started the simulation from the potential field. Moreover, the continuous energy accumulation and release process has been produced in a more self-consistent way, by applying a more realistic condition, i.e., the rotation of the positive sunspot at the bottom boundary. The key magnetic structure in favor of initiating the eruption can form by sunspot rotation directly. In addition, a pre-eruption MFR that emerged through the photosphere as described in \cite{2011ApJ...740...68F} is not necessary in our simulation for triggering the eruption. The sunspot rotation can form an MFR by fast reconnection in the pre-formed PIL-CS during E1 and E2, but not before. Furthermore, most researches of AR 10930 are only focused on the X3.4 flare on December 13th and few of them have studied the relationship of the two flares (the other X1.5 flare on December 14th). Our results show the two events can be triggered in the same way by fast reconnection in the CS formed in a recurrent manner by sunspot rotation as described in Section~\ref{Results}. Since sunspot rotation is a persistent motion for days, our result suggests an efficient way of continuous energy injection, which can reproduce the homologous eruption in AR 10930. The importance of sunspot rotation has also been taken into consideration in some previous researches while the corresponding numerical models were established in different ways. To investigate the effect of sunspot rotation in AR 10898, \cite{2013SoPh..286..453T} rotated a envelop field of a pre-existing MFR. As the the envelop field expanded progressively, the MFR became unstable and triggered to erupt by torus instability. \cite{2021ApJ...922..108J} rotated the reconstructed potential field of AR 12665 along with flux emergence at the PIL. As a consequence, a sigmoidal structure formed with an overlying MFR created and rose to erupt like a CME. Compared with these rotation-driven models, our model is much simpler with only sunspot rotation and a potential field as the initial state, and the MFRs in our simulation could form spontaneously at the onset time of reconnection in the pre-formed PIL-CS and erupted as a CME. Both of the formation of MFR and PIL-CS were not related to the flux emergence. The quasi-static evolution and impulsive eruption process can be obtained solely by the rotation of the initial potential field. Owing to the simple settings of our simulation in many aspects, more realistic consideration should be taken in future improvements of the model for reproducing the eruptions. For example, the smoothing of magnetogram have weakened the magnetic gradient near the main PIL. As a consequence the eruption strength will decrease, because according to \cite{2022A&A...658A.174B}, the eruption strength is highly correlated with the magnetic gradient of the main PIL. A more realistic velocity field at photosphere, including rotational, shearing and converge motions, derived from observation could be applied as the boundary condition to get a more self-consistent and realistic evolution (which has been shown in \citealt{2021FrP.....9..224J} and \citealt{pub.1147837699}). Another key point we have not considered in the current model is the flux emergence process, for which the normal velocity (i.e. velocity in $z$-direction) at photosphere ought to be used to mimic the emergence process of sunspot. To summarize, the whole process from potential field to eruption has been reproduced, showing the full MHD evolution of slow energy accumulation to fast release. Our simulation reveals the importance of sunspot rotation and magnetic reconnection in eruption initiation mechanism. The homologous eruptions as driven by persistent photospheric motion and initiated by the fundamental mechanism \citep{2021NatAs...5.1126J} may be common in solar ARs. Future works will be carried out with the aforementioned improvements for more realistic modeling of solar eruptions that can be potentially applied to the space weather forecast. ~\\ This work is jointly supported by National Natural Science Foundation of China (NSFC 41731067, 42030204, 42174200), the Fundamental Research Funds for the Central Universities (Grant No. HIT.OCEF.2021033), Shenzhen Science and Technology Program (Grant No. RCJC20210609104422048 and JCYJ20190806142609035). The computational work was carried out on TianHe-1(A), National Supercomputer Center in Tianjin, China. \bibliographystyle{aasjournal}

Title: 3D orbital architecture of a dwarf binary system and its planetary companion

Abstract: Because of the diversity of stellar masses and orbital sizes of binary systems, and the complex interaction between star-star, star-planet and planet-planet, it has been difficult to fully characterize the planetary systems associated with binary systems. Here, we report high-precision astrometric observations of the low-mass binary system GJ 896AB, revealing the presence of a Jupiter-like planetary companion (GJ 896Ab). The planetary companion is associated to the main star GJ 896A, with an estimated mass of 2.3 Jupiter masses and an orbit period of 284.4 days. A simultaneous analysis of the relative astrometric data obtained in the optical and infrared with several telescopes, and the absolute astrometric data obtained at radio wavelengths with the Very Long Baseline Array (VLBA), reveals, for the first time, the fully characterized three-dimensional (3D) orbital plane orientation of the binary system and the planetary companion. The planetary and binary orbits are found to be in a retrograde configuration and with a large mutual inclination angle ($\Phi$ = 148 deg) between both orbital planes. Characterizing the 3D orbital architecture of binary systems with planets is important in the context of planet formation, as it could reveal whether the systems were formed by disk fragmentation or turbulence fragmentation, as well as the origin of spin-orbit misalignment. Furthermore, since most stars are in binary or multiple systems, our understanding of systems such as this one will help to further understand the phenomenon of planetary formation in general.

https://export.arxiv.org/pdf/2208.14553

\title{3D orbital architecture of a dwarf binary system and its planetary companion} \correspondingauthor{Salvador Curiel} \email{scuriel@astro.unam.mx} \author[0000-0003-4576-0436]{Salvador Curiel$^{*}$} \affil{Instituto de Astronom{\'\i}a, Universidad Nacional Aut\'onoma de M\'exico (UNAM), Apdo Postal 70-264, Ciudad de M\'exico, M\'exico. scuriel@astro.unam.mx} \author[0000-0002-2863-676X]{Gisela N. Ortiz-Le\'on$^{1}$} \affil{Max Planck Institut f\"ur Radioastronomie, Auf dem H\"ugel 69, D-53121 Bonn, Germany} \author[0000-0002-2564-3104]{Amy J. Mioduszewski} \affiliation{National Radio Astronomy Observatory, P.O. Box 0, Socorro, NM 87801, USA } \author[0000-0002-9723-0421]{Joel Sanchez-Bermudez$^{1}$} \keywords{Exoplanets (498); Exoplanet systems (484); Exoplanet dynamics (490); Extrasolar gaseous giant planets (509); Astrometric exoplanet detection (2130); Low mass binary systems} \section{Introduction} \label{sec:intro} Numerical simulations suggest two main channels for the formation of multiple stellar systems: disk fragmentation \citep[][]{adams89}, which produces secondaries through gravitational instability within a massive accretion disk, and turbulent fragmentation \citep[e.g.,][]{goodwin04, fisher04}, where turbulence in the original molecular core leads to multiple density enhancements, which independently collapse. Binary systems formed by turbulence fragmentation are expected to have initial separations larger than 500 astronomical units (au), which corresponds to the rough boundary between the disk size and the molecular core scale \citep[e.g.,][]{offner10}, while disk fragmentation suggest the formation of compact ($<$ 500 au) binaries \citep[e.g.,][]{kratter10}. However, dynamical evolution may quickly modify the separation, making that initially wide binaries migrate to separations $<$200 au in time spans of $\sim$0.1 Myr \citep[][]{offner10, offner16}. If substantial orbital evolution occurs during the main accretion phase, which lasts $\sim$0.5 Myr \citep[][]{dunham14}, then binary systems will end up much closer binary systems. Theoretical works have addressed the formation of planets around single stars of different masses. The core-accretion theory predicts that the formation of giant-mass planets scales with the mass of the central star; thus, it is expected that very few Jovian-mass planets are formed around low-mass stars \citep[e.g.,][]{laughlin04,kennedy08,mercer20}. The core-accretion theory indicates that these planets would be formed in orbits far from the star, at several au. On the other hand, it is expected that disk fragmentation may also be able to form giant-mass planets around low-mass stars \citep[e.g.,][]{boss06}. In this case, the orbit of the planet is expected to be relatively closer to the star, from a few to several au. Only a very small fraction of the detected extrasolar planets (less than 4\%) are known to be associated to stars in binary systems. This is probably, at least in part, due to strong observational biases. For instance, the radial velocity (RV) technique is in general limited to binary systems with separations larger than 2$''$, and the transit technique is limited to high inclination angles of the orbital plane of the planets and the binary systems \citep[e.g.,][and references therein]{marzari19}. In addition, the presence of a stellar companion has adverse effects on planet formation. It is possible that the stellar companion strongly influence both the formation of planets and their subsequent dynamical evolution. For example, the presence of a close stellar companion can tidally truncate the protoplanetary disk \citep[e.g.,][]{savonije94, kraus12}. Observational surveys have shown that, even when present, protoplanetary disks are on average less massive in tight binaries \citep[e.g.,][]{harris12}. In addition, the evolution of truncated disks under the action of viscous forces is faster and should produce more short-lived disks, with less time available for planet formation \citep[e.g.,][]{muller12}. This could particularly affect the formation of giant gaseous planets, which need to accrete vast amount of gas before the primordial disk disperses. There is observational evidence that suggests that the typical life time of protoplanetary disks in close binaries separated by $\leq$40 au is less than 1 Myr, as compared with the typical disk life time of 5$-$10 Myrs in single stars \citep[e.g.,][]{marzari19}. Furthermore, in the case of close binaries, there might not be enough mass left in the truncated disks to form jovian planets. Only a small fraction (about 15\%) of the planets associated with binaries have been found associated to binary systems with separations $\leq$40 au \citep[e.g.,][]{marzari19, fontanive21}, and according to the Catalogue of Exoplanets in Binary Systems \citep[][]{schwarz16} only 7 planets are associated to M dwarf binary systems. The formation of planets in binary systems is not well understood. Current planetary formation models take into account only the formation of planets around single stars. To our knowledge, there are no theoretical models that take into account, for instance, the simultaneous formation of planetary and sub-stellar or stellar companions in a single protoplanetary disk, or the formation of planetary systems during the formation of a binary system through turbulent fragmentation. A variety of mechanisms has been proposed to explain spin-orbit misalignment observed in several planetary systems. Dynamical interactions between stars and/or planets \citep[][]{nagasawa08} help explaining small and large differences in the inclination angle of close-in giant-planets. The chaotic star formation environment during the accretion phase \citep[][]{bate10} and perturbations from a stellar companion \citep[][]{thies11, batygin13, lai14} also help explaining large misalignment between the stellar spin and the planetary orbit. Thus, studying the spins of individual members in binary systems (including sub-stellar and planetary companions) could reveal whether they were formed by disk fragmentation or turbulence fragmentation, as well as the origin of the spin-orbit misalignment \citep[e.g.,][]{offner16}. To accomplish this, the three-dimensional (3D) orbital plane orientation of both, the binary system and the planetary companions would need to be known. However, in most cases, not all angles of the orbital plane of the binary system and their planetary companions are known. In all known binary systems with planetary companions, the position angle of the line of nodes ($\Omega$) is unknown, and in several cases even the inclination angle of the orbits is also unknown. Binaries formed within the same accretion disk are likely to have common angular momenta and, therefore, aligned stellar spins, whereas binaries formed via turbulent fragmentation are likely to possess independent angular momentum vectors and, thus, have randomly oriented spins. It is expected that the planets have orbital spins similar to the stellar spin of their host star. However, in the case of binary systems, the orientation and eccentricity of the planetary orbits are expected to be affected by the presence of the stellar companion. Astrometry is the only technique capable of directly giving all three angles (longitude of the periastron $\omega$, position angle of the line of nodes $\Omega$, and the inclination angle $i$) of the orbital planes. In particular, radio interferometric astrometry, with milli-arc-second (mas) angular resolution and very high astrometric precision (usually better than 60 micro-arc-second ($\mu$as)), can be used to search for planetary companions associated to binary systems with wide orbits, as well as close binary systems with separations $\leq$50 au. \subsection{GJ~896 binary system} \label{sec:bsys} GJ~896AB (a.k.a. EQ Peg, BD+19~5116, J23318+199, HIP 116132) is a nearby low-mass M dwarf binary system with an estimated age of 950 Myr \citep[][]{parsamyan95}, at a distance of 6.25 pc, and a separation of 5.4 arcsec with a PA of 78$^\circ$ \citep[e.g.,][]{heintz84, liefke08, bower11, pearce20}. However, some observations also suggest that the age of this binary system is $\lesssim$ 100 Myr \citep[][]{riedel11, zuckerman13}. If this latter age is confirmed, these stars would be the closest pre-main-sequence stars known. This binary system is part of a quadruple system, where the other two stellar companions are further away from the main binary. Recent observations suggest that both stars have companions, but theirs orbits have not been characterized \citep[e.g.,][]{delfosse99, winters21}. Both stars have been observed to have radio outbursts \citep[e.g.,][]{pallavicini85, jackson89, benz95, gagne98, crosley18a, crosley18b, villadsen19, davis20} and are flaring X-ray emitters \citep[e.g.,][]{robrade04, liefke08}. GJ~896A has been detected at several epochs with the VLBA showing compact and variable flux emission \citep[][]{bower09, bower11}. An early determination of the orbital motion of the binary GJ~896AB suggested an orbital period of about 359 yrs and a semi-major axis of 6.87 arcsec \citep[e.g.,][]{heintz84}. More recent, and more accurate, relative astrometric fits indicate that the orbital parameters of this binary system are $P$ $\sim$234 yr, $a$ $\sim$ 5.3 arcsec, $i$ $\sim$ 126$^\circ$, $\Omega$ $\sim$ 77$^\circ$, $e$ $\sim$ 0.11 and $\omega$ $\sim$ 97$^\circ$ \citep[][]{mason01}. However, since the time baseline used for these orbital determinations is $\lesssim$34\% the estimated period, the orbital parameters were not well constrained. The more massive star, GJ~896A, is an M3.5 star with an estimated mass of 0.39 M$_\odot$, a radius of 0.35 R$_\odot$, a rotation period of 1.061 days, and a rotational inclination angle of 60$^{\circ}$, while the less massive star, GJ~896B, is an M4.5 star with an estimated mass of 0.25 M$_\odot$, a radius of 0.25 R$_\odot$, a rotation period of 0.404 days, and a rotational inclination angle of about 60$^{\circ}$$\pm$20$^{\circ}$ \citep[e.g.,][]{delfosse99, morin08, davison15, pearce20}. Here we present the discovery of a Jovian-mass planetary companion to the young close-by (6.25 pc away from the Sun) M3.5 dwarf GJ~896A, which is the more massive star in the low-mass M dwarf binary system GJ~896AB with a separation of about 31.6 au. Section~\ref{sec:obs} describes the observations and data reduction. In Section~\ref{sec:procid} we present the methodology to fit the astrometric data. Section~\ref{sec:results} presents the results, including the fitted 3D orbital architecture of this system. The results are discussed in Section~\ref{sec:discusion} and our conclusions are given in Section~\ref{sec:conclusions}. In Appendix~\ref{sec:flares} we discuss the possible contributions of the variability of the main star to the expected $``$jitter$"$ of the star, and in Appendix~\ref{sec:mcmc} we present the posterior sampling of the combined astrometric fit solution using an MCMC sampler. \section{Observations and Data Reduction} \label{sec:obs} We analyzed archival and new Very Long Baseline Array (VLBA) observations taken at 8.4~GHz toward the M dwarf binary system GJ~896AB. Observations of GJ~896A were carried out in fourteen epochs between March, 2006 and November, 2011 as part of the Radio Interferometric Planet (RIPL) survey and its precursor (\citealt{bower09,bower11}; program IDs: BB222 and BB240). New observations of the two stars GJ~896A and GJ~896B were obtained as part of our own program BC264 (PI: S.\ Curiel) in three epochs between August, 2020 and October, 2020. The RIPL observations recorded four 16-MHz frequency bands in dual-polarization mode. For the most recent observations, we recorded four 128-MHz frequency bands, also in dual polarization mode, using the new 4~Gbps recording rate of the VLBA. The observing sessions consisted of switching scans between the target and the phase reference calibrator, J2328+1929, spending about 1~min on the calibrator and 3~min on the target. Scans on secondary calibrators (J2334+2010, J2334+1843 and J2328+1956) were obtained every $\approx$30--60 min. In addition, fringe calibrators were observed occasionally during the sessions. For program BC264, additional 30-min blocks of calibrators, the so-called geodetic-like blocks, distributed over a wide range of elevations were included at the beginning and end of the observing run. The RIPL observations included the 100-m Green Bank Telescope added to the VLBA array. In project BC264 the typical on source time is of 2 hours, spread over a full track of 3 hours, plus 1 hour for the two geodetic-like blocks. On the other hand, RIPL alternated between two targets in a 8 hours track, and no geodetic-like blocks were observed. To determine the orbital motion of the binary system, we used archival relative astrometric measurements of GJ~896AB taken from the Washington Double Star Catalog \citep{mason01}, maintained by the US Naval Observatory. This data set includes a total of 73 measurements starting in the year 1941 until 2017. We discarded the data points from 1950 and 1952.66 since they largely deviate from the rest of the observations due to possible systematics. The VLBA data were reduced with the Astronomical Imaging System \citep[AIPS;][]{greisen03}, following standard procedures for phase-referencing observations \citep{torres07,ortizleon17}. First, corrections for the ionosphere dispersive delays were applied. We then corrected for post-correlation updates of the Earth orientation parameters. Corrections for the digital sampling effects of the correlator were also applied. The instrumental single-band delays caused by the VLBA electronics, as well as the bandpass shape corrections were determined from a single scan on a strong fringe calibrator and then applied to the data. Amplitude calibration was performed by using the gain curves and system temperature tables to derive the system equivalent flux density of each antenna. We then applied corrections to the phases for antenna parallactic angle effects. Multi-band delay solutions were obtained from the geodetic-like blocks, which were then applied to the data to correct for tropospheric and clock errors. The final step consisted of removing global frequency- and time-dependent residual phase errors obtained by fringe-fitting the phase calibrator data, assuming a point source model. In order to take into account the non-point-like structure of the calibrator, this final step was repeated using a self-calibrated image of the calibrator as a source model. Finally, the calibration tables were applied to the data and images of the stars were produced using the CLEAN algorithm. We used a pixel size of 50~$\mu$as and pure natural weighting. The synthesized beam of these images are, on average, $2.7\times 1.1$~mas, and the achieved rms noise levels are in the range between 11 and 130~$\mu$Jy~beam$^{-1}$ (Table \ref{tab_1}). The large range of sensitivities is expected because of the wide range in observing strategies. The assumed position for the primary phase calibrator J2328+1929 during correlation changed between epochs due to multiple updates of the VLBA calibrator position catalogs. Thus, before deriving any calibration, the position of J2328+1929 was corrected to the value assumed in the observations of 2020, R.A.$=$23:28:24.874773 and Dec.$=$+19:29:58.03010. The assumed positions for the observations performed between 2006 and 2011 were taken from the correlator files available in the VLBA file server at {http://www.vlba.nrao.edu/astro/VOBS/astronomy/}. GJ~896A was detected in thirteen epochs of RIPL and in the three epochs of program BC264. GJ~896B was only detected in two epochs of BC264. To obtain the positions of the centroid in the images of GJ~896A and B, we used the task {\tt MAXFIT} within AIPS, which gives the position of the pixel with the peak flux density. The position error is given by the astrometric uncertainty, $\theta_{\rm res}/(2\times {\rm S}/{\rm N})$, where $\theta_{\rm res}$ is the full width at half maximum (FWHM) size of the synthesized beam, and ${\rm S}/{\rm N}$ the signal-to-noise ratio of the source \citep{thompson17}. Furthermore, we quadratically added half of the pixel size to the position error. In order to investigate the magnitude of systematic errors in our data, we obtain the positions of the secondary calibrator, J2334+2010, in all observed epochs. The rms variation of the secondary calibrator position is (0.14, 0.14)~mas. The angular separation of J2334+2010 relative to the main calibrator is $1\rlap.{^{\rm o}}6$, while the target to main calibrator separation is $0\rlap.{^{\rm o}}97$. The main calibrator, target and secondary calibrator are located in a nearly linear arrangement (see Fig.\ 1 in \citealt{bower11}). Since systematic errors in VLBI phase-referenced observations scale linearly with the source-calibrator separation \citep{pradel06,reid14}, we scale the derived rms value for J2334+2010 with the ratio of the separations from the target and J2334+2010 to the main calibrator. This yields a systematic error of (0.09, 0.08)~mas, which was added in quadrature to the position errors in each coordinate. Table \ref{tab_1} summarizes the observed epochs, the positions, the associated uncertainties, and the integrated flux densities of GJ~896A and B. Figure~\ref{fig_1} shows the intensity maps of both stars for one of the two epochs when both stars were detected with the VLBA. The time of the observations (in Julian day) included in Table \ref{tab_1} corresponds to the average time of the time span of each observed epoch (typically of about 4 hours, including the geodetic-like blocks in BC264, and 8 hours in RIPL). The integrated flux densities were obtained by fitting the source brightness distribution with a Gaussian model. \section{Fitting of the Astrometric Data.} \label{sec:procid} We followed the same fitting procedure presented by \citet[][]{curiel19, curiel20}. We used two astrometric fitting methods: non-linear Least-squares algorithm \citep[][]{curiel19, curiel20} and the asexual genetic algorithm AGA \citep[][]{canto09, curiel11, curiel19, curiel20}. Both fitting codes include iterative procedures that search for the best fitted solution in a wide range of posible values in the multi-dimensional space of parameters. These iterative procedures help the fitting codes not be trapped in a local minimum, and to find the global minimum. In addition, we fit the data using different initial conditions to confirm that the best fitted solution corresponds to the global minimum solution. These algorithms can be used to fit absolute astrometric data (e.g., planetary systems), relative astrometric data (e.g., binary systems), and combined (absolute plus relative) astrometric data (e.g., a planetary companion associated to a star in a binary system). To fit the astrometric data, we model the barycentric two-dimensional position of the source as a function of time ($\alpha(t)$, $\delta(t)$), accounting for the (secular) effects of proper motions ($\mu_\alpha$, $\mu_\delta$), accelerations terms ($a_\alpha$, $a_\delta$) due to a possible undetected companion with very long orbital period, the (periodic) effect of the parallax $\Pi$, and the (Keplerian) gravitational perturbation induced on the host star by one or more companions, such as low-mass stars, substellar companions, or planets (mutual interactions between companions are not taken into account). We searched for the best possible model (a.k.a, the closest fit) for a discrete set of observed data points ($\alpha(i)$, $\delta(i)$). The fitted function has several adjustable parameters, whose values are obtained by minimizing a $''$merit function$''$, which measures the agreement between the observed data and the model function. We minimize the $\chi^{2}$ function to obtain the maximum-likelihood estimate of the model parameters that are being fitted \citep[e.g.,][]{curiel19, curiel20}. We also use the recursive least-squares periodogram method with a circular orbit (RLSCP) presented by \citet[][]{curiel19, curiel20} to search for astrometric signals that indicate the presence of possible companions. We start the search by comparing the least-squares fit of the basic model (proper motions and parallax only) and a one-companion model (proper motions, parallax, and Keplerian orbit of a single companion). If a signal is found, and it is confirmed by the two astrometric fitting models, we remove this signal by comparing the least-squares fits of a one- and a two-companion model (proper motions, parallax, and Keplerian orbits of one and two companions, respectively), and so on. The RLSCP periodogram follow a Fisher F$-$distribution with $k_{p} – k_{0}$ and $N_{obs} – k_{p}$ degrees of freedom, where $k_{p}$ and $k_{0}$ are the number of parameters that are fitted when the model includes a planetary companion and without a planetary companion, respectively. $N_{obs}$ is the number of observed epochs. Thus, an astrometric signal found in the periodogram indicates a high probability that a planetary companion is orbiting the star. In addition, we also use the open-source package {\tt lmfit} \citep[][]{newville20}, which uses a non-linear least-squares minimization algorithm to search for the best fit of the observed data. This python package is based on the {\tt scipy.optimize} library \citep[][]{newville20}, and includes several classes of methods for curve fitting, including Levenberg-Marquardt minimization and emcee \citep[][]{foremanmackey13}. In addition, {\tt lmfit} includes methods to calculate confidence intervals for exploring minimization problems where the approximation of estimating parameter uncertainties from the covariance matrix is questionable. \section{Results} \label{sec:results} By combining our new data with the VLBA data from the archive \citep[][]{bower09,bower11}, we were able to search for sub-stellar companions associated to the main star GJ~896A in this binary system. These multi-epoch astrometric observations covered about 5317 days (16.56 yr), with an observational cadence that varies during the time observed. The observations were not spread regularly over the 16.56 years, the gaps between observations ranged from weekly to monthly to 7 years (see Sec.~\ref{sec:obs}). There are a total of 16 epochs in the analysis presented here. The time span and cadence of the observations are more than adequate to fit the proper motions and the parallax of this binary system, and to search for possible companions with orbital periods between several days and a few years. In the following sections we present the resulting astrometric parameters for the fit with no companions (Sec.~\ref{sec:ssa}), a fit including a single (new planetary) companion (Sec.~\ref{sec:sca}), the fit of the relative astrometry of the binary (Sec.~\ref{sec:bsys}), a fit combining the absolute and relative astrometry (Sec.~\ref{sec:caf}), and finally a fit combining the absolute and relative astrometry plus a planetary companion (Sec.~\ref{sec:fcaf}). \subsection{Single-source Astrometry } \label{sec:ssa} First, both the least-squares and the AGA algorithms were used to fit the proper motions and the parallax of GJ~896A, without taking into account any possible companion. The results of the absolute astrometric fit are shown in column 1 of Table~\ref{tab_2} and Figure~\ref{fig_2}. We find that the residuals are quite large, and present an extended temporal trend that indicates the presence of a companion with an orbital period larger than the time span of the observations. The recursive least-squares periodogram with a circular orbit (RLSCP) of the astrometric data (see Figure~\ref{fig_3}, top panel) shows a very strong signal that extent beyond the extend of the plot. A blind search for the orbital period of the signal indicates that the orbital period of this astrometric signal could not be constrained. This suggests that the signal may be due to a companion with an orbital period much larger than the time span of the observations ($\gg$ 16 yr). As we will see below (see Secs.~\ref{sec:caf} and \ref{sec:fcaf}), this long term signal is due to the gravitational interaction of GJ~896A with its very low-mass stellar companion GJ~896B. Since the orbital period of the binary system ($>$~200 yrs) is much larger than the time span of the VLBA observations ($\sim$16 yrs), the variation trend in the residuals can be taken into account by using accelerations terms in the astrometric fit. The astrometric data was then fitted with the least-squares and AGA algorithms, including acceleration terms, which take into account the astrometric signature due to the low-mass stellar companion (single-source solution). The results of this single-source solution are shown in column 2 of Table~\ref{tab_2} and Figures~\ref{fig_4}a. The astrometric solution shows that GJ~896A has proper motions of $\mu_\alpha$ = 574.171 $\pm$ 0.014 mas yr$^{-1}$ and $\mu_\delta$ = $-$60.333 $\pm$ 0.014 mas yr$^{-1}$. The fitted parallax is $\Pi$ = 160.027 $\pm$ 0.094 mas, which corresponds to a distance of 6.2489 $\pm$ 0.0037 pc. These values are similar to those obtained from the astrometric fit where no acceleration terms were taken into account. The fitted acceleration terms are relatively large ($a_{\alpha}$ = 0.8893 $\pm$ 0.0034 mas yr$^{-2}$ and $a_{\delta}$ = 0.1402 $\pm$ 0.0035 mas yr$^{-2}$). We obtain now a better fit to the astrometric data, with smaller residuals (rms $\sim$ 0.24 mas in both R.A. and Dec.; see column 2 of Table~\ref{tab_2}). The total residuals (rms $\sim$ 0.34) are a factor of 25 smaller than those obtained with the astrometric fit without acceleration terms (rms $\sim$ 8.64). However, the residuals are large compared to the mean noise present in the data (rms $\sim$0.11 and 0.10 mas for R.A. and Dec., respectively) and the astrometric precision expected with the VLBA ($<$70 $\mu$as). Below we investigate the possibility that these residuals are due to a planetary companion orbiting around GJ~896A. \subsection{Single-companion Astrometry } \label{sec:sca} We carried out a RLSCP of the astrometric data, including accelerations terms. The RLSCP shows now at least two significant peaks (see Figure~\ref{fig_3}, middle panel), the most prominent one with a period of about 280 days, and the weaker signal with a period of about 237 days. The stronger signal appears to be well constrained, with a false alarm probability (FAP) of about 0.90\%. This low FAP suggests that the main signal in the periodogram is real and due to the presence of a companion in a compact orbit. We then used both the least-squares and AGA algorithms to fit the astrometric observations of GJ~896A, now including acceleration terms and a possible single companion in a compact orbit (single-companion solution). The solution of the fit is shown in column 3 of Table~\ref{tab_2} and Figures~\ref{fig_4}b and \ref{fig_5}. The fit of the astrometric data clearly improves when including a companion, as seen by the $\chi^{2}_{red}$. The $\chi^{2}_{red}$ is now a factor 2.4 smaller than that obtained with the single-source solution. Table~\ref{tab_2} and Figure~\ref{fig_4} show that the residuals of the single-companion solution (rms $\sim$0.14) are a factor of 1.7 smaller than in the case of the single-source solution (rms $\sim$0.24). Column 3 in Table~\ref{tab_2} summarizes the best fit of the VLBA astrometric data, including this new companion in a compact orbit. The single-companion solution shows that GJ~896A has proper motions of $\mu_\alpha$ = 574.142 $\pm$ 0.019 mas yr$^{-1}$ and $\mu_\delta$ = $-$60.354 $\pm$ 0.020 mas yr$^{-1}$. The fitted parallax is $\Pi$ = 159.83 $\pm$ 0.13 mas, which corresponds to a distance of 6.2565 $\pm$ 0.0051 pc. The fitted acceleration terms are $a_{\alpha}$ = 0.8871 $\pm$ 0.0047 mas yr$^{-2}$ and $a_{\delta}$ = 0.1400 $\pm$ 0.0049 mas yr$^{-2}$. These values are similar to those obtained from the single-source solution plus acceleration. Figure~\ref{fig_5} shows the orbital motion of the star GJ~896A due to the gravitational pull of the companion. The orbit of the main star around the barycenter has an orbital period $P$ = 281.56 $\pm$ 1.67 days, an eccentricity $e = 0.30 \pm 0.11$, a longitude of the periastron $\omega = 344.0^\circ \pm 13.1^\circ$, a position angle of the line of nodes $\Omega = 47.7^\circ \pm 12.1^\circ$, a semi-major axis $a_{A} = 0.52 \pm 0.11$ mas, and an inclination angle $i = 66.0^\circ \pm 15.0^\circ$, which indicates that the orbit is prograde ($i < 90^\circ$). The orbit of the star around the barycenter due to the companion is well constrained, which is consistent with the narrow signal observed in the periodogram (see Figure~\ref{fig_3}, middle panel). With this astrometric fit alone we can not estimate the mass of the companion. However, we used the estimated mass of the planetary system GJ~896A that we obtain below, using the full combined astrometric fit of this binary system (see Sec.~\ref{sec:fcaf}). Column 3 in Table~\ref{tab_2} summarizes the parameters of the new companion, here after GJ~896A$b$. We find that its mass is 0.00223 $\pm$ 0.00047 M$_\odot$, which is consistent with a planetary companion with a mass of 2.35 $\pm$ 0.49 M$_{J}$. The orbit of the planetary companion around the barycenter has a semi-major axis $a_{b}$ = 0.6352 $\pm$ 0.0018 au (or 101.53 $\pm$ 0.42 mas). There is an ambiguity with the position angle of the line of nodes between $\Omega$ and $\Omega + 180^{\circ}$. This ambiguity can be solved by radial velocity (RV) observations. However, since this planetary companion has not been detected with RV observations, we will leave the fitted $\Omega$ angle in Table~\ref{tab_2}. We further discuss this ambiguity below (see section~\ref{sec:mutinc}). The orbit of the planetary companion is well constrained. However, the rms of the residuals ($\sim$0.14 mas) are still large compared to the mean noise present in the data and the astrometric precision expected with the VLBA. To investigate the possibility of a second possible planetary companion, we obtained a new RLSCP, including acceleration terms and a signal with a period of 281.6 days. The new RLSCP does not show a significant signal (see Figures~\ref{fig_3}, bottom panel). We also carried out a blind search for a second planetary companion using the least-squares and AGA algorithms. We did not find a possible candidate. We will need further observations to investigate whether or not this source has more planetary companions. \subsection{Binary-system Astrometry: Relative fit} \label{sec:bsys} GJ~896AB is a visual binary that contains two M dwarf stars that has been observed in the optical and near infrared for more than 80 yr. The Washington Double Star \citep[WDS,][]{mason01} Catalog provides separation and position angle measurements of this binary system spanning these decades and go back as far as 1941. Since the stellar companion GJ~896B was detected in two of our recent VLBA observations of this binary system, we include the separation and the position angle of these two epochs in the relative astrometric fit. We performed an astrometric fit to the relative astrometric measurements with both, the least-squares and the AGA algorithms (binary-system solution). Since the WDS catalog does not provide estimated error bars for most of the observed epochs, we applied a null weight to the astrometric fit, which translate into a uniform weight for all the observed epochs. The results of the binary-system solution are presented in column 1 of Table~\ref{tab_3} and Figure~\ref{fig_6}. We find that the relative astrometric fit of this binary system is well constrained when assuming a circular orbit. When the eccentricity of the binary system is taken into account, the solution does not converge to a stable solution. The orbit of the low-mass stellar companion GJ~896B around the main star GJ~896A has an orbital period $P_{AB}$ = 96606.13 $\pm$ 2.24 days (264.5 yr), a position angle of the line of nodes $\Omega_{AB} = 79.7083^\circ \pm 0.0054^\circ$, a semi-major axis $a_{AB} = 5378.79 \pm 0.51$ mas, and an inclination angle $i_{AB} = 130.6592^\circ \pm 0.0087^\circ$, which indicates that the orbit of the stellar companion is retrograde ($i > 90^\circ$). Using the distance to the source that we obtain from the full combined astrometric fit (see Sec.~\ref{sec:fcaf}), we obtain that the mass and the semi-major axis of this binary system are $m_{AB}$ = 0.544304 M$_\odot$ and a$_{AB}$ = 33.64265 au, respectively. As it was mentioned before, there is an ambiguity with the position angle of the line of nodes between $\Omega$ and $\Omega + 180^{\circ}$. In this case, the observed RV of the two M dwarfs can be used to find the correct angle. From the GAIA catalog \citep[][]{lindegren18}, the observed mean radial velocity of GJ~896A is $-0.02 \pm 0.31$ km s$^{-1}$. The GAIA catalog does not contain the radial velocity of GJ~896B. However, recent RV observations of this source indicate that its RV is about 3.34 km s$^{-1}$ \citep[][]{morin08}. This means that GJ~896B is red shifted with respect to GJ~896A. Thus, the position angle of the line of nodes that is consistent with these RVs is $\Omega_{AB} = 259.7083^\circ \pm 0.0054^\circ$. This correction to the position angle of the line of nodes is applied in all the astrometric fits presented in Table~\ref{tab_3}. \subsection{Binary-system Astrometry: Combined Astrometric fit} \label{sec:caf} The optical/infrared relative astrometry of the binary system GJ~896AB \citep[][]{mason01}, obtained in a time interval of about 80 yrs, was also combined with the absolute radio astrometry to simultaneously fit: the orbital motion of the two stars in GJ~896AB; the orbital motion of the planetary companion around GJ~896A (see Sec.~\ref{sec:fcaf}); and the parallax and proper motion of the entire system. However, since most of the relative astrometric observations do not have estimated errors, for the relative astrometry part we carried out a uniform weighted fit (without errors) of the observed data (73 epochs, including the two relative positions of GJ~896B obtained with the VLBA), while for the absolute astrometry of the VLBA observations of the main star GJ~896A (16 epochs) and the secondary star GJ~896B (2 epochs), we applied a weighted fit using the estimated error of the observed position at each epoch. The results presented here and in Sec.~\ref{sec:fcaf} were obtained by fitting simultaneously the absolute astrometric observations of both stars, and the relative astrometry of this binary system. The results are limited by: a) the lack of error bars of the relative astrometric data, b) the relative astrometric data covers about 35$\%$ of the orbital period of the binary system ($\sim$229.06 yrs; see Sec.~\ref{sec:fcaf}), and c) the absolute astrometry covers only a time span of about 16.56 yrs. Thus, all errors reported below are statistical only. Fitting the proper motions and parallax of a binary system is complex due to the orbital motions of each component around the center of mass, especially since both stars have different masses ($m_{B}/m_{A} <$ 1). The proper motion and parallax of a binary system can be obtained with high precision when the orbital period of the system is small compared with the time span of the astrometric observations. In contrast, when the time span of the astrometric observations is small compared with the orbital period of the binary system, it is difficult to separate the orbital motion of each component and the proper motion of the system since both spatial movements are blended. The best way to separate both movements is to simultaneously fit the proper motions, the parallax, and the orbital motion of the binary system. Our combined astrometric fit includes all these components, and thus gives an accurate estimate of the proper motions and the parallax of this binary system. We carried out a combined fit of the relative and absolute astrometric data and found that the astrometric fit improves substantially. The solution of the combined astrometric fit is similar to that obtained from the relative astrometric fit. However, using the combined astrometric fit, we are able to obtain the parallax and the proper motions of the binary system, as well as improved orbits of the two stars around the center of mass and the masses of the system and the individual stars. The results of this combined fit are shown in Figures~\ref{fig_7}a and \ref{fig_8}a, and summarized in column 2 of Table~\ref{tab_3}. Although, Figure~\ref{fig_8}a shows the full combined astrometric solution (including a planetary companion; see Sec~\ref{sec:fcaf}), the solution for the relative astrometric part of the binary system is basically the same as that obtained from the combined astrometric solution (see columns 2 and 3 of Table~\ref{tab_3}). Thus, we present only one figure showing both fits. In contrast to the case of the relative astrometric fit (see~Sec.~\ref{sec:bsys}), we find here that the combined astrometric fit is well constrained when considering that the orbit of the binary system could be eccentric (e $\ge$ 0). The proper motions of the binary system are $\mu_\alpha$ = 571.515 $\pm$ 0.019 mas yr$^{-1}$ and $\mu_\delta$ = $-$37.750 $\pm$ 0.019 mas yr$^{-1}$. The parallax of the binary system is $\Pi$ = 159.98 $\pm$ 0.14 mas, which corresponds to a distance of 6.2506 $\pm$ 0.0055 pc. The orbit of the binary system has an orbital period $P_{AB}$ = 83665.80 $\pm$ 1.64 days (229.06 yr), a position angle of the line of nodes $\Omega_{AB} = 255.0891^\circ \pm 0.0028^\circ$, a semi-major axis of the binary system $a_{AB} = 5057.96 \pm 0.36$ mas, a semi-major axis of the main source $a_{AB}(A) = 1381.015 \pm 0.069$ mas, and an inclination angle $i_{AB} = 130.0664^\circ \pm 0.0085^\circ$. The estimated mases of the binary system and the two components are $m_{AB}$ = 0.602253 $\pm$ 0.000020 M$_\odot$, $m_{AB}(A)$ = 0.43782 $\pm$ 0.00054 M$_\odot$, and $m_{AB}(B)$ = 0.16444 $\pm$ 0.00020 M$_\odot$, respectively. The semi-major axis of the binary system and the two stellar components are $a_{AB}$ = 31.615 $\pm$ 0.028, $a_{AB}(A)$ = 8.6322 $\pm$ 0.0075, and $a_{AB}(B)$ = 22.983 $\pm$ 0.020 au, respectively. Column 2 of Table~\ref{tab_3} indicates that there is a relatively small improvement of about 17\% in the $\chi^{2}_{red}$. This is because the residuals of the relative data of the binary system are much larger that those of the absolute astrometry of the main star, and this dominates the residuals. However, Table~\ref{tab_3} and Figure~\ref{fig_7} show that the residuals of the relative astrometric data of the binary system from the combined astrometric solution (rms $\sim$116.38 mas) are a factor of 1.3 smaller than in the case of the relative astrometry solution (rms $\sim$152.0 mas). Thus, the combined astrometric solution is an improvement to the solution obtained from the pure relative astrometric fit. In addition, the residuals of the absolute astrometric data of the main source GJ~896A (rms $\sim$ 0.27 and 0.47 mas for R.A. and Dec.) are large compared to the mean noise present in the data. Comparing the residuals that were obtained from the single-source fit (see~Sec.~\ref{sec:ssa}) of the astrometric observations of the main star GJ~896A, including acceleration terms, and the residuals of the combined astrometric fit (see column 2 of Table~\ref{tab_2} and column 2 of Table~\ref{tab_3}, and Figures~\ref{fig_4} and \ref{fig_7}), we find that the rms of the residuals of the absolute astrometry are similar. In addition, we find that the temporal distribution of the residuals are also very similar. However, the residuals of the last three epochs from the combined astrometric fit are considerable larger that those obtained from the single-source plus acceleration fit. We do not find an explanation for this discrepancy. But, we notice that two out of these three epochs are the ones where the secondary star was detected, and the coordinates of this star were used in the fit of the relative astrometry of the binary system and in the absolute astrometry fit of the secondary star GJ~896B. Since the secondary star seems to be a close binary (see Secs.~\ref{sec:intro} and ~\ref{sec:pmot}) and it was only detected in two epochs, this probably affects the astrometric fit (see Figure~\ref{fig_7}). Further observations will be needed to improve the astrometric fit of this binary system. As an independent check of the derived parameters, we performed a combined fit of the astrometric data with the {\tt lmfit} package \citep[][]{newville20}, which uses a non-linear least-squares minimization algorithm to search for the best fit of the observed data (see~Appendix~\ref{sec:mcmc} for further details). We find that the astrometric fit is well constrained, and that the solution and the residuals of the fit are in good agreement with those derived with the combined astrometric fit (see~Sec.~\ref{sec:caf}). These results suggest that the main star may have at least one companion. Below we investigate further the possibility that these residuals are due to a planetary companion orbiting around the main star GJ~896A (as in the case of the single-companion fit, see Sec.~\ref{sec:sca}). \subsection{Binary-system plus planet Astrometry: Full Combined Astrometric fit} \label{sec:fcaf} We carried out a combined astrometric fit of the relative orbit of the binary and the absolute astrometric data of the main and the secondary stars, now including the orbital motion of a possible companion in a compact orbit (full combined astrometric fit), and found that the astrometric fit improves substantially. The results of this full combined astrometric fit are presented in Figures~\ref{fig_7}b, \ref{fig_8} and \ref{fig_9}. We find that the full combined astrometric fit is well constrained. Column 3 of Table~\ref{tab_3} summarizes the best fit parameters of the full combined astrometric fit. By combining relative and absolute astrometric data of the binary system, we are able to obtain the full dynamical motion of the system, including the close companion. From the best fit of the full combined astrometric data, we obtained proper motions $\mu_\alpha$ = 571.467$\pm$0.023 mas yr$^{-1}$ and $\mu_\delta$ = $-$37.715$\pm$0.023 mas yr$^{-1}$, and a parallax $\Pi$ = 159.88$\pm$0.17 mas for the binary system (see column 3 of Table~\ref{tab_3}). The estimated proper motions and parallax are very similar to those obtaned from the Combined astrometric fit. However, as expected, the proper motions differ from those obtained by GAIA for both stars GJ~896A and GJ~896B, which where obtained from independent linear fits of both stars \citep[][]{lindegren18}. The parallax, which corresponds to a distance of $d$ = 6.2545$\pm$0.0066 pc, is an improvement to that obtained by GAIA for both stars ($\Pi_{A}$ = 159.663$\pm$0.034 mas, and $\Pi_{B}$ = 159.908$\pm$0.051 mas). The parallax estimated by us is for the binary system, while GAIA's parallaxes are for the individual stars. The part of the solution that corresponds to the astrometric fit of the binary system is very similar to that obtained from the combined astrometric fit (see columns 2 and 3 of Table~\ref{tab_3}). The fit of the binary system does not improve in this case. However, the best full combined fit of the astrometric data indicates that the M3.5 dwarf GJ~896A has at least one planetary companion, GJ~896A$b$. The orbit of the star around the barycenter, due to this planetary companion, has an orbital period $P_{Ab}$ = 284.39 $\pm$ 1.47 days, an eccentricity $e_{Ab} = 0.35 \pm 0.19$, a longitude of the periastron $\omega_{A} = 353.11^\circ \pm 11.81^\circ$, a position angle of the line of nodes $\Omega_{Ab} = 45.62^\circ \pm 9.60^\circ$, a semi-major axis $a_{A} = 0.51 \pm 0.15$ mas, and an inclination angle $i_{Ab} = 69.20^\circ \pm 25.61^\circ$, which indicates that the orbit is well constrained and prograde ($i_{Ab} < 90^\circ$). However, the rms of the residuals ($\sim$0.21 and $\sim$0.45 mas in R.A and Dec., respectively) are still large compared to the mean noise present in the data ($\sim$0.15 mas) and the astrometric precision expected with the VLBA ($<$70 $\mu$as). This may explain the large errors of the orbital parameters. The large residuals may also indicate the presence of a second planetary companion. We carried out a blind search for a possible astrometric signal due a second planetary companion using the least-squares and the AGA algorithms, however, we did not find a candidate. Using the fitted solution, we obtain the total mass of the system ($m_{AB}$ = 0.603410$\pm$0.000025 $M_{\odot}$), the masses of the two stars ($m_{AB}(A)$ = 0.43814$\pm$0.00065 $M_{\odot}$ and $m_{AB}(B)$ = 0.16527$\pm$0.00025 $M_{\odot}$). In addition, we find that the mass of the star GJ~896A is $m_{A}$ = 0.43599$\pm$0.00092 $M_{\odot}$, and that the mass of the planetary companion is $m_{b}$ = 0.00216$\pm$0.00064 $M_{\odot}$, which is consistent with being planetary in origin with an estimated mass of 2.26$\pm$0.57 $M_{J}$. The semi-major axis of the orbit of the binary system is $a_{AB}$ = 31.635 $\pm$ 0.033 au. The semi-major axis of the two stars around the center of mass are $a_{AB}(A)$ = 8.6646 $\pm$ 0.0091 au and $a_{AB}(B)$ = 22.971 $\pm$ 0.024 au, and the semi-major axis of the orbit of the planetary companion is $a_{b}$ = 0.63965 $\pm$ 0.00067 au (or 102.27 $\pm$ 0.15 mas). This is the first time that a planetary companion of one of the stars in a binary system has been found using the astrometry technique. The solution of the full combined astrometric fit of the planetary companion is similar to that obtained from the single-companion solution (see~Sec.~\ref{sec:sca} and Tables~\ref{tab_2} and \ref{tab_3}). The fact that the astrometric signal appears in the periodogram of the absolute astrometric observations of GJ~896A, and in both, the single-companion astrometric fit and the full combined astrometric fit obtained, in both cases with two different algorithms (least-squares and AGA), supports the detection of an astrometric signal due to a companion. Furthermore, the similar astrometric fit solution obtained from the single-companion astrometric fit and the full combined astrometric fit further supports the planetary origin of the astrometric signal. \section{Discussion} \label{sec:discusion} \subsection{Proper Motions and Orbital Acceleration} \label{sec:pmot} As mentioned before, the estimation of the proper motions of a binary system is complex due to the orbital motions of each component around the center of mass, specially when both stars have different masses ($m_{B}/m_{A} <$ 1) and the time span of the observations cover only a fraction of the orbital period of the binary system. In such a case, it is difficult to separate the orbital motion of each component and the proper motion of the system, both movements are blended. The best way to separate both movements is to simultaneously fit the proper motions, the parallax, and the orbital motion of the binary system. The full combined astrometric fit that we carried out (see Sec.~\ref{sec:fcaf}) includes all these components, and thus gives good estimates of the proper motion and the orbital motion of the binary system (see Table~\ref{tab_3}). The full combined astrometric solution (see column 3 of Table~\ref{tab_3}) shows that the orbital motions of the two stars around the center of mass and of the planetary companion GJ~896A$b$ around the main star GJ~896A are well constrained. In addition, we found a similar solution for the orbital motion of the planetary companion using the full combined astrometric fit (see Sec.~\ref{sec:fcaf} and column 3 of Table~\ref{tab_3}) and the single-companion astrometric fit (see Sec.~\ref{sec:sca} and column 3 of Table~\ref{tab_2}). In the former case, the astrometric fit includes intrinsically the orbital motion of GJ~896A around the center of mass of the binary system, while in the latter case, the included acceleration terms take into account this orbital motion. Tables~\ref{tab_2} and \ref{tab_3} show that the proper motions of the system obtained with the single-source, single-companion and the full combined astrometric fits differ significantly. Here we discuss the origin of this difference. The single-source astrometric solution gives $\mu_\alpha$ = 576.707 mas yr$^{-1}$ and $\mu_\delta$ = $-$59.973 mas yr$^{-1}$ (see Sec.~\ref{sec:ssa} and column 1 of Table~\ref{tab_2}), where we include only the proper motions and the parallax to the absolute astrometric fit of the main star GJ~896A. Including acceleration terms (single-source plus acceleration astrometric solution), to take into account the orbital motion of the binary system, the estimated proper motions of the system are $\mu_\alpha$ = 574.171 mas yr$^{-1}$ and $\mu_\delta$ = $-$60.333 mas yr$^{-1}$ (see Sec.~\ref{sec:sca} and column 2 of Table~\ref{tab_2}). On the other hand, from the full combined astrometric solution, the proper motions of the system are $\mu_\alpha$ = 571.467 mas yr$^{-1}$ and $\mu_\delta$ = $-$37.715 mas yr$^{-1}$ (see Sec.~\ref{sec:fcaf} and column 3 of Table~\ref{tab_3}). The absolute astrometric solutions presented in Table~\ref{tab_2} show that by including the acceleration terms in the fit, we improve the estimated proper motions of the system, and that the inclusion of an astrometric signal due to a companion in the fit does not affect the estimation of the proper motions and the estimated acceleration terms. Comparing the single-source and the full combined astrometric solutions, we find a significative difference in the estimated proper motions, particularly in the north-south direction ($\Delta\mu_\alpha$ = 5.24 mas yr$^{-1}$ and $\Delta\mu_\delta$ = $-$22.26 mas yr$^{-1}$). This is consistent with the fact that the VLBA astrometric observations of GJ~896A were obtained when the source was crossing the ascending node of the binary orbit and mainly in the north-south direction (see discussion below, and Figures~\ref{fig_8} and \ref{fig_9}). These results indicate that the orbital motion of GJ~896A around the center of mass of the system blends with the proper motions of the binary system, and in second order with the acceleration terms included in the absolute astrometric fit. Thus, we conclude that the best estimates of the proper motions of this binary system are those obtained with the full combined astrometric fit (see Table~\ref{tab_3}). We can also obtain an estimation of the velocity on the plane of the sky of the center of mass $\mu(CM)$ of this binary system, as follows: \begin{equation} \mu_{\alpha}(A) = \mu_{\alpha}(CM) + v_{\alpha}(A), \\ \end{equation} \begin{equation} \mu_{\delta}(A) = \mu_{\delta}(CM) + v_{\delta}(A), \end{equation} \noindent and \begin{equation} v_{\alpha}(A) = (2 \pi a_{A}/P_{AB}) \times sin(\theta_{rot}) \times cos(\phi), \end{equation} \begin{equation} v_{\delta}(A) = (2 \pi a_{A}/P_{AB}) \times cos(\theta_{rot}) \times cos(\phi), \end{equation} \noindent where $\mu_{\alpha}(A)$ and $\mu_{\delta}(A)$ are the observed proper motion of the main source GJ~896A (single-source solution; see column 1 of Table~\ref{tab_2}), $\mu_{\alpha}(CM)$ and $\mu_{\delta}(CM)$ are the proper motions of the center of mass of the binary system, $v_{\alpha}(A)$ and $v_{\delta}(A)$ are the projected rotational velocity of GJ~896A on its orbital motion around the center of mass, $a_{A}$ is semi-major axis of the orbital motion of GJ~896A around the center of mass, P$_{AB}$ is the orbital period of the binary system, $\theta_{rot}$ $\approx$ $\Omega_{AB} - 90^\circ$ is the position angle of the projected rotation velocity (tangential velocity) of GJ~896A around the center of mass of the system (see Figures~\ref{fig_8}b and \ref{fig_9}), $\phi = 180^{\circ} - i_{AB}$ is the complementary angle of the orbital plane of the binary system, and $ i_{AB}$ is the fitted inclination angle of the orbital plane of the binary system (see column 3 of Table~\ref{tab_3}). Using the fitted parameters presented in column 1 of Table~\ref{tab_2} and column 3 of Table~\ref{tab_3}, $\mu_{\alpha}(A)$ = 576.707 mas yr$^{-1}$ and $\mu_{\delta}(A)$ = $-$59.973 mas yr$^{-1}$ ($\mu_{A}$ = 579.8170 mas yr$^{-1}$ with PA = 95.94$^\circ$), we obtain $v_{\alpha}(A)$ = 6.2925 mas yr$^{-1}$ and $v_{\delta}(A)$ = $-$23.6355 mas yr$^{-1}$ ($v_{rot}$ = 24.4588 mas yr$^{-1}$ with PA = 165.09$^\circ$), and thus $\mu_{\alpha}(CM)$ = 570.4145 mas yr$^{-1}$ and $\mu_{\delta}(CM)$ = $-$36.3375 mas yr$^{-1}$ ($\mu(CM)$ = 571.5707 mas yr$^{-1}$ with PA = 93.65$^{\circ}$). We find that the estimated proper motions of the center of mass are very similar to those obtained with the full combined astrometric fit $\mu_{\alpha}$ = 571.467 mas yr$^{-1}$ and $\mu_{\delta}$ = $-$37.715 mas yr$^{-1}$ ($\mu$ = 572.72 mas yr$^{-1}$ with PA = 93.79$^\circ$; see Table~\ref{tab_3}). The difference in the estimated proper motions of the center of mass is about 1 mas yr$^{-1}$. This further confirms that the full combined astrometric fit gives the best estimate for the proper motions of the center of mass of a binary system. We can estimate the acceleration, $acc(A)$, of GJ~896A due to the gravitational pull of the low-mass stellar companion GJ~896B, as follows: \begin{equation} acc(A) = (2\pi/P_{AB})^{2} a_{A}, \end{equation} \noindent where $acc(A)$ is the mean acceleration of GJ~896A, $P_{AB}$ is the orbital period of the binary system, and $a_{A}$ is the semi-major axis of the orbital motion of GJ~896A around the center of mass. Using the full combined solution (see column 3 of Table~\ref{tab_3}), we obtain that $acc(A)$ $\simeq$ 1.04234 mas yr$^{-2}$. In the case of the single-companion solution (see column 3 of Table~\ref{tab_2}), the fitted acceleration terms are $a_\alpha$ = 0.8873 mas yr$^{-2}$ and $a_\delta$ = 0.1402 mas yr$^{-2}$ , and thus the acceleration of the main star in the plane of the sky is $acc(A)$ = $\sqrt{a_\alpha ^{2} + a_\delta^{2}}$ = 0.89831 mas yr$^{-2}$. Thus, the estimated acceleration of GJ~896A due to GJ~896B is consistent with the acceleration found in the single-companion astrometric fit. The acceleration $acc(A)$ obtained from the single-source astrometric fit of the VLBA data allows us to place a mass estimate for the stellar companion, $m_{B}$, using \begin{equation} \left(\frac{m_{B}}{M_\odot}\right) = 0.02533 \left(\frac{acc(A)}{AU yr^{-2}}\right) \left(\frac{a_{AB}}{AU}\right)^{2}, \end{equation} \noindent or \begin{equation} \left(\frac{m_{B}}{M_\odot}\right) = 0.29368 \left[\left(\frac{acc(A)}{AU yr^{-2}}\right) \left(\frac{a_{A}}{AU}\right)^{2} \left(\frac{M_{AB}}{M_\odot}\right)^{2} \right]^{1/3}, \end{equation} \noindent where $acc(A)$ = $\sqrt{a_{\alpha}^{2} + a_{\delta}^{2}}$ is the estimated acceleration needed to fit the absolute astrometric data (see column 3 of Table~\ref{tab_2}), $a_{AB}$ = $a_{A}$ + $a_{B}$ is the semi-major axis of the orbit of GJ~896B around GJ~896A, and m$_{AB}$ is the total mass of the binary system (see column 3 of Table~\ref{tab_3}). From equation~7 we find that the estimated mass of the stellar companion GJ~896B is $m_{B}$ $\approx$ 0.15728 M$\odot$. This estimated mass is consistent with the mass of the stellar companion ($m_{B}$ = 0.16527 M$\odot$) obtained from the full combined astrometric fit (see column 3 of Table~\ref{tab_3}). \subsection{Expected Radial Velocities} \label{sec:radvel} The solution of the full combined astrometric fit can be used to estimate an expected induced maximum RV of the star due to the gravitational pull of a companion as follows \citep[e.g.,][]{canto09,curiel20}: \begin{equation} K =\left(\frac{2 \pi G}{T}\right)^{1/3} \frac{m_{c} sin(i)} {(M_{*} + m_{c})^{2/3}} \frac{1} {\sqrt{1 - e^{2}}}, \end{equation} \noindent where G is the gravitational constant, and $T$, $M_{*}$, $m_{c}$, and $e$ are the estimated orbital period, star and companion masses, and the eccentricity of the orbit of the companion. Using the full combined astrometric solution (see column 3 of Table~\ref{tab_3}), the maximum RV of GJ~896A induced by the planetary companion GJ~896A$b$ is $K_{A}(b)$ $\sim$ 121 m s$^{-1}$, and the maximum RV induced by the stellar companion GJ~896B is $K_{A}(B)$ $\sim$ 867 m s$^{-1}$. These RVs can in principle be observed with modern high-spectral resolution spectrographs. The maximum radial velocity of both stars occurred in October 2013, when GJ~896A and GJ~896B passed through the ascending and descending nodes, respectively, of their orbits around the center of mass of the binary system (see Figure~\ref{fig_8}). The RV signal due to GJ~896A$b$ can be measured with modern high-spectral resolution spectrographs on a time span shorter than one year. Recent radial velocity observations of GJ~896A show a radial velocity variability of $\Delta V$ $\sim$ 175$\pm$37 m s$^{-1}$ \citep[e.g.,][]{gagne16}, which is consistent with the expected radial velocity of this source, induced by the planetary companion GJ~896A$b$. However, these observations were taken in just 7 epochs within a time span of about 4 years. Further observations will be needed to confirm whether the RV variability observed on GJ~896A is due to this planetary companion. The RV signal of GJ~896A (and GJ~896B) due to the stellar companion will be difficult to measure due to the binary's very long orbital period of 229.06 yr. It will be very difficult to separate this signal from the systemic velocity $V_{0}$ of the binary system. However, we can use the observed mean RV of GJ~896A \citep[][]{lindegren18}, and the estimated maximum RV of this star due to its stellar companion to obtain a raw estimate of the systemic radial velocity of the binary system, as follows: \begin{equation} V_{obs}(A) = K_{A}(B) + V_{0}, \end{equation} \noindent where $V_{obs}$ is the observed radial velocity, $K_{A}(B)$ is the expected maximum RV of GJ~896A due to GJ~896B, and $V_{0}$ is the barycentric RV of the binary system. From the GAIA catalog, the observed radial velocity of GJ~896A is $V_{obs}(A)$ = $-$0.02$\pm$0.31 km s$^{-1}$ \citep[][]{lindegren18}. Since these RV observations of GJ~896A were take in a time span of about 2 years, the RV signal from GJ~896A$b$ is averaged out in this mean radial velocity. The reference epoch of the GAIA DR2 observations is J2015.5, which is close to orbital location where the radial velocity of GJ~896A is maximum and negative (see Figure~\ref{fig_8}). Thus, $K_{A}(B)$ = $-$867 m s$^{-1}$, and the resulting barycentric RV of the binary system is $V_{0}$ $\sim$ 847 m s$^{-1}$. A similar calculation can be carried out for GJ~896B. In this case, the observed RV is $V_{obs}(B)$ = 3340 m s$^{-1}$ \citep[][]{morin08}, which was obtained also close to the maximum RV of this star. The reference epoch of the RV observations is 2006.0, which is close to orbital location where the radial velocity of GJ~896B is maximum and positive (see Figure~\ref{fig_8}). We estimate that the maximum RV of GJ~896B induced by the main star GJ~896A is $K_{B}(A)$ $\sim$ 2299 m s$^{-1}$. Thus, in this case we obtain a barycenter RV of $V_{0}$ $\sim$ 1041 m s$^{-1}$. This barycentric RV is different to that obtained from the RV observation of GJ~896A (847 m s$^{-1}$). There is a difference of about 194 m s$^{-1}$. The RV measurements of GJ~896B were obtained in a time period of only a few days, and therefore, the observed RV would include the contribution due to possible companions associated to this M4.5 dwarf. This suggests that GJ~896B may have at least one companion. Such radial velocity signature should be easily detected using modern high-spectral resolution spectrographs. Recent observations also suggest that the M4.5 dwarf GJ~896B may be an unresolved binary system \citep[][]{winters21}. Further observations will show whether this very low mass M dwarf star has a companion. \subsection{Habitable zone and Snow Line} A simple estimate of the habitable zone (HZ) can be obtained as follows. The inner and outer boundaries of the HZ around a star depends mainly on the stellar luminosity. Thus, combining the distance dependence of the HZ as function of the luminosity of the star and the mass-luminosity relation, the inner $a_{i}$ and outer $a_{o}$ radius of the HZ are: \begin{equation} a_{i} = \sqrt{ \left(\frac{L_\star}{L_\odot}\right) \frac{ 1}{ 1.1}}, ~~~~~ a_{o} = \sqrt{ \left(\frac{L_\star}{L_\odot}\right) \frac{ 1}{ 0.53}} \end{equation} \noindent \citep[e.g.,][]{selsis07, kopparapu13}, where $L_{\star}$ is the luminosity of the star in solar luminosities. In the case of M dwarfs with masses below 0.43 M$_\odot$, the mass-stellar luminosity relation can be approximated as: \begin{equation} \left(\frac{L_\star}{L_\odot}\right) = 0.23 \times \left(\frac{M_\star}{M_\odot}\right)^{2.3} \end{equation} \noindent \citep[e.g.,][]{kuiper38, duric12}, where $M_{\star}$ is the mass of the star in solar masses. Using the estimated mass of GJ~896A ($m_{A}$ = 0.436 M$\odot$ ; see Table~\ref{tab_3}), the limits of the habitable zone around this M3.5 dwarf are $a_{i}$ $\sim$ 0.18 au and $a_{o}$ $\sim$ 0.26 au. This is considerably smaller than the semi-major axis of the orbit of the planetary companion GJ~896A$b$ ($a_{b}$ = 0.6397 au). Since the planetary companion is in an eccentric orbit ($e_{b}$ = 0.35), the minimum and maximum distance between the planet and the star are 0.42 and 0.86 au, respectively. Thus, the orbit of the planet lyes outside the habitable zone of this M3.5 dwarf star. We can also estimate if the planetary companion GJ~896A$b$ is located inside the snow line, $a_{snow}$. The snow line is located at an approximate distance of \citep[e.g.,][]{kennedy08} \begin{equation} a_{snow} = 2.7 \left(\frac{M_\star}{M_\odot}\right)^{2}. \end{equation} \noindent Using the estimated mass of GJ~896A, the snow line in this planetary system is located at a radius of $a_{snow}$ $\sim$ 0.51 au. Thus the estimated orbit of the planetary companion GJ~896A$b$ is located outside the snow line. However, given the eccentricity of the orbit, the planet moves around the estimated snow line distance, but most of the time is located outside the snow line. Recent results suggest that stars with a mass of about 0.4 $M_{\odot}$, such as GJ~896A, have a 1\% probability of having at least one Jovian planet \citep[e.g.,][]{kennedy08}. Even when the probability of having a Jovian planet is very low, the results presented here show that the main star GJ~896A in the M-dwarf binary system GJ~896AB has at least one Jupiter-like planet. \subsection{Flux variability of the source.} The radio continuum flux density of GJ~896A is clearly variable in time. Figure~\ref{fig_10} shows that the source goes through a large variability in short periods of time. The flux variability does not seem to have a clear pattern. The flux density of the source has changed in nearly two orders of magnitud during the last 16 years of the radio observations, with variations at scales of months and at scales of a few years. However, further observations will be required to find if the variability has a defined temporal period. \subsection{Mutual Inclination Angle} \label{sec:mutinc} Characterizing the full 3D orbital architecture of binary systems containing a planetary companion can aid to investigate the importance of the star-star and star-planet mutual interaction. Combining relative and absolute astrometric data of the binary system, we are able to obtain the 3D orbital architecture of the system, including the planetary companion (see Figure~\ref{fig_9}). Since the full combined astrometric fit (relative plus absolute astrometry) provides the inclination angle and the position angle of the line of nodes of the orbital planes of both the planetary companion GJ~896$b$ and the binary system GJ~896AB, we can directly measure the mutual inclination angle of this system. A first approach would be to estimate the inclination difference ($\Delta i$) between the inclination angles of the orbital planes of the planet and the binary system, assuming that their position angle of the line of nodes is equal to 0$^{\circ}$. The inclination angles are measured with respect to the plane of the sky (such that $i$=0$^{\circ}$ corresponds to a face-on orbit, and it increases from the East toward the observer). Using the fitted inclination angles we obtain that $\Delta i =$ 60.9$^{\circ}$$\pm$22.5$^{\circ}$ (see Table~\ref{tab_3} and Figure~\ref{fig_9}), which is a very large difference. The mutual inclination angle $\Phi$ between the two orbital planes can be determined through \citep[e.g.,][]{kopal59, muterspaugh06}: \begin{equation} cos~\Phi = cos~i_{Ab} ~cos~i_{AB} + sin~i_{Ab} ~sin~i_{AB} ~cos(\Omega_{Ab} - \Omega_{AB}), \end{equation} \noindent where $i_{Ab}$ and $i_{AB}$ are the orbital inclination angles, and $\Omega_{Ab}$ and $\Omega_{AB}$ are the position angles of the line of nodes. The position angle of the line of nodes is measured anti-clockwise from the North toward the ascending node. Table~\ref{tab_3} contains the inclination angle and the position angle of the line of nodes for the planet GJ~896A$b$ and the binary system GJ~896AB. There is an ambiguity in the position angles of the line of nodes ($\Omega$ or $\Omega$ + 180$^{\circ}$, where $\Omega$ is the fitted angle), which can be disentangled by RV measurements. For the orbital motion of the binary system, recent RV measurements of both stellar components show that GJ~896B is receding ($V_{obs}(B)$ = $+$3.34 km s$^{-1}$; \citet[][]{morin08}) and GJ~896A is approaching to us ($V_{obs}(A)$ = $-$0.02$\pm$0.31 km s$^{-1}$; \citet[][]{lindegren18}), thus the correct position angle of the line of nodes is $\Omega_{AB}$ = 255.09$^{\circ}$ ($\Omega_{AB}$ + 180$^{\circ}$) (see Table~\ref{tab_3}). However, the planetary companion has no measured RV, so the fitted value of the position angle of the line of nodes of the orbital plane of GJ~896A$b$ could be either $\Omega_{Ab}$ = 45.6$^{\circ}$ or 225.6$^{\circ}$, the former is the fitted angle. From these position angles we calculate a mutual inclination angle between the two orbital planes of $\Phi$ = 148$^\circ$ for the fitted $\Omega_{Ab}$, and $\Phi$ = 67$^\circ$ for the second possibility ($\Omega_{Ab}$ + 180$^{\circ}$). Taking into account the long term stability of the system (see Sec.~\ref{sec:orbitstab}), we found that the former solution ($\Phi$ = 148$^\circ$) is stable in a very long period of time, while the latter solution ($\Phi$ = 67$^\circ$) is unstable in a short period of time. This result indicates that there is a large mutual inclination angle of $\Phi$ = 148$^\circ$ between both orbital planes. Recent observations suggest that the rotation axis of GJ~896A has an inclination angle of about 60$^{\circ}$$\pm$20$^{\circ}$ with respect to the line of sight \citep[][]{morin08}, and thus, an inclination angle of $i_{s}$ $\sim$ 210$^{\circ}$ with respect to the plane of the sky (see Figure~\ref{fig_9}). On the other hand, the inclination angle of the rotation axis of the orbital motion of GJ~896A$b$ is $i_{p}$ = 159.2$^{\circ}$$\pm$25.61$^{\circ}$ ($i_{b}$ + 90$^{\circ}$). Thus these rotation planes have a difference in their inclination angles of $\Delta_{s-p}$ $\sim$ 51$^{\circ}$ (see Figure~\ref{fig_9}). This indicates that the orbital motion of the planetary companion and the rotation plane of the star are far from being coplanar. We can also compare the angle of the rotation axis of the host star GJ~896A and the inclination angle, $i_{bs}$, of the rotation axis of orbital motion of the binary system GJ~896AB. In this case the inclination angle is $i_{bs}$ = 220.065$^{\circ}$$\pm$0.010$^{\circ}$ ($i_{AB}$ + 90$^{\circ}$). Thus the rotation planes of the star GJ~896A and the binary system have a difference in their inclination angles of $\Delta i_{bs-s}$ $\sim$ 10$^{\circ}$. In addition, the rotation axis of GJ~896B, with respect to the line of sight, is also about 60$^{\circ}$$\pm$20$^{\circ}$ \citep[][]{morin08}, which is basically the same as that of GJ~896A, and thus, the difference between the inclination angle of the rotation axis of GJ~896B, with respect of the plane of the sky, compared with the rotation axis of the binary system is the same as that found for the star GJ~896A ($\sim$ 10$^{\circ}$). Thus, this suggests that the rotation planes of both stars are nearly parallel to the orbital plane of the binary system ($\Delta i$ $\sim$ 10$^{\circ}$), while the orbital planes of the planet and the binary system have a mutual inclination angle of $\Phi$ $\simeq$ 148$^{\circ}$. However, having similar inclination angles does not necessarily imply alignment between the rotation axis of both stars since the position angles of the line of nodes are unknown. The rotation axis of the stars could be different, and thus, they could have larger and different mutual inclination angles with respect to the orbital plane of the binary system. \subsection{Orbital Stability} \label{sec:orbitstab} We applied direct N-body integrations to the full combined orbital solution obtained from the Keplerian orbits of the binary system GJ~896AB and the planetary companion GJ~896A$b$ (see Table~\ref{tab_3}). We integrated the orbits for at least 100 Myrs using the hybrid integrator in Mercury~6 \citep[][]{chambers99}, which uses a mixed-variable symplectic integrator \citep[][]{wisdom91} with a time step approximately equal to a hundredth ($\simeq 1/100$) of the Keplerian orbital period of the planetary companion. During close encounters, Mercury uses a Bulrich-Stoer integrator with an accuracy of $10^{-12}$. We identify an unstable system if: a) the two companions (the planetary companion and low-mass stellar companion) collide; b) the planet is accreted onto the star (astrocentric distance less than 0.003 au); and c) the planet is ejected from the system (astrocentric distance exceeds 200 au). The integration time of 100 Myrs long exceeds the 10,000 binary periods that is considered as a stability criterion. The simulation using the fitted Keplerian orbital parameters proved to be stable for at least 100 Myrs. However, the simulation of the alternative position angle of the line of nodes of the planetary orbit ($\Omega = \Omega_{Ab} + 180^{\circ}$) turned out to be unstable in very short times, the planet collided with the host star GJ~896A after a few tens of thousand years. In this case, the eccentricity of the planetary orbit increases rapidly due to the interaction between the stellar companion and the planet. After a few tens of thousand years the eccentricity reaches an extreme value of 1, and thus, the planet collides against the main star. This indicates that the fitted solution contains the correct angle of the line of nodes of the planetary orbit. To complement our stability analysis, we also performed N-body long-term integrations using REBOUND \citep[][]{rein12}. We tested the combined orbital solutions using two different integrators: IAS15 \citep[][]{rein15} and Mercurius \citep[][]{rein19}. The first one is a 15th order high-precision non-symplectic integrator. The second one is a hybrid symplectic integrator. These two integrators allow us to corroborate the results obtained with Mercury 6. For both integrators, we integrate over 10,000 orbits of the binary system GJ~896AB using 20,000 points per orbital period (i.e., one sampling point every ~4 days). This allows us to monitor the changes in the orbital parameters of the planet GJ~896A$b$ with reliable accuracy. Both, the best-fit solution reported in Table~\ref{tab_3} and the complementary one with $\Omega$ = $\Omega_{Ab}$ + 180$^{\circ}$ were tested. Our results are in agreement with those obtained with Mercury 6. Our best-fit solution is stable over the whole integration time, while the alternative solution becomes unstable in a short period of time. A detailed discussion about the 3D orbital stability of this binary system is out of the scope of this paper and it will be presented elsewhere. \subsection{Binary System Formation and Stability} \label{sec:binaryform} The present separation between the two stars ($\sim$31.64 au) in this binary system suggests that they were most likely formed in a massive accretion disk by disk fragmentation, and not by turbulent fragmentation of the original molecular core. Binary systems formed by turbulent fragmentation are expected to have separations of hundreds, or even thousands, of au, which better explain the formation of binary systems with wide orbits \citep[e.g.,][]{offner16}. On the other hand, in the case of disk fragmentation, the lower mass stellar companion is formed in the outer parts of the disk (probably at a distance of a few hundred au) and then, during the early evolution of the binary system, the stellar companion migrates inwards to a closer orbit \citep[e.g.,][]{tobin16}. The apparently similar spin angles of the two stars ($\sim$ 210$^{\circ}$), and the relatively small, but significant, difference between the spin axis of both stars and the rotation axis of the binary system ($\ga$ 10$^{\circ}$), are consistent with this formation mechanism. Since we do not know the position angles of the line of nodes of the rotation plane of both stars, it is possible that the rotation planes are not coplanar, and thus, they probably have different mutual inclination angles with respect to the orbital plane of the binary system. Either way, the difference in the inclination angle between the spin axis of the stars and the orbital axis of the binary system ($\ga$ 10$^\circ$) suggests that during the evolution of this binary system, the star-star interaction between both stars may have significantly changed the orientation of the rotation axes of both stars. In addition, the large mutual inclination angle ($\Phi$ = 148$^\circ$) that we find between the orbital planes of the planetary companion GJ~896A$b$ and the binary system GJ~896AB indicates that something changed the orientation of the orbital plane of the planet, probably from initially being coplanar to be in a retrograde configuration at present time. The most likely origin of this large mutual inclination angle is the star-planet interaction, where the low mass stellar companion GJ~896B produces some important torque over the orbit of the planetary companion of GJ~896A. Recent studies have shown that in the case of planet-planet interaction (planetary systems with at least one external Jovian planet), the mutual inclination angles are $\la$10$^{\circ}$ \citep[see, e.g.,][]{laughlin02}. In a few cases there is evidence of an inclination between the orbits of the planets of up to $\sim$40$^{\circ}$ \citep[see, e.g.,][]{mcarthur10, dawson14, mills17}. However, since these studies are based on planetary systems detected with the RV and/or transit techniques, they lack of information about the position angle of the line of nodes, and thus, the mutual inclination angles calculated this way are lower limits. In addition, according to the catalogue of Exoplanets in Binary Systems \citep[][]{schwarz16}, 160 planets have been found associated with 111 binary systems, 79 of which are in S-type orbits \citep[i.e., the planet is orbiting one of the stars; see, also][]{marzari19}. Most of these planets were found using RV and transit techniques, and some of the other planets were found using other techniques, such as imaging and microlensing. In all of them, some of the orbital parameters are missing, in particular, the position angle of the line of nodes of the orbital plane has not being obtained, and in several cases, the inclination angle is also unknown. This is the first time that the remarkable full 3D orbital architecture of a binary-planetary system has been determined. \subsection{Planetary origin of the astrometric signal.} The presence of a Jovian planet associated to the main star in a binary system, such as the one we present here, would produce secular perturbations on the lower mass stellar companion. Such interaction would produce long-term periodic variations in the orbital motion of the secondary (including the eccentricity and the inclination angle), as well as in the orbital parameters of the planetary companion. Given the orbital period of about 229 yrs of this binary system, several decades of absolute and relative astrometric observations of the binary system are probably needed to fully constrain the orbital motion of the binary system. As we have mentioned above, we are limited by the time span of the astrometric observations we present here (80 years for the relative astrometry and 16.6 yrs in the case of the absolute astrometry), which is much smaller than the orbital period of the binary system. However, in the analysis we present here, we show that by combining the absolute astrometric observations of the primary and the secondary stars, as well as the relative astrometry of the binary system, in the astrometric fit, the orbital motion of the binary system and the planetary companion are well constrained. We have found that the orbits of the binary system and the planetary companion are somewhat eccentric. A similar astrometric signal due the planetary companion was found using different methods. We obtained a similar astrometric signal using both the periodogram of the astrometric data, and the single-companion astrometric fits of the absolute astrometric data obtained with two different algorithms (least-squares and AGA). This indicates that the astrometric signal is real. The detection of a similar astrometric signal with the full combined astrometric fit (relative plus absolute astrometry) further support the detection of the astrometric signal. In addition, we complemented the analysis of the astrometric observations with the non-linear least squares minimization package {\tt lmfit} \citep[][]{newville20}, finding a similar astrometric solution (see Appendix). This strongly indicates that the astrometric signal is real. Furthermore, the fact that the astrometric solution from the different methods is similar indicates that the astrometric signal is consistent with the companion being planetary in origin. A detailed study of the stability of the orbital motions in this system can give important information about the star-star and star-planet interactions. A detailed discussion about the 3D orbital stability of this binary system is out of the scope of this paper and it will be presented elsewhere. The astrometric signal that we find is consistent with a planetary companion associated to the M3.5 Dwarf GJ~896A. However, this astrometric signal could be contaminated by the expected astrophysical $``$jitter$"$ added to the true source position due to stellar activity. It has been estimated that GJ~896A has a radius of $\sim$0.35 $R_\odot$ \citep[e.g.,][]{morin08, pearce20}. Thus, the radius of GJ~896A at a distance of 6.2567 pc is about 0.260 mas. This result indicates that the astrometric signal due to the planetary companion (0.51 mas) is about twice the radius of the host star. In addition, assuming that the radio emission originates within $\sim$0.194 stellar radius above the photosphere \citep[e.g.,][]{liefke08, crosley18a}, the size of the expected $``$jitter$"$ is about 0.31 mas. However, the analysis that we carried out for the variability of the radio emission of the main star GJ~896A shows that the $``$flares$"$ seen in several of the observed epochs contribute only with less that 0.12 mas to the expected $``$jitter$"$ (see Appendix~\ref{sec:flares} and panels (d) in Figure~\ref{fig_flares}), compared to the expected 0.31 mas contribution to the $``$jitter$"$ that we have estimated here. Thus, the expected $``$jitter$"$ due to the variability of the star is about a factor of 4.3 smaller than the astrometric signal of 0.51 mas observed in GJ~896A (see Table~\ref{tab_3}). This result supports the detection of the planetary companion GJ~896A$b$. To further investigate the validity of the planetary astrometric signal, we have applied several statistical tests comparing the astrometric solutions obtained without and with a planetary companion (see Table~\ref{tab_2}). The solutions of the best astrometric fits show that when a planetary companion is not included the residuals of the fit have a rms scatter of 0.34 mas and a $\chi^2$ = 190.91, compared to a rms scatter of 0.20 mas and $\chi^2$ = 57.41 when the planetary companion is included in the fit. An F-test shows that the probability of the $\chi^2$ dropping that much (due to the inclusion of the planetary companion) is less than 4\% by mere fluctuations of noise. Using the Bayesian Information Criterion (BIC), we find that the inclusion of the planetary companion is preferred with a $\Delta$BIC = $-$109.24, this is a significant difference between our best fit model with the planetary companion and the one without it. Statistically, an absolute value of Delta BIC of more than ten implies that the model with the planetary companion better reproduces the data without overfitting it by including more free parameters in the model. Similar results were obtained using the Akaike Information Criterion (AIC) with a $\Delta$AIC = $-$119.50. We therefore conclude that there is a very high probability that the planetary companion GJ~896Ab is orbiting the main star GJ~896A. The large proper motions of the binary system may also contribute to the expected $``$jitter$"$ of the star. The change in position of GJ~896A due to the proper motions of the binary system, $\Delta_{PM}(A)$ in mas, can be estimated by: \begin{equation} \Delta_{PM}(A) = \sqrt{\mu_\alpha^{2} + \mu_\delta^{2}} \times \left(\delta t / 8766\right), \end{equation} \noindent where $\delta t$ is the time span of each observed epoch in hours (between 3 and 5 hours on source). Using the estimated proper motions of the binary system ($\mu$ = 572.71 mas yr$^{-1}$), we obtain that the maximum expected contribution to the $``$jitter$"$ by the proper motions of the system is $\Delta_{PM}(A)$ = 0.16 mas in a time span of 2.5 hours (half the observing time), which is about three times smaller than the observed astrometric signal. We also obtain that the contribution of the orbital motion of GJ~896A around the center of mass to the expected $``$jitter$"$ of the star (about 0.009 mas in a time span of 2.5 hours) is smaller than that estimated from the proper motions of the star. Adding in quadrature all these possible contributions, the total expected $``$jitter$"$ is about 0.2 mas, which is still about a factor of 2.6 smaller than the astrometric signal due to the planetary companion GJ~896A$b$. The highest contribution to the expected $``$jitter$"$ is that due to the proper motions of the star. However, the astrometric signal is observed in both direction (R.A. and Dec.), while the proper motion of the star is basically in the east direction. In addition, the contribution of the proper motions of the star to the expected $``$jitter$"$ probably averages out when we integrate over the time span of the observations. In addition, given the synthesized beam of the images (about 2.7$\times$1.1 mas, in average), the magnitude of the proper motions is in fact too small to even affect the estimated size of the source. This suggests that the contribution of the proper motions to the expected $``$jitter$"$ is probably much smaller than we have estimated here. Thus, the expected $``$jitter$"$ that we estimate here is most likely an upper limit. This further indicates that the astrometric signal is real, and planetary in origin. \section{Conclusions and Final Remarks} \label{sec:conclusions} The combined (relative plus absolute) fit of the astrometric observations of this binary system show that the main star GJ~896A has at least one planetary companion. This is the first time that a planetary companion of one of the stars in a binary system has been found using the astrometry technique. The astrometric solution indicates that the planetary companion has an orbital period of 284.39 $\pm$ 1.47 days, an estimated mass of 2.26 $\pm$ 0.57 M$_{J}$, a relatively eccentric orbit ($e_{Ab}$ = 0.35 $\pm$ 0.19), and a semi-major axis of $a_{b}$ = 0.63965 $\pm$ 0.00067 au (or 102.27 $\pm$ 0.15 mas). In addition, the full combined astrometric fit also shows that the binary system has an estimated orbital period of 83664.63 $\pm$ 1.98 days (or 229.06 yrs), and the two stars have estimated masses of $m_{AB}$ = 0.603410 $\pm$ 0.000025 $M_{\odot}$, $m_{AB}(A)$ = 0.43814 $\pm$ 0.00065 $M_{\odot}$ and $m_{AB}(B)$ = 0.16527 $\pm$ 0.00025 $M_{\odot}$, respectively. The astrometric solution also indicates that the binary system and the stars have semi-major axis of $a_{AB}$ = 31.635 $\pm$ 0.033 au, $a_{AB}(A)$ = 8.6646 $\pm$ 0.0091 au and $a_{AB}(B)$ = 22.971 $\pm$ 0.024 au, respectively. Thanks to the proximity of the binary system GJ~896AB, the Jovian-like planetary companion GJ~896A$b$ $-$ one of the nearest to Earth yet found $-$ is well suited for a detailed characterization (for example, direct imaging and spectroscopy) that could give important constrains on the nature and formation mechanisms of planetary companions in close binary systems. Combining the relative and absolute astrometric observations, we have found the 3D orbital architecture of the binary system GJ~896AB and the planetary companion GJ~896A$b$. We have performed long-term numerical integrations to test the stability of the orbital solution of this binary system, using both posible position angles of the line of nodes of the planetary companion. We find that only one solution is stable. The second solution of the Keplerian orbit, using $\Omega$ = $\Omega_{Ab}$ + 180$^{\circ}$, turns out to be unstable in very short timescales, the planetary companion collides with the host star GJ~896A after a few thousand years. On the other hand, the fitted solution ($\Omega = \Omega_{Ab}$) proved stable for at least 100 Myrs. This indicates that the position angle of the line of nodes of both orbital planes are $\Omega_{AB}$ = 255.1$^{\circ}$ and $\Omega_{Ab}$ = 45.6$^{\circ}$, and their mutual inclination angle is $\Phi$ = 148$^\circ$. This result is consistent with both orbits being retrograde. This is the first time that the full 3D orbital architecture of a binary system with a planetary companion has been obtained using astrometric observations. This kind of results can not be achieved with other exoplanet methods. Astrometry gives important complementary information to other exoplanet detection techniques. In addition, high-angular resolution radio astrometry is becoming a powerful technique, capable of giving the full 3D orbital architecture of planetary systems and planetary systems in binary and multiple stellar system. Our results demonstrate that astrometric observations have the potential to fully characterize the orbital motions of individual and multiple planetary systems, as well as the 3D orbital architecture of binary systems, and binary systems with planets associated to them. The discovery of gaseous planets associated to low-mass stars poses a great challenge to all current planetary formation scenarios. For instance, core accretion models face serious problems to explain giant planets orbiting around M dwarfs with masses below 0.4 M$_\odot$ \citep[e.j.,][]{laughlin04,ida13,burn21}. In order to explain such planets, these models need to include extraordinary conditions, such as increasing the efficiency of core-accretion planet formation, using high mass protoplanetary disks, which are inconsistent with observational results, and/or slowing down (reducing) their migration speed. It is not clear if gravitational instability of the protoplanetary disk \citep[e.j.,][]{mercer20,boss21} could more naturally produce giant planets around low-mass stars. The formation of this kind of planetary systems through disk fragmentation also requires high-mass protoplanetary disks (with high accretion rate from an envelope). Furthermore, the discovery of Jovian planets associated with low-mass binary systems, such as the one we have found, is even more challenging to current formation scenarios. Particularly, in the case of close binary systems with separation $<$40 au, where it is expected that the stellar companion would truncate the protoplanetary disk, Jovian planets will be very difficult to form. Further theoretical models will be required to understand the formation of giant-mass planets, such as the one we found associated to the main star of the low-mass binary system GJ~896AB. In addition, since most stars are in binary or multiple systems, our understanding of systems such as this one will be crucial to understand the phenomenon of planetary formation in general. ~~ ~~ ~~ \begin{acknowledgements} \noindent We are grateful to the anonymous referee for the useful comments and suggestions, which helped improve this paper. We thank L. F. Rodr{\'\i}guez for valuable comments on an early version of the paper. S.C. acknowledges support from UNAM, and CONACyT, M\'exico. The authors acknowledge support from the UNAM-PAPIIT IN103318 and IN104521 grants. The observations were carried out with the Very Long Baseline Array (VLBA), which is part of the National Radio Astronomy Observatory (NRAO). The NRAO is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This research has made use of the Washington Double Star Catalog maintained at the U.S. Naval Observatory. This research has made use of the Catalogue of Exoplanets in Binary Systems maintained by Richard Schwarz and \'A. Bazs\'o at {https://www.univie.ac.at/adg/schwarz/bincat$_{-}$binary.html}. This publication makes use of the SIMBAD database operated at the CDS, Strasbourg, France. This work has made use of data from the European Space Agency (ESA) mission Gaia ({https://www.cosmos.esa.int/gaia}), processed by the Gaia Data Processing and Analysis Consortium (DPAC, {https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. \end{acknowledgements} \vspace{5mm} \facilities{VLBA} \software{AIPS \citep[][]{greisen03}, astropy \citep{astropycol13,astropycol18}, corner\citep{foremanmackey16}, emcee \citep{foremanmackey13}, lmfit \citep[][]{newville20}, scipy \citep[][]{virtanen20}, matplotlib,\citep{hunter07}, numpy\citep{vanderwalt11}. PyAstronomy\citep{czesla19}. } ~~ ~~ ~~ {~~~~~~~~~~~~~~~~~~~~~{\bf ORCID iDs}} Salvador Curiel {https:/orcid.org/0000-0003-4576-0436} Gisela N. Ortiz-Le\'on {https:/orcid.org/0000-0002-2863-676X} Amy J. Mioduszewski {https:/orcid.org/0000-0002-2564-3104} Joel Sanchez-Bermudez {https:/orcid.org/0000-0002-9723-0421} ~~ ~~ ~~ \appendix \section{Variability of the radio emission}\label{sec:flares} In this section, we investigate the level of intra-epoch variability of the radio emission detected with the VLBA and how it affects the astrometry of GJ~896A. To that end, the real part of the $(u,v)$ data was plotted as a function of time, averaged over the $(u,v)$ plane. At twelve epochs, we found short-timescale ($\sim$7--40 min) variability or flares in the light curves of the real part of the visibility. During these flares, the peak flux density of the source increases by about 1.2 to 6 times above the mean flux density. From these flux curves we determine the time interval where the flare is observed, and construct maps using only the time range of the flare. Using these images, we then obtain the position of the flare with the AIPS task {\tt MAXFIT}. We also construct maps using only the time range outside the flare (which we will call ``quiescent'' emission) and measure the source position. Panel (a) in Figure \ref{fig_flares} shows residuals obtained for GJ~896A after removing parallax, proper motion and orbital motion due to the stellar companion GJ~896B from the full combined astrometric fit solution (see Section~\ref{sec:caf} and Figure \ref{fig_7}a). Here, it is important to note that the positions of GJ~896A reported in Table~\ref{tab_1} were obtained using the maps constructed for the full observing time at each epoch (typically 3--5 hours), which we will call the ``average'' source position. In the next three panels of Figure \ref{fig_flares}, we plot offsets between (b) the position of the flare and the source position without the flare, (c) the position of the flare and the average source position, and (d) the source position without the flare and the average source position. We see in panel (b) that at some epochs there are big offsets between the flare and the source position without the flare, i.e., the quiescent emission. These offsets have rms values of 0.22 and 0.16~mas, in R.A.\ and Dec., respectively. The plots in panel (d), on the other hand, suggest that when we average the data over the full observing time, the ``average'' position is dominated by the quiescent emission and not by the flare. In this case, the differences between the position of the quiescent emission and the average position are small (with rms values of 0.09~mas in both R.A.\ and Dec.) compared to the offsets observed during the flare events (see panel (b)). In addition, the differences plotted in panel (d) are also small compared to the residuals from the astrometric fit shown in panel (a), which correspond to the astrometric signal of the main star due to the planet. The rms of the differences plotted in panel (d) are 2.9 and 5.8 times smaller, in R.A.\ and Dec., respectively, than the rms of the residuals. We also note that these differences do not follow the same temporal trend as the residuals, which indicate that the residuals of the astrometric fit of our data cannot be induced by the flare activity. From this analysis, we conclude that the flares occurring at short-timescales do not affect the astrometry of GJ~896A. This is because the position of the radio emission obtained by averaging over the full observing time of each observation is dominated by quiescent emission. Furthermore, this indicates that the astrometric residuals have a different origin, such as the presence of one or more companions. The flare events observed in GJ~896A are short$-$ and long$-$duration bursts with time scales between 7 and 40 minutes. The radio emission shows right-circular polarization (RCP) during the outbursts, with a degree of polarization between 10$\%$ and 80$\%$, except one epoch where the radio emission shows left-circular polarization (LCP). These characteristics suggest that the origen of the busts may be associated to electron cyclotron maser emission (ECM). This tentative interpretation is consistent with the interpretation of the outbursts events observed in this magnetically active M dwarf star with the Jansky Very Large Array \citep[e.g.,][]{crosley18a, crosley18b, villadsen19}. However, there may also be multiple phenomena responsible for the short-duration bursts that we observed in GJ~896A. A detailed discussion of the possible origin of the observed radio flares is out of the scope of this paper and it will be presented elsewhere. \section{Posterior Sampling} \label{sec:mcmc} We used the open-source package $lmfit$ \citep[][]{newville20}, which includes several minimization algorithms to search for the best fit of observational data. In particular, we used the default Levenberg-Marquardt minimization algorithm, which uses a non-linear least-squares minimization method to fit the data. This gives an initial fit solution to the astrometric bidimensional data. {\tt lmfit} also includes a wrapper for the Markov Chain Monte Carlo (MCMC) package {\tt emcee} \citep[][]{foremanmackey13}. When fitting the combined astrometric data (absolute astrometry of both stars GJ~896A and GJ~896B and the relative astrometry of the binary system GJ~896AB), we weighted the data by the positional errors of both coordinates ($\alpha$ and $\delta$). We used 250 walkers and run the MCMC for 30000 steps with a 1000 step burn-in, at which point the chain length is over 50 times the integrated autocorrelation time. The fitted solution is listed in Table~\ref{tab_A1}, and Figure~\ref{fig_emcee} shows the correlation between the fitted parameters. The fitted solution is very similar to that obtained from the full combined astrometric fit (see column (2) in Table~\ref{tab_3}). The $\chi^2$ and reduced $\chi^2$ are also very similar to those obtained from the full combined astrometric fit. The residuals of the fit are shown in Figure~\ref{fig_emcee_resid}, which are very similar to those obtained with the non-linear least-squares and AGA algorithms (see Figure~\ref{fig_7}a). {} \begin{deluxetable}{cccccccc} \centering % \tablecaption{Properties of the VLBA detections. \label{tab_1} } \tablewidth{0pt} \tablehead{ \colhead{Julian day} & \colhead{$\alpha(J2000)$} & \colhead{$\sigma_\alpha$} & \colhead{$\delta(J2000)$} & \colhead{$\sigma_\delta$} & \colhead{Flux density} & \colhead{rms} \\ & \colhead{(h:m:s)}& \colhead{(s)} & \colhead{($^{\rm o}$:$'$:$''$)} & \colhead{($''$)} & \colhead{(mJy)} & \colhead{($\mu$Jy~beam$^{-1}$)} \\ \colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} & \colhead{(6)} & \colhead{(7)} } \startdata \sidehead{GJ~896A} \hline 2453818.30167 & 23:31:52.4337116 & 0.0000130 & +19:56:13.712431 & 0.000193 & 0.23 $\pm$ 0.06 & 37 \\ 2453821.29347 & 23:31:52.4345726 & 0.0000063 & +19:56:13.715392 & 0.000092 & 3.16 $\pm$ 0.16 & 63 \\ 2454482.38363 & 23:31:52.4967295 & 0.0000066 & +19:56:13.570044 & 0.000097 & 0.31 $\pm$ 0.02 & 11 \\ 2454679.86273 & 23:31:52.5343755 & 0.0000067 & +19:56:13.704862 & 0.000098 & 0.44 $\pm$ 0.04 & 18 \\ 2454792.59143 & 23:31:52.5301869 & 0.0000061 & +19:56:13.570540 & 0.000089 & 2.69 $\pm$ 0.08 & 40 \\ 2454978.01521 & 23:31:52.5703946 & 0.0000061 & +19:56:13.606475 & 0.000089 & 8.82 $\pm$ 0.24 & 131 \\ 2455100.68189 & 23:31:52.5719964 & 0.0000062 & +19:56:13.599894 & 0.000090 & 1.26 $\pm$ 0.04 & 24 \\ 2455226.34873 & 23:31:52.5809688 & 0.0000066 & +19:56:13.443962 & 0.000096 & 0.44 $\pm$ 0.04 & 20 \\ 2455466.68189 & 23:31:52.6126263 & 0.0000062 & +19:56:13.537918 & 0.000091 & 3.55 $\pm$ 0.16 & 71 \\ 2455591.34853 & 23:31:52.6215688 & 0.0000063 & +19:56:13.383583 & 0.000092 & 2.51 $\pm$ 0.14 & 90 \\ 2455730.96293 & 23:31:52.6547197 & 0.0000061 & +19:56:13.508659 & 0.000090 & 4.56 $\pm$ 0.13 & 49 \\ 2455829.69210 & 23:31:52.6534315 & 0.0000091 & +19:56:13.480797 & 0.000135 & 0.18 $\pm$ 0.03 & 17 \\ 2455879.54625 & 23:31:52.6518749 & 0.0000063 & +19:56:13.402746 & 0.000093 & 0.56 $\pm$ 0.03 & 17 \\ 2459072.90975 & 23:31:53.0249869 & 0.0000060 & +19:56:12.983043 & 0.000088 & 2.57 $\pm$ 0.04 & 17 \\ 2459105.81954 & 23:31:53.0231982 & 0.0000062 & +19:56:12.956925 & 0.000090 & 0.82 $\pm$ 0.03 & 13 \\ 2459135.73763 & 23:31:53.0214293 & 0.0000063 & +19:56:12.915578 & 0.000091 & 0.73 $\pm$ 0.04 & 16 \\ \hline \sidehead{GJ~896B} \hline 2459105.81954 & 23:31:53.3848609 & 0.0000076 & +19:56:14.482490 & 0.000112 & 0.18 $\pm$ 0.03 & 13 \\ 2459135.73763 & 23:31:53.3828962 & 0.0000103 & +19:56:14.447438 & 0.000154 & 0.11 $\pm$ 0.03 & 15 \\ \enddata \end{deluxetable} \startlongtable \begin{deluxetable}{lccc} \centering % \tabletypesize{\scriptsize} \tablewidth{0pt} \tablecolumns{4} \tablecaption{Absolute Astrometric Fits\tablenotemark{a} \label{tab_2}} \tablehead{ % \\ Parameters & Single-source & Single-source$+$acceleration & Single-companion % } \decimals \startdata & (1) & (2) & (3) \\ \hline % & & Fitted Parameters & \\ \hline % $\mu_{\alpha}$ (mas yr$^{-1}$) & 576.707 $\pm$ 0.014 & 574.171 $\pm$ 0.014 & 574.142 $\pm$ 0.019 \\ $\mu_{\delta}$ (mas yr$^{-1}$) & $-$59.973 $\pm$ 0.014 & $-$60.333 $\pm$ 0.014 & $-$60.354 $\pm$ 0.020 \\ $a_{\alpha}$ (mas yr$^{-2}$) & $...$ & 0.8893 $\pm$ 0.0034 & 0.8871 $\pm$ 0.0047 \\ $a_{\delta}$ (mas yr$^{-2}$) & $...$ & 0.1402 $\pm$ 0.0035 & 0.1400 $\pm$ 0.0049 \\ $\Pi$ (mas) & 161.136 $\pm$ 0.094 & 160.027 $\pm$ 0.094 & 159.83 $\pm$ 0.13 \\ $P$ (days) & $...$ & $...$ & 281.56 $\pm$ 1.67 \\ $T_{0}$ (Julian day) & $...$ & $...$ & 2,455,696.5 $\pm$ 10.4 \\ $e$ & $...$ & $...$ & 0.30 $\pm$ 0.11 \\ $\omega$ (deg) & $...$ & $...$ & 344.1 $\pm$ 13.1 \\ $\Omega$ (deg) & $...$ & $...$ & 47.7 $\pm$ 12.1 \\ $a_{A}$ (mas) & $...$ & $...$ & 0.52 $\pm$ 0.11 \\ $i$ (deg) & $...$ & $...$ & 66.0 $\pm$ 15.0 \\ \hline % & & Other Parameters & \\ \hline % $D$ (pc) & 6.2059 $\pm$ 0.0036 & 6.2489 $\pm$ 0.0037 & 6.2565 $\pm$ 0.0051 \\ $m_{Ab}$ ($M_\odot$)\tablenotemark{b} & $...$ & $...$ & 0.43814 (fixed) \\ $m_{A}$ ($M_\odot$) & $...$ & $...$ & 0.43589 $\pm$ 0.00047 \\ $m_{b}$ ($M_\odot$). & $...$ & $...$ & 0.00225 $\pm$ 0.00047 \\ $m_{b}$ ($M_{J}$) & $...$ & $...$ & 2.35 $\pm$ 0.49 \\ $a_{Ab}$ (au) & $...$ & $...$ & 0.6386 $\pm$ 0.0025 \\ $a_{A}$ (au) & $...$ & $...$ & 0.00328 $\pm$ 0.00070 \\ $a_{b}$ (au) & $...$ & $...$ & 0.6353 $\pm$ 0.0018 \\ $a_{b}$ (mas) & $...$ & $...$ & 101.53 $\pm$ 0.42 \\ $\Delta$$\alpha$ (mas)\tablenotemark{c} & 8.51 & 0.24 & 0.14 \\ $\Delta$$\delta$ (mas)\tablenotemark{c} & 1.51 & 0.24 & 0.13 \\ $\chi^2$, $\chi^2_{red}$\tablenotemark{d} & 105728.53, 4066.48 & 190.91, 7.95 & 57.43, 3.38 \\ BIC\tablenotemark{e} & 105,745.86 & 215.17 & 105.93 \\ AIC\tablenotemark{e} & 105,738.53 & 204.91 & 85.41 \\ \enddata \tablenotetext{a}{The parameters presented here were obtained with AGA. Very similar results were obtained with the least-squares fitting method. Column (1) contain the astrometric fit of the VLBA data without acceleration terms. Column (2) includes acceleration terms. Column (3) corresponds to the full astrometric fit, where a single planetary companion is included.} \tablenotetext{b}{The mass of GJ~896A, including possible companions, was taken from the combined fit presented in Table 3.} \tablenotetext{c}{The rms dispersion of the residual.} \tablenotetext{d}{$\chi^2$ and reduced $\chi^2$ of the astrometric fit. The residuals clearly improve when we include acceleration terms and the companion.} \tablenotetext{e}{Bayesian Information Criteria (BIC) and Akaike Information Criterion (AIC) are the criteria used to choose the best fitted model \citep[e.g.,][]{liddle07}. The smaller the BIC and AIC values, the better the model. } \end{deluxetable} \startlongtable \begin{deluxetable}{lccc} \centering \tabletypesize{\scriptsize} \tablewidth{0pt} \tablecolumns{4} \tablecaption{Relative and Combined Astrometry Fits\tablenotemark{a} \label{tab_3}} % \tablehead{ \\ \colhead{Parameter} & \colhead{Relative\tablenotemark{b}} & \colhead{Combined\tablenotemark{c}} & \colhead{Full Combined\tablenotemark{d}} % } \startdata & (1) & (2) & (3) \\ \hline % $\mu_{\alpha}$ (mas yr$^{-1}$) & ... & 571.515 $\pm$ 0.019 & 571.467 $\pm$ 0.023 \\ $\mu_{\delta}$ (mas yr$^{-1}$) &. ... & $-$37.750 $\pm$ 0.019 & $-$37.715 $\pm$ 0.023 \\ $\Pi$ (mas) & 159.88 (fixed) & 159.98 $\pm$ 0.14 & 159.88 $\pm$ 0.17 \\ \hline % & & Binary System & \\ \hline % $P_{AB}$ (days) & 96606.13 $\pm$ 2.24 & 83665.80 $\pm$ 1.64 & 83664.63 $\pm$ 1.98 \\ $T_{0}$$_{AB}$ (Julian day)\tablenotemark{e} & 2,504,877.56 $\pm$ 1.48 & 2,401,894.57 $\pm$ 1.00 & 2,401,891.34 $\pm$ 1.19 \\ $e_{AB}$ & 0.0 (fixed) & 0.108047 $\pm$ 0.000044 & 0.108047 $\pm$ 0.000053 \\ $\omega_{AB}(A)$ (deg) & $ ...$ & 127.1531 $\pm$ 0.0037 & 127.1416 $\pm$ 0.0045 \\ $\omega_{AB}(B)$ (deg) & 0.0 (fixed) & 307.1531 $\pm$ 0.0037 & 307.1416 $\pm$ 0.0045 \\ $\Omega_{AB}$ (deg) & 259.7083 $\pm$ 0.0054 & 255.0891 $\pm$ 0.0028 & 255.0919 $\pm$ 0.0034 \\ $a_{AB}(A)$ (mas) & ... & 1381.015 $\pm$ 0.069 & 1385.328 $\pm$ 0.083 \\ $a_{AB}(B)$ (mas) & ... & 3676.95 $\pm$ 0.35 & 3672.64 $\pm$ 0.42 \\ $a_{AB}$ (mas) & 5378.79 $\pm$ 0.51 & 5057.96 $\pm$ 0.36 & 5057.97 $\pm$ 0.43 \\ $i_{AB}$ (deg) & 130.6592 $\pm$ 0.0087 & 130.0664 $\pm$ 0.0085 & 130.065 $\pm$ 0.010 \\ $Q_{B/A}$ & ... & 0.375588 $\pm$ 0.000030 & 0.377202 $\pm$ 0.000037 \\ \hline % & & Companion & \\ \hline % $P_{Ab}$ (days) & ... & ... & 284.39 $\pm$ 1.47 \\ $T_{0}$$_{Ab}$ (Julian day) & ... & ... & 2,455,702.65 $\pm$ 17.26 \\ $e_{Ab}$ & ... & ... & 0.35 $\pm$ 0.19 \\ $\omega_{A}$ (deg) & ... & ... & 353.11 $\pm$ 11.81 \\ $\Omega_{Ab}$ (deg) & ... & ... & 45.62 $\pm$ 9.60 \\ $a_{A}$ (mas) & ... & ... & 0.51 $\pm$ 0.15 \\ $a_{b}$ (mas) & ... & ... & 102.27 $\pm$ 0.15 \\ $i_{Ab}$ (deg) & ... & ... & 69.20 $\pm$ 25.61 \\ \hline % & & Other Parameters & \\ \hline % $D$ ~(pc) & ... & 6.2506 $\pm$ 0.0055 & 6.2545 $\pm$ 0.0066 \\ $m_{AB} ~(M_\odot)$ & 0.544304 $\pm$ 0.000023 & 0.602253 $\pm$ 0.000020 & 0.603410 $\pm$ 0.000025 \\ $m_{AB}(A) ~(M_\odot)$\tablenotemark{f} & ... & 0.43782 $\pm$ 0.00054 & 0.43814 $\pm$ 0.00065 \\ $m_{AB}(B) ~(M_\odot)$ & ... & 0.16444 $\pm$ 0.00020 & 0.16527 $\pm$ 0.00025 \\ $a_{AB} ~(AU)$ & 33.64265 $\pm$ 0.00052 & 31.615 $\pm$ 0.028 & 31.635 $\pm$ 0.033 \\ $a_{AB}(A) ~(AU)$ & ... & 8.6322 $\pm$ 0.0075 & 8.6646 $\pm$ 0.0091 \\ $a_{AB}(B) ~(AU)$ & ... & 22.983 $\pm$ 0.020 & 22.971 $\pm$ 0.024 \\ $m_{A} ~(M_\odot)$\tablenotemark{g} & ... & ... & 0.43599 $\pm$ 0.00092 \\ $m_{b} ~(M_\odot)$ & ... & ... & 0.00216 $\pm$ 0.00064 \\ $m_{b} ~(J)$ & ... & ... & 2.26 $\pm$ 0.57 \\ $a_{Ab} ~(AU)$ & ... & ... & 0.64282 $\pm$ 0.00068 \\ $a_{A} ~(AU)$ & ... & ... & 0.00317 $\pm$ 0.00093 \\ $a_{b} ~(AU)$ & ... & ... & 0.63965 $\pm$ 0.00067 \\ $\Delta$$\alpha$$_{AB}$ ~(mas)\tablenotemark{h} & 123.11 & 89.60 & 89.60 \\ $\Delta$$\delta$$_{AB}$ ~(mas)\tablenotemark{h} & 89.14 & 74.28 & 74.28 \\ $\Delta$$\alpha$$_{A}$ ~(mas)\tablenotemark{h} & ... & 0.27 & 0.21 \\ $\Delta$$\delta$$_{A}$ ~(mas)\tablenotemark{h} & ... & 0.47 & 0.45 \\ $\Delta$$\alpha$$_{B}$ ~(mas)\tablenotemark{h} & ... & 0.20 & 0.11 \\ $\Delta$$\delta$$_{B}$ ~(mas)\tablenotemark{h} & ... & 0.12 & 0.042 \\ $\chi^2$, $\chi^2_{red}$\tablenotemark{i} & 980846.81, 6956.36 & 977729.63, 5785.38 & 977566.99, 6034.36 \\ \enddata \tablenotetext{a}{The parameters presented here were obtained with AGA. Very similar results were obtained with the least-squares fitting method. The subindex A, B and AB correspond to the main star (GJ~896A), the low mass stellar companion (GJ~896B) and the binary system (GJ~896AB), respectively.} \tablenotetext{b}{Relative astrometric fit of secondary star GJ~896B around the main star GJ~896A. In this case the parallax is fixed using the solution of the full combined astrometric fit (column 3).} \tablenotetext{c}{The combined astrometric fit is obtained by fitting simultaneously the relative astrometry of the binary system and the absolute astrometry of the main star GJ~896A and the secondary star GJ~896B (see text). All the free parameters are fitted simultaneously.} \tablenotetext{d}{The full combined astrometric fit is obtained by fitting simultaneously the relative astrometry of the binary system and the absolute astrometry of the two stars, including a companion associated to GJ~896A (see text). All the free parameters are fitted simultaneously.} \tablenotetext{e}{Time of the periastron passage. In the case of the relative astrometry (column 1), $T_{0AB}$ corresponds to the time of the passage through the ascending line of nodes of the orbit of the secondary star around the main star. In the case of the combined astrometry (columns 2 and 3), $T_{0AB}$ corresponds to the time of the periastron passage of the primary star.} \tablenotetext{f}{Dynamical mass of the star GJ~896A including any posible companions.} \tablenotetext{g}{Dynamical mass of the star GJ~896A removing the mass of the planetary companion.} \tablenotetext{h}{RMS dispersion of the residuals. The first two terms correspond to the residuals of the binary system part of the fit, and next two term correspond to the residuals of the absolute astrometry part of the fit of the main star, and the last two terms correspond to the residuals of the secondary star.} \tablenotetext{i}{$\chi^2$ and reduced $\chi^2$ of the astrometric fit. In all three cases the residuals of the relative astrometry dominates the residuals of the fit.} \end{deluxetable} \startlongtable \begin{deluxetable}{lc} \centering % \tabletypesize{\scriptsize} \tablewidth{0pt} \tablecolumns{2} \tablecaption{Mean values and 68$\%$ confidence intervals for the fitted parameters from the $lmfit$ analysis \label{tab_A1}} \tablehead{ % \\ Parameters & Fitted Parameters % } \decimals \startdata $\mu_{\alpha}$ (mas yr$^{-1}$) & 571.472 $\pm$ 0.018 \\ $\mu_{\delta}$ (mas yr$^{-1}$) & $-$37.681 $\pm$ 0.075 \\ $\Pi$ (mas) & 159.971 $\pm$ 0.036 \\ $P$ (days) & 83159.87 $\pm$ 209.51 \\ $T_{0}$ (Julian day) & 2,402,034.71 $\pm$ 147.15 \\ $e$ & 0.1132 $\pm$ 0.0020 \\ $\omega_{A}$ (deg) & 126.48 $\pm$ 0.11 \\ $\Omega_{A}$ (deg) & 254.855 $\pm$ 0.089 \\ $a_{A}$ (mas) & 1383.42 $\pm$ 5.26 \\ $i$ (deg) & 129.959 $\pm$ 0.015 \\ $Q(m_{B}/m_{A})$ & 0.3773 $\pm$ 0.0017 \\ $\chi^2$, $\chi^2_{red}$ & 977668.32, 5785.02 \\ \enddata \end{deluxetable}

Title: TOI-1452 b: SPIRou and TESS reveal a super-Earth in a temperate orbit transiting an M4 dwarf

Abstract: Exploring the properties of exoplanets near or inside the radius valley provides insights on the transition from the rocky super-Earths to the larger, hydrogen-rich atmosphere mini-Neptunes. Here, we report the discovery of TOI-1452 b, a transiting super-Earth ($R_{\rm p} = 1.67 \pm 0.07$ R$_{\oplus}$) in an 11.1--day temperate orbit ($T_{\rm eq} = 326 \pm 7$ K) around the primary member ($H = 10.0$, $T_{\rm eff} = 3185 \pm 50$ K) of a nearby visual binary M dwarf. The transits were first detected by TESS, then successfully isolated between the two $3.2^{\prime\prime}$ companions with ground-based photometry from OMM and MuSCAT3. The planetary nature of TOI-1452 b was established through high-precision velocimetry with the near-infrared SPIRou spectropolarimeter as part of the ongoing SPIRou Legacy Survey. The measured planetary mass ($4.8 \pm 1.3$ M$_{\oplus}$) and inferred bulk density ($5.6^{+1.8}_{-1.6}$ g/cm$^3$) is suggestive of a rocky core surrounded by a volatile-rich envelope. More quantitatively, the mass and radius of TOI-1452 b, combined with the stellar abundance of refractory elements (Fe, Mg and Si) measured by SPIRou, is consistent with a core mass fraction of $18\pm6$ % and a water mass fraction of $22^{+21}_{-13}$%. The water world candidate TOI-1452 b is a prime target for future atmospheric characterization with JWST, featuring a Transmission Spectroscopy Metric similar to other well-known temperate small planets such as LHS 1140 b and K2-18 b. The system is located near Webb's northern Continuous Viewing Zone, implying that is can be followed at almost any moment of the year.

https://export.arxiv.org/pdf/2208.06333

\title{TOI-1452\,b: SPIRou and TESS reveal a super-Earth in a temperate orbit transiting an M4 dwarf} \correspondingauthor{Charles Cadieux} \email{charles.cadieux.1@umontreal.ca} \suppressAffiliations \author[0000-0001-9291-5555]{Charles Cadieux} % \affiliation{Universit\'e de Montr\'eal, D\'epartement de Physique, IREX, Montr\'eal, QC H3C 3J7, Canada} \author[0000-0001-5485-4675]{Ren\'e Doyon} % \affiliation{Universit\'e de Montr\'eal, D\'epartement de Physique, IREX, Montr\'eal, QC H3C 3J7, Canada} \affiliation{Observatoire du Mont-M\'egantic, Universit\'e de Montr\'eal, Montr\'eal H3C 3J7, Canada} \author[0000-0002-9479-2744]{Mykhaylo Plotnykov} % \affiliation{Department of Physics, University of Toronto, Toronto, ON M5S 3H4, Canada} \author[0000-0001-5450-7067]{Guillaume H\'ebrard} % \affiliation{Institut d'astrophysique de Paris, UMR7095 CNRS, Sorbonne Universit\'e, 98 bis bd Arago, 75014 Paris, France} \author[0000-0003-0029-2835]{Farbod Jahandar} % \affiliation{Universit\'e de Montr\'eal, D\'epartement de Physique, IREX, Montr\'eal, QC H3C 3J7, Canada} \author[0000-0003-3506-5667]{\'Etienne Artigau} % \affiliation{Universit\'e de Montr\'eal, D\'epartement de Physique, IREX, Montr\'eal, QC H3C 3J7, Canada} \affiliation{Observatoire du Mont-M\'egantic, Universit\'e de Montr\'eal, Montr\'eal H3C 3J7, Canada} \author[0000-0003-3993-4030]{Diana Valencia} % \affiliation{Department of Physical \& Environmental Sciences, University of Toronto at Scarborough, Toronto, ON M1C 1A4, Canada} \affiliation{David A. Dunlap Dept.\ of Astronomy \& Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario, M5S 3H4, Canada} \author[0000-0003-4166-4121]{Neil J. Cook} % \affiliation{Universit\'e de Montr\'eal, D\'epartement de Physique, IREX, Montr\'eal, QC H3C 3J7, Canada} \author[0000-0002-5084-168X]{Eder Martioli} % \affiliation{Institut d'astrophysique de Paris, UMR7095 CNRS, Sorbonne Universit\'e, 98 bis bd Arago, 75014 Paris, France} \affiliation{Laborat\'orio Nacional de Astrof\'isica, Rua Estados Unidos 154, Itajub\'a, MG 37504-364, Brazil} \author[0000-0002-5922-8267]{Thomas Vandal} % \affiliation{Universit\'e de Montr\'eal, D\'epartement de Physique, IREX, Montr\'eal, QC H3C 3J7, Canada} \author[0000-0001-5541-2887]{Jean-Fran\c cois Donati} % \affiliation{Universit\'e de Toulouse, CNRS, IRAP, 14 Avenue Belin, 31400 Toulouse, France} \author[0000-0001-5383-9393]{Ryan Cloutier} % \altaffiliation{Banting Fellow} \affiliation{Center for Astrophysics $\vert{}$ Harvard \& Smithsonian, 60 Garden Street, Cambridge, MA, 02138, USA} \author[0000-0001-8511-2981]{Norio Narita} % \affiliation{Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan} \affiliation{Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan} \affiliation{Instituto de Astrof\'{i}sica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain} \author[0000-0002-4909-5763]{Akihiko Fukui} % \affiliation{Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan} \affiliation{Instituto de Astrof\'{i}sica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain} \author[0000-0003-3618-7535]{Teruyuki Hirano} % \affiliation{Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan} \affiliation{National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan} \author[0000-0002-7613-393X]{Fran\c cois Bouchy} % \affiliation{Departement d’astronomie, Universit\'e de Gen\`eve, Chemin Pegasi, 51, CH-1290 Versoix, Switzerland} \author[0000-0001-6129-5699]{Nicolas B. Cowan} % \affiliation{Department of Earth \& Planetary Sciences, McGill University, 3450 rue University, Montréal, QC H3A 0E8, Canada} \affiliation{Department of Physics, McGill University, 3600 rue University, Montréal, QC H3A 2T8, Canada} \author[0000-0002-9329-2190]{Erica J. Gonzales} % \affiliation{University of California Santa Cruz, Santa Cruz CA 95065, USA} \author[0000-0002-5741-3047]{David R. Ciardi} % \affiliation{NASA Exoplanet Science Institute-Caltech/IPAC, Pasadena, CA 91125 USA} \author[0000-0002-3481-9052]{Keivan G.\ Stassun} % \affiliation{Department of Physics and Astronomy, Vanderbilt University, 6301 Stevenson Center Ln., Nashville, TN 37235, USA} \affiliation{Department of Physics, Fisk University, 1000 17th Avenue North, Nashville, TN 37208, USA} \author[0000-0002-0111-1234]{Luc Arnold} % \affiliation{Canada-France-Hawaii Telescope, CNRS, Kamuela, HI 96743, USA} \author[0000-0001-5578-1498]{Bj\"orn Benneke} % \affiliation{Universit\'e de Montr\'eal, D\'epartement de Physique, IREX, Montr\'eal, QC H3C 3J7, Canada} \author[0000-0002-1024-9841]{Isabelle Boisse} % \affiliation{Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France} \author[0000-0001-9003-8894]{Xavier Bonfils} % \affiliation{Univ.\ Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France} \author[0000-0003-2471-1299]{Andr\'es Carmona} % \affiliation{Univ.\ Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France} \author[0000-0002-6174-4666]{P\'ia Cort\'es-Zuleta} % \affiliation{Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France} \author[0000-0001-5099-7978]{Xavier Delfosse} % \affiliation{Univ.\ Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France} \author[0000-0003-0536-4607]{Thierry Forveille} % \affiliation{Univ.\ Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France} \author[0000-0002-1436-7351]{Pascal Fouqu\'e} % \affiliation{Canada-France-Hawaii Telescope, CNRS, Kamuela, HI 96743, USA} \affiliation{Universit\'e de Toulouse, CNRS, IRAP, 14 Avenue Belin, 31400 Toulouse, France} \author[0000-0001-8056-9202]{Jo\~ao Gomes da Silva} % \affiliation{Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762, Porto, Portugal} \author[0000-0002-4715-9460]{Jon~M.~Jenkins} % \affiliation{NASA Ames Research Center, Moffett Field, CA 94035, USA} \author[0000-0001-9129-4929]{Flavien Kiefer} % \affiliation{Institut d'astrophysique de Paris, UMR7095 CNRS, Sorbonne Universit\'e, 98 bis bd Arago, 75014 Paris, France} \author[0000-0001-7157-6275]{\'Agnes K\'osp\'al} % \affiliation{Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, E\"otv\"os Lor\'and Research Network (ELKH), Konkoly-Thege Mikl\'os \'ut 15-17, 1121 Budapest, Hungary} \affiliation{Max Planck Institute for Astronomy, K\"onigstuhl 17, 69117 Heidelberg, Germany} \affiliation{ELTE E\"otv\"os Lor\'and University, Institute of Physics, P\'azm\'any P\'eter s\'et\'any 1/A, 1117 Budapest, Hungary} \author[0000-0002-6780-4252]{David Lafreni\`ere} % \affiliation{Universit\'e de Montr\'eal, D\'epartement de Physique, IREX, Montr\'eal, QC H3C 3J7, Canada} \author[0000-0002-1532-9082]{Jorge H. C. Martins} % \affiliation{Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762, Porto, Portugal} \author[0000-0002-2842-3924]{Claire Moutou} % \affiliation{Universit\'e de Toulouse, CNRS, IRAP, 14 Avenue Belin, 31400 Toulouse, France} \author[0000-0001-7804-2145]{J.-D.~do~Nascimento,~Jr.} % \affiliation{Universidade Federal do Rio Grande do Norte (UFRN), Departamento de F\'isica, 59078-970, Natal, RN, Brazil} \affiliation{Center for Astrophysics $\vert{}$ Harvard \& Smithsonian, 60 Garden Street, Cambridge, MA, 02138, USA} \author{Merwan Ould-Elhkim} % \affiliation{Universit\'e de Toulouse, CNRS, IRAP, 14 Avenue Belin, 31400 Toulouse, France} \author[0000-0002-8573-805X]{Stefan Pelletier} % \affiliation{Universit\'e de Montr\'eal, D\'epartement de Physique, IREX, Montr\'eal, QC H3C 3J7, Canada} \author[0000-0002-6778-7552]{Joseph D. Twicken} % \affiliation{SETI Institute, Mountain View, CA 94043, USA} \affiliation{NASA Ames Research Center, Moffett Field, CA 94035, USA} \author[0000-0002-0514-5538]{Luke~G.~Bouma} % \affiliation{Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA} \author{Scott~Cartwright} % \affiliation{Proto-Logic Consulting LLC, Washington DC, 20009 USA} \author[0000-0002-7786-0661]{Antoine Darveau-Bernier} % \affiliation{Universit\'e de Montr\'eal, D\'epartement de Physique, IREX, Montr\'eal, QC H3C 3J7, Canada} \author[0000-0001-5707-8448]{Konstantin Grankin} % \affiliation{Crimean Astrophysical Observatory, Department of Stellar Physics, Nauchny, 298409, Crimea} \author[0000-0002-5658-5971]{Masahiro Ikoma} % \affiliation{National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan} \author[0000-0002-5331-6637]{Taiki Kagetani} % \affiliation{Department of Multi-Disciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan} \author[0000-0003-1205-5108]{Kiyoe Kawauchi} % \affiliation{Instituto de Astrof\'{i}sica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain} \affiliation{Departamento de Astrof\'{i}sica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain} \author[0000-0001-9032-5826]{Takanori Kodama} % \affiliation{Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan} \author[0000-0001-6181-3142]{Takayuki Kotani} % \affiliation{Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan} \affiliation{National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan} \affiliation{Department of Astronomy, School of Science, The Graduate University for Advanced Studies (SOKENDAI), 2-21-1Osawa, Mitaka, Tokyo, Japan} \author[0000-0001-9911-7388]{David~W.~Latham} % \affiliation{Center for Astrophysics $\vert{}$ Harvard \& Smithsonian, 60 Garden Street, Cambridge, MA, 02138, USA} \author{Kristen Menou} % \affiliation{Department of Physical \& Environmental Sciences, University of Toronto at Scarborough, Toronto, ON M1C 1A4, Canada} \affiliation{David A. Dunlap Dept.\ of Astronomy \& Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario, M5S 3H4, Canada} \affiliation{Department of Physics, University of Toronto, Toronto, ON M5S 3H4, Canada} \author[0000-0003-2058-6662]{George~Ricker} % \affiliation{MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA} \affiliation{MIT Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA} \author[0000-0002-6892-6948]{Sara~Seager} % \affiliation{MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA} \affiliation{Earth and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA} \affiliation{Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA} \author[0000-0002-6510-0681]{Motohide Tamura} % \affiliation{Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan} \affiliation{Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan} \affiliation{National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan} \author[0000-0001-6763-6562]{Roland~Vanderspek} \affiliation{MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA} \author[0000-0002-7522-8195]{Noriharu Watanabe} \affiliation{Department of Multi-Disciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan} \section{Introduction} \label{sec:intro} Over the past decade, it has become increasingly clear that the typical extrasolar planetary system is quite different from our Solar System. Exoplanets are usually found in a much more compact orbital configuration \citep{Howard_2010} and the majority of systems have at least one planet with a size intermediate between the Earth and Neptune (\citealt{Howard_2012}; \citealt{Fressin_2013}). Population studies based on the \textit{Kepler} sample have shown that the occurrence rate distribution of close-in ($P < 100$\,days) exoplanets displays a valley/gap near 1.5--2.0\,R$_{\oplus}$ (\citealt{Fulton_2017}; \citealt{Fulton-Petigura_2018}; \citealt{Mayo_2018}; \citealt{Hardegree-Ullman_2020}). This radius valley most likely separates scaled-up, rocky versions of the Earth (super-Earths) and hydrogen-rich planets reminiscent of Neptune, but smaller (mini-Neptunes). This transition is known to be period-dependent (\citealt{VanEylen_2018}; \citealt{Martinez_2019}) and to vary with the host star properties such as metallicity (\citealt{Petigura_2018}; \citealt{Owen_2018}), mass (\citealt{McDonald_2019}; \citealt{Cloutier-Menou_2020}), and age (\citealt{Berger_2020}; \citealt{David_2021}). The existence of a radius valley was rapidly attributed to total or partial photoevaporation of the atmosphere by highly energetic photons during the first 100 Myr, when the host star is more active (\citealt{Owen_2013}, \citeyear{Owen_2017}; \citealt{Lopez_2014}; \citealt{Lopez_2018}; \citealt{Wu_2019}). However, another atmospheric erosion mechanism is plausible, involving mass loss caused by the release of energy from the planet core, accumulated during formation and slowly cooling down over Gyr timescales (\citealt{Ginzburg_2018}; \citealt{Gupta_2019}, \citeyear{Gupta_2020}). More recently, \cite{Lee_2021} have shown that the radius valley can be sculpted as a feature of formation, involving gas-poor accretion and supporting the hypothesis of a primordial bimodal distribution, rather than the result of subsequent atmospheric erosion. In order to identify which mechanism dominates, \cite{Rogers_2021} predict that the number of well-characterized small exoplanets must reach $\gtrsim 5000$. Such characterization requires the precise knowledge of planetary radii ($\lesssim5\%$ uncertainty) and, if possible, the planet mass. The combination of the two measurements leads to the mean density of the objects, a way to determine whether their internal structure is compatible with a rocky, gaseous, or intermediate bulk composition. Identifying new small planets transiting nearby bright stars is the primary objective of the ongoing NASA Transiting Exoplanet Survey Satellite (TESS) mission \citep{Ricker_2015}. In operation since 2018, TESS has observed 85\% of the celestial sphere, staring for at least $\sim$27\,days at over 50 sectors covered so far ($24\arcdeg\times 96\arcdeg$ per sector). The TESS survey has already unveiled more than 5000 candidate exoplanets of which more than two hundred have been confirmed as new transiting planetary systems, including small planets around M-dwarf hosts (e.g., TOI-270, \citealt{Gunther_2019}; LP 791-18, \citealt{Crossfield_2019}; L~98-59, \citealt{Cloutier_2019}; LTT~1445\,A, \citealt{Winters_2019}; LTT~3780, \citealt{Cloutier_2020a}; TOI-1235, \citealt{Cloutier_2020b}; TOI-700, \citealt{Gilbert_2020}; TOI-1266, \citealt{Demory_2020}; LP 714-47, \citealt{Dreizler_2020}; TOI-776, \citealt{Luque_2021}). M dwarfs represent prime targets not only for TESS, but in exoplanetary science in general. They are the most abundant stars in the solar neighborhood \citep{Reyle_2021} and host on average 2.5$\pm$0.2 planets per M dwarf with radii 1--4 R$_\oplus$ \citep{Dressing_2015}. Their smaller size compared to Sun-like stars facilitate the detection and characterization of new exoplanets by producing deeper transits for planets of a given size. The larger planet-to-star mass ratio amplifies the planetary radial velocity (RV) signal, allowing easier mass determination. Lastly, their lower luminosity results in a closer-in Habitable Zone (HZ), with orbital periods typically of one or two weeks adequately sampled by TESS. The James Webb Space Telescope (JWST) is poised to revolutionize the field of exoplanet atmospheres \citep{Bean_2018} by offering a collecting area more than 6 times larger than the Hubble Space Telescope (HST) and spectral coverage from the visible to the mid-infrared (0.6--28\,$\mu$m). JWST will allow simultaneous identification of many chemical species with large absorption cross section in the infrared (e.g., H$_2$O, CH$_4$, CO, CO$_2$, NH$_3$), as well as probe the atmosphere of terrestrial planets with unprecedented sensitivity. One key objective of TESS is to discover the best transiting exoplanets amenable for atmospheric characterization with JWST \citep{Kempton_2018}. Here, we report the discovery of a new small exoplanet around the nearby M dwarf TOI-1452. The planet was first detected by TESS, then characterized via follow-up efforts including RV monitoring with the SPIRou spectropolarimeter. The complete set of observations is described in Section~\ref{sec:obs}. The host star properties and physical parameters are derived in Section~\ref{sec:stellar_char}. Our data analysis and results are presented in Section~\ref{sec:analysis}. The implications of this discovery and prospects for follow-up characterization are discussed in Section~\ref{sec:discussion}, followed by concluding remarks in Section~\ref{sec:conclusion}. \section{Observations} \label{sec:obs} \subsection{TESS photometry} \label{sec:phot_tess} TOI-1452 (TIC 420112589) was observed by TESS in sectors 14 through 26 (except 18), thus almost continuously from July 18, 2019 to July 4, 2020, in sectors 40--41 from June 25 to August 20, 2021, and finally in sector 47 from December 31, 2021 to January 27, 2022 (details in Table~\ref{table:tessobs}). TOI-1452 was sampled at the TESS 2-minute ``short'' cadence, as the star is part of the Cool Dwarf List \citep{Muirhead_2018}, a specially curated list of high-priority late-K and M dwarfs added to the TESS Input Catalog (TIC, \citealt{Stassun_2018b}, \citeyear{Stassun_2019}). We used the publicly available\footnote{Mikulski Archive for Space Telescopes (MAST): \href{https://archive.stsci.edu/tess/}{\texttt{archive.stsci.edu/tess/}}} per-sector light curves produced by the TESS Science Processing Operations Center (SPOC, \citealt{Jenkins_2016}) at NASA Ames, more specifically their Presearch Data Conditioning Simple Aperture Photometry (\texttt{PDCSAP}, \citealt{Smith_2012}; \citealt{Stumpe_2012}, \citeyear{Stumpe_2014}). The \texttt{PDCSAP} light curves are corrected for both instrumental systematic trends seen across stars in the same sector/camera/CCD and for flux contamination from nearby stars located within a few TESS pixels (21$\arcsec$). Flux dilution reduces the observable transit depth, resulting in an underestimation of the planetary radius if not accounted for. This was particularly important for TOI-1452 because a companion star (TIC 420112587, see Sect.\ \ref{sec:bound}) is separated by only $3\farcs2$ and has a similar magnitude in the TESS band ($\Delta T = 0.204$). A new background correction was implemented for the TESS extended mission (starting with sector 27). We followed the procedure outlined in the TESS Data Release 38 notes\footnote{\href{https://archive.stsci.edu/tess/tess_drn.html}{\texttt{archive.stsci.edu/tess/tess\_drn.html}}} to correct our \texttt{PDCSAP} fluxes from the primary mission (sectors 14--26), adjusting the baseline level and reducing the inferred transit depth by $\sim$1.7\%. This ensures that the primary and extended mission data produce the same estimate of the planetary radius. Figure~\ref{fig:TESS_fov} shows a 11$\times$11 pixels sub-region around TOI-1452 from TESS sector 14 and the same region of the sky observed from the ground. This illustrates how TESS alone cannot resolve the source of a transit between TOI-1452 and the nearby companion TIC 420112587. The normalized \texttt{PDCSAP} light curve of TOI-1452 from sectors 14 and 21 is presented in Figure~\ref{fig:TESS_lc_phase}, while the remaining sectors are shown in Figure~\ref{fig:TESS_lc_complete}. \begin{deluxetable}{ccccc} \tablecaption{TESS observations of TOI-1452} \tablehead{ \colhead{Sector} & \colhead{Camera} & \colhead{CCD} & \colhead{UT Start Date} & \colhead{UT End Date} } \startdata 14 & 3 & 2 & 2019-07-18 & 2019-08-14\\ 15 & 3 & 2 & 2019-08-15 & 2019-09-10\\ 16 & 2 & 1 & 2019-09-12 & 2019-10-06\\ 17 & 4 & 2 & 2019-10-08 & 2019-11-02\\ 19 & 4 & 1 & 2019-11-28 & 2019-12-23\\ 20 & 4 & 1 & 2019-12-24 & 2019-01-20\\ 21 & 4 & 1 & 2020-01-21 & 2020-02-18\\ 22 & 4 & 4 & 2020-02-19 & 2020-03-17\\ 23 & 4 & 4 & 2020-03-19 & 2020-04-15\\ 24 & 3 & 4 & 2020-04-16 & 2020-05-12\\ 25 & 3 & 3 & 2020-05-14 & 2020-06-08\\ 26 & 3 & 3 & 2020-06-09 & 2020-07-04\\ 40 & 3 & 2 & 2021-06-25 & 2021-07-23\\ 41 & 3 & 2 & 2021-07-24 & 2021-08-20\\ 47 & 4 & 1 & 2021-12-31 & 2022-01-27 \enddata \tablenocomments{} \label{table:tessobs} \end{deluxetable} A search of the sectors 14--16 with an adaptive, wavelet-based matched filter (\citealt{Jenkins_2002}; \citealt{Jenkins_2010}, \citealt{Jenkins_2020}) first identified transit signatures for TOI-1452. The Data Validation Reports (DVR; \citealt{Twicken_2018}; \citealt{Li_2019}) fitted a limb-darkened transit model with a signal-to-noise ratio (SNR) of 8.0, a period of 11.06409 days, and an average uncontaminated transit depth of 3.778 parts per thousand (ppt), corresponding to a preliminary planetary radius of $1.83 \pm 0.30$\,R$_{\oplus}$. This led to the announcement of the planet candidate TOI-1452.01 \citep{Guerrero_2021} by the TESS Science Office on October 26, 2019. Simultaneously, the TESS mission announced the candidate TOI-1760.01 around the companion star TIC 420112587 sharing the same ephemeris as TOI-1452.01. Ground-based photometry was able to isolate the transit signal, originating from TOI-1452 (see Sect.\ \ref{sec:psf}). The latest available DVR from sectors 14--41 includes 30 transits and reports a period of 11.06196 days along with a radius $R_{\rm p} = 1.60 \pm 0.42$\,R$_{\oplus}$. Our complete reanalysis of the TESS light curve presented in Section~\ref{sec:analysis} has resulted in a more precise planetary radius, in agreement with previous estimates. Figure~\ref{fig:TESS_lc_phase} shows the phase-folded 32 transits from currently available sectors. We note that TESS is expected to continue monitoring TOI-1452 during 2022. \subsection{OMM-PESTO transit monitoring} \label{sec:phot_ground} Due to the coarse image sampling of TESS (21\arcsec \ per pixel), the origin of a transit signal may be ambiguous when several stars are located inside the aperture (e.g., Fig.\ \ref{fig:TESS_fov}). For this reason, TESS planet candidates are prone to false positives, occasionally attributed to a nearby eclipsing binary (NEB) contaminating the light curve \citep{Sullivan_2015}. Ground-based follow-up with arc-second angular resolution is therefore necessary to validate on-target transit and reject the NEB scenario. For TOI-1452, a particular challenge was to determine the signal's provenance between the target and its $3\farcs2$ companion. Two transit events of TOI-1452.01 were observed using the PESTO camera installed on the 1.6\,m telescope of the Observatoire du Mont-Mégantic (OMM), Québec, Canada. PESTO features a $1024\times 1024$ pixel \hbox{EMCCD} detector with a pixel scale of $0\farcs466$, providing a field of view (FOV) of $7\farcm95\times7\farcm95$. We scheduled the two observing sequences with the \texttt{TESS Transit Finder} (\texttt{TTF}), a customized version of the \texttt{Tapir} software package \citep{Jensen_2013}, and have used \texttt{AstroImageJ} (\texttt{AIJ}; \citealt{Collins_2017}) to perform image calibrations, including bias subtractions and flat field corrections, and differential aperture photometry. A first full transit was observed on February 22, 2020 in the $i^{\prime}$ filter with a sequence of 30\,s exposure time. As seen in the lower panel of Figure~\ref{fig:TESS_fov}, \hbox{TOI-1452} and TIC\,420112587 were partially resolved. Using a circular aperture of 7$\farcs$9 containing both stars, the transit was detected 53\,min earlier than predicted by the \texttt{TTF} (2.2$\sigma$ early), causing us to miss observing a proper pre-ingress baseline (see Fig.\ \ref{fig:OMM-PESTO_lc}, upper left). Additional TESS data later confirmed that the period was slightly overestimated by SPOC (sectors 14--16 only) at the time of observations, explaining why the transit arrived ``early''. The transit timing was also later confirmed by TESS sector 22 data, which was contemporaneous to this dataset. Even without a proper baseline, this transit was particularly valuable because it allowed us to reject the NEB false positive scenario and to determine, using point spread function (PSF) fitting (see Sect.\ \ref{sec:psf}), that the signal originated from TOI-1452. A second full transit of TOI-1452.01 was observed on March 4, 2021 in $i^{\prime}$, using a 10\,s exposure time sequence. With a combined aperture of 8$\farcs$4, the transit was detected on time according to the \texttt{TTF}. In addition to a standard airmass linear detrending (also performed for the first transit), we used the Width (mean of the x- and y-direction FWHM) detrending option in \texttt{AIJ}. This was necessary to account for flux loss when the seeing was worse for certain exposures in the sequence, without increasing the aperture radius and dealing with flux contamination from a third star. This made sure that the transit depth was consistent with the one derived from the first PESTO observation, when the overall seeing was better (see Table~\ref{table:gbobs}). The OMM-PESTO observations are summarized in Table~\ref{table:gbobs}. The resulting aperture photometry transits are shown in Figure~\ref{fig:OMM-PESTO_lc}, and were included with the 32 TESS transits in our joint analysis (transit and RV datasets) presented in Section~\ref{sec:jointfit}. \subsection{MuSCAT3 transit monitoring} \label{sec:phot_ground_muscat} A full transit of TOI-1452.01 was observed on September 8, 2021 with the multiband imager MuSCAT3 \citep{2020SPIE11447E..5KN} on the 2\,m Faulkes Telescope North (FTN) of Las Cumbres Observatory (LCO) at Haleakala observatory, Hawaii. MuSCAT3 has four optical channels, each of which is equipped with a 2k $\times$ 2k CCD camera with a pixel scale of 0\farcs266 pixel$^{-1}$, enabling $g^{\prime}$-, $r^{\prime}$-, $i^{\prime}$-, and $z_{\rm s}$-band simultaneous imaging. For transit monitoring, significant chromaticity in the transit depths could indicate a diluted eclipsing binary. The exposure times were set at 35, 12, 6, and 3\,s for the $g^{\prime}$, $r^{\prime}$, $i^{\prime}$, and $z_{\rm s}$ bands, respectively. The observations were performed in-focus to spatially resolve the host star from the nearby companion star at $3\farcs2$, resulting in the FWHM of stellar point spread function of 3--8 pixels ($0\farcs8$--$2\farcs0$) depending on the airmass and band (see Table~\ref{table:gbobs}). The obtained images were calibrated by the {\tt BANZAI} pipeline \citep{curtis_mccully_2018_1257560}. We performed aperture photometry on the calibrated images using a custom pipeline \citep{2011PASJ...63..287F} with aperture radii of 5 pixels, or $1\farcs3$, for all bands, which is almost free from flux contamination from the nearby companion star. For each band, we extracted the light curve using different sets of comparison stars, but have found that using only the companion (TIC 420112587) as a reference produced the minimum point-to-point dispersion. Since both the companion and the target stars have a similar color and are close to each other, the attenuation by the atmosphere is almost identical, so that we can safely assume that any second-order extinction effect (airmass dependent) is almost negligible. We clearly detected the transit on the target star in all bands, as shown in Figure~\ref{fig:MuSCAT3_lc}, providing further unambiguous evidence that TOI-1452 hosts a transiting object. A summary of this dataset is provided in Table~\ref{table:gbobs}. \begin{deluxetable}{ccccc} \tablecaption{Summary of the ground-based transit monitoring of TOI-1452} \tablehead{ \colhead{UT Date} & \colhead{Camera} & \colhead{Filter} & \colhead{PSF} & \colhead{Aperture}\\[-0.1cm] & & & FWHM ($^{\prime \prime}$) & Size ($^{\prime \prime}$) } \startdata \multicolumn{5}{c}{\textit{OMM 1.6\,m}}\\ 2020-02-22 & PESTO & $i^{\prime}$ & 2.9 & 7.9$^\dag$\\ 2021-03-04 & PESTO & $i^{\prime}$ & 3.9 & 8.4$^\dag$\\ \hline \multicolumn{5}{c}{\textit{LCO-FTN 2\,m}}\\ 2021-09-08 & MuSCAT3 & $g^{\prime}$ & 1.8 & 1.3\\ 2021-09-08 & MuSCAT3 & $r^{\prime}$ & 1.3 & 1.3\\ 2021-09-08 & MuSCAT3 & $i^{\prime}$ & 1.3 & 1.3\\ 2021-09-08 & MuSCAT3 & $z_{\rm s}$ & 1.0 & 1.3\\ \enddata \tablecomments{$^\dag$Using an aperture containing TOI-1452 and TIC 420112587} \label{table:gbobs} \end{deluxetable} \subsection{Keck\,II/NIRC2 high-resolution imaging} \label{sec:imaging} One or more unresolved sources not in \textit{Gaia} EDR3 could still be located close to TOI-1452, whether gravitationally bound or not. A blended eclipsing binary (BEB) could indicate a false positive detection, and any other flux source would lead to underestimate the size of the transiting object in the TESS, PESTO and MuSCAT3 light curves. For these reasons, we searched for sub-arcsecond sources around TOI-1452 with the NIRC2 adaptive optics imaging camera installed on the 10\,m Keck\,II telescope. The images were acquired on May 28, 2020 in the $K$ band with a spatial resolution of 0$\farcs$01 per pixel, integration time per coadd of 1.6\,s, mean PSF FWHM of 0$\farcs$061, and airmass of 1.69. Figure~\ref{fig:Keck_imaging} shows the 5$\sigma$ contrast curve of TOI-1452, revealing that no additional companion is detected with a contrast ratio $\Delta K \leq 5.429$ for separation greater than $0\farcs5$. Although in the $K$ band, this contrast limit is similar to the difference in magnitude required ($\Delta T = 5.55$) for a 50\% depth BEB to mimic a 3\,ppt transit in the TESS light curve. Following the procedure of \cite{Lillo-Box_2014}, we calculated the probability of contamination from a blended source due to a random alignment inside $0\farcs5$. For this, we simulated the galactic stellar population in a region near the target with \texttt{TRILEGAL} \citep{Girardi_2012}, using their default bulge, halo, disk (thin and thick) parameters and the log-normal initial mass function of \cite{Chabrier_2001}. The probability of an undetected source with $\Delta K \leq 5.55$ inside $0\farcs5$ is less than 0.04\%, so we can safely assume that the transit signal is not produced by a BEB or significantly diluted by a background star. \subsection{SPIRou velocimetry} \label{sec:nIR_rv} TOI-1452 was observed at 53 epochs from June 4, 2020 to October 8, 2020 with the near-infrared (0.98--2.5\,$\mu$m) SPIRou spectropolarimeter (\citealt{Donati_2018}; \citeyear{Donati_2020}) mounted on the 3.6\,m Canada-France-Hawaii telescope (CFHT). The observations were conducted as part of the ongoing SPIRou Legacy Survey (SLS; \citealt{Donati_2020}), more precisely its Transit Follow-up program (SLS-WP2), which aims to characterize exoplanets orbiting low mass stars revealed by photometric surveys such as TESS. SLS-WP2 has thus far allowed the characterization of the brown dwarf TOI-1278\,B \citep{Artigau_2021}, the sub-Neptune TOI-1759\,b \citep{Martioli_2022} transiting M dwarfs, and the studies of the transiting planets HD 189733\,b (\citealt{Moutou_2020}; \citealt{Boucher_2021}) and AU Mic\,b \citep{Martioli_2020}. SPIRou offers simultaneous high resolution spectroscopy and polarimetry, with a spectral resolving power $R \sim 70\,000$. Each epoch measurement consisted of four consecutive 15-min exposures, i.e., a polarimetric sequence, with two rotating Fresnel rhombs varying positions between the exposures. During such a sequence, the two science fibers, A and B, each receive orthogonal polarization states, giving access to the circular polarization and total intensity of the light beam (Stokes V and I). A total of 212 spectra were collected, with SNR per spectral element ($\sim$2.2\,km/s/pixel for SPIRou) between 20 and 65 (median of 55) near 1.6\,$\mu$m. Four individual spectra were rejected; one due to loss of guiding, three others because of high extinction (clouds). A single polarimetric sequence of four 15-min exposures was also acquired on the $3\farcs2$ companion (TIC 420112587) on April 22, 2021, principally to check its rotation profile and magnetic activity level (see Section~\ref{sec:polarimetry}). The diameter of the SPIRou fiber is $1\farcs33$ and the typical seeing during the observations of TOI-1452 was $0\farcs8$. We measure no correlation between the radial velocity residuals (Keplerian and activity models described in Sect.\ \ref{sec:jointfit}) and the seeing, suggesting that any effect of contamination from the companion was negligible. Per-epoch RV measurements consisted of taking the error weighted mean of the individual observations within a polarimetric sequence. The data analysis presented in Section~\ref{sec:jointfit} was performed on the unbinned RVs, but we show the per-epoch average to facilitate visualization. The SPIRou data were reduced with \texttt{APERO} v0.7.194 (Cook et al., in prep.). In brief, \texttt{APERO} starts by correcting known H4RG infrared detector defects \citep{Artigau_2018}, then proceeds to identify bad pixels, locate each spectral order on the image, calculate the shape of the instrument pupil slicer \citep{Micheau_2018}, and finally determine from nightly calibration sequences the flat and blaze corrections to apply. Once this preprocessing and calibration step is completed, \texttt{APERO} performs an optimal flux extraction \citep{Horne_1986} in both science channels, separately (fibers A and B) and together (AB), as well as in the simultaneous calibration channel (fiber C). The extracted 2D images (49 orders by 4088 pixels) are then spectral flat fielded, as well as corrected for thermal background and for any leakage from the calibration channel to the science ones. A nightly pixel-to-wavelength solution is applied using a combination of a UNe hollow-cathode lamp and a Fabry-Perot (FP), as described in \cite{Hobson_2021}. \texttt{APERO} uses the simultaneous FP measurements from fiber C to calculate drifts between individual science frames relative to the nightly wavelength solution (typically below 2\,m/s). Finally, a telluric absorption and night-sky emission correction is applied in a two-step process. The science frames are first pre-cleaned with a TAPAS \citep{Bertaux_2014} absorption model that leaves percent-level residuals for deep ($>$50\%) H$_2$O and dry absorption features (e.g., CH$_4$, O$_2$, CO$_2$, N$_2$O and O$_3$). Then, a telluric residuals model with 3 degrees of freedom per pixel (optical depths for the H$_2$O and dry components and a constant) is fitted to the pre-cleaned spectra. The grid of telluric models was generated from a set of rapidly rotating hot stars observed with SPIRou at various airmass, water columns, and dry absorptions, producing telluric corrected spectra with final residuals at the level of the PCA-based method of \cite{Artigau_2014}. Radial velocity measurements were obtained from the telluric-corrected spectra using the novel line-by-line (LBL) method \citep{Artigau_2022}. The LBL formalism is based on the \cite{Bouchy_2001} framework, in which Doppler shifts are inferred for individual spectral lines ($\sim$16\,000 for an M dwarf observed with SPIRou) as opposed to a given spectral range. As in \cite{Bouchy_2001}, such calculations require a noiseless template since velocities are derived from the comparison between the residuals (observed spectrum minus template) and the derivative of the template. For a given observed star, one uses in practice a high SNR combined spectrum as a template, so that any remaining noise is small compared to that of an individual spectrum. For TOI-1452, the combined spectrum produced by \texttt{APERO} did not reach a SNR as high as other bright standard stars observed in the SLS. Moreover, TOI-1452 is located near the North ecliptic pole, meaning its yearly Barycentric Earth Radial Velocity (BERV) variation is small. Our observations with SPIRou covered BERV excursions between 1.7 and 4.8\,km/s, which is not ideal to filter out tellurics lines (i.e., stellar lines do not move a lot with respect to the telluric lines), producing a template that still contains some telluric artefacts. For these reasons, we used the template of Gl 699, a standard star monitored with SPIRou for 2.5\,years with a spectral type (M4V) similar to that of TOI-1452 (M4$\pm$0.5, see Sect.\ \ref{sec:toi_1452}) and a good BERV coverage ($\pm 26$\,km/s). For each spectrum, the LBL algorithm combines thousands of per-line velocities into a single RV measurement, with per-line uncertainties varying from 50 m/s for the strongest features to tens of km/s for the shallow ones. This is achieved using a simple mixture model: per-line velocities either originate from a Gaussian distribution centered on the mean velocity, with a standard deviation derived from \cite{Bouchy_2001}, or they arise from another distribution, namely that of high-sigma outliers, whose plausible causes are diverse (persisting bad pixels, cosmic rays, telluric residuals, etc.). Lastly, the LBL RVs are corrected for the instrumental day-to-day drift measured by the FP and for a long-term zero point obtained with a Gaussian process regression using the most observed stars in the SLS. This zero point calibration is similar to \cite{Courcol_2015} for the SOPHIE spectrograph, but will be described in more details in a forthcoming publication (Vandal et al., in prep.). The comparison between the LBL and other methods such as the cross-correlation function and template matching is discussed in \cite{Martioli_2022} and in \cite{Artigau_2022}. The final SPIRou radial velocities of TOI-1452 are listed in Table~\ref{table:spirou_rv}, with typical precision of 8.0\,m/s per exposure, or 4.0\,m/s per epoch. \subsection{IRD velocimetry} \label{sec:IRD_rv} Seven high-resolution spectra of TOI-1452 were obtained with the InfraRed Doppler (IRD) spectrograph on the Subaru 8.2\,m telescope \citep{2012SPIE.8446E..1TT, 2018SPIE10702E..11K} between September 26, 2020 and June 25, 2021. IRD covers the near-infrared wavelengths between 970\,nm and 1730\,nm, with a spectral resolution $R \sim 70\,000$. For accurate RV measurements, stellar spectra were obtained simultaneously with the reference spectra of the laser-frequency comb (LFC). The integration times were set to 600--1500 sec, depending on the available observing time slots and sky conditions. The IRD fiber has a $0\farcs48$ diameter, so that flux contamination from the companion star is not an issue. The raw IRD data were reduced following the standard procedure of \cite{2020PASJ...72...93H}. We extracted wavelength-calibrated one-dimensional spectra for TOI-1452, as well as for the simultaneously injected LFC. The typical SNR of the TOI-1452 extracted spectra was 60--70 per pixel around 1000\,nm. To measure precise RVs for TOI-1452, the reduced spectra were put into the RV analysis pipeline for IRD \citep{2020PASJ...72...93H}. This pipeline fits each small spectral segment of the observed spectra by the forward-modeling technique, taking into account the instantaneous variations of Earth's atmospheric features as well as the instrumental profile of the spectrograph (which is estimated based on each laser-comb spectrum). The seven IRD RV measurements have an overall precision of 4.0\,m/s and are given in Table~\ref{table:spirou_rv}. \section{Stellar Characterization} \label{sec:stellar_char} \subsection{TOI-1452 (TIC 420112589)} \label{sec:toi_1452} The star TOI-1452 (TIC 420112589) is a nearby M dwarf at a distance of 30.504 $\pm$ 0.013 pc \citep{Gaia_Collaboration_2021}. This star does not belong to any known young stellar moving groups, with a very high probability ($>$99.9\%) of being a field star \citep{Gagne_2018}. The presence of flares and short-period sinusoidal signal in the TESS \texttt{PDCSAP} data (see Fig.\ \ref{fig:TESS_lc_phase} and \ref{fig:TESS_lc_complete}) cannot be attributed with certainty to TOI-1452, due to flux contamination from multiple nearby objects. An analysis of the photometric variations is presented in Section~\ref{sec:tess_analysis}, but we note that the polarimetric data from SPIRou reveal no important surface magnetic field variations (see Sect.\ \ref{sec:polarimetry}), suggesting that TOI-1452 is relatively quiet, with a rotation period probably much longer than the modulation seen in the TESS light curve. As discussed in Section~\ref{sec:stellar_char_spirou}, we measure an effective temperature of $3185 \pm 50$\,K for TOI-1452 using the SPIRou combined spectrum, from which a spectral type (SpT) between M4 and M4.5 is inferred based on Table 5 of \cite{Pecaut_Mamajek_2013}. We also considered the \textit{Gaia} DR2 color to SpT relation of \cite{Kiman_2019}, more specifically the $G$\,--\,$G_{\rm RP}$ relationship, for an independent SpT determination. From this relationship, the \textit{Gaia} magnitudes and their respective uncertainties, we obtain a SpT of M$3.7 \pm 0.6$. The same M4 spectral type was derived through a visual comparison of the SPIRou combined spectrum, degraded to a lower resolution ($R\sim5\,000$), with spectral type standards of the IRTF spectra library (\citealt{Cushing_2005}; \citealt{Rayner_2009}). Considering all these estimates, we adopt a SpT of M4 $\pm$ 0.5. The mass of TOI-1452 was inferred from the \cite{Mann_2019} absolute $K_{\rm s}$ magnitude ($M_{K_{\rm s}}$) to $M_{\star}$ relation for M dwarfs. Taking into consideration the dispersion of this relation, the $K_{\rm s}$ magnitude, the distance, and their corresponding uncertainties, a mass of $0.249 \pm 0.008$\,M$_{\odot}$ is obtained. A similar approach was used to measure the stellar radius, this time using the $M_{K_{\rm s}}$--$R_{\star}$ relationship of \cite{Mann_2015}, from which we derive $R_{\star} = 0.275 \pm 0.009$\,R$_{\odot}$. Other physical parameters such as the surface gravity ($\log g$), the mean density ($\rho_{\star}$), and the luminosity ($L_{\star}$) were determined from the $M_{\star}$, $R_{\star}$, and $T_{\rm eff}$ estimates. The stellar parameters of TOI-1452 are summarized in Table~\ref{table:stellarparams}. \begin{deluxetable}{ccc} \tablecaption{TOI-1452 stellar properties} \tablehead{ \colhead{Parameter} & \colhead{Value} & \colhead{Ref.} } \startdata \multicolumn{3}{c}{\textit{Designations}}\\ TIC & 420112589 & 1\\ TOI & 1452 & 1\\ 2MASS & J19204172+7311434 & 2\\ UCAC4 & 816-023943 & 3\\ \textit{Gaia} EDR3 & 2264839957167921024 & 4\\ \hline \multicolumn{3}{c}{\textit{Astrometry}}\\ RA (J2016.0) & 19:20:41.75 & 4\\ DEC (J2016.0) & +73:11:42.35 & 4\\ $\mu_{\alpha} \cos \delta$ (mas/yr) & 7.800 $\pm$ 0.017 & 4\\ $\mu_{\delta}$ (mas/yr) & -74.076 $\pm$ 0.017 & 4\\ $\pi$ (mas) & 32.7823 $\pm$ 0.0140 & 4\\ Distance (pc) & 30.5043 $\pm$ 0.0130 & 4\\ \hline \multicolumn{3}{c}{\textit{Stellar parameters}}\\ $T_{\rm eff}$ (K) & 3185 $\pm$ 50 & 5\\ SpT & M4 $\pm$ 0.5 & 5\\ $\left[ {\rm M/H} \right]$ & $-0.07$ $\pm$ 0.02 & 5\\ $M_{\star}$ (M$_{\odot}$) & 0.249 $\pm$ 0.008 & 5\\ $R_{\star}$ (R$_{\odot}$) & 0.275 $\pm$ 0.009 & 5\\ log $g$ (dex) & 4.95 $\pm$ 0.03 & 5\\ $\rho_{\star}$ (g/cm$^3$) & 16.8 $\pm$ 1.9 & 5\\ $L_{\star}$ (L$_{\odot}$) & 0.0070 $\pm$ 0.0006 & 5\\ \hline \multicolumn{3}{c}{\textit{Photometry}}\\ $B$ & 15.94 $\pm$ 0.03 & 1\\ $V$ & 14.35 $\pm$ 0.12 & 1\\ $G_{\rm BP}$ & 15.222 $\pm$ 0.004 & 4\\ $G$ & 13.598 $\pm$ 0.003 & 4\\ $G_{\rm RP}$ & 12.362 $\pm$ 0.004 & 4\\ $T$ & 12.295 $\pm$ 0.007 & 1\\ $g$ & 15.580 $\pm$ 0.002 & 6\\ $r$ & 14.383 $\pm$ 0.007& 6\\ $i$ & 12.873$^*$ & 6\\ $z$ & 12.272$^*$ & 6\\ $y$ & 11.875 $\pm$ 0.020& 6\\ $J$ & 10.604 $\pm$ 0.058 & 2\\ $H$ & 10.026 $\pm$ 0.058 & 2\\ $K_{\rm s}$ & 9.740 $\pm$ 0.046 & 2\\ $W1$ & 8.938 $\pm$ 0.023$^\dag$ & 7\\ $W2$ & 8.760 $\pm$ 0.019$^\dag$ & 7\\ $W3$ & 8.686 $\pm$ 0.023$^\dag$ & 7\\ $W4$ & 8.46 $\pm$ 0.29$^\dag$ & 7\\ \enddata \tablecomments{$^*$The uncertainty was not indicated.\\ $^\dag$WISE magnitudes include the flux from TOI-1452 and TIC 420112587.} \tablerefs{(1) TIC \citep{Stassun_2019}. (2) 2MASS \citep{Skrutskie_2006}. (3) UCAC4 \citep{Zacharias_2013}. (4) \textit{Gaia} EDR3 \citep{Gaia_Collaboration_2021}. (5) This work. (6) Pan-STARRS1 DR2 \citep{Chambers_2016}. (7)\ AllWISE \citep{Wright_2010}.} \label{table:stellarparams} \end{deluxetable} \subsection{Bound companion (TIC 420112587)} \label{sec:bound} TOI-1452 has a resolved companion (TIC 420112587) with several comparable photometric and astrometric measurements. The two objects have similar \textit{Gaia} EDR3 magnitudes of \hbox{$G = 13.598 \pm 0.003$} and \hbox{$G = 13.830 \pm 0.003$} for \hbox{TOI-1452} and TIC 420112587 respectively. Their \textit{Gaia} EDR3 parallaxes are identical (within the errors), $32.782 \pm 0.014$\,mas for TOI-1452 and $32.791 \pm 0.014$\,mas for TIC 420112587, indicating a very similar distance to these stars. Their projected angular separation is $3\farcs182$, which corresponds to a projected physical separation of $\sim$97\,au, using a common approximate distance of 30.5\,pc. The proper motion of TIC 420112587 is similar to that of TOI-1452, with $\mu_{\alpha} \cos \delta$ and $\mu_{\delta}$ within 15\% for the two stars (see Tables~\ref{table:stellarparams} and \ref{table:stellarparams2}). TOI-1452 and TIC 420112587 most likely form a visual binary, i.e., a resolved gravitationally bound system, which was previously reported in the TOI visual-binary catalog of \cite{Mugrauer_2020}, as well as in the binary catalog based on \textit{Gaia} EDR3 of \cite{El-Badry_2021}. From our single visit on the companion star with SPIRou, we measure an RV offset between TOI-1452 and TIC 420112587 of $-6.9$\,km/s. The SPIRou template spectrum of TIC 420112587 combining only four individual spectra at the same epoch and BERV does not allow for a similar spectral analysis as the one presented for TOI-1452 in Section~\ref{sec:stellar_char_spirou}. Using the empirical relationships of \cite{Mann_2015, Mann_2019}, we obtain a mass of 0.226 $\pm$ 0.006\,M$_{\odot}$ and a radius of 0.254 $\pm$ 0.008\,R$_{\odot}$ for TIC 420112587. The mass ratio of the binary system is close to unity ($q = 0.91 \pm 0.04$), with TOI-1452 as the primary member. The projected physical separation and masses of the two stars imply an orbital period of about 1400 years. The radial velocity variation expected from such an orbital motion and for a circular orbit is under $\sim$1.5\,m/s over the span of our SPIRou RV observations. According to Table 5 of \cite{Pecaut_Mamajek_2013} and a $T_{\rm eff} = 3060 \pm 50$\,K derived from the spectral energy distribution (see analysis below), TIC 420112587 has an M5 spectral type. The \cite{Kiman_2019} $G$\,--\,$G_{\rm RP}$ relationship yields a SpT of M$4.0\pm0.6$, so we adopt an intermediate spectral type of M4.5 $\pm$ 0.5. A summary of the stellar properties of TIC 420112587 is presented in Table~\ref{table:stellarparams2}. \begin{deluxetable}{ccc} \tablecaption{TIC 420112587 stellar properties} \tablehead{ \colhead{Parameter} & \colhead{Value} & \colhead{Ref.} } \startdata \multicolumn{3}{c}{\textit{Designations}}\\ TIC & 420112587 & 1\\ TOI & 1760 & 1\\ 2MASS & J19204172+7311467 & 2\\ \textit{Gaia} EDR3 & 2264839952875245696 & 3\\ \hline \multicolumn{3}{c}{\textit{Astrometry}}\\ RA (J2016.0) & 19:20:41.76 & 3\\ DEC (J2016.0) & +73:11:45.53 & 3\\ $\mu_{\alpha} \cos \delta$ (mas/yr) & 6.845 $\pm$ 0.017 & 3\\ $\mu_{\delta}$ (mas/yr) & -82.216 $\pm$ 0.017 & 3\\ $\pi$ (mas) & 32.7913 $\pm$ 0.0141 & 3\\ Distance (pc) & 30.4959 $\pm$ 0.0131 & 3\\ \hline \multicolumn{3}{c}{\textit{Stellar parameters}}\\ $T_{\rm eff}$ (K) & 3060 $\pm$ 50 & 4\\ SpT & M4.5 $\pm$ 0.5 & 4\\ $M_{\star}$ (M$_{\odot}$) & 0.226 $\pm$ 0.006 & 4\\ $R_{\star}$ (R$_{\odot}$) & 0.254 $\pm$ 0.008 & 4\\ log $g$ (dex) & 4.98 $\pm$ 0.03 & 4\\ $\rho_{\star}$ (g/cm$^3$) & 19.5 $\pm$ 1.8 & 4\\ $L_{\star}$ (L$_{\odot}$) & 0.0051 $\pm$ 0.0005 & 4\\ \hline \multicolumn{3}{c}{\textit{Photometry}}\\ $B$ & 15.76 $\pm$ 0.17 & 1\\ $V$ & 13.99 $\pm$ 0.2 & 1\\ $G_{\rm BP}$ & 15.512 $\pm$ 0.005 & 3\\ $G$ & 13.830 $\pm$ 0.003 & 3\\ $G_{\rm RP}$ & 12.576 $\pm$ 0.004 & 3\\ $T$ & 12.499 $\pm$ 0.008 & 1\\ $g$ & 15.890 $\pm$ 0.002 & 5\\ $r$ & 14.659 $\pm$ 0.003 & 5\\ $i$ & 13.153 $\pm$ 0.002 & 5\\ $z$ & 12.456 $\pm$ 0.021 & 5\\ $y$ & 12.111 $\pm$ 0.007 & 5\\ $J$ & 10.795 $\pm$ 0.027 & 2\\ $H$ & 10.257 $\pm$ 0.031 & 2\\ $K_{\rm s}$ & 9.944 $\pm$ 0.023 & 2\\ \enddata \tablerefs{(1) TIC \citep{Stassun_2019}. (2) 2MASS \citep{Skrutskie_2006}. (3) \textit{Gaia} EDR3 \citep{Gaia_Collaboration_2021}. (4) This work. (5) Pan-STARRS1 DR2 \citep{Chambers_2016}.} \label{table:stellarparams2} \end{deluxetable} \subsection{Spectral energy distribution fit} \label{sec:sed_fit} As an independent determination of the basic stellar parameters, as well as to estimate the contaminating flux from the nearby companion star, we performed an analysis of the broadband spectral energy distribution (SED) of the stars together with the \textit{Gaia} EDR3 parallax \citep[with no systematic offset applied; e.g.,][]{Stassun_2021}, following the procedures described in \cite{Stassun_2016} and \cite{Stassun_2017,Stassun_2018a}. For both stars, we pulled the $JHK_S$ magnitudes from 2MASS, the W1--W4 magnitudes from WISE, and the $grizy$ magnitudes from Pan-STARRS. Together, the available photometry spans the full stellar SED over the wavelength range 0.4--10~$\mu$m (see Figure~\ref{fig:SED}). We excluded the WISE photometry from the initial fitting because the two stars are blended in WISE, such that the catalog photometry in fact represents the sum of the fluxes of both stars. For each star, we performed a fit using NextGen stellar atmosphere models \citep{Hauschildt_1999}, with the free parameters being the effective temperature ($T_{\rm eff}$) and metallicity ([Fe/H]). The remaining free parameter is the extinction $A_V$, which we fixed at zero due to the stars' proximity. The resulting fit for TOI-1452 (Figure~\ref{fig:SED}) has a reduced $\chi^2$ of 1.8 with $T_{\rm eff} = 3100 \pm 50$~K and [Fe/H] = $0.0 \pm 0.5$. Integrating the model SED gives the bolometric flux at Earth, $F_{\rm bol} = 2.34 \pm 0.11 \times 10^{-11}$ erg~s$^{-1}$~cm$^{-2}$. Taking the $F_{\rm bol}$ and $T_{\rm eff}$ together with the {\it Gaia\/} parallax, gives the stellar radius, $R_\star = 0.286 \pm 0.011$~R$_\odot$. This independent radius measurement is consistent, although slightly less precise, with the one derived using \citealt{Mann_2015} ($R_\star = 0.275 \pm 0.009$~R$_\odot$). Similarly, the resulting parameters for the companion star from the SED fit are $T_{\rm eff} = 3060 \pm 50$\,K, [Fe/H]~$ = 0.0 \pm 0.5$, and $R_\star = 0.263 \pm 0.010$\,R$_\odot$. This radius estimate is again fully consistent with the value derived from empirical relation ($R_\star = 0.254 \pm 0.008$\,R$_\odot$). The sum of the two stellar models is compared to the combined WISE fluxes in Figure~\ref{fig:SED}, showing good agreement. Integrating the companion SED within the TESS bandpass yields a flux ratio (companion relative to TOI-1452) of $0.77 \pm 0.03$. Note that the flux ratio derived strictly from the $T$ magnitudes from the TIC is $0.829 \pm 0.002$. In the event that the \texttt{PDCSAP} overestimated the dilution correction for TIC 420112587, this difference in flux ratio would imply a $\sim$1.7\% overestimation of the planetary radius. \subsection{Stellar parameters from SPIRou spectra} \label{sec:stellar_char_spirou} The high-resolution combined spectrum of TOI-1452 from SPIRou lets us determine $T_{\rm eff}$ and the abundance of several elements with relatively good accuracy. This work follows the methodology of Jahandar et al.\ (in prep.), which we briefly summarize here. Because models and observations can often show significant discrepancies (e.g., continuum mismatch in the $Y$ and $J$ bands), we only select for the fitting analysis a subset of relatively strong lines that are matching the models. The selected lines are then divided into several groups of 15 lines, each analyzed independently through a chi-squared fitting routine to infer both $T_{\rm eff}$ and [M/H] for all groups. The spectrum is compared with a grid of ACES models (\citealt{allard2012models}; \citealt{husser2013new}). The advantage of this method is that it yields several (typically 15) independent measurements that can be used to characterize the inherent uncertainties associated with the fitting procedure. This analysis applied to the TOI-1452 spectrum yields $T_{\rm eff} = 3185\pm50$\,K and [M/H]=$-0.07\pm0.02$, in good agreement with the parameters derived from the SED fitting analysis. The quoted uncertainty for $T_{\rm eff}$ is internal to our fitting methodology and ignore potential systematic differences with bolometric $T_{\rm eff}$ estimates based on interferometric measurements. While our $T_{\rm eff}$ estimates have yet to be calibrated with bolometric $T_{\rm eff}$, it is empirically demonstrated that 50--60\,K is a typical uncertainty derived from atmosphere models (e.g., \citealt{Mann2013}; \citeyear{Mann_2015}). We thus adopt 50\,K for our $T_{\rm eff}$ uncertainty, a conservative value given that the temperature derived from the SPIRou spectrum is inferred from several independent measurements. An illustration of the temperature and abundance sensitivity for an Al I line (at 1675.514\,nm) is shown in Figure~\ref{fig:teff_met}. In practice, several tens of lines are used to derive $T_{\rm eff}$. Once $T_{\rm eff}$ is determined, one can then proceed, through a similar procedure, to determine the abundance of all individual lines of a given element. The high-resolution near-infrared spectrum of an M dwarf is characterized by several hundreds of relatively strong OH lines. By selecting only those that are well isolated, i.e., with no known spectral features within a few pixels using the PHOENIX/BT-Settl (\citealt{Allard_2012}; \citeyear{Allard_2013}) and NIST \citep{Ralchenko_2010} line lists, we find 72 OH lines, whose individual abundance can be used to quantify the inherent, per-line uncertainty of this method. This uncertainty obviously does not consider any possible systematic errors associated with the ACES atmosphere models. The 72 independent OH abundance measurements are presented in Figure~\ref{fig:OHdist}, showing a good match with a Gaussian distribution with standard deviation 0.13\,dex. For all elements and molecules detected in TOI-1452, we list the average abundance of all lines in Table~\ref{table:abundances} (see also Figure~\ref{fig:abundances}). For chemical species with only one line, we adopt an uncertainty of 0.13\,dex from the OH distribution. We report abundances for Fe, Mg and Si that constitute the bulk material of an exoplanet core and mantle. The overall metallicity ([M/H]) and its corresponding error are determined by averaging the final abundance of each element in Table~\ref{table:abundances}, assuming a common uncertainty for all elements taken as the median of all individual uncertainties. This approach is chosen to avoid putting too much weight on the oxygen abundance characterized by a small uncertainty. \begin{deluxetable}{cccc} \tablecaption{Stellar abundance of TOI-1452 for various chemical species measured by SPIRou} \tablehead{ \colhead{Element} & \colhead{[X/H]} & \colhead{$\sigma$} & \colhead{\# of lines}} \startdata Fe I & $-0.07$ & 0.03 & 38 \\ Al I & 0.07 & 0.11 & 4 \\ Mg I & $0.02$ & 0.07 & 5 \\ Si I & $0.11$ & 0.13 & 1 \\ Ti I & $-0.31$ & 0.06 & 10 \\ Ca I & $0.01$ & 0.12 & 2 \\ Cr I & $0.04$ & 0.06 & 4 \\ K I & 0.03 & 0.13 & 1 \\ O I$^*$ & $-0.24$ & 0.02 & 72 \\ C I & $-0.17$ & 0.05 & 11 \\ N I & $-0.12$ & 0.13 & 1 \\ Na I & $-0.22$ & 0.06 & 2 \\ $<>^{\dag}$ & $-0.07$ & 0.02 & -- \enddata \tablecomments{$^*$The oxygen abundance is inferred from OH lines.\\ $^{\dag}$Average abundance of all elements.} \label{table:abundances} \end{deluxetable} \subsection{Spectropolarimetry with SPIRou} \label{sec:polarimetry} The combination of the four exposures within a polarimetric sequence obtained with SPIRou yields the circular polarization profile at the surface of the star \citep{Donati_2020}. The intensity (Stokes I), circular (Stokes V), and null polarization spectra were generated in \texttt{APERO} following the \texttt{spirou-polarimetry} code\footnote{\href{https://github.com/edermartioli/spirou-polarimetry}{\texttt{github.com/edermartioli/spirou-polarimetry}}}. We applied the Least-Square Deconvolution (LSD) method of \cite{Donati_1997}, also outlined in \cite{Martioli_2020}, to compute the average I and V profiles. We used the VALD database \citep{Piskunov_1995} and a MARCS atmosphere model \citep{Gustafsson_2008} with $T_{\rm eff} = 3000$\,K and log $g$ = 5.0\,dex to search for valid atomic features. Lines deeper than 3\,\% and with a known Landé factor were selected to produce a line mask of 955 atomic lines, used in this LSD analysis of TOI-1452. An estimate of the longitudinal magnetic field ($B_{\ell}$) at the stellar surface can then be obtained using Equation\,5 of \cite{Donati_1997}, combining the Stokes I and V LSD profiles, the mean Landé factor of 1.24, and the mean wavelength of 1604.59\,nm. By doing this over multiple epochs, one can monitor the large-scale surface magnetic field, expected to vary with the rotation of the star. The polarimetric capabilities of SPIRou can thus serve as a useful activity tracer simultaneous to the RV measurements, as demonstrated in \cite{Martioli_2022}, where the rotation period of the moderately active M0 star TOI-1759 ($P_{\rm rot} = 35.65^{+0.17}_{-0.15}$\,days) was determined from the $B_{\ell}$ time series. We obtained independent and consistent values for the $B_{\ell}$ of TOI-1452 using the Libre-Esprit pipeline (\citealt{Donati_1997}, \citealt{Donati_2020}), but present below the values from the \texttt{APERO} pipeline. The $B_{\ell}$ time series of TOI-1452 is presented in Figure~\ref{fig:Blong}. A simple Lomb-Scargle periodogram analysis shows no obvious periodicity. The $B_{\ell}$ data do not favor a sinusoidal model, which could be associated with stellar rotation, over a constant magnetic field (mean $B_{\ell} = -3.8 \pm 1.8$\,G). The small variation of $B_{\ell}$ suggests that the field is intrinsically weak (quiet star), or that it is strongly axisymmetric with respect to the rotation axis. Alternatively, the rotation period of TOI-1452 could be longer than the 4-month span of our observations, but this is close to the largest known period for M dwarfs \citep{Newton_2018}. This LSD analysis was also applied on the companion star TIC 420112587 using the single polarimetric sequence acquired with SPIRou. We report a polarimetric signal and a $B_{\ell}$ consistent with a null value, indicating that the companion is also probably inactive. \section{Data Analysis \& Results} \label{sec:analysis} \subsection{Determining the transit origin with PSF photometry} \label{sec:psf} The objective of the first OMM-PESTO transit follow-up was to establish the origin of the TESS signal, particularly between the target (TOI-1452) and its companion (TIC 420112587). Standard aperture photometry ruled out any NEB in the FOV, but was unable to isolate the transit between the two stars, as they were only partially resolved. We therefore had to rely on a different method using point spread function (PSF) fitting to extract the relative flux of both stars. We used the \texttt{photutils} \citep{Larry_Bradley_2020} package to perform the DAOPHOT \citep{Stetson_1987} PSF photometry algorithm. This was achieved by fitting the PSFs with an effective PSF model (ePSF) generated in \texttt{photutils} using the 6 stars with the highest SNR in the FOV (excluding our targets), then integrating the best-fit models over pixels containing the stars' signal. The intent here was not to produce a precise uncontaminated light curve, but rather to detect any flux deficit (or excess) that would indicate from which star the transit originates. We thus inspected the TOI-1452 to TIC 420112587 flux ratio as a function of time, normalized to unity outside of transit. The resulting relative light curve is presented in Figure~\ref{fig:PSFphot} and shows a flux deficit on TOI-1452 during transit. We did not fit a transit model on this light curve, as it is less precise than the one obtained using a combined circular aperture (Fig.\ \ref{fig:OMM-PESTO_lc}). We nonetheless measure a mean relative flux deficit of $2.33 \pm 0.43$\,ppt, which is an approximation of the uncontaminated transit depth. This flux deficit is comparable in amplitude to the diluted corrected TESS depth ($3.31 \pm 0.19$\,ppt) and was detected with a confidence level sufficiently high ($>$5$\sigma$) to conclude that TOI-1452 was the source of the transit and justify an RV monitoring campaign on this star, starting with SPIRou in June 2020. Later, the MuSCAT3 photometry was able to resolve TOI-1452 and TIC 420112587 and unambiguously identify that the former star hosts a transiting object. \subsection{TESS light curve analysis} \label{sec:tess_analysis} The TOI-1452 \texttt{PDCSAP} light curve (Fig.\ \ref{fig:TESS_lc_phase} and \ref{fig:TESS_lc_complete}) features stellar flares with amplitude of a few percents and ppt-level sinusoidal variations. A strong peak at 0.93\,days appears in the Lomb-Scargle periodogram of the multi-year light curve, as well as in all individual sectors. However, computing the autocorrelation function, which is more reliable for accurate photometric rotation period determination \citep{McQuillan_2013}, would often find a period of 1.9\,days (2$\times$0.93\,days) depending on the sector. Since the \texttt{PDCSAP} data are corrected for systematic trends, it is unlikely that such corrections significantly perturb those short-term flares and sinusoids. Regardless of the origin of these signals (TOI-1452, TIC 420112587, or any contaminating star), it is crucial to remove the periodic variations to accurately measure the transit parameters. To accomplish this, we adopted a sequential approach where we first correct the \texttt{PDCSAP} data using a Gaussian Process (GP), then fit the 32 corrected transits with a model. The details of the GP regression are presented below, while the transit modeling is described in Section~\ref{sec:jointfit}. We started by masking the epochs of transit and removing outliers from the \texttt{PDCSAP} light curve with sigma clipping. It was determined that a 3.5\,$\sigma$ clipping was robust enough to remove both obvious outliers and stellar flares. This sigma clipping removed less than 0.2\% of the out-of-transit data. Parts of sectors 21 and 47 coinciding with TESS momentum dump events show large amplitude variations; those were considered to be non-astrophysical and were manually rejected. We also rejected data points in sectors 40 and 41, as they are isolated and have a median considerably different than unity. The data not considered in this analysis are either displayed in blue (transits) or in red (rejected) in Figures~\ref{fig:TESS_lc_phase} and \ref{fig:TESS_lc_complete}. The \textit{cleaned} out-of-transit \texttt{PDCSAP} dataset was too large ($N = 237\,634$) to be efficiently modeled with a GP. We therefore binned the data and instead used the corresponding 1-hour effective cadence light curve ($N = 7924$). The GP regression was done with \texttt{celerite2} (\citealt{celerite1_2017}; \citeyear{celerite2_2018}). We selected its \texttt{RotationTerm} kernel because it was specifically designed to model a range of quasi-periodic variability, from stellar rotation to pulsations. This kernel is the sum of two stochastically-driven, damped harmonic oscillator (SHO) terms (\texttt{SHOTerm}) capturing both primary ($P_{\rm GP}$) and secondary ($P_{\rm GP}/2$) modes in Fourier space. The Fourier transform of the covariance function, known as the power spectral density (PSD), takes the following form: \begin{equation} \begin{split} S(\omega) = \sqrt{\frac{2}{\pi}} & \frac{S_1\,\omega_1^4} {(\omega^2-\omega_1^2)^2 + \omega_1^2\,\omega^2/Q_1^2}\\ & + \sqrt{\frac{2}{\pi}} \frac{S_2\,\omega_2^4} {(\omega^2-\omega_2^2)^2 + \omega_2^2\,\omega^2/Q_2^2} \end{split} \end{equation} where each \texttt{SHOTerm} PSD is described by their respective power $S_{1}$, $S_{2}$ at $\omega = 0$, their undamped angular frequency $\omega_{1}$, $\omega_{2}$, and their own quality factor $Q_{1}$, $Q_{2}$. Since the periods of the two oscillators are separated by a factor of 2 ($\omega_{2} = 2 \omega_{1}$), the parametrization below reduces by one the number of free parameters: \begin{gather} \sigma_{1} = \sqrt{S_1 \omega_1 Q_1}\\ \sigma_{2} = \sqrt{S_2 \omega_2 Q_2}\\ \tau_1 = \frac{2 Q_1}{\omega_1}\\ \tau_2 = \frac{2 Q_2}{\omega_2}\\ P_{\rm GP} = \frac{2 \pi}{\omega_1} = \frac{4 \pi}{\omega_2} \end{gather} where $\sigma_1$, $\sigma_2$ are the standard deviations (amplitudes) of the primary and secondary modes, $\tau_1$, $\tau_2$ are the damping timescales of the primary and secondary oscillations, and $P_{\rm GP}$ is the undamped period of the primary mode. Note that these parameters differ slightly from the default \texttt{RotationTerm} kernel parametrization by making no assumptions on the relative amplitudes and quality factors between the two modes. Our \texttt{PDCSAP} GP model consisted of the five hyperparameters above, plus an excess white noise term $s$. We sampled the posterior distributions of the parameters in their logarithmic form $\{\ln \sigma_1$, $\ln \sigma_2$, $\ln \tau_1$, $\ln \tau_2$, $\ln P_{\rm GP}$, $\ln s \}$ using the Markov chain Monte Carlo (MCMC) package \texttt{emcee} \citep{Foreman-Mackey_2013} and a Bayesian formalism. We employed 100 walkers and performed 100\,000 steps with a burn-in of 10\,000. The number of steps was greater than 50 times the autocorrelation timescale for each parameter, which usually indicates a sufficient level of convergence (\citealt{Sokal_1997}; \citealt{Foreman-Mackey_2019}). The adopted prior distributions and the posteriors median, 16$^{\rm th}$ and 84$^{\rm th}$ percentiles are reported in Table~\ref{table:lightcurveparams}. The resulting mean GP prediction is shown in Figures~\ref{fig:TESS_lc_phase} and \ref{fig:TESS_lc_complete} superimposed on the original \texttt{PDCSAP} cadence. Even though the sinusoidal variations visually appear to repeat every $\sim$0.93 day, our model converged to a very well constrained primary oscillation of $1.8680 \pm 0.0004$\,days, thus indicating significant power at the second harmonic. \begin{deluxetable}{ccc} \tablecaption{Prior and posterior distributions of the quasi-periodic GP model of the TOI-1452 TESS light curve (details in Sect.\ \ref{sec:tess_analysis})} \tablehead{ \colhead{Parameter} & \colhead{Prior} & \colhead{Posterior}} \startdata $\ln \sigma_1$ & $\mathcal{U}\left(-10, 0\right)$ & $-7.5^{+0.5}_{-0.5}$\\ $\ln \sigma_2$ & $\mathcal{U}\left(-10, 0\right)$ & $-7.1^{+0.5}_{-0.4}$\\ $\ln \left[\tau_1 / \textrm{days} \right]$ & $\mathcal{U}\left(-10, 10\right)$ & $-1.42^{+0.13}_{-0.14}$\\ $\ln \left[\tau_2 / \textrm{days} \right]$ & $\mathcal{U}\left(-10, 10\right)$ & $7.1^{+1.1}_{-0.8}$\\ $\ln \left[P_{\rm GP} / \textrm{days} \right]$ & $\mathcal{U}\left(-2, 5\right)$ & 0.6248$^{+0.0002}_{-0.0002}$\\ $\ln s$ & $\mathcal{U}\left(-15, 0\right)$ & $-12.7^{+1.6}_{-1.5}$\\ \enddata \tablecomments{$\mathcal{U}\left(a,b \right)$ is the uniform distribution between value $a$ and $b$.} \label{table:lightcurveparams} \end{deluxetable} It is beyond the scope of this study to assess the exact cause of this strong and persistent signal, but we showed earlier that the SPIRou magnetic field constraints of TOI-1452 are inconsistent with a fast rotator and active object. Moreover, a 1.9-day rotation period for TOI-1452 would correspond to a $v \sin i$ of $\sim$7\,km/s, readily detectable in the SPIRou combined spectrum. Instead, the mean line profile FWHM measured from the cross-correlation function (CCF) calculated in \texttt{APERO} suggests a slow rotator (i.e., $v \sin i < 2$\,km/s). We repeated this step for the companion star from the single visit with SPIRou, and also measured a FWHM consistent with $v \sin i < 2$\,km/s. Thus, the rotation of the companion star is also most probably not causing this photometric signal. \subsection{Joint transit-RV fit} \label{sec:jointfit} In order to constrain the physical and orbital parameters of TOI-1452\,b, we conducted a joint analysis of the transits (TESS, OMM-PESTO, and MuSCAT3) and the RV data (SPIRou and IRD). The joint fit was performed using the \texttt{juliet} \citep{Espinoza_2019} package, which utilizes \texttt{batman} \citep{Kreidberg_2015} to generate transit models and \texttt{radvel} \citep{Fulton_2018} to compute Keplerian RV models. The \texttt{juliet} framework implements nested sampling algorithms to sample posterior distributions, while also enabling model comparison via evaluations of the Bayesian log-evidence ($\ln Z$). We chose the \texttt{dynesty} \citep{Speagle_2020} dynamic nested sampling option in \texttt{juliet}. Standard nested sampling \citep{Skilling_2006} was designed to estimate evidences, not posteriors, and thus struggles with parameter estimation for complex distributions. Dynamic nested sampling \citep{Higson_2019}, on the other hand, adapts the number of live points based on the structure of the posteriors, providing parameter estimation comparable to MCMC algorithms. The transit and RV components of the joint fit have four parameters in common: the orbital period $P$, the time of inferior conjunction $t_0$, the eccentricity $e$, and the argument of periastron $\omega$. For the transit modeling, we followed the parametrization from \cite{Espinoza_2018} of the impact parameter $b$ and the planet-to-star radius ratio $p = R_{\rm p} / R_{\star}$ to efficiently sample physically plausible values in ($b$,~$p$) space. Instead of fitting the scaled semi-major axis $a/R_{\star}$, we used the stellar density $\rho_{\star}$ parameterization available in \texttt{juliet}. Fitting $\rho_{\star}$ takes into account any prior information on the stellar mass and radius. We adopted a Gaussian prior on $\rho_{\star}$ using the value and uncertainty in Table~\ref{table:stellarparams}. Stellar limb-darkening effects in TESS, OMM-PESTO, and MuSCAT3 transits were modeled using per-instrument and per-filter quadratic $q_1$ and $q_2$ parameters defined in \cite{Kipping_2013}. For each instrument, we included in \texttt{juliet} a flux dilution factor $D$, a baseline flux $M$, and an extra jitter term $\sigma$. We set $D_{\rm TESS}$ to 1 (no dilution), as the \texttt{PDCSAP} data are already corrected for crowding effects. The OMM-PESTO light curve combines the flux of TOI-1452 and TIC 420112587, which requires an adequate $D_{\rm PESTO}$ factor to compensate for contamination. We thus constructed a Gaussian prior on $D_{\rm PESTO}$ with a mean value calculated with Equation 6 of \cite{Espinoza_2019} and flux ratio derived from TOI-1452 and TIC 420112587 magnitudes in the $i$ band (see Tables~\ref{table:stellarparams} and \ref{table:stellarparams2}). The adopted prior on $D_{\rm PESTO}$ was $\mathcal{N}\left(0.564, 0.0564^2\right)$, that is with a 10\% standard deviation to account for errors on the magnitudes and deviations between $i$ and $i^{\prime}$. We also explored fixing $D_{\rm PESTO}$ to 0.564, while letting $D_{\rm TESS}$ vary freely between 0 and 2. Both approaches yielded a consistent measurement of the planetary radius (within 1-$\sigma$), indicating that the \texttt{PDCSAP} fluxes were in all likelihood properly corrected for contamination. The dilution in the MuSCAT3 light curves was a priori unknown. However, it is expected that the $g^{\prime}$ transit was more affected by dilution, as the seeing was worse for this filter (see Table \ref{table:gbobs}). We adopted a conservative approach where a different $D_{\rm MuSCAT3}$ is applied for each filter, with uniform priors between 0.5 (twice the flux) and 1. The parameters specific to the RV Keplerian component were the semi-amplitude $K$, per-instrument offsets $\gamma$ and extra white noise terms $\sigma$. We explored adding a global GP to model common stellar activity signal in the SPIRou and IRD data. For this, we used the GP implementation in \texttt{juliet} that runs \texttt{celerite} \citep{celerite1_2017}. We chose the Matérn-3/2 approximation kernel, which takes the following form: \begin{equation} k_{i,j}\left( \tau \right) = A_{\rm GP}^2 \left[ \left( 1 + 1/\epsilon \right) e^{-\left(1 - \epsilon \right) w} \left(1 - 1/\epsilon \right) e^{-\left(1 + \epsilon \right) w}\right] \end{equation} where $\tau = |t_i - t_j|$ is the time interval between data points $i$ and $j$, $A_{\rm GP}$ is the amplitude of the GP, $w = \sqrt{3} \tau/\ell_{\rm GP}$, with $\ell_{\rm GP}$ the timescale of the GP, and $\epsilon$ is set to 0.01 (when $\epsilon \rightarrow 0$, $k_{i,j}$ converges to a Matérn-3/2 kernel). We did not fit a per-instrument $A_{\rm GP}$ and $\ell_{\rm GP}$ due to the limited number of RV measurements from IRD. We also considered choosing a quasi-periodic kernel in \texttt{celerite} instead (Equation 56 of \citealt{celerite1_2017}). Since no clear periodicity was detected in the $B_{\ell}$ time series, or other activity indicators from the LBL such as the dLW metric \citep{Zechmeister_2018} or chromatic velocity slope changes, we applied a uniform prior on the stellar rotation period, namely $\mathcal{U}(0.1, 120)$\,days. We found that the Matérn-3/2 kernel gave equivalent results with fewer hyperparameters needed (2 instead of 4) and that the quasi-periodic GP did not converge to a specific rotation period, showing no preference for a period of 0.93\,days (or 2$\times$0.93\,days) as seen in TESS photometry. This is another indication that the sinusoidal signal in the out-of-transit \texttt{PDCSAP} data is probably not associated with TOI-1452 stellar activity. We examined the change in Bayesian log-evidence for a suite of joint models ($\mathcal{M}$), all having an identical transit component. The ``zero'' planet model ($\mathcal{M}_{\rm 0p}$) has a $K$ fixed to 0\,m/s, with only the RV offsets and extra white noise terms allowed to vary. This model tests whether the RV dispersion can be fully explained by white noise only, without questioning the transit detection. Single planet models can either be with circular ($\mathcal{M}_{\rm 1cp}$; $e = 0$, $\omega = 90^{\circ}$) or eccentric orbits ($\mathcal{M}_{\rm 1ep}$; free $e, \omega$). Two additional models include a global RV activity GP ($\mathcal{M}_{\rm 1cp+GP}$ and $\mathcal{M}_{\rm 1ep+GP}$). To objectively assess the contribution from the IRD observations, we decided to apply this framework first on the SPIRou data individually, then using the full RV dataset (SPIRou + IRD). For two competing models, the difference in log-evidence ($\Delta \ln Z$) informs on the probability that one model matches the data better than the other. To interpret the significance of the $\Delta \ln Z$ and select the ``best'' model, we followed the empirical scale introduced in Table 1 of \cite{Trotta_2008}. A $\Delta \ln Z > 5$ translates into ``strong" evidence towards the model with the highest $\ln Z$. A $2.5 < \Delta \ln Z < 5$ corresponds to ``moderate'' evidence, while $\Delta \ln Z \leq 2.5$ shows ``weak'' evidence at best, i.e., neither model should be favoured in that case. Figure~\ref{fig:modelselect} shows the Bayesian log-evidence for different joint models and datasets. Note that the typical errors on the $\ln Z$ computed by \texttt{dynesty} were 0.5, so that the $\Delta \ln Z$ presented in Figure~\ref{fig:modelselect} have associated uncertainties of 0.7. We first observe that all planetary models are strongly favoured ($\Delta \ln Z > 5$) compared to the ``zero'' planet solution ($\mathcal{M}_{\rm 0p}$), providing quantitative evidence that the TOI-1452\,b Keplerian signal is detected in velocimetry, in phase with transit. There is also compelling evidence for models with an RV activity GP ($\mathcal{M}_{\rm 1cp+GP}$ and $\mathcal{M}_{\rm 1ep+GP}$), increasing $\ln Z$ by approximately 10 relative to $\mathcal{M}_{\rm 1cp}$ and $\mathcal{M}_{\rm 1ep}$. However, considering only the SPIRou data yields similar or slightly larger $\Delta \ln Z$ for all $\mathcal{M}$ compared to joint fits that include the seven IRD RV measurements. This suggests that the IRD observations do not significantly contribute to improve the Keplerian solution for TOI-1452\,b. The median RV uncertainty from IRD (4.03\,m/s) is nearly identical to SPIRou (4.00\,m/s), but the point-to-point scatter (RMS) is much larger: respectively 12.71\,m/s for IRD and 5.76\,m/s for SPIRou. The planetary models fail to capture the extra RMS in the IRD data, and are instead converging to solutions with white noise term comparable to the overall scatter (see Table~\ref{table:rvcomponent}). This is apparent in Figure~\ref{fig:RV_analysis} showing the RV component of the joint fit (model $\mathcal{M}_{\rm 1cp+GP}$) using the full dataset, with each instrument having their original error bar plotted. The IRD radial velocities were produced using the template spectrum of TOI-1452. As previously mentioned, this star has a small BERV excursion, which is not ideal for filtering out telluric lines. This may explain the increased dispersion in the resulting RVs, in this case, at a level much larger than the Keplerian signal. For this reason, we opted to present below the results using only the SPIRou RVs. We nonetheless provide all the relevant parameters of the RV modeling for the SPIRou only and SPIRou+IRD datasets in Table~\ref{table:rvcomponent}. The eccentric model $\mathcal{M}_{\rm 1ep+GP}$ produced the highest $\ln Z$ (Fig.\ \ref{fig:modelselect}), but with a Bayesian evidence indistinguishable from the circular model $\mathcal{M}_{\rm 1cp+GP}$ ($\Delta \ln Z = 0.9$). We report an eccentricity of $0.12^{+0.12}_{-0.08}$, with $e < 0.32$ at a 95\% confidence, but argue that the simpler, circular model should be preferred at this point. The adopted priors and resulting posteriors of the $\mathcal{M}_{\rm 1cp+GP}$ fit are summarized in Table~\ref{table:modelparams}. The MuSCAT3 photometric parameters are given in another table (Table~\ref{table:muscat3params}) to facilitate comparison between filters. We measure a dilution factor $D$ consistent with no dilution for the $i^{\prime}$ transit, with moderate level of contamination ($\sim$30\%) in the $g^{\prime}$ band. Even if we assume instead that the flux dilution was exactly zero for all MuSCAT3 filters, the uncorrected transit depths ($\delta_{\rm uncorr.}$) presented in Table~\ref{table:muscat3params} show no sign of strong chromaticity. The best-fit transit models of the TESS, OMM-PESTO, and MuSCAT3 photometry are shown in Figures~\ref{fig:TESS_lc_phase}, \ref{fig:OMM-PESTO_lc}, and \ref{fig:MuSCAT3_lc} respectively. The best circular ($\mathcal{M}_{\rm 1cp+GP}$) and eccentric ($\mathcal{M}_{\rm 1ep+GP}$) RV orbital fits of TOI-1452\,b are depicted in Figure~\ref{fig:RV_phase} in a phase-folded format. \begin{deluxetable*}{lccr} \tablecaption{Prior and posterior distributions of the joint transit-RV fit for model $\mathcal{M}_{\rm 1cp+GP}$ (details in Sect.\ \ref{sec:jointfit}) using only the SPIRou radial velocities} \tablehead{ \colhead{Parameter} & \colhead{Prior$^{\rm a}$} & \colhead{Posterior} & \colhead{Description} } \startdata \multicolumn{4}{c}{\textit{Fitted parameters}}\\ $\rho_{\star}$ (g/cm$^3$) & $\mathcal{N}\left(16.8, 1.9^2\right)$ & 16.8$^{+1.0}_{-1.4}$ & Stellar density\\ $P$ (days) & $\mathcal{U}\left(11.0, 11.1\right)$ & 11.06201 $\pm$ 0.00002 & Orbital period\\ $t_0$ (BJD - 2\,457\,000) & $\mathcal{U}\left(1691.4, 1691.6\right)$ & 1691.5321 $\pm$ 0.0015 & Time of inferior conjunction\\ $r_1$ & $\mathcal{U}\left(0, 1\right)$ & 0.46$\pm$0.08 & Parametrization$^{\rm b}$\,for $R_{\rm p}/R_{\star}$ and $b$\\ $r_2$ & $\mathcal{U}\left(0, 1\right)$ & 0.0555 $\pm$ 0.0014 & Parametrization$^{\rm b}$\,for $R_{\rm p}/R_{\star}$ and $b$\\ $K$ (m/s) & $\mathcal{U}\left(0, 10\right)$ & 3.50 $\pm$ 0.94 & RV semi-amplitude\\ $e$ & 0 (fixed) & 0 & Orbital eccentricity\\ $\omega$ ($^{\circ}$) & 90 (fixed) & 90 & Argument of periastron\\ $A_{\rm GP}$ (m/s) & $\mathcal{LU}\left(10^{-2}, 100\right)$& 4.5$^{+2.0}_{-1.2}$ & Amplitude of the GP\\ $\ell_{\rm GP}$ (days) & $\mathcal{LU}\left(10^{-2}, 100\right)$ & 11.3$^{+12.0}_{-6.4}$ & Timescale of the GP\\ $\gamma_{\rm SPIRou}$ (m/s) & $\mathcal{U}\left(-33995,-33975\right)$ & $-33985$ $\pm$ 2 & SPIRou RV systemic component\\ $\sigma_{\rm SPIRou}$ (m/s) & $\mathcal{U}\left(0, 10\right)$ & 2.3 $\pm$ 1.3 & SPIRou RV extra white noise$^{\rm c}$ \\ $q_{\rm 1,TESS}$ & $\mathcal{U}\left(0, 1\right)$ & 0.35$^{+0.27}_{-0.19}$ & TESS limb-darkening parameter$^{\rm d}$\\ $q_{\rm 2,TESS}$ & $\mathcal{U}\left(0, 1\right)$ & 0.37$^{+0.33}_{-0.24}$ & TESS limb-darkening parameter$^{\rm d}$\\ $D_{\rm TESS}$ & 1.0 (fixed) & 1.0 & TESS dilution factor\\ $M_{\rm TESS}$ & $\mathcal{N}\left(0, 0.1^2\right)$ & -0.00032 $\pm$ 0.00010 & TESS baseline flux\\ $\sigma_{\rm TESS}$ (ppm)& $\mathcal{LU}\left(1, 10\,000\right)$ & 15$^{+70}_{-12}$ & TESS extra white noise\\ $q_{\rm 1,PESTO}$ & $\mathcal{U}\left(0, 1\right)$ & 0.67$^{+0.21}_{-0.26}$ & PESTO limb-darkening parameter$^{\rm d}$\\ $q_{\rm 2,PESTO}$ & $\mathcal{U}\left(0, 1\right)$ & 0.46$^{+0.28}_{-0.25}$ & PESTO limb-darkening parameter$^{\rm d}$\\ $D_{\rm PESTO}$ & $\mathcal{N}\left(0.564, 0.0564^2\right)$ & 0.586 $\pm$ 0.040 & PESTO dilution factor\\ $M_{\rm PESTO}$ & $\mathcal{N}\left(0, 0.1^2\right)$ & 0.00013 $\pm$ 0.00018 & PESTO baseline flux\\ $\sigma_{\rm PESTO}$ (ppm)& $\mathcal{LU}\left(1, 10\,000\right)$ & 2287 $\pm$ 67 & PESTO extra white noise \\ \multicolumn{4}{c}{$\mathbf{\vdots}$}\\ \multicolumn{4}{c}{MuSCAT3 photometric parameters in Table~\ref{table:muscat3params}}\\ \hline \multicolumn{4}{c}{\textit{Derived parameters}}\\ $R_{\rm p}$ (R$_{\oplus}$) & --- & 1.672 $\pm$ 0.071 & Planetary radius\\ $M_{\rm p}$ (M$_{\oplus}$) & --- & 4.82 $\pm$ 1.30 & Planetary mass\\ $\rho$ (g/cm$^3$) & --- & 5.6$^{+1.8}_{-1.6}$ & Planetary bulk density\\ $a$ (au) & --- & 0.061 $\pm$ 0.003 & Orbital semi-major axis\\ $\delta \equiv \left(R_{\rm p}/R_{\star} \right)^2$ & --- & 3.09 $\pm$ 0.16 & Transit depth\\ $b$ & --- & 0.19 $\pm$ 0.13 & Transit impact parameter\\ $i$ ($^{\circ}$) & --- & 89.77 $\pm$ 0.16 & Orbital inclination\\ $T_{\rm eq}$ (K) & & & Equilibrium temperature\\ \hspace{0.25cm} $\left[A_{\rm B} = 0\right]$ & --- & 326 $\pm$ 7 & \\ \hspace{0.25cm} $\left[A_{\rm B} = 0.3\right]$ & --- & 298 $\pm$ 6 & \\ \hspace{0.25cm} $\left[A_{\rm B} = 0.77\right]$ & --- & 226 $\pm$ 5 & \\ $S$ (S$_{\oplus}$) & --- & 1.8 $\pm$ 0.2 & Insolation\\ \enddata \tablecomments{$^{\rm a}\mathcal{U}\left(a,b \right)$ is the uniform distribution between value $a$ and $b$. $\mathcal{LU}\left(a,b \right)$ is the log-uniform (Jeffreys) distribution between value $a$ and $b$. $\mathcal{N}\left(\mu, \sigma^2 \right)$ is the normal distribution with mean $\mu$ and variance $\sigma^2$.\\ $^{\rm b}$Parametrization from \cite{Espinoza_2018}.\\ $^{\rm c}$White noise term for single exposures within polarimetric sequences. \\ $^{\rm d}\{q_1, q_2\}$ are linked to the quadratic limb-darkening coefficients $\{u_1, u_2\}$ through the transformations outlined in \cite{Kipping_2013}.} \label{table:modelparams} \end{deluxetable*} \begin{deluxetable*}{cccccc} \tablecaption{Prior and posterior distributions of the MuSCAT3 photometric parameters for model $\mathcal{M}_{\rm 1cp+GP}$ (details in Sect.\ \ref{sec:jointfit}) using only the SPIRou radial velocities} \tablehead{ \colhead{Parameter} & \colhead{Prior} & \colhead{$g^{\prime}$} & \colhead{$r^{\prime}$} & \colhead{$i^{\prime}$} & \colhead{$z_{\rm s}$}} \startdata \multicolumn{6}{c}{\textit{Fitted parameters}}\\ $q_1$ & $\mathcal{U}\left(0, 1\right)$ & 0.60$^{+0.26}_{-0.31}$ & 0.22$^{+0.27}_{-0.15}$ & 0.46$^{+0.32}_{-0.27}$ & 0.24$^{+0.29}_{-0.16}$\\ $q_2$ & $\mathcal{U}\left(0, 1\right)$ & 0.59$^{+0.27}_{-0.33}$ & 0.37$^{+0.35}_{-0.24}$ & 0.24$^{+0.28}_{-0.16}$ & 0.27$^{+0.33}_{-0.19}$ \\ $D$ & $\mathcal{U}\left(0.5, 1\right)$ & 0.71 $\pm$ 0.12 & 0.88$^{+0.07}_{-0.09}$ & 0.96$^{+0.03}_{-0.05}$ & 0.85 $\pm$ 0.07\\ $M$ & $\mathcal{N}\left(0, 0.1^2\right)$ & -0.00141 $\pm$ 0.00036 & -0.00106 $\pm$ 0.00016 & -0.00130 $\pm$ 0.00014 & -0.00131 $\pm$ 0.00013\\ $\sigma$ (ppm)& $\mathcal{LU}\left(1, 10\,000\right)$ & 2748 $\pm$ 319 & 46$^{+427}_{-43}$ & 1837$^{+191}_{-206}$ & 29$^{+227}_{-26}$ \\ \hline \multicolumn{6}{c}{\textit{Derived parameters}}\\ $\delta_{\rm uncorr.}$ (ppt) & --- & 2.18$^{+0.40}_{-0.36}$ & 2.72$^{+0.21}_{-0.24}$ & 2.94 $\pm$ 0.17 & 2.64 $\pm$ 0.20 \enddata \tablenocomments{} \label{table:muscat3params} \end{deluxetable*} \section{Discussion} \label{sec:discussion} \subsection{Planet composition} \label{sec:composition} The analysis of the transit and RV data yields a mass of $4.82 \pm 1.30$\,M$_{\oplus}$ and a radius of $1.672 \pm 0.071$\,R$_{\oplus}$, which together convert into a planetary bulk density of 5.6$^{+1.8}_{-1.6}$\,g/cm$^3$. A density similar to that of the Earth (5.5\,g/cm$^3$) for a planet that has more mass is indicative of an object composed of lighter material. We placed these measurements in a mass-radius diagram (Fig.\ \ref{fig:MR}), along with various theoretical composition curves obtained by the interior structure model of \citealt{Valencia2007, Valencia2013,Plotnykov2020}. We populated this diagram, and the figures in the discussion below, with data from the NASA Exoplanet Archive \citep{Akeson_2013} using the \texttt{exofile} tool\footnote{\href{https://github.com/AntoineDarveau/exofile}{\texttt{github.com/AntoineDarveau/exofile}}}. By comparing the mass and size of TOI-1452\,b to theoretical M-R curves in Figure~\ref{fig:MR}, we see three possibilities for the nature of this planet: (1) an ocean planet, (2) a bare rock with an iron content less than that of Earth, or (3) a terrestrial planet with a thin, low molecular weight atmosphere (e.g., H-He). The water world hypothesis is supported by a temperate equilibrium temperature for TOI-1452\,b of $298 \pm 6$\,K assuming an Earth-like Bond albedo ($A_{\rm B} = 0.3$), and between 226 and 326\,K for extreme $A_{\rm B}$ of 0.77 (Venus-like) and 0 (pure absorber). The insolation level of TOI-1452\,b is about 80\% higher than the Earth ($S = 1.8 \pm 0.2$\,S$_{\oplus}$), similar to Venus ($S = 1.91$\,S$_{\oplus}$). Focusing on the first two possibilities, we used an MCMC approach (\texttt{emcee}, \citealt{Foreman-Mackey_2013}) coupled to an interior structure model \citep{Valencia2007} to obtain the distribution of mass fractions of iron and water that are consistent with the data. The details of this modeling can be found in \citet{Plotnykov2020}. The H$_2$O layer is described by the equation of state from \citealt{Hemley1987, Stewart_2005}. We report below the 16$^{\rm th}$, 50$^{\rm th}$, and 84$^{\rm th}$ percentiles of the posterior distributions (available in Appendix~\ref{mcmc:summary}). From the chemical analysis of the star (Sect.\ \ref{sec:stellar_char_spirou}), we obtain chemical ratios relevant to the planetary interior. Notably, TOI-1452 might have a slightly lower Fe/Mg weight ratio compared to the Sun (see Table~\ref{table:ratios}), but in fair agreement with that of a sample of $\sim$1000 M dwarfs from the APOGEE (DR16) catalog (\citealt{Majewski_2016}; \citealt{Ahumada_2020}) with known chemical ratios (see Fig.\ \ref{fig:Fe2Mg}). The APOGEE abundances are derived from high-resolution near-infrared spectroscopy ($R \sim 22\,000$), with typical uncertainty on [Fe/H] and [Mg/H] of 0.02\,dex. Obtaining the Si ratios for TOI-1452 was difficult given the scarcity of spectral lines. In addition, we derive a C/O weight ratio consistent with the solar value. Our chemical abundance ratios for TOI-1452 are summarized in Table~\ref{table:ratios}. To infer the planet's composition for scenarios 1 and 2, we can either make no assumptions on the refractory ratios, and thus, obtain all possible compositions that fit the mass-radius data; or, assume that the refractory ratios of planet and star are related, and use the star's ratios as priors in the Bayesian analysis. Given that the refractory ratios of super-Earths seem to span a larger range than that of stars \citep{Plotnykov2020}, we applied both methods here. In both cases, we kept the Mg/Si ratio in the planetary mantle the same as the star. This assumption should not affect the results considerably, given that the mantle minerals formed by different Mg/Si ratios have similar equations of state and thus, the relative content of Mg to Si is not constrained by planetary mass and radius data \citep{Plotnykov2020}. In the case where we make no assumptions, we obtain a water-mass fraction (WMF) of $0.27^{+0.20}_{-0.15}$, and core-mass fraction (CMF) of $0.30^{+0.20}_{-0.17}$. These values translate to $ \mathrm{Fe/Si} = 2.9^{+5.4}_{-1.9}$ and $\mathrm{Fe/Mg} = 3.4 ^{+6.3}_{-2.2}$ by weight. The planetary refractory ratios are particularly large primarily due to large uncertainty in planetary mass but also the degeneracy that ensues when considering water. The tail of the Fe/Mg distribution is long because this case allows for little to no mantle in the planet (see Fig.\ \ref{fig:corner}). If instead, we assume the refractory ratios of the star as priors, we obtain a lower water and core mass fractions: WMF = $0.22^{+0.21}_{-0.13}$, CMF = $0.18 \pm 0.06$, resulting in Fe/Si = $1.3 \pm 0.4$ and Fe/Mg = $1.5 \pm 0.4$. Thus, both abundance scenarios yield a non-zero, yet poorly-constrained, WMF. The large uncertainty on both the CMF and WMF is rooted to the modest mass constraint (3.7$\sigma$). Better mass measurements are needed to confirm that TOI-1452\,b has a significant WMF. A different possibility (scenario 2 discussed above) is that this planet is a bare rock with no significant atmosphere, perhaps because it lost any acquired water through atmospheric evaporation during the high insolation phase of the M dwarf host star (\citealt{Bolmont2017,Barnes2013}). In this case, we constrain the physical model solutions but make no assumptions on the refractory ratios. The results show a $\mathrm{CMF} = 0.19^{+0.18}_{-0.12}$, with Fe/Si = $1.0^{+1.0}_{-0.5}$ and Fe/Mg = $1.2^{+1.2}_{-0.6}$ by weight. This CMF indicates a planet not as dense as the Earth, with refractory ratios still consistent at the 1-$\sigma$ level with those of its host star (see Table \ref{table:ratios} and Fig.\ \ref{fig:Fe2Mg}). However, the maximum a posteriori estimate of Fe/Mg ($\sim$0.8) for this bare rock scenario is lower than that of its host star. Forming planets that are iron poor with respect to their host star is difficult \citep{Scora2020}. Therefore, based on our current knowledge of planet formation and the chemical characteristics of the star, this scenario is less likely. A summary of this interior modeling is presented in Table~\ref{table:ratios} and Figure~\ref{fig:Fe2Mg}. The posterior distributions for the three models (no assumptions, stellar priors and bare rock) are shown in Appendix~\ref{mcmc:summary}. Outside of these scenarios, our observations do not rule out other structures with a low molecular weight atmosphere (scenario 3) such as an Earth-like interior surrounded by an H-He envelope at $T = 300\,K$ containing $\sim$0.1\% of the total mass (up to 0.5\% H-He at 1-$\sigma$). One way to firmly break the degeneracy in planetary internal structures would be to characterize the atmosphere of TOI-1452\,b. \begin{deluxetable*}{cccccc} \tablecaption{Chemical ratios by weight for the TOI-1452 system} \tablehead{ \colhead{Chemical} & \colhead{TOI-1452} & \colhead{Sun$\dag$} & TOI-1452\,b & TOI-1452\,b & TOI-1452\,b \\[-0.1cm] Ratios & & & No assumptions & Stellar priors & Bare rock \\ } \startdata Fe/Mg & $1.48^{+0.36}_{-0.29}$ & 1.83$\pm$0.25 & $3.4^{+6.3}_{-2.2}$ & $1.5^{+0.4}_{-0.4}$& $1.2^{+1.2}_{-0.6}$\\ Mg/Si & $0.86^{+0.37}_{-0.26}$ & 1.06$\pm$0.13 & 0.86$^*$ & 0.86$^*$& 0.86$^*$ \\ C/O & $0.48^{+0.11}_{-0.09}$ & 0.41$\pm$0.07 & ${\hat{}}$ & ${\hat{}}$ & ${\hat{}}$ \\ \hline CMF & -- & -- & $0.30^{+0.20}_{-0.17}$ & $0.18^{+0.06}_{-0.06}$ & $0.19^{+0.18}_{-0.12}$ \\ WMF & -- & -- & $0.27^{+0.20}_{-0.15}$ & $0.22^{+0.21}_{-0.13}$ & -- \\ \enddata \tablecomments{ $^{\dag}$ Photospheric abundance ratios from \citet{Asplund2009}.\\ $^{*}$ Mg/Si ratio of the planets are fixed to the star TOI-1452 ratio.\\ $\hat{}\ $ The interior models assume no carbon compounds. \\ } \label{table:ratios} \end{deluxetable*} \subsection{Atmospheric characterization prospect} \label{sec:atmo_char} TOI-1452\,b is a prime target for follow-up transit spectroscopy with JWST. The system is located near Webb's Continuous Viewing Zone (CVZ), more precisely at a few degrees ($\sim$10$^{\circ}$) off the Northern CVZ, which means that it could be observed most of the year. Moreover, TOI-1452\,b is one of the few identified super-Earths in a temperate regime ($T_{\rm eq}$ between 200 and 400\,K) orbiting a relatively bright star amenable to transmission spectroscopy observations (see Figure~\ref{fig:TSM} and Table~\ref{table:temperate_planets}). The expected strength of the atmospheric signal is characterized by the Transmission Spectroscopy Metric (TSM, \citealt{Kempton_2018}), which is proportional to the host star's $J$ magnitude and the planet's atmospheric scale height. Figure~\ref{fig:TSM} displays the TSM as a function of equilibrium temperature for known small exoplanets with available mass measurements. The sample is restricted to systems with well-determined masses (relative uncertainty $< 30$\%) since a constraint on surface gravity is essential to correctly interpret the transmission spectrum of an exoplanet \citep{Batalha_2019}. The temperate subset of Figure~\ref{fig:TSM} is detailed in Table~\ref{table:temperate_planets}. The TSM of TOI-1452\,b (39.9) is similar to well-known temperate systems such as LHS 1140\,b (50.0) and K2-18\,b (40.8), while being 60\% below the highest listed target in this subset, L231-32\,d (TOI-270\,d, 104.0). All seven host stars in Table~\ref{table:temperate_planets} have $T_{\rm eff}<4000$\,K (or average $T_{\rm eff}$ of 3225\,K), confirming the high interest of M dwarfs for planetary atmospheric characterization. Note that the high-value target L~98-59\,d (TSM above 200, \citealt{Cloutier_2019}; \citealt{Demangeon_2021}) was just barely excluded from Table~\ref{table:temperate_planets} due to its $T_{\rm eq}$ (409\,K) being slightly above our 400\,K cut. Our subset also excludes intriguing planets with plausible temperate environment, but deprived of mass measurement (or imprecise mass), such as TOI-700\,c (TSM = 79.7, \citealt{Gilbert_2020}), TOI-1266\,c (TSM = 48.8, \citealt{Demory_2020}, and K2-3\,c (TSM = 25.5, \citealt{Damasso_2018}). The TOI-1452 system is a unique target to explore the atmospheric properties of temperate planets within the radius valley. This paper provides the first mass determination needed for the interpretation of future transmission spectra. \begin{deluxetable*}{ccccccccccc} \centering \tablecaption{Transmission Spectroscopy Metric (TSM) for a subset of well-characterized small exoplanets in a temperate equilibrium temperature regime (200\,K $\leq T_{\rm eq} \leq$ 400\,K)} \tablehead{\colhead{Planet} & \colhead{$P$} & \colhead{$M_{\rm p}$} & \colhead{$R_{\rm p}$} & \colhead{$T_{\rm eq}$\,$[A_{\rm B} = 0]$} & \colhead{$J$} & \colhead{$T_{\rm eff}$} & \colhead{$M_{\star}$} & \colhead{$R_{\star}$} & \colhead{TSM} & \colhead{Ref.}\\[-0.1cm] & (days) & (M$_{\oplus}$) & (R$_{\oplus}$) & (K) & (mag) & (K) & (M$_{\odot}$) & (R$_{\odot}$) & & } \startdata L231-32\,d & 11.380 & 4.78 & 2.133 & 388 & 9.10 & 3506 & 0.39 & 0.38 & 104.0 & (1) \\ TOI-1231\,b & 24.246 & 15.4 & 3.65 & 331 & 8.88 & 3553 & 0.48 & 0.48 & 97.6 & (2)\\ LTT~3780\,c & 12.252 & 8.6 & 2.30 & 353 & 9.01 & 3331 & 0.40 & 0.37 & 72.6 & (3)\\ LHS 1140\,b & 24.739 & 6.38 & 1.635 & 214 & 9.61 & 2988 & 0.19 & 0.21 & 50.0 & (4)\\ K2-18\,b & 32.940 & 8.63 & 2.610 & 279 & 9.76 & 3457 & 0.50 & 0.44 & 40.8 & (5)\\ \textbf{TOI-1452\,b} & 11.062 & 4.82 & 1.672 & 326 & 10.60 & 3185 & 0.25 & 0.28 & 39.9 & (6)\\ TRAPPIST-1\,b & 1.511 & 1.374 & 1.116 & 399 & 11.35 & 2566 & 0.09 & 0.12 & 28.6 & (7)\\ TRAPPIST-1\,d & 4.049 & 0.388 & 0.788 & 287 & 11.35 & 2566 & 0.09 & 0.12 & 25.6 & (7)\\ TRAPPIST-1\,c & 2.422 & 1.308 & 1.097 & 341 & 11.35 & 2566 & 0.09 & 0.12 & 24.3 & (7)\\ TRAPPIST-1\,e & 6.101 & 0.692 & 0.920 & 251 & 11.35 & 2566 & 0.09 & 0.12 & 19.9 & (7)\\ LHS 1140\,c & 3.777 & 1.76 & 1.169 & 400 & 9.61 & 2988 & 0.19 & 0.21 & 18.7 & (4)\\ TRAPPIST-1\,f & 9.208 & 1.039 & 1.045 & 218 & 11.35 & 2566 & 0.09 & 0.12 & 17.0 & (7)\\ \enddata \tablerefs{(1) \cite{VanEylen_2021}. (2) \cite{Burt_2021}. (3) \cite{Cloutier_2020a}. (4) \cite{Lillo-Box_2020}. (5) \cite{Benneke_2019}. (6) This work. (7) \cite{Agol_2021}.} \label{table:temperate_planets} \end{deluxetable*} \subsection{Implications for the emergence of the M dwarf radius valley} \label{sec:radius_valley} Planets on either side of the radius valley differ by their composition, typically `rocky' for the smaller super-Earths, and `gaseous' for the larger mini-Neptunes. This transition occurs as a consequence of a varying envelope mass fraction: adding an H-He envelope up to a few percents of the total mass of a planet essentially doubles its observable radius (\citealt{Lopez_2014}; \citealt{Chen_2016}). Thermally-driven atmospheric escape processes such as photoevaporation (\citealt{Owen_2013}, \citeyear{Owen_2017}; \citealt{Lopez_2014}; \citealt{Lopez_2018}; \citealt{Wu_2019}) and core-powered mass loss (\citealt{Ginzburg_2018}; \citealt{Gupta_2019}, \citeyear{Gupta_2020}) have been proposed as radius valley emergence mechanisms. In these models, super-Earths and mini-Neptunes originate from the same population of planets that form with an extended H-He envelope around Earth-like core, with the population of rocky super-Earths emerging after losing their primordial atmospheres to hydrodynamic escape. Another possible scenario is to assemble rocky super-Earths at late times after most or all of the gas has been dissipated from the protoplanetary disk (\citealt{Lee_2014}; \citealt{Lopez_2018}; \citealt{Lee_2021}). The two classes of planets would form on different timescales, resulting in a bimodal distribution without relying on any subsequent atmospheric escape. Each of the aforementioned mechanisms predicts that the rocky-to-gaseous transition ($R_{\rm valley}$) varies with parameters such as orbital period $P$ and stellar mass $M_{\star}$. Photoevaporation, core-powered mass loss, and gas-poor accretion models predict a negative slope in $R_{\rm p}$--$P$ space, respectively $R_{\rm valley} \propto P^{-0.25 \textendash 0.15}$ (\citealt{Owen_2017}; \citealt{Lopez_2018}; \citealt{Mordasini_2020}), $R_{\rm valley} \propto P^{-0.11}$ \citep{Gupta_2019}, and $R_{\rm valley} \propto P^{-0.08}$ \citep{Lee_2021}. Conversely, the formation of super-Earths strictly by the merging of planetary embryos in a gas-depleted environment, analogous to the formation of terrestrial planets in the Solar System, would produce a positive slope ($R_{\rm valley} \propto P^{0.11}$, \citealt{Lopez_2018}). One way to test the proposed models is to compare these predictions to the real population of exoplanets. From occurrence rate calculations of small close-in planets around Sun-like stars, \cite{Martinez_2019} measured a $d \log R_{\rm valley} / d \log P = -0.11 \pm 0.02$, consistent with thermally-driven mass loss and gas-poor formation. Using a similar methodology but for planets around low-mass stars with $T_{\rm eff} < 4700$\,K, \cite{Cloutier-Menou_2020} obtained a $d \log R_{\rm valley} / d \log P = 0.058 \pm 0.022$, where the positive sign suggests that the gas-depleted formation of super-Earths may be dominant around M dwarfs. These distinct slope measurements carve out regions in the $R_{\rm p}$--$P$ parameter space where the models make opposing predictions regarding the bulk composition of a planet (i.e., either rocky or gaseous). This framework to test radius valley emergence models around M dwarfs was introduced in \cite{Cloutier-Menou_2020} and has since been applied to a number of transiting planets (TOIs 776\,b; \citealt{Luque_2021}, 1235\,b; \citealt{Cloutier_2020b}, 1634\,b; \citealt{Cloutier_2021}, 1685\,b; \citealt{Bluhm_2021}). Figure~\ref{fig:RP} presents the period--radius diagram for exoplanets around M-dwarf hosts ($T_{\rm eff} < 4000$\,K). Each planet is color-coded by its bulk density relative to the Earth-like structure model of \citealt{Valencia2007} (see Figure~\ref{fig:MR}). TOI-1452\,b sits on or slightly above the empirical \textit{valley} of \cite{Cloutier-Menou_2020}, while being considerably below the slope measured by \cite{Martinez_2019}, scaled down to match the median stellar mass of the \cite{Cloutier-Menou_2020} sample (using Equation 11 therein). The locus of TOI-1452\,b in Figure~\ref{fig:RP}, combined with our density estimate, are incompatible with the photoevaporation and core-powered mass loss models. However, the likely intermediate nature of TOI-1452\,b cannot strongly support the alternative gas-depleted formation scenario either as the dominant mechanism for the emergence of the M-dwarf radius valley. A volatile-rich interior for TOI-1452\,b could indicate a different formation pathway, e.g., one without significant gas accretion during the disk lifetime. Figure~\ref{fig:RP} also highlights three other systems presenting similarities with TOI-1452\,b, namely TOI-1235\,b \citep{Cloutier_2020b}, L~98-59\,d (\citealt{Cloutier_2019}; \citealt{Demangeon_2021}) and LHS 1140\,b (\citealt{Dittmann_2017}; \citealt{Lillo-Box_2020}). All four planets have a similar size, while spanning a large interval in periods. TOI-1235\,b ($P = 3.445$\,days, $R_{\rm p} = 1.738$\,R$_{\oplus}$) and LHS 1140\,b ($P = 24.737$\,days, $R_{\rm p} = 1.635$\,R$_{\oplus}$) have densities compatible with bona fide super-Earths; their position in Figure~\ref{fig:RP} indicates that they are probably examples of the largest terrestrial planets that can be assembled around M dwarfs without accreting a substantial hydrogen envelope. On the other hand, L~98-59\,d ($P = 7.451$\,days, $R_{\rm p} = 1.521$\,R$_{\oplus}$) is a likely water-rich ($\sim$30\%) planet that may be approaching, like TOI-1452\,b, the minimum size for volatile-rich objects around a low-mass star. These four systems constitute benchmarks for understanding the formation and evolution of planets within the radius valley. \section{Summary \& Conclusion} \label{sec:conclusion} This paper reports the discovery and characterization of the transiting temperate exoplanet TOI-1452\,b. A joint analysis of transit observations from TESS and other ground-based telescopes combined with radial velocity measurements from SPIRou and IRD, yields a mass of $4.82 \pm 1.30$\,M$_{\oplus}$ and a radius of $1.672 \pm 0.071$\,R$_{\oplus}$. These physical parameters are consistent with either a rocky world with a Fe/Mg ratio similar to the host star (Fe/Mg $=1.2^{+1.2}_{-0.6}$ by weight), a water-rich interior (either $22^{+21}_{-13} \%$ H$_2$O by weight, if stellar priors are assumed for the planetary refractory ratios, or $27^{+20}_{-15} \%$ H$_2$O if no assumptions are made) or a terrestrial planet surrounded by a $\lesssim $1\% H-He atmosphere. Orbiting its M4 host star ($T_{\rm eff} = 3185 \pm 50$\,K) every $11.06201 \pm 0.00002$\,days, the planet receives about twice as much radiation than the Earth ($S = 1.8 \pm 0.2$\,S$_{\oplus}$), corresponding to a blackbody temperature of $326\pm7$\,K. The results of our interior modeling and the fact that the planet receives modest irradiation make TOI-1452\,b a good candidate water world. TOI-1452\,b is a prime target for upcoming atmospheric characterization efforts with JWST, featuring a high Transmission Spectroscopy Metric compared to other known temperate exoplanets. Transit spectroscopy observations with JWST should reveal the true nature of this intriguing exoplanet lying within the radius valley, whether this is a rocky world or one with a volatile envelope. Being observable with JWST most of the year, TOI-1452\,b is a unique system for studying exoplanets at the transition between super-Earths and mini-Neptunes. \acknowledgments We acknowledge the use of public TESS Alert data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. This paper includes data collected by the TESS mission that are publicly available from the Mikulski Archive for Space Telescopes (MAST). \par Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT) which is operated from the summit of Maunakea by the National Research Council of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientique of France, and the University of Hawaii. The observations at the Canada-France-Hawaii Telescope were performed with care and respect from the summit of Maunakea which is a signicant cultural and historic site. \par Based on observations obtained at the Observatoire du Mont-Mégantic, financed by Université de Montréal, Université Laval, the Canada Economic Development program and the Ministère de l'Économie et de l'Innovation. \par This paper is also based on observations made with the MuSCAT3 instrument, developed by the Astrobiology Center and under financial supports by JSPS KAKENHI (JP18H05439) and JST PRESTO (JPMJPR1775), at Faulkes Telescope North on Maui, HI, operated by the Las Cumbres Observatory. \par This work is partly supported by the Natural Science and Engineering Research Council of Canada and the Institute for Research on Exoplanets through the Trottier Family Foundation \par This work is partly supported by MEXT/JSPS KAKENHI Grant Numbers JP22000005, JP15H02063, JP17H04574, JP18H05439, JP18H05442, JP19K14783, JP21H00035, JP21K13975, JP21K20376, JST CREST Grant Number JPMJCR1761, and the Astrobiology Center of National Institutes of Natural Sciences (NINS) (Grant Numbers AB031010, AB031014). \par We thank Dr.\ Martin Turbet for the suggestions to improve the discussion section. \par We acknowledge very useful feedback and discussion from Dr.\ Ansgar Reiners, regarding the importance to properly check the contamination from the nearby companion in the SPIRou and IRD spectra and its effect on the final radial velocities. \par JFD acknowledges funding from the European Research Council (ERC) under the H2020 research \& innovation programme (grant agreement \#740651 NewWorlds). \par This work has been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation. \par PC thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant \#1829740, the Brinson Foundation, and the Moore Foundation; her participation in the program has benefited this work. \par A.C. and X.D. acknowledges funding from the ANR of France under contract number ANR\-18\-CE31\-0019 (SPlaSH). This work is supported by the French National Research Agency in the framework of the \textit{Investissements d'Avenir} program (ANR-15-IDEX-02), through the funding of the ``Origin of Life" project of the Grenoble-Alpes University. \par F.K. acknowledge the ANR project [SPlaSH] from the French Agence Nationale de Recherche with reference ANR-18-CE31-0019-02 \par J.H.C.M. is supported in the form of a work contract funded by Fundação para a Ciência e Tecnologia (FCT) with the reference DL 57/2016/CP1364/CT0007; and also supported from FCT through national funds and by FEDER-Fundo Europeu de Desenvolvimento Regional through COMPETE2020- \textit{Programa Operacional Competitividade e Internacionalização} for these grants UIDB/04434/2020 \& UIDP/04434/2020, PTDC/FIS-AST/32113/2017 \& POCI-01-0145-FEDER-032113, PTDC/FIS-AST/28953/2017 \& POCI-01-0145-FEDER-028953, PTDC/FIS-AST/29942/2017. \par This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 716155 (SACCRED). \par KG acknowledges the partial support from the Ministry of Science and Higher Education of the RF (grant 075-15-2020-780). \par TV acknowledges funding from the \textit{Fonds de Recherche du Qu\'ebec - Nature et Technologie} (FRQNT). \facilities{TESS, OMM-PESTO, MuSCAT3, KeckII/NIRC2, CFHT/SPIRou, Subaru/IRD} \software{\texttt{emcee} \citep{Foreman-Mackey_2013}; \texttt{Astropy} \citep{Astropy_2018}; \texttt{radvel} \citep{Fulton_2018}; \texttt{matplotlib} \citep{Hunter_2007}; \texttt{celerite} \citep{celerite1_2017}; \texttt{celerite2} (\citealt{celerite1_2017}; \citeyear{celerite2_2018}); \texttt{juliet} \citep{Espinoza_2019}; \texttt{batman} \citep{Kreidberg_2015}; \texttt{SciPy} \citep{Virtanen_2020}; \texttt{NumPy} \citep{Harris_2020}; \texttt{photutils} \citep{Larry_Bradley_2020}; \texttt{Tapir} \citep{Jensen_2013}; \texttt{AstroImageJ} \citep{Collins_2017}}; \texttt{TRILEGAL} \citep{Girardi_2012}. \bibliography{TOI-1452.bib}{} \bibliographystyle{aasjournal.bst} \includeAffiliations \allauthors \appendix \counterwithin{table}{section} \counterwithin{figure}{section} \section{TESS Light Curve} We present the TESS multi-sector \texttt{PDCSAP} light curve in Figure~\ref{fig:TESS_lc_complete}, with the exception of sectors 14 and 21, which were previously shown in Figure~\ref{fig:TESS_lc_phase}. \setcounter{figure}{0} \section{Supplementary material regarding the joint transit-RV fit} In this appendix, we summarize the RV component of the joint transit-RV models introduced in Section~\ref{sec:jointfit}. The main RV parameters of the joint fits are reported in Table~\ref{table:rvcomponent}. All models and datasets detect the planetary signal with a coherent semi-amplitude $K$. Models with an activity GP ($\mathcal{M_{\rm 1cp+GP}}$ and $\mathcal{M_{\rm 1ep+GP}}$) produced the highest Bayesian log-evidence (Fig.~\ref{fig:modelselect}) and needed the smallest amount of additional white noise ($\sigma_{\rm SPIRou}$, $\sigma_{\rm IRD}$). We ultimately adopted the results of the SPIRou only joint fit because including the seven IRD RV measurements yields similar or smaller Bayesian log-evidences (Fig.~\ref{fig:modelselect}), with extra white noise term $\sigma_{\rm IRD}$ about three times the level of the planetary signal (see Table~\ref{table:rvcomponent}). \begin{table}[b!] \end{table} \begin{deluxetable}{lcccc} \tablecaption{RV component of the joint transit-RV fit for different models ($\mathcal{M}$) and datasets} \tablehead{\colhead{Parameter} & \colhead{1cp} & \colhead{1cp+GP} & \colhead{1ep} & \colhead{1ep+GP} } \startdata \multicolumn{5}{c}{\textit{SPIRou only}}\\ $K$ (m/s) & 4.2$\pm$0.9 & 3.5$\pm$0.9 & 4.7$\pm$0.9 & 3.6$\pm$0.9\\ $e$ & --- & --- & 0.20$\pm$0.09 & 0.12$^{+0.12}_{-0.08}$\\ $A_{\rm GP}$ (m/s) & --- & 4.5$^{+2.0}_{-1.2}$ & --- & 4.4$^{+1.9}_{-1.2}$\\ $\ell_{\rm GP}$ (days) & --- & 11.3$^{+12.0}_{-6.4}$ & --- & 11.6$^{+12.4}_{-6.4}$\\ $\sigma_{\rm SPIRou}$ (m/s) & 4.9$\pm$0.9 & 2.3$\pm$1.3 & 4.7$^{+0.9}_{-1.0}$ & 2.2$\pm$1.3\\ \hline \multicolumn{5}{c}{\textit{SPIRou\,+\,IRD}}\\ $K$ (m/s) & 4.1$\pm$0.9 & 3.5$\pm$0.9 & 4.7$\pm$0.9 & 3.6$\pm$0.9\\ $e$ & --- & --- & 0.19$\pm$0.09 & 0.12$^{+0.12}_{-0.08}$\\ $A_{\rm GP}$ (m/s) & --- & 4.7$^{+2.3}_{-1.3}$ & --- & 4.6$^{+2.2}_{-1.2}$\\ $\ell_{\rm GP}$ (days) & --- & 11.1$^{+11.2}_{-6.2}$ & --- & 11.7$^{+11.2}_{-6.2}$\\ $\sigma_{\rm SPIRou}$ (m/s) & 4.9$\pm$0.9 & 2.3$\pm$1.3 & 4.7$^{+0.9}_{-1.0}$ & 2.3$^{+1.2}_{-1.3}$\\ $\sigma_{\rm IRD}$ (m/s) & 13.6$^{+5.4}_{-3.6}$ & 11.5$^{+5.7}_{-4.3}$ & 14.2$^{+5.7}_{-3.8}$ & 11.9$^{+5.5}_{-4.1}$\\ \enddata \tablecomments{$\mathcal{M_{\rm 1cp}}$: single circular orbit planet\\ $\mathcal{M_{\rm 1cp+GP}}$: single circular orbit planet and activity GP\\ $\mathcal{M_{\rm 1ep}}$: single eccentric orbit planet\\ $\mathcal{M_{\rm 1ep+GP}}$: single eccentric orbit planet and activity GP} \label{table:rvcomponent} \end{deluxetable} \clearpage \section{Summary of interior parameters} \label{mcmc:summary} We present the summary plot for the interior analysis from MCMC modeling. \vspace{2cm} \clearpage \section{Radial Velocity Measurements} \setcounter{table}{0} \renewcommand{\thetable}{D.\arabic{table}} We present the radial velocity measurements of TOI-1452 from SPIRou and IRD in the online Table \ref{table:spirou_rv}. \begin{table}[b!] \end{table} \begin{deluxetable}{lccc} \tablecaption{SPIRou and IRD RV measurements \label{table:spirou_rv}} \tablehead{ \colhead{Instrument} & \colhead{BJD - 2\,400\,000} & \colhead{RV (m/s)} & \colhead{$\sigma_{\rm RV}$ (m/s)} } \startdata SPIRou & 59004.995291 & -33983.32 & 8.39 \\ SPIRou & 59005.006067 & -33975.16 & 8.12 \\ SPIRou & 59005.016836 & -33980.21 & 8.25 \\ SPIRou & 59005.027551 & -33995.65 & 8.28 \\ SPIRou & 59009.008115 & -33975.18 & 8.02 \\ SPIRou & 59009.018950 & -33974.66 & 8.20 \\ SPIRou & 59009.029725 & -33975.25 & 7.96 \\ \ldots & \ldots & \ldots & \ldots \\ IRD & 59118.766592 & 15.17 & 4.08 \\ IRD & 59122.769496 & 17.14 & 3.86 \\ IRD & 59122.787278 & 14.54 & 3.77 \\ IRD & 59156.837456 & -14.4 & 4.86 \\ IRD & 59372.001844 & -9.95 & 3.71 \\ IRD & 59373.904952 & 2.00 & 5.70 \\ IRD & 59390.850628 & 12.31 & 4.03 \\ \enddata \tablecomments{Table \ref{table:spirou_rv} is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.} \end{deluxetable}

Title: The ALMaQUEST Survey X: What powers merger induced star formation?

Abstract: Galaxy mergers are known to trigger both extended and central star formation. However, what remains to be understood is whether this triggered star formation is facilitated by enhanced star formation efficiencies, or an abundance of molecular gas fuel. This work presents spatially resolved measurements of CO emission collected with the Atacama Large Millimetre Array (ALMA) for 20 merging galaxies (either pairs or post-mergers) selected from the Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) survey. Eleven additional merging galaxies are selected from the ALMA MaNGA QUEnching and STar formation (ALMaQUEST) survey, resulting in a set of 31 mergers at various stages of interaction and covering a broad range of star formation rates (SFR). We investigate galaxy-to-galaxy variations in the resolved Kennicutt-Schmidt relation (rKS: $\Sigma_{H_2}$ vs. $\Sigma_{SFR}$), the resolved molecular gas main sequence (rMGMS: $\Sigma_{\star}$ vs. $\Sigma_{H_2}$), and the resolved star-forming main sequence (rSFMS: $\Sigma_{\star}$ vs. $\Sigma_{SFR}$). We quantify offsets from these resolved relations to determine if star formation rate, molecular gas fraction, and/or star formation efficiency (SFE) is enhanced in different regions of an individual galaxy. By comparing offsets in all three parameters we can discern whether gas fraction or SFE powers an enhanced $\Sigma_{SFR}$. We find that merger-induced star formation can be driven by a variety of mechanisms, both within a galaxy and between different mergers, regardless of interaction stage.

https://export.arxiv.org/pdf/2208.06426

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} galaxies: interactions -- galaxies: star formation -- galaxies: evolution \end{keywords} \section{Introduction} Evidence has consistently demonstrated that galaxy-galaxy mergers can trigger star formation: from the bluer colour of peculiar galaxies \citep{Larson1978StarGalaxies,BartonGillespie2003Tidally-TriggeredAges,Lambas2012GalaxyInteractions} and excess H$\alpha$ emission noted in interacting pairs \citep{Kennicutt1987THERATES,Knapen2009THESTARBURSTS}, to large single-fibre spectroscopic surveys revealing an excess of star formation as a merger event progresses \citep{Ellison2008GalaxyRelation, Woods2010TRIGGERED0.08-0.38, Scudder2012GalaxyKpc,Patton2013GalaxyGalaxies,Knapen2015InteractingFormation}. Hydrodynamical simulations illustrate how induced star formation likely stems from gas losing angular momentum and fueling centralized star formation, as a result of the non-axisymmetric structures generated by gravitational forces in an interaction \citep{Barnes1991FuelingMergers,Mihos1996GasdynamicsMergers,Iono2004RadialObservations,Hopkins2013StarMedium}. The degree to which star formation is enhanced can vary drastically depending on the properties of an interaction, with orbital parameters \citep{DiMatteo2007StarStudy,DOnghia2010QUASI-RESONANTINTERACTIONS,Moreno2015MappingFormation}, mass ratio between constituents \citep{Cox2006FeedbackMergers,Cox2008TheStarbursts} and gas fraction of the interacting disk \citep{Bournaud2011HYDRODYNAMICSSPHEROIDS,Perez2011ChemicalInteractions,Scudder2015GalaxyInteractions,Fensch2017High-redshiftFormation} all leading to varied star formation strengths. In simulations, star formation tends to peak when the interacting galaxies are either close to pericentric passage or at the moment of coalescence \citep{Lotz2008GalaxyMergers,Scudder2012GalaxyKpc, Hani2020InteractingStage}. Such results are supported by observational evidence, which has consistently demonstrated that global star formation rate values are greatest for interactions with equal mass ratios and small projected separations \citep{Nikolic2004StarSurvey,Lin2007AEGIS:1,Ellison2008GalaxyRelation,Woods2010TRIGGERED0.08-0.38,Scudder2012GalaxyKpc,Ellison2013GalaxyGalaxies,Bickley2022StarUNIONS}. Resolved spectroscopic studies have revealed further complexity of merger-induced star formation on a local scale. Integral Field Spectroscopy (IFS) surveys such as the MaNGA Survey (\citealt{Bundy2015OverviewObservatory}), the Calar Alto Legacy Integral Field Area Survey (CALIFA, \citealt{Sanchez2012CALIFASurvey}), and the Sydney-Australian Astronomical-Observatory Multi-object Integral field spectrograph Survey (SAMI, \citealt{Allen2015TheRelease}) have allowed for the collection of resolved spectroscopic data of large samples of galaxies from which robust statistical results can be attained. Investigations of interacting galaxy pairs in IFS surveys have further corroborated the central enhancement of star formation noted in global studies, but often find a diversity of behaviours in the outskirts of the galaxy where star formation can be unaffected, enhanced, or suppressed \citep{Barrera-Ballesteros2015CentralGalaxies,Pan2019SDSS-IVInteractions,Steffen2021SDSS-IVPairs}. Such variation may be linked to the global galaxy properties, with only the higher mass galaxy in a pair showing enhanced star formation in the outskirts \citep{Steffen2021SDSS-IVPairs}. However, evidence also suggests only galaxies at pericentre and coalescence have elevated star formation at large radii \citep{Pan2019SDSS-IVInteractions}. Significant variations in spatial star formation enhancement are therefore apparent even in a single stage of interaction. Whereas pairs of galaxies represent the pre-merger regime, post-merger galaxies allow us to study the late stages of the interaction sequence. On average, post-mergers have central star formation enhancements that moderately fall off with radius, but galaxies with similar central star formation enhancements can have enhanced, normal, or suppressed star formation in the outskirts \citep{Thorp2019SpatiallyMaNGA}. Just as the unique properties of a galaxy and its interaction parameters can lead to a diverse range of global star formation enhancement, there is equal if not greater complexity when star formation is examined on the resolved scale. A stepping stone towards understanding the variation of star formation seen in merging galaxies is to characterize the gas which fuels star formation on a kpc-scale. The tight correlation between star formation rate (SFR) and molecular gas mass (\Mgasend) which exists on a global scale (often called the Kennicutt-Schmidt relation for its flagship publications \citealt{Schmidt1959TheFormation} and \citealt{Kennicutt1989THEDISKS}) exists on a local scale as well, resulting in the resolved Kennicutt-Schmidt (rKS) \citep{Bigiel2008THESCALES,Leroy2008THEEFFECTIVELY,Schruba2011AGALAXIES}. A high star formation efficiency (SFE$=$SFR$/$\Mgasend) can lead to regions notably above the rKS from a local boost in star formation \citep{Leroy2013MOLECULARGALAXIES,Bolatto2017TheCARMA,Utomo2017THEGALAXIES}. Offsets from the rKS are the primary driver of offsets in the resolved star-forming main sequence (rSFMS, the strong correlation between \SigSFR and \Sigstarend; \citealt{Sanchez2013Mass-metallicityRate,Cano-Diaz2016Spatially-ResolvedSurvey,GonzalezDelgado2016AstrophysicsGalaxies}), demonstrating that although the star formation rate surface density (\SigSFRend) is regulated by molecular gas mass surface density (\Siggasend) as predicted by the rKS, more varied behaviour in star formation stems from changes in SFE \citep{Ellison2020TheEfficiency.}. However, there are other ways star formation can be augmented, such as a high gas surface density to fuel stellar growth \citep{Bigiel2008THESCALES,Leroy2013MOLECULARGALAXIES,Schruba2011AGALAXIES}. A surplus of molecular gas manifests as an offset from the correlation between \Siggas and the stellar mass surface density (\Sigstarend), otherwise known as the resolved molecular gas main sequence (rMGMS, \citealt{Wong2013CARMAPROPERTIES,Lin2019TheSequence,Ellison2021TheThem}). Environmental effects on the \Sigstar profile can also boost star formation \citep{Usero2015VARIATIONSGALAXIES,Gallagher2018DenseGalaxies,Jimenez-Donaire2019EMPIRE:Galaxies}. Examining how mergers deviate from the rKS and rMGMS will help discern what drives enhancements in star formation on a kpc-scale. Several attempts have been made to pinpoint the relative importance of the total molecular gas fraction (\fgasend$=$\Mgasend/\Mstarend) and the SFE to driving enhanced star formation in mergers. Early studies measuring total molecular gas mass with single dish telescopes found conflicting results concerning whether total molecular gas fraction or star formation efficiency drives enhanced star formation in mergers \citep{Braine1992AGalaxies, Casasola2004TheSystems,Huchtmeier2008InteractingSurvey}. More recent studies have leaned towards an enhanced gas fraction driving star formation in both interacting pair \citep{Violino2018GalaxyMergers,Pan2018TheProperties} and post-merger galaxies \citep{Ellison2018EnhancedExhaustion}. SFEs for merging galaxies are mostly normal, except for very close pairs and equal-mass systems whose violent interactions lead to enhanced SFEs \citep{Pan2018TheProperties}. What remains unclear is how these global properties drive kpc-scale variations in star formation. Only recently have observations of resolved molecular gas properties been collected for large samples to complement large optical IFS surveys, allowing for the combined analysis of \Siggas and \SigSFRend. The Extragalactic Database for Galaxy Evolution (EDGE) - CALIFA survey \citep{Bolatto2017TheCARMA} targeted 126 CALIFA galaxies with the Combined Array for Millimeter-wave Astronomy (CARMA) to investigate depletion time gradients within galaxies \citep{Utomo2017THEGALAXIES,Colombo2018TheSequence} and the processes which regulate star formation on a kpc-scale \citep{Barrera-Ballesteros2021EDGE-CALIFAScales}. The ALMA-MaNGA QUEnching and STar formation (ALMaQUEST) survey \citep{Lin2020ALMAQUEST:SURVEY} observed MaNGA galaxies with ALMA to not only confirm that the key scaling relations between star formation rate, molecular gas, and stellar mass that exist on a global scale stem from a kpc-scale relationship, but also that the resolved star-forming main sequence is likely the result of the two other relations \citep{Lin2019TheSequence, Ellison2021TheThem, Baker2022TheSequence}. ALMaQUEST has also revealed that even though the absolute star formation rate in a galaxy is primarily driven by the amount of molecular gas, the scatter in the resolved star formation efficiency is mostly driven by local changes in SFE \citep{Ellison2020The0}. Given that variations in star formation in the outskirts of mergers are by definition scatter from the rSFMS, such a result might imply that SFE may drive star formation for individual regions of galaxy, even if globally gas reservoir is the dominant driver. Further investigation of the interplay between star formation and molecular gas has been done on the scale of molecular clouds with the Physics at High Angular resolution in Nearby Galaxies (PHANGS)-ALMA survey \citep{Leroy2021PHANGS-ALMA:Galaxies}. With resolution on the order of $\sim$100pc, PHANGS-ALMA found greater scatter in all three scaling relations compared to lower-resolution studies, revealing significant variation in star formation and molecular gas content even within similar morphological environments \citep{Pessa2021AstronomyScale,Querejeta2021StellarGalaxies}. Although EDGE-CALIFA, the largest of surveys of this kind, does have a small number of mergers that may be responsible for changes in SFR and depletion times \citep{Bolatto2017TheCARMA,Utomo2017THEGALAXIES,Chown2019LinkingGalaxies}, none of the surveys with both IFS and molecular gas data have a sufficient number of mergers to make a dedicated study of interaction induced physics. A handful of detailed case-studies of the resolved star-formation efficiency of pre-merger galaxies have been completed, revealing significant diversity of depletion time on a resolved scale \citep{Tomicic2018Two2276,Bemis2019Kiloparsec-Scale4038/39}. A study dedicated to a diverse sample of merging and post-merger galaxies is required to better understand the most extreme variations in star formation and gas properties. In the present work we have observed a sample of 31 merging galaxies with a broad range of interaction progressions and star formation rates, with the specific goal of investigating how merger properties effect the mechanisms which drive star formation. Eleven of these mergers are taken from the main ALMaQUEST sample \citep{Lin2020ALMAQUEST:SURVEY}, plus we present observations for 20 additional galaxies with new CO(1-0) data obtained from ALMA. Together, we refer to this sample of 31 galaxies as the ALMaQUEST merger sample. In this paper, we aim to distinguish whether star formation efficiency or molecular gas fraction drives spatial changes in star formation rate. In Pan et al. (in prep) we will further investigate how the resolved star formation and gas properties vary with merger configuration. In Section \ref{sec:Data} we describe our methods for selecting a sample of mergers, as well as the MaNGA and ALMA observations utilized in this investigation. Section \ref{sec: analysis} presents our main results, comparing individual resolved scaling relations for our merger sample, as well as comparing offsets from all three scaling relations. We summarize the impact of this work in Section \ref{sec:summary}. Throughout the work we adopt a cosmology in which $\textrm{H}_{\textrm{0}}=70$ km/s/Mpc, $\Omega_M=0.3$, and $\Omega_{\Lambda}=0.7$. \section{Data} \label{sec:Data} \subsection{Merger Sample Selection} \label{subsec:selection} Our sample of merging galaxies is collected from the MaNGA data release 15 (DR15), which was the most recent publicly available release at the time of our observations and for the duration of this project. We visually classify post-merger and interacting galaxies from the $\sim$4800 galaxies in DR15 using the Sloan Digital Sky Survey (SDSS) Sky Server \emph{gri}-images (\emph{r}-band half-light surface brightness limit of 23.0 mag arcsec$^{-2}$; \citealt{Strauss2002SpectroscopicSample}). Post-merger galaxies are distinguished by clear morphological disturbances indicating a recent interaction, such as tidal tails or shells, but with no obvious companion. Galaxies in an interacting pair are identified with similar indicators, with the addition of a clear visual connection to a second disturbed galaxy (such as tidal bridges). Beyond the visually selected sample, we also identify a group of spectroscopic pairs, where a visible connection between a galaxy and a possible companion is unclear, but spatial and redshift information reveal the two galaxies may be interacting. Spectroscopic pairs are selected using the \cite{Patton2016GalaxySeparations} catalogue, which provides the closest companion for each galaxy in SDSS data release 7 (DR7), with a companion boundary at projected separation $r_p=$ 1 Mpc and a maximum difference in velocity between the two of $\Delta$v $=$ 1000 km/s. We assume any MaNGA galaxies with a projected separation less than 2'' from the \cite{Patton2016GalaxySeparations} galaxy position are the same object. We limit our spectroscopic pairs to those with a mass ratio between 0.1 and 10 (meaning the smaller of the pair is more than 10$\%$ the mass of the larger), and only include galaxies whose companion is within $r_p<$100 kpc and $\Delta$v$<$500 km/s. For simplicity, the rest of this work will not distinguish visually and spectroscopically selected pairs; we will simply refer to ``pairs''. The combination of the pair and post-merger galaxies yields a total sample of 903 galaxies in DR15 which we hence refer to as the parent merger sample, from which we can select a subsample for ALMA observations. \begin{table*} \centering \caption{Summary of global properties of post-merger and pair galaxies observed with ALMA, with post-mergers first (no $r_p$ values), followed by pairs ordered by $r_p$. Key global properties derived from the MaNGA VAC are included (\Mstarend, SFR, $\Delta$SFR, z), as well as merger properties such as $r_p$, $\Delta v$, and mass ratio (the mass of the galaxy divided by that of its companion). Post-mergers have empty columns for merger properties, which require information about a companion galaxy. Pair galaxies that were visually identified only have $r_p$ values, since spectrscopic information is not available for the companion. Also included are spaxel counts for each galaxy that meet our various star-forming and CO S/N cuts, along with the overlap between these regions. Galaxies with less than 10 spaxels of CO+SF(H$\alpha$+D4000) overlap are excluded for plots of individual galaxies.} \label{tab:properties} \begin{tabular}{lcccccccccccc} % \hline plate-ifu & $\log \textrm{M}_{\star}$ & $\log$ SFR & $\Delta$ SFR & z & $r_p$ & $\Delta$ v & Mass & SF & SF & CO S/N$>$3 & CO+SF & CO+SF\\ & & & & & & & Ratio & (H$\alpha$) & (H$\alpha$+D4000) & & (H$\alpha$) & (H$\alpha$+D4000) \\ & $\log M_{\odot}$ & $\log M_{\odot}/yr$ & dex & & kpc & km/s & & \# Spaxels & \# Spaxels & \# Spaxels & \# Overlap & \# Overlap \\ \hline \hline 9195-3702 & 11.14 & 1.10 & 0.62 & 0.064 & - & - & - & 0 & 53 & 204 & 0 & 5 \\ 9194-3702 & 11.06 & 0.97 & 0.52 & 0.075 & - & - & - & 0 & 811 & 379 & 0 & 328 \\ 8083-9101 & 11.14 & 0.59 & 0.11 & 0.038 & - & - & - & 331 & 1163 & 322 & 60 & 186 \\ 8952-12701 & 10.73 & $-$0.38 & $-$0.23 & 0.029 & - & - & - & 21 & 37 & 354 & 18 & 28 \\ 8084-3702 & 10.23 & 0.43 & 0.60 & 0.022 & - & - & - & 240 & 1007 & 552 & 150 & 531 \\ 8156-3701 & 10.52 & 0.87 & 0.66 & 0.053 & - & - & - & 778 & 941 & 279 & 279 & 279 \\ 8081-9101 & 10.60 & 0.32 & 0.23 & 0.028 & - & - & - & 321 & 600 & 436 & 255 & 335 \\ 8615-3703 & 10.19 & 0.40 & 0.45 & 0.018 & - & - & - & 324 & 997 & 538 & 276 & 519 \\ 9512-3704 & 10.70 & 0.14 & $-$0.03 & 0.055 & - & - & - & 168 & 552 & 136 & 33 & 40 \\ 7977-12705 & 10.85 & 0.47 & 0.26 & 0.027 & - & - & - & 981 & 1472 & 538 & 197 & 201 \\ 8623-1902 & 10.17 & $-$1.00 & $-$0.84 & 0.025 & - & - & - & 4 & 175 & 257 & 2 & 87 \\ 8616-12702 & 10.76 & $-$0.24 & $-$0.40 & 0.031 & - & - & - & 469 & 1113 & 258 & 78 & 104 \\ 9195-3703 & 10.39 & 0.47 & 0.42 & 0.027 & - & - & - & 225 & 833 & 504 & 110 & 460 \\ 8615-1901 & 9.68 & $-$0.08 & 0.51 &0.020 & - & - & - & 461 & 588 & 2 & 2 & 2 \\ 8153-12702 & 9.90 & $-$0.74 & $-$0.45 & 0.038 & 0.46 & - & - & 333 & 1119 & 0 & 0 & 0 \\ 7975-6104 & 11.01 & 0.61 & 0.17 & 0.079 & 7.04 & - & - & 209 & 588 & 391 & 209 & 320 \\ 8241-12705 & 10.40 & 0.00 & $-$0.05 & 0.027 & 8.23 & 80.0 & 3.14 & 847 & 1678 & 592 & 381 & 508 \\ 8082-9102 & 10.74 & 0.73 & 0.53 & 0.037 & 12.24 & - & - & 554 & 1197 & 668 & 422 & 591 \\ 7977-9102 & 10.93 & 0.83 & 0.41 & 0.063 & 13.78 & - & - & 524 & 1129 & 380 & 287 & 354 \\ 9195-9101 & 10.74 & 0.39 & 0.16 & 0.057 & 15.36 & - & - & 678 & 1797 & 722 & 455 & 675 \\ 8616-9101 & 11.10 & 0.75 & 0.28 & 0.092 & 19.89 & 276.0 & 4.19 & 112 & 1044 & 363 & 84 & 249 \\ 8078-12703 & 10.81 & 0.14 & $-$0.02 & 0.028 & 20.20 & 102.0 & - & 1208 & 2597 & 1128 & 444 & 796 \\ 7968-12705 & 11.30 & $-$1.41 & $-$1.37 & 0.086 & 20.45 & 13.0 & 0.80 & 0 & 93 & 8 & 0 & 0 \\ 8078-6104 & 10.41 & 0.23 & 0.16 & 0.044 & 26.15 & 109.0 & 0.42 & 197 & 1039 & 433 & 153 & 431 \\ 8085-12701 & 10.43 & 0.59 & 0.61 &0.030 & 29.51 & - & - & 2293 & 2601 & 458 & 456 & 458 \\ 8085-6101 & 11.07 & $-$0.73 & $-$1.09 & 0.052 & 31.48 & - & - & 0 & 0 & 2 & 0 & 0 \\ 8083-12703 & 10.46 & 0.32 & 0.24 & 0.025 & 45.24 & 60.6 & - & 822 & 2201 & 1264 & 641 & 1171 \\ 8082-12704 & 11.42 & 0.77 & 0.05 & 0.132 & 51.28 & - & - & 3 & 1617 & 451 & 3 & 244 \\ 8085-3704 & 10.72 & 0.66 & 0.43 & 0.037 & 58.73 & 37.0 & 0.89 & 319 & 727 & 608 & 299 & 552 \\ 8450-6102 & 10.43 & 0.64 & 0.54 & 0.042 & 75.35 & 102.0 & 1.81 & 1162 & 1550 & 536 & 503 & 536 \\ 8728-3701 & 10.64 & $-$0.37 & $-$0.49 & 0.028 & 89.75 & 113.0 & 1.96 & 0 & 31 & 156 & 0 & 0 \\ \hline \end{tabular} \end{table*} To select candidate mergers for ALMA observations we first exclude all mergers with declination greater than 20\textdegree, leaving only those with positions that overlap with ALMA's observational range. From the initial parent merger sample only 143 galaxies meet this criterion. We next select mergers for which our target CO line S/N can be achieved in less than 5 hours, to ensure we can observe as many mergers as possible within a competitive proposal (see Section \ref{subsec:ALMA} for more details). What remains is a sample of 6 post-mergers and 14 pairs, which are observed as part of an ALMA Cycle 7 program (2019.1.00260.S, P.I.: Hsi-An Pan, details in Section \ref{subsec:ALMA}). We also make use of eleven galaxies from the ALMaQUEST survey that show clear signs of an interaction within our classification scheme: 8 post-mergers and 3 pairs \citep{Lin2020ALMAQUEST:SURVEY,Ellison2020The0}. The SDSS \emph{gri}-images of three of the galaxies in the final ALMaQUEST merger sample are included in the left panels of Figure \ref{fig:ALMaQUEST_Ex} (continued for all mergers in Appendix \ref{app:DP}). The final sample of 14 post-merger and 17 pair galaxies is summarized in Table \ref{tab:properties}. Included in the table are key global properties from the MaNGA {\sc pipe3d} Value Added Catalogue (VAC) including total stellar mass (\Mstarend), total star formation rate (SFR), and redshift (z). We also include the offset from the global star-forming main sequence $\Delta$SFR, where $\Delta \textrm{SFR}= \log \textrm{SFR}_{\textrm{galaxy}} - \log \textrm{SFR}_{\textrm{control}}$. $\textrm{SFR}_{\textrm{control}}$ is the median SFR value of galaxies within 2$\sigma$ of a fit to the star-forming main sequence (controls are also matched within 0.1 dex in \Mstar and 0.005 in z). Merger properties are listed as well, including $r_p$, $\Delta v$, and mass ratio. Post-merger galaxies have empty $r_p$, $\Delta v$, and mass ratio values since there is no companion with which to compare. Pair galaxies that were visually identified do not have measured $\Delta v$ and mass ratio values, since there is no spectroscopic information from their companion. The merger sample covers a broad range of \Mstar and SFR. Figure \ref{fig:Merger_SFMS} compares these properties for our post-merger (orange circles) and pair (triangles) galaxies with respect to the rest of MaNGA, shown as grey hexbins. Pairs are colour-coded by the projected separation from their closest companion. Our sample is representative of a typical merger sample, containing galaxies with both greatly elevated and comparatively low SFRs. The inset histogram of Figure \ref{fig:Merger_SFMS} shows the distribution of the offset from the star-forming main sequence $\Delta$SFR of our post-merger (orange) and pair (blue) galaxies with respect to the rest of MaNGA (grey). Although the pair and post-merger samples have on average larger $\Delta$SFR values than non-interacting galaxies, we still cover a broad range of $\Delta$SFR values to reflect the overall variety of behaviours seen in larger galaxy merger studies and to probe any corresponding diversity in the molecular gas properties of those mergers. We note that there may be some selection bias in the sample, given some post-mergers come from the ALMaQUEST starburst sample of \cite{Ellison2020The0}, and the rest still need to have enough gas for detectable CO as described in the following section. \subsection{ALMA Observations} \label{subsec:ALMA} Observations of CO $J=1 \rightarrow 0$ (CO hereafter), rest frame 115 GHz, were completed for 6 post-merger and 14 pair galaxies as part of Cycle 7 ALMA program 2019.1.00260.S (P.I.: Hsi-An Pan). The remaining 11 mergers were already available from the original ALMaQUEST sample \citep{Lin2020ALMAQUEST:SURVEY}. Cycle 7 observations were designed to replicate the methodology used in the original ALMaQUEST survey, the details of which can be found in the survey paper by \cite{Lin2020ALMAQUEST:SURVEY}. We provide a brief summary of those techniques below. Observations were carried out using the Band 3 receiver, taken in the C43-2 configuration to achieve a synthesized beam full-width half maximum (FWHM) of 2.5'' in order to match the effective resolution of the MaNGA survey \citep{Law2015ObservingSurvey}. The ALMA field of view in our chosen configuration is $\sim$50'' which is sufficient to cover the MaNGA footprint for all of our galaxies. The spectral set up includes one high resolution spectral window targeting the CO emission ($\sim$10 km/s), and three low-resolution spectral windows around the target line to detect the continuum ($\sim$90 km/s). To reach a CO S/N greater than 3 for 50$\%$ of spaxels with H$\alpha$ S/N$>$3, the on-target integration time varied between 0.1-3.3 hours. The same methodology is used for the main ALMaQUEST sample, ensuring consistent data quality between ALMaQUEST data and that acquired in Cycle 7 \citep{Lin2017SDSS-IVMaNGA,Lin2020ALMAQUEST:SURVEY}. Data were calibrated using the ALMA data reduction software Common Astronomy Software Applications (CASA, \citealt{McMullin2007CASAApplications}) using version 5.4 for all but 3 galaxies (which were observed in earlier cycles and thus used CASA version 4.5), along with the standard ALMA reduction pipeline. In ALMA's Band 3 the systematic flux uncertainty inherent with calibration is roughly 5-10$\%$ \citep{2019acpg.rept.....D}. Continua were subtracted in the visibility domain for a handful of galaxies both in the original ALMaQUEST and the Cycle 7 galaxies. Once the continuum was subtracted, the task CLEAN was used to clean data down to 1$\sigma$ and produce spectral line data cubes with a Briggs weighting (robust parameter=0.5), resulting in a native effective beam size ranging from 1.6''-2.8'' depending on the target. The data were then re-imaged with a user-specified spaxel size (0.5'') and a restoring beamsize (2.5'' $\times$ 2.5'') to match the MaNGA image grid (prior) and spatial resolution (latter). The final cubes have channel widths of 11 km/s and $\sigma_{\textrm{rms}}\sim$0.2-2 mJy/beam. We applied the CASA task IMMOMENTS to these datacubes to determine moment 0 (integrated intensity) maps. IMMOMENTS can also generate moment 1 (intensity-weighted velocity) and moment 2 (intensity-weighted velocity dispersion) maps, but those are not relevant to the analysis of this work and are saved for future projects. Moment 0 maps are constructed by integrating the CO emission from a set velocity range without any clipping in signal. Table \ref{tab:properties} lists the number of spaxels in each galaxy where the CO line emission from the moment 0 map has S/N$>3$. Four of the merger galaxies observed as part of Cycle 7 (8615-1901, 8153-12702, 7968-12705, 8085-6101, all four early-type galaxies) have less than ten spaxels with CO S/N$>$3, and are thus excluded from any further analysis that looks at individual galaxies (rather than spaxels). The CO luminosity per spaxel determined from the moment 0 map is converted to molecular gas surface density \Siggas ($\textrm{M}_{\odot}/\textrm{kpc}^2$) using a constant conversion factor of $\alpha_{\textrm{CO}}$=4.35 ($\textrm{M}_{\odot} (\textrm{K } \textrm{km/s } \textrm{pc}^2)^{-1}$) \citep{Bolatto2013THEFACTOR}. All \Siggas values are then inclination corrected using the b/a axial ratios from the NASA Sloan Atlas (NSA) catalogue, themselves determined from single Sérsic fits. Three example \Siggas maps are provided in Figure \ref{fig:ALMaQUEST_Ex}, with the rest of the \Siggas maps from the sample available in Appendix \ref{app:DP}. There has been extensive research into the viability of a constant conversion factor, and many suggest using a metallicity dependent \citep{Accurso2017DerivingRelations,Sun2020DynamicalGalaxies} or metallicity and line intensity dependent \citep{Narayanan2012ALaw} conversion instead. The requirements needed to determine accurate metallicity measurements would limit our spaxel count significantly, given the sparse overlap between high S/N star-forming spaxels and CO measurements (see Table \ref{tab:properties}), as we will discuss further in Subsection \ref{subsec:SFR-D4000}. To minimize the loss of spaxels we choose to use a constant conversion factor. In Appendix \ref{app:alpha_CO} we characterize the difference in \Siggas measurements when a metallicity dependent $\alpha_{\textrm{CO}}$ is used (for spaxels where that is possible), and confirm that key results from this work cannot result from inaccuracies of conversion factor. \subsection{MaNGA Data Products} \label{subsec:MANGA} The work presented here primarily uses MaNGA data products from the {\sc pipe3d} spectral fitting pipeline, described in detail in \cite{Sanchez2016Pipe3DFIT3D, Sanchez2016Pipe3DDataproducts}. Along with global values provided by the {\sc pipe3d} VAC mentioned previously, we also make extensive use of the {\sc pipe3d} stellar mass surface density (\Sigstarend) and emission line fluxes. We correct all emission line fluxes for dust using a Milky Way dust extinction curve \citep{Cardelli1989THEEXTINCTION}, assuming an intrinsic H$\alpha$/H$\beta$ ratio of 2.85. Star formation rate surface densities (\SigSFRend) are determined from the dust corrected H$\alpha$ luminosity using the \cite{Kennicutt1994PastGalaxies} relation and assuming a Salpeter intitial mass function \citep{Salpeter1955THEEVOLUTION}. Both \Sigstar and \SigSFR are inclination corrected using the b/a axial ratios (derived from single Sérsic fits) provided in the NSA catalogue. The same b/a ratio is used to compute an inclination corrected galactocentric radius from the V-band centre of the MaNGA map. We also use the 4000\AA \textrm{ } break strength (D4000) provided by {\sc pipe3d}, as it serves a crucial role in expanding our star formation rate measurements described in the next section. \vfill \subsubsection{Star Formation Rates - D4000 vs H$\alpha$} \label{subsec:SFR-D4000} The work presented here requires that both SFRs and molecular gas surface densities are measured in a given spaxel. Ideally we would use SFRs determined from H$\alpha$ and only consider the \SigSFR values of star-forming spaxels defined by the \cite{Kauffmann2003TheAGN} designation on a Baldwin, Phillips \& Terlevich diagram (BPT; \citealt{Baldwin1981CLASSIFICATIONOBJECTS}). Along with a \cite{Kauffmann2003TheAGN} star-forming cut, we also impose a S/N$>3$ cut for the flux of each diagnostic emission line, as well as requiring an H$\alpha$ equivalent width (EW)$>$6\AA{ }limit to ensure H$\alpha$ flux stems from a young stellar population. Of the 13,046 spaxels with CO detections S/N$>3$, only 44.5\% pass all of these star-forming criteria. Table \ref{tab:properties} provides for each galaxy the total count of spaxels which pass the star-forming criteria cut for H$\alpha$, which have CO S/N$>3$, and which pass both criteria. To maximize the number of spaxels with both CO and SFR measurements, we elect to approximate \SigSFR for spaxels that are not star-forming (based on our BPT and EW criteria) using the relationship between sSFR and D4000 found for both global \citep{Brinchmann2004TheUniverse} and local \citep{Spindler2018SDSS-IVEnvironment, Wang2019OnGalaxies,Bluck2020ArePhenomena} scales. We adopt an empirical approach similar to that in \cite{Bluck2020ArePhenomena}, who tested the validity of this approximation in the MaNGA {\sc pipe3d} data products. For bins of D4000 we compute a median sSFR value based on all star-forming spaxels (using our previous criteria). For any spaxel that does not meet our star-forming criteria, either due to low signal-to-noise or AGN contamination based on the BPT cut, we assign the median sSFR from the closest D4000 bin, and convert that to \SigSFR by multiplying sSFR and \Sigstarend. Figure \ref{fig:sSFR-D4000} displays the complete sSFR-D4000 distribution for star-forming spaxels in MaNGA along with a red line showing the median sSFR value, up to D4000=1.45, used to approximate \SigSFR when a spaxel is not star forming. Spaxels with D4000>1.45 are generally quenched and therefore the sSFR-D4000 relation is no longer viable \citep{Bluck2020ArePhenomena}. As can be seen from Figure \ref{fig:sSFR-D4000}, the D4000-sSFR relation turns steeply downwards at D4000$>$1.4. We therefore only estimate \SigSFR from D4000 when D4000$<$1.4, which represents a slightly stricter cut than that used in \cite{Bluck2020ArePhenomena}. Table \ref{tab:properties} provides the increased spaxel count when both H$\alpha$- and D4000-\SigSFR values are employed. Of the 13,046 spaxels with CO S/N>3, now 78.8\% of spaxels have measurable \SigSFR, almost doubled from the 44.5\% of our previous criteria. 9195-3702 and 8450-6102, despite having a considerable number of spaxels with CO S/N>3, still have less than 10 spaxels of overlap between good CO and \SigSFR measurements. These two galaxies are thus excluded from individual galaxy analysis later in this work, along with the other four galaxies previously mentioned. Thus we have 25 galaxies which can be studied on an individual basis, and we limit our studies to these 25 for the rest of the work. Multiple tests are performed to check how using D4000 approximated \SigSFR might impact our results, details of which are included in Appendix \ref{app:sSFR-d4000}. Adopting this method of measuring \SigSFR sacrifices accuracy for completeness in our analysis; to assess how this will impact our results we repeat key parts of our analysis using only H$\alpha-$\SigSFR (for galaxies where that is possible). Unless specified otherwise, for the rest of this work \SigSFR and any derived products, use the combined H$\alpha$+D4000 \SigSFR values for both the merger and isolated sample. Figure \ref{fig:ALMaQUEST_Ex} summarizes the various data products from MaNGA and ALMA used within this work for three galaxies from our sample: a post-merger galaxy (top), an interacting pair (middle), and a spectroscopic pair (bottom). We display maps of H$\alpha$-\SigSFRend, as well as our combined H$\alpha$+D4000-\SigSFR map. The middle galaxy is a clear case where we are able to recover a significant number of spaxels using the combined \SigSFR values. In particular, a significant fraction of spaxels in the outskirts of the galaxy are recovered to maximize the spatial coverage of our analysis. Though the middle galaxy demonstrates how the combined H$\alpha$+D4000-\SigSFR method recovers lower \SigSFR values, the method also recovers AGN dominated spaxels. Thus the main limitation of our analysis becomes the CO signal-to-noise, and the overlap between CO and \SigSFR. \subsubsection{Spaxel Offsets from Resolved Relations} \label{subsec: offsets} Following \cite{Ellison2020The0}, we compute offsets from the rKS, rSFMS, and rMGMS to quantify how an individual spaxel may deviate from the average spaxel behaviour. All offsets are computed as the log difference between a spaxel value and the median value of a set of control spaxels that defines the average behaviour, i.e. $\Delta X = \log X - <\log X_{\textrm{control}}>_{\textrm{median}}$. For example, we define an offset from the SFMS as \dsigsfrend, with positive \dsigsfr values corresponding to enhanced star formation compared to the control sample. The sample of control spaxels is collected from all DR15 MaNGA spaxels from non-merging galaxies (as classified in Subsection \ref{subsec:selection}) with b/a$>$0.34 (excluding galaxies with inclination greater than 70\textdegree) and which are star-forming based on our cuts for the H$\alpha$-\SigSFR measurements described in Section \ref{subsec:SFR-D4000}. All MaNGA \SigSFR values are included, not just ALMaQUEST, such that the requirement for CO detections in ALMaQUEST spaxels does not bias our control spaxel set to higher \SigSFR. Note that we exclude D4000-\SigSFR values from the control sample, given we want to know the difference from the star-forming population that defines the resolved star-forming main sequence. The majority of spaxels on the rSFMS for which we can measure accurate D4000-\SigSFR tend to not meet our star-forming criteria due to low S/N in BPT diagnostic emission lines, rather than being classified as AGN or composite spaxels. Including D4000-\SigSFR in the control would lead to slightly lower \dsigsfr values as a result of the contribution from low S/N spaxels. We have tested whether the inclusion of D4000-\SigSFR in our control would alter any major conclusions of this work, and find our results remain unchanged. From the set of star-forming controls we select a subset that is matched within 0.1 dex \Sigstarend, 0.1 $R/R_{e}$ (where R is the inclination corrected galactocentric radius, and the effective radius $R_{e}$ is the \emph{r}-band half-light radius from the NSA catalogue), and 0.1 dex \Mstar of the merger spaxel. The median of this control set defines our ``regular'' star-forming behaviour. Similar offsets can be determined for the other scaling relations. An offset from the resolved KS is referred to as \dSFEend; a value above the KS would have a larger star formation rate given the molecular gas in a spaxel, i.e. an enhancement in the efficiency at which gas is converted to stars. Rather than rely on S/N and BPT cuts to define star-forming spaxels, as was achieved with \dsigsfrend, we instead make use of the \dsigsfr value to construct the star-forming rKS. The control sample is also limited to galaxies in ALMaQUEST, rather than all of DR15, given \Siggas is required to compute \dSFEend. Thus the control sample is selected from non-interacting ALMaQUEST galaxies with b/a$>$0.34, with the additional cut of -0.5< \dsigsfrend <0.5 to select star-forming spaxels. A subset of the control is found by matching to the merger spaxel within 0.1 dex \Siggasend, 0.1 $R/R_{e}$, and 0.1 dex \Mstarend. Using the same control spaxel sample as for \dSFE, we determine offsets from the rMGMS, referred to as \dfgas. The control spaxels are again taken from non-interacting ALMaQUEST galaxies and matched within 0.1 dex \Siggasend, 0.1 $R/R_{e}$, and 0.1 dex \Mstarend. Unlike the other two offset metrics, \dfgas can be computed for any spaxels with CO S/N>3. Since there is no dependence on whether \SigSFR is measurable, or if measured \SigSFR values overlap with good CO measurements, maps of \dfgas tend to be more complete than the other two offsets. Maps of offset parameters for the merger sample are available in Appendix \ref{app:maps}. One key difference between \cite{Ellison2020The0}, who first introduced these offset methods, and the methods used here is we do not match control spaxels by metallicity. Limiting our analysis to spaxels with valid metallicity measurements drastically diminishes our total spaxel count, as described in detail in Section \ref{subsec:ALMA}, limiting the spatial coverage of some galaxies and removing three from viable examination entirely. Nonetheless, we have repeated the analysis described in this section with a metallicity control (similar to that used in \citealt{Ellison2020The0}), and find little change in our key results for those galaxies on which this check can be performed (i.e., those with large CO+SFR(H$\alpha$) overlap). \vfill \section{Analysis} \label{sec: analysis} \subsection{Resolved Scaling Relations} \label{subsec: Relations} The stellar mass surface density (\Sigstarend), SFR surface density (\SigSFRend), and molecular gas surface density (\Siggasend) are all interconnected in three well established resolved relations: the resolved Kennicutt-Schmidt relation and the resolved molecular gas main sequence, which together drive the resolved star-forming main sequence\citep{Lin2019TheSequence, Ellison2021TheThem, Baker2022TheSequence}. We investigate these three scaling relations both for merger populations, and for individual galaxies in our merger sample. Figure \ref{fig:All_scaling_relations} compares the resolved scaling relations for post-merger (orange) and pair (blue) galaxies with respect to the relatively isolated galaxies in the rest of ALMaQUEST (grey). Histograms of the \SigSFRend, \Siggasend, and \Sigstar are provided for context as well. Note that the grey histogram does not represent the control spaxel population used to compute various offset parameters as described in Subsection \ref{subsec: offsets}, rather all spaxels where \SigSFR can be measured for ALMaQUEST. Given the selection of the original ALMaQUEST sample, including a sample of starburst galaxies, the non-interacting spaxels may be biased to high \SigSFRend. This is likely why the median \SigSFR for non-interacting galaxies (shown as a dashed line in the \SigSFR histogram) is slightly larger than the median \SigSFR of the pair sample. We perform Kolmagorov-Smirnov (KS) tests between all three samples to ascertain whether the pair or post-merger \SigSFRend/\Siggasend/\Sigstarend values could be drawn randomly from the non-interacting spaxel sample. We find the probability of this hypothesis to be approximately zero for all three properties ($P_{KS}\approx0$). The pair sample has a tail towards lower \Siggasend, which seems to be driven by two individual galaxies with uniquely low \Sigstar and \Siggas values (8078-6104 and 8083-12703). The post-merger sample also has a tail towards large \SigSFRend, \Siggasend, and \Sigstar values, which manifests at the upper end of each scaling relation. Along with the clear lack of post-merger spaxels with \Siggas$<6.5$log $\textrm{M}_{\odot}/\textrm{kpc}^2$, it is clear that our post-merger sample probes regions of heightened SFR and molecular gas properties. There is significant diversity on a galaxy-per-galaxy basis to consider, as has been seen in other studies (\citealt{Vulcani2019GASP.Galaxies,Ellison2021TheThem,Pessa2021AstronomyScale}, Brown et al. in prep). Figure \ref{fig:SFMS_resolved} shows the rSFMS for each post-merger and pair galaxy. The star-forming spaxels are colour-coded by radius, while the D4000-\SigSFR are a plain yellow if they did not meet our S/N cuts or brown if they are AGN based on the \cite{Kauffmann2003TheAGN} criteria. The non-interacting ALMaQUEST spaxels are shown as a grey histogram for comparison (including both H$\alpha$-\SigSFR and D4000-\SigSFRend). By examining galaxies as individuals we see clear divergence from the spaxels in non-interacting galaxies, as well as a diversity of behaviour within post-merger and pair classifications. Many mergers show a large population of spaxels above the star-forming main sequence, as is expected if mergers trigger star formation, such as 8156-3701 (1st row, 5th column), 8615-3703 (2nd row, 2nd column), and 8085-12701 (5th row, 1st column). We note that three post-mergers (8084-3702, 8156-3701, 8615-3703) and one pair (8450-6102) were selected from the starburst sample as part of the original ALMaQUEST (ALMA program 2018.1.00541.S, PI: Ellison), which could bias our merger sample towards strongly star-forming galaxies. Yet many spaxels and at times entire mergers are also below the star-forming main sequence, such as 9194-3702 (1st row, 1st column), 7975-6104 (3rd row, 3rd column), and 8078-12703 (4th row, 4th column). In particular, both galaxies with concentrations of ``AGN'' spaxels as defined by the BPT diagram tend to lie below the non-interacting rSFMS (see 9194-3702 (1st row, 1st column) and 9195-3703 (3rd row, 2nd column)), as has been observed previously with global quantities \citep{Ellison2016TheGalaxies,Leslie2016QuenchingSequence,Mcpartland2019DissectingUniverse}. This range of enhanced and suppressed star formation rates drives much of the overall scatter of our sample on the global star-forming main sequence (refer back to Figure \ref{fig:Merger_SFMS}). Mergers present diverse behaviour in the rKS as well, as shown in in Figure \ref{fig:KS_resolved}. Many post-mergers and pairs have overall greater scatter in the rKS than the non-interacting sample (see 8615-3703 (2nd row, 2nd column), 8085-12701 (5th row, 1st column), 8083-12703 (5th row, 2nd column)), looking ``puffier''; a similar effect as was seen in \cite{Ellison2021TheThem}. The increased scatter could represent smaller regions of enhanced SFE in a merger that, on average, has a low or normal global SFE. Note that although D4000-\SigSFR values tend to be lower on the rKS, many still overlap with H$\alpha$-\SigSFR spaxels. Spaxels which are classified as ``AGN'' by a BPT diagram (brown in the figure) tend to exist at larger \Siggas and \SigSFR values, reflecting their central location in the galaxy and the overall radial dependence of the rKS (smaller radii predominantly fill the upper end of the rKS). However, AGN spaxels at large \Siggas could also support a scenario where the infall of molecular gas fuels a central AGN. The two largest AGN spaxel populations belonging to post-mergers appears to support such a scenario, though two galaxies alone are not enough to determine if one scenario is more likely than another. Such a query can be further investigated by looking for enhancements in the molecular gas main sequence at high \Siggas values. Figure \ref{fig:MGMS_resolved} appears to confirm that variations in the rMGMS are the least drastic of the three relations, as was found for non-interacting galaxies by \cite{Ellison2021TheThem}. Interestingly, the AGN spaxels (shown in brown) are not offset to below the rMGMS, as would be implied by the comparatively low gas fractions found for AGN spaxels with MaNGA \citep{Sanchez2018SDSSGalaxies} and EDGE-CALIFA \citep{Ellison2021TheQuenching}. Rather the AGN spaxels in the merger sample are comparable to star-forming isolated spaxels, or even enhanced in the case of 9194-3702 (1st row, 1st column) and 9195-3703 (3rd row, 2nd column). What seems to be an inconsistency may stem from the unique nature of the merger stage. Rather than capture central gas depletion triggered by an AGN, as in the isolated galaxies of \cite{Ellison2021TheQuenching}, mergers may have more recently funnelled molecular gas to the galaxy's centre which has yet to be consumed by the AGN. In the next section we will investigate what drives a merger galaxy to be offset from the resolved scaling relations by comparing spaxel offsets from scaling relations. \subsection{Efficiency vs. Fuel Driven Enhanced Star Formation} \label{subsec: which mechanism?} We can use maps of spaxel offsets from resolved scaling relations (\dsigsfrend, \dSFEend, and \dfgasend, as described in Subsection \ref{subsec: offsets}), to discern whether changes in SFE or \fgas drive enhancements in star formation rate. Figure \ref{fig:offset_maps} shows the offset maps for an example post-merger galaxy, maps for the entire sample are available in Appendix \ref{app:maps}, for which we can attempt to discern whether star formation is driven by enhanced SFE or enhanced \fgasend. Star formation in this post-merger is generally enhanced (\dsigsfrend$>0$), as is expected given the average spatial enhancement of post-mergers (see \citealt{Thorp2019SpatiallyMaNGA}). That enhancement corresponds to a global enhancement in star formation efficiency (positive \dSFEend). Interestingly, this post-merger mostly has a deficit in molecular gas (negative \dfgasend), with some smaller regions of surplus gas. Figure \ref{fig:offset_maps} thus implies that the boost of star formation in this post-merger in particular is predominantly driven by an enhanced efficiency at which gas is converted to stars, not an enhanced amount of gas to fuel that star formation. From the offset maps alone we can infer some interesting behaviour. Many pair galaxies, for example, have large \dfgas values along spiral arms (see 9195-9101, 7977-9102, and 8241-12705 in Appendix \ref{app:maps}). Some pair galaxies have central enhancements in \Siggas that correspond to a central burst of star formation, such as 9195-3703 and 8078-6104. These two galaxies support a scenario where inflow of molecular gas fuels merger triggered central star formation. Post-merger galaxies 8615-3703 and 8084-3702 have an excess of molecular gas across the galaxy's surface, rather than a central concentration. Yet that is not a universal scenario; within our sample, 8156-3701 has one of the strongest enhancements in SFR, but has suppressed \Siggas across the galaxy. Although the distribution of these offset parameters can reveal interesting results for individual galaxy behaviour, it is difficult to extract general trends for merging and post-merger galaxies from visual examination alone. In particular it is difficult to parse whether enhanced SFE or enhanced \Siggas drives any merger-triggered star formation. To discern which mechanism is likely more influential over star formation enhancements, we implement an analysis which includes all three offset parameters in a single diagram as pioneered by \cite{Ellison2020The0} using ALMaQUEST and \cite{Moreno2021SpatiallyInteractions} using simulations. Figure \ref{fig:SFMS_mechanism} shows the offsets in star formation efficiency versus the offsets in molecular gas, for all spaxels with enhanced star formation (\dsigsfrend$>0$). We separate the galaxies into three categories: the non-interacting set from ALMaQUEST, pair galaxies, and post-merger galaxies. For these diagrams we plot the line of equality which is used to distinguish an ``efficiency driven'' and ``fuel driven'' regime, i.e. where one offset is greater than the other. If a spaxel lies above this line of equality, then \dSFE$>$\dfgasend; if \dsigsfrend$>$0 this would imply enhanced efficiency is prompting the enhanced star formation, more so than the gas fraction. All three galaxy populations have a relatively equal percentage of spaxels in the efficiency driven and fuel driven regime, implying that enhanced star formation is equally driven by enhanced fuel and SFE when all spaxels are examined together. There is a slight bias towards efficiency driven star formation for the post-merger spaxels, with 63\% of spaxels in the efficiency driven regime. We remind the reader that the original ALMaQUEST sample includes 11 galaxies that were deliberately selected to be starbursts \citep{Ellison2020The0}. Three of these starbursts are in our post-merger sample, one is in the pairs sample, and 12 are in our isolated sample. Since \cite{Ellison2020The0} showed that starbursts tend to be dominated by high SFEs, the isolated sample in Figure \ref{fig:SFMS_mechanism} is not statistically representative of a normal galaxy distribution. Figure \ref{fig:SFMS_mechanism} might lead us to believe that enhancements in both SFE and \fgas are equally important for driving SFR enhancements in both post-mergers and pairs. However, in the previous subsection, we found that ensemble spaxel distributions showed considerable diversity when plotted on a galaxy-by-galaxy basis. We now investigate whether the same is true for distributions of offset properties. Figure \ref{fig:Example_Mechanism} replicates the combined offset diagram from Figure \ref{fig:SFMS_mechanism}, but for a single post-merger galaxy (the same one that is shown in Figure \ref{fig:offset_maps}). 96\% of spaxels with \dsigsfrend$>0$ in this galaxy are in the efficiency driven regime, confirming that the star formation in this galaxy is driven by enhanced SFE as is evident from visually inspecting Figure \ref{fig:offset_maps}. Figure \ref{fig:Example_Mechanism} highlights that whilst the ensemble of spaxels across all galaxies might have a relatively even split between those whose star formation is enhanced by SFE or gas fraction (Figure \ref{fig:SFMS_mechanism}), individual galaxies may be strongly driven by one process or the other. Offset diagrams like that in Figure \ref{fig:Example_Mechanism} are provided for all mergers which have more than 20 spaxels where \dsigsfrend$>0$ in Figure \ref{fig:delta_plots_all}, including the percentage of spaxels in the fuel/efficiency driven regimes. There are cases of both post-merger and pair galaxies that are indisputably driven by a single mechanism, such as 9194-3702 (fuel driven) or 8085-12701 (efficiency driven), but for the majority the dominant star formation driver is less clear. In order to classify the dominance of fuel or efficiency in driving star formation enhancements on a galaxy-by-galaxy basis, we quantify the percentage of spaxels in the fuel/efficiency driven regime of our offset diagrams. If more than 60$\%$ of a galaxy's \dsigsfrend$>0$ spaxels are in the fuel-driven regime (i.e., \dfgasend$>$\dSFEend), then we classify the galaxy as ``fuel-driven''. If less than 40$\%$ of a galaxy's \dsigsfrend$>0$ spaxels are in the fuel-driven regime, then we classify the galaxy as ``efficiency driven''. If the percentage of spaxels in the fuel-driven regime is between 40$\%$ and 60$\%$, we can assume that enhanced amounts of gas and enhanced SFE are approximately equally contributing to the enhanced star formation. Note that this method of classification gives equal weight to all spaxels, regardless of the strength of star-formation in each spaxel. By doing so our analysis focuses on what mechanism drives star-formation for most regions in a galaxy, not necessarily the mechanism which drives ``the most'' star-formation. \begin{table} \centering \caption{The fraction of \dsigsfrend$>0$ spaxels in the fuel driven regime (i.e., \dfgasend$>$\dSFEend) for each merger with more than 20 \dsigsfrend$>0$ spaxels. The fraction of fuel driven spaxels is computed using the combined H$\alpha$+D4000 \SigSFRend, as well as just the H$\alpha$ star formation rates (with appropriate cuts). Note some galaxies do not have fractions computed for H$\alpha$, because they do not have more than 20 spaxels that meet our star-forming cuts. Fractions greater than 0.6, what we consider a fuel driven merger, are coloured blue. Fractions less than 0.4, what we consider an efficiency driven merger, are coloured red. Those between 0.4 and 0.6 are left black, since both mechanisms contribute relatively equally.} \label{tab:mechanism} \begin{tabular}{ccc} % \hline plate-ifu & Fraction of & Fraction of\\ & Fuel Driven Spaxels & Fuel Driven Spaxels\\ & & (H$\alpha$ only)\\ \hline \hline 9195-3702 & - & - \\ 9194-3702 & \textcolor{blue}{0.97} & - \\ 8083-9101 & 0.59 & 0.50 \\ 8952-12701 & - & - \\ 8084-3702 & \textcolor{blue}{0.78} & \textcolor{blue}{0.75} \\ 8156-3701 & \textcolor{red}{0.04} & \textcolor{red}{0.04} \\ 8081-9101 & 0.50 & 0.53 \\ 8615-3703 & \textcolor{blue}{0.60} & \textcolor{blue}{0.64} \\ 9512-3704 & - & - \\ 7977-12705 & \textcolor{red}{0.06} & \textcolor{red}{0.06} \\ 8623-1902 & 0.52 & - \\ 8616-12702 & - & - \\ 9195-3703 & \textcolor{red}{0.33} & \textcolor{red}{0.05} \\ \hline 7975-6104 & \textcolor{blue}{0.97} & \textcolor{blue}{0.97} \\ 8241-12705 & 0.52 & 0.52 \\ 8082-9102 & \textcolor{blue}{0.64} & \textcolor{blue}{0.62} \\ 7977-9102 & \textcolor{blue}{0.69} & \textcolor{blue}{0.68} \\ 9195-9101 & 0.51 & 0.55 \\ 8616-9101 & \textcolor{red}{0.34} & \textcolor{blue}{0.60} \\ 8078-12703 & \textcolor{red}{0.10} & \textcolor{red}{0.14} \\ 8078-6104 & \textcolor{red}{0.16} & \textcolor{red}{0.21} \\ 8085-12701 & \textcolor{red}{0.05} & \textcolor{red}{0.05} \\ 8083-12703 & \textcolor{red}{0.38} & \textcolor{red}{0.39} \\ 8082-12704 & - & - \\ 8085-3704 & \textcolor{blue}{0.60} & \textcolor{blue}{0.67} \\ 8450-6102 & \textcolor{red}{0.24} & \textcolor{red}{0.24} \\ \hline \end{tabular} \end{table} The fraction of fuel driven spaxels for every merger is listed in Table \ref{tab:mechanism}. We calculate this fraction for both our combined \SigSFR values and for the H$\alpha$-\SigSFR only, although some mergers do not have enough H$\alpha$-\SigSFR spaxels to meet our criteria. For galaxies which have enough H$\alpha$-\SigSFR spaxels to measure the fuel-driven fraction, we find adding D4000-\SigSFR to the analysis does not change the classification of the galaxy as fuel or efficiency driven (except for 8616-9101). Interestingly, those galaxies we are only able to analyze with the inclusion of D4000-\SigSFR tend to have large fractions of fuel driven spaxels, i.e. the star formation is driven by an excess of fuel. All four recovered galaxies miss our H$\alpha$-\SigSFR criteria based on low signal-to-noise, implying that the star formation in these galaxies is truly low (and thus a small or negative \dSFE is to be expected). Further discussion on how D4000-\SigSFR impacts the fraction of fuel driven spaxels is included in Appendix \ref{app:sSFR-d4000}, where we find the inclusion of D4000-\SigSFR does not drastically change our results. For both \SigSFR measurement methods there is a relatively similar number of fuel and efficiency driven galaxies, with slightly fewer galaxies driven by neither mechanism. We also consider whether variations in the fraction of fuel driven spaxels could be driven by inaccuracies resulting from using a constant $\alpha_{\textrm{CO}}$ conversion factor. We investigate the error in \Siggas from using a constant conversion factor, and its impact on the fraction of fuel driven spaxels, in Appendix \ref{app:alpha_CO}, and find our results are robust to variations in $\alpha_{\textrm{CO}}$. Figure \ref{fig:FF_hist} displays the distribution of the fraction of fuel driven spaxels, both for the merger sample as a whole (grey) and the pair (blue) and post-merger (orange) sub-samples. Both the post-merger and pair sample show relatively equal distribution in fuel fraction implying a diversity of star formation drivers throughout the interaction process. There is an intriguing excess of extremely low fuel fractions (below 0.1), compared to the more uniform distribution above a fuel fraction of 0.5, implying that ``efficiency driven'' mergers have \dSFEend$>$\dfgas in most spaxels. However, given only 3 post-mergers are below a fuel fraction of 0.4, this could be the result of small number statistics. Both observations \citep{Ellison2013GalaxyGalaxies, Patton2013GalaxyGalaxies,Knapen2015InteractingFormation} and simulations \citep{Cox2008TheStarbursts,Torrey2012THEGALAXIES,Moreno2019InteractingContent,Patton2020InteractingContext} have demonstrated that merger induced star formation peaks at coalescence. Despite the diversity demonstrated in Table 2, it is interesting to note that 2 of the 3 post-mergers with a global $\Delta$SFR $>0.5$ are fuel driven enhancements (9194-3702, 8084-3702). We might wonder, then, if there is some correlation between the fraction of fuel driven spaxels and the interaction stage, with the largest fraction of fuel driven spaxels occurring at coalescence when the star formation is strongest. The higher percentage of efficiency driven galaxies in the pair sample (50\% as opposed to 33\% in the post-mergers) would agree with a scenario where efficiency driven star-formation becomes less important towards coalescence. Such a correlation would fit with the hypothesis that merger-induced star formation stems from non-axisymmetric structures causing a loss of angular momentum for gas, which then funnels into the centre of the galaxy towards the end of an interaction. Figure \ref{fig:Fraction_mechanism} is a visual representation of Table \ref{tab:mechanism}, showing the fraction of fuel driven spaxels as a function of projected separation between pairs, with each point colour-coded by global offset from the star-forming main sequence. The post-mergers are shown with a projected separation as arbitrary values below zero, in the grey bar. Figure \ref{fig:Fraction_mechanism} reveals no clear correlation with the fraction of fuel driven spaxels and the projected separation (Pearson's correlation coefficient=$-$0.355). Thus there is no evidence from our sample that the merger stage is strongly linked to whether enhanced star formation is driven by elevated fuel or efficiency. It is important to note that projected separation is not a perfect representation of merger stage due to both uncertainty on whether a galaxy has already completed one or more pericentric passages, as well as projection effects. Morphological signs can serve as an additional indicator for interaction stage given galaxies are most disturbed at first passage and right before coalescence \citep{Lotz2004AClassification,Lotz2008GalaxyMergers,Hambleton2011AdvancedGalaxies}. Unfortunately most of our mergers have some degree of visual disturbance, making it difficult to directly link morphology to a distinct interaction stage (outside of pre- and post-coalescence). We attempt to account for this by highlighting pair galaxies which are connected to their companion by a tidal arm on Figure \ref{fig:Fraction_mechanism}, as a more concrete indicator that first passage has already occurred. Even with this additional category, no trend with merger stage and fraction of fuel driven spaxels arises, further affirming that interaction stage plays a secondary role influencing how star formation is enhanced. Figure \ref{fig:Fraction_mechanism} encapsulates one of the main conclusions of this paper: interacting galaxies show a broad diversity of the relative contributions of fuel and efficiency, with no obvious dependence on the merger stage we adopted. This sample suggests that the properties of the galaxies themselves (e.g. gas fractions, morphologies) as well as the details of the interaction (orbit and mass ratio) play a larger role in how star formation is enhanced. We have checked to see if the fraction of fuel driven spaxels correlates with \Mstarend, \Mgasend, \fgasend, SFR, and $\Delta$SFR, and found only \Mgas and \fgas have a Pearson's correlation coefficient greater than 0.6 (note that \fgas and \Mgas are themselves correlated, with a Pearson's correlation coefficient of 0.69). However, a correlation between the fraction of spaxels driven by enhanced amounts of molecular gas and large gas fractions is not a particularly noteworthy result. More likely it is the combined properties of the galaxy and its companion that drive the diversity of star formation properties observed. Recent simulations have found that mass ratio and orbital geometry can both impact how star formation is powered in mergers \citep{Moreno2021SpatiallyInteractions}. It is possible that, with a larger sample, more pronounced correlations with merger stage or integrated galaxy property would emerge, but for this sample such trends are not obvious. \subsection{What drives resolved changes in star formation?} \label{sec:Discussion} We have shown that, for both pair and post-merger galaxies, either enhanced SFE or enhanced \fgas can drive the subsequent star formation enhancement that results from an interaction. The merger sample is split evenly between galaxies whose star formation driven by SFE, by \fgasend, and those equally driven by both. An underlying issue yet to be addressed is how the mechanism which powers star formation relates to the spatial variations in star formation. Figure \ref{fig:profiles} displays radial profiles of \dsigsfr for the merger sample separated into distinct categories of efficiency driven, fuel driven, or driven by both based on the fraction of fuel driven spaxels in Table \ref{tab:mechanism} (profiles are additionally colour-coded by the exact fuel fraction computed). By creating radial profiles of offsets in \SigSFR we can investigate whether the fuel fraction we have computed can help explain the diversity of merger radial profiles seen with previous MaNGA studies (i.e., \citealt{Thorp2019SpatiallyMaNGA,Pan2019SDSS-IVInteractions,Steffen2021SDSS-IVPairs}). A more detailed analysis of merger radial profiles for different ALMaQUEST data products will be included in Pan et al. (in prep); what is discussed here is simply to assess how our global fuel fraction might relate to variations in \SigSFR already noted by previous works. The efficiency driven and fuel driven radial profiles in Figure \ref{fig:profiles} have clearly distinct behaviour. Fuel driven mergers tend to have greater \dsigsfr at small radii, though that does not always lead to a suppression of star formation in the outskirts. Efficiency driven mergers have both greater peaks in \dsigsfr (5 out of 9 surpass 0.5 dex enhancement at one point) and a greater diversity in profile behaviour. Profiles for the efficiency driven mergers can both increase \dsigsfr with radius, decrease \dsigsfr with radius, or have consistent \dsigsfr out to 2$R_e$. Thus it is likely a diversity of fuel fraction values could explain some of the diversity of post-merger profiles noted in \cite{Thorp2019SpatiallyMaNGA}. Interestingly, the fuel driven mergers show similar profiles to the starburst \dsigsfr profiles found in \cite{Ellison2020The0}, despite non-interacting starbursts being primarily driven by enhanced SFE. It is possible that mergers provide one route to a starburst by funnelling gas to a centralized starburst, and the resulting SFR enhancement profile is indistinguishable from secular starbursts. Efficiency driven mergers, on the other hand, are more likely to have stronger and continuous SFR enhancements that distinguish them from isolated galaxies of a similar $\Delta$SFR. In the case of the later, the strongest bursts of star formation cannot be driven by enhanced gas reservoirs alone. There are still issues with applying a single ``fuel driven'' prescription to what is clearly quite variable on the resolved scale. Simulations of the resolved gas properties of mergers have demonstrated that mergers with overall suppressed star formation can still have appreciable amounts of centralized cold-dense gas \citep{Moreno2021SpatiallyInteractions}. We can see this in our handful of fuel-driven mergers that have suppressed star formation beyond 1$R_e$, despite centralized boosts in star-formation. It is possible that, though a merger is primarily fuel or efficiency driven, there are regions that deviate from this norm. After all, the mergers which are driven by ``both'' are lacking any trend in radial profile unlike the other two categories. It is clearly worthwhile to explore how the dominant mechanism might vary within a galaxy. We attempt to pinpoint spatial variations in the dominant star-forming mechanism by examining the difference between \dSFE and \dfgasend. The difference between two offset parameters provides a rough approximation of which offset is dominant in an individual spaxel by simply quantifying which is larger. The difference between \dSFE and \dfgas is a bit like a fuel fraction for each spaxel, though we note that the number attributed to each has no physical meaning beyond comparing offset values. We can thus construct maps of \dSFEend$-$\dfgas to demonstrate which mechanism is more important within a certain region of a galaxy. We only consider this for spaxels where \dsigsfrend, \dSFEend, and \dfgas are positive, to avoid interpreting the contribution of suppressed SFE or \fgasend. Figure \ref{fig:Sub_method} displays maps of \dSFEend$-$\dfgas for three example galaxies, in conjunction with the \dSFE vs. \dfgas diagram colour-coded by \dSFEend$-$\dfgasend. Note how the magnitude of \dSFEend$-$\dfgasend scales with distance from the line of equality, where the contribution from SFE and \fgas is equal. Regions where \dSFE is greater are shown in purple, and regions where \dfgas is greater are shown in brown. We specifically chose three galaxies that are predominantly efficiency or fuel driven, to demonstrate how this clear-cut classification may break down on a spaxel-per-spaxel basis. Figure \ref{fig:Sub_method} demonstrates that, even within a given galaxy, the dominant driver of enhanced star formation can vary spatially. The first row of Figure \ref{fig:Sub_method} shows a post-merger galaxy with a global fuel fraction of 0.33, indicating that SFE is the dominant driver of enhanced star formation. However, a more complicated picture emerges when we look at the spatially resolved map of \dSFEend$-$\dfgas in the right column. Although the extended spiral arms are indeed dominated by enhanced SFE, the core is dominated by enhanced gas fractions. Similar results were found for the Antennae merging galaxies, which had central enhancements in dense gas along with lowered SFE \citep{Bemis2019Kiloparsec-Scale4038/39}. By examining maps of \dSFEend$-$\dfgas for all mergers (included with offset parameter maps in Appendix \ref{app:maps}), we find that most of our merger sample follows a similar trend of \dfgas dominating the central regions of the galaxy. To quantify whether \dfgas is more important in the centre of a merging galaxy, we compute a gradient in \dSFEend$-$\dfgas for each merger, subtracting the median \dSFEend$-$\dfgas within 1$R_e$ from the median where $R>1R_e$. All but two mergers have a positive gradient (i.e., \dfgas is more important in the centres than the outskirts), confirming that similar to previous case studies \dfgas plays a key role in triggering central bursts of star formation. Positive gradients are found both within efficiency driven and fuel driven mergers, with no correlation to the global fraction of fuel driven spaxels (Pearson's correlation coefficient=$-$0.19). Efficiency driven mergers have mostly efficiency driven spaxels (positive values in \dSFEend$-$\dfgas maps) with fuel driven central starbursts; but given there are more spaxels at large radii than small, those efficiency driven spaxels dominate the percentage of star-forming regions overall. The two mergers with negative gradients, 8156-3701 and 8081-9101, are not unique in their global galaxy properties, but have distinct \dsigsfr profiles. 8156-3701 (a post-merger with fuel fraction=0.04) has \dsigsfr$>0.5$ out to two effective radii, powered by predominantly $\Delta$SFE. In contrast, 8081-9101 (a post-merger fuel fraction=0.5) has brief suppressions in star formation at the centre and edge of the galaxy, but enhancements $\sim$0.25 dex for the majority of the galaxy. Though no substantial result can be derived from two galaxies alone, it is interesting that mergers without central fuel enhancements display two of the most chaotic \dsigsfr profiles. However, on average, our sample confirms that central bursts of star formation in mergers require centralized enhancements in gas fraction. Clearly the consistency of positive gradients in \dSFEend$-$\dfgas does not negate further variations between galaxies, as can be seen in the examples from Figure \ref{fig:Sub_method} alone. The middle row of Figure \ref{fig:Sub_method} shows a post-merger with a negative gradient in \dSFEend$-$\dfgas. Though star formation in this merger is clearly driven by enhanced SFE (it has a fuel fraction of 0.04), \fgas plays a slightly more important role in the outskirts than in the centre (though \dSFE is always greater). But even galaxies with a positive gradient, as expected, can have local variations from the norm. The last row of Figure \ref{fig:Sub_method} shows a fuel driven post-merger (fuel fraction 0.78) with a positive \dSFEend$-$\dfgas gradient. However, regions where \dSFEend$-$\dfgas$<0$ persist out to large radii as well, following a central extended structure within the galaxy. In this case fuel-driven star formation is not limited to the centre of the galaxy (though \dSFE becomes more important at large radii as well). Such variations are similar to those found in studies of isolated galaxies, where the slope and offset of the rSFMS, rKS, and rMGMS can vary significantly depending on the local galaxy environment (particularly for spiral arms, \citealt{Pessa2021AstronomyScale}). In summary, we find that there is not only considerable diversity from galaxy-to-galaxy in terms of the relative importance of fuel and efficiency, but even within a given galaxy, different regions show considerable variation. \section{Summary \& Conclusion} \label{sec:summary} We have presented MaNGA+ALMA CO(1-0) observations of 31 merging galaxies from the ALMaQUEST merger sample in order to study whether interaction-induced star formation is driven primarily by enhancements in SFE or gas fraction. The sample includes 14 post-merger galaxies and 17 pairs with projected separations up to 90 kpc. We compare the rSFMS, rKS, and rMGMS of the merger galaxies to relatively isolated galaxies in the ALMaQUEST survey. Although the ensemble of spaxels in the post-mergers, pairs and isolated galaxies have similar scaling relations (Figure \ref{fig:All_scaling_relations}), significant variation can be seen when examining the resolved relations for individual mergers (see Figures \ref{fig:SFMS_resolved}, \ref{fig:KS_resolved}, and \ref{fig:MGMS_resolved}). In order to better understand the spatial variations in each merger, we construct maps of offsets from these resolved relations (\dsigsfrend, \dSFEend, and \dfgasend). By directly comparing \dSFE and \dfgas for all spaxels with enhanced star formation, we can identify whether the star formation is driven by an enhanced gas reservoir, or an enhanced efficiency at which gas is converted into stars. We find that when all spaxels in all galaxies are considered together, \fgas and SFE contribute equally to enhanced star formation (Figure \ref{fig:SFMS_mechanism}). However, the approximately equal importance of fuel and efficiency across the entire sample is misleading.. When we examine galaxies on an individual basis, a different picture emerges. Some mergers are clearly dominated by either efficiency or fuel driven star formation, with all points lying in one regime of Figure \ref{fig:delta_plots_all}. About a third of the merger sample is predominantly efficiency driven, a third is fuel driven, and a third is driven equally by both. We investigate whether the dominant star formation mechanism might be correlated with global galaxy properties (SFR, \Mstarend, \Mgasend, \fgasend, $\Delta$SFR), and find only an unsurprising correlation with fuel driven star formation and large gas fractions. The driving star formation mechanism does not depend on the stage of an interaction either. The percentage of fuel driven post-mergers is somewhat more than the percentage of fuel driven pair galaxies. But fuel fraction does not correlate with other indicators of interaction stage, such as projected separation and tidal connections (Figure \ref{fig:Fraction_mechanism}). The progression of the interaction does not lead to clear evolution in the fuel fraction, which implies the unique properties of the progenitors are more influential on which mechanisms powers enhancements in star-formation. In section \ref{sec:Discussion} we investigate how the dominant star-forming mechanism leads to variations on a spatial scale. Radial profiles of \dsigsfr for the merger sample are shown in Figure \ref{fig:profiles}, separated into three categories based on fuel fraction. We find that ``fuel driven'' mergers (with fuel fractions > 0.6) have relatively distinct \dsigsfr profiles compared to ``efficiency driven'' mergers (fuel fraction < 0.4), which often have stronger and more extended \dsigsfr enhancements. Thus a range of fuel fractions in mergers could explain the diversity of star formation offset profiles seen in previous merger studies \citep{Thorp2019SpatiallyMaNGA,Pan2019SDSS-IVInteractions,Steffen2021SDSS-IVPairs}. The range of \dsigsfr profiles in these three categories implies more internal variation within a galaxy than a single fuel fraction can capture. Figure \ref{fig:Sub_method} demonstrates this variation in the dominant mechanism within a galaxy by presenting 2D maps of \dSFEend$-$\dfgasend, indicating which offset parameter is largest in each spaxel. Even galaxies which are driven predominantly by a single mechanism on a global scale can exhibit internal deviations from the dominant mechanism. For example, both efficiency and fuel driven mergers tend to have \dfgasend$>$\dSFE in their centre, confirming that enhanced amounts of molecular gas are crucial to merger-induced central starbursts, even if the rest of the galaxy has efficiency driven star formation. The work presented here is the first to investigate the resolved molecular gas and star formation properties for a relatively large set of mergers spanning a wide range of sSFRs and stages of interaction, with the explicit goal of understanding how gas and star formation evolve with an interaction. Our work adds to the growing evidence that, despite following overall scaling relationships, galaxies are diverse in their details. Moreover, we have shown that in addition to diversity on a global scale (i.e. some mergers have their star formation driven by fuel, others by efficiency), there is significant internal variation on kpc-scales. \section*{Acknowledgements} We acknowledge and respect the Lekwungen peoples on whose traditional territories the University of Victoria stands and where the majority of this work was conducted. We strive to honour the Songhees, Esquimalt, and WSÁNEĆ peoples who were the first astronomers of this land and whose continued stewardship is crucial to its preservation. MDT thanks Jorge Moreno, Shoshannah Byrne-Mamahit, Salvatore Quai, Robert Bickley, Scott Wilkinson and Joanna Woo for discussion throughout the creation of this work, as well as William Baker and Fangting Yuan for key editorial remarks. SLE and DRP gratefully acknowledges NSERC of Canada for Discovery Grants which helped to fund this research. HAP acknowledges support by the Ministry of Science and Technology of Taiwan under grant 110-2112-M-032-020-MY3. SLE, MDT, LL and HAP also gratefully acknowledge grant MOST 107-2119-M-001-024 and 108-2628-M-001-001-MY3 for travel funding that facilitated both the ALMA data reduction and analysis of ALMaQUEST survey. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2015.1.01225.S, ADS/JAO.ALMA\#2017.1.01093.S, ADS/JAO.ALMA\#2018.1.00558.S, ADS/JAO.ALMA\#2018.1.00541.S, ADS/JAO.ALMA\#2019.1.00260.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Tai- wan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSSIV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatory of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University. \section*{Data Availability} The MaNGA data cubes used in this work are publicly available at https://www.sdss.org/dr15/. The ALMA data used in this work are publicly available after the standard one year proprietary period via the ALMA archive: http://almascience.nrao.edu/aq/. \bibliographystyle{mnras} \bibliography{references.bib} \appendix \section{ALMaQUEST - Merger Set} \label{app:DP} Included in figure \ref{fig:ALMaQUEST_DP_1}-\ref{fig:ALMaQUEST_DP_5} are data product maps for the ALMaQUEST merger set. Post-mergers are shown first, followed by pairs in order of increasing pair separation. All 31 mergers are included, even the 6 galaxies with insufficient overlap between CO $S/N>3$ and H$\alpha$+D4000-\SigSFR measurements which are excluded from individual galaxy analysis. Data products include, from left to right: the SDSS \emph{gri}-image, inclination corrected stellar mass surface density (from {\sc pipe3d}), inclination corrected molecular gas surface density (computed from CO luminosity), inclination corrected star formation rate surface density (computed from H$\alpha$ luminosity), inclination corrected star formation rate surface density (computed from H$\alpha$ luminosity and sSFR-D4000 fit), molecular gas fraction (\Siggasend/\Sigstarend), and star formation efficiency (\SigSFRend/\Siggasend). \section{Metallicity dependent CO conversion factor} \label{app:alpha_CO} Throughout this work we elect to use a constant $\alpha_{\textrm{CO}}$ rather than a metallicity dependent one. This is highly beneficial to our analysis in that we can include spaxels which do not meet the requirements to measure accurate gas-phase metallicities (requiring high signal-to-noise in multiple emission lines). As a result our \Siggas measurements will be imperfect, leading to the possibility that variations \dSFE and \dfgas (both of which require \Siggasend) could be the result of inaccuracies in $\alpha_{\textrm{CO}}$. Depending on the degree of that inaccuracy this could sully our final conclusions concerning the role enhanced efficiency and fuel play in increasing star formation activity. Though we cannot compute a metallicity dependent $\alpha_{\textrm{CO}}$ ($\alpha_{\textrm{CO, met}}$ from here on) for every spaxel, we can characterize the change in \Siggas for spaxels where $\alpha_{\textrm{CO, met}}$ is measurable. We compute a $\alpha_{\textrm{CO, met}}$ for all spaxels that are star-forming based on the criteria described in Subsection \ref{subsec:SFR-D4000}, for which we can also measure the gas phase metallicity using the O3N2 calibration of \cite{Pettini2004Oiii/NiiRedshift}. We set $\alpha_{\textrm{CO, met}}=4.35 \times Z'^{-1.6} (\textrm{M}_{\odot} (\textrm{K } \textrm{km/s } \textrm{pc}^2)^{-1})$ where $Z'$ is the metallicity normalized to solar metallicity ($Z'=10^{\textrm{PP04}_{\textrm{O3N2}}-8.69}$) \citep{Sun2020DynamicalGalaxies}. We find the median difference between \Siggas and \Siggasmet to be $-$0.068 dex, with a standard deviation of 0.124 dex. If we want to be conservative, we can say that \dfgas and \dSFE both have an uncertainty of 2$\sigma$ on the difference in \Siggasend, or 0.248 dex. Figure \ref{fig:delta_plot_buffer} is a replication of Figure \ref{fig:delta_plots_all}, with a grey bar representing 0.248 dex of uncertainty; points within this range could change from fuel to efficiency driven, or vice versa, if one switched from a constant to a metallicity dependent $\alpha_{\textrm{CO}}$. We can thus exclude spaxels within this grey bar from our calculation of the fuel fraction to achieve a value that is independent of variations in $\alpha_{\textrm{CO}}$. By excluding borderline spaxels, most galaxies tend to be more ``fuel'' or ``efficiency'' driven. Figure \ref{fig:FF_change_met} directly compares the original fuel fraction, using a constant $\alpha_{\textrm{CO}}$, to this new one which incorporates error from not using $\alpha_{\textrm{CO, met}}$. Ideally all points would be along the line of equality, confirming that the fraction of fuel driven spaxels does not change. Instead galaxies which are fuel driven become more fuel driven, rising above the line of equality, and galaxies which are efficiency driven become more efficiency driven. As a result very few galaxies change whether we would consider them ``efficiency'' or ``fuel'' driven. We highlight regions where both the new and old fuel fractions agree on the dominant mechanism: where efficiency dominates (red), fuel fraction dominates (blue), or both equally influece star formation(grey). Only two mergers change which mechanism dominates: both are mergers where efficiency and fuel are equally important with a constant $\alpha_{\textrm{CO}}$, but then change to fuel driven. This shift is not the result of a significant change in fuel fraction, rather the two mergers were already at the border between categories. As discussed in the main analysis, these categories of ``efficiency'' and ``fuel'' driven are less robust than the continuously changing fuel fraction measurement. In summary, our measured fuel fraction changes little when uncertainties in $\alpha_{\textrm{CO}}$ are considered. Moreover, our results concerning \dSFE being more important to star formation than \dfgas (or vice versa) are unlikely to stem from local variations in conversion factor. Rather they characterize the true relationship between star-formation and molecular gas in a merger event. \section{sSFR-D4000} \label{app:sSFR-d4000} The SFRs presented herein include those that have been calibrated from D4000. Here, we demonstrate that our conclusions are not significantly affected by the inclusion of SFRs computed in this way. Figure \ref{fig:sSFR_D4000_dSFR} shows the distribution of sSFR versus D4000 for all star forming spaxels in MaNGA, based on our star forming criteria described in Subsection \ref{subsec:SFR-D4000}, with contours representing the density of the distribution of hexbins colour-coded by the median \dsigsfr value in that bin. The asymptotic nature of the sSFR-D4000 relation as D4000 approaches 1.45 can lead to drastic difference in approximated \SigSFR for spaxels with very similar D4000 values. Therefore, only spaxels with D4000>1.4 are used for SFR calculations. There is a non-negligible scatter in the sSFR-D4000 relation which is strongly correlated with \dsigsfrend. This is to be expected; by definition a spaxel with high sSFR given the D4000 index would have a higher \SigSFR given its \Sigstarend. The red line is a median sSFR in a bin so it makes sense that scatter significantly above the line would corresponds to large, positive \dsigsfrend. Thus by approximating \SigSFR from the median sSFR, we will always underestimate the offset of the star formation from regular behaviour (both enhancements and deficits). That bias works in favour of this analysis, where we are looking for strong offsets in \SigSFR. Thus for D4000-\SigSFR values, any enhancement or suppression in star formation is a lower limit on the true offset; the same can be said for enhancements and deficits in SFE. When D4000-\SigSFR is used the absolute value of \dsigsfr and \dSFE will be underestimated, but \dfgas (which does not depend on a \SigSFR measurement) will not be changed. The dependence of \dSFE on \SigSFR is crucial when comparing \dSFE and \dfgas to determine which mechanism drives enhanced star formation. When using a D4000-\SigSFR the value of \dSFE is a lower limit, so \dSFEend$>$\dfgas will be consistent even if D4000-\SigSFR is less accurate than H$\alpha$-\SigSFRend. However if \dSFEend$<$\dfgasend, \dSFE could be underestimated and the statement might not be true if we could measure a H$\alpha$-\SigSFRend. We therefore have to consider that galaxies with a large fraction of fuel driven spaxels (i.e., \dSFEend$<$\dfgasend) might have a smaller fraction if we were not dependent on D4000-\SigSFR measurements. To check how this bias might alter our results, we calculate the fraction of fuel driven spaxels both with our combined H$\alpha$+D4000 \SigSFR values, as well as those which only have H$\alpha$-\SigSFRend. Four galaxies do not have at least 20 spaxels with viable H$\alpha$-\SigSFRend, so can not be part of the test. Both fractions are included in Table \ref{tab:mechanism} and discussed in the text, though we provide additional comparison here. Figure \ref{fig:sSFR_D4000_dSFR_change} directly compares the fuel fraction determined from only H$\alpha$ spaxels to the fuel fraction when both H$\alpha$ and D4000 spaxels are included. Ideally these two values would be equal and all galaxies would lie on the line of equality, but as expected there is some difference between the two. Difference between the two fuel fractions can be acceptable so long as it does not change which mechanism predominantly drives star formation. To better demonstrate this distinction we highlight regions where both fractions agree: that star formation is efficiency drive (red), fuel driven (blue), or both (grey). Only one galaxy is outside these acceptable regions: 8616-9101, which is efficiency driven when both H$\alpha$ and D4000 spaxels are included, but fuel driven when only H$\alpha$ spaxels are considered. 8616-9101 is an interesting case given we expect spaxels that use D4000-\SigSFR to underestimate \dSFE and bias towards large fuel fractions, but in this case including D4000 spaxels leads to a smaller fuel fraction. Considering only one galaxy changes dominant mechanism, we are assured that our results are not dependent on the inclusion of D4000-\SigSFR values. \section{Offset Maps for Entire Merger Sample} \label{app:maps} Figures \ref{offset_cat1}-\ref{offset_cat4} are a complete catalogue of the \dsigsfrend, \dSFEend, and \dfgas distributions for the merger galaxy sample examined in this work. Diverging colourbars are used such that blue spaxels represent enhancements in \SigSFRend, SFE, or \fgasend. Red spaxels represent a supression in either variable. The SDSS \emph{gri}-image is included as well, to provide context for the galaxy and the MaNGA IFU coverage. One will note that some galaxies have \dfgas measurements where no \dsigsfr or \dSFE is measured. This results from the stricter cuts required to measure \SigSFRend, a limitation not imposed on the calculation of \dfgasend. The fourth column provides a map of \dSFEend$-$\dfgas for all spaxels where \dsigsfrend$>0$, as described in Section \ref{sec:Discussion}. Note that most mergers have negative \dSFEend$-$\dfgas in the centre of the galaxy, implying gas fractions are more centrally enhanced than SFE. \bsp % \label{lastpage}

Title: E and B modes of the CMB y-type distortions: polarised kinetic Sunyaev-Zeldovich effect from the reionisation and post-reionisation eras

Abstract: We study the E and B mode polarisation of the cosmic microwave background (CMB) originating from the transverse peculiar velocity of free electrons, at second order in perturbation theory, during the reionisation and post-reionisation eras. Interestingly, the spectrum of this polarised kinetic Sunyaev-Zeldovich (SZ) effect can be decomposed into a blackbody part and a y-type distortion. The y-distortion part is distinguishable from the primary E and B modes and also the lensing B modes. Furthermore, it is also differentiable from the other y-type signals, such as the thermal SZ effect, which are unpolarised. We show that this signal is sensitive to the reionisation history, in particular to how fast reionisation happens. The E and B modes of y-type distortion provide a way to beat the cosmic variance of primary CMB anisotropies and are an independent probe of the cosmological parameters. The blackbody component of the pkSZ effect would be an important foreground for the primordial tensor modes for tensor to scalar ratio $r \lesssim 3\times10^{-5}$.

https://export.arxiv.org/pdf/2208.02270

\flushbottom \section{Introduction \label{sec_intro}} In addition to the primary anisotropies created during recombination, several other physical processes at later redshifts can generate secondary anisotropies in the cosmic microwave background (CMB) at linear and higher orders in perturbations. A large number of ongoing and future experiments will measure the polarised CMB anisotropies with progressively higher sensitivity. These experiments will also have a larger number of frequency bands compared to the past experiments. The future experiments (funded and proposed) include ground-based experiments such as the Simons Observatory \cite{ade2019simons}, CMB-S4 \cite{abazajian2016cmb}, and CMB-HD \cite{sehgal2019cmb} and satellite-based missions such as LiteBIRD \cite{matsumura2014mission}, PIXIE \cite{kogut2011primordial}, PRISM \cite{andre2013prism}, PICO \cite{PICO}, and CMB-Bharat \cite{CMB_Bharat}. This opens up the exciting possibility of having a new window into the Universe using \textit{polarised spectral distortion anisotropies} of the CMB. The secondary anisotropies contain a wealth of information, but detecting them is challenging because of their small amplitude. Even if a CMB experiment has sufficient sensitivity, distinguishing the secondary from the primary anisotropies is difficult if they have the same spectrum. In particular, we are limited by the cosmic variance of the primary CMB anisotropies. The situation becomes more promising if the secondary anisotropies have a different spectrum and thus can evade the cosmic variance limit of the primary and other secondary anisotropies. One such physical process is the second order polarisation of the CMB due to the kinetic Sunyaev-Zeldovich (pkSZ) effect which is the focus of this paper. This pkSZ effect was first predicted in 1980 by Rashid Sunyaev and Yakov Zeldovich \cite{SZ_80}. The pkSZ effect from reionisation as well as from galaxy clusters has been studied previously \cite{Sazonov1999,Hu_2000, BAUMANN2003, roebber2014polarization,Pierpaoli2016}. Our work differs from the previous studies in several ways. Instead of a flat sky approximation, we derive the full sky exact expressions of the E and B mode power spectrum of the pkSZ effect. We also study the effect of different reionisation histories on the power spectrum, instead of assuming instantaneous reionisation. We show that the pkSZ effect is sensitive to the reionisation history and future observations can in principle extract information about various cosmological parameters, beating the cosmic variance limit of primary anisotropies. A similar analysis was previously done by Renaux-Petel et al. \cite{renaux2014spectral}. We compare our results with theirs in section \ref{sec_result}. During the era of reionisation (z $\sim 6-20$) \cite{adam2016planck}, the free electrons that are produced have some peculiar velocity with respect to the CMB rest frame. As a result, in the electron rest frame, the CMB is no longer isotropic \cite{kamionkowski2003aspects, peebles_fireball}. In addition to the dipole, multipoles of all higher orders are present in the intensity of incoming radiation in the electron rest frame. This occurs due to the non-linear nature of the relativistic Doppler boost, as well as the non-linear relation between the temperature and the intensity in the Planck spectrum. In particular, a quadrupolar anisotropy gets generated. Thomson scattering of this quadrupolar anisotropic radiation by the electrons produces linear polarisation \cite{dodelson2003modern, durrer2020cosmic, KOSOWSKY199649,chandrasekhar2013radiative}. The polarisation strength is proportional to the square of the transverse velocity of the electrons. More importantly, the spectral signature of this polarised signal is different from that of primary CMB polarisation as well as the lensing B modes. It can be shown that the intensity quadrupole consists of a blackbody spectrum along with a y-type distortion \cite{Sazonov1999,kamionkowski2003aspects,Sunyaev_2013, Chluba_superposition_BB}. The y-type distortion (SZ spectrum) provides a unique signature to this polarisation signal which makes it possible to detect using component separation techniques, provided the required sensitivity is achieved in future. The y-distortion part is also free from the cosmic variance of the primary polarisation as they do not have the same spectrum. Since the signal is generated at higher order in perturbation theory \cite{fidler2014intrinsic}, we expect to get both E and B modes even when the velocity fields are sourced by purely scalar perturbations. We perform a full sky numerical calculation of the y-type angular power spectrum of both the E and B modes. We assume a homogeneous electron density during reionisation which evolves with redshift. We include both symmetric and asymmetric reionisation \cite{lewis2008cosmological, adam2016planck}. For completeness, we also include the second reionisation of helium \cite{heinrich2017complete}. We show that the pkSZ effect is sensitive to the central redshift as well as the duration of reionisation. If detected, this signal can be instrumental in distinguishing different reionisation histories. It can also act as an independent probe to measure large-scale velocity fields and thus constrain other cosmological parameters. The blackbody part of the signal will also act as a foreground for the primordial B modes. A precise measurement of the intrinsic B mode polarisation has important implications for understanding the physics of the early universe \cite{baumann2009probing, krauss2010primordial}. In order to correctly measure the primordial signal, an accurate prediction of these foreground signals is necessary. We show the E and B mode power spectrum, for our fiducial symmetric reionisation history chosen to be consistent with current observations, in figure \ref{fig:summary plot}. Also shown is the Poisson noise contribution due to the galaxy clusters. For comparison, the primordial B modes for tensor to scalar ratios $r$ of $10^{-4}$ and $10^{-5}$ are given. We have ignored the spatial variation in the electron density field. As we see, the low redshift contribution from the Poisson noise from galaxy clusters is $\sim$ 2 orders of magnitudes smaller than the reionisation signal. The PRISM sensitivity curve is plotted to provide an idea about the detectability of this signal in future experiments. More details are given in section \ref{sec_result}. We will assume a flat $\mathrm{\Lambda CDM}$ universe with baryon and matter density parameters $\mathrm{\Omega_b = 0.0490}$ and $\mathrm{\Omega_m = 0.3111}$, Hubble constant, $\mathrm{H_0} =$ 100h $\mathrm{kms^{-1} Mpc^{-1}}$ with $\mathrm{h = 0.6766}$, spectral index of primordial curvature perturbations $\mathrm {ns = 0.9665}$, its amplitude $\mathrm {log(As)} =-8.678 $ and helium mass fraction $\mathrm{X_{He} = 0.24}$ \cite{aghanim2020planck}. We used publicly accessible codes CAMB \cite{lewis2000efficient} and Colossus \cite{diemer2018colossus} for our numerical analysis and Vegas \cite{lepage1978new} for multidimensional adaptive Monte-Carlo integration. We will be using units with the speed of light in vacuum $c=1$. \section{Polarised kinetic Sunyaev-Zeldovich effect \label{sec_polfield}} A general photon distribution can be characterised by a set of Stokes parameters, $\mathrm{I,\;Q}$, $\mathrm{U}$, and $\mathrm{V}$, where I is the intensity, Q and U measure the linear polarisation, and V is the measure of the circular polarisation. Since the Thomson scattering does not generate circular polarisation \cite{rybicki1991radiative,dodelson2003modern,chandrasekhar2013radiative}, we can define a triad $\mathcal{T}$ to describe the incoming and outgoing radiation in a Thomson scattering process, \bea \mathcal{T}=\left\{\frac{\delta \mathrm{I}}{\mathrm{I}}, \frac{ \mathrm{Q}}{\mathrm{I}},\frac {\mathrm{U}}{\mathrm{I}}\right\}=\{\mathcal{I},\mathcal{Q},\mathcal{U}\}, \eea where $\mathcal{I}$ is the average background intensity and $\delta \mathrm{I}$ is the difference in intensity with respect to the background. In the case of the CMB, we are interested in the polarisation anisotropies and must work with the polarisation field. The Stokes parameters now are dependent on position, time, and the momentum of the photons. We will use the combination $\left(\mathcal{Q}\pm i\mathcal{U}\right)\left(\mathbf{ r},\mathbf{p},\eta\right)\equiv P_{\pm}\left(\mathbf{ r},\mathbf{p},\eta\right)$ to describe the 3D polarisation field, where $\mathbf{ r}$ is the position vector, $\mathbf{ p}$ is the momentum vector, and $\eta$ is the conformal time. The momentum vector is given by $\mathbf{ p}=\mathrm{p}\mathbf{\hat{n}}$, where p is the magnitude and $\mathbf{\hat{n}}$ is the direction of propagation. The full photon distribution can be decomposed into a spectral shape and an amplitude. For the CMB, the spectral dependence on the momentum p is known separately and can be factored out of the Boltzmann equation \cite{dodelson2003modern, Pitrou_y_sky, Chluba_2by2}. Hence, we only need to evolve the amplitude part of a particular spectral shape. Therefore, the polarisation field becomes a function of $\mathbf{ r}, \mathbf{\hat{n}}$ and $\eta$. The time evolution of this polarisation field is given by the Boltzmann equation \cite{Seljak_1996, durrer2020cosmic} \bea \frac{dP_{\pm}(\mathbf{ r},\mathbf{\hat{n}},\eta)}{d\eta}=C[P_{\pm}],% \eea where $\hat{n}$ is the direction of incoming photons and $\eta$ is the conformal time and $C[P_\pm]$ is the Thomson collision term which can be written as \cite{Hu_White_97,2013_Tram} \bea C[P_{\pm}]=\tau'P_{\pm}(\mathbf{ r},\mathbf{\hat{n}},\eta)-\tau'P^{\pm}_{\mathrm{sc}}\left(\mathbf{ r},\mathbf{\hat{n}},\eta\right), \eea where the first term on the right hand side accounts for the photons that are scattered out of the line of sight, while the second term is the source term. The differential optical depth or the scattering rate is given by \bea \tau'=\frac{d\tau}{d\eta}=-n_\mathrm{e}a\sigma_{\mathrm{T}}, \eea where $n_\mathrm{e}$ is the electron number density, $a$ is the expansion scale factor, and $\sigma_{\mathrm{T}}$ is the Thomson scattering cross section. The source term $P_{\mathrm{sc}}\left(\mathbf{ r},\mathbf{\hat{n}},\eta\right)$ is given by \cite{chandrasekhar2013radiative, durrer2020cosmic, piattella2018lecture} \bea \label{p_in} P^{\pm}_{\mathrm{sc}}\left(\mathbf{ r},\mathbf{\hat{n}},\eta\right)=-\frac{\sqrt{6}}{10}\sum_{\lambda =-2}^{2}\,_{\pm2}Y_{2 \lambda}\left(\mathbf{ \hat{n}}\right)\int d^{2}\mathbf{ \hat{n}}\;Y_{2 \lambda}^{*}\left(\mathbf{ \hat{n}'}\right)\mathcal{I}_{\mathrm{sc}}\left(\mathbf{r},\mathbf{ \hat{n}'},\eta\right), \eea where $\mathbf{ \hat{n}'}$ is the incoming photon direction in the rest frame of the electrons and $\mathcal{I}_{\mathrm{sc}}\left(\mathbf{r},\mathbf{ \hat{n}'},\eta\right)$ is the corresponding intensity of the incoming unpolarised blackbody CMB radiation. The spin-2 and spin-0 spherical harmonic functions are $\,_{\pm2}Y_{2 \lambda}$ and $Y_{2 \lambda}$ respectively. The orthogonality condition of spherical harmonics ensures that the integral in the above equation is non-zero only if $\mathcal{I}_{\mathrm{sc}}\left(\mathbf{r},\mathbf{ \hat{n}'},\eta\right)$ has a quadrupolar anisotropy. When solving the Boltzmann equation, we integrate along our line of sight direction. This implies that the position vector $\mathbf{r}$ along the line of sight is a function of $\mathbf{ \hat{n}}$ and $\eta$. We consider $\eta_0$ to be the conformal time today and $\eta_{i}$ to be the conformal time at some early epoch before reionisation. Also, at $\eta_0$, $\tau(\eta_0)=0$, by the definition of $\tau$, \bea \label{tau_chi} \tau(\eta)=\int_{\eta}^{\eta_0}n_\mathrm{e}(\eta)\sigma_{\mathrm{T}}a\,d\eta. \eea Therefore, the polarisation field today can be written as \cite{Khatri_crinkles} \bea \label{b_soln} P_{\pm}\left(\mathbf{ \hat{n}};\eta_0\right)=e^{-\tau(\eta_{i})}P_{\pm}\left(\mathbf{ \hat{n}};\eta_{i}\right)-\int_{\eta_{i}}^{\eta_0}e^{-\tau(\eta)}\tau'P^{\pm}_{\mathrm{sc}}\left(\mathbf{ r},\mathbf{ \hat{n}},\eta\right)d\eta, \eea where the integral is performed along the line of sight, $\mathbf{ r}=\mathbf{ r}(\eta,\mathbf{\hat{n}})$. We are interested in the scattering of initially unpolarised CMB radiation. Thus we have, $P\left(\mathbf{ \hat{n}};\eta_{i}\right)=0$. Therefore, using eq.(\ref{p_in}) in eq.(\ref{b_soln}) we get \bea P_{\pm}\left(\mathbf{ \hat{n}};\eta_0\right)=\int_{\eta_{i}}^{\eta_0}d\eta \;\frac{\sqrt{6}}{10}\tau'e^{-\tau(\eta)}\sum_{\lambda =-2}^{2}\,_{\pm2}Y_{2 \lambda}\left(\mathbf{ \hat{n}}\right)\int d^{2}\mathbf{ \hat{n}'}\;Y_{2 \lambda}^{*}\left(\mathbf{ \hat{n}'}\right)\mathcal{I}_{\mathrm{sc}}\left(\mathbf{r},\mathbf{ \hat{n}'},\eta\right). \eea We can do a change of variables from conformal time $(\eta)$ to comoving distance $(\chi=\eta_0-\eta)$, to get the polarisation field at $\chi=0$, \begin{align} \label{pol_sem_final} P_{\pm}\left(\hat{\mathbf{ n}}\right)=-\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)a(\chi)\,_{\pm2}Y_{2 \lambda}\left(\mathbf{ \hat{n}}\right)\int d^{2}\mathbf{ \hat{n}'}\;Y_{2 \lambda}^{*}\left(\mathbf{ \hat{n}'}\right)\mathcal{I}_{\mathrm{sc}}\left(\mathbf{r},\mathbf{ \hat{n}'},\chi\right), \end{align} where $(\chi_{i}=\eta_0-\eta_{i})$. Since we are considering homogeneous reionisation, the electron density field has no spatial variation. It is a function of redshift (comoving distance) only. The explicit form of reionisation history will be described in section \ref{sec_EB_pow}. To complete the discussion, we need to know the spectrum of the polarised radiation after scattering. \subsection{Spectral signature \label{sec_spectral}} The origin of the quadrupole and the spectral distortion of incoming CMB intensity in the electron frame, $\mathcal{I}_{\mathrm{sc}}\left(\mathbf{r},\mathbf{ \hat{n}'},\chi\right)$, can be understood by looking at the Doppler boost when we shift to electron rest frame from the CMB rest frame. In the electron frame, the CMB photons coming from different directions follow a blackbody spectrum with temperature in direction $\mathbf{\hat{n}}$ given by \cite{rybicki1991radiative, Sunyaev_2013} \begin{align} \label{ksz_temp} T\left(\mathbf{ r},\mathbf{ \hat{n}'},\chi\right)=\frac{T_0(\chi)}{\gamma\left(1+\mathbf{ v}(\mathbf{r},\chi)\cdot\mathbf{ \hat{n}'}\right)}&=T_0(\chi)\left[1\underbrace{-\mathbf{ v}\cdot\mathbf{ \hat{n}'}+\frac{1}{2}v^2+\left(\mathbf{ v}\cdot\mathbf{ \hat{n}'}\right)^2+\mathcal{O}\left(v^3\right)+\cdot\cdot\cdot}_{\theta(\mathbf{ \mathbf{r}, \hat{n}'},\chi)}\right]\nonumber\\ &\equiv T_0(\chi)\left[1+\theta(\mathbf{ \mathbf{r}, \hat{n}'},\chi)\right], \end{align} where $T_0(\chi)$ is the average temperature of the CMB, the velocity field of the electrons with respect to the CMB rest frame is given by $\mathbf{ v}(\mathbf{r},\chi)$, and $\gamma=1/\sqrt{\left(1-v^2\right)}$ is the Lorentz factor associated with the transformation from the CMB rest frame to the electron rest frame. Due to the non-linearity of the relativistic Doppler shift, if we expand the temperature in a Taylor series, we see that the multipole moments of all orders are present. The spectrum still remains a blackbody in each direction. The blackbody spectrum however is also a non-linear function of the temperature. Further expanding the intensity in a Taylor series and subtracting the average background blackbody spectrum, we see that the spectrum at second order is no longer a pure blackbody but has a y-type component too. This is essentially the y-type distortion produced by the mixing of blackbodies in the Thomson scattering \cite{Illarionov_mixing_BB_1972, Chluba_superposition_BB, khatri_mixing_BB}. Therefore, the difference in intensity $\delta I_{\nu}=I_{\nu}-\bar{I}_{\nu}$ or the difference in occupation number $\delta n_{\nu}=n_{\nu}-\bar{n}_{\nu}$ with respect to the average background as seen by the electron is given by \begin{align} \delta n_{\nu}=\frac{1}{2h\nu^{3}}\delta I_{\nu}=\left(\theta+\theta ^2\right)\left(T\frac{\partial n_{pl}}{\partial T}\right)\bigg|_{T_0}+\frac{\theta^2}{2}\left(T^4\frac{\partial}{\partial T}\left(\frac{1}{T^2}\frac{\partial n_{pl}}{\partial T}\right)\right)\bigg|_{T_0}+\mathcal{O}(\theta^3)\;\cdots, \end{align} where $I_{\nu}$ and $n_{\nu}$ are the intensity and occupation number in the electron rest frame and $\bar{I}_{\nu}$ and $\bar{n}_{\nu}$ are the intensity and occupation number of the average blackbody spectrum with temperature $T_0$. The resultant fractional relative intensity is given by \begin{align} \label{temp_quadpole} \frac{\delta I}{I}\equiv \mathcal{I}_{\mathrm{sc}}=\frac{\delta n_{\nu}}{n_{\nu}}=\left(\theta+\theta ^2\right)g(x)+\frac{\theta^2}{2}y(x)+\mathcal{O}(\theta^3)\:\:\cdots, \end{align} where $g(x)=\frac{xe^x}{(e^x-1)}$ is the differential blackbody spectrum, $y(x)=\frac{xe^x}{(e^x-1)}\left(x\frac{e^x+1}{e^x-1}-4\right)$ is the y-type distortion spectrum, and $x=\left(\frac{h\nu}{k_BT_o}\right)$ is the dimensionless frequency. We note that $g(x)$ is also the spectrum of the primordial CMB anisotropies for all the CMB experiments which make a differential measurement of the CMB. We are interested in the quadrupolar component which will contribute to the polarisation. Collecting the terms from eq.(\ref{temp_quadpole}) which will contribute to the quadrupolar moment we get \bea \label{k_quadrupole} \mathcal{I}_{\mathrm{sc}}\big|_{(\mathrm{quadrupolar})}=2\left(\mathbf{ v}\cdot\mathbf{ \hat{n}'}\right)^2g(x)+\frac{1}{2}\left(\mathbf{ v}\cdot\mathbf{ \hat{n}'}\right)^2y(x). \eea It is important to note that the quadrupole consist of a blackbody spectrum with an amplitude, $2\left(\mathbf{ v}\cdot\mathbf{ \hat{n}'}\right)^2$ along with a y-type distortion with an amplitude (y-amplitude), $\frac{1}{2}\left(\mathbf{ v}\cdot\mathbf{ \hat{n}'}\right)^2$ \cite{zeldovich1969interaction}. It is because of this y-type (SZ type) distortion we will be able to distinguish the pkSZ effect from other signals. We should emphasize that there are two different sources of this quadrupole. One contribution is due to the non-linear nature of the relativistic Doppler shift itself which creates a temperature quadrupole. This only contributes to the blackbody part of the scattered polarised spectrum. The second quadrupole arises due to the non-linear relation between the intensity and the temperature. This contributes to both the blackbody part and the SZ spectrum. Since eq.(\ref{k_quadrupole}) fixes the spectrum of the scattered radiation, we only need to calculate the amplitude. Therefore, we can replace $\mathcal{I}_{\mathrm{sc}}\left(\mathbf{r},\mathbf{ \hat{n}'},\chi\right)$ by $\left(\mathbf{ v}(\mathbf{r},\chi)\cdot\mathbf{ \hat{n}'}\right)^2$ in eq.(\ref{pol_sem_final}) with an understanding that the spectrum is given by $2g(x)+\frac{1}{2}y(x)$. So, the expression for polarisation field becomes, \begin{align} \label{pol_final} P_{\pm}\left(\hat{\mathbf{ n}}\right)=-\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi)\,_{\pm2}Y_{2 \lambda}\left(\mathbf{ \hat{n}}\right)\int d^{2}\mathbf{ \hat{n}'}\;Y_{2 \lambda}^{*}\left(\mathbf{ \hat{n}'}\right)\left(\mathbf{ v}(\mathbf{r},\chi)\cdot\mathbf{ \hat{n}'}\right)^2. \end{align} The expression $\sum_{\lambda =-2}^{2}\,_{2}Y_{2\lambda}\left(\mathbf{ \hat{n}}\right)\int d^{2}\mathbf{ \hat{n}'}\;Y_{2 \lambda}^{*}\left(\mathbf{ v}\cdot\mathbf{ \hat{n}'}\right)^2$ reduces to the square of transverse velocity $v_t$ along with some numerical factors and a phase (as shown in Appendix \ref{App:Quad_dep}). Thus only the transverse to the line of sight component of the velocity contributes to the polarisation signal and is proportional to the square of the transverse velocity field as expected \cite{SZ_80}. Having checked this explicitly we proceed with our calculations with the total electron velocity field. We will mostly be interested in the y-type part of the signal and will present all our results as E and B mode power spectrum of the y-type distortion. In order to compare with CMB polarisation signals, we will present our results in temperature units using the conversion $\Delta T=$2(y-amplitude)$T_{\mathrm{CMB}}$ for the y-distortion valid in the Rayleigh-Jeans (RJ) part of the spectrum. The blackbody part of the spectrum is twice this value, i.e. $\Delta T_{\mathrm{BB}}=2 \Delta T$. We can now proceed to extract the harmonic coefficients of E and B modes. \section{E and B mode harmonic coefficients \label{sec_EB_harm}} In the helicity basis, the pair $\left(\mathcal{Q}\pm i\mathcal{U}\right)\left(\mathbf{\hat{n}}\right)\equiv P_{\pm}\left(\mathbf{\hat{n}}\right)$ transforms as a spin-2 field under rotation about $\mathbf{\hat{n}}$. Thus, on a 2-sphere, we can decompose $ P_{+}\left(\mathbf{\hat{n}}\right)$ as \bea \label{defn_Q+iU} P_{+}\left(\mathbf{\hat{n}}\right)=\sum_{\ell,m}a_{\ell m}\,\,_{2}Y_{\ell m}\left(\mathbf{\hat{n}}\right). \eea Therefore \bea \label{harm_coeff} a_{\ell m}=\int P_{+}\left(\mathbf{\hat{n}}\right)\,_{2}Y^*_{\ell m}\left(\mathbf{\hat{n}}\right)d^2{\hat{\mathbf{n}}}, \eea where $\,_{\pm2}Y_{\ell m}$ are the spin-2 spherical harmonic functions. The sum over $\ell$ starts from $\ell=2$ as the spin weighted spherical harmonics, $\,_{\pm s}Y_{\ell m}$ vanishes for $|s|>l$. We can obtain $ P_{-}\left(\mathbf{\hat{n}}\right)$ by complex conjugation of $ P_{+}\left(\mathbf{\hat{n}}\right)$. \begin{align} P_{-}\left(\mathbf{\hat{n}}\right)&=\sum_{\ell,m}a^{*}_{\ell m}\,\,_2Y^{*}_{\ell m}\left(\mathbf{\hat{n}}\right) =\sum_{\ell,m}a^{*}_{\ell -m}\,(-1)^{m}\,_{-2}Y_{\ell m}\left(\mathbf{\hat{n}}\right). \end{align} We can now define the E and B mode coefficients as \bea \label{e_b_coeff} e_{\ell m}=\frac{1}{2}\left(a_{\ell m}+(-1)^{m}a^{*}_{\ell -m}\right),\hspace{0.5in}b_{\ell m}=-\frac{i}{2}\left(a_{\ell m}-(-1)^{m}a^{*}_{\ell -m}\right). \eea Therefore we can write, \bea P_{\pm}\left(\mathbf{\hat{n}}\right)=\sum_{\ell,m}\left(e_{\ell m}\pm i\,b_{\ell m}\right)\,\,_{\pm2}Y_{\ell m}\left(\mathbf{\hat{n}}\right). \eea Since the Stokes parameters $\mathcal{Q}$ and $\mathcal{U}$ are not coordinate invariant, we define scalar fields which, like temperature perturbation, will be coordinate invariant quantities. This can be achieved using spin raising ($\cancel{\partial}$) and lowering operators ($\cancel{\partial}^{*}$) which creates ordinary spherical harmonics from spin weighted harmonic functions \cite{durrer2020cosmic}. \\ \bea \cancel{\partial}^{2}(\,_{-2}Y_{\ell m})=\sqrt{\frac{(\ell +2)!}{(\ell -2)!}}\;Y_{\ell m}\;, \hspace{1in} \left(\cancel{\partial}^{*}\right)^{2}(\,_{2}Y_{\ell m})=\sqrt{\frac{(\ell +2)!}{(\ell -2)!}}\;Y_{\ell m}. \eea The $\mathcal{E}$ and $\mathcal{B}$ fields are related to $\mathcal{Q}$ and $\mathcal{U}$ as \begin{align} \mathcal{E}(\mathbf{\hat{n}})=\frac{1}{2}\left[\left(\cancel{\partial}^{*}\right)^{2}P_{+}\left(\mathbf{\hat{n}}\right)+\left(\cancel{\partial}\right)^{2}P_{-}\left(\mathbf{\hat{n}}\right)\right] =\sum_{\ell,m}e_{\ell m}\sqrt{\frac{(\ell +2)!}{(\ell -2)!}}\,Y_{\ell m}\left(\mathbf{\hat{n}}\right) \label{Emode_real} \end{align} and \begin{align} \mathcal{B}(\mathbf{\hat{n}})=-\frac{i}{2}\left[\left(\cancel{\partial}^{*}\right)^{2}P_{+}\left(\mathbf{\hat{n}}\right)-\left(\cancel{\partial}\right)^{2}P_{-}\left(\mathbf{\hat{n}}\right)\right] =\sum_{\ell,m}b_{\ell m}\sqrt{\frac{(\ell +2)!}{(\ell -2)!}}\,Y_{\ell m}\left(\mathbf{\hat{n}}\right)\label{Bmode_real}. \end{align} To find the E and B modes, we first convert eq.(\ref{pol_final}) to Fourier space. Since the velocity fields are sourced by scalar modes, we can write $\mathbf{v}\left(\mathbf{r},\chi\right)=\nabla u\left(\mathbf{r},\chi\right)$, where $u$ is the velocity potential. Therefore in Fourier space we have, $\mathbf{\tilde{v}}(\mathbf{k},\chi)=-i\,\mathbf{\hat{k}}\,\tilde{u}(\mathbf{k},\chi)$, where \begin{align} \label{window_0} \mathbf{v}\left(\mathbf{r},\chi\right)=\int \frac{d^3\mathbf{k}}{(2\pi)^3}\mathbf{\tilde{v}}(\mathbf{k},\chi)\;e^{i\mathbf{k}\cdot \mathbf{r}}. \end{align} From here onward, we will suppress the $\chi$ dependence in $\mathbf{\tilde{v}}$ and $\tilde{u}$. The scalar product between the electron velocity and the incoming photon direction transforms as \begin{align} \label{vdotn_sq_f_space} \left(\mathbf{\tilde{v}}(\mathbf{k_{1}})\cdot\mathbf{\hat{n}}'\right)\left(\mathbf{\tilde{v}}(\mathbf{k_{2}})\cdot\mathbf{\hat{n}}'\right)=- \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\left(\mathbf{\hat{k}_{1}}\cdot\mathbf{\hat{n}}'\right)\left(\mathbf{\hat{k}_{2}}\cdot\mathbf{\hat{n}}'\right). \end{align} We can now perform the integrals over $d^{2}\mathbf{ \hat{n}'}$, using relations between spherical harmonics and scalar product of two vectors to get \begin{align} \label{n'_integral} \int d^{2}\mathbf{\hat{n}'}\; Y_{2\lambda}^{*}\left(\mathbf{\hat{n}'}\right)\left(\mathbf{\hat{k}_1}\cdot\mathbf{\hat{n}'}\right)\left(\mathbf{\hat{k}_2}\cdot\mathbf{\hat{n}'}\right)=(-1)^\lambda \left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\; \sum_{p_1,p_2}\left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}}), \end{align} where $\left(\begin{array}{ccc} l_2& l_3 & l_1\\ m_2& m_3 & -m_1 \end{array}\right)$ is the Wigner 3j symbol \cite{varshalovich1988quantum}. Using eq.(\ref{vdotn_sq_f_space}) and eq.(\ref{n'_integral}) in eq.(\ref{pol_final}) we obtain the following expression for $P_{+}\left(\hat{\mathbf{ n}}\right)$: \begin{align} \label{pol_ksp} P_{+}\left(\hat{\mathbf{ n}}\right)=\left(\frac{4\pi}{3}\right)^2&\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}(-1)^\lambda \int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi)\,_{2}Y_{2 \lambda}\left(\mathbf{ \hat{n}}\right)\nonumber\\ &\int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6} e^{i\left(\mathbf{k_1}+\mathbf{k_2}\right)\cdot \mathbf{r}}\; \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\sum_{p_1,p_2} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}}). \end{align} Since $P_{-}\left(\hat{\mathbf{ n}}\right)$ is related to $P_{+}\left(\hat{\mathbf{ n}}\right)$ through a complex conjugation, we only need to consider $P_{+}\left(\hat{\mathbf{ n}}\right)$ for our calculation. Defining $\mathbf{k}=\mathbf{k_1}+\mathbf{k_2}$, we expand the exponential in eq.(\ref{pol_ksp}) into spherical harmonics and spherical Bessel functions $j_\ell(x)$ using the identity, \bea \label{exp_ylm} \exp\left(i\mathbf{k}\cdot\mathbf{r}\right)=4\pi\sum_{L,M}i^L\,Y_{LM}^{*}(\hat{\mathbf{k}})Y_{LM}(\mathbf{\hat{n}})\,j_{L}(k\chi). \eea Substituting in eq.(\ref{harm_coeff}) and using \cite{varshalovich1988quantum, durrer2020cosmic} \begin{align} \label{2ylm} \int Y_{LM}(\mathbf{\hat{n}})\,_{2}Y_{2 \lambda}\left(\mathbf{\hat{n}}\right)\,_{2}&Y^*_{\ell m}(\mathbf{\hat{n}})\,d^2{\hat{\mathbf{n}}}=\nonumber\\ &\sqrt{\frac{{(2L+1)}(2.2+1)(2\ell +1)}{4\pi}}(-1)^{(m)} \left(\begin{array}{ccc} L& 2 & \ell\\ 0& -2 & 2 \end{array}\right) \left(\begin{array}{ccc} L& 2& \ell\\ M& \lambda & -m \end{array}\right), \end{align} we get \begin{align} \label{harm_coeff_start} a_{\ell m}=&4\pi\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}(-1)^\lambda \int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi)\int\int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6}\tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\nonumber\\ &\hspace{1.5cm} \sum_{{L, M}}i^L Y_{LM}^{*}(\hat{\mathbf{k}})\,j_{L}(k\chi)\sum_{p_1,p_2} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}})\;A^{\lambda L M}_{\ell m}, \end{align} where \begin{align} A^{\lambda L M}_{\ell m}=\sqrt{\frac{5(2L+1)(2\ell +1)}{4\pi}}(-1)^{(m)}\: \left(\begin{array}{ccc} L& 2 & \ell\\ 0& -2 & 2 \end{array}\right) \left(\begin{array}{ccc} L& 2& \ell\\ M& \lambda & -m \end{array}\right). \end{align} Using eq.(\ref{e_b_coeff}) we can now obtain the E mode and B mode harmonic coefficients. To simplify further, we use the following properties of Wigner 3j symbols, \bea \left(\begin{array}{ccc} l_1& l_2& l_3\\ -m_1& -m_2 & m_3 \end{array}\right)=(-1)^{(l_1+l_2+l_3)} \left(\begin{array}{ccc} l_1& l_2& l_3\\ m_1& m_2 & -m_3 \end{array}\right). \eea After some algebraic manipulations, (see Appendix \ref{App:EB_coeff_power} for more details) we get the following expression for the E and B mode harmonic coefficients, \begin{align} \label{e_harmonic_coeff} \hspace{-0.3cm}e_{\ell m}=&\frac{1}{2}(4\pi)\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}(-1)^\lambda \int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi)\int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6} \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\nonumber\\ & \sum_{{L, M}}i^L Y_{L M}^{*}(\hat{\mathbf{k}})\,j_{L}(k\chi)\sum_{p_1,p_2} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}})\;A^{\lambda L M}_{\ell m}\left(1+(-1)^{(L+\ell)}\right) \end{align} and \begin{align} \label{b_harmonic_coeff} \hspace{-0.3cm} b_{\ell m}=&-\frac{i}{2}(4\pi)\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}(-1)^\lambda \int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi)\int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6} \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\nonumber\\ & \sum_{{L, M}}i^L Y_{L M}^{*}(\hat{\mathbf{k}})\,j_{L}(k\chi)\sum_{p_1,p_2} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}})\;A^{\lambda L M}_{\ell m}\left(1-(-1)^{(L+\ell)}\right). \end{align} \section{Power spectrum of E and B modes \label{sec_EB_pow}} Taking the ensemble average, denoted by the angular brackets $\langle\:\:\rangle$, gives us the auto-spectra of the E mode $\langle e_{\ell m}e^{*}_{\ell' m'}\rangle$ and the B mode $\langle b_{\ell m}b^{*}_{\ell' m'}\rangle$ polarisation. We have explicitly checked that they are diagonal in the harmonic space, i.e. \bea \langle e_{\ell m}e^{*}_{\ell' m'}\rangle=C^{EE}_{\ell}\;\delta_{\ell,\ell'}\;\delta_{m,m'} \hspace{0.4in} \mathrm{and}\hspace{0.4in} \langle b_{\ell m}b^{*}_{\ell' m'}\rangle=C^{BB}_{\ell}\;\delta_{\ell,\ell'}\;\delta_{m,m'}, \eea as expected from statistical homogeneity and isotropy. Thus, we a priori choose $\ell=\ell'$ and $m=m'=0$ for our numerical calculations. From eq.(\ref{e_harmonic_coeff}) and eq.(\ref{b_harmonic_coeff}) we get \begin{align} \label{cl_ee_1} C^{EE}_{\ell}=& \frac{T^{2}_{\mathrm{CMB}}}{4}\left[(4\pi)\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\right]^2\sum_{\lambda,\lambda' =-2}^{2}(-1)^{(\lambda+\lambda')}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\, a(\chi)n_\mathrm{e}(\chi)\nonumber\\ &\int_{0}^{\chi_{i}}d\chi'\;e^{-\tau(\chi')} a(\chi')n_\mathrm{e}(\chi')\int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6} \int \int \frac{d^3\mathbf{k_1'}d^3\mathbf{k_2'}}{(2\pi)^6} \,\Big\langle \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\tilde{u}^{*}(\mathbf{k_{1}'})\tilde{u}^{*}(\mathbf{k_{2}'})\Big\rangle \nonumber\\ &\sum_{{L, M}\atop{L',M'}}i^{(L-L')}Y_{L M}^{*}(\hat{\mathbf{k}})Y_{L' M'}(\hat{\mathbf{k}}')\;j_{L}(k\chi)\,j_{L'}(k'\chi')\sum_{{p_1,p_2}\atop{p_1',p_2'}} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right) \left(\begin{array}{ccc} 1& 1 & 2\\ p_1'& p_2' & -\lambda' \end{array}\right)\nonumber\\ &Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}})Y_{1p_1'}(\mathbf{{\hat{k}}_1'})Y_{1p_2'}(\mathbf{{\hat{k}_2'}})A^{\lambda L M}_{\ell m}A^{\lambda' L' M'}_{\ell m}\left(1+(-1)^{(L+\ell)}\right)\left(1+(-1)^{(L'+\ell)}\right) \end{align} and \begin{align} \label{cl_bb_1} C^{BB}_{\ell}=& \frac{T^{2}_{\mathrm{CMB}}}{4}\left[(4\pi)\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\right]^2\sum_{\lambda,\lambda' =-2}^{2}(-1)^{(\lambda+\lambda')}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\, a(\chi)n_\mathrm{e}(\chi)\nonumber\\ &\int_{0}^{\chi_{i}}d\chi'\;e^{-\tau(\chi')}a(\chi')n_\mathrm{e}(\chi')\int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6}\int \int \frac{d^3\mathbf{k_1'}d^3\mathbf{k_2'}}{(2\pi)^6} \,\Big\langle \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\tilde{u}^{*}(\mathbf{k_{1}'})\tilde{u}^{*}(\mathbf{k_{2}'})\Big\rangle \nonumber\\ &\sum_{{L, M}\atop{L',M'}}i^{(L-L')}Y_{L M}^{*}(\hat{\mathbf{k}})Y_{L' M'}(\hat{\mathbf{k}}')\;j_{L}(k\chi)\,j_{L'}(k'\chi')\sum_{{p_1,p_2}\atop{p_1',p_2'}} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right) \left(\begin{array}{ccc} 1& 1 & 2\\ p_1'& p_2' & -\lambda' \end{array}\right) \nonumber\\ &Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}})Y_{1p_1'}(\mathbf{{\hat{k}}_1'})Y_{1p_2'}(\mathbf{{\hat{k}_2'}})A^{\lambda L M}_{\ell m}A^{\lambda' L' M'}_{\ell m}\left(1-(-1)^{(L+\ell)}\right)\left(1-(-1)^{(L'+\ell)}\right), \end{align} where we have multiplied the expressions by $T^{2}_{\mathrm{CMB}}$ ( $T_{\mathrm{CMB}}=2.725$ K), to give the results in temperature units. We need to calculate the ensemble average over the velocity potentials. These are Gaussian random fields. We use the Isserlis theorem to break the 4 point function, \begin{align} \label{corr_expansion_0} \Big\langle \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\tilde{u}^{*}(\mathbf{k_{1}'})\tilde{u}^{*}(\mathbf{k_{2}'})\Big\rangle=&\Big\langle \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\Big\rangle\Big\langle \tilde{u}^{*}(\mathbf{k_{1}'})\tilde{u}^{*}(\mathbf{k_{2}'})\Big\rangle+\Big\langle \tilde{u}(\mathbf{k_{1}}) u^{*}(\mathbf{k_{1}'})\Big\rangle\Big\langle \tilde{u}(\mathbf{k_{2}})u^{*}(\mathbf{k_{2}'})\Big\rangle\nonumber\\ &+\Big\langle \tilde{u}(\mathbf{k_{1}}) u^{*}(\mathbf{k_{2}'})\Big\rangle\Big\langle \tilde{u}(\mathbf{k_{2}})u^{*}(\mathbf{k_{1}'})\Big\rangle. \end{align} The ensemble average over the velocity potentials is given by \begin{align} \label{ensemble_def} \Big\langle\tilde{u}(\mathbf{k})\tilde{u}^{*}(\mathbf{k'})\Big\rangle=(2\pi)^3P_{uu}(k)\,\delta(\mathbf{k}-\mathbf{k'})\hspace{0.2in}\mathrm{and}\hspace{0.2in} \Big\langle\tilde{u}(\mathbf{k})\tilde{u}(\mathbf{k'})\Big\rangle=(2\pi)^3P_{uu}(k)\,\delta(\mathbf{k}+\mathbf{k'}), \end{align} where \bea P_{uu}(k)=\frac{\left(aH(a)f(a)\right)^2}{k^2}P_{L}(k,a,a'), \eea $P_{L}$ is the linear matter power spectrum and f is the growth rate \cite{dodelson2003modern}, \bea \left(f= \frac{d\,\ln D_+(a)}{d\,\ln a}\simeq \left[\Omega_m(a)\right]^{0.55}\right). \eea We have used the linear matter power given by Colossus \cite{diemer2018colossus} which uses the model given in \cite{1998Eisenstein_Hu} to calculate the transfer function. It can be easily shown on doing the angular integrals, that the first term in eq.(\ref{corr_expansion_0}) does not contribute to the power spectrum (see Appendix \ref{App:4point}). The contributions from the second and the third terms are equal as eq.(\ref{cl_ee_1}) and eq.(\ref{cl_bb_1}) are symmetric under the exchange of $\mathbf{k_1}$ and $\mathbf{k_2}$. Hence, we only need to consider one of the terms, $\Big\langle \tilde{u}(\mathbf{k_{1}}) u^{*}(\mathbf{k_{1}'})\Big\rangle\Big\langle \tilde{u}(\mathbf{k_{2}})u^{*}(\mathbf{k_{2}'})\Big\rangle$. Using eq.(\ref{ensemble_def}), we get (in temperature units), \begin{align} \label{cl_ee_2} C^{EE}_{\ell}=& \frac{T^{2}_{\mathrm{CMB}}}{2}\left[(4\pi)\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\right]^2\sum_{\lambda,\lambda' =-2}^{2}(-1)^{(\lambda+\lambda')}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\, a(\chi) n_\mathrm{e}(\chi)\nonumber\\ &\int_{0}^{\chi_{i}}d\chi'\;e^{-\tau(\chi')}a(\chi')n_\mathrm{e}(\chi')\sum_{{L, M}\atop{L',M'}}\sum_{{p_1,p_2}\atop{p_1',p_2'}}i^{(L-L')} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right) \left(\begin{array}{ccc} 1& 1 & 2\\ p_1'& p_2' & -\lambda' \end{array}\right)\nonumber\\ &\int \int \frac{dk_1k^2_{1}\,dk_2k^2_{2}}{(2\pi)^6}P_{uu}(k_1)P_{uu}(k_2)j_{L}(k\chi)j_{L'}(k'\chi')\int d\Omega_{\mathbf{k_1}}\int d\Omega_{\mathbf{k_2}}\;Y_{L M}^{*}(\hat{\mathbf{k}})Y_{L'M'}(\hat{\mathbf{k}})\nonumber\\ &Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}})Y_{1p_1'}(\mathbf{{\hat{k}}_1})Y_{1p_2'}(\mathbf{{\hat{k}_2}})A^{\lambda L M}_{\ell m}A^{\lambda' L' M'}_{\ell m}\left(1+(-1)^{(L+\ell)}\right)\left(1+(-1)^{(L'+\ell)}\right) \end{align} and \begin{align} \label{cl_bb_2} C^{BB}_{\ell}=& \frac{T^{2}_{\mathrm{CMB}}}{2}\left[(4\pi)\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\right]^2\sum_{\lambda,\lambda' =-2}^{2}(-1)^{(\lambda+\lambda')}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\, a(\chi) n_\mathrm{e}(\chi)\nonumber\\ &\int_{0}^{\chi_{i}}d\chi'\;e^{-\tau(\chi')}a(\chi')n_\mathrm{e}(\chi')\sum_{{L, M}\atop{L',M'}}\sum_{{p_1,p_2}\atop{p_1',p_2'}}i^{(L-L')} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right) \left(\begin{array}{ccc} 1& 1 & 2\\ p_1'& p_2' & -\lambda' \end{array}\right)\nonumber\\ &\int \int \frac{dk_1k^2_{1}\,dk_2k^2_{2}}{(2\pi)^6}P_{uu}(k_1)P_{uu}(k_2)j_{L}(k\chi)j_{L'}(k'\chi')\int d\Omega_{\mathbf{k_1}}\int d\Omega_{\mathbf{k_2}}\;Y_{L M}^{*}(\hat{\mathbf{k}})Y_{L'M'}(\hat{\mathbf{k}})\nonumber\\ &Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}})Y_{1p_1'}(\mathbf{{\hat{k}}_1})Y_{1p_2'}(\mathbf{{\hat{k}_2}})A^{\lambda L M}_{\ell m}A^{\lambda' L' M'}_{\ell m}\left(1-(-1)^{(L+\ell)}\right)\left(1-(-1)^{(L'+\ell)}\right). \end{align} We should note that the above expressions for the angular power spectra is derived by taking $\left(\mathbf{v}\cdot\mathbf{\hat{n}'}\right)^2$ as the source term from eq.(\ref{k_quadrupole}) in eq.(\ref{pol_final}) with unit normalisation. The actual signal depends on the frequency of observation with spectra given by eq.(\ref{k_quadrupole}). From component separation perspective, we want to decompose the spectrum into a differential blackbody part $g(x)$ and y-type distortion part, $y(x)$. In particular, so that we are not affected by the cosmic variance of the primary CMB E-modes and the lensing B-modes, we want a strategy which will eliminate the blackbody part while preserving the y-distortion part. In the Rayleigh-Jeans (small frequency) limit, we see that the expressions of y(x) in eq.(\ref{temp_quadpole}) becomes equal to $-2$, i.e. $\lim_{x\rightarrow0} y(x)=-2$, which cancels the factor of 1/2 multiplying $y(x)$ in eq.(\ref{k_quadrupole}). Therefore eq.(\ref{cl_ee_2}) and eq.(\ref{cl_bb_2}) give the y-type E and B mode power spectra in the RJ limit in temperature units. For the blackbody part the amplitude is equal to $2\left(\mathbf{v}\cdot\mathbf{\hat{n}'}\right)^2$. Thus, the blackbody power spectra are actually 4 times the above expressions in temperature units. \bea C^{BB \;\mathrm{y-type}}_{\ell}\Big|_{\mathrm{RJ}}=C^{BB}_{\ell} \;\;\mathrm{and} \hspace{1cm} C^{EE\; \mathrm{y-type}}_{\ell}\Big|_{\mathrm{RJ}}= C^{EE}_{\ell}. \eea and just for the blackbody part for both E and B modes we have, \bea C^{\; \mathrm{y-type}}_\ell\Big|_{\mathrm{RJ}}=\frac{1}{4}C^{\mathrm{Blackbody}}_{\ell}. \eea We can now integrate these expressions numerically to obtain the final results. For the integrals over radial $k_1$ and $k_2$ modes, we integrate from $10^{-5}\;\mathrm{Mpc^{-1}}$ to $1.5\;\mathrm{Mpc^{-1}}$ in comoving coordinates. The line of sight integration over comoving distances $\chi$ and $\chi'$ are from $z=20$ before the reionisation starts until $z=0$. We consider two models for the reionisation history. A redshift symmetric model defined by hyperbolic tangent function \cite{lewis2000efficient, heinrich2017complete} and a redshift asymmetric model \cite{DouspisAsymmetric}. The ionisation fraction is defined as the ratio between the electron number density and the total hydrogen number density at that redshift, $\mathrm{X_e}(z)=\left(\frac{n_\mathrm{e}(z)}{n_\mathrm{H}(z)}\right)$, where $n_\mathrm{H}(z)=n_\mathrm{H}(0)\left(1+z\right)^3$, $n_\mathrm{H}(0)$ is the hydrogen number density at $z=0$ assuming primordial abundance. We assume the first helium reionisation to proceed identically. Thus, for the symmetric model the ionisation fraction is given as \begin{align} \label{reion_history} \mathrm{X_e}^{\mathrm{Sym}}(z)=\left[\frac{(1+f)}{2}\left\{1+\tanh\left(\frac{q_{\mathrm{re}}-q}{\Delta q_{\mathrm{re}}}\right)\right\}+\frac{f}{2}\left\{1+\tanh\left(\frac{q^{\mathrm{HeII}}_{\mathrm{re}}-q}{\Delta q^{\mathrm{HeII}}_{\mathrm{re}}}\right)\right\}\right] \end{align} and for asymmetric case, \begin{align} \label{reion_history_asym} \mathrm{X_e}^{\mathrm{Asym}}(z)=\left[\left\{{(1+f) \hspace{2.9cm}z<z_{\mathrm{end}}\atop (1+f)\left(\frac{z_{\mathrm{early}}-z}{z_{\mathrm{early}}-z_{\mathrm{end}}}\right)^\alpha\hspace{0.5cm}z>z_{\mathrm{end}}}\right\} +\frac{f}{2}\left\{1+\tanh\left(\frac{q^{\mathrm{HeII}}_{\mathrm{re}}-q}{\Delta q^{\mathrm{HeII}}_{\mathrm{re}}}\right)\right\}\right], \end{align} where $q(z)=\left(1+z\right)^{1.5}$, $q_{\mathrm{re}}=q(z_{\mathrm{re}})$, $\Delta q_{\mathrm{re}}=1.5(\sqrt{1+z_{\mathrm{re}}})\beta_{\mathrm{re}}$, and $f=\left(\frac{\mathrm{m_H}}{\mathrm{m_{He}}}\frac{\mathrm{X_{He}}}{1-\mathrm{X_{He}}}\right)\simeq0.079$. The central redshift of reionisation is given by $z_{\mathrm{re}}$ and $\beta_{\mathrm{re}}$ is a parameter characterising how fast reionisation happens. We also define the duration of reionisation as $\Delta z_{\mathrm{re}}=z_{10\%}-z_{99\%}$, where $z_{x \%} $ is the redshift when $\frac{X_e}{(1+f)}=\frac{x}{100}$, i.e. the hydrogen is $x \%$ ionised. For the $\mathrm{2^{nd}}$ reionisation of Helium, we always use a hyperbolic tangent function. It happens at redshift $\sim\, 3$. We have chosen $\beta^{\mathrm{HeII}}_{\mathrm{re}}=3.5$ and $z^{\mathrm{HeII}}_{\mathrm{re}}=0.5$. We note that $\mathrm{HeII}$ reionisation gives a negligible contribution to the signal, but we have included it for completeness. We have also fixed $z_{\mathrm{early}}=20$ and $z_{\mathrm{end}}=6$ in the asymmetric model. In this case, the exponent $\alpha$ determines the rapidity with which reionisation takes place. The reionisation histories for different reionisation parameters are shown in figure \ref{fig:reion_history} in appendix \ref{App:Reion_history}. \begin{table}% \centering \begin{subtable}{0.6\textwidth} \vspace{0.0cm} \hspace{-0.05cm} \begin{minipage}[c]{0.9\textwidth} \begin{tabular}{|ccc|} \hline \multicolumn{3}{|c|}{At $\beta_{\mathrm{re}}= 0.5$} \\ \hline \multicolumn{1}{|c|}{\begin{tabular}[c]{@{}c@{}}Central \\Redshift $(z_{\mathrm{re}})$\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}} Duration\\ $\left(z_{10\%}-z_{99\%}\right)$\end{tabular}} & Optical depth $\tau$ \\ \hline \multicolumn{1}{|c|}{7.5} & \multicolumn{1}{c|}{1.7323} & 0.05534 \\ \hline \multicolumn{1}{|c|}{8.5} & \multicolumn{1}{c|}{1.7283} & 0.06587 \\ \hline \multicolumn{1}{|c|}{9.5} & \multicolumn{1}{c|}{1.7251} & 0.07697 \\ \hline \end{tabular} \caption{For different central redshift, fixing $\beta_{\mathrm{re}}= 0.5$.} \label{tab: table1 } \end{minipage} \end{subtable}% \begin{subtable}{0.5\textwidth} \vspace{0.4cm} \hspace{-0.49cm} \begin{minipage}[c]{0.87\textwidth} \begin{tabular}{|ccc|} \hline \multicolumn{3}{|c|}{At central redshift $z_{\mathrm{re}}=8.5$} \\ \hline \multicolumn{1}{|c|}{$\beta_{\mathrm{re}}$} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Duration\\ $\left(z_{10\%}-z_{99\%}\right)$\end{tabular}} & Optical depth $\tau$ \\ \hline \multicolumn{1}{|c|}{0.1} & \multicolumn{1}{c|}{0.340} & 0.065871 \\ \hline \multicolumn{1}{|c|}{0.5} & \multicolumn{1}{c|}{1.728} & 0.065872 \\ \hline \multicolumn{1}{|c|}{1.3} & \multicolumn{1}{c|}{4.670} & 0.065875 \\ \hline \end{tabular} \caption{For different width of reionisation, fixing central redshift at $z_{\mathrm{re}}=8.5$.} \label{tab: table2 } \end{minipage} \end{subtable} \begin{subtable}{0.5\textwidth} \vspace{0.4cm} \hspace{0.0in} \begin{minipage}[c]{0.85\textwidth} \begin{tabular}{|ccc|} \hline \multicolumn{3}{|c|}{At $\mathrm{z_{early}}=20$ and $\mathrm{z_{end}}=6$} \\ \hline \multicolumn{1}{|c|}{$\alpha$} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Duration\\ $\left(z_{10\%}-z_{99\%}\right)$\end{tabular}} & Optical depth $\tau$ \\ \hline \multicolumn{1}{|c|}{3} & \multicolumn{1}{c|}{7.454} & 0.07891 \\ \hline \multicolumn{1}{|c|}{5} & \multicolumn{1}{c|}{5.138} & 0.06516 \\ \hline \multicolumn{1}{|c|}{12} & \multicolumn{1}{c|}{2.432} & 0.05137 \\ \hline \end{tabular} \caption{For different rapidity parameter $\alpha$, fixing $\mathrm{z_{early}}=20$ and $\mathrm{z_{end}}=6$.} \label{tab: table3 } \end{minipage} \end{subtable}% \caption{The total optical depth and duration of reionisation for different reionisation model parameters.} \end{table} \section{Results \label{sec_result}} We performed the integrals in eq.(\ref{cl_bb_2}) and eq.(\ref{cl_ee_2}) numerically using multidimensional Adaptive Monte Carlo integration, Vegas algorithm \cite{lepage1978new}. We have checked the saturation of our numerical results by increasing the Monte Carlo steps until the result converges and the standard deviation is at least an order of magnitude less than the mean value. The angular power spectrum of the y-type E and B modes in the RJ limit of the spectrum in temperature units, for a symmetric reionisation history with central redshift $z_{\mathrm{re}}= 8.5$ and $\beta_{\mathrm{re}}=0.5$, is plotted in figure \ref{fig:summary plot}. The primordial B modes at tensor to scalar ratio $r$ of $10^{-4}$ and $10^{-5}$, the PRISM sensitivity curve, and the PICO sensitivity curve are also plotted for comparison. This figure shows that detecting the y-type E and B modes will be challenging. The blackbody component of the pkSZ B modes is 4 times larger compared to the y-type power spectrum in figure \ref{fig:summary plot} and will start to be an important foreground for the detection of primordial B modes at tensor to scalar ratio $r\lesssim 3\times10^{-5}$. We also study the sensitivity of the pkSZ effect to the reionisation history. In table \ref{tab: table1 }, table \ref{tab: table2 }, and table \ref{tab: table3 }, we show the values for different reionisation parameters used in the analysis and the corresponding total Thomson optical depths as defined by eq.(\ref{tau_chi}). The power spectrum for these different reionisation histories are shown in figure \ref{fig:effect_redshift}, figure \ref{fig:effect_redshift_asym} and figure \ref{fig:effect_width}. For all the cases of symmetric reionisation, we have compared the power spectrum curves with the fiducial case, having parameters $ \beta_{\mathrm{re}}=8.5$ and $ \beta_{\mathrm{re}}=0.5$. \subsection{Dependence on optical depth \label{subsec_central_redshift}} The main effect of changing the central redshift of reionisation in the symmetric model is the obvious change in the total optical depth as shown in table \ref{tab: table1 }. With the increase in optical depth, as we see in figure \ref{fig:effect_redshift}, the polarisation signal also increases as expected. More scatterings between electrons and photons generate more polarisation, increasing the power spectrum at all scales smaller than the horizon size at reionisation. Changing the central redshift from 8.5 to 9.5 increases the optical depth by 17\% and it decreases by the same amount when the central redshift is changed from 8.5 to 7.5. This get reflected in the power spectrum of the E and B modes. At the peak position, the power changes by the same amount as the optical depth, as seen in figure \ref{fig:bmode_frac_sym} and figure \ref{fig:emode_frac_sym}. In the case of asymmetric reionisation, with an increase in $\alpha$, reionisation happens later and faster as shown in figure \ref{fig:asym}. The total optical depth and the power spectra decrease as $\alpha$ increases. When $\alpha$ is changed from 3 to 12 the optical depth decreases by 34\%. We see the same percentage decrease in the power spectrum at the peak position in figure \ref{fig:bmode_frac_asym} and figure \ref{fig:emode_frac_asym}. \subsection{Dependence on duration of reionisation \label{subsec_duration}} When we keep the central redshift fix but change the duration of reionisation, the total optical depth changes negligibly. However, the power spectrum still changes by a significant amount. As shown in figure \ref{fig:Bmode_width} and figure \ref{fig:Emode_width}, the y-type angular power spectrum decreases with the increase in $\beta_{\mathrm{re}}$. When the optical depth remains almost constant, the result of increasing the duration of reionisation is an increase in cancellation of polarisation created by uncorrelated velocity fields along the line of sight. This cancellation is greater on small scales as the number of uncorrelated regions within a given length is larger for modes with a smaller coherence scale. We thus expect that modes with scales smaller than the width of reionisation will be most significantly affected, with the suppression increasing with decreasing scale. For example, consider $z_{\mathrm{re}}=8.5$, $\beta_{\mathrm{re}}=0.5$, and $\Delta z_{\mathrm{re}}=1.7283$, the comoving distance corresponding to the duration is, \bea \label{duration} \Delta \chi=\chi\left(z=z_{10\%}\right)-\chi \left(z=z_{99\%}\right)=\chi\left(z=9.041\right)-\chi \left(z=7.313\right)\simeq 495 \mathrm{Mpc}. \eea This corresponds to a wave number of $\sim0.012 \mathrm{{Mpc}^{-1}}$, which in-turn corresponds to multipole of $\ell\sim 80$. Therefore, we expect power at $\ell \gtrsim 80$ to be suppressed compared to the case of almost instantaneous reionisation corresponding to $\beta_{\mathrm{re}}=0.1$. As the duration increases, this effect should become important at lower multipoles. Indeed, this is what we observe from figure \ref{fig:bmodesfrac} and figure \ref{fig:emodesfrac}. The pkSZ effect from reionisation was previously studied by Renaux-Petel et al. \cite{renaux2014spectral}. They reported not observing any noticeable difference in the power spectrum when they changed the duration of reionisation. They observed at most a 2\% difference at the peak of the spectrum when they changed the value of $\Delta q_{\mathrm{re}}$ from 0 to 3. They used the Planck 2013 parameters with best fit reionisation optical depth of 0.925 and central reionisation redshift of 11.4. Changing the value of $\Delta q_{\mathrm{re}}$ from 0 to 3 with $z_{\mathrm{re}}=11.4$ amounts to changing $\beta_{\mathrm{re}}$ from 0 to 0.56. At higher redshifts, the same reionisation width, $\Delta z_{\rm re}$, in redshift corresponds to a much smaller comoving distance. Thus the power spectrum is less sensitive to the change in reionisation duration $\Delta z_{\rm re}$ for higher reionisation redshift $z_{\rm re}$. However, the current Planck data \cite{adam2016planck} allows a parameter space where $\beta_{\mathrm{re}}$ can reach almost 1.3 for $z_{\mathrm{re}}=8.5$. We observe an almost 3-10 \% difference at $\ell>10$ when we change $\beta_{\mathrm{re}}$ from 0.1 to 1.3 ($\Delta q_{\mathrm{re}} \rightarrow$ 0 to 6) with $\sim8\%$ difference at the peak at $\ell\sim200$. Compared to the fiducial case of $\beta_{\mathrm{re}}=0.5$, the power spectra for $\beta_{\mathrm{re}}=0.1$ and $\beta_{\mathrm{re}}=1.3$ show a deviation of 5\% at small scales as shown in figure \ref{fig:bmodesfrac} and figure \ref{fig:emodesfrac}. It is easy to understand why the polarisation signal should be sensitive to the width of the reionisation, as explained above, by looking at the cancellation of the polarisation coming from randomly oriented velocities of electrons along the line of sight. In fact, this cancellation is similar to the cancellation of the linear kSZ effect due to random orientation of the line of sight component of the velocities \cite{1987Vishniac}. We should therefore have similar sensitivity to reionisation in the linear kSZ effect which is much simpler to calculate. We explicitly calculate the effect of changing reionisation duration on the kSZ signal for comparison. The kSZ signal is proportional to the line of sight velocity field. The angular power spectrum is much simpler and has been calculated in the past \cite{Hernandez_Monteagudo_2006, hernandez2009peculiar}. Considering the same reionisation history and ignoring the spatial fluctuations in the electron density, the kSZ signal depends on the velocity-velocity two-point correlations. The angular power spectrum is given by \cite{1987Vishniac,hernandez2009peculiar}: \begin{align} C^{\mathrm{kSZ}}_{\ell}=&\frac{\pi}{2}\sigma^2_{\mathrm{T}}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\, a(\chi)\int_{0}^{\chi_{i}}d\chi'\;e^{-\tau(\chi')}\,a(\chi)n_\mathrm{e}(\chi)n_\mathrm{e}(\chi')\int dkk^2 \, P_{uu}(k)\, j'_{\ell}(k\chi)j'_{\ell}(k\chi') \end{align} where $j_{\ell}'(x)=\frac{dj_{\ell}(x)}{dx}$. We have again considered the velocity field to be sourced by the scalar perturbations. We expect to see the same kind of suppression in the power spectrum at smaller scales when we increase the duration. Figure \ref{fig:ksz_effects} shows our results. Indeed we find very similar effects (in fact the effects are accentuated) on the power spectrum as seen in the polarisation signal. The power spectrum decreases by $\sim$ 80\% on small scales when we increase $\beta_{\mathrm{re}}$ from 0.1 to 1.3. Unfortunately, the linear kSZ signal is much weaker than the primary CMB signal and is overwhelmed by the cosmic variance of the primary CMB. So the linear kSZ power spectrum can never be observed. On the other hand, as mentioned before, the pkSZ effect has a different spectrum and is therefore not affected by the cosmic variance of primary CMB or the lensing E and B modes. \section{Contribution from Galaxy clusters shot noise \label{sec_shotnoise}} At low redshifts, $z\lesssim2$, most of the free electrons are in the intra cluster medium (ICM). We have so far calculated the contribution from the average number density of electrons and ignored the spatial fluctuations in the electron density. However, at $z\lesssim2$, there will be an extra contribution to the polarisation power spectrum due to the discrete nature of clusters of galaxies. Even if we assume that clusters are randomly distributed in space with some average number density, it should be noted that it is not a continuous density field. In any given small volume the number density of cluster and hence number density of free electrons will fluctuate following a Poisson distribution. In the case of Poisson contribution, the spatial correlation decouples from the velocity correlation and we just modulate the expressions already derived for the polarisation power spectra. To model the cluster number density we consider a stochastic field $\Phi(\mathcal{M},\mathbf{r'},\eta)$ \cite{Hernandez_Monteagudo_2006, hernandez2009peculiar}, where $\mathcal{M}$ is the mass of the cluster, $\mathbf{r'}$ is the cluster center and $\eta$ is the conformal time. This field is Poisson distributed. Therefore, the correlation function becomes, \begin{align} \Big\langle\Phi(\mathcal{M}_1,\mathbf{r_1'},\eta_1)\Phi(\mathcal{M}_2,\mathbf{r_2'},\eta_2)\Big\rangle=\bar{n}(\eta_1)\,\delta(\mathcal{M}_1-\mathcal{M}_2)\,\delta^{3}(\mathbf{r_1'}-\mathbf{r_2'}), \end{align} where $\bar{n}(\eta)$ is the mean number density at a conformal time $\eta$. It is given by the Sheth and Tormen mass function \cite{sheth2001ellipsoidal}. We also need to model the cluster gas profile by some window function $\mathrm{W}(\mathbf{r}-\mathbf{r'},\mathcal{M})$, where $\mathbf{r}$ is the comoving position vector. Since we are interested in the Poisson contributions on larger scales compared to cluster sizes, the exact profile is not important and we choose a Gaussian profile (see Appendix \ref{App:E_density}) for the window function. We can now write the electron number density as a function of both spatial and time coordinates, \bea \label{e_no_den_cluster} n_\mathrm{e}\left(\mathbf{ r},\eta \right) =\int d\mathbf{ r}{'}\int d\mathcal{M}\; n_\mathrm{e}^{0}(\mathcal{M}) \;\mathrm{W}\left(|\mathbf{ r}-\mathbf{ r}{'}|,\mathcal{M}\right)\Phi(\mathcal{M},\mathbf{ r}{'},\eta), \eea where $n_\mathrm{e}^{0}(\mathcal{M})$ is the central electron number density. Replacing the electron number density in eq.(\ref{pol_final}) using eq.(\ref{e_no_den_cluster}), we obtain the corresponding polarisation signal. Here, the density-density and velocity-velocity correlations are decoupled. After doing the same exercise as in the previous case we get the expressions for the Poisson contributions to the polarisation power spectra. They are given in Appendix \ref{App:Poisson}. The contribution from these terms is shown in figure \ref{fig:summary plot}. We observe that the signal from the cluster shot noise is on average 2 orders of magnitude less than the contributions from reionisation on large scales but becomes comparable on small scales. We have only considered the Poisson contribution from the clusters. The clusters of galaxies are however also clustered spatially. We have also ignored the fact that reionisation is expected to be patchy as there can be large fluctuation in the electron density field during reionisation. Taking the electron number density fluctuations fully into account will give rise to terms which are formally $3^{\mathrm{rd}}$ order in perturbation theory and the power spectra will involve 6-point correlation functions. Our focus in this paper is on $2^{\mathrm{nd}}$ order terms and we leave the higher order terms for future work. We should however point out that even though patchy reionisation is formally a $3^{\mathrm{rd}}$ order effect, it does not mean that the contribution would be small. The electron density fluctuations can be of order unity and some of the cancellations in the $2^{\mathrm{nd}}$ order terms can be avoided, similar to patchy kSZ effect \cite{1987Vishniac}. Thus patchy reionisation may give a comparable contribution. Calculating the power spectra, however, requires dealing with the product of 3 perturbation variables instead of 2 and is more complicated and numerically challenging. We plan to study these effects in a separate paper in the near future. \section{Scalar, Vector, and Tensor contributions to the power spectrum \label{subsec_SVT_contribution} } From all the plots of the E and B modes, it is evident that the E modes are always greater than the B modes. A way to investigate why this is the case is to look at individual contributions from scalar, vector and tensor components of the polarisation field. It should be noted that although the velocity field is sourced by purely scalar field, at second order, scalar, vector and tensor components are all present. Different components corresponds to choosing different values of $\lambda$ in eq.(\ref{pol_final}). In this way, we can also check whether a cross correlation between different components producing a specific type of polarisation exists or not and if it does then to what extent? To see this we choose different values of $\lambda$ and $\lambda'$ in eq.(\ref{cl_ee_2}) and eq.(\ref{cl_bb_2}) and plot the corresponding power spectrum. The results are shown in figure \ref{fig:Diff_lambda}. As expected, the B-modes from the scalar modes, i.e. $\lambda$ or $\lambda'=0$ vanish. In fact the integrand vanishes identically when $\lambda$ or $\lambda'=0$ in the case of B modes. We also observe that the cross correlations between scalar, vector and tensor modes are much smaller than the auto-correlations. For B modes, the vector and the tensor modes contribute equally whereas for E modes the maximum contribution is from the scalar modes and the least from the vector modes. Also, we see that the tensor modes have almost an equal contribution to both the E and B modes, while the vector modes primarily contribute to the B modes. We refer the reader to \cite{HU1997primer} for a more detailed discussion. \section{Conclusion \label{sec_conclusion}} We have calculated the y-type E and B mode polarisation angular power spectrum arising from the transverse velocity of free electrons during the reionisation and post-reionisation eras. We have shown that the polarisation signal is not only sensitive to the central redshift of reionisation or optical depth as expected but more interestingly to the duration of reionisation also. Our results, through a more detailed study, show that the conclusion drawn by Renaux-Petel et al. \cite{renaux2014spectral} about the dependence of polarisation power spectra on the duration of reionisation is not complete and it stems from the fact that the whole allowed parameter space was not explored. We point out the close relationship between the linear kSZ effect, which probes the line of sight component of the electron velocity and the pkSZ effect. The response of the pkSZ effect to the duration of reionisation is similar to the kSZ effect. % We want to mention that as we were completing this work another paper on the pkSZ effect appeared on arXiv \cite{pksZ_Kamion}. They also studied the polarisation signal from reionisation, but their work was focused on using this signal to probe cosmic birefringence and non-Gaussianity rather than reionisation. Our numerical results for reionisation agree qualitatively and are of similar magnitude, although they do not specify the exact reionisation history they have used. Our expressions for the E and B mode power spectrum are equivalent, although written in a different form. We show the equivalence in Appendix \ref{App:Quad_dep}. We have not included the spatial fluctuations in the electron density field, in particular patchy reionisation, in our analysis. This will formally include $3^{\mathrm{rd}}$ order terms. We leave the higher order calculations for our future work. Similarly, for the contributions from galaxy clusters, we also need to include the effects of spatial clustering of galaxy clusters. We however expect the contribution from galaxy clusters to be sub-dominant in analogy with the linear kSZ effect. The spectrum of the pkSZ effect can be decomposed into a sum of differential blackbody spectrum (identical to the primary CMB anisotropies) and a y-type spectrum. This is very important from the component separation perspective. We can extract the pkSZ effect from the multi-frequency CMB data by separating the y-type signal while suppressing the blackbody signal. This strategy can enable us to detect the pkSZ signal unencumbered by the cosmic variance of the primary CMB anisotropies. Moreover, the other dominant y-type distortion anisotropies, primary as well as secondary such as the thermal SZ effect, are unpolarised. Thus we can, in principle, measure this signal as precisely as the primary CMB signal if sufficient sensitivity is reached in future. The pkSZ effect has important cosmological information. It is sensitive to the matter velocity power spectrum in addition to the parameters of reionisation. The E and B modes of the y-type distortions thus have the potential to measure the cosmological parameters beyond the cosmic variance limit of the blackbody CMB anisotropies. \acknowledgments This work is supported by the Department of Atomic Energy, Government of India, under Project Identification Number RTI 4002. This work is also supported by Max Planck Partner Group for the cosmology of Max Planck Institute for Astrophysics Garching at Tata Institute of Fundamental Research funded by Max-Planck-Gesellschaft. We acknowledge the use of computational facilities of the Department of Theoretical Physics at Tata Institute of Fundamental Research, Mumbai. AKG is thankful to Carlos HernГЎndez-Monteagudo, Aseem Paranjape, Subhabrata Majumdar, and Anoma Ganguly for useful discussions. \appendix \section{Reionisation models \label{App:Reion_history}} We show in figure \ref{fig:reion_history} the ionisation fraction, $\mathrm{X_e}(z)$, for the different sets of reionisation parameters that were used in our analysis. Figure \ref{fig:sym_centralz} and figure \ref{fig:sym_duration} show the ionisation fraction for the case of symmetric reionisation, for different central redshift $z_{\mathrm{re}}$ keeping $\beta_{\mathrm{re}}$ fixed and vice-versa. Also shown are the redshifts corresponding to $z_{10\%}$ and $z_{99\%}$. For the case of a fixed $\Delta z_{\mathrm{re}}$, though the redshift interval of the duration is fixed, it corresponds to a different physical time interval as an equal redshift interval centred around earlier epochs correspond to shorter physical time interval. The case of asymmetric reionisation is shown in figure \ref{fig:asym} for different rapidity parameters $\alpha$. \section{Derivation of harmonic coefficients and angular power spectrum \label{App:EB_coeff_power}} This section contains all the steps to derive the harmonic coefficients. We begin with eq.(\ref{harm_coeff_start}). \begin{align} a_{\ell m}=&4\pi\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}(-1)^\lambda \int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi)\int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6} \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\nonumber\\ & \sum_{{L, M}}i^L Y_{L M}^{*}(\hat{\mathbf{k}})\,j_{L}(k\chi)\sum_{p_1,p_2} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}})\;A^{\lambda L M}_{\ell m}. \end{align} To find the E and B modes coefficient we need to know $a^{*}_{\ell m}$. Taking the complex conjugate of $a_{\ell m}$ we get \begin{align} a^{*}_{\ell m}=&4\pi\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}(-1)^\lambda \int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi)\int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6} \tilde{u}^{*}(\mathbf{k_{1}})\tilde{u}^{*}(\mathbf{k_{2}})\nonumber\\ &\sum_{{L, M}}(-i)^L Y_{L M}(\hat{\mathbf{k}})\,j_{L}(k\chi)\sum_{p_1,p_2} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y_{1p_1}(\mathbf{{\hat{k}}_1})Y_{1p_2}(\mathbf{{\hat{k}_2}})\;A^{\lambda L M}_{\ell m}. \end{align} Since the velocity potentials are Gaussian random fields, $\tilde{u}^{*}(\mathbf{k})=\tilde{u}(\mathbf{-k})$. Now, we can change all the $\mathbf{k}$ vectors to $-\mathbf{k}$ vectors. Since the integral is over all the $\mathbf{k}$ space, the limits of integration do not change. Doing so we get \begin{align} a^{*}_{\ell m}=&4\pi\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}(-1)^\lambda \int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi)\int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6} \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\nonumber\\ &\sum_{{L, M}}(-i)^L Y_{L M}(-\hat{\mathbf{k}})\,j_{L}(k\chi)\sum_{p_1,p_2} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y_{1p_1}(-\mathbf{{\hat{k}}_1})Y_{1p_2}(-\mathbf{{\hat{k}_2}})\;A^{\lambda L M}_{\ell m}. \end{align} Now, using the formula: \bea \label{y_lm_prop} Y_{\ell m}(-\mathbf{\hat{k}})=(-1)^{l+m}\,Y^{*}_{\ell -m}(\mathbf{\hat{k}}) \eea and \begin{align} \left(\begin{array}{ccc} \label{m_reln} \ell_{1} & \ell_{2} & \ell_3\\ m_1 & m_2 & m_3 \end{array}\right)\neq 0 \;\;\text{ if $(m_1 +m_2 +m_3=0)\:\:\text{\&}\:\:(|\ell_1-\ell_2|\leq\ell_3\leq \ell_1+\ell_2)$}, \end{align} we get \begin{align} a^{*}_{\ell m}=&4\pi\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}(-1)^\lambda \int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi)\int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6}\, \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\nonumber\\ &\sum_{{L, M}}(i)^L Y^{*}_{L-M}(\hat{\mathbf{k}})\,j_{L}(k\chi)\sum_{p_1,p_2}(-1)^{(M+\lambda)} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y^{*}_{1-p_1}(\mathbf{{\hat{k}}_1})Y^{*}_{1-p_2}(\mathbf{{\hat{k}_2}})\;A^{\lambda L M}_{\ell m}. \end{align} Again, since $M$, $\lambda $, $p_1$, and $p_2$ are dummy variables which run from $-L\rightarrow L$, $-2\rightarrow2$, $-1\rightarrow1$, and $-1\rightarrow1$ respectively, we can change $M$ to $-M$, $\lambda$ to $-\lambda$, $p_1$ to $-p_1$, and finally $p_2$ to $-p_2$ without changing the final results. Finally we get \begin{align} a^{*}_{\ell m}=&4\pi\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}(-1)^\lambda \int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi)\int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6} \, \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\nonumber\\ &\sum_{{L, M}}(i)^L Y^{*}_{LM}(\hat{\mathbf{k}})\,j_{L}(k\chi)\sum_{p_1,p_2}(-1)^{(M+\lambda)} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y^{*}_{1p_1}(\mathbf{{\hat{k}}_1})Y^{*}_{1p_2}(\mathbf{{\hat{k}_2}})\;A^{-\lambda L -M}_{\ell m}. \end{align} Next, we use the following properties of the Wigner 3j symbols to simplify further, \bea \left(\begin{array}{ccc} l_1& l_2& l_3\\ -m_1& -m_2 & m_3 \end{array}\right)=(-1)^{(l_1+l_2+l_3)} \left(\begin{array}{ccc} l_1& l_2& l_3\\ m_1& m_2 & -m_3 \end{array}\right). \eea Note, that there are Winger 3j symbols in $A^{\lambda L M}_{\ell m}$, which needs to be simplified too. After simplification we get, \begin{align} a^{*}_{\ell m}=&4\pi\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}(-1)^\lambda \int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi)\int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6}\, \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\nonumber\\ &\sum_{{L, M}}(i)^L Y^{*}_{LM}(\hat{\mathbf{k}})\,j_{L}(k\chi)\sum_{p_1,p_2}(-1)^{(L+\ell+m)} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y^{*}_{1p_1}(\mathbf{{\hat{k}}_1})Y^{*}_{1p_2}(\mathbf{{\hat{k}_2}})\;A^{\lambda L M}_{\ell -m}. \end{align} Now, to get the E and B mode coefficients we use eq.(\ref{e_b_coeff}). Therefore, for E mode we get \begin{align} \hspace{-0.2cm}e_{\ell m}=&\frac{1}{2}(4\pi)\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}(-1)^\lambda \int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi) \int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6} \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\nonumber\\ &\sum_{{L, M}}i^L Y_{L M}^{*}(\hat{\mathbf{k}})\,j_{L}(k\chi)\sum_{p_1,p_2} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}})\;A^{\lambda L M}_{\ell m}\left(1+(-1)^{(L+\ell)}\right) \end{align} and similarly for B mode, \begin{align} \hspace{-0.2cm}b_{\ell m}=&-\frac{i}{2}(4\pi)\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}(-1)^\lambda \int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)\,a(\chi)\int \int \frac{d^3\mathbf{k_1}d^3\mathbf{k_2}}{(2\pi)^6}\tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}}) \nonumber\\ &\sum_{{L, M}}i^L Y_{L M}^{*}(\hat{\mathbf{k}})\,j_{L}(k\chi)\sum_{p_1,p_2} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right)Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}})\;A^{\lambda L M}_{\ell m}\left(1-(-1)^{(L+\ell)}\right). \end{align} \section{Calculation of the 4-point function \label{App:4point}} In this section we will show that out of the three terms that we get when we break the 4 point function using Isserlis theorem, only one term contributes. We begin with eq.(\ref{corr_expansion_0}). \begin{align} \Big\langle \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\tilde{u}^{*}(\mathbf{k_{1}'})\tilde{u}^{*}(\mathbf{k_{2}'})\Big\rangle=&\Big\langle \tilde{u}(\mathbf{k_{1}})\tilde{u}(\mathbf{k_{2}})\Big\rangle\Big\langle \tilde{u}^{*}(\mathbf{k_{1}'})\tilde{u}^{*}(\mathbf{k_{2}'})\Big\rangle+\Big\langle \tilde{u}(\mathbf{k_{1}}) u^{*}(\mathbf{k_{1}'})\Big\rangle\Big\langle \tilde{u}(\mathbf{k_{2}})u^{*}(\mathbf{k_{2}'})\Big\rangle\nonumber\\ &+\Big\langle \tilde{u}(\mathbf{k_{1}}) u^{*}(\mathbf{k_{2}'})\Big\rangle\Big\langle \tilde{u}(\mathbf{k_{2}})u^{*}(\mathbf{k_{1}'})\Big\rangle. \end{align} Let us look at the first term. \begin{align} \label{corr_1st_term} \Big\langle u(\mathbf{k_{1}})u(\mathbf{k_{2}})\Big\rangle\Big\langle u^{*}(\mathbf{k_{1}'})u^{*}(\mathbf{k_{2}'})\Big\rangle&=(2\pi)^6\;P_{uu}(k_1)P_{uu}(k_1')\;\delta(\mathbf{k_1}+\mathbf{k_2})\;\delta(\mathbf{k_1'}+\mathbf{k_2'}). \end{align} Therefore, using eq.(\ref{corr_1st_term}) in the angular integral part over $k_1$, $k_2$, $k_1'$, and $k_2'$ of eq.(\ref{cl_ee_1}) and eq.(\ref{cl_bb_1}) we get \begin{align} &\int d\Omega_{\mathbf{k_{1}}}d\Omega_{\mathbf{k_{1}'}}\;Y^{*}_{1p_1}(\mathbf{{\hat{k}}_1})Y^{*}_{1p_2}(-\mathbf{{\hat{k}_1}})\;Y_{1p_1'}(\mathbf{{\hat{k}}_1'})Y_{1p_2'}(-\mathbf{{\hat{k}_1'}}),\nonumber\\ &\hspace{1.6in}=(-1)^{(1+p_2)}\int d\Omega_{\mathbf{k_{1}}}\;Y^{*}_{1p_1}(\mathbf{{\hat{k}}_1})Y_{1-p_2}(\mathbf{{\hat{k}_1}})\times\nonumber\\ &\hspace{1.8in}(-1)^{(1+p_2')}\int d\Omega_{\mathbf{k_{1}'}}Y_{1p_1'}(\mathbf{{\hat{k}}_1'})Y^{*}_{1-p_2'}(\mathbf{{\hat{k}_1'}}),\\ &\hspace{1.6in}=(-1)^{(p_2+p_2')}\:\delta_{p_1,-p_2}\;\delta_{p_1',-p_2'}. \end{align} But we see from the Wigner 3j coefficients present in eq.(\ref{cl_ee_1}) and eq.(\ref{cl_bb_1}) that, \begin{align} \sum_{p_1,p_2}C^{2 \lambda }_{1p_1 1p_2}\;\delta_{p_1,-p_2} =\sum_{p_1}C^{2 \lambda }_{1p_1 1-p_1}=0. \end{align} Hence, the contribution from the first term is zero. \section{Quadratic dependence of the polarisation field on electron's transverse velocity \label{App:Quad_dep}} We begin with eq.(\ref{ksz_temp}). As we have shown using Taylor expansion, \bea \theta(\mathbf{ \hat{n}'})=\frac{1}{2}v^2-\mathbf{ v}\cdot\mathbf{ \hat{n}'}+\left(\mathbf{ v}\cdot\mathbf{ \hat{n}'}\right)^2+\mathcal{O}\left(\left(\mathbf{ v}\cdot\mathbf{ \hat{n}'}\right)^3\right)+\cdot\cdot\cdot \eea From eq.(\ref{k_quadrupole}), we saw that the contribution to the intensity for the SZ part of the spectrum is proportional to $\left(\theta(\mathbf{ \hat{n}'})\right)^2$. Therefore squaring and rearranging the terms we get: \bea \left(\theta(\mathbf{ \hat{n}'})\right)^2=\left(\mathbf{ v}\cdot\mathbf{ \hat{n}'}\right)^2-\frac{1}{3}v^2+ \frac{1}{3}v^2 +\mathcal{O}(v^4) +\cdots \eea Therefore, the contribution to the quadrupolar moment will just be $\frac{1}{3}v^2\left(3\cos^2\zeta-1\right)$, where $\zeta$ is the angle between $\mathbf{v}$ and $\mathbf{\hat{n}'}$. The monopole term $\propto v^2$ will not contribute anyway when we integrate over $d^{2}\mathbf{ \hat{n}'}$. Now, we can write, \bea \frac{1}{3}v^2\left(3\cos^2\zeta-1\right)=\frac{4}{3}\sqrt{\frac{\pi}{5}}\;v^2\;Y_{20}(\hat{\mathbf{v}};\mathbf{\hat{n}'}), \eea where $Y_{20}(\hat{\mathbf{v}};\mathbf{\hat{n}'})$ is defined by considering the z-direction to be along $\mathbf{\hat{n}'}$. Now we can use the following property of spherical harmonics to split it as a product of two spherical harmonics \cite{durrer2020cosmic}, \bea Y_{20}(\hat{\mathbf{v}};\mathbf{\hat{n}'})=\sqrt{\frac{4\pi}{5}}\sum_{m'}\;Y^{*}_{2m'}(\hat{\mathbf{v}};\mathbf{\hat{e}})\,Y_{2m'}(\mathbf{\hat{n}'};\mathbf{\hat{e}}), \eea where $\mathbf{\hat{e}}$ is some general direction which is our new z-direction. Therefore, from eq.(\ref{pol_sem_final}) we observe \bea \int d^{2}\mathbf{ \hat{n}'}\;Y_{2 \lambda}^{*}\left(\mathbf{ \hat{n}'}\right)\mathcal{I}_{\mathrm{sc}}\left(\mathbf{r},\mathbf{ \hat{n}'}\right)= \frac{8\pi}{15}\;v^2\;Y^{*}_{2\lambda}(\hat{\mathbf{v}};\mathbf{\hat{e}}). \eea So, finally we get, \begin{align} P_{+}\left(\hat{\mathbf{ n}}\right)&=- \frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\sum_{\lambda =-2}^{2}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)a(\chi)\,_{2}Y_{2 \lambda}\left(\mathbf{ \hat{n}}\right)\left[\frac{8\pi}{15}\;v^2\;Y^{*}_{2\lambda}(\hat{\mathbf{v}};\mathbf{\hat{e}})\right],\\ &=-\frac{8\pi}{15}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)a(\chi)v^2\left[\sum_{\lambda =-2}^{2}\,_{2}Y_{2 \lambda}\left(\mathbf{ \hat{n}}\right)\;Y^{*}_{2\lambda}(\hat{\mathbf{v}};\mathbf{\hat{e}})\right],\\ &=-\frac{8\pi}{15}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)a(\chi)v^2\sqrt{\frac{5}{4\pi}}\;Y_{2-2}(\mathbf{\hat{v}};\mathbf{\hat{n}}), \end{align} To simplify this further we consider $\hat{\mathbf{n}}$ as the new $\hat{\mathbf{z}}$ direction. Doing so, we can split the spherical harmonics $Y_{2-2}(\mathbf{\hat{v}};\mathbf{\hat{n}})$ in terms of angle $\theta$ and $\phi$ of the spherical polar coordinate system. Thus we get \begin{align} P_{+}{\left(\hat{\mathbf{n}}\equiv \hat{\mathbf{z}}\right)}&=-\frac{\sigma_{\mathrm{T}}}{10}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)a(\chi)v^2\,\sin^2\theta\;e^{-2i\phi},\\ &=-\frac{\sigma_{\mathrm{T}}}{10}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\,n_\mathrm{e}(\chi)a(\chi)v_t^2\;e^{-2i\phi}, \end{align} where $v_t=v\sin\theta$ is the transverse to $\hat{\mathbf{ n}}$. Thus, we have shown that the polarisation signal is proportional to the square of the transverse velocity field. \section{Electron number density profile for galaxy clusters \label{App:E_density}} We considered a Gaussian profile for the gas present in the ICM. One may consider a more realistic profile, such as given in \cite{2002MNRAS.336.1256K}, but for scales much larger than the halo sizes, the exact nature of the profile is unimportant. What matters is the volume occupied by the gas regardless of the detailed shape. This makes the calculation faster without decreasing the accuracy of our final results. We start with a halo profile of the form \begin{align} \rho(r)=\rho_0\exp\left(-\frac{4r^2}{R^2}\right). \end{align} where R is some scale radius. For a halo with mass = $\mathcal{M}$, the scale radius is so chosen that $R(\mathcal{M}_\mathrm{200m}) =0.95\mathcal{M}$. Using the normalisation, $\int 4\pi\rho(r)r^2dr=\mathcal{M}$, we get $\rho_0$ as \bea \rho_0=\frac{8\mathcal{M}}{\pi^{3/2}R^3}. \eea The gas density can be written as \bea \rho_{\mathrm{gas}}(r)=\frac{\Omega_b}{\Omega_m}\rho(r)=\frac{\Omega_b}{\Omega_m}\frac{8\mathcal{M}}{\pi^{3/2}R^3}\exp\left(\frac{-4r^2}{R^2}\right). \eea From the gas density we can easily find the electron number density by dividing it by mean gas mass per electron \begin{align} n_\mathrm{e}(r)=\frac{\rho_{\mathrm{gas}}(r)}{1.14m_\mathrm{p}}&=\frac{\Omega_b}{\Omega_m}\frac{1}{1.14\pi^{3/2}}\frac{8\mathcal{M}}{m_\mathrm{p}R^3}\exp\left(\frac{-4r^2}{R^2}\right),\\ &=n_\mathrm{e}^{0}\;W(r). \end{align} where $n_\mathrm{e}^{0}=\frac{\Omega_b}{\Omega_m}\frac{1}{1.14\pi^{3/2}}\frac{8\mathcal{M}}{m_\mathrm{p}R^3}$, $W(r)=\exp\left(\frac{-4r^2}{R^2}\right)$ and $m_\mathrm{p}$ is the mass of proton. So if we take the Fourier transform of the electron number density we get \begin{align} n_\mathrm{e}(k)&=n_\mathrm{e}^{0}\int d\mathbf{r}\exp\left(-i\mathbf{k}\cdot\mathbf{r}\right)W(r),\nonumber\\ &=n_\mathrm{e}^{0} \int dr\,4\pi r^2\;\frac{\sin(kr)}{kr}\exp\left(\frac{-4r^2}{R^2}\right),\nonumber\\ &=n_\mathrm{e}^{0}\frac{R^3}{8}\,\pi^{3/2}\exp\left(\frac{-k^2\,R^2}{16}\right),\nonumber\\ &=\frac{\Omega_b}{1.14\Omega_m}\frac{\mathcal{M}}{m_\mathrm{p}}\exp\left(\frac{-k^2\,R^2}{16}\right). \end{align} \section{Poisson contribution to polarisation power spectra. \label{App:Poisson}} In this section, we derive the power spectrum of E and B modes from the Poisson term. We have to repeat the same process as shown in the case of uniform electron number density field to obtain the power spectra. The density-density correlations and the velocity-velocity correlations can be calculated separately. Using eq.(\ref{e_no_den_cluster}) in place of electron number density in eq.(\ref{pol_final}) and doing a variable change from conformal time to comoving distance as shown earlier, we get in temperature units \begin{align} \hspace{0cm} \label{cl_ee_poi} C^{EE\,(\mathrm{Poi})}_{\ell}=& \frac{T^{2}_{\mathrm{CMB}}}{2}\left[(4\pi)\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}n_\mathrm{e}^{0}\right]^2\sum_{\lambda,\lambda' =-2}^{2}(-1)^{(\lambda+\lambda')}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\, a(\chi)\nonumber\\ &\int_{0}^{\chi_{i}}d\chi'e^{-\tau(\chi')}\,a(\chi')\int d\mathcal{M}\,\bar{n}(\mathcal{M},\chi)\sum_{{L, M}\atop{L',M'}}\sum_{{p_1,p_2}\atop{p_1',p_2'}}i^{(L-L')} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right) \left(\begin{array}{ccc} 1& 1 & 2\\ p_1'& p_2' & -\lambda' \end{array}\right)\nonumber\\ &\int \int \frac{dk_1dk_2}{(2\pi)^6}k^2_{1}k^2_{2}\; P_{uu}(k_1)P_{uu}(k_2)\int \frac{dk_3k^2_{3}}{(2\pi)^3}\mathrm{W}\left(k_3,\mathcal{M}\right)^2j_{L}(k_3\chi)\,j_{L'}(k_3\chi')\nonumber\\ &\int d\Omega_{\mathbf{k_1}}\;Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_1'}(\mathbf{{\hat{k}}_1})\int d\Omega_{\mathbf{k_2}}Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}})Y_{1p_2'}(\mathbf{{\hat{k}_2}})\int d\Omega_{\mathbf{k_3}}\;Y_{L M}^{*}(\mathbf{\hat{k}_3}) Y_{L' M'}(\mathbf{\hat{k}_3})\,\times\nonumber\\ &\hspace{1.5in}A^{\lambda L M}_{\ell m}A^{\lambda' L' M'}_{\ell m}\left(1+(-1)^{(L+\ell)}\right)\left(1+(-1)^{(L'+\ell)}\right) \end{align} and similarly, \begin{align} \label{cl_bb_poi} C^{BB\,(\mathrm{Poi})}_{\ell}=& \frac{T^{2}_{\mathrm{CMB}}}{2}\left[(4\pi)\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\,n_\mathrm{e}^{0}\right]^2\sum_{\lambda,\lambda' =-2}^{2}(-1)^{(\lambda+\lambda')}\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\, a(\chi)\nonumber\\ & \int_{0}^{\chi_{i}}d\chi'e^{-\tau(\chi')}\,a(\chi')\int d\mathcal{M}\,\bar{n}(\mathcal{M},\chi)\sum_{{L, M}\atop{L',M'}}\sum_{{p_1,p_2}\atop{p_1',p_2'}}i^{(L-L')} \left(\begin{array}{ccc} 1& 1 & 2\\ p_1& p_2 & -\lambda \end{array}\right) \left(\begin{array}{ccc} 1& 1 & 2\\ p_1'& p_2' & -\lambda' \end{array}\right)\nonumber\\ &\int \int \frac{dk_1dk_2}{(2\pi)^6}k^2_{1}k^2_{2}\;P_{uu}(k_1)P_{uu}(k_2) \int \frac{dk_3k^2_{3}}{(2\pi)^3}\mathrm{W}\left(k_3,\mathcal{M}\right)^2j_{L}(k_3\chi)\,j_{L'}(k_3\chi')\nonumber\\ &\int d\Omega_{\mathbf{k_1}}\;Y_{1p_1}^{*}(\mathbf{{\hat{k}}_1})Y_{1p_1'}(\mathbf{{\hat{k}}_1}) \int d\Omega_{\mathbf{k_2}}Y_{1p_2}^{*}(\mathbf{{\hat{k}_2}})Y_{1p_2'}(\mathbf{{\hat{k}_2}})\int d\Omega_{\mathbf{k_3}}\;Y_{L M}^{*}(\mathbf{\hat{k}_3}) Y_{L' M'}(\mathbf{\hat{k}_3})\,\times\nonumber\\ &\hspace{1.5in}A^{\lambda L M}_{\ell m}A^{\lambda' L' M'}_{\ell m}\left(1-(-1)^{(L+\ell)}\right)\left(1-(-1)^{(L'+\ell)}\right). \end{align} The angular integrals can be performed analytically in this case. We can also sum over the Wigner 3j symbols. After doing these simplifications, we finally get, \begin{align} \label{cl_ee_poi_final} C^{EE\,(\mathrm{Poi})}_{\ell}=&T^{2}_{\mathrm{CMB}}\, (2\pi)\left[\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\,n_\mathrm{e}^{0}\right]^2\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\, a(\chi)\int_{0}^{\chi_{i}}d\chi'\;e^{-\tau(\chi')}\, a(\chi')\nonumber\\ &\hspace{0.1in}\int d\mathcal{M}\,\bar{n}(\mathcal{M},\chi)\sum_{L}(2L+1) \left[ \left(\begin{array}{ccc} L& 2& \ell\\ 0& -2 & 2 \end{array}\right)\right]^2\int \int \frac{dk_1dk_2}{(2\pi)^6}\;k^2_{1}k^2_{2}P_{uu}(k_1) P_{uu}(k_2)\nonumber\\ &\hspace{0.9in}\int \frac{dk_3k^2_{3}}{(2\pi)^3}\mathrm{W}\left(k_3,\mathcal{M}\right)^2j_{L}(k_3\chi)\,j_{L}(k_3\chi')\;\left(1+(-1)^{(L+\ell)}\right)^2 \end{align} and similarly, \begin{align} \label{cl_bb_poi_final} C^{BB\,(\mathrm{Poi})}_{\ell}=&T^{2}_{\mathrm{CMB}}\, (2\pi)\left[\left(\frac{4\pi}{3}\right)^2\sqrt{\frac{3}{2\pi}}\frac{\sqrt{6}\sigma_{\mathrm{T}}}{10}\,n_\mathrm{e}^{0}\right]^2\int_{0}^{\chi_{i}}d\chi\;e^{-\tau(\chi)}\, a(\chi)\int_{0}^{\chi_{i}}d\chi'\;e^{-\tau(\chi')}\, a(\chi')\nonumber\\ &\hspace{0.1in}\int d\mathcal{M}\,\bar{n}(\mathcal{M},\chi)\sum_{L}(2L+1) \left[ \left(\begin{array}{ccc} L& 2& \ell\\ 0& -2 & 2 \end{array}\right)\right]^2\int \int \frac{dk_1dk_2}{(2\pi)^6}\;k^2_{1}k^2_{2}P_{uu}(k_1)P_{uu}(k_2)\nonumber\\ &\hspace{0.9in} \int \frac{dk_3k^2_{3}}{(2\pi)^3}\mathrm{W}\left(k_3,\mathcal{M}\right)^2j_{L}(k_3\chi)\,j_{L}(k_3\chi')\;\left(1-(-1)^{(L+\ell)}\right)^2. \end{align} We choose the mass range of the clusters to be between $10^{13} \mathrm{M_{\odot}}$ to $10^{17} \mathrm{M_{\odot}}$. \bibliographystyle{unsrtads} \bibliography{references.bib}

Title: Black hole hyperaccretion in collapsars. III. GRB timescale

Abstract: Gamma-ray bursts (GRBs) are classified into long and short populations (i.e., LGRBs and SGRBs) based on the observed bimodal distribution of duration $T_{90}$. Multimessenger observations indicated that most SGRBs and LGRBs should be powered by ultrarelativistic jets launched from black hole (BH) hyperaccretion in compact object mergers and massive collapsars, respectively. However, the duration criterion sometimes cannot correctly reflect the physical origin of a particular GRB. In the collapsar scenario, a GRB can be observed when the jet breaks out from the envelope and circumstellar medium successfully. The observed GRB duration reflects only the time that the engine operates after the jet breaks out. This work studies the propagation of jets driven by the neutrino annihilation or Blandford-Znajek mechanism in massive collapsars. The signatures of the progenitors for producing LGRBs, SGRBs, and failed GRBs in the collapsar scenario are exhibited. The competition between the mass supply onto the BH hyperaccretion and jet propagation into the envelope are definitely dependent on the density profiles of the collapsars. We show that duration and isotropic energy $E_{\rm{\gamma,iso}}$ of GRBs can help constrain the density profiles of collapsars. Finally, we propose that a collapsar-origin SGRB, GRB 200826A, might originate from a neutrino-annihilation-dominated jet launched by a $\sim 10~M_\odot$ collapsar whose progenitor's envelope has been stripped.

https://export.arxiv.org/pdf/2208.09952

\title{Black hole hyperaccretion in collapsars. III. GRB timescale} \correspondingauthor{Tong Liu} \email{tongliu@xmu.edu.cn} \author[0000-0002-9130-2586]{Yun-Feng Wei} \affiliation{Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, China} \author[0000-0001-8678-6291]{Tong Liu} \affiliation{Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, China} \keywords{accretion, accretion disks - black hole physics - gamma-ray burst: general} \section{Introduction} Gamma-ray bursts (GRBs) are among the most luminous explosions in the universe. According to the observed bimodal distribution of duration $T_{90}$ (the time interval during which a detector observes $5\%$ to $95\%$ of the total gamma-ray fluence), GRBs are classified into long and short populations \citep{Kouveliotou1993}. Short-duration GRBs (SGRBs, $T_{90}<2 ~\rm{s}$) and long-duration GRBs (LGRBs, $T_{90}>2 ~\rm{s}$) are generally considered to be linked to binary neutron star (NS) or NS-black hole (BH) mergers \citep[e.g.,][]{Abbott2017} and deaths of massive stars \citep[e.g.,][]{Woosley2006}, respectively. However, the duration is sometimes not a reliable indicator of the GRB physical origin. At present, some LGRBs have the statistical properties of SGRBs and are suggested to come from compact object mergers \citep[e.g.,][]{Gal-Yam2006,Gehrels2006}. Meanwhile, some SGRBs have been suggested to originate from massive collapsars \citep[e.g.,][]{Zhang2007,Levesque2010,Thone2011,Xin2011}. Therefore, \citet{Zhang2006} and \citet{Zhang2009} suggested that GRBs should be classified physically into two classes: Type I (compact-object-merger origin) and Type II (massive-collapsar origin). There are two ways that the duration of Type II GRBs may be less than $2 ~\rm{s}$. First, they can be naturally caused by the ``tip-of-iceberg" effect \citep[e.g.,][]{Lu2014}. In this case, a real LGRB may be observed as a ``short" one if the majority of gamma-ray emission episodes are below the detection threshold of GRB detectors. Second, massive stars indeed produce SGRBs. In the collapsar scenario \citep[e.g.,][]{Woosley1993,MacFadyen1999,Woosley2002,Zhang2004,Woosley2012}, the collapsed core of a massive star forms a fast-rotating BH surrounded by a hyperaccretion disk, which launches the relativistic jets along its rotation axis. If one of the jets can break out from the envelope and circumstellar medium in the prompt emission phase and is along the line of sight, then an observable GRB is triggered. Subsequently, the jet will be decelerated in the medium or winds to produce the multi-band afterglows. Thus, the observed GRB duration reflects only the duration of the central engine after the jet breaks out. Once the observed timescale is less than $2~\rm{s}$, this event is classified as an SGRB. If the jet fails to break out, then no GRB can be detected. Therefore, the duration of GRB mainly depends on jet propagation in the envelope. Jet propagation in collapsars has been investigated in both analytical works \citep[e.g.,][]{Matzner2003,Bromberg2011a,Bromberg2012,Suwa2011,Matsumoto2015,Liu2018,Liu2019,Song2019} and numerical works \citep[e.g.,][]{Zhang2003,Mizuta2006,Mizuta2009,Mizuta2011,Morsony2007,Nagakura2011,Nagakura2012,Nagakura2013,Nakauchi2012,Tominaga2007}. These studies found that the propagation of jets in various types of massive stars, such as Wolf-Rayet (WR) stars and Population III (Pop III) stars, can produce GRBs. The detection of GRB 200826A confirms that collapsars can produce SGRBs, and this burst appears to sit on the brink between a successful and a failed GRB \citep[e.g.,][]{Ahumada2021,Zhang2021}. The purpose of this work is to systematically study the dependence of GRB durations on jet propagation in the collapsar scenario. The paper is organized as follows. In Sections 2.1 and 2.2, we introduce the progenitor and jet models. In Section 2.3, we review the essential features of jet propagation and the corresponding characteristics of GRBs, such as the duration and energy of prompt emission. The results are shown in Section 3. In Section 4, we constrain the collapsar density profile of GRB 200826A. A brief summary is given in Section 5. \section{Models} \subsection{Progenitor model} In this paper, we construct a series of collapsar density profiles to investigate jet propagation. Note that these collapsars are evolved massive stars (from zero-age main sequence to the stage that the core begins to collapse) and have experienced mass loss. We assume that the precollapse star has a core-envelope structure \citep[see e.g.,][]{Kumar2008}. Then, the density profiles are divided into two parts: star core and envelope. As shown in Figure 1, the density profiles of the core and the envelope can be approximated as $\rho_{\rm{cor}} (r) \propto r^{-k_1}$ and $\rho_{\rm{env}} (r) \propto r^{-k_2}$, respectively. Note that $r_1$ is the boundary between the core and envelope. The outer boundary of collapsar $r_2$ is defined as the radius at which the density decreases to $10^{-10} ~\rm{g}~\rm{cm}^{-3}$. Actually, the exact stellar surface is difficult to calculate for the simulations of stellar evolution \citep[e.g.,][]{Suwa2011}. For different stellar models in \citet{Woosley2002}, the densities of the outermost layer of collapsars are different. Following the previous studies of jet propagation \citep[e.g.,][]{Nakauchi2012,Nakauchi2013}, we adjust them to the same value. Here, we adopt the point of $10^{-10} ~\rm{g}~\rm{cm}^{-3}$ as the effective collapsar surface in calculating the jet propagation. This density is low enough that the accretion of the envelope cannot ignite the central engine \citep[e.g.,][]{Liu2018}. $r_0$ is the radius where the enclosed mass reaches $3 ~M_{\odot}$, and $\rho _0$ is the density at this radius. Then the mass of collapsar can be calculated as: \beq \frac{M_{\rm{col}}}{M_{\odot }} = 3+ \int_{r_{0}}^{r_{1}}\rho_{\rm{cor}}4\pi r^{2} dr+\int_{r_{1}}^{r_{2}}\rho_{\rm{env}}4\pi r^{2} dr. \eeq Referring to the presupernova model \citep[e.g.,][]{Woosley2002,Woosley2007,Heger2010}, we adopt $r_0=5.5\times 10^{8}~\rm{cm}$ and $\rho_0=6\times 10^{6}~\rm{g}~\rm{cm}^{-3}$ as typical parameters in our calculations. By changing the values of $r_{\rm{1}}$ and $k_{\rm{2}}$, we can construct a series of density profiles. The minimum mass of the collapsar is set to $M_{\rm col}=10 ~M_{\odot}$, which corresponds to the typical mass of a carbon-oxygen WR star \citep{Matzner2003}. $r_{1}$ determines the size of the core. The core becomes larger as $r_{1}$ becomes larger. The value of $r_{\rm{1}}$ varies from $10^{9}$ to $10^{10}~ \rm{cm}$. $k_{1}$ and $k_{2}$ determine the structure of the core and envelope of the collapsar, respectively. The value of $k_{\rm{2}}$ varies from $5$ to $40$. When $k_{2}$ is small, the collapsar will have a thick envelope. The progenitor of such a collapsar might be a low metallicity massive star, which experiences little mass loss during evolution and thereby has a large envelope at the end of its life. Here the lower limit of $k_{2}$ is set to $5$. In fact, for some types of stars in their precollapse phase, $k_{2}$ is less than $5$. For example, the envelope of the red supergiant (RSG) is very thick ($\rho_{\rm{env}}\propto r^{-3/2}$). However, the jets cannot break out from these RSGs \citep[e.g.,][]{Suwa2011,Matsumoto2015}, which suggests that these RSGs are not the progenitors of GRBs. Therefore, we consider steeper envelope density profiles here. As discussed in Section 3, our results show that collapsars are unlikely to produce SGRBs when $k_{2}<5$. Thus, we set the lower limit of $k_{2}$ to $5$. When $k_{2}$ is very large, the collapsar almost loses its envelope. Such a collapsar might be a WR star, and jets can easily break out of them. According to all cases of the density profiles of collapsars, when $k_{2}>40$, the mass ratio of the envelope to collapsar is less than $1\%$, which suggests that the collapsar almost loses its envelope. Thus, we set the upper limit of $k_{2}$ to $40$. Then, we calculate the mass supply rate of collapsars with different density profiles. The accretion timescale of matter at a radius $r$ is estimated as a free-fall timescale \citep[e.g.,][]{Woosley2012,Matsumoto2015}: \beq t_{\rm{ff}}=\sqrt{\frac{3\pi}{32G\bar{\rho}}}=\frac{\pi}{2}\sqrt{\frac{r^{3}}{2G M_r}}, \eeq where $\bar{\rho}=3M_{r}/(4\pi r^{3})$ is the mean density within $r$ and $M_{r}$ is the mass coordinate. Then, the mass supply rate can be evaluated as \citep[e.g.,][]{Suwa2011}: \beq \dot{M}_{\rm{pro}}=\frac{dM_{r}}{dt_{\rm{ff}}}=\frac{dM_{r}/dr}{dt_{\rm{ff}}/dt}=\frac{2M_{r}}{t_{\rm{ff}}}(\frac{\rho}{\bar{\rho}-\rho }). \eeq where $\rho$ is the mass density of the collapsar. We roughly set the above mass supply rate equals to the accretion rate $\dot{M}$ of the BH hyperaccretion system in the center of the collapsar \citep[e.g.,][]{Kashiyama2013,Nakauchi2013}. We neglect the mass loss in the process. Due to the angular momentum redistribution, an outflow is launched when the matter falls onto the outer boundary of the disk. The accretion of rotating gas through the disk is significantly reduced compared with the accretion of nonrotating gas \citep[e.g.,][]{Igumenshchev2003,Proga2003}. Moreover, the outflows might be launched from the disk when the mass accretion rate is high \citep[e.g.,][]{Liu2018,Song2019}. As a result, only a fraction of supplied mass is eventually accreted into the BH and the accretion rate we calculated is overestimated. In other words, the larger $r_1$ or $k_2$ might be required in the below results if the outflows are considered. \subsection{Jet model} The relativistic jets are launched from a BH hyperaccretion system in the center of a massive collapsar. The mechanism for converting the accretion energy or BH rotation energy into directed relativistic outflow remains uncertain. There are two well-known candidate mechanisms: neutrino annihilation and magnetohydrodynamic mechanisms, including the Blandford-Znajek (BZ) process \citep{Blandford1977}. For the neutrino annihilation, if the accretion rate is very high, then photons are trapped in the disk, and only generous neutrinos can escape from the disk surface and annihilate in the space out of the disk to produce relativistic electron-positron jets. This accretion disk is called a neutrino-dominated accretion flow (NDAF), which has been widely investigated in previous works \citep[e.g.,][]{Popham1999,Narayan2001,Kohri2002,Lee2005,Gu2006,Chen2007,Janiuk2007,Kawanaka2007,Liu2007,Liu2015a,Liu2015b,Liu2017,Lei2009,Xue2013,Song2016,Nagataki2018}. For the BZ mechanism, the BH rotation energy can be converted into the Poynting flux jet by a surrounding magnetic field \citep[e.g.,][]{Lee2000a,Lee2000b,Mizuno2004,McKinney2004,Barkov2008,Nagataki2009,Lei2013,Lei2017,Liu2015a,Wu2013}. For the same BH spin parameter and accretion rate, the neutrino annihilation luminosity is approximately two orders of magnitude smaller than the BZ luminosity \citep[e.g.,][]{Liu2015a,Liu2017}. Here we adopt two jet models for jet producing mechanisms in this work. We assume that the jet is formed when the mass of BH reaches 3 $M_{\odot }$ since it has little effect on the luminosity \citep[e.g.,][]{Qu2022} and set $t=0$ at this time. First, we assume that the jet is driven by neutrino annihilation. The jet luminosity is estimated as the neutrino annihilation luminosity: $L_{\rm{j}}=L_{\rm{\nu }\rm{\bar{\nu}}}$. By investigating one-dimensional global solutions of NDAFs in the Kerr metric, \citet{Xue2013} derived fitting formulae for the annihilation luminosity, which is written as \beq \log L_{{\nu }{\bar{\nu}}}({\rm erg\,s^{-1}})\approx 49.50+2.45a_{*}+2.17\log\dot{m}, \eeq where $a_{*}$ is the mean dimensionless BH spin parameter, and $\dot{m}=\dot{M}/M_\odot ~\rm s^{-1}$ is the dimensionless accretion rate. Here, we adopt $a_{*}=0.9$ \citep[e.g.,][]{Song2020,Wei2021}. Second, the jet is assumed to be driven by the BZ process and neutrino annihilation together. If the jet is produced by the BZ process, the required large-scale magnetic field may need to be maintained by the NDAF. These two mechanisms may coexist in a BH hyperaccretion system \citep[e.g.,][]{Liu2015a,Liu2017,Liu2018}. The neutrino annihilation luminosity mainly plays a role in the early stage of accretion. When the accretion rate decreases, the neutrino related process will be terminated and the BZ mechanism dominates the jet luminosity. Therefore, the duration of the central engine for the second model is obviously longer than that for the first model. For the BZ process, the jet luminosity is given by \citep[e.g.,][]{Komissarov2010,Suwa2011,Matsumoto2015} \beq L_{j}=\eta \dot{M}c^{2}. \eeq where $\eta$ is the efficiency parameter. The value of $\eta$ is still uncertain in the collapsar scenario. Here, we assume that $\eta =6.2\times 10^{-4}$, which is a typical value for a WR star progenitor model \citep[e.g.,][]{Suwa2011}. Then, the jet luminosity can be written as \beq L_{j}=1.10\times 10^{51}\dot{m}+10^{49.50+2.45a_{*}+2.17\log\dot{m}}. \eeq Here, the BH spin parameter is also adopted as $a_{*}=0.9$. \subsection{Jet propagation} The essential features of jet propagation in an envelope were described in previous works. After a relativistic jet is launched, it pushes the collapsar matter, and two shocks are formed at the jet head. One is the forward shock, which sweeps the collapsar matter. The other is the reverse shock, which would decelerate the head of the jet. The velocity of the jet head can be calculated from the pressure balance at the interface of the jet and the envelope as \citep[e.g.,][]{Matzner2003,Bromberg2011b,Matsumoto2015,Liu2018} \beq \beta _{h}=\frac{1}{1+\tilde{L}^{-1/2}}, \eeq where \beq \tilde{L}\equiv \frac{L_{j}(t-r_{\rm{h}}/c)}{\pi r_{\rm{h}}^{2}\theta _{j}^{2}\rho (r_{\rm{h}})c^{3}} \eeq and $\theta _{j}$ is the jet half-opening angle. We adopt $\theta _{j}=5^{\circ}$ in our calculations. $r_{\rm h}(t)=\int_{0}^{t}c\beta_{\rm{h}} dt$ is the radius of the jet head. The jet breakout time $t_{b}$ is defined as $r_{\rm{h}}(t_{b})=r_{\rm{2}}$. Only after a jet breaks out from the collapsar can a GRB be observed. In the rest frame, the duration of GRB is $t_{\rm{GRB}}=t_{\rm{eng}}-t_{b}$, where $t_{\rm{eng}}$ is the duration of the central engine. We assume that the central engine operates until the jet luminosity decreases to $10^{47}~\rm{erg}~\rm{s}^{-1}$. When the velocity of the jet head is nonrelativistic, the shocked material is pushed sideways to form a hot cocoon around the jet \citep{Matzner2003}. We assume that the jet energy goes through the shocked region into the cocoon before the jet reaches the collapsar surface. If the jet head can break out from the envelope successfully and the velocity of the jet head is larger than that of the cocoon, then the jet emission can be seen as a GRB \citep{Matzner2003,Toma2007}. The efficiency for converting the jet energy to radiation energy remains uncertain. Here following \citet{Nakauchi2012}, we assume the efficiency $\zeta$ is 10 $\%$. The energy of prompt emission can be estimated by \beq E_{\gamma}=\int_{t_{\rm{b}}}^{t_{\rm{eng}}}\zeta L_{\rm{j}}(t)dt. \eeq Considering the beaming correction, the isotropic energy of prompt emission can be calculated by \citep[e.g.,][]{Yi2017} \beq E_{\gamma,\rm iso }=E_{\gamma}/f_{\rm{b}}, \eeq where \beq f_{\rm{b}}=1-\rm{cos}\theta _{\rm{j}}\simeq \theta _{\rm{j}}^{2}/2. \eeq In this paper, we study the dependence of jet propagation on the collapsar density profile. Note that we consider the propagation of a jet in the stationary envelope in this work. The fallback process and jet propagation are separately calculated. Therefore, the feedback of the jet on the accretion is neglected. According to \citet{Nagakura2012}, some portion of the envelope ceases to fall due to jet propagation. However, a larger amount of matter can still be accreted into the BH. Thus, even considering the feedback of the jet, the jet may break out successfully and create a GRB. We also neglect the effect of the star rotation. \citet{Nagakura2013} investigated jet propagation through a rotating collapsing WR star. They found that the neutrino-driven jet can break out of the star by the different progenitor rotations and suggested that rapidly rotating stars are more likely to produce GRBs. \section{Results} The GRB durations $t_{\rm{GRB}}$ and isotropic energy of prompt emission $E_{\rm{\gamma,iso}}$ of different collapsars for the first jet model are displayed in Figures 2(a) and 2(b), respectively. At the initial accretion stage, the mass supply to the BH hyperaccretion is from the core. The core becomes larger as $r_{1}$ becomes larger. Thus, a larger $r_{1}$ corresponds to a longer duration of the central engine. We assume that the central engine operates until the jet luminosity decreases to $10^{47}~\rm{erg}~\rm{s}^{-1}$. Therefore, the accretion of the envelope contributes to the duration of the central engine, especially when the envelope is thick. This contribution may be important in metal-free Pop III star. \citet{Suwa2011} investigated jet propagation in the envelope and showed that massive Pop III stars ($>100~M_{\odot}$) can produce GRBs even though they have large hydrogen envelopes due to the long-lasting accretion of the envelope itself. In addition, the jet propagation in light Pop III stars ($\sim 40~M_{\odot}$) has been investigated \citep[e.g.,][]{Nakauchi2012,Nagakura2012,Liu2018}. These studies found that light Pop III stars has the possibility of producing GRBs even though it keeps a massive hydrogen envelope. Some Pop III GRBs from high redshift might be detected as long-duration X-ray-rich GRBs by \emph{EXIST} or long-duration X-ray flashes by \emph{Lobster}. $k_{\rm{2}}$ determines the size and density of the envelope. There is competition between the mass supply onto the BH hyperaccretion and jet propagation into the envelope. Generally, a thick envelope can enhance the mass supply onto the BH hyperaccretion and increase the duration of the central engine, while a thin envelope allows the jet to break out of the collapsar quickly. When $k_{\rm{2}}$ is small, the collapsar has a thick envelope. Therefore, a smaller $k_{\rm{2}}$ corresponds to a longer duration of the central engine and a longer duration of jet propagation. In our work, the collapsar with a small $k_{\rm{2}}$ may come from a light Pop III star. When $k_{\rm{2}}$ is large, the collapsar almost loses its envelope, and both the duration of the central engine and the duration of jet propagation are short. As a result, collapsars with different density profiles can produce GRBs with different durations and $E_{\rm{\gamma,iso} }$ values. The parameter space for GRBs with different durations is shown in Figure 2(a). The blank region corresponds to the case in which the jet fails to break out of the collapsar. The black line corresponds to $t_{\rm{GRB}}=2 ~\rm{s}$. Above this line, collapsars produce LGRBs. Obviously, a collapsar with a large core and a thin envelope is more likely to produce a LGRB. Below this line, collapsars would produce SGRBs or failed GRBs. When $k_{\rm{2}}$ is small, the jet can break out of the collapsar only when $r_{1}$ is large. This is because it takes a jet more time to break out of the collapsar when its envelope is thick. Therefore, the duration of the central engine should be long, i.e., $r_{1}$ should be large. We note that $t_{\rm{GRB}}=2 ~\rm{s}$ can be achieved regardless of the thickness of the envelope. However, these two situations are different. When $k_{\rm{2}}$ is small, a SGRB is mainly caused by the fact that the duration of the jet propagation is similar to the duration of the central engine. For a large $k_{\rm{2}}$, the jet can break out of the collapsar immediately, and an SGRB is caused by the fact that the duration of the central engine itself is short. We note that when $k_{\rm{2}}>10$, the envelope is already very thin and the jet can easily break out of the collapsar. $E_{\rm{\gamma,iso} }$ of different collapsars for the first model are shown in Figure 2(b). We can see that $E_{\rm{\gamma,iso} }$ increases as $k_{\rm{2}}$ increases. When the envelope is thick, it takes a long time to break out of the collapsar. Thus, a large part of the jet energy is consumed in the envelope, and the corresponding $E_{\rm{\gamma,iso} }$ is relatively small. As a result, GRBs from collapsars would have a specific duration and $E_{\rm{\gamma,iso} }$, which can help constrain the density profiles of collapsars. The GRB duration $t_{\rm{GRB}}$ of different collapsars for the second jet model are displayed in Figure 3(a). We find that collapsars are more likely to produce LGRBs for the second jet model. SGRBs are produced only when $k_{\rm{2}}$ is large, i.e., the duration of the central engine itself is short. For the second jet model, even if the envelope is thick, the jet can easily beak out of the collapsar. Although a larger envelope may achieve $t_{\rm{eng}}\sim t_{b}$, the jet is nonrelativistic after breaking out of a very thick envelope. According to our result, the jet is nonrelativistic when $k_{\rm{2}}<5$. Therefore, for both jet models, collapsars are unlikely to produce SGRBs when $k_{\rm{2}}<5$. Figure 3(b) shows $E_{\rm{\gamma,iso} }$ of different collapsars for the second model. For the same parameters, $E_{\rm{\gamma,iso} }$ in Figure 3(b) is larger than that in Figure 2(b). Similarly, $E_{\rm{\gamma,iso} }$ increases as $k_{\rm{2}}$ increases. For both jet models, collapsars with small $k_{\rm{2}}$ can produce GRBs. As a result, our calculations support the results of previous works \citep[e.g.,][]{Nakauchi2012,Nagakura2012} that light Pop III stars may produce GRBs. The Pop III GRBs from the early universe are expected to be detected in the future. \section{Applications to GRB 200826A} GRB 200826A was detected by the Gamma-ray Burst Monitor (GBM) onboard the \emph{Fermi} Gamma-ray Space Telescope on August 26, 2020. The duration of this burst is $1.14\pm 0.13~\rm{s}$ in the $50-300$ keV energy range \citep{Ahumada2021}. \citet{Zhang2021} measured the ``amplitude parameter $f$" of this burst, which is defined as the ratio between the peak flux and the average background flux of the GRB lightcurve. They obtained $f=7.58\pm 1.23$ for GRB 200826A. According to \citet{Lu2014}, to make a LGRB a ``tip-of-iceberg" SGRB, the effective $f$ value is typically $<1.5$. The enormous $f$ value suggested that GRB 200826A is a genuine SGRB and cannot be the tip of iceberg of an LGRB. The optical afterglow of GRB 200826A was detected by the Large Binocular Telescope Observatory, and it helps identify the redshift $z \approx 0.7481$ \citep{Rothberg2020}. Therefore, the rest-frame duration of GRB 200826A should be $\sim 0.65~\rm{s}$, and the isotropic energy is $E_{\rm{\gamma,iso} }\sim7.1\times 10^{51}~\rm{erg}~\rm{s}^{-1}$. An supernova (SN) bump in the afterglow light curve was confirmed by \citet{Ahumada2020}. Meanwhile, additional optical observations of the afterglow and the SN have been reported \citep{Ahumada2021,Zhang2021,Rossi2022}. According to the spectral properties, host galaxy offset, total energy, and association with SN, GRB 200826A is considered to be the shortest LGRB from a collapsar. It appears to sit on the brink between a successful and a failed collapsar. Here, we use our model to roughly constrain the collapsar density profile of GRB 200826A. According to our results, $t_{\rm{GRB}}<2 ~\rm{s}$ can be achieved by different density profiles. Therefore, we mainly constrain the density profile of GRB 200826A by isotropic energy, which is set as $E_{\rm{\gamma,iso} }=7.1\times 10^{51}~\rm{erg}~\rm{s}^{-1}$. First, we consider the first jet model, and the result is displayed in Figure 4(a). The black line corresponds to $t_{\rm{GRB}}=2 ~\rm{s}$, and the blue line corresponds to $E_{\rm{\gamma,iso} }=7.1\times 10^{51}~\rm{erg}~\rm{s}^{-1}$. The red line corresponds to collapsar mass $M_{\rm col}=10 ~M_{\odot}$. We mark the location of GRB 200826A in the figure with a green pentagram. The result shows that this burst might be produced by a collapsar whose envelope has been stripped. The mass of this collapsar is approximately $10~M_{\odot}$. Figure 4(b) displays the result for the second jet model. The black and blue lines correspond to $t_{\rm{GRB}}=2 ~\rm{s}$ and $E_{\rm{\gamma,iso} }=7.1\times 10^{51}~\rm{erg}~\rm{s}^{-1}$, respectively. We can see that the $E_{\rm{\gamma,iso} }$ of SGRBs from collapsars are generally higher than the $E_{\rm{\gamma,iso} }$ of GRB 200826A. Therefore, for the second jet model, the collapsar is unlikely to produce GRB 200826A even there is an efficiency of jet luminosity for the BZ mechanism or the outflows are considered. Moreover, SGRBs with relatively low $E_{\rm{\gamma,iso} }$ values are unlikely to be produced by collapsars for the second jet model. As a result, we suggest that GRB 200826A might come from a $\sim 10 ~M_{\odot}$ collapsar whose progenitor's envelope has been stripped and whose jets are driven by the annihilation of neutrinos. \section{Summary} In this paper, we investigated the propagation of jets in collapsars with different density profiles. We adopt two jet models. For the first jet model, the jet is driven by neutrino annihilation. For the second jet model, the jet is driven by the BZ process and neutrino annihilation together. For the same collapsar, the jet luminosity of the second model is obviously larger than that of the first model. There is competition between mass supply onto the BH hyperaccretion and jet propagation into the envelope. Although a thick envelope can increase the duration of the central engine, it also increases the duration of the jet propagation. As a result, the duration and $E_{\rm{\gamma,iso}}$ of GRBs from collapsars are determined by mass supply and jet propagation together. We found that collapsars can produce LGRBs, SGRBs, and failed GRBs for both models. The density profiles for producing GRBs with different durations and $E_{\rm{\gamma,iso}}$ values are exhibited. Generally, a massive collapsar with a thin envelope is more likely to produce LGRBs. For the first jet model, both thick and thin envelopes can result in the production of SGRBs. For the second model, jets can easily break out of collapsars. We note that only collapsars with thin envelopes can give rise to SGRBs. Although a thick envelope can lead to $t_{\rm{eng}}\sim t_{\rm b}$, the jet would be nonrelativistic after it breaks out of the collapsar. Thus, for the second model, collapsars are more likely to produce LGRBs. The thickness of the envelope can significantly affect $E_{\rm{\gamma,iso}}$. $E_{\rm{\gamma,iso}}$ increases as the envelope becomes thinner. GRB 200826A is considered to be an SGRB from a collapsar. We show that this burst might be produced by a $\sim 10~M_{\odot}$ collapsar whose envelope has been stripped, and the jets should be launched by the neutrino annihilation process. Note that the propagation of the jet is very complicated. Many factors, such as the jet opening angle, star rotation, circumstellar medium, jet feedback, and disk outflow feedback \citep{Liu2019} affect the propagation of the jet and GRB production. Therefore, it is inadequate to constrain the properties of collapsars solely according to the duration and $E_{\rm{\gamma,iso}}$ of GRBs. The joint multimessenger observations, i.e., MeV neutrinos and gravitational waves (GWs), can help investigate GRB physics. In \citet{Wei2019} and \citet{Wei2020}, we proposed that whether the jets are chocked in the envelopes or not, the activities of the central compact objects will produce the detectable MeV neutrinos and GWs. Furthermore, we presented that the strong GW signals from collapsars, jets, and central engines in the Local Group can be detected by the operational or planned GW detectors \citep[e.g.,][]{Wei2020}. \acknowledgments This work was supported by the National Natural Science Foundation of China under grant 12173031.

Title: Into the darkness: Ultra-high energy neutrinos from high-redshift electromagnetic cascades

Abstract: We study the impact of the muon pair production and double pair production processes induced by ultra-high energy photons on the cosmic microwave background. Although the muon pair production cross section is smaller than the electron pair production one, the associated energy loss length is comparable or shorter than the latter (followed by inverse Compton in the deep Klein-Nishina regime) at high-redshift, where the effect of the astrophysical radio background is expected to be negligible. By performing a simulation taking into account the details of $e/\gamma$ interactions at high energies, we show that a significant fraction of the electromagnetic energy injected at $E\gtrsim 10^{19}\,$eV at redshift $z\gtrsim 5$ is channeled into neutrinos. The double pair production plays a crucial role in enhancing the multiplicity of muon production in these electromagnetic cascades. The ultra-high energy neutrino spectrum, yet to be detected, can in principle harbour information on ultra-high energy sources in the young universe, either conventional or exotic ones, with weaker constraints from the diffuse gamma ray flux compared to their low redshift counterparts.

https://export.arxiv.org/pdf/2208.06440

\begin{flushleft} LAPTH-039/22 \end{flushleft} \title{Into the darkness: \\Ultra-high energy neutrinos from high-redshift electromagnetic cascades} \author{AmirFarzan Esmaeili} \email{a.farzan.1993@aluno.puc-rio.br} \affiliation{Departamento de FГ­sica, PontifГ­cia Universidade CatГіlica do Rio de Janeiro, Rio de Janeiro 22452-970, Brazil} \author{Antonio Capanema} \email{antoniogalvao@aluno.puc-rio.br} \affiliation{Departamento de FГ­sica, PontifГ­cia Universidade CatГіlica do Rio de Janeiro, Rio de Janeiro 22452-970, Brazil} \author{Arman Esmaili} \email{arman@puc-rio.br} \affiliation{Departamento de FГ­sica, PontifГ­cia Universidade CatГіlica do Rio de Janeiro, Rio de Janeiro 22452-970, Brazil} \affiliation{The Abdus Salam ICTP, Strada Costiera 11, 34151, Trieste, Italy} \affiliation{LAPTh, CNRS, USMB, F-74940 Annecy, France} \author{Pasquale Dario~Serpico} \email{serpico@lapth.cnrs.fr} \affiliation{LAPTh, CNRS, USMB, F-74940 Annecy, France} \section{Introduction\label{sec:intro}} The high-redshift and high-energy universe remains largely mysterious. The hadronic component of ultra-high energy cosmic rays (UHECRs) suffers significant energy losses, besides lacking directionality due to deflections in extragalactic magnetic fields. Photons and electrons undergo fast interactions with the cosmic microwave background (CMB), quickly degrading their energy, eventually contributing only to the diffuse extragalactic background below the TeV. Even gravitational wave signals at high-redshift will have to wait next generation detectors in order to be properly explored~\cite{Kalogera:2019sui}, and linking them to high-energy counterparts in other messengers is far from evident. Neutrinos are thus the most promising messengers, especially after the detection of astrophysical neutrinos by IceCube~\cite{Aartsen:2013bka,Schneider:2019ayi,Stettner:2019tok,Aartsen:2020aqd}, currently up to $\sim {\cal O}(10)$ PeV~\cite{Aartsen:2021glw}, paving the way to explore the remote and violent universe through these elusive particles. The near future improvements in the sensitivity of neutrino telescopes will further promote this exploration, notably searching for ultra-high-energy $\nu$'s ($E\gtrsim 10^{17}$\,eV): Their presence is guaranteed at least from interactions of UHECRs with the CMB (so-called {\it cosmogenic neutrinos}~\cite{Berezinsky:1969erk}), but could also originate from yet unknown astrophysical or exotic processes~\cite{Abbasi:2021vjr,Fang:2017cqe,Batista:2019nhf}. Of particular interest for diagnostics is the interplay between these messengers, especially between the photons and neutrinos. Gamma rays have long been used to constrain ultra-high energy (UHE) neutrinos, see for instance~\cite{Waxman:1998yy,Mannheim:1998wp,Kalashev:2002kx}. In the following, we revisit the link between these particles at high-$E$ and high-$z$, since differing in some peculiar aspects from the standard expectations. In particular, we will argue that electromagnetic cascades drain some significant energy into the neutrino channel, altering the usual expectation for the ratio of energy into neutrinos vs gamma-rays, besides obviously modifying the expected spectra of UHE neutrinos. The fact that the propagation of UHE photons/electrons may be quantitatively different at high-$z$ was studied in~\cite{Kusenko:2000fk,Postma:2001na}, where the key process responsible for the drainage into the neutrino channel was thought to be $e\gamma\to e\mu\mu$. Soon after, this process was reassessed and found of negligible importance in~\cite{Athar:2001sm}. The same article also suggested that the $\gamma\gamma\to\mu\mu$ process may however play a similar role~\footnote{An early mention of this process can also be found in~\cite{Berezinsky:1970nm}.}. This process has been studied within some approximations in~\cite{Wang:2017phf}, where it was concluded that at low redshift $0\leq z<5$, and due to the interplay with the diffuse universal radiation background (URB)~\footnote{Furthermore, the $e^\pm$ in the cascade could also quickly lose energy via synchrotron emission in intergalactic magnetic fields, if these are close to the current upper limits at the nG level.}, only a relatively small fraction of the initial energy of electromagnetic cascades ($\lesssim 10\%$) channels into neutrinos. Similar considerations were also briefly exposed in~\cite{Li:2007ps}. Here we complement these studies by expanding the redshift range and particularly improving on the microphysics of the cascade development at high energies, leading us to somewhat different conclusions. The basic idea is that muon pair production (MPP) is non-negligible in the interaction of UHE photons with CMB, where the subsequent decay of muons generates neutrinos. The MPP introduces an important deviation from the course of well-studied electromagnetic cascade of high energy photons/electrons, where it is the chain of electron pair production (EPP) and inverse-Compton scattering (ICS) which leads to the degradation of initial photon/electron energy and the production of a lower-energy photon spectrum. The large inelasticity in both EPP and ICS renders the MPP feasible since effectively the energy loss length in electromagnetic cascade is larger than the interaction length of MPP. The picture is further altered by the role of the double pair production (DPP), which constitutes an important energy-loss channel and is responsible for lowering the drainage of energy via MPP, while at the same time increasing the muon multiplicity in the cascade. In section~\ref{sec:interactions} we introduce the microphysics ingredients used to describe the aforementioned processes, and justify qualitatively the importance of MPP. A Monte Carlo simulation of processes and the corresponding distributions of energy drainage and neutrino spectrum are given in section~\ref{sec:results}. Section~\ref{sec:conc} is devoted to the discussions and conclusions, summarizing the take-home message and commenting on possible applications. For the ease of reference, well-known analytical formulae of neutrino spectra emitted in muon decay are reported in Appendix~\ref{nuspectra}. \section{Photon and electron interactions\label{sec:interactions}} The propagation of UHE photons and electrons in intergalactic space is hindered by the interaction with background photon fields ($\gamma_{\rm b}$) that permeate the Universe. Starting with an UHE photon, the main relevant interactions are EPP ($\gamma\gamma_{\rm b}\to e^+e^-$) and ICS ($e\gamma_{\rm b}\to e\gamma$) where their successive iteration develops the conventional ``electromagnetic cascade". At variance with low-$E$ and low-$z$ cascades, MPP ($\gamma\gamma_{\rm b}\rightarrow\mu^-\mu^+$) and DPP ($\gamma\gamma_{\rm b}\rightarrow e^+e^-e^+e^-$) are also of interest for our analysis and will significantly contribute to the cascade development. At low-$z$, UHE photons are dominantly interacting with the URB and electrons are possibly affected by syncrhotron emission on extragalactic magnetic fields, if close to the allowed upper limits of nG strength~\cite{Lee:1996fp}. However, the URB is expected to drop at $z\gtrsim 2$, and by $z\sim 5$ it should be vanishingly small compared to the CMB~\cite{Lee:1996fp, Kusenko:2000fk, Postma:2001na}. While the exact redshift at which the URB can be neglected is poorly known, it is robust to assume that such a redshift exists, since the URB is of astrophysical origin and, while the CMB density of photons grows with $z$ as $(1+z)^3$, the density of astrophysical sources drops at $z>2$ and should eventually be vanishing at $z\gtrsim 15$. Based on models in the literature, we estimate that $z=5$ is a rather conservative assumption, hence we will show results considering interactions solely with the CMB only above this redshift. The total and differential cross sections of EPP (also dubbed Breit-Wheeler process~\cite{Breit:1934zz}) are respectively \cite{Berestetskii:1965xsa} \begin{multline}\label{eq:PP_tot_xsec} \sigma_{\rm EPP} = \sigma_{\rm T}\,\frac{3}{16}(1-\beta^2) \bigg[ (3-\beta^4)\ln \frac{1+\beta}{1-\beta} \\ - 2\beta(2-\beta^2) \bigg]~, \end{multline} and \begin{multline}\label{eq:PP_diff_xsec} \frac{{\rm d}\sigma_{\rm EPP}}{{\rm d}E_e} = \sigma_{\rm T}\,\frac{3}{4}\,\frac{m_e^2}{s}\,\frac{1}{E_\gamma} \left[ \frac{E_e}{E_\gamma - E_e} + \frac{E_\gamma - E_e}{E_e} \right. \\ + E_\gamma\left(1-\beta^2\right)\left( \frac{1}{E_e} + \frac{1}{E_\gamma - E_e} \right) \\ \left. - \frac{E_\gamma^2\left(1-\beta^2\right)^2}{4}\left( \frac{1}{E_e} + \frac{1}{E_\gamma - E_e} \right)^2 \right]~, \end{multline} where $m_e$ is the electron mass, $\sigma_T=8\pi\alpha^2/(3m_e^2)$ is the Thomson cross section (in the whole paper, natural units are used), $\alpha$ the fine structure constant, $\beta = \sqrt{1-4m_e^2/s}$ is the velocity of outgoing electron in the CM frame, $s =2E_\gamma\epsilon(1-\mu)$ is the squared CM energy, $\epsilon$ and $E_\gamma$ are respectively the energies of the target (here CMB photon) and high energy photons, $\mu$ is the cosine of the angle between the momenta of the incoming photons, and $E_e$ is the energy of the produced electron (or positron) whose allowed range is $(1-\beta)/2\leq E_e/E_\gamma\leq(1+\beta)/2$. The ICS total and differential cross sections are given by \cite{Aharonyan:1981csr} \begin{multline}\label{eq:ICS_tot_xsec} \sigma_{\rm ICS} = \sigma_{\rm T}\,\frac{3}{8}\,\frac{m_e^2}{\tilde{s}\tilde{\beta}} \left[\frac{2}{\tilde{\beta}(1+\tilde{\beta})}(2 + 2\tilde{\beta} - \tilde{\beta}^2 - 2\tilde{\beta}^3) \right. \\ \left. - \frac{1}{\tilde{\beta}^2}(2 - 3\tilde{\beta}^2 - \tilde{\beta}^3) \ln \frac{1+\tilde{\beta}}{1-\tilde{\beta}} \right]~, \end{multline} and \begin{multline}\label{eq:ICS_diff_Xsec_CM} \frac{{\rm d}\sigma_{\rm ICS}}{{\rm d}E_e'} = \sigma_{\rm T}\,\frac{3}{8}\,\frac{m_e^2}{\tilde{s}}\,\frac{1}{E_e}\,\frac{1+\tilde{\beta}}{\tilde{\beta}} \left[ \frac{E'_e}{E_e} + \frac{E_e}{E'_e} \right. \\ \left. +\frac{2(1-\tilde{\beta})}{\tilde{\beta}} \left( 1 - \frac{E_e}{E'_e} \right) + \frac{1-\tilde{\beta}^2}{\tilde{\beta}^2} \left( 1 - \frac{E_e}{E'_e} \right)^2 \right]~, \end{multline} where $\tilde{\beta} =(\tilde{s}-m_e^2)/(\tilde{s}+m_e^2)$ is the velocity of the outgoing electron in the CM frame, $\tilde{s}=m_e^2+2\epsilon(E_e-\mu\sqrt{E_e^2-m_e^2})$ is the squared CM energy, $E_e$ is the energy of the initial high-energy electron, and $E'_e$ is the energy of the outgoing electron, whose allowed range is $(1-\tilde{\beta})/(1+\tilde{\beta})\leq E'_e/E_e\leq 1$. The threshold energy for EPP derives from the condition $s= 4m_e^2$. For $s\geq 4m_\mu^2$ ($m_\mu$ being the muon mass), MPP ($\gamma\gamma_{\rm b}\rightarrow\mu^-\mu^+$) also becomes accessible. The MPP's total cross section, $\sigma_{\rm MPP}$, and differential cross section, ${\rm d}\sigma_{\rm MPP}/{\rm d}E_\mu$, can be obtained by replacing $m_e\to m_\mu$ in the formulae for EPP. For the MPP on the bulk of the CMB at redshift $z$, the threshold energy is $E_{\rm MPP}^{\rm thr}=m_\mu^2/\langle\epsilon \rangle\simeq 1.8\times10^{19}/(1+z)$~eV, for the benchmark value $\langle\epsilon \rangle \simeq 2.7 (1+z)T_0$, where $T_0=2.35\times 10^{-4}$~eV is the current CMB temperature. At high energies the DPP becomes relevant. While a fully analytic expression for $\sigma_{\rm DPP}$ is quite involved, it rapidly converges to the constant value $\sigma_{\rm DPP}^\infty\simeq 6.45\times 10^{-30}~{\rm cm}^2$ above its threshold at $s=16m_e^2$, and its energy dependence can be closely approximated by $\sigma_{\rm DPP} \approx (1-4m_e^2/E_\gamma \epsilon)^6 \sigma_{\rm DPP}^\infty$ \cite{Brown:1973dpp}. We will not employ the full expression for the DPP differential cross section, but simply approximate the process in assuming that one of the pairs carries the quasi-totality of the projectile photon energy, sharing it equally. This captures the main quantitative effect of DPP on the cascade development~\cite{Demidov:2009dpp}. Although MPP is suppressed with respect to EPP (for example, at $s=10^{18}~\rm eV^2$, $\sigma_{\rm MPP}/\sigma_{\rm EPP}\approx0.26$), muon production is definitely relevant since EPP and ICS have large inelasticities at high energies. The inelasticity of a given process, i.e. the average fraction of energy transferred from the initial leading particle to the produced leading particle, is given by \begin{equation}\label{eq:elasticity} \eta(s) = \frac{1}{\sigma(s)}\int {\rm d}E'\,\frac{E'}{E_0}\, \frac{{\rm d}\sigma}{{\rm d}E'}(E',s)~, \end{equation} where $E'$ is the energy of the produced leading particle and $E_0$ is the energy of the initial leading particle. Due to the large inelasticities in EPP and ICS, the initial UHE photon (or electron) undergoes a sequence of EPP+ICS, at each step of which the leading particle emerges with an energy close to the initial one. If MPP happens, this sequence is greatly altered since the final-state $e^\pm$ from muon decay carry a comparatively small fraction of the parent photon energy. For the DPP, we assume that each one of the leptons in a pair $e^+e^-$ carries half of the initial photon energy, which is very close to the actual (and much more involved) calculation~\cite{Demidov:2009dpp}. To quantify the relative prominence of these processes, let us define the {\it interaction length} (or mean free path) \begin{equation} \lambda_p(E) = \frac{1}{\int {\rm d}\epsilon \int{\rm d}\mu\, P(\mu)\, n_{\rm CMB}(\epsilon)\,\sigma_p(s)}~,\label{Intlength} \end{equation} where $p=$~EPP, MPP or DPP, $n_{\rm CMB}$ is the number density of CMB photons per unit energy, $s=s(E,\epsilon,\mu)$, $P(\mu)=(1-\mu)/2$ is the distribution of collision angles (or flux factor), and the integral over $\mu$ extends up to $1-2m^2/E_\gamma\epsilon$, with $m$ being either $m_e$ or $m_\mu$. Similarly, one can define the {\it energy loss length}~\footnote{A more correct definition in terms of stopping power ${\rm d} E/{\rm d} x$ would be $\Lambda=\int {\rm d} x\, E/(-{\rm d} E/{\rm d} x)$, but for moderate energy dependence of the integrand it leads to comparable results. We content ourselves with the simplest definition of eq.~(\ref{Elosslength}), given the mere illustrative purpose of this quantity in this section.} \begin{equation} \Lambda_p(E) = \frac{1}{\int {\rm d}\epsilon \int{\rm d}\mu\, P(\mu)\, n_{\rm CMB}(\epsilon)\,\sigma_p(s)\,[1-\eta_p(s)]}~.\label{Elosslength} \end{equation} For a catastrophic event like MPP, $\Lambda\simeq \lambda$, but whenever only a small fraction of energy is lost at each interaction, as it is for the leading particle in EPP/ICS cycles, $\Lambda\gg \lambda$. Figure~\ref{fig:length_comparison} compares $\Lambda_{\rm EPP}$ (blue color), $\lambda_{\rm MPP}\simeq \Lambda_{\rm MPP}$ (green color) and $\lambda_{\rm DPP}$ (black color) as functions of the UHE photon's energy at the observation point. The vertical and horizontal axes are scaled by $(1+z)^3$ and $(1+z)$, respectively, thus the curves are valid for any redshift \footnote{Since the differential density $n_{\rm CMB}(\epsilon)$ scales as $(1+z)^2$, lengths get contracted by $(1+z)$ and energies increase by $(1+z)$ with $z$, this implies that $\lambda(E,z)=(1+z)^{-3}\,\lambda_0[E(1+z)]$. Ditto for $\Lambda$.}. Remarkably, $\lambda_{\rm MPP} < \Lambda_{\rm EPP}$ at high energies indicates that we expect MPP to happen before the photons/electrons have lost a significant fraction of their initial energy via the EPP/ICS cycle. On the other hand, since $\lambda_{\rm DPP} \lesssim \Lambda_{\rm EPP}$ at high energies, DPP affects the cascade development; similarly, $\lambda_{\rm DPP} < \lambda_{\rm MPP}$ signals the relevance of DPP inclusion in the study of MPP. Qualitatively, starting from a very high energy photon, we expect the role of DPP is to split the initial photon energy into a pair $e^\pm$ almost equally. This is followed by ICS events, where the upscattered photons initiate a new multiplicative process via DPP and so on, until the photon energies end-up close to the minimum of the MPP interaction length, around $E_\gamma(1+z)\simeq 10^{20}{\rm eV}$. At that point, the particles are only a factor $2-3$ less likely to undergo muon generation via MPP than to degrade below MPP threshold via a final DPP event, or to start a ``conventional'' cascade via EPP; this explains why MPP matters. Since in a MPP event about 65\% of the energy is carried by the neutrinos (see Appendix~\ref{nuspectra}), a rough expectation is that, away from threshold effects, on average slightly below $65\%/2\sim 30\%$ of $E_\gamma$ is drained into the neutrino flux. We also expect that the higher the energy, the larger is the multiplicity of muons through which the drainage is happening, with this number scaling proportionally to $E_\gamma(1+z)/10^{20}{\rm eV}$. Finally, we can anticipate that a significant spread around the average should be present due to the stochastic nature of these events. Also note that, as shown in Fig.~\ref{fig:length_comparison}, these interaction lengths are well below the Hubble length~\footnote{$H(z)\simeq H_0\sqrt{\Omega_\Lambda+\Omega_M(1+z)^{3}}$ being the Hubble expansion rate, $H_0\simeq 70~$km/s/Mpc its current value, and $\Omega_M\simeq 0.3$ the matter density of the universe in terms of the critical one. Instead, $\Omega_\Lambda\simeq 1-\Omega_M$ is the dimensionless energy density of the cosmological constant, playing a negligible role at redshifts of interest here. } $H(z)^{-1}$. Hence, particle dynamics rather than cosmology rules the evolution in $E$-space, with the cascade development that can be considered almost instantaneous in $z$. These qualitative arguments motivate a more quantitative study of the effect of this process, which we embark on in the next section. Before moving to that, let us mention our rationale for ignoring some additional processes (a synoptic description of which can be found in~\cite{Lee:1996fp}). Charged pion production ($\gamma\gamma_{\rm b}\rightarrow \pi^+\pi^-$) becomes possible at $s\geq 4m_{\pi^\pm}^2$, but its cross section is only comparable to EPP and MPP in a small window of energies (corresponding to the $f_2(1270)$ resonance)~\cite{Boyer:1990opp,Morgan:1987cej} and is otherwise sub-leading. Including this process would only mildly strengthen the conclusions of this article. The production of neutral pions, kaons and heavier hadrons in $\gamma\gamma_{\rm b}$ scattering is even more suppressed~\cite{Whalley:2001tpr,Morgan:1994vne}, justifying that we neglect them. Triplet pair production ($e\gamma_{\rm b}\rightarrow e e^+ e^-$), has a cross section comparable to that of ICS already at $\sim 10^{17}$~eV, but leads to a negligible energy loss below $\sim10^{22}$~eV since the energy fraction carried by the produced pair is very small ($\sim 10^{-3}$)~\cite{Sigl:2001wih} and its inclusion is expected to change our conclusions at the few percent level at most. Finally, we also neglect the synchrotron energy losses of UHE electrons onto extragalactic magnetic fields. While these may be of importance at low redshift, see~\cite{Wang:2017phf}, unless the fields are of primordial origin, their role with respect to losses on CMB should vanish going to high-$z$, with an argument qualitatively similar to what we discussed for the URB. Note that, despite limited information on extragalactic magnetogenesis, current evidence suggests indeed that extragalactic fields grow at low-$z$ via an astrophysical dynamo mechanism, rather than being primordial~\cite{Pomakov:2022poi} (or implying much smaller primordial seeds), consistent with the hypothesis done here. \section{Simulation and results\label{sec:results}} To quantitatively assess the role of MPP at high energies, we proceed with a Monte Carlo simulation. This is unavoidable if one is to take into account the discrete and stochastic nature of the processes. As previously discussed, it turns out that the mean free path between interactions is so short compared to the cosmological scales that the change in the redshift of two successive processes can be safely ignored. Thus, starting with a photon~\footnote{Starting with an electron would not lead to appreciable differences.} with specified energy $E_\gamma$ and redshift $z$, only the evolution in $E-$space is relevant, described as a sequence of interactions where the leading electromagnetically interacting particle's energy degrades, until the MPP process is no longer kinematically open. At each photon interaction, we compare a random number in [0,1] with the probability to yield a MPP \begin{equation}\label{eq:pmpp} p_{\rm MPP}= \frac{\lambda_{\rm MPP}^{-1}}{\lambda_{\rm EPP}^{-1} + \lambda_{\rm MPP}^{-1} + \lambda_{\rm DPP}^{-1}}\,. \end{equation} As an example, at $z=15$ we estimate $p_{\rm MPP}\approx(0.07, 0.07, 0.02)$ respectively for $E_\gamma=(10^{19},10^{20},10^{21})$~eV. The probability of DPP can be defined similarly to Eq.~(\ref{eq:pmpp}) by replacing $\lambda_{\rm MPP}^{-1}$ with $\lambda_{\rm DPP}^{-1}$ in the numerator, leading to $p_{\rm DPP}\approx(0.18, 0.62, 0.93)$ for the same $E_\gamma$'s and $z$. Obviously, for EPP we have $p_{\rm EPP} = 1-p_{\rm DPP}-p_{\rm MPP}$. The cascade development depends on the selected interaction at each step. When MPP is chosen, the $e^\pm$ from the $\mu^\pm$ decay are injected again into the simulation by performing ICS on CMB and creating an UHE photon which starts a new branch of cascade. If DPP is chosen, an $e^+e^-$ pair will be followed (the other pair carrying negligible energy); each member of the pair, assumed to carry energy $E_\gamma/2$, initiates a new branch after a single ICS event. Finally, the EPP case will be followed by ICS. The photon energy coming from ICS events, or the $e^+/e^-$ ones from EPP events are sampled from the corresponding differential cross sections reported in the previous section. The quantities of main interest for phenomenology are the fraction of the initial photon energy channeled into neutrinos, {$f_\nu$}, and the neutrino spectra resulting from this process. Figure~\ref{fig:MCdist} shows the distribution of $f_\nu$ for $10^4$ injected photons with energy $E_\gamma=10^{19}$~eV, $10^{20}$~eV and $10^{21}$~eV in panels (a), (b) and (c), respectively, from top to bottom; the panels in the leftmost column report results at $z=5$, in the middle one at $z=10$ and in the rightmost column at $z=15$. The plots show that at 10$^{19}\,$eV and $z=5$ only about 12\% of the photons experience MPP. This fraction grows to about 25\% at $z=10$ and 35\% at $z=15$. At 10$^{20}\,$eV, well above 70\% of photons experience MPP at $z\geq 5$, with this fraction exceeding 94\% at $z= 15$. Eventually, for $E_\gamma=10^{21}\,$eV, basically every cascade involves one or more MPP events. This behaviour makes sense once realising that, at lower energy, threshold effects reducing the importance of MPP are important. At the highest energy, as discussed, the multiplicity of energetic $e^\pm$ due to DPP makes the probability that none of them initiate a MPP vanishingly small. Note how the distribution of $f_\nu$ are broad (and skewed), reflecting the stochastic nature of the processes. The mean value of $f_\nu$ is a strongly dependent function of energy near the threshold, while being almost constant with energy at high-$E$, as reported in Figure~\ref{fig:Efrac}, for the initial photon energies $E_\gamma=10^{19}$~eV (green), $10^{20}$~eV (red), and $10^{21}$~eV (blue). The bar around each curve shows the standard deviation, calculated from the distributions in Figure~\ref{fig:MCdist}. It mildly shrinks with $E_\gamma$, since high multiplicities make the process ``more deterministic''. Figure~\ref{fig:MPPnumberDist} illustrates the point that, especially at high-$E_\gamma$ and high-$z$, the multiplicity of muons via MPP events is considerable, for the reasons described in the previous section. For instance, at $E_\gamma=10^{21}$~eV, on average $\sim 6,11$ and 15 MPPs will be realized for injection at $z=5,10$ and 15, respectively. Even if the MPP process typically intervenes only when the particles have degraded to energies significantly lower than the injected ones, its multiplicity makes its impact on the energy budget not negligible. Note that in the early study~\cite{Li:2007ps} this aspect was completely missed ``by construction'', since no follow-up of the leptons produced via DPP was performed. Their estimate of only $\sim$~few percent of the electromagnetic energy drainage into neutrinos is thus not only due to the different conditions relevant at low-$z$, but also to the fact that they did not include this important effect. The average all-flavor neutrino spectrum {\it at the Earth} from a photon injected at $z=10$ with energy $E_\gamma=10^{20}$~eV is depicted in Figure~\ref{fig:nuSpec} by the blue solid curve. This is based on the well-known analytical descriptions of neutrino spectra from muon decay (see the formulae in~\cite{Lipari:1992lse}, also summarised in Appendix~\ref{nuspectra}) which have been averaged over the $10^4$ injected photons in our simulation. The wiggles at the peak come from multiple MPPs which for the case of Figure~\ref{fig:nuSpec} can be up to five MPPs, with $\sim 46\%$ of cases leading to two or three MPPs. In a conventional scenario, UHE photons are the product of decays of $\pi^0$'s, that are unavoidably accompanied by $\pi^\pm$'s, whose decays produce neutrinos. In Figure~\ref{fig:nuSpec} we also show, by the black dotted curve, the neutrino spectrum from the decay chain $\pi^\pm \to \mu^\pm\nu_\mu(\bar{\nu}_\mu)\to e^\pm \nu_\mu\bar{\nu}_\mu \nu_e (\bar{\nu}_e)$ (see the formulae in Appendix~\ref{nuspectra}) with the energy of $\pi^\pm$ equal to $2\times10^{20}$~eV. The little discontinuity in the dotted curve comes from the contribution of the neutrino emitted directly from the pion decay $\pi^\pm \to \mu^\pm\nu_\mu(\bar{\nu}_\mu)$. Note how the neutrinos from MPP emerge over those from $\pi^\pm$ in the low-energy part of the distribution, where they dominate the flux by one order of magnitude. At higher $E_\gamma$ and $z$, where the number of MPPs grows, yet more pronounced features are expected in the neutrino spectrum, as can be seen in Figure~\ref{fig:nuSpec-2} which shows the case of $E_\gamma=10^{21}$~eV injected at $z=15$. The same features of Figure~\ref{fig:nuSpec} are now present in a more exacerbated form. This clearly illustrates the relevance of the MPP process in shaping the UHE neutrino spectra from high-$z$/high-$E$ sources. Another implication worth commenting upon is that the process discussed here alters the multimessenger $\gamma$-$\nu$ correlation. Conventional production scenarios arising from $pp$ or $p\gamma$ interactions in an UHE astrophysical source predict that the neutrino and gamma-ray emission spectra are related by \begin{equation}\label{eq:nugamma} \varepsilon_\gamma\, \frac{{\rm d}N_\gamma}{{\rm d}\varepsilon_\gamma} = \frac{1}{3K_\pi} \left[\varepsilon_\nu\, \frac{{\rm d}N_\nu}{{\rm d}\varepsilon_\nu}\right]_{\varepsilon_\nu = \varepsilon_\gamma/2}~, \end{equation} where $K_\pi\approx 1\,(1/2)$ is the charged to neutral pion ratio in the $pp\,(p\gamma)$ process. Integrating both sides of eq. (\ref{eq:nugamma}) over energy implies that, at the source, the ratio of total energies in $\gamma$'s and $\nu$'s obeys ${\cal E}_\gamma \simeq 2/3 \,(4/3)\,{\cal E}_\nu$. The net effect of MPP is to alter this ratio towards the neutrino sector: For example, from Figure \ref{fig:Efrac} we read that for a source emitting $10^{21}$~eV gammas/neutrinos at $z=5$, approximately $24\%$ of the initial photon energy is transferred to neutrinos during the cascade above MPP threshold; naively, the new balance would be ${\cal E}_\gamma' \to 0.76 {\cal E}_\gamma$, ${\cal E}_\nu' \to \left(1+0.24\times 2/3 \,(4/3)\right) {\cal E}_\nu$, hence ${\cal E}_\gamma'/{\cal E}_\nu'\simeq 0.44\, (0.77)$, i.e. a ratio changed by about 40\%. The actual energy budget ending up in the low energy diffuse photon flux is more complicated to compute, since one must account for the contribution seeded by $e^\pm$ from charged pion decays, as well as the fraction of the electromagnetic cascade channelled away by $e^\pm$. However, this simple calculation shows that the role of MPP is to make UHE sources at high redshift {\it darker} than their low-$z$ counterparts even in indirect electromagnetic signals, while making them correspondingly {\it brighter} in the UHE neutrino signal. \section{Discussion and conclusions\label{sec:conc}} Despite ongoing searches via instruments like IceCube~\cite{IceCube:2015gsk} or the Pierre Auger Observatory~\cite{PierreAuger:2015ihf}, ultra-high energy neutrinos remain elusive. However, the existence of a guaranteed neutrino flux of cosmogenic origin around $E\sim 10^{17}\,$eV and of viable technology to measure it has stimulated a number of designs to achieve this goal. Even in absence of an earlier serendipitous discovery, an instrument like GRAND~\cite{GRAND:2018iaj} would eventually open up this window in neutrino astronomy, and stimulate further questions. In this article, we studied some microphysics aspects associated to the UHE $\nu$ flux production which has been largely neglected: The role of muon pair production in draining energy from electromagnetic messengers at UHE and altering the UHE $\nu$ flux, and its interplay with double pair production. We argued that MPP is expected to be relevant at $E\gtrsim 10^{19}\,$eV and at high redshift $z\gtrsim 5$. The resulting flux would fall at the Earth in the $E\gtrsim 10^{17}\,$eV range of interest for an instrument like GRAND. The physics of the process would somewhat loosen constraints from the diffuse gamma ray flux, and spectral features in the neutrino flux, such as the transition between muon and pion channel production around $10^{18}\,$eV, visible in Figures~\ref{fig:nuSpec} and \ref{fig:nuSpec-2}, may be the least elusive of their signatures. Naive expectations from energetics would suggest that the UHE neutrino signal is dominated by relatively low-$z$ sources (which suffer less degradation due to redshift), making hard in this case to dig such a signal out of a larger flux. However, so little is known about UHE sources at high-$z$ that one cannot exclude that new classes of very energetic UHE emitters could be unveiled, for which our considerations are particularly relevant. One conceivable example is provided by the processes associated to the birth and growth of supermassive black holes~\cite{Woods:2018lty}, which are still unsolved astrophysical problems~\cite{Inayoshi:2019fun}. Another flux that would have likely escaped detection at low-$z$ could be due to exotic supermassive relics produced in the early universe, if decaying with a lifetime shorter than the Hubble time. Similar scenarios were considered in the past as ``top-down'' models of UHECRs~\cite{Kachelriess:2008bk} and are still considered in relation to dark matter candidates~\cite{Guepin:2021ljb}. Related models would generally leave a major imprint in the UHE neutrino flux. We have limited ourselves to generic considerations, to be as model-independent as possible. For instance, we did not attempt to link the electromagnetic particles injected to primary UHECR fluxes. Of course, for definite scenarios where our results apply, it would be interesting to perform specific calculations, perhaps including also sub-leading microphysics processes, and moving to full-fledged multimessenger studies. Such tasks are left for future investigations. In conclusion, if the past is of any guidance, it is wise to be ready to possible surprises from the opening of any new astrophysical window. For the UHE sky at high-$z$, one should be aware that differences are present with respect to naive expectations valid at low-$z$, which is without doubts the most important message of this article. \begin{acknowledgments} A.F.~E. thanks the support received by the CNPq scholarship No. 140315/2022-5 and by the CAPES/PROEX scholarship No. 88887.617120/2021-00. A.~C. thanks the support received by the CNPq scholarship No. 140316/2021-3 and by the CAPES/PROEX scholarship No. 88887.511843/2020-00. During his stay at LAPTh, A.~E. has been partially supported by the CNRS invited researcher program. A.~E. would like to thank ICTP for hospitality during the completion phase of this project. A.~E. thanks partial financial support by the Brazilian funding agency CNPq (grant 407149/2021). \end{acknowledgments} \appendix \section{Neutrino spectra}\label{nuspectra} Muon decay generates a neutrino spectrum whose shape depends on the energy distribution of produced $\mu^+$ and $\mu^-$; for a monoenergetic UHE photon, their distribution is given by ${\rm d}N_{\mu^\pm}/{\rm d}E_{\mu^\pm}\equiv (1/\sigma_{\rm MPP}){\rm d}\sigma_{\rm MPP}/{\rm d}E_{\mu^\pm}$. The total (all flavors) neutrino spectrum ${\rm d}N_\nu/{\rm d}E_\nu$ from the decay of $\mu^\pm$ with spectrum ${\rm d}N_{\mu^\pm}/{\rm d}E_{\mu^\pm}$ can be written as~\cite{Lipari:1992lse} \begin{multline}\label{eq:Convolution} \frac{{\rm d}N_\nu}{{\rm d}E_\nu}(E_\nu)=\int_{E_{\mu,\rm min}}^{E_{\mu,\rm max}}{\rm d}E_\mu\,\frac{{\rm d}N_{\mu^\pm}}{{\rm d}E_{\mu^\pm}}(E_{\mu}) \\ \times \left[F_{\mu^{\pm}\rightarrow\overset{(-)}{\nu}_\mu}(E_\mu;E_\nu)+F_{\mu^{\pm}\rightarrow\overset{(-)}{\nu}_e}(E_\mu;E_\nu)\right]~, \end{multline} where \begin{equation}\label{eq:LipariNuFlux3} F_{a\rightarrow b}(E_a;E_b)=\frac{1}{E_a}F_{a\rightarrow b}\left(\frac{E_b}{E_a}\right)~, \end{equation} and for unpolarized muons one has \begin{equation}\label{eq:LipariNuFlux1} F_{\mu^{\pm}\rightarrow\overset{(-)}{\nu}_\mu}(y)=\frac{5}{3}-3y^2+\frac{4}{3}y^3~, \end{equation} and \begin{equation}\label{eq:LipariNuFlux2} F_{\mu^{\pm}\rightarrow\overset{(-)}{\nu}_e}(y)=2-6y^2+4y^3~. \end{equation} From these relations we can estimate the total energy drainage from the initial photon to neutrinos. By a simple inspection of the above formulae, on the average $\sim 65\%$ of the energy of a photon at the time of MPP goes to neutrinos. In Figures \ref{fig:nuSpec} and \ref{fig:nuSpec-2}, we also show the neutrino spectra from a monoenergetic charged pion decay. Given a pion spectrum ${\rm d}N_{\pi^\pm}/{\rm d}E_{\pi^\pm}$, there are two contributions to the final neutrino flux: (i) the (anti)muon neutrino emitted directly from the pion decay $\pi^\pm \to \mu^\pm\nu_\mu(\bar{\nu}_\mu)$, \begin{equation} \frac{{\rm d}N_\nu}{{\rm d}E_\nu}(E_\nu)=\int_{E_{\pi,\rm min}}^{E_{\pi,\rm max}} {\rm d}E_\pi\,\frac{{\rm d}N_{\pi^\pm}}{{\rm d}E_{\pi^\pm}}\,F_{\pi^\pm \rightarrow\overset{(-)}{\nu}_\mu}(E_\pi;E_\nu)~, \end{equation} where \begin{equation} F_{\pi^\pm \rightarrow\overset{(-)}{\nu}_\mu}(x) = \frac{1}{1-r_\pi}[1-\theta(x-1+r_\pi)]~, \end{equation} obeys the scaling (\ref{eq:LipariNuFlux3}) and $r_\pi = (m_\mu/m_\pi)^2$, and (ii) the neutrinos emitted in the subsequent muon decay. The latter can be obtained by convoluting \begin{equation} \frac{{\rm d}N_{\mu^\pm}}{{\rm d}E_{\mu^\pm}}(E_{\mu})= \int_{E_{\pi,\rm min}}^{E_{\pi,\rm max}} {\rm d}E_\pi\,\frac{{\rm d}N_{\pi^\pm}}{{\rm d}E_{\pi^\pm}}\,F_{\pi^\pm \rightarrow\mu^\pm}(E_\pi;E_\mu)~, \end{equation} in Eq.~(\ref{eq:Convolution}), where \begin{equation} F_{\pi^\pm \rightarrow\mu^\pm}(x)=\frac{1}{1-r_\pi}\theta(x-r_\pi)~. \end{equation} \bibliography{references}

Title: Lobster Eye X-ray Optics

Abstract: This chapter describes the history, principles, and recent developments of large field of view X-ray optics based on lobster eye designs. Most of grazing incidence (reflective) X-ray imaging systems used in astronomy and other applications, are based on the Wolter 1 (or modified) arrangement. But there are also other designs and configurations proposed for future applications for both laboratory and space environments. Kirkpatrick-Baez (K-B) based lenses as well as various types of lobster eye optics serve as an example. Analogously to Wolter lenses, all these systems use the principle that the X-rays are reflected twice to create focal images. Various future projects in X-ray astronomy and astrophysics will require large optics with wide fields of view. Both large Kirkpatrick-Baez modules and lobster eye X-ray telescopes may serve as solutions as these can offer innovations such as wide fields of view, low mass and reduced costs. The basic workings of lobster eye optics using Micro Pore Optics (MPOs) and their various uses are discussed. The issues and limiting factors of these optics are evaluated and current missions using lobster eye optics to fulfil their science objectives are reviewed. The Multi Foil Optics (MFO) approach represents a promising alternative. These arrangements can also be widely applied in laboratory devices. The chapter also examines the details of alternative applications for non-Wolter systems in other areas of science, where some of these systems have already demonstrated their advantages such as the K-B systems which have already found wide applications in laboratories and synchrotrons.

https://export.arxiv.org/pdf/2208.07149

\title*{Lobster Eye X-ray Optics} \author{Rene Hudec \thanks{corresponding author} and Charly Feldman} \institute{Rene Hudec \at Czech Technical Universiuty in Prague, Technicka 2, CZ-160 00 Praha 2, Czech Republic \email{hudecren@fel.cvut.cz} \and Charly Feldman \at School of Physics and Astronomy University of Leicester, University Road, LE1 7RH, UK \email{chf7@leicester.ac.uk}} \abstract{This chapter describes the history, principles, and recent developments of large field of view X-ray optics based on lobster eye designs. Most of grazing incidence (reflective) X-ray imaging systems used in astronomy and other applications, are based on the Wolter 1 (or modified) arrangement. But there are also other designs and configurations proposed for future applications for both laboratory and space environments. Kirkpatrick-Baez (K-B) based lenses as well as various types of lobster eye optics serve as an example. Analogously to Wolter lenses, all these systems use the principle that the X-rays are reflected twice to create focal images. Various future projects in X-ray astronomy and astrophysics will require large optics with wide fields of view. Both large Kirkpatrick-Baez modules and lobster eye X-ray telescopes may serve as solutions as these can offer innovations such as wide fields of view, low mass and reduced costs. The basic workings of lobster eye optics using Micro Pore Optics (MPOs) and their various uses are discussed. The issues and limiting factors of these optics are evaluated and current missions using lobster eye optics to fulfill their science objectives are reviewed. The Multi Foil Optics (MFO) approach represents a promising alternative. These arrangements can also be widely applied in laboratory devices. The chapter also examines the details of alternative applications for non-Wolter systems in other areas of science, where some of these systems have already demonstrated their advantages such as the K-B systems which have already found wide applications in laboratories and synchrotrons.} \section{Keywords} X--ray astronomy, X--ray astrophysics, X--ray optics, X-ray telescope, lobster eye optic, Micro Pore Optics \section{Introduction} In this chapter, we focus on the non-Wolter grazing incidence X-ray imaging systems with emphasis on lobster eye (LE) wide Field of View (FoV) systems. Many scientific achievements over the last two decades in X-ray astronomy are closely related to the use of imaging X-ray telescopes. In astronomy and astrophysics, it was the use of imaging X-ray telescopes based on grazing incidence X-ray optics that opened a completely new window into the Universe and has lead to great discoveries during past decades. To acknowledge these achievements, the 2002 Nobel Prize for physics was awarded to Professor Riccardo Giacconi who significantly contributed to the construction of the first astronomical X-ray telescopes in the 60's and 70's. These telescopes achieve much better signal to noise ratio than X-ray experiments without optics, which allows for the detection of faint sources for example. The use of X-ray optics further allows imaging, precise localization, photometry, spectroscopy, variability studies, and an estimation of physical parameters of X-ray emitting regions (temperature, electron density...). Space borne X-ray optics are also well suited for monitoring the X-ray sky for variable and transient objects including X-ray novae, X-ray transients, X-ray flares of stars and AGNs, galactic bulge sources, X-ray binaries, SGRs (Soft Gamma Ray Repeaters) and X-ray afterglows of GRBs (Gamma Ray Bursts). X-ray optics are an important part of numerous past, recent, and future space projects (EXOSAT, ROSAT, Einstein, RT-4M Salyut 7, Fobos, AXAF/Chandra, XMM-Newton, ABRIXAS, BeppoSAX, ASCA, XEUS, Athena...). In the laboratory, there are numerous applications for X-ray optics, e.g. in plasma physics, laser plasma, synchrotron analyses, biology, crystallography, medicine, material and structure testing, X-ray lithography, etc. This chapter reviews and discusses X-ray imaging mirrors based on grazing incidence reflections which are alternatives to Wolter systems, with an emphasis on lobster eye optics. These systems are described elsewhere in the literature, but their space and astronomy applications are still marginally discussed. We review and discuss these systems, their past and recent ground-based applications, and their potential for future X-ray astronomy and laboratory applications. The grazing incidence reflecting X-ray lenses discussed in this work typically reflect from the optical to soft X-rays of 2 to 10 keV, depending on the reflecting surface material and on the angle of incidence. Since there are science goals that require higher energies, recent efforts have focused on various improvements and additional surface layers, such as multi-layers, to meet this. We give a brief introduction of the various types of the lobster eye X-ray optics in Section \ref{intro} where both Schmidt and Angel optics are discussed. In Section \ref{secmpo}, the lobster eye systems in the Angel arrangement, based on Micro Pore Optics (MPOs) are described. In Section \ref{secsch}, the wide-field systems of lobster eye optics in the Schmidt arrangement based on the Multi Foil Technology (MFO) are discussed in detail, with emphasis on prototypes already designed, developed and tested. Section \ref{KBop} addresses K-B X--ray optics and finally in Section \ref{sum}, we give a summary of the chapter including a comparison between the MPO and MFO technologies. \section{Lobster eye X-ray optics} \label{intro} \subsection{Introduction} Crustaceans eyes such as lobsters, shrimps and crayfish, provide an excellent opportunity for biomimicking and creating novel X-ray optics. Instead of a standard lens, a lobster's eye consists of a large number of square pores evenly distributed across a sphere, with each pore pointed towards a common centre. The light reflects off the very smooth walls of each pore and is focused onto the curved retinal surface. The retina and the pores form concentric spheres with the radius of curvature of the retina being half that of the pores. The ratio of the width of the pores to their length requires the light to be reflected at very shallow angles - as is required to focus X-rays (see Figure \ref{lob}). The lobster eye geometry for X-ray imaging was first introduced by Angel in 1979\cite{ang}, where he discussed the possibility of creating an X-ray telescope with an extraordinarily large FoV, and even an all sky imager consisting of a full spherical optic and a spherical detector. The design is based on a modification of the K-B \cite{kirk} (see section \ref{KBop}) arrangement, but using successive orthogonal surfaces which are coincident in space to reflect the X-rays instead of separate successive surfaces. This dictates the need for pores with a square cross section - exactly mimicking a lobster's eye. Angel's premise was that by replicating the geometry of crustaceans such as lobsters or crayfish, it would be possible to form an effective X-ray optic. The optics would need to be thin (1-2 mm thick), made up of regular, square cross section pores, which were evenly distributed over a spherical surface, with highly polished pore walls and an X-ray reflective coating. An alternative, but similar, arrangement had already been proposed by Schmidt in 1975\cite{Schmidt} (shown in Figure \ref{figsch}). Both of these arrangements will be discussed in full detail in the following sections. The wide-field mirror modules offer advantageous application in astrophysics. The major scientific achievements of X-ray astronomy in the recent past are closely related to the use of large X-ray imaging telescopes based mostly on the Wolter 1 X-ray objectives. These systems usually achieve excellent angular resolution as well as very high sensitivity, but are quite limited in the FoV available, which is less than 1 degree in most cases. However, the future of X-ray astronomy and astrophysics requires not only detailed observations of particular triggers, but also precise and highly sensitive X-ray sky surveys, patrol and monitoring. The confirmed X-ray counterparts of GRBs may serve as an excellent example. Detailed investigations of GRBs, in almost all cases, indicate the presence of variable and/or fading X-ray counterparts/afterglows. The X-ray identification of GRBs has lead to great improvements in our study and understanding of these sources. This has enabled identifications at other wavelengths due to better localization accuracy provided in X-ray, compared with gamma ray observations. Since most GRBs seem to be accompanied by X-ray emission, the future systematic monitoring of these X-ray transients/afterglows is extremely important. However, these counterparts are faint in most cases, requiring powerful wide FoV telescopes. An obvious alternative seems to be the use of wide FoV X-ray optics allowing the signal to noise ratio to be increased if compared to non-focusing optical systems. The desired limiting sensitivity for a lobster eye telescope is roughly $10^{-12} erg^{-2}s^{-1}$ which would allow for the detection of an order of magnitude greater number of sources to be observed over the course of a day \cite{Sve04}. This is consistent with the fluxes associated with X-ray afterglows of GRBs. Furthermore, the wide field X-ray telescopes play an important role in monitoring faint variable X-ray sources to provide better statistics of such objects (note e.g. the occurrence of two faint fading X-ray sources inside the gamma ray error box of GRB970616) as well as in other fields of X-ray astrophysics. The recent hunting for faint fading X-ray afterglows of GRBs has indicated that there is a large number of faint and/or variable X-ray sources worthy of detailed study. There have been many attempts to increase the available FoV coverage of Wolter and analogous X-ray telescopes. To avoid any confusion, we suggest to restrict the term “wide-field X-ray optics” only to optical systems with a FoV significantly larger than 1 degree, whilst using the term “narrow-field system” for systems with FoV less than 1 degree. In the following sections we describe in detail the two alternative approaches to the wide-field lobster eye optics, namely the Angel arrangement (based on MPO optics) and the Schmidt arrangement (based on MFO optics). \section{Lobster eye telescopes using Micro Pore Optics} \label{secmpo} \subsection{Introduction} As Angel\cite{ang} described, optics based on a lobster eye design could be used to create X-ray point-to-point imaging, focusing, collimation and beam splitters\cite{martin}. With a flat optic and equal optic-to-detector and optic-to-source distances, point-to-point imaging\cite{fraspie} of an unmagnified X-ray source is achievable. In this instance, the pores are parallel to one another and are arranged such that they are parallel to the line defined by the centre of the source to the centre of the detector. In this geometry, the expanding cone of X-rays from the source is focused onto the detector from a region of the optic defined by the critical angle for X-ray reflection. For imaging of sources, focusing is achieved when the optic is slumped with a spherical Radius of Curvature (RoC) and the detector is placed at the focal length where $F=RoC/2$. The convex surface of the optic is positioned towards the source\cite{fraspie2}. This geometry works for both finite and infinite source distances (e.g. planets, stars etc.) but where the optic to source distance is larger than the optic to detector distance. Collimation can be achieved by chemically roughening the channel walls for example, in this configuration a small FoV is defined, and rays from outside this filed of view are efficiently blocked, reducing the sky background\cite{Mineo}. Beam splitters are created by rotating a slumped optic so that the concave surface of the optic is towards the X-ray source. By setting the correct optic-to-source distance, a series of narrow parallel beams can be created, or by shortening the distance further, the beam can be radially and uniformly diverged. Micro Pore Optics (MPOs) are glass optics, typically 1-3 mm in thickness, with regular square pores. These pores are slumped to a spherical figure and coated to reflect X-rays. Usually, each MPO is 40 mm by 40 mm in size with 20 $\mu$m or 40 $\mu$m wide pores and with an open fraction of 60\% or higher. An example is shown on the left of Figure \ref{figmpo}. Using MPOs in a lobster eye geometry, in comparison to a traditional Wolter telescope\cite{wolt}, can provide a very large FoV with very low mass. As the pore apertures are so small the channels do not have to be long to achieve the required grazing angles and therefore reduces the mass of the optic. The size of the FoV of a lobster eye telescope depends only on the angular extent of the spherical optic and detector. A large lobster eye geometry can be realised by tessellating an array of smaller optic tiles, such as MPOs, over a larger spherical optic frame. If the frame has the same spherical figure as the slumped optics, then all channels will point to the centre of that sphere. Providing the optic is constructed with very small gaps between the tessellated MPOs for support structures, there is little to no vignetting and no change in the PSF over the FoV of the assembled optic. This optic design has been used and adapted for X-ray telescope missions such as the MXT on board SVOM\cite{gotz}, the SXI instrument on SMILE\cite{smile} and the WXT on the Chinese Einstein Probe\cite{ep} mission, and proposed missions including TAP\cite{tap} and Gamow\cite{gam}. The trade off with using this technology is the more modest angular resolution compared to traditional telescopes such as Chandra\cite{changman} and XMM-Newton\cite{2}. The Mercury Imaging X-ray Spectrometer (MIXS) instrument on BepiColombo\cite{mixs} used the same technology but in a different way. MIXS comprises of both a collimator and a telescope. The telescope uses optics with radially packed square pores, in concentric rings to approximate a Wolter geometry. Further details on MIXS are discussed in Section \ref{bep}. Section \ref{secmpo} gives a description of the workings of MPOs; detailing production, the formation of the unique characteristic Point Spread Function (PSF) and the calculation of the fundamental parameters for the mission specific energy band. The details and methodology of how to formulate a narrow field lobster eye optic arrangement are explained. Some of the issues and limiting factors of these optics are evaluated and finally, the current missions using lobster eye optics to fulfil their science objectives are reviewed. \subsection{MPOs - Production and design} At the time of Angel's paper\cite{ang}, Micro Channel Plates (MCPs), thin square glass plates with regular and smooth circular pores, were in use as photo multipliers\cite{woodhead,ruggi}, electron detectors and X-ray detectors\cite{wiza,mcp}. The similarities between the proposed optics needed by Angel to form lobster eye optics and MCPs was recognised and has since been extensively pursued by several authors \cite{theory, wilks, fraspie, kaa}. In order to differentiate between the glass plates used for detectors and those used for optics, the term MPOs has been adopted for the latter, whilst the former retained the name MCPs. The geometry of each MPO comprises a square packed array of microscopic pores, each with a square cross-section. The pores are arranged over a spherical surface with a RoC $R_{slump}$, where $F$ is the focal length of the MPO, as shown on the right of Figure \ref{geom1}. The ideal geometry is such that all the pores within the MPO point towards a common centre of curvature, with reflections from the walls of the pores producing an image on a spherical focal surface which is concentric with the spherical optic. The RoC of the spherical focal surface is equal to $F$, or half the RoC of the optic. Whilst the technology of MPOs for X-ray optics (collimation or focusing) differs from MCP technology in several ways, they are both formed from lead glass and the manufacturing of the two is very similar. A full description of the production of MCPs and MPOs can be found in Feldman et al \cite{thspie}. MPOs consisting of square pores with widths of 10 $\mu$m up to 720 $\mu$m can be fabricated but for the majority of applications, they tend to be either 20 $\mu$m or 40 $\mu$m and are either square packed and slumped like a lobster eye, e.g. SVOM's MXT and SMILE's SXI, or radially packed, e.g. BepiColombo's MIXS-T. The individual pore fibres are stacked to form square multifibres, the size of which are determined by the pore size. For example, an MPO made up of 40 $\mu$m pores will be stacked into square multifibres of 25 pores by 25 pores, which in turn will be stacked with $\sim$31 multifibres across in each direction to form an MPO of 40 mm by 40 mm. A single MPO formed as described with an aluminium film on the convex surface, is shown on the left of Figure \ref{figmpo}, and a microscope image of a complete multifibre is shown on the right. The lead glass used to form the MPOs can effectively reflect incoming X-rays with a flat MPO providing a reflection efficiency of $\sim$60-80\% at Al-K (1.49 keV). However, a slumped bare glass MPO, which is required for X-ray focussing, only reflects $\sim$20-40\% (measured at Al-K, 1.49 keV, and an angle of 55 arcmin off-axis) of incomming X-rays. It is not fully understood why the reflection efficiency decreases during the slumping process. In order to improve the reflection of incident X-rays, the pores are coated with iridium (10 $\pm$ 5 nm thick for MIXS and 25 $\pm$ 5 nm for the MXT) by a process of atomic layer deposition. Studies are being carried out to investigate nickel, platinum and multilayer coatings, to improve the X-ray reflection efficiency at specific energy bands relevant to different mission's science goals. On the front (convex) surface of the MPO, an aluminium film of 60-100 nm thickness is applied. This layer acts as both an optical blocking filter and as a thermal control surface. The PSF produced by a single MPO, and an array of MPOs, is very distinctive and unique, as shown in Figure \ref{geom2}. The cross feature formed by these optics comprises a focused spot, horizontal and vertical cross arms and a diffuse patch. The focused spot is created by rays which undergo 2 grazing incidence reflections off orthogonal sides of a single pore, as shown on the left of Figure \ref{geom1}. The vertical and horizontal cross-arms are caused by rays which undergo single or successive odd numbers of reflections off the pore walls, with single reflections off a pore wall creating the cross-arms before the first minima. The arms beyond the first minima are much more faint and are formed by multiple reflections at higher angles. A diffuse patch is created by rays which pass straight-through the MPO, or have undergone multiple even numbers of reflection. Figure \ref{geom2} demonstrates a simulated PSF generated by a single perfect MPO. For X-ray applications, a classic lobster eye telescope working in the photon energy range 0.2-10 keV, has an optimum $L/d$ ratio (length of pore $L$ and pore width $d$) of \textgreater 50, as the median grazing angle (in radians) for reflection within the pores is given by \begin{equation} \theta_{g}=d/L \sim 1. \end{equation} Willingale et al.\cite{willf} empirically gives the critical angle of the bare lead glass MPOs as: \begin{equation} \theta_{c}(E)=aE^{-1.04} \label{eq:crit} \end{equation} where $E$ is the photon energy and $a$=2.4 for $\theta_{c}$ in degrees and $E$ in keV. Equation \ref{eq:crit} demonstrates that the critical reflection angle is a function of the X-ray energy, as the energy increases, the critical angle decreases. This means that the area being used to focus X-rays of both individual MPOs and assembled optics, decreases as the energy increases. This is a significant consideration, as the energy being focused may influence the point spread function if there are variations in the quality of individual MPOs. At C-K (0.28 keV) $\theta_{c}$ is $\sim$9$^{o}$ whereas at Ti-K$_{\alpha}$ (4.51 keV), $\theta_{c}$ is $\sim$0.5$^{o}$, which is very significant, especially for short focal length optics. One example to consider is a 40 mm by 40 mm MPO with 40 $\mu$m pores coated in iridium, 1.2 mm thick and a radius of curvature of 600 mm, assuming an infinite source and the optic is on-axis. At C-K the angle at the edge of the MPO is 1.9$^{o}$ so the whole MPO will contribute to the PSF, but at Ti-K$_{\alpha}$ only the central $\sim$5 mm by 5 mm will. As the corners of the MPO tend to have the poorest form (see Section \ref{prod}) then the on-axis PSF Full Width Half Maximum (FWHM) tends to improve as the energy increases, as the influence of the corners reduces. The amount of an MPO which contributes to the PSF is also dependent on the length of the channels. If, in the example above, the MPO is 2.4 mm thick, or 1.2 mm thick with 20 $\mu$m channels, then the open angle of the channels is only 0.95$^{o}$. In this case only the central $\sim$10 mm x 10 mm area will provide significant flux in the PSF, as X-rays interacting outside this area will require twice as many reflections to contribute to the PSF. These effects must be taken into consideration when designing the optic for the mission specific energy band. \subsection{Design of a narrow-field optimised lobster eye telescope} For wide-field lobster eye designs, $L/d$ is constant across the whole optic aperture, with no specific on-axis point. In comparison, a narrow-field-optimised telescope, which has a varying thickness across the aperture, as described by Angel\cite{ang}, will provide the maximum effective area on-axis. An example of a narrow-field lobster eye optic is the SVOM Microchannel X-ray Telescope (MXT)\cite{gotz}. Ideally, the full optic would be formed of a continuous series of pores, slumped to a specific radius of curvature, with the length of the pores tapering towards the edges. However, it is not possible to manufacture a single MPO of the required size, nor with a varying thickness. Therefore the optic aperture is formed of smaller tessellated MPOs, with the pore length of each MPO determined by its position in the optic aperture. To create such an optic, the required pore length (or thickness) for each MPO needs to be calculated, where $r$ is the radial position of the centre of each individual MPO from the centre of the assembled optic aperture. \begin{equation} L = 2 \xi dF/r \label{eq:optThick} \end{equation} $L$ is calculated using Equation \ref{eq:optThick}. As $F$, the focal length, and $d$, pore diameter, are constant, the thickness of the MPO, or length of the pore, is determined by $r$, the radial position from the centre of the optic, and $\xi$, a scaling factor which is dependent on the energy and the quality (pore alignment, surface roughness etc.) of the MPOs. This indicates that within the optic aperture, the maximum $L/d$ of a single MPO is at the centre of the array, where $r$ is smallest, and reduces towards the outer edge of the assembled optic, where $r$ is largest. In order to calculate the required thickness, $L$, of each MPO at a given radial position across such an assembled optic, the value of $\xi$ is required. Ray traced simulations were completed at 1 keV to determine the optimum value of $\xi$, assuming 40 $\mu$m pores. The results are shown in Figure \ref{digr}. The top two panes of Figure \ref{digr} show that if $\xi$ is too low and therefore the MPO is too thin, then the number of double reflections decreases and are replaced by single reflections - reducing the area in the double reflection spot and increasing the FWHM. Similarly, if $\xi$ is large and the MPO is too thick, then the X-rays suffer many more multiple (\textgreater 2) reflections, again reducing the area in the double reflection spot. Figure \ref{digr} demonstrates how the variation of the value of $\xi$, and therefore the thickness of the MPOs, effects the area, gain and FWHM of the MPO. Here, the gain is a measure of the focusing power of the optic and is the ratio between the total collecting area and the area in the double reflection spot. As shown in Figure \ref{digr}, the optimum value of $\xi$ at 1 keV is 1.25. This simplifies Equation \ref{eq:optThick}, at 1 keV, to: \begin{equation} L = 2.5dF/r \label{eq:optThick2} \end{equation} \subsection{Limitations of MPOs} \label{prod} If you have a perfect MPO, which is perfectly spherical, with perfectly smooth pores all pointing to exactly the same position on a curved focal plane at exactly the correct focal distance, the minimum PSF size you could achieve would be the width of the pores at that optic-to-detector distance. The angular size of the pore at the focal length of the optic is the fundamental limit of the resolution of an MPO. If every pore is identical and all point to the same position on the focal plane, then the beam from each pore will pileup on top of each other, perfectly, to the width of a pore. In reality this is not the case and there are many deformations in the form of the MPO which limit the resolution. The full details of the majority of the deformations can be found in Willingale et al.\cite{dickspie}, but they are summarised here. The three intrinsic aberrations associated with the lobster eye geometry, which limit the angular resolution performance of the optic, independent of the technology used to construct the pore array, are; spherical aberrations, the geometric pore size and diffraction limits. Non intrinsic aberrations include slumping and formation of the multifibres. Slumping introduces additional radial tilt and shear errors as the pores are stretched and compressed to form the correct profile. Misalignment of multifibres to one another, deformations at the multifibre boundaries and within the multifibres themselves, contribute to the total angular resolution. In addition, the pore surface roughness further increases the angular resolution of the MPO. The combination of the above errors imposes a theoretical limit on the angular resolution of $\sim$2 arcmin for a single MPO, however, the majority of MPOs have an angular resolution far larger than this. The MPOs are slumped by a technique that sandwiches the MPOs between convex and concave diamond turned mandrels of the appropriate radius of curvature. Equal pressure and heat is applied to both mandrels and across the full surface in order to prevent the shearing of the channels with respect to each other. After slumping, the MPOs and mandrels are left to cool to room temperature which keeps the MPOs form. Unfortunately, trying to slump a square MPO onto a sphere causes deformations in the form of the optic. If you think of trying to wrap a basketball with a square piece of paper, at the centre the fit is very good but towards the corners you get crinkles and folds which distort your piece of paper. This is similar to what happens to an MPO and the end result is that the form of the corner regions of the optics is not as good as at the centre. You can also end up with an astigmatism in the optic where the radius of curvature in one axis does not match that in the other axes. Both of these effects have a massive influence on the net focal length of the optic and the PSF size and shape. At lower energies, the structure of the corners has a strong influence on the PSF but the astigmatism can affect the PSF at all energies. In addition to the effect of the slumping on individual MPOs, the variation of RoC between the MPOs combined within an assembly will have an affect on the full optic assembly PSF. The optic-to-detector distance of best focus for the optic assembly is governed by the RoC and the form of the frame, but the size of the PSF of the assembly is governed by the individual MPOs. If none of the MPOs have the same RoC as the frame, then they will all be out of focus by varying amounts and this will increase the size of the PSF. \subsection{Current Missions} Several missions over the next few years are using this technology in order to take advantage of the large FoV and light-weight nature of these optics for various scientific goals, including planetary science and astronomy. Below is a description of some of the current selected missions. \subsubsection{BepiColombo} \label{bep} The first instrument is the Mercury Imaging X-ray Spectrometer (MIXS)\cite{newmixs} on board the ESA-JAXA mission BepiColombo. Although it was launched in October of 2018, it will not insert into its scientific orbit around Mercury, its destination, until late 2025 - early 2026. MIXS consists of two instruments, the telescope MIXS-T and the wide field collimator MIXS-C, shown on the left and right respectively on the MIXS optical bench in Figure \ref{bepi}. MIXS-T uses the radial packing of 20 $\mu$m square pores and two consecutive sector MPOs, slumped with different radii of curvature to simulate a Wolter geometry\cite{willf}. In order to create the 1 m focal length, the front sectors have a RoC of 4 m and the rear sectors have a RoC of 1.3 m. The FoV of MIXS-T is $\sim$1.1$^{\circ}$ and consists of 36 tandem, sector pairs. The inner ring sectors have a thickness of 2.2 mm, the middle ring optics are 1.3 mm thick and the outer ring optics are just 0.9 mm thick. The MIXS-C instrument uses 20 $\mu$m, square pore, square packed MPOs which are 40 mm by 40 mm in size and 1.2 mm thick. These MPOs have been slumped to a radius of curvature of 550 mm and give a FoV of $\sim$10$^{\circ}$. The complete MIXS instrument on its optical bench weighs $\sim$11 kgs. By using these two instruments side by side, an elemental map of the Mercurian surface using X-ray fluorescence from the solar wind\cite{fras2} will be created. \subsubsection{SVOM} The Space-based multi-band astronomical Variable Objects Monitor (SVOM)\cite{kari} is a Chinese – French mission to be launched in 2023. It is comprised of four space borne instruments, including the Microchannel X-ray Telescope (MXT)\cite{gotz}. The MXT’s main goal is to precisely localize, and spectrally characterize X-ray afterglows of GRBs. The MXT is a narrow-field-optimised, lobster eye X-ray focusing telescope, consisting of an array of 25 square MPOs, with a focal length of 1.14 m and working in the energy band 0.2 - 10 keV. The design of the MXT optic (MOP) is optimised to give a 1$^{o}$ detector limited FoV but the optic has the unique characteristics of a lobster eye design, with a wide FoV $>$ 6$^{o}$, and a PSF which is constant over the entire FoV. The MPOs on the Flight Module (FM) MOP have a pore size of 40 $\mu$m giving the optimum thicknesses across the aperture of 2.4 mm in the centre and 1.2 mm at the edges. The left of Figure \ref{mxt} shows the completed FM MOP. Each MPO is 40 mm by 40 mm square and there is a 2 mm gap between each MPO on the frame. The total mass of the fully assembled optic was measured to be 1.43 kg. \subsubsection{Einstein Probe} Einstein Probe\cite{ep} is a Chinese Academy of Science (CAS) mission due for launch in 2023, with its primary goals to discover high-energy transients and monitor variable objects. The mission consists of two instruments, the Wide field X-ray Telescope (WXT), a lobster eye X-ray telescope consisting of twelve identical modules; and the Follow-up X-ray Telescope (FXT)\cite{fxtspie}, which is a traditional Wolter X-ray telescope. The FXT has been jointly developed by CAS, the European Space Agency (ESA) and the Max Planck Institute for Extraterrestrial Physics (MPE). Each of the WXT modules is comprised of 36 MPOs in a 6 by 6 array (left of Figure \ref{figep}), with a 375 mm focal length, a total FoV of more than 3600 square degrees, an angular resolution goal of 5 arcmin per module and working in the energy range of 0.5-4 keV. Each of the twelve WXT modules, has a focal plane comprised of 4 CMOS detectors in a 2 by 2 array. The modules are aligned so that each 3 by 3 quadrant of MPOs focuses onto a single CMOS detector, thus creating 4 discrete telescopes per module with overlapping FoVs (right of Figure \ref{figep}). \subsubsection{SMILE} Solar wind Magnetosphere Ionosphere Link Explorer (SMILE)\cite{smile} is a joint mission between ESA and CAS to investigate the dynamic response of the Earth's magnetosphere to the impact of the solar wind. From an elliptical polar orbit it will combine soft X-ray imaging of the Earth's magnetopause and magnetospheric cusps with simultaneous UV imaging of the Northern aurora, and will monitor in situ the solar wind and magnetosheath plasma conditions so as to set the imaging data into context. It is due for launch in late 2024 or early 2025 with 4 separate instruments on board, including the Soft X-ray Imager (SXI). The SXI is an elongated lobster eye telescope with an array of 4 by 8 MPOs. Each MPO is 40 mm by 40 mm, with iridium coated 40 $\mu$m pores and a focal length of 300 mm. The high charge state solar wind ions in collision with hydrogen produce photons at soft X-ray (and EUV) energies within the 0.2 keV to 2.5 keV band. The focal plane consists of 2 CCDs and the instrument has a FoV of 26.5$^{o}$ by 15.5$^{o}$. The wide FoV enables SXI to spectrally map the location, shape, and motion of Earth's magnetospheric boundaries. Figure \ref{figsmile} shows an exploded CAD diagram of the SXI instrument on the left, the structural thermal model of the full instrument during vibration testing on the top right, and a simulation of the data expected on the bottom right. \section{Lobster eye optics in MFO/Schmidt arrangement} \label{secsch} \subsection{Schmidt objectives} The lobster eye geometry X-ray optics offer an excellent opportunity to achieve very wide fields of view. One dimensional lobster eye geometry was originally suggested by Schmidt \cite{Schmidt}, based upon flat reflectors. The device consists of a set of flat reflecting surfaces. The plane reflectors are arranged in an uniform radial pattern around the perimeter of a cylinder of radius R. X-rays from a given direction are focused to a line on the surface of a cylinder of radius R/2 (Fig. 2). The azimuthal angle is determined directly from the centroid of the focused image. At glancing angle of X-rays of wavelength 1 nm and longer, this device can be used for the focusing of a sizable portion of an intercepted beam of parallel incident X-rays. Focusing is not perfect and the image size is finite. On the other hand, this type of focusing device offers a wide FoV, of up to a maximum of the half sphere of the coded aperture. It is possible to achieve an angular resolution on the order of one tenth of a degree or better. Two such systems in sequence, with orthogonal stacks of reflectors, form a double-focusing device. Such a device offers a FoV of up to 1000 square degrees at a moderate angular resolution. It is obvious that this type of wide-field X-ray telescope could play an important role in future X-ray astrophysics. These innovative very wide field X-ray telescopes have only recently been suggested for space-based applications. One of the first proposals was the All Sky Supernova and Transient Explorer (ASTRE, Gorenstein \cite{Goren79}\cite{Goren87}). This proposal included a cylindrical coded aperture outside of the reflectors, which provide angular resolution along the cylinder axis. The coded aperture contains circumferential open slits that are 1 mm wide and are in a pseudo-random pattern. The line image is modulated along its length by the coded aperture. The image is cross-correlated with the coded aperture to determine the polar angle of one or more sources. The FoV of this system can be, in principle, up to 360$^{\circ}$ in the azimuthal direction and nearly 90\% of the solid angle in the polar direction. To create this mirror system, a development of double-sided flats is necessary. There is also potential for extending the wide field imaging system to higher energy with the application of multi-layers or other coatings in analogy to those described for flat reflectors in the K-B geometry. The angular resolution of the lobster eye optics in the Schmidt arrangement is a function of spacing between the reflecting plates and focal length. In the Schmidt arrangement, the lobster eye consists of plates of thickness $t$, and depth $d$ (Fig. 12). Spacing between plate planes is $s$, focal length $f$, radius $r$, focal point $F$, and $\beta$ is the angle between optical axis and focused photons. Beam A (Fig. 12) shows the situation where the plate is fully illuminated and the crossection of the plate is maximal (effective reflection). Beam B is the last beam that can be reflected into the focal point. Beams that are further from the optical axis reflect more than once (critical reflection). If reflected twice from the same set of plates, the photon does not reach the focal point and continues parallel to the incoming photon direction \cite{Sve03}. If $t \ll s \ll d \ll f$ we can derive the following simple equations\cite{Sve03} \cite{Inne01}, where $\alpha$ is the estimate of the angular resolution. \begin{center} $f =\frac{r}{2}$ $\beta _E = \frac{(s-t)}{d}$ $\beta _L = 2 \beta _E$ $\alpha \sim\frac{2s}{r} =\frac{s}{f}$ \end{center} The design concept is different for lobster eye systems based on two reflections, a single reflection on a horizontally oriented surface (pore wall or mirror) and a single reflection on a horizontally oriented surface. Particularly, this is a case of Schmidt lobster eye. A paper by Tichy et al. \cite{Tichy2019} presents analytical formula allowing direct computing of the effective collecting area for those systems by the formula \begin{equation} L(r,s,t,\zeta) = 2r \frac{s}{s+t} \frac{ {\widetilde{\cal R}}(2 \zeta) - 2 {\widetilde{\cal R}}(\zeta) + {\widetilde{R}}(0) } { \zeta}, \end{equation} where ${\widetilde{\cal R}}(\theta):= \int \int {\cal R}(\theta) \mbox{d} \theta \mbox{d} \theta =\int {\bar{\cal R}}(\theta) \mbox{d} \theta $ is an arbitrary second antiderivative of $\cal R$. Radius of the system measured to mirror center is denoted $r$, $s$ represents mirror spacing (or pore width) and $t$ is mirror (pore wall) thickness. The effective collecting area equals $L^2$ for the Angel system and $L_1L_2$ for the Schmidt system, where $L_1$ and $L_2$ are related to individual mirror stacks as they have different radii and they may differ in other parameters. The value $\zeta$ is the ratio between mirror (pore) depth $d$ and $s$. The optimal value of this ratio is given by the reflectivity function for given surface and photon energy only. The paper by Tichy et al. \cite{Tichy10b} presents the detailed procedure for how the optimal value of this ratio can be analytically calculated. In addition, a paper by Tichy and Willingale \cite{Tichy2018} presents a formula for the optimal value of $s$ as \begin{equation} s=-t+ \sqrt{2Rt \zeta +t^2} \end{equation} Here, $R=(r+d/2)$ is the radius of the system measured to the front aperture (d is the mirror depth). This solves a common problem when focal length is limited e.g. by available space in a spacecraft. Mirror (pore wall) thickness $t$ should be as small as possible but must be large enough to achieve sufficient stress endurance, etc. The 1D and 2D lobster eye Schmidt modules are illustrated in Figs. 14 and 17. To test the design and assembly of lobster eye modules in Schmidt geometry, various test modules were manufactured and tested (Table 1). The first lobster eye X-ray Schmidt telescope prototype (midi) consisted of two perpendicular arrays of flats (36 and 42 double-sided flats 100 mm x 80 mm each). The flats were 0.3 mm thick and gold-coated. The focal distance was 400 mm from the midplane. The FoV was about 6.5 degrees (Fig. 13). The results of optical and X-ray tests indicated a performance close to those provided by mathematical modelling (ray-tracing). X-ray testing was carried out in the test facility of the X-ray astronomy group at the University of Leicester. At a later date, test modules with a Schmidt geometry were designed and developed using 0.1 mm thick gold coated glass plates that were 23 mm x 23 mm, with a 0.3 mm spacing. The aperture/length ratio is 80. A single module has 60 plates. Two analogous modules represent the 2D system for laboratory tests, providing focus to focus imaging with focal distances of 85 and 95 cm. The innovative gold coating technique resulted in a final surface micro roughness rms to 0.2-0.5 nm. Various modifications of this arrangement have been designed both for imaging sources at final distances (for laboratory tests) as well as for distant sources (the corresponding double--focusing array has f = 250 mm and FoV = 2.5 deg). In parallel, numerous ray-tracing simulations have been performed, allowing for a comparison between theoretical and experimental results. \begin{table}[h!] \centering \begin{tabular}{||c c c c c c c c c c||} \hline Module & Size & Plate thickness & Distance & Length & Eff. angle & f & Resolution & FoV & Energy \\ [0.5ex] \hline\hline Macro & 300 & 0.75 & 10.8 & 300 & 0.036 & 6 000 & 7 & 16 & 3 \\ Middle & 80 & 0.3 & 2 & 80 & 0.025 & 400 & 20 & 12 & 2 \\ Mini 1 & 24 & 0.1 & 0.3 & 30 & 0.01 & 900 & 2 & 5 & 5 \\ Mini 2 & 24 & 0.1 & 0.3 & 30 & 0.01 & 250 & 6 & 5 & 5\\ Micro & 3 & 0.03 & 0.07 & 14 & 0.005 & 80 & 4 & 3 & 10 \\ [1ex] \hline \end{tabular} \caption{The parameters of selected test Schmidt lobster eye modules assembled and tested. The distance parameter means the separation between reflecting foils. The parameters size, plate thickness, distance, length and focal distance f are given in mm, resolution in arcmin, FoV in degrees and optimal energy in keV.} \label{table:1} \end{table} Following the aforementioned developments, even smaller (micro) lobster eye modules were constructed and tested in both visible light and X-rays. As an example, we show X-ray test results for the mini and micro lobster eye modules (Fig. 21). These results show the on-axis and off-axis imaging performance of the lobster eye module tested. For mosaics of X-ray test images for various energies see Fig. 22 and for various off-axis angles at 4.5 keV see Fig. 23. \subsubsection{Substrates for lobster eye lenses in Schmidt/MFO arrangement} In general, there is growing need for large segmented X-ray foil telescopes of various geometries and geometrical arrangements. The requirement of minimizing the weight of future large X-ray space telescopes and at the same time achieving large collecting area for future large astronomical telescopes can be met with thin X-ray-reflecting foils (i.e., thin, lightweight, multiple layers that can be easily nested to form precise high-throughput mirror assemblies). This includes the large modules of the Wolter 1 geometry, the large Kirkpatrick-Baez (further referred as K-B) modules (as they can play an important role in future X-ray astronomy projects as a promising and less laborious to produce alternative) as well as the large lobster eye modules in the Schmidt arrangements. Although these particular X-ray optics modules differ in the geometry of foils/shells arrangements, they do not differ much from the point of the view of the foils/shells production and assembly, and also share all the problems of calculations, design, development, weight constraints, manufacture, assembling, testing, etc. It is evident that these problems are common and rather important for the majority of the large aperture X-ray astronomy space-based observatories. Most of the future space projects require light material alternatives \cite{Hud2011}. We (Czech team with participation of the first author of this chapter) have developed various prototypes of the above mentioned X-ray optics modules based on high quality X-ray reflecting gold coated float glass foils \cite{Hud00}. The glass represents a promising alternative to electroformed nickel shells used in Wolter optics, the main advantage being much lower specific weight (typically 2.2 g cm$^{-3}$ if compared with 8.8 g cm$^{-3}$ for nickel). For the large prototype modules of dimensions equal to or exceeding 30 x 30 x 30 cm, mostly glass foils of thickness of 0.75 mm have been used, although in the future this thickness can be further reduced down to 0.3 mm and perhaps even less (we have successfully designed, developed and tested systems based on glass foils as thin as 30 microns, albeit for much smaller sizes of the modules). More recently, Silicon wafers with superior flatness and micro-roughness are serving as alternative substrates for lobster eye MFO modules. The recent HORUS experiment can serve as an example. HORUS has 4 modules, 2 modules with Au surface, 2 modules with Ir surface, each module has 17 silicon foils, i.e. in total 4 x 17 Si wafers 0.625 mm thick, with an aperture of 85 x 65 mm f=2 m. The goal is to experimentally compare different reflective layers (Fig. 24). These substrates, both glass foils and silicon wafers, can be used in various X-ray optics arrangements using MFO technology, mostly lobster eye and K-B \cite{Hud2009}\cite{Hud2016c}. \subsubsection{The application and the future of lobster eye telescopes in Schmidt arrangements} It is obvious that the lobster eye Schmidt MFO prototypes confirm the feasibility to design and develop these telescopes with currently available technologies. Considerations for fabricating and assembling a wide-field space-based X-ray observatory include: (1) Reduction of the micro-roughness and slope errors of the reflecting surfaces to optimize the angular resolution and reflectivity/efficiency of the system. The past development has already led to significant micro roughness improvement (to 0.2-0.5 nm for glass substrates and 0.1 nm for silicon substrates) (2) The design and construction of larger or multiple modules to achieve a larger FoV (of order of 1000 square degrees and/or more) and enhance the collecting area (3) Reduction in the spacing and plate thickness (Schmidt arrangement) to improve imaging performance (angular resolution and system efficiency) and (4) Advanced, alternative layer applications, and/or other approaches applied to the reflecting surfaces to improve the reflectivity and to extend the energy bandpass to higher energies. The application of very wide field Schmidt MFO X-ray imaging systems could be without doubt very valuable in many areas of X-ray and gamma-ray astrophysics. Results of analyses and simulations of lobster eye X-ray telescopes have indicated that they will be able to monitor the X-ray sky at an unprecedented level of sensitivity, an order of magnitude better than any previous X-ray all-sky monitor. Limits as faint as 10$^{-12}$ erg cm$^{-2}$ s$^{-1}$ for daily observation in the soft X-ray range (typically 1-10 keV) are expected to be achieved, allowing monitoring of all classes of X-ray sources, including X-ray binaries, fainter classes such as AGNs, coronal sources, cataclysmic variables, as well as fast X-ray transients including GRBs and the nearby Type II supernovae\cite{Hud2012}\cite{Hud2018c}\cite{Hudec2006c} . For pointed observations, limits better than 10$^{-14}$ erg cm$^{-2}$ s$^{-1}$ (0.5 to 3 keV) could be obtained, sufficient enough to detect X-ray afterglows to GRBs \cite{Sve04}\cite{Hud2013}. \subsubsection{Lobster eye Laboratory Modifications} The lobster eye soft X-ray optics, originally proposed and designed for astronomical (space) applications, has potential for numerous laboratory applications. As an example, lobster eye optics can be modified for efficient collection of laser-plasma radiation for wavelengths longer than 8 nm \cite{Bart}. The optics for this application consist of two orthogonal stacks of ellipsoidal mirrors forming a double-focusing device \cite{Bart}. The ellipsoidal surfaces were covered by a layer of gold that has relatively high reflectivity at the wavelength range that is 8-20 nm up to an incident angle of around 10 degrees. % The width of the mirrors forming the optics assemblies is 40 mm. As can be noticed, the spacing between adjacent mirrors increases with the distance from the axis. The curvature of the mirrors and the spacing between them were optimized using ray tracing simulations to maximize the optics aperture and to minimize the size of the focal spot. \subsubsection{Hybrid lobster eye} The lobster eye Schmidt MFO configuration described in the previous sections is a wide-field, relatively low angular resolution optics. Achieving finer angular resolution is challenging given the current limitations of the technological limitation related to the mirror thickness and minimum spacing \cite{Sve05}. One possible solutions to improving angular resolution is to invoke the typical use case of the standard lobster eye configuration as an All Sky Monitor (ASM) for X-ray Astronomy. The lobster eye is used onboard a space-based platform and will continuously scan the sky. If an area of the sky is outside the FoV of the optics, it will be inside the FoV sometime later because of scanning. This operational scenario allows for a smaller FoV in the scanning direction, which in-turn permits finer angular resolution. The desired optics would have a wide FoV and moderate angular resolution in one direction, and a smaller FoV and better angular resolution in another. It is necessary to use curved mirrors to achieve better angular resolution. However, this puts constraints on the mirror dimensions. A combination of the standard one-dimensional lobster eye optics in one direction and K-B parabolic mirrors in the other direction meet the desired requirements \cite{Sve05}, shown in Fig. 25. Preliminary results of this configuration indicate that the Hybrid lobster eye works as intended, i.e. it improves the angular resolution in one direction while still having a wide FoV in another. However, the blurring increases rapidly with the off-axis distance in the direction where there is focusing from the parabolic mirrors. Consequently, it is reasonable to think about such optics for pointed observations if the source and/or image are expected to be highly asymmetric. The effect of blurring is reduced for scanning observations, hence the increase in angular resolution is achievable. There is a loss of sensitivity with this configuration, which translates to a significant decrease in the limiting flux. This fact, combined with manufacturing difficulties, makes this configuration of limited use for space-based applications. However, there is potential for use in laboratory applications\cite{Sve06b}. \subsection{Space experiments with lobster eye MFO X-ray optics} The lobster eye optics in the Schmidt/MFO arrangement was placed on-board the Czech nanosatellite VZLUSAT-1 and on-board the NASA Water Recovery Rocket experiment. More systems are in study and/or in preparation for future space missions. For example, the HORUS double test module was designed and tested recently in order to compare modules with various reflective layers, see Fig. 24 \cite{Stehlik21}. \subsubsection{VZLUSAT-1} The small lobster eye telescope onboard the VZLUSAT–1 nanosatellite uses the first lobster eye MFO Schmidt X–ray optics in space. The first Czech technological CubeSat satellite VZLUSAT-1 was designed and built during the 2013 to 2016 period. It was successfully launched into Low Earth Orbit at an altitude of 505 km on June 23, 2017 as part of international mission QB50 onboard a PSLV C38 launch vehicle. The satellite was developed in the Czech Republic by the Czech Aerospace Research Centre, in cooperation with Czech industrial partners and universities \cite{Daniel16}. The payload fits into a 2U CubeSat (extended to 3U in space) and includes a 1D \cite{Pina15} \cite{Pina16} miniature X-ray telescope with a Timepix detector in its focal plane \cite{Baca16}. The main mission goal is the technological verification of the system \cite{Urban17} \cite{Daniel16}. However, there is potential for science as the telescope will view bright celestial sources as part of its mission \cite{Blazek}. The satellite represents the 5th satellite in space with Czech X–ray optics onboard. The 1D lobster eye module onboard VZLUSAT–1 has focal length of 250 mm and is composed of 116 wedges and 56 reflective double-sided gold-plated foils (thickness 145 microns). The input aperture is 29×19 mm$^{2}$, outer dimensions are 60×28×31mm$^{3}$. The active part of the foils is 19mm in width and 60 mm in length and the energy range is 3 to 20 keV. Images of the optics are shown in Fig. 26. \subsubsection{REX Rocket Experiment} The Rocket EXperiment 1 (REX1) was a secondary payload instrument on the Water Recovery X-ray Rocket (WRX-R) experiment. WRX-R was launched from the Kwajalein Atoll in the Marshall Islands on 4 April 2018. WRX-R was the first astrophysics sounding rocket mission to use a newly developed NASA water recovery system for astronomical payloads as an cost effective alternative to typical land recoveries that also may result in payload damage \cite{WRX}. The WRX-R was led by the Pennsylvania State University (PSU), USA. The primary payload was the soft X-ray spectroscope of PSU. WRX-R's primary instrument was a grating spectrometer that consisted of a mechanical collimator, an X-ray reflection grating array, a grazing incidence mirrors, and a hybrid CMOS detector. The Czech team provided the REX1 optical instrument as a secondary payload \cite{Urban21} \cite{Daniel17} \cite{Daniel19}. It was the first time that an X-ray lobster eye telescope was flown in a rocket experiment to observe an astrophysical object. The design of the REX1 instrument for the WRX-R was based on the concept of an optical baffle, which is normally used for NASA Sounding rocket experiments. This is a simple construction of a quill-shaped boulder with the anchor on one side of the block base, where the baffle is attached to the sounding rocket. The REX1 optical instrument consisted of two parts - vacuum chamber and hermetically sealed box. The vacuum part contained two (one 1D and one 2D) X-ray telescopes with Timepix pixel detectors \cite{Pina19}. The modules were assembled using Multi-Foil Technology (MFT). The material of the housing of the optical module was an aluminum alloy. The 1D X-ray lobster eye system with a focal length of 250 mm, had a FoV of 3.3 x 2.0 degrees and spanned the spectral range from 3 keV to 20 keV. The 1D lobster eye module was composed of 116 wedges and 56 reflective double-sided gold-plated glass foils (thickness of 145 $\mu$m). The gold coating allows the material to reflect incoming X-ray photons that have shallow incident angles of 0.5 deg or less. The input aperture was 29 x 19 mm$^{2}$, while the outer dimensions were 60 x 28 x 31 mm. The active area of the module was 19 mm in width and 6 mm in length and the energy range was 3 to 20 keV. The second lobster eye telescope was a 2D X-ray system with a focal length of 1065 mm. The FoV of this system was 0.8 x 0.8 deg with spectral range from 3 keV to 10 keV. The 2D lobster eye X-ray optics of REX was composed of two 1D sub-modules where one-sided gold-plated glass foils were in the vertical plane of the horizontal arrangement. Each sub-module consisted of 55 pieces of thin at glass foils (thickness of 0.34 mm) which were arranged so that the focal length was around 1.0 meter. The external dimensions of the module was approximately 80 x 80 x 170 mm. Both REX1 lobster eye modules can be seen in the Fig. 27. The 2nd generation of the optical system for the Rocket Experiment (REX2) is currently under study \cite{Pina21}. This optical device is based on the successful mission REX1 described above. The purpose of REX2 is to verify the X-ray optical system that consists of a wide-field 2D X-ray lobster eye assembly with an uncooled Quad Timepix3 detector (512x512 px @ 55 microns and spectrometer (active area 7 mm$^{2}$, resolution 145 eV @ 5.9 keV ). The 2D X-ray lobster eye optics is a combination of two 1D lobster eye modules with a focal length of up to 1 m and a FoV larger than 4.0 x 4.0 deg. The proposed optical system has imaging capabilities (2.5 to 20 keV) and spectroscopy capabilities (0.2 to 10 keV). The optical system was recently tested in an X-ray vacuum chamber \cite{Pina21}. \section{Kirkpatrick-Baez Optics} \label{KBop} In this section we briefly describe Kirkpatrick-Baez (K-B) X-ray optics. From the standpoint of manufacturing, there is significant number of similarities to lobster eye optics in MFO Schmidt arrangements as both are based on multiple thin foils. Although the Wolter systems are generally well known, Hans Wolter was not the first who proposed X-ray imaging systems based on the reflection of X-rays. In fact, the first grazing incidence system to form a real image was proposed by Kirkpatrick and Baez in 1948\cite{kirk}. This system consists of a set of two orthogonal parabolas in the configuration shown in Figure 28. The first reflection focuses to a line, which the second surface focuses to a point. This was necessary to avoid the extreme astigmatism suffered by a single mirror but was still not free from geometric aberrations. The system is nevertheless attractive for the ease of constructing the reflecting surfaces. These surfaces can be produced as flat plates and then mechanically bent to the required curvature. In order to increase the aperture, a number of mirrors can be nested together, but it should be noted that such nesting introduces additional aberrations. This configuration is used mostly in experiments not requiring large collecting area (solar, laboratory). Recently, however, large modules of K-B mirrors have been suggested also for stellar X-ray experiments \cite{Hud18} \cite{Hud2018b}. Despite this fact, astronomical X-ray telescopes flown so far on satellites mostly used the Wolter 1 type optics. However, K-B was used in several rocket experiments in the past, and in addition to that, they were proposed and discussed for use on several satellite experiments. Alternately, in the lab, K-B systems are in frequent use, e.g. at synchrotron facilities. In order to increase the collecting area (the frontal area), a stack of parabolas of translation can be constructed for astrophysical applications. However, in contrast to the single double-plate system, the image of a point-like source starts to become increasingly extended in size as the number of plates involved increases. Wolter type I telescopes bend the incident ray direction two times in the same plane, whereas the two bendings in K-B systems occur in two orthogonal planes, which for the same incidence angle on the primary mirror requires a longer telescope \cite{Aschen}. \subsection{K-B systems in astronomical applications} As an alternative to Wolter optics based instruments, Van Speybroeck et al.\cite{Van} designed several K-B telescope configurations that focus the x rays with sets of two orthogonal parabolas of translation. According to Van Speybroeck et al.\cite{Van}, the crossed parabola systems should find application in astronomical observations such as high sensitivity surveys, photometry, and certain kinds of spectroscopy where a large effective area rather than high angular resolution is the most important factor. The design of a K-B grazing incidence X-ray telescope to be used to scan the sky, would allow for the distribution of the reflected X rays and spurious images over the FoV to be analyzed. Kast\cite{Kast} has shown that in order to obtain maximum effective area over the FoV, it is necessary to increase the spacing between plates for a scanning telescope as compared to a pointing telescope. Spurious images are necessarily present in this type of lens, but they can be eliminated from the FoV by adding properly located baffles or collimators. X-ray telescopes of the type suggested by Kirkpatrick and Baez \cite{kirk} have several advantages over other types of X-ray telescopes for a general sky survey for low-energy X-ray sources. These telescopes use two orthogonal sets of nested parabolas of translation (perpendicular to one another) to provide 2D focusing of an X-ray image. Although their angular resolution for axial rays is somewhat worse compared with telescopes using successive concentric figures of revolution, they can be constructed more easily and have greater effective area \cite{Van}. Note that more recent papers give somewhat different findings, namely that the K-B Si stacks provide an alternative solution with a reduced on-axis collecting area but wider field of view and comparable angular resolution \cite{Willi10}. In either case, these telescopes, in general, can be constructed more easily. The design of K-B-type telescopes has been discussed by several authors e.g. \cite{Van} \cite{Goren73}\cite{Weissk}, and results have been reported from several experiments using 1D focusing from a single set of plates \cite{Goren71}\cite{Catura}\cite{Borken}. For a more recent status see \cite{Hud10} \cite{Hud18}. \subsubsection{K-B as a segmented mirror} Segmentation can also be applied to an array of K-B stacked orthogonal parabolic reflectors (Figure 29). As shown in Figure 29, a large K-B mirror can be segmented into rectangular modules of equal size and shape \cite{Goren96}. A segmented K-B telescope has the advantage of being highly modular on several levels. All segments are rectangular boxes with the same outer dimensions. Along a column, the segments are nearly identical and many are interchangeable with each other. All reflectors deviate from flatness only slightly. On the other hand, the Wolter reflectors are highly curved in the azimuthal direction and the curvature varies over a wide range. Furthermore, within a segment, the K-B reflectors themselves can be segmented along the direction of the optical axis. As shown in Figure 29, a K-B mirror system can be folded more easily than the Wolter mirror into a compact volume for launch and deployment in space. The examples of assembled K-B modules based on superior quality gold coated Si wafer substrates are illustrated in Figure 30. \subsubsection{K-B in Astronomical Telescopes: Recent Status and Future Plans} The first attempt to create an astronomical K-B module with silicon wafers was reported by Joy et al. \cite{Joy}. A telescope module that consisted of 94 silicon wafers with diameter of 150 mm, uncoated, with thickness of 0.72 mm was constructed. The device was tested both with optical light and with X-rays. The measured FWHM was 150 arc-seconds, which was dominated by large-scale flatness. It should be noted that the surface quality and flatness of Si wafers has improved since this time. Recent efforts towards supporting future larger and precise imaging astronomical X-ray telescopes require re-considering both the technologies and mirror assembly design. Future large X-ray telescopes require new light-weight and thin materials/substrates such as glass foils and/or silicon wafers\cite{Hud15c}. Their shaping to small radii, as required in Wolter designs, is not an easy task. While the K-B arrangements have potential to represent a less laborious and hence less expensive alternative because of (i) no need of mandrels (ii) no need of polishing and (iii) no need of bending to small radii. The use of K-B arrangement for the proposed IXO project (the proposed joint NASA/ESA/JAXA International X-ray Observatory) was suggested and investigated by Marsikova et al.\cite{Marsik09}, Hudec et al. \cite{Hud2011}, and by Willingale and Spaan \cite{Willi10}. These investigations indicate that if superior quality reflecting plates were used and the focal length is large, an angular resolution of order of a few arcsec could be achieved. Recent simulations further indicate that in comparison to Wolter arrangement, the K-B optics exhibit reduced on axis collecting area but larger FoV, at comparable angular resolution \cite{Willi10}. A very important factor is the ease of constructing highly segmented modules based on multiply nested thin reflecting substrates if compared with Wolter design. While e.g. the Wolter design for future large space X--ray telescopes such as Athena requires the substrates to be precisely formed with curvatures as small as 0.25 m, the alternative K-B arrangement uses almost flat or only slightly bent sheets. Hence, the feasibility to construct a K-B module with the required Athena 5 arc-second HEW resolution at an affordable cost is, in principle, lower than the cost of a Wolter arrangement. Note however that in order to achieve the comparable effective area, the focal length of K-B system is required to be about twice of the focal length of Wolter system\cite{Marsik09}\cite{Hud10}\cite{Hud2011}. \section{Summary} \label{sum} The grazing incidence X-ray optical elements of non-Wolter type (lobster eye and Kirkpatrick-Baez) offer alternative solutions for many future space- and lab-based applications. They can offer cheaper, and/or lighter alternatives, and also a much larger FoV. At the same time, new computer-based systems allow us to consider alternative designs and arrangements \cite{Nentv17}. Although both Angel and Schmidt designs were suggested in the 70's, both have seen rapid development over the past few years with MPO optics in an Angel arrangement already on selected missions and the Schmidt design using MFOs being proven on rocket and CubeSat experiments. A direct and reliable comparison between MFO and MPO designs of lobster eye X-ray optics is difficult, as in both cases the real optics performance deviates from the theoretical. The necessary slumping of the MPOs introduces additional sources of error\cite{Bannister,dickspie}, whilst the MFO design is harder to assemble. Both designs differ in geometry using both Angel and Schmidt designs, and require different manufacturing and assembling technology. The MFO technology enables a larger effective area with easy deposition of reflective layers, whilst the MPOs are lighter and are easier to assemble into a large array. The effective area at 10 keV for MFOs is higher than for MPOs although alternative coatings are being investigated for MPOs to improve the higher energy response. The prototypes developed and tested for both arrangements confirm that these light weight telescopes are fully feasible and can achieve angular resolutions of several arcmin or better over a very wide FoV. While both provide a more modest angular resolution compared to Chandra\cite{1} and XMM-Newton\cite{2} for example, they can still be used to help solve pressing questions in X-ray astrophysics, and can also be used for other applications such as within laboratories. K-B optics have already found wide applications in synchrotrons, and have demonstrated their performance and advantages. \section{Acknowledgements} The authors wish to thank the other members of their research groups. The research leading to these results has received funding from the European Union’s Horizon 2020 Programme under the AHEAD2020 project (grant agreement n. 871158)

Title: Listening to Celestial Algebras

Abstract: In this essay, we immerse into the framework of normed division algebras as a suitable arena to accommodate the standard model of elementary particles, and we explore some applications to cosmology. Remarkably, they permit interesting non-trivial realisations of the cosmological principle with an interplay between the symmetry groups of the quaternions and octonions. We also argue how these realisations give rise to potentially observational signatures in gravitational waves astronomy.

https://export.arxiv.org/pdf/2208.05267

\newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{eqnarray}} \newcommand{\ea}{\end{eqnarray}} \newcommand{\R}{{\mathbb R}} \newcommand{\Z}{{\mathbb{C}}} \newcommand{\Ha}{{\mathbb H}} \newcommand{\Oc}{{\mathbb O}} \newcommand{\algebras}{\R\otimes\Z\otimes\Ha\otimes\Oc} \newcommand{\eh}{\text{e}} \newcommand{\eo}{\text{f}} \newcommand{\ii}{\text{i}} \newcommand{\norm}[1]{\| #1\|} \def\d{\mathrm{d}} \newcommand{\nuM}{\hat{\nu}} \newcommand{\cM}{\hat{C}} \newcommand{\pM}{\hat{N}} \newcommand{\mM}{\hat{M}} \newcommand{\ssl}{\mathfrak{sl}} \newcommand{\highlight}[1]{\colorbox{yellow}{#1}} \firstpage{1} \makeatletter \setcounter{page}{\@firstpage} \makeatother \pubvolume{8} \issuenum{8} \articlenumber{0407} \pubyear{2022} \copyrightyear{2022} \datereceived{19 May 2022} \dateaccepted{28 July 2022} \datepublished{3 August 2022} \hreflink{} % \pdfoutput=1 \Title{Listening to Celestial Algebras} \newcommand{\orcidauthorA}{0000-0002-3053-9213} % \newcommand{\orcidauthorB}{0000-0002-9403-8565} % \Author{{Jose} Beltr\'an Jim\'enez $^{1}$\orcidA{} and Tomi S. Koivisto $^{2,3}$\orcidB{} } \address{% $^{1}$ \quad Departamento de F\'isica Fundamental and IUFFyM, Universidad de Salamanca, {37008} Salamanca, Spain;\\ $^{2}$ \quad Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia.\\ $^{3}$ \quad National Institute of Chemical Physics and Biophysics, Rävala pst. 10, 10143 Tallinn, Estonia.} \corres{\href{mailto:jose.beltran@usal.es}{jose.beltran@usal.es}; \href{mailto:tomi.koivisto@ut.ee}{tomi.koivisto@ut.ee}} \abstract{\textls[-25]{In this essay, we immerse into the framework of normed division algebras as a suitable arena to accommodate the standard model of elementary particles, and we explore some applications to cosmology. Remarkably, they permit interesting non-trivial realisations of the cosmological principle with an interplay between the symmetry groups of the quaternions and octonions. We also argue how these realisations give rise to potentially observational signatures in gravitational waves astronomy.}} \keyword{division algebras; quaternions; octonions; cosmology; gravitational waves} \begin{document} \section{Introduction} One of the wonders of the natural world resides in its compliance to being described in a mathematical language. In~this language, numbers form the channel to communicate with Nature, and one could even assert that science (and physics in particular) is about obtaining theoretical predictions expressed in terms of numbers that can eventually be compared to observational measurements. Natural numbers allow us to count things, and that is why they were the first ones to make an appearance as a practical construct. Soon, negative natural and rational numbers claimed their position for practical purposes. Beyond~this point, it would seem that nothing more was necessary for the primordial mathematical needs of the first societies. However, when geometry had to be included in the mathematical machinery, new numbers such as $\sqrt{2}$ and $\pi$ also called their role to measure lengths and surfaces. Subsequent needs and mathematical consistency gave the leading role in this play to the real numbers $\R$ and, not coincidentally, they were the first field to be discovered. They are a continuum completion of the more intuitive rational numbers and form an algebra that possesses all the nice properties that one could ask for, namely: it is ordered, commutative and associative. Moreover, one can define a norm and a well-defined division, thus making it a normed division algebra. Nevertheless, the reals fail in one important aspect: polynomial equations with real coefficients do not necessarily admit real solutions. This innocent observation prompted the introduction of the imaginary unit $\ii$ and led to the discovery of the complex numbers $\Z$ that extend $\R$ into a closed field so that all polynomial equations now find solutions within the same field, which is a~result known as the fundamental theorem of algebra proven by Carl Friedrich Gauss. The~complex numbers share the commutativity and associativity properties of the reals as well as maintaining the status of a normed division algebra. They lose the order of the reals, but~this is a hitch we are willing to accept in view of all the additional gifts brought about by them. Despite appearing as a mathematical construct, it is undeniable that Nature embraces the complex numbers, and they play a paramount role in fundamental aspects as quantum mechanics,\footnote{We can allude to the Aharonov--Bohm effect as a {\it real} manifestation of complex numbers in Nature.} but also in very practical situations such as the description of electric~circuits. After discovering the reals and the complex numbers and their suitability to describe Nature, a~pertinent question would be: are there more types of numbers that could help us in our understanding of the natural world? If so, is there an infinite class of {\it{ numbers}} that will appear in our ever-increasingly fundamental comprehension of the natural laws? The answer to these questions is intriguingly precise: there are exactly four distinctive types of numbers (conforming normed division algebras). However,~then, does this mean that they suffice to have a complete fundamental description of Nature? In the following, we will occupy ourselves with the first questions and defer this more ambitious last question to another~occasion. \section{The neglected but valuable~quaternions} The search for numbers beyond the complex realm commenced from a pure mathematical curiosity in an endeavour undertaken by William Rowan Hamilton, who discovered the quaternions $\Ha$ as a four-dimensional extension of the complex numbers: a~discovery that was literally carved in stone for the posterity in the Brougham Bridge of Dublin. These numbers still form a normed division algebra, but~they leave the commutativity of $\Z$ behind. The~quaternions extend the complex numbers with two additional imaginary units so an arbitrary $q\in\Ha$ can be expressed as \be q=q_0+q_1 \eh_1+q_2 \eh_2+q_3 \eh_3\,, \ee where $\eh_i$ belong to a set that generates, together with the unit element $1$, the~quaternionic algebra $\Ha=\text{Span}\{1,\eh_1,\eh_2,\eh_3\}$ via the following product rules: \be \eh_i\eh_j=-\delta_{ij}+\epsilon_{ijk}\eh_k, \label{eq:productquat} \ee that give rise to the commutation relations \be [\eh_i,\eh_j]=2\epsilon_{ijk}\eh_k. \ee These commutation relations already hint at the suitability of quaternions to describe three-dimensional rotations in terms of multiplication of pure imaginary quaternions with $q_0=0$. We can also introduce the conjugation that consists in changing the sign of the imaginary units: $\bar{q}\equiv q_0q_0-q_1 \eh_1-q_2 \eh_2-q_3 \eh_3$, so a quaternion is said to be purely imaginary if it satisfies $\bar{q}=-q$. The~norm of a quaternion is then defined as $\norm{q}^2=q\bar{q}=q_0^2+q_1^2+q_2^2+q_3^2$ and its inverse $q^{-1}=\bar{q}/\norm{q}$. At the time, the~quaternions were a very fashionable subject, and physics made extensive use of them (e.g., Maxwell equations were written with quaternions). When Gibbs noticed that quaternions could be expressed in the three-dimensional vector space endowed with a dot and a cross product, they became a proper contender. Quaternions eventually lost the battle and vectors arouse as the standard framework for physics, displacing quaternions to a marginal place. They however have always lurked in different corners of physics, finding their way to provide insightful applications. One can easily understand why they are an appealing groundwork for physics after noticing that they admit a representation in $\text{M}(2,\Z)$ via \be q\mapsto A(q)=\begin{pmatrix} q_0+\ii q_1 & q_2+\ii~q_3 \\ -q_2+\ii q_3 & q_0-\ii q_1 \end{pmatrix}\,, \label{eq:Ahom} \ee where multiplication in $\Ha$ simply becomes matrix product with $A(q_1q_2)=A(q_1) A(q_2)$, i.e.,~the above gives an homomorphism between quaternions and $2\times2$ complex matrices. One can observe that imaginary quaternions can be expressed in terms of Pauli matrices, thus corroborating their relation to rotations. Furthermore, the~homomorphism also shows that $\norm{q}^2=\det A(q)$, which further unveils the isomorphism between unit quaternions and $SU(2)$. It is then direct to uncover that rotations can be realised as unitary quaternions $r$ (which satisfy $\bar{r}=r^{-1}$) acting on pure imaginary quaternions $x$ (that satisfy $\bar{x}=-x$ by definition) via $x\mapsto r\,x\,\bar{r}$. It is certainly very appealing that pure imaginary quaternions provide a realisation of the real three-vector space where rotations is simply realised by quaternionic multiplication as well as their intimate relation to $SU(2)$. In~view of these properties and noticing that quaternions are actually a four-dimensional algebra, which coincides with the spacetime dimension, one cannot help but wonder if the spacetime Lorentz symmetry can find a dwell within the quaternionic algebra. The~fascinating answer is that it indeed does! Perhaps even more fascinating is that Lorentz symmetry rightfully finds its place not in $\Ha$ but in $\Z\otimes\Ha$. After~all, why should we leave our old friend $\Z$ out of the function? The application of complex quaternions to special relativity did not take long since its inception by Einstein in 1905. In~two independent works by A. Conway in 1911~\cite{Conway1911} and L. Silberstein a year later~\cite{SilbersteinLXXVIQF}, special relativity and Lorentz transformations were presented to quaternions, and it was explored in subsequent years by different authors (see e.g.,~\cite{Synge:1972zz} and references therein). We can show how to realise spacetime Lorentz symmetry by identifying the spacetime position with the quaternion \be x=-\ii x^0+x^i\eh_i, \ee that is antihermitian $x+x^{\dagger}=0$, where $x^\dagger$ is the complex and quaternion conjugate of $x$. It is straightforward to check that this condition is preserved under the quaternion transformation $x\mapsto \hat{q}\,x\,\hat{q}^{\dagger}$ with $\hat{q}$ a unit complex quaternion. Moreover, the~norm \linebreak $\norm{x}^2=-(x^0)^2+\vec{x}^2$, that gives the spacetime interval, is also invariant. This permits realising Lorentz transformations of the spacetime coordinates as a multiplication by unit complex quaternions in the space of antihermitian complex quaternions (see e.g.,~\cite{Rao1983} for an explicit construction). Restricting to real unit quaternions, we recover the spatial rotations, so boosts are naturally associated to the antihermitian (pure quaternionic imaginary) part of $\hat{q}$. Very much like pure imaginary quaternions over the reals realise rotations in the Euclidean space, pure imaginary quaternions with complex coefficients generate the Lie algebra $\ssl(2,\Z)$, which precisely corresponds to the double cover of the Lorentz group. This can be easily seen from \eqref{eq:Ahom} with $q_0=0$. If~we introduce the basis \be \left\{\hat{\eh}_1=\ii \eh_1,\;\hat{\eh}_2=\frac{1}{\sqrt{2}}(\eh_2+\ii\eh_3),\;\hat{\eh}_3=-\frac{1}{\sqrt{2}}(\eh_2-\ii\eh_3)\right\}\,, \ee it is immediate to obtain \be q=z_1\hat{\eh}_1+z_2\hat{\eh}_2+z_3\hat{\eh}_3\quad\mapsto \quad A(q)=\begin{pmatrix} z_1 & z_2 \\ z_3 & -z_1 \end{pmatrix},\quad\quad z_1,z_2,z_3\in\Z, \label{eq:Ahomtosl} \ee that establishes the realisation of $\ssl(2,\Z)$ with unit quaternions. This should be sufficiently convincing and stimulating to embrace the algebra $\Z\otimes\Ha$ as a promising framework to describe the Lorentz group. Of~course, a number of authors have engaged this venture with interesting results. A~related subject that deserves attention is that once we have a proper characterisation of the Lorentz group within $\Z\otimes\Ha$, the~possibility of describing gravity as a localisation of this algebra emerges as~well as new avenues to exploring theories of gravity. This is a speculation we should not dismiss and should receive a more meticulous~scrutiny. \section{The surprisingly cooperative character of the untamed~octonions} Having met the quaternions and its interesting applications, our curiosity eagerly craves the exploration of more algebras and their suitability to describe physics. The~task of obtaining higher-dimensional extensions of the quaternions was tackled by John T. Graves, who showed the existence of an eight-dimensional algebra that he called {\it octaves} but~are now known as {\it octonions}. This algebra is generated by \be \Oc={\text{Span}}\{1,\eo_1,\eo_2,\eo_3,\eo_4,\eo_5,\eo_6,\eo_7\} \ee that satisfy the following~rules: \begin{itemize} \item $\eo_a^2=-1$. \item Anticommutativity: $\{\eo_a,\eo_b\}=0$ for $a\neq b$. \item Cycling identity: If $\eo_a\eo_b=\eo_c$, then $\eo_{a+1}\eo_{b+1}=\eo_{c+1}$ mod 7. \item Index doubling identity: If $\eo_a\eo_b=\eo_c$, then $\eo_{2a}\eo_{2b}=\eo_{2c}$ mod 7. \end{itemize} The octonion multiplication table is not particularly illuminating. For~a detailed and excellent review of the properties and some applications of the octonions, we refer to~\cite{Baez:2001dm}. In~analogy with the quaternions, we can write the octonions multiplication as \be \eo_a\eo_b=-\delta_{ab}+f_{abc}\eo_c \ee where $f_{abc}$ represents the structure constants of the octonions algebra, which are completely antisymmetric. A~common representation is the Cartan--Schouten--Coxeter given by \be f_{abc}=1\quad\text{for}\quad(abc)\in\{(124), (235), (346), (457), (561), (672), (713)\}, \ee while the remaining ones can be obtained from the aforementioned properties. The~commutator of two octonions is \be [\eo_a,\eo_b]=2f_{abc}\eo_c. \ee Unlike its lower dimensional relatives, the~octonions are not associative, and this property is encoded in the so-called associator \be [\eo_a,\eo_b,\eo_c]=(\eo_a\eo_b)\eo_c-\eo_a(\eo_b\eo_c)\neq0. \ee In the seven-dimensional space of the pure imaginary octonions, we can also introduce the dual of the structure constants \be \tilde{f}_{abcd}=\frac{1}{6!}\epsilon_{abcdefg}f_{efg} \ee which are related to the non-associative character of $\Oc$. The~analogous dual of the structure constants of $\Ha$ vanish identically by virtue of the Jacobi identity $[\eh_i,[\eh_j,\eh_k]]=0$ that reflects the associative property of the quaternionic algebra. For~$\Oc$, we instead have $[\eo_a,[\eo_b,\eo_c]]=3\tilde{f}_{abcd}\eo_d$ that reveals the untamed character of octonions who do not even comply with~associativity. Due to their non-associative nature, octonions have remained even more neglected than their more amicable relatives the quaternions. For~this, the~unit octonions do not even form a group, while unit quaternions are keen to be related to physically sound groups such as rotations or Lorentz transformations. They however hide a beautiful gem inside their more intricate algebraic structure that is unveiled once we \footnote{With a little help from E. Cartan~\cite{Cartan1915}.} note that the automorphism group of the octonions is the exceptional Lie group $G_2$, so the structure constants $f_{abc}$ and their dual $\tilde{f}_{abcde}$ are invariants of $G_2$. In~this sense, the~octonions follow an analogy with the quaternions since the Lie algebra of $G_2$ can be represented in terms of pure imaginary octonions (and $G_2$ can be obtained via exponentiation). Furthermore, one can construct an $SU(3)$ subgroup of $G_2$ as the little group of some imaginary octonionic unit. This is the joyful moment when we realise that octonions may be willing to cooperate for describing color, as~explored for instance by G\"uydin and G\"ursey in~\cite{GuydinandGursey}. After discovering the octonions and their potential suitability to describe quarks, nobody can blame us for further pursuing our algebra hunt. A~useful result at this point is that $\R$, $\Z$, $\Ha$ and $\Oc$ can be sequentially generated by means of the Cayley--Dickson algorithm. However, the~application of the Cayley--Dickson algorithm to $\Oc$ leads to the so-called sedenions that lack a well-defined division. While it is true that in each iteration we give up one important property (order, commutativity and associativity for $\Z$, $\Ha$ and $\Oc$ respectively), not having a division may seem an excessive concession so one could be a bit more reluctant to include them in the class of {\it{ sensible numbers} }. \footnote{Of course, this has not prevented to find physical applications for the sedenions as recently explored in e.g.,~\cite{Masi:2021cgm}.} Thus, it seems we have to content ourselves with the reals, the~complex, the~quaternions and the octonions as our possible numbers. This is supported by Hurwitz's theorem that states that the only normed division algebras are indeed $\R$, $\Z$, $\Ha$ and $\Oc$. This is a very remarkable result and, thus, it is very tempting to find their appropriate place to describe physics. Delving into their respective group structures, we have seen that the real unit quaternions are nicely related to $SU(2)$, which is a~result that resembles the relation of unit complex numbers to $U(1)$, while complex quaternions naturally give representations of the Lorentz group. On~the other hand, octonions are related to $G_2$, which contains $SU(3)$ as a subgroup. Thus, we conclude that the only four normed division algebras are intimately related to the fundamental groups of the standard model (possibly even including gravity). It is then extremely appealing to employ \footnote{\textls[-25]{It is clear that the factor $\R$ is redundant and considering $\Z\otimes\Ha\otimes\Oc$ would be sufficient. We prefer to keep it there to emphasise that the general framework of the only four normed division algebras is employed}.} $\algebras$ as the algebraic framework to formulate the fundamental laws of physics (see Figure \ref{fig:algebras}). Exploring these speculations has led to remarkable insights, and it is an increasingly viable hypothesis that the elementary forces and particles are numbers (see e.g.,~\cite{Dixon1994,Furey:2015yxg,Gording:2019srz}). \section{Algebras in the~Sky} \textls[-25]{We have discussed how the algebras can provide a very appealing framework for the standard model of elementary particles and, in~particular, the~intriguing suitability of $\algebras$ to describe the particles. In~this essay, we want to explore the potential relevance for cosmology. The~first two factors of the algebra have been extensively applied to cosmological models where real and complex fields have a long history. We will thus focus on the last two factors that remain nearly unexplored (see e.g.,~\cite{Gunaydin:2020ric} for a recent application). The~interest is not only in that they have been seldom studied in cosmological scenarios, but, as~we will argue in the following, they happen to permit more interesting realisations of the cosmological symmetries dictated by the cosmological principle as the usual homogeneity and isotropy \footnote{Cosmological models violating these symmetries have also been considered as in the Bianchi or Lema\^itre-Tolman-Bondi cosmologies.}. This requirement in turn forces our universe to have maximally symmetric spatial sections so they could be described by the groups $ISO(3)$, $SO(3,1)$ and $SO(4)$ for the flat, open and closed universes, respectively. Our universe seems to prefer being flat, so we will assume $ISO(3)$. Thus, in~order to develop cosmological models, we need to have a residual $ISO(3)$ symmetry for our background configuration of fields. The~reals and the complex numbers give trivial realisations of this symmetry group because both real and complex scalar fields comply with these symmetries by simply taking a homogeneous vacuum state. In~that case, homogeneity and isotropy are obvious and they are generated by the usual linear and angular~momentum. } Moving on to the higher dimensional algebras, things become more interesting. Before~proceeding, it is convenient to pause for a moment and describe the mechanism of the cosmological scenarios based on quaternions and octonions with a more familiar framework. The~underlying idea that we will exploit consists of realising the cosmological principle not directly in terms of the Euclidean subgroup of the Poincar\'e symmetry but~as a diagonal subgroup of $ISO(3,1)\times G$, where $G$ is some internal group. Thus, homogeneity and isotropy are generated by linear and angular momenta that arise as some linear combinations of those within $ISO(3,1)$ and some generators of $G$. Of~course, not every $G$ allows for this construction, but~it should contain some subgroup isomorphic to rotations plus translations. There are many different manners in which this symmetry breaking pattern can be realised, and we refer to~\cite{Nicolis:2015sra} for an exhaustive and comprehensive classification. In~this respect, we only need to resort to the natural symmetry groups that these structures are endowed with. For~the quaternions over the reals, this requirement means that the resulting theory will naturally have an $SU(2)$ symmetry, since this is the symmetry group of unit quaternions as well as the symmetry group of automorphisms of $\Ha$. With~this rudimentary algebra, we can already start considering interesting cosmological applications. Let us proceed to explore some of them: \begin{itemize} \item {\it Quaternionic solid inflation}. A~scenario with a scalar quaternionic field $\phi$ can adopt a profile of the form $\langle\phi\rangle=x^i\eh_i$ that breaks isotropy as well as the quaternionic $SU(2)$ symmetry. However, there is a linear combination that is preserved. Regarding homogeneity, we need to make an additional assumption of some internal Abelian symmetry $T$ (e.g., a~shift symmetry) that could restore homogeneity. The~symmetry breaking pattern would then be $ISO(3,1)\times SU(2)\times T\rightarrow ISO_{\text{D}}(3)$. This symmetry breaking pattern appears in solid inflation~\cite{Endlich:2012pz}, and our construction provides a quaternionic formulation of this scenario. {Let us elaborate on the quaternionic formulation of the solid. We can resort to a pure imaginary quaternionic field $\phi(x)$, thus satisfying $\bar{\phi}=-\phi$, so that (global quaternionic) rotations can be realised with unit quaternions. Thus, the~simultaneous action of a spatial rotation $x^i\to R^i{}_jx^j$ and a quaternionic rotation $\phi(x)\to r\,\phi(x)\,\bar{r}$ on $\langle\phi\rangle$ yields \be \langle\phi\rangle=x^i\eh_i\to \langle\tilde{\phi}\rangle=r\left(R^i{}_jx^j\right)\eh_i\bar{r}=x^j\left(\,r\,R^i{}_j\,\eh_i\,\bar{r}\right). \ee Now, we can choose $r$ so that the imaginary quaternion $R^i{}_j\,\eh_i\,$ is rotated to $\eh_j$ and, therefore, $\langle\phi\rangle$ remains invariant thanks to the cooperation of $R^i{}_j$ and $r$. In~order to formulate the solid theory, we need to write down a Lagrangian that is required to be real and enjoy Lorentz invariance and both (global quaternionic) rotations and shift symmetry. We will not delve much into the procedure to systematically construct the allowed terms. Instead, we will simply quote that the required conditions are fulfilled by the quaternionic operators: \be X=\partial_\mu\phi\partial^\mu\bar{\phi},\quad Y=\partial_\mu\phi\partial_\nu\bar{\phi} \partial^\mu\phi\partial^\nu\bar{\phi}\quad \text{and}\quad Z=\partial_\mu\phi\partial_\nu \bar{\phi} \partial_\rho\phi\partial^\mu\bar{\phi}\partial^\nu\phi\partial^\rho\bar{\phi}. \ee It is straightforward to see that they are real, shift symmetric, Lorentz invariant, and~have the symmetry $\phi\to r\,\phi\,\bar{r}$. It is less trivial to see that they exhaust all the possibilities, so any other operator satisfying the desired properties is a function of the above three. In~fact, $X$, $Y$ and $Z$ encode the three independent invariants of the matrix $\hat{B}$ with components $B^{ij}\equiv \partial_\mu\phi^i\partial^\mu\phi^j$ where $\phi(x)=\phi^i(x)\eh_i$. It can be shown, with~a tedious but simple direct computation, that $X$, $Y$ and $Z$ are linear combinations of the traces of $\hat{B}$, $\hat{B}^2$ and $\hat{B}^3$, which are equivalent to the three fundamental objects employed in~\cite{Endlich:2012pz}.} \item {\it Triad cosmology}. Let us now consider a real quaternionic \footnote{Let us pedantically clarify that by real quaternionic field we refer to a vector field over $\Ha$ with real coefficients.} vector field $\mathcal{Q}_\mu$ that takes a background configuration of the form $\langle\mathcal{Q}\rangle=Q(t)\delta^a_{i} \eh_a \text{d} x^i$ with $Q(t)$ some time-dependent function. Again, the~quaternionic sector naturally introduces an $SU(2)$ symmetry that can conspire with the spacetime Lorentz symmetry to preserve a diagonal $SO(3)$. The~scenario is now analogous to models with multiple vector fields featuring internal non-Abelian (global or local) symmetries (see e.g.,~\cite{BeltranJimenez:2018ymu} and references therein). \item {$\algebras$ \it{cosmology}}. In~the previous examples, it was necessary to resort to some external elements that could assist us in developing the non-trivial realisations of homogeneity and/or isotropy. For~the quaternionic solid, an~additional shift symmetry was necessary, while the quaternionic triad is required to use quaternionic vector fields. It is however possible to overcome these seemingly assisting structures and simply embrace the full generality of $\algebras$. As~explained above, the~complex quaternionic sector conveniently contains Lorentz, while the octonionic sector can account for internal symmetries such as color. Thus, a~theory using this full algebra will have a symmetry group $G$ that will contain, at~least, the~natural groups of the algebra. We have seen that the group of automorphisms of the quaternions and the octonions are $SU(2)$ and $G_2$ respectively. Furthermore, the~complex quaternions provide representations of the Lorentz group. In~this case, we can envision having \linebreak $G\supset SO(3,1)\times G_2$ as the symmetry group, so it is clear that a symmetry breaking pattern $G\rightarrow ISO_{\rm D}(3)$ is possible, where the linear and angular momenta generating this residual three-dimensional Euclidean group is a combination of the generators in $SO(3,1)$ and $G_2$. This group contains several $SU(2)$ subgroups so, in~particular, the~triad cosmology explained above in terms of vector quaternions is possible. However, other realisations are also possible. \end{itemize} Let us point out that our reasoning is fully general and the mechanism does not necessitate any specific Lagrangian, but~it is the own structural properties of the algebras that allows the non-trivial realisations of the cosmological principle. These structural properties are enough to obtain observational~signatures. \section{The sound of cosmic~algebras} The non-trivial realisations of the cosmological principle provided by quaternions and octonions is not of purely mathematical interest, but~they can have an observational impact for gravitational waves (GWs) astronomy. The~underlying reason is that the background rotational symmetry now combines the usual spatial rotations of space and the internal symmetries of the scalar quaternion and octonion fields. This means that the perturbations will organise themselves into irreps of this symmetry and, as~a consequence, we can have additional helicity-2 perturbations even if the only properly spin-2 field in the theory corresponds to the usual GWs. These additional helicity-2 modes will exhibit a mixing with GWs mediated by the cosmological background configuration of the non-commutative sector of $\algebras$. Depending on the structure of the original group, there could be several additional helicity-2 modes, but~we will focus on the case where only one extra species arises. The~helicity-2 sector will then be conformed by the usual GWs $h_{(\lambda)}$ together with the additional guy $t_{(\lambda)}$ where $\lambda$ stands for the two polarisation modes of each~perturbation. In the described framework, we can test the presence of a non-trivial background for the non-commutative sector of the algebra $\algebras$ by studying the GWs signal emitted by a binary black hole system. The~scenario we envision is depicted in Figure~\ref{fig:propagation} and consists of two black holes that are assumed to live on an uncharged sector concerning the non-commutative piece of $\algebras$. In~that situation, the~inspiral black holes will only emit the usual GWs $h_{(\lambda)}$, but~no emission in the $t_{(\lambda)}-$channel will be present. As~these GWs travel toward the Earth, they propagate on the non-trivial background of $\algebras$ that will mediate an oscillation into $t_{(\lambda)}$ modes, thus modulating the received signal in our GWs interferometers. We thus arrive at the leitmotif of this essay: {\it {the cosmic presence of $\algebras$ can be heard through GWs}.} Under very general assumptions, the~propagation from the source to the receiver will be governed by a system of equations that can be parameterised in Fourier space and in conformal time $\eta$ as \be \label{eq:generalequation} \left[\frac{\d^2}{\d\eta^2} + \nuM\frac{\d}{\d\eta}+\cM k^2 + \pM k +\mM\right]\begin{pmatrix} h_{(\lambda)} \\ t_{(\lambda)} \end{pmatrix}=0\,, \ee with $k$ representing the Fourier mode and $\nuM$, $\cM$, $\pM$ and $\mM$ representing some matrices encoding the non-trivial cosmological background of $\algebras$. These matrices will typically evolve in time over cosmological time-scales and could also depend on the helicity mode $\lambda$. The~off-diagonal components of these matrices describe the {\it flavor oscillations} that can occur in a variety of manners and lead to different effects, all of which will give rise to distinctive modulations of the GWs signals measured by the interferometers. A~detailed quantitative analysis of the different effects derived from Equation~\eqref{eq:generalequation} can be found in~\cite{BeltranJimenez:2019xxx,Ezquiaga:2021ler}, and some particular cases of GWs oscillations have also been explored for cosmological gauge fields~\cite{Caldwell:2017sto,Caldwell:2018feo} and in massive gravity~\cite{Narikawa:2014fua,Max:2017flc}, though~in this latter case, there is an additional spin-2 field in the theory. Qualitatively, the~flavour oscillations produce interesting effects, some of which we mention in the following (see also Figure~\ref{fig:signatures}): \begin{itemize} \item Anomalous propagation speed of GWs. If~$\cM$ is different from the identity, then the oscillations will induce an anomalous propagation speed of GWs even if $\cM_{hh}=1$. The~reason is that the oscillation will make the GWs propagate as $t$-modes for some time, thus modifying the effective propagation speed throughout its path. \item Generation of chirality. If~$\pM$ is different from 0, then the different helicities will oscillate in a different manner, thus generating chirality for the GWs. The~reason is that $\pM$, governing a term linear in $k$, typically arises from violations of parity. \item Oscillations in the GWs luminosity distance. Due to the oscillations and the presence of the friction matrix $\nuM$, the~luminosity distance of GWs $d^{\text{GW}}_{L}$ will be affected. Comparing this quantity with the electromagnetic counterpart $d^{\text{EM}}_{L}$, we can also hear some non-trivial cosmic $\algebras$. \end{itemize} There are other interesting observational effects that will be visible in GWs astronomy such as echoes, wave distortions or birrefringence that we will not explain in detail but~are comprehensively analysed in~\cite{BeltranJimenez:2019xxx,Ezquiaga:2021ler}. The~remarkable result is that the subtle whisper of the cosmic algebras will be imprinted in the GWs, opening the possibility to test their presence through a variety of~effects. \unskip \section{Conclusions} Normed division algebras are interesting fellows from a pure mathematical viewpoint since they conform {\it {sensible}} types of numbers. Numbers are allegedly our way of communicating with Nature, and the reals and complex numbers have repeatedly proved their appropriateness to that purpose and so they are hardwired in our description of the physical laws. Quaternions have also claimed their position in this endeavour, but~they have remained largely marginalised. They are starting to receive well-deserved attention due to their suitability to describe some fundamental aspects of particles. Octonions are by far the most obscure member of this family, although~they hide extremely remarkable properties that not only are suitable for color but~also permit giving a fresh new look at known intriguing results in, e.g.,~string theory and supersymmetry. This essay has been devoted to reviewing this family and some of its applications in physics. If~this family does play a fundamental role in the foundations of our standard model, including gravity, it could also plausibly participate in the cosmological evolution of our universe. In~this respect, we have argued how the two non-commutative members naturally lead to interesting realisations of the cosmological principle that arise from their structural properties. If~that is the case, they can be probed with different distinctive signatures in GWs astronomy so the future observations of GWs could unveil the presence of cosmic quaternions and octonions and bring exciting news about these undeservedly neglected~acquaints. We will conclude with a wild surmise. Fundamental aspects of the standard model are encoded in scattering amplitudes and, in~particular, the~analytical properties of physical observables such as the $S-$matrix. In~this respect, complex analysis emerges as a fundamental tool, where analiticity, poles, branch cuts, etc., have precise physical meanings. We cannot resist speculating that quaternionic and octonionic analysis (or analysis in $\algebras$ more generally) could eventually bring out new insights on our comprehension of the most fundamental laws of physics and could resolve long-standing difficulties with the gravity~sector. \vspace{6pt} \funding{J.B.J. was supported by the {\it Atracci\'on del Talento Cient\'ifico en Salamanca} programme and the Project PGC2018-096038-B-100 funded by the Spanish ``Ministerio de Ciencia e Innovación''. T.S.K. was supported by the Estonian Research Council grants PRG356 ``Gauge Gravity'', MOBTT86 and by the EU through the European Regional Development Fund CoE program TK133 ``The Dark Side of the Universe''.} \end{paracol} \newpage \reftitle{References} \bibliography{Refs}

Title: Double Mode Cepheids from the Zwicky Transient Facility Survey

Abstract: Multi-mode Cepheids pulsate simultaneously in more than one mode of oscillation. They provide an independent means to test stellar models and pulsation theories. They can also be used to derive metallicities. In recent years, the number of known multi-mode Cepheids has increased dramatically with the discovery of a large number of Galactic double-mode Cepheids. To date, 209 double-mode Cepheids have been detected in the Galactic bulge and disk, mostly based on the Optical Gravitational Lensing Experiment's (OGLE) catalog. In this paper, we conduct a comprehensive search for double-mode Cepheids in the northern sky based on Zwicky Transient Facility Data Release 5. We found 72 such objects in the Milky Way. The periods of the 30 sample objects already included in the OGLE catalog show excellent agreement with the OGLE periods. The period ratios of our new Cepheids are consistent with those of known double-mode Cepheids, as evidenced by their loci in the so-called `Petersen diagram'. Compared with OGLE, the completeness of our double-mode Cepheid sample is around 71\%. The much improved temporal sampling of the Zwicky Transient Facility offers significant scope to find more double-mode Cepheids, especially at the distribution's short-period end.

https://export.arxiv.org/pdf/2208.03080

\thispagestyle{plain} \newcommand{\btx}{\textsc{Bib}\TeX} \newcommand{\thestyle}{\texttt{\filename}} \begin{center}{\bfseries\Large Reference sheet for \thestyle\ usage}\\ \large(Describing version \fileversion\ from \filedate) \end{center} \begin{quote}\slshape For a more detailed description of the \thestyle\ package, \LaTeX\ the source file \thestyle\texttt{.dtx}. \end{quote} \head{Overview} The \thestyle\ package is a reimplementation of the \LaTeX\ |\cite| command, to work with both author--year and numerical citations. It is compatible with the standard bibliographic style files, such as \texttt{plain.bst}, as well as with those for \texttt{harvard}, \texttt{apalike}, \texttt{chicago}, \texttt{astron}, \texttt{authordate}, and of course \thestyle. \head{Loading} Load with |\usepackage[|\emph{options}|]{|\thestyle|}|. See list of \emph{options} at the end. \head{Replacement bibliography styles} I provide three new \texttt{.bst} files to replace the standard \LaTeX\ numerical ones: \begin{quote}\ttfamily plainnat.bst \qquad abbrvnat.bst \qquad unsrtnat.bst \end{quote} \head{Basic commands} The \thestyle\ package has two basic citation commands, |\citet| and |\citep| for \emph{textual} and \emph{parenthetical} citations, respectively. There also exist the starred versions |\citet*| and |\citep*| that print the full author list, and not just the abbreviated one. All of these may take one or two optional arguments to add some text before and after the citation. \begin{quote} \begin{tabular}{l@{\quad$\Rightarrow$\quad}l} |\citet{jon90}| & Jones et al. (1990)\\ |\citet[chap.~2]{jon90}| & Jones et al. (1990, chap.~2)\\[0.5ex] |\citep{jon90}| & (Jones et al., 1990)\\ |\citep[chap.~2]{jon90}| & (Jones et al., 1990, chap.~2)\\ |\citep[see][]{jon90}| & (see Jones et al., 1990)\\ |\citep[see][chap.~2]{jon90}| & (see Jones et al., 1990, chap.~2)\\[0.5ex] |\citet*{jon90}| & Jones, Baker, and Williams (1990)\\ |\citep*{jon90}| & (Jones, Baker, and Williams, 1990) \end{tabular} \end{quote} \head{Multiple citations} Multiple citations may be made by including more than one citation key in the |\cite| command argument. \begin{quote} \begin{tabular}{l@{\quad$\Rightarrow$\quad}l} |\citet{jon90,jam91}| & Jones et al. (1990); James et al. (1991)\\ |\citep{jon90,jam91}| & (Jones et al., 1990; James et al. 1991)\\ |\citep{jon90,jon91}| & (Jones et al., 1990, 1991)\\ |\citep{jon90a,jon90b}| & (Jones et al., 1990a,b) \end{tabular} \end{quote} \head{Numerical mode} These examples are for author--year citation mode. In numerical mode, the results are different. \begin{quote} \begin{tabular}{l@{\quad$\Rightarrow$\quad}l} |\citet{jon90}| & Jones et al. [21]\\ |\citet[chap.~2]{jon90}| & Jones et al. [21, chap.~2]\\[0.5ex] |\citep{jon90}| & [21]\\ |\citep[chap.~2]{jon90}| & [21, chap.~2]\\ |\citep[see][]{jon90}| & [see 21]\\ |\citep[see][chap.~2]{jon90}| & [see 21, chap.~2]\\[0.5ex] |\citep{jon90a,jon90b}| & [21, 32] \end{tabular} \end{quote} \head{Suppressed parentheses} As an alternative form of citation, |\citealt| is the same as |\citet| but \emph{without parentheses}. Similarly, |\citealp| is |\citep| without parentheses. Multiple references, notes, and the starred variants also exist. \begin{quote} \begin{tabular}{l@{\quad$\Rightarrow$\quad}l} |\citealt{jon90}| & Jones et al.\ 1990\\ |\citealt*{jon90}| & Jones, Baker, and Williams 1990\\ |\citealp{jon90}| & Jones et al., 1990\\ |\citealp*{jon90}| & Jones, Baker, and Williams, 1990\\ |\citealp{jon90,jam91}| & Jones et al., 1990; James et al., 1991\\ |\citealp[pg.~32]{jon90}| & Jones et al., 1990, pg.~32\\ |\citetext{priv.\ comm.}| & (priv.\ comm.) \end{tabular} \end{quote} The |\citetext| command allows arbitrary text to be placed in the current citation parentheses. This may be used in combination with |\citealp|. \head{Partial citations} In author--year schemes, it is sometimes desirable to be able to refer to the authors without the year, or vice versa. This is provided with the extra commands \begin{quote} \begin{tabular}{l@{\quad$\Rightarrow$\quad}l} |\citeauthor{jon90}| & Jones et al.\\ |\citeauthor*{jon90}| & Jones, Baker, and Williams\\ |\citeyear{jon90}| & 1990\\ |\citeyearpar{jon90}| & (1990) \end{tabular} \end{quote} \head{Forcing upper cased names} If the first author's name contains a \textsl{von} part, such as ``della Robbia'', then |\citet{dRob98}| produces ``della Robbia (1998)'', even at the beginning of a sentence. One can force the first letter to be in upper case with the command |\Citet| instead. Other upper case commands also exist. \begin{quote} \begin{tabular}{rl@{\quad$\Rightarrow$\quad}l} when & |\citet{dRob98}| & della Robbia (1998) \\ then & |\Citet{dRob98}| & Della Robbia (1998) \\ & |\Citep{dRob98}| & (Della Robbia, 1998) \\ & |\Citealt{dRob98}| & Della Robbia 1998 \\ & |\Citealp{dRob98}| & Della Robbia, 1998 \\ & |\Citeauthor{dRob98}| & Della Robbia \end{tabular} \end{quote} These commands also exist in starred versions for full author names. \head{Citation aliasing} Sometimes one wants to refer to a reference with a special designation, rather than by the authors, i.e. as Paper~I, Paper~II. Such aliases can be defined and used, textual and/or parenthetical with: \begin{quote} \begin{tabular}{lcl} |\defcitealias{jon90}{Paper~I}|\\ |\citetalias{jon90}| & $\Rightarrow$ & Paper~I\\ |\citepalias{jon90}| & $\Rightarrow$ & (Paper~I) \end{tabular} \end{quote} These citation commands function much like |\citet| and |\citep|: they may take multiple keys in the argument, may contain notes, and are marked as hyperlinks. \head{Selecting citation style and punctuation} Use the command |\bibpunct| with one optional and 6 mandatory arguments: \begin{enumerate} \item the opening bracket symbol, default = ( \item the closing bracket symbol, default = ) \item the punctuation between multiple citations, default = ; \item the letter `n' for numerical style, or `s' for numerical superscript style, any other letter for author--year, default = author--year; \item the punctuation that comes between the author names and the year \item the punctuation that comes between years or numbers when common author lists are suppressed (default = ,); \end{enumerate} The optional argument is the character preceding a post-note, default is a comma plus space. In redefining this character, one must include a space if one is wanted. Example~1, |\bibpunct{[}{]}{,}{a}{}{;}| changes the output of \begin{quote} |\citep{jon90,jon91,jam92}| \end{quote} into [Jones et al. 1990; 1991, James et al. 1992]. Example~2, |\bibpunct[; ]{(}{)}{,}{a}{}{;}| changes the output of \begin{quote} |\citep[and references therein]{jon90}| \end{quote} into (Jones et al. 1990; and references therein). \head{Other formatting options} Redefine |\bibsection| to the desired sectioning command for introducing the list of references. This is normally |\section*| or |\chapter*|. Define |\bibpreamble| to be any text that is to be printed after the heading but before the actual list of references. Define |\bibfont| to be a font declaration, e.g.\ |\small| to apply to the list of references. Define |\citenumfont| to be a font declaration or command like |\itshape| or |\textit|. Redefine |\bibnumfmt| as a command with an argument to format the numbers in the list of references. The default definition is |[#1]|. The indentation after the first line of each reference is given by |\bibhang|; change this with the |\setlength| command. The vertical spacing between references is set by |\bibsep|; change this with the |\setlength| command. \head{Automatic indexing of citations} If one wishes to have the citations entered in the \texttt{.idx} indexing file, it is only necessary to issue |\citeindextrue| at any point in the document. All following |\cite| commands, of all variations, then insert the corresponding entry to that file. With |\citeindexfalse|, these entries will no longer be made. \head{Use with \texttt{chapterbib} package} The \thestyle\ package is compatible with the \texttt{chapterbib} package which makes it possible to have several bibliographies in one document. The package makes use of the |\include| command, and each |\include|d file has its own bibliography. The order in which the \texttt{chapterbib} and \thestyle\ packages are loaded is unimportant. The \texttt{chapterbib} package provides an option \texttt{sectionbib} that puts the bibliography in a |\section*| instead of |\chapter*|, something that makes sense if there is a bibliography in each chapter. This option will not work when \thestyle\ is also loaded; instead, add the option to \thestyle. Every |\include|d file must contain its own |\bibliography| command where the bibliography is to appear. The database files listed as arguments to this command can be different in each file, of course. However, what is not so obvious, is that each file must also contain a |\bibliographystyle| command, \emph{preferably with the same style argument}. \head{Sorting and compressing citations} Do not use the \texttt{cite} package with \thestyle; rather use one of the options \texttt{sort} or \texttt{sort\&compress}. These also work with author--year citations, making multiple citations appear in their order in the reference list. \head{Long author list on first citation} Use option \texttt{longnamesfirst} to have first citation automatically give the full list of authors. Suppress this for certain citations with |\shortcites{|\emph{key-list}|}|, given before the first citation. \head{Local configuration} Any local recoding or definitions can be put in \thestyle\texttt{.cfg} which is read in after the main package file. \head{Options that can be added to \texttt{\char`\\ usepackage}} \begin{description} \item[\ttfamily round] (default) for round parentheses; \item[\ttfamily square] for square brackets; \item[\ttfamily curly] for curly braces; \item[\ttfamily angle] for angle brackets; \item[\ttfamily colon] (default) to separate multiple citations with colons; \item[\ttfamily comma] to use commas as separaters; \item[\ttfamily authoryear] (default) for author--year citations; \item[\ttfamily numbers] for numerical citations; \item[\ttfamily super] for superscripted numerical citations, as in \textsl{Nature}; \item[\ttfamily sort] orders multiple citations into the sequence in which they appear in the list of references; \item[\ttfamily sort\&compress] as \texttt{sort} but in addition multiple numerical citations are compressed if possible (as 3--6, 15); \item[\ttfamily longnamesfirst] makes the first citation of any reference the equivalent of the starred variant (full author list) and subsequent citations normal (abbreviated list); \item[\ttfamily sectionbib] redefines |\thebibliography| to issue |\section*| instead of |\chapter*|; valid only for classes with a |\chapter| command; to be used with the \texttt{chapterbib} package; \item[\ttfamily nonamebreak] keeps all the authors' names in a citation on one line; causes overfull hboxes but helps with some \texttt{hyperref} problems. \end{description}

Title: The SHARDDS survey: limits on planet occurrence rates based on point sources analysis via the Auto-RSM framework

Abstract: In the past decade, HCI surveys provided new insights about the frequency and properties of substellar companions at separation larger than 5 au. In this context, our study aims to detect and characterise potential exoplanets and brown dwarfs within debris disks, by considering the SHARDDS survey, which gathers 55 Main Sequence stars with known bright debris disk. We rely on the AutoRSM framework to perform an in-depth analysis of the targets, via the computation of detection maps and contrast curves. A clustering approach is used to divide the set of targets in multiple subsets, in order to reduce the computation time by estimating a single optimal parametrisation for each considered subset. The use of Auto-RSM allows to reach high contrast at short separations, with a median contrast of 10-5 at 300 mas, for a completeness level of 95%. Detection maps generated with different approaches are used along with contrast curves, to identify potential planetary companions. A new planetary characterisation algorithm, based on the RSM framework, is developed and tested successfully, showing a higher astrometric and photometric precision for faint sources compared to standard approaches. Apart from the already known companion of HD206893 and two point-like sources around HD114082 which are most likely background stars, we did not detect any new companion around other stars. A correlation study between achievable contrasts and parameters characterising HCI sequences highlights the importance of the strehl, wind speed and wind driven halo to define the quality of high contrast images. Finally, planet detection and occurrence frequency maps are generated and show, for the SHARDDS survey, a high detection rate between 10 and 100 au for substellar companions with mass >10MJ.

https://export.arxiv.org/pdf/2208.09204

\titlerunning{SHARDDS Survey} \authorrunning{Dahlqvist et al.} \title{The SHARDDS survey: limits on planet occurrence rates based on point sources analysis via the Auto-RSM framework\thanks{Based on observations collected at the European Southern Observatory under ESO programmes 096.C-0388(A) and 097.C-0394(A)} } \author{C.-H. Dahlqvist\inst{1}, J. Milli\inst{2}, O. Absil\inst{1}\fnmsep\thanks{F.R.S.-FNRS Senior Research Associate}, F. Cantalloube\inst{3}, L. Matra\inst{8}, E. Choquet\inst{3}, C. del Burgo\inst{9}, J. P. Marshall\inst{4,5}, M.~Wyatt\inst{10}, S. Ertel\inst{6,7}} \institute{STAR Institute, Universit\'{e} de Li\`{e}ge, All\'{e}e du Six Ao\^{u}t 19c, 4000 Li\`{e}ge, Belgium\\ \email{carl-henrik.dahlqvist@uliege.be} \and Universit\'{e} Grenoble-Alpes, CNRS, IPAG F-38000 Grenoble, France \and Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France \and Academia Sinica, Institute of Astronomy and Astrophysics, 11F Astronomy-Mathematics Building, NTU/AS campus, No. 1, Section 4, Roosevelt Rd., Taipei 10617, Taiwan \and Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD 4350, Australia \and Large Binocular Telescope Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA \and Steward Observatory, Department of Astronomy, University of Arizona, 993 N. Cherry Ave, Tucson, AZ, 85721, US \and School of Physics, Trinity College Dublin, the University of Dublin, College Green, Dublin 2, Ireland \and Instituto Nacional de Astrof\'{\i}sica, \'Optica y Electr\'onica, Luis Enrique Erro 1, Sta. Ma. Tonantzintla, Puebla, Mexico \and Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK} \date{} \abstract {In the past decade, high contrast imaging allowed the detection and characterisation of exoplanets, brown dwarfs, and circumstellar disks. Large surveys provided new insights about the frequency and properties of massive sub-stellar companions with separations from 5 to 300 au. } {In this context, our study aims to detect and characterise potential exoplanets and brown dwarfs within debris disks, considering a diverse population of stars with respect to stellar age and spectral type. We present in this paper the analysis of a set of H-band images taken by the VLT/SPHERE instrument in the context of the SHARDDS survey. This survey gathers 55 main-sequence stars within 100\,pc, known to host a high-infrared-excess debris disk, allowing us to potentially better understand the complex interactions between substellar companions and disks.} {We rely on the Auto-RSM framework to perform an in-depth analysis of the considered targets, via the computation of detection maps and contrast curves. A clustering approach is used to divide the set of targets into multiple subsets, in order to reduce the computation time by estimating a single optimal parametrisation for each considered subset. Detection maps generated with different approaches are used along with contrast curves to identify potential planetary companions. Planet detection and planet occurrence frequencies are derived from the generated contrast curves, relying on two well-known evolutionary models, namely AMES-DUSTY and AMES-COND. Finally, we study the influence of the observing conditions and observing sequence characteristics on the performance measured in terms of contrast.} {The use of Auto-RSM allows us to reach high contrast at short separations, with a median contrast of $10^{-5}$ at 300 mas, for a completeness level of 95\%. A new planetary characterisation algorithm, based on the RSM framework, is developed and tested successfully, showing a higher astrometric and photometric precision for faint sources compared to standard approaches. Apart from the already known companion of HD206893 and two point-like sources around HD114082 which are most likely background stars, we did not detect any new companion around other stars. A correlation study between achievable contrasts and parameters characterising high contrast imaging sequences highlights the importance of the Strehl ratio, wind speed at a height of 30 meters, and presence of wind-driven halo to define the quality of high contrast images. Finally, planet detection and occurrence rate maps are generated and show, for the SHARDDS survey, a high sensitivity between 10 and 100 au for substellar companions with masses >10$M_J$. } {} \keywords{surveys-methods: data analysis-methods: statistical-techniques: image processing-techniques: high angular resolution-planetary systems-planets and satellites: detection} \section{Introduction} \label{sec:intro} In our current understanding of planetary system formation, gas giant planets form in gas-rich protoplanetary disks that dissipate in a few million years \cite[e.g.][]{Williams2011}, leaving behind one or several planets as well as belts of smaller rocky bodies that never managed to grow to full-sized planets \cite[e.g. ][]{Krivov2010}. These belts, also known as debris disks because of their collisional activity, are composed of all sub-planetary rocky bodies, ranging from kilometer-sized planetesimals to micron-sized dust \cite[see for example][for a review]{Wyatt2008}. These dust particles are detectable by their reflected light or thermal emission, creating an infrared excess above the stellar photosphere. Current far-infrared surveys can detect debris disks with an infrared excess above $10^{-6}$ and identify debris disks in around 30\% of A stars and 20\% of FGK stars \cite[e.g. ][]{Eiroa13}, but the real occurrence rate could be much higher \citep{Pawellek2021}. Those disks are a natural place to look for exoplanets because planet formation succeeded at least to form large planetesimals in those systems. This is one of the reasons why direct imaging surveys generally include many debris disk host stars, such as in the SPHERE-SHINE survey \citep[SPHERE infrared survey for exoplanets, ][]{Desidera2021} or the GPI-GPIES survey \citep[Gemini Planet Imager Exoplanet Survey, ][]{Nielsen2019}. \citet{Meshkat17} found indeed a tentative evidence that giant planets have a higher occurrence rate in debris disks hosts, and the first emblematic directly imaged planets were found in the massive debris disks system $\beta$ Pic \citep{Lagrange2009} or HR\,8799 \citep{Marois2008}. Following this strategy, we present in this study a direct imaging survey of a sample of 55 main-sequence stars hosting high-infrared excess debris disks: the SPHERE High-Angular Resolution Debris Disks Survey (SHARDDS). This survey already revealed debris disks resolved for the first time in scattered light: HD\,114082 \citep{Wahhaj16,Engler2022}, 49 Ceti \citet{Choquet2017}, HD\,105 \citep{Marshall2018} as well as a substellar companion (HD\,206893 B) close to the deuterium burning limit \citep{Milli16b,Delorme2017_206,Romero2021}. Here, we use the homogeneous observations made in the context of this high-contrast survey to search for companions with the Regime Switching Model (hereafter RSM) post-processing algorithm \citep{Dahlqvist20} and provide detection maps and contrast curves. The RSM method focuses on the detection of point sources within high-contrast images, by making use of the angular diversity introduced via pupil tracking mode observations. The concept behind RSM is to model the spatio-temporal evolution of the pixel intensities contained in the cubes of residuals generated by several PSF-subtraction techniques. As each PSF-subtraction technique models the speckle field differently, combining multiple techniques helps to average out residual speckle noise while preserving potential planetary signals. The RSM approach relies on a two-state Markov chain to model annulus-wise the pixel intensities and estimate the probability to be either in a speckle noise regime or a planetary regime. The probability associated to the planetary regime is then used to compute a detection map. Compared to other state-of-the-art post-processing methods dedicated to high-contrast imaging, the \textit{Exoplanet Imaging Data Challenge}\footnote{\url{https://exoplanet-imaging-challenge.github.io/}} has shown that the RSM technique has a very low false positive rate and is among the best algorithms in terms of detection capabilities\citep{Cantalloube20}. More recently, the Auto-RSM framework \citep{Dahlqvist21b} was developed to reduce the burden of parameter selection and further optimise the performance of the RSM algorithm. This optimisation framework consists of three main steps: (i) the definition of the optimal set of parameters for the PSF-subtraction techniques, (ii) the optimisation of the RSM algorithm, and (iii) the selection of the optimal set of PSF-subtraction techniques and ADI sequences \citep[Angular Differential Imaging, ][]{Marois08} used to generate the final RSM probability map. The Auto-RSM framework being computationally expensive, a clustering approach is used to divide the set of targets into multiple subsets. For each subset, the cluster center is identified and the Auto-RSM framework is applied onto it to provide the optimal parametrisation for the entire cluster. The obtained optimal parametrisations are also compared to unveil potential commonalities and understand their relationship with the ADI sequence characteristics. Detection maps are then computed via the RSM approach, relying on these optimal parametrisations. The detection maps are used to identify potential planetary companions, and a new companion characterisation framework based on the RSM approach is introduced. The detection maps are also used to compute contrast curves, which are used together to estimate detection probability maps and occurrence rate maps, based on well-known evolutionary models. The relationship existing between reachable contrasts and parameters characterising HCI observing sequences is also investigated. The remainder of this paper is organised as follows. Section 2 describes the target selection for the SHARDDS survey. In Section~3, we present our data reduction pipeline involving the definition of clusters along with cluster centres on which the Auto-RSM optimisation procedure is applied. The computation of detection maps and contrast curves follows the estimation of the optimal parametrisations. Section 4 is devoted to the characterisation of potential planetary candidates. In Section~5, we consider the contrast curve as a performance metric and analyse the potential drivers of this performance. Section 6 focuses on the estimation of the planetary detection probability from which we derive an estimated planetary occurrence rate associated to the SHARDDS survey. Finally, Section~7 concludes this work. \section{Survey description} \label{sec:Survey} The SHARDDS survey was designed to image circumstellar disks around bright nearby stars (within 100 pc from the Earth) in the near-infrared using the VLT/SPHERE instrument \citep[Very Large Telescope/Spectro-Polarimetric High-contrast Exoplanet REsearch,][]{Beuzit19}. The aim of the survey is to better understand the scarcity of debris disks detection in scattered light, by targeting disks without any scattered-light detection at the time of the survey design (2014), either because the target was not observed with high-contrast instruments, or because the disk might be too compact and faint to be accessible with first-generation high-contrast instruments such as HST/NICMOS \citep[Hubble Space Telescope/Near Infrared Camera and Multi-Object Spectrometer, ][]{Thompson98} or VLT/NaCo \citep[Very Large Telescope/Nasmyth Adaptive Optics System Near-Infrared Imager and Spectrograph, ][]{Lenzen03,Rousset2003}, having poor performance below $0.5"$. The underlying goals are to characterise the disks architecture and properties, and statistically link these properties to the stellar age, spectral type, and potential presence of companions. This paper contributes to the achievement of these objectives by applying the RSM detection algorithm \citep{Dahlqvist21b} on the datasets, to detect potential planetary candidates. The RSM detection algorithm was designed to unveil point-like sources and is therefore not fitted to detect extended features such as debris disks. The detection of companions can bring valuable information to better understand the secular interactions between debris disks and companions, and whether such interactions are always needed to explain particular signatures in disks such as azimuthal asymmetries, warps or sharp edges \citep[see ][ for emblematic examples of signatures within debris disks attributable to a companion]{Mouillet97,Lagrange12,Lestrade15}. The SHARDDS survey includes 55 main-sequence stars visible from the Southern hemisphere, covering spectral types A-M and ages 10 Myr - 6 Gyr . This diverse sample of debris systems aims to provide a comprehensive view of planetary system properties and their time evolution. These stars were selected for the expected brightness of their disks (fractional luminosity above $10^{-4}$) and because they were not yet resolved in scattered light. All stars that were not observable from Paranal with an airmass below 2, were excluded from the sample. The SPHERE/IRDIS instrument \citep[InfraRed Dual-band Imager and Spectrograph]{Dohlen08} was used with the broad-band H filter ($\lambda=1.625\mu m, \Delta \lambda=0.290\mu m$), as well as an apodised Lyot coronagraph with a radius of 92 mas (N\_ALC\_YJH\_S) to reach a high contrast in the innermost regions. The broad-band H filter was selected for its wide spectral band-pass allowing to collect more disk photons, but also because the performance of the extreme adaptive optics system improves at longer wavelengths and the dust from debris disks typically displays a red colour, while the thermal background is not as high as in the K band and does not dominate the noise budget at large separations. The observations were made in pupil-stabilised mode, using the Angular Differential Imaging observing strategy. The targets were observed around meridian passage to ensure a large rotation of the field of view, with about 40 minutes long coronagraphic images. The observations were grouped in two programs, 46 sources were imaged during P96 (1 October 2015 - 31 March 2016) and 9 during P97 (1 April 2016 - 30 September 2016). Due to adverse observing conditions, multiple observation sessions were required for some targets, leading to an actual dataset of 73 ADI sequences. Table \ref{Datasets} provides details on the set of targets, including the number of observation sequences acquired for each target (epoch). The distances, magnitudes, and spectral types were taken from the Hippparcos and GAIA catalogues \citep{HIPPARCOS,GAIA}. The target Fomalhaut C, part of the SHARDDS sample, was excluded from our analysis because of poor observing conditions for all three epochs \cite[dataset published in][]{Cronin-Coltsmann2021}. \begin{table*}[!htbp] \caption{Name, coordinates, magnitude distribution, spectral-type, age and distance, along with the number of ADI sequences for each SHARDDS target. } \label{Datasets} \centering \footnotesize \begin{tabular}{lcccccccc} \hline Name & RA & DEC & V mag & H mag & Sp. type & Age (My) & Distance (pc) & \# Epochs \\ \hline HD\,105 & 00:05:53 & -41:45:11 & 7.53 & 6.19 & G0V & 30$^1$ & 38.85 & 1 \\ HD\,203 & 00:06:50 & -23:06:27 & 6.17 & 5.33 & F3V & 23$^2$ & 39.97 & 1 \\ HD\,377 & 00:08:26 & +06:37:00 & 7.59 & 6.15 & G2V & 170$^3$ & 38.52 & 1 \\ HD\,3003 & 00:32:44 & -63:01:53 & 5.09 & 5.16 & A0V & 30$^1$ & 45.89 & 1 \\ HD\,3670 & 00:38:57 & -52:32:03 & 8.21 & 7.15 & F5V & 30$^4$ & 77.58 & 1 \\ HD\,9672 & 01:34:38 & -15:40:34 & 5.61 & 5.53 & A1V & 40$^6$ & 57.08 & 1 \\ HD\,10472 & 01:40:24 & -60:59:56 & 7.61 & 6.69 & F2IV/V & 30$^7$ & 71.17 & 2 \\ HD\,10638 & 01:44:23 & +32:30:57 & 6.73 & 6.19 & A3 & 100$^8$ & 68.68 & 1 \\ HD\,13246 & 02:07:26 & -59:40:45 & 7.50 & 6.30 & F7V & 40$^9$ & 45.60 & 1 \\ HD\,14082B & 02:17:25 & +28:44:30 & 7.74 & 6.36 & G2V & 21$^9$ & 39.75 & 1 \\ AG-Tri & 02:27:29 & +30:58:24 & 10.12 & 7.24 & K8 & 23$^1$ & 41.05 & 4 \\ HD\,15257 & 02:28:10 & +29:40:09 & 5.29 & 4.82 & F0III & 1000$^8$ & 49.93 & 1 \\ HD\,16743 & 02:39:08 & -52:56:05 & 6.77 & 5.97 & F1III/IV & 200$^8$ & 57.94 & 1 \\ HD\,17390 & 02:46:45 & -21:38:22 & 6.47 & 5.63 & F3IV/V & 610$^{10}$ & 48.19 & 1 \\ HD\,21997 & 03:31:54 & -25:36:50 & 6.37 & 6.12 & A3IV/V & 30$^{11}$ & 69.64 & 1 \\ HD\,22179 & 03:35:30 & +31:13:37 & 8.93 & 7.49 & G5IV & 63$^{12}$ & 70.37 & 1 \\ HD\,24636 & 03:48:11 & -74:41:38 & 7.13 & 6.22 & F3IV/V & 30$^{13}$ & 57.05 & 1 \\ HD\,25457 & 04:02:37 & -00:16:08 & 5.38 & 4.34 & F6V & 70$^{13}$ & 18.77 & 1 \\ HD\,31392 & 04:54:04 & -35:24:16 & 7.61 & 5.89 & G9V & 3690$^{10}$ & 25.77 & 1 \\ HD\,35650 & 05:24:30 & -38:58:10 & 9.05 & 6.11 & K6V & 70$^1$ & 17.48 & 1 \\ HD\,274255 & 05:30:14 & -42:41:50 & 9.71 & 6.47 & M0V & 1000$^{14}$ & 19.15 & 1 \\ HD\,37484 & 05:37:40 & -28:37:34 & 7.25 & 6.29 & F3V & 30$^{15}$ & 59.10 & 2 \\ HD\,38207 & 05:43:21 & -20:11:21 & 8.47 & 7.55 & F2V & 534$^{16}$ & 110.99 & 1 \\ HD\,38206 & 05:43:22 & -18:33:26 & 5.73 & 5.84 & A0V & 30 $^{15}$ & 71.43 & 2 \\ HD\,40540 & 05:57:53 & -34:28:34 & 7.54 & 6.93 & A8IV & 170$^3$ & 88.26 & 1 \\ HD\,53842 & 06:46:14 & -83:59:29 & 8.62 & 6.40 & F5V & 30$^{17}$ & 57.87 & 1 \\ HD\,60491 & 07:34:26 & -06:53:48 & 8.14 & 6.14 & K2V & 500$^{18}$ & 23.51 & 1 \\ HD\,69830 & 08:18:24 & -12:37:55 & 5.95 & 4.36 & G8V & 5670$^{16}$ & 12.56 & 1 \\ HD\,71722 & 08:26:25 & -52:48:26 & 6.04 & 5.91 & A0V & 324$^3$ & 69.35 & 1 \\ HD\,73350 & 08:37:50 & -06:48:24 & 6.73 & 5.32 & G5V & 600$^{19}$ & 24.34 & 1 \\ HD\,76582 & 08:57:35 & +15:34:52 & 5.68 & 5.21 & F0IV & 538$^{20}$ & 48.80 & 1 \\ HD\,80950 & 09:17:28 & -74:44:04 & 5.86 & 5.92 & A0V & 138$^{20}$ & 77.34 & 1 \\ HD\,82943 & 09:34:51 & -12:07:46 & 6.53 & 5.25 & F9V & 430$^3$ & 27.61 & 4 \\ HD\,84075 & 09:36:18 & -78:20:41 & 8.59 & 7.24 & G2V & 40$^{13}$ & 64.10 & 1 \\ HD\,107649 & 12:22:25 & -51:01:34 & 8.78 & 7.76 & F5V & 17$^{21}$ & 108.34 & 1 \\ HIP\,63942 & 13:06:15 & +20:43:45 & 9.40 & 6.21 & K5 & 4500$^{22}$ & 18.80 & 1 \\ HD\,114082 & 13:09:16 & -60:18:30 & 8.21 & 7.23 & F3V & 17$^{21}$ & 95.69 & 1 \\ HD\,120534 & 13:50:40 & -31:12:23 & 7.02 & 6.33 & A5V & 320$^{17}$ & 86.81 & 3 \\ HD\,122652 & 14:02:32 & +31:39:39 & 7.15 & 5.94 & F8 & 500$^8$ & 39.54 & 2 \\ HD\,133803 & 15:07:15 & -29:30:16 & 8.12 & 7.36 & A9V & 16$^{21}$ & 110.74 & 2 \\ HD\,135599 & 15:15:59 & +00:47:46 & 6.91 & 5.12 & K0V & 1300$^{23}$ & 15.82 & 2 \\ HD\,138965&15:40:11& -70:13:40 & 6.42 & 6.34 & A1V & 348$^{27}$ & 78.08 & 1\\ HD\,145229 &16:09:26& +11:34:28&7.44 & 6.06 &G0& 650$^{26}$ &33.74&1\\ HD\,157728 & 17:24:06 & +22:57:37 & 5.72 & 5.22 & A7V & 100$^{17}$ & 42.74 & 1\\ HD\,164249A & 18:03:03 & -51:38:56& 7.01& 6.02 &F6V&1800$^{25}$ &49.60&1\\ HD\,172555 &18:45:26 &-64:52:16& 4.77 &4.25 &A7V& 20$^{28}$ &28.79 &1\\ HD\,181296 & 19:22:51 & -54:25:26 & 5.02& 5.15 &A0V&12$^{30}$& 47.37&1\\ HD\,182681 & 19:26:56 &-29:44:35 & 5.64 & 5.66 & B8.5V & 107$^{31}$ & 71.42 & 1\\ HD\,192758 & 20:18:16 & -42:51:36 & 7.03 & 6.30 & A5V & 45$^{17}$ & 66.53 & 2 \\ HD\,201219 &21:07:56&+07:25:58 & 0 & 46.5 &G5& 5370$^{10}$ & 37.89&1\\ HD\,205674 & 21:37:21 & -18:26:28 & 7.17 & 6.25 & F4IV & 850$^{10}$ & 56.40 & 2 \\ HD\,206893 & 21:45:22 & -12:47:00 & 6.67 & 5.69 & F5V & 250$^{29}$ & 40.80 & 1 \\ HD\,218340 & 23:08:12 & -63:37:41 & 8.44 & 7.07 & G3V & 2050$^{24}$ & 56.18 & 1 \\ HD\,221853 & 23:35:36 & +08:22:57 & 7.34 & 6.44 & F0 & 20$^{17}$ & 65.40 & 1 \\ \hline \end{tabular} \tablefoot{ For the definition of the star age multiple papers have been used: $^1$ \citep{Zuckerman04}, $^2$ \citep{Zuckerman01}, $^3$ \citep{Chen_2014}, $^4$ \citep{Moor11}, $^6$ \citep{Rodriguez12}, $^7$ \citep{Fernandez2008}, $^8$ \citep{Rhee_2007}, $^9$ \citep{Malo13}, $^{10}$ \citep{Casagrande11}, $^{11}$ \citep{Torres08}, $^{12}$ \citep{Metchev_2009}, $^{13}$ \citep{Zuckerman11}, $^{14}$ \citep{Meshkat17}, $^{15}$ \citep{dasilva09}, $^{16}$ \citep{Vican_2012}, $^{17}$ \citep{Moor06}, $^{18}$ \citep{King03}, $^{19}$ \citep{Taberno12}, $^{20}$ \citep{Zorec12}, $^{21}$ \citep{Mamajek02}, $^{22}$ \citep{West08}, $^{23}$ \citep{Mamajek08}, $^{24}$ \citep{Delgado14}, $^{25}$ \citep{HIPPARCOS}, $^{26}$ \citep{Kim05}, $^{27}$ \citep{Matthews18}, $^{28}$ \citep{Mamajek14},$^{29}$ \citep{Delorme2017}, $^{30}$ \citep{Smith08}, $^{31}$ \citep{Gullikson16}. } \end{table*} \section{Data reduction} \subsection{Pre-processing and extraction of environmental data} The first reduction steps consist in applying standard calibrations to the raw IRDIS images (sky subtraction, flat-field correction, and bad-pixel correction), and registering the frames. This was done using a dedicated pipeline in python \footnote{available at \href{https://github.com/jmilou/sphere\_pipeline.git}{https://github.com/jmilou/sphere\_pipeline.git}}. The frame registration was done using the four satellite spots imprinted on the IRDIS images by a specific waffle pattern applied on the deformable mirror of SPHERE \citep{Delorme2017,Galicher2018}. % The ouput of the pre-processing consists of a temporal cube of frames (individual detector integrations), cosmetically cleaned and recentered, called hereafter an ADI sequence. This cube is accompanied by the corresponding list of parallactic angles for the dedicated high-contrast image processing steps (see section \ref{subsec:Imageproc}). For the clustering of data and to guide the interpretation, we also extracted environmental data from either the adaptive optics telemetry\footnote{The SPHERE real time controller called SPARTA stores a summary of the adaptive optics telemetry during each observation. Those files are available on the ESO archive as described in \citet{Milli2017_SPHERE}. We developed an automatic script to query and analyse the SPARTA and ASM data available at \href{https://github.com/jmilou/sparta.git}{https://github.com/jmilou/sparta.git}} or the Astronomical Site Monitor (ASM) of the Cerro Paranal Observatory\footnote{\href{http://archive.eso.org/cms/eso-data/ambient-conditions.html}{http://archive.eso.org/cms/eso-data/ambient-conditions.html}}. We collected, among other, data on the seeing, coherence time, relative humidity, temperature, wind speed, and direction at various heights above the platform, Strehl ratio, precipitable water vapour. \subsection{Image processing} \label{subsec:Imageproc} The resulting corrected sets of ADI sequences have been cropped to a 199 $\times$ 199 pixels size, corresponding to the innermost region of the field of view (FOV). We consider angular separations below 1.25 arcsec to take advantage of the higher sensitivity of the RSM map algorithm in the region near the host star, while limiting the computation time. Indeed the increased performance of the RSM map algorithm compared to other PSF-subtraction techniques reduces above 1 arcsec \citep[see ][]{Dahlqvist20,Dahlqvist21,Cantalloube20}, which makes it less suitable for larger angular distances when considering its high computational cost. The computation time is also reduced by limiting the size of the ADI sequences to a maximum of 300 frames, relying on image binning when necessary. The binning procedure consists in the computation of a pixel-wise moving average of the derotated cube. The noise content of these ADI sequences should be reduced by the binning procedure via partial time-averaging. \subsection{Clustering} \label{subsec:Clustering} In order to take full advantage of the RSM algorithm, we rely on the Auto-RSM optimisation framework \citep[see ][]{Dahlqvist21b} to define the optimal sets of parameters for the PSF-subtraction techniques and the RSM algorithm itself. This optimisation pipeline being computationally expensive, we propose to apply it on a subset of targets representative of the whole dataset. The obtained optimal parametrisations can then be used to compute the RSM detection maps for all targets. \cite{Dahlqvist21b} showed a relatively high degree of similarity in the optimal parametrisations of both the PSF-subtraction techniques and the RSM algorithm, when using ADI sequences generated with the VLT SPHERE instrument. Dividing the SHARDDS dataset into multiple subsets should nevertheless allow us to account for small variations in the optimal parametrisations depending on the ADI sequence characteristics. The subdivision of the SHARDDS dataset in multiple subsets is based on a set of observables characterising the ADI sequences. The subdivision itself is done via the K-means algorithm \citep{Macqueen67}, a centroid-based clustering procedure aiming to find the centroids that minimise the within-cluster sum-of-squares, also called inertia. The K-means algorithm was selected as it provides a good estimate of the centroids position. This is a key element to define properly which ADI sequence within a cluster is the most representative. These centroids being often not associated to a sample, we define the most representative ADI sequences as the ones closest to the cluster centroids. Once defined, the Auto-RSM optimisation framework is applied on the selected set of ADI sequences. The optimal parametrisations are then used to compute the RSM detection maps for the remaining ADI sequences of each cluster, following the standard RSM map procedure. \subsubsection{Clustering parameters} The K-means algorithm needs to be applied on a set of parameters that characterise the properties of the ADI sequences. For our cluster analysis, we chose metrics providing information about the sequence, the observing conditions, and the noise distribution within the set of frames. This set of observables consists in the mean seeing, the Strehl ratio, the mean coherence time, the number of images, the total field rotation in term of parallactic angle, the raw contrast at 200, 500, and 700 mas, the autocorrelation timescale between images, the mean wind speed at 30 meters, and the wind driven halo strength and asymmetry\citep{Cantalloube20a}. The seeing, Strehl ratio, and coherence time are commonly used performance indicators to assess the observing conditions. Considering the 40 minutes integration time used for the SHARDDS survey, the number of images contained in the ADI sequence affects the sampling frequency, and therefore both the performance and the parametrisation of the PSF-subtraction algorithm (e.g. optimal number of principal components). The field rotation also impacts the performance because of the higher self-subtraction of the signal in the case of small field rotation. When mitigating self-subtraction, it translates into a reduced set of available images to compute the reference PSF. The raw contrasts were estimated by placing apertures of 1 Full Width at Half Maximum (FWHM, 43\,mas) diameter in the selected annuli and computing the ratio between the mean encircled flux and the stellar flux. The autocorrelation timescale between the ADI sequence images was estimated by considering the region between 300-600 mas, where the adaptive optics is affecting the most the performance. The flux within a one FWHM aperture was computed for each pixel in the selected region and for each image. An exponential function was then fitted on the temporal autocorrelation of these fluxes and its exponential factor was kept as a measure of the autocorrelation decay rate. We expect that a slower autocorrelation decay will result in lower performance. The wind-driven halo (WHD) strength and asymmetry were computed using the method presented in \cite{Cantalloube20a}. The WDH is a bright elongated structure centred on the coronagraph in high contrast images, due to uncompensated atmospheric turbulence. The WDH cannot be easily treated with standard PSF-subtraction techniques and affects therefore the achievable contrast at small angular separations (below 1000 mas). Along with the WDH, the low wind effect \cite[LWE,][]{Milli2018_LWE} is also a wind-driven phenomenon degrading the performance of high contrast imaging. LWE arises from uncorrected wavefront aberrations due to air temperature inhomogeneities in large telescope pupil, caused by the radiative cooling of the spiders, which dominates in the absence of wind. We included the wind speed at 30 m to account for this potential effect. The number of images included in the ADI sequences was identified as a key metrics for the definition of the optimal parametrisation during the development of the Auto-RSM framework. We have therefore decided to divide the SHARDDS dataset into two subsets before applying the clustering algorithm. We defined a threshold of 151 frames to separate the two subsets, as this value ensures that the standard deviation of the number of images within each subset is equivalent. This ensures a similar distribution in terms of sequence size within the two subsets. \subsubsection{Application and results of the K-means clustering} The K-means algorithm being based on Euclidean distance, the selected set of metrics must be standardised before applying the clustering algorithm, to avoid that metrics with larger values dominate the calculation. Before applying the K-means algorithm, we looked for possible multicollinearity between the selected set of observables. Relying on the variance inflation factor \citep[VIF][]{Belsley05} and Pearson correlations \citep{benesty2009pearson}, we removed the contrast at 200 and 700 mas, which led to multicollinearity, affecting potentially the definition of the clusters. The initialisation of the K-means algorithm consisting in the random selection of initial centroids, the results may lack consistency and differ from one estimation to another. The algorithm can also be affected by the order of the observables. In order to tackle these two issues, we initialised our estimation by running the K-means algorithm 100 times, selecting at each iteration a different permutation of the parameters. We then took the mean of these centroids positions to initialise the final cluster definition. We finally defined the number of clusters. This definition was based on the analysis of the evolution of the total squared distance between cluster members and their centroid when changing the number of clusters. Looking at Fig. \ref{clustdist}, we see that the largest fraction of the total squared distance reduction occurs between one and four clusters. We therefore selected for both subsets a number of clusters equal to four, implying a total of eight ADI sequences on which Auto-RSM will be applied. The eight cluster centroids, as well as the composition of their respective clusters are presented in Table \ref{Clusters}. \begin{table}[t] \caption{Subdivision of the SHARDDS dataset into eight clusters.} \label{Clusters} \centering \begin{tabular}{ll} \hline Cluster center & Cluster members \\ \hline \textbf{Cluster 1-1} &\\ HD\,192758 & HD\,38207, HD\,37484, HD\,10472, AG Tri,\\ & HD\,84075,HD\,192758 $2^{nd}$ epoch, HD\,274255 \\ \textbf{Cluster 1-2} &\\ HD\,3670 & HD\,37484 $2^{nd}$ epoch , HD\,22179, \\ & AG Tri $2^{nd}$ epoch, AG Tri $3^{rd}$ epoch, \\ &HD\,82943 $3^{rd}$ epoch ,HD\,114082 \\ \textbf{Cluster 1-3} &\\ HD\,201219 & HD\,53842, AG Tri $4^{th}$ epoch, HD\,218340, \\ &HD\,221853\\ \textbf{Cluster 1-4} &\\ HD\,14082B & HD\,82943, HD\,107649\\ \textbf{Cluster 2-1} &\\ HD\,21997 & HD\,24636, HD\,15257, HD\,10472 $2^{nd}$ epoch,\\ & HD\,145229 $2^{nd}$ epoch\\ \textbf{Cluster 2-2} &\\ HD\,206893 & HD\,40540, HD\,35650, HD\,31392, HD\,25457,\\ & HD\,17390, HD\,16743, HD\,9672, HD\,105, \\ & HD\,69830, HD\,71722,HD\,120534, HD\,182681,\\ & HD\,120534 $2^{nd}$ epoch, HD\,164249A\\ \textbf{Cluster 2-3} &\\ HD\,181296 & HD\,14082B $2^{nd}$ epoch, HD\,13246, HD\,203, \\ & HD\,60491, HD\,122652, HD\,135599 $3^{rd}$ epoch,\\ & HD\,145229, HD\,172555, HD\,181296\\ \textbf{Cluster 2-4} &\\ HD\,3003& HD\,377, HD\,73350, HD\,76582, HD\,80950,\\ &HD\,82943 $2^{nd}$ epoch, HD\,82943 $4^{th}$ epoch,\\ &HD\,138965, HD\,157728, HIP63942,\\ & HD\,122652 $2^{nd}$ epoch\\ \hline \end{tabular} \end{table} After the subdivision of the dataset into eight clusters, we made several consistency checks by relying on principal component analysis to reduce the dimensionality of our set of observables and eliminate residual correlations between the variables. We tested the K-means algorithm with different numbers of principal components and retrieved almost every time the same set of clusters. Figure \ref{ClusterProj} illustrates the repartition between the different clusters in the space formed by the first two principal components. As can be seen, the different clusters are relatively well defined except for cluster 2-2 and 2-4, for which a larger set of principal components are necessary to make a clear distinction. We finally applied a Gaussian mixture model instead of the K-means algorithm as a last consistency check. The Gaussian mixture model considers on top of the number of clusters and the centroid position, the standard deviation of the distance between cluster members to characterise clusters. The obtained cluster repartitions were very close although not exactly the same. Two targets were excluded from these clusters, HD\,133803 and HD\,205674. They were treated separately as they were imaged at two epochs separated by only a couple of days. We therefore took advantage of the ability of the RSM algorithm to deal with multiple ADI sequences at once to generate a single detection map per target. This was not possible for the other multi-epoch targets due to the longer time span separating the image sequences, implying a potential movement of planetary candidates. \subsection{High contrast image processing} \label{subsec:Detectmap} This section is devoted to the computation of RSM detection maps for all the targets included in the SHARDDS survey, as well as the computation of the contrast curves. This computation starts with the optimisation of the model parameters via the Auto-RSM framework for the eight selected targets (see cluster center in Table \ref{Clusters}). The Auto-RSM framework requires the selection of the PSF-subtraction techniques as well as the definition of the parameter ranges to be considered during the optimisation. We considered in this paper six different PSF-subtraction techniques: annular PCA \citep[APCA,][]{Gonzalez17}, non-negative matrix factorisation \citep[NMF,][]{Ren18}, the local low rank plus sparse plus Gaussian decomposition \citep[LLSG,][]{Gonzalez16}, locally optimised combination of images \citep[LOCI,][]{Lafreniere07}, and forward-model versions of KLIP \citep{Soummer12,Pueyo16}, and LOCI \citep[see ][for more details]{Dahlqvist21}. % The considered ranges of principal components for APCA, NMF, and KLIP, the ranks for LLSG, and the tolerance for LOCI are selected by a new function of the PyRSM python package\footnote{\url{https://github.com/chdahlqvist/RSMmap}}, which regroups the different functions of the Auto-RSM framework. This function studies the evolution of the contrast at different angular separations when modifying the number of principal components, ranks, or tolerance. The upper boundary of the considered ranges is defined as the value for which the contrast, averaged over the different angular separations, reaches a peak. In the case of APCA, NMF, LLSG, and KLIP, the obtained ranges were divided in two equal size ranges, to form two separate models. This should provide more diversity to the RSM algorithm and increase the framework's performance as planetary signals and residual speckle noise evolve differently with the number of principal components used to generate the reference PSF. Regarding the other parameters of the PSF-subtraction techniques, a single range was defined for all cluster centroids. The range for the number of segments was fixed to $[1,4]$, the FOV rotation threshold to $[0.25,1]$ and the crop size to [3,5] for standard PSF-subtraction techniques and [7,9] for the forward model versions to account for the side lobes due to self-subtraction \citep[see ][ for more details about these parameters]{Dahlqvist21b}. The computation of the PSF forward model being computationally very intensive and side lobes due to self-subtraction becoming fainter for increasing angular separation, we considered the forward model versions for only the first 400 mas. % Having defined all the parameters, the Auto-RSM optimisation framework was applied on each centroid, using the full frame mode to optimise the PSF-subtraction techniques and RSM algorithm parameters, the forward model to compute the RSM detection maps, and the bottom-up approach to select the optimal set of likelihoods \citep[see ][]{Dahlqvist21b}. Following the original Auto-RSM framework, the parameters optimisation was performed using the reversed parallactic angles. Considering the low probability of detecting a planet, we also tried to use the original parallactic angle to optimise the parameters, but it did not lead to a performance increase in terms of contrast. We therefore relied on reversed parallactic angles to avoid any potential planetary signal suppression during the optimisation process. We investigate in Appendix \ref{common} the similarities existing between the optimal parametrisations obtained for the eight cluster centroids, as well as the relationships between these optimal parameters and the set of metrics characterising the ADI sequences. The comparison of the optimal parametrisations is done via the computation of dissimilarity measures between cluster centroids, for both the PSF-subtraction techniques and the RSM algorithm. The results demonstrate a relatively high degree of similarity between the different parametrisations, confirming the conclusions drawn in \cite{Dahlqvist21b} about the high stability of the ADI sequence imaged by the VLT/SPHERE instrument. The Pearson correlations between the ten observables characterising our ADI sequences, and the PSF-subtraction techniques parameters show a sensible correlation for some observables, with the contrast at 500 mas showing the highest average correlation rate, and the exponent of the autocorrelation function the lowest one. \subsubsection{Detection maps} Following the definition of the optimal set of parameters for the cluster centroids, we computed the RSM detection maps for every target of the SHARDDS survey. Two sets of detection maps were computed using the original and the reverse parallactic angles. The detection maps with the reversed parallactic angles allowed the computation of a radially dependent residual noise estimate, which is subtracted from the detection map to account for the noise angular evolution (see Appendix~\ref{thresh} for more details about the radial threshold computation and its interpretation). The resulting detection maps were then analysed to uncover potential planetary signals or other bright structures. From this analysis, we rejected HD\,107649 due to the presence of extended speckle-like bright structures. For other targets, some redundant epochs presenting a high degree of residual noise were also removed\footnote{These ADI sequences include AG Tri, AG Tri $2^{nd}$ epoch, AG Tri $3^{rd}$ epoch, HD\,82943 and HD\,82943 $3^{rd}$ epoch}. From the remaining ADI sequences, we identified 16 targets containing a point-like source or an extended bright structure above a probability threshold of 0.05. To insure that these detections were not the result of a sub-optimal parametrisation of the RSM algorithm, we applied the Auto-RSM algorithm to 15 of these targets. From the set of 16 targets including detections above a 0.05 probability threshold, one was a cluster centroid (HD\,206893) for which we kept the original RSM detection maps. We performed a correlation analysis similar to the one made in Appendix \ref{common} on these 15 targets, in order to assess the influence of a stronger speckle field on the optimal parametrisations. We found much lower correlation rates between these optimal parameters and the set of metrics characterising the ADI sequences. We also observed a higher degree of dissimilarity between the parametrisations of these 15 targets, especially for the PSF-subtraction techniques parameters. These results highlight the limits of a clustering approach based solely on the parameters characterising the ADI sequence, when facing noisier samples. They also demonstrate the necessity to adopt an empirical approach, such as the Auto-RSM optimisation framework, to optimise the parametrisation when the samples noise structure cannot be well captured by the set of ADI sequence characteristics. However, the low residual noise level in the detection maps shown in Figures \ref{Empty_map1}-\ref{Empty_map3}, as well as the large fraction of the survey dataset (70\%) that did not require the use of Auto-RSM, still favour the use of a limited number of optimal parameter sets computed for well chosen targets. Following this individual optimisation, the analysis of the resulting 16 detection maps allowed the detection of three already known point-like sources that will be further analysed in the next section (see Figure \ref{Target_map}). The detection maps containing no plausible planetary candidates are shown in Appendix \ref{detmap}. As can be seen from Figures \ref{Empty_map1}-\ref{Empty_map3} , the residual noise level is most of the time very low, except for bright structures observed in HD\,53842 and HD\,80950. These structures are diffraction patterns due to the presence of a bright companion just outside the 199 $\times$ 199 pixels window considered in this analysis. For HD\,80950, the companion is situated at a projected separation of 130 au with an apparent magnitude in H band of 9.97. HD\,53842 is a very young binary system, with a primary spectral type F5 star and a secondary M-dwarf situated at a projected separation of 82 au, with an estimated orbital period of 300 years (C. del Burgo, in prep). \subsubsection{Contrast curves} \label{subsec:Contrastcurve} Following the computation of the detection maps, we relied on an optimised version of the approach proposed in \cite{Dahlqvist21} to compute contrast curves for every target. When relying on probability detection maps, standard signal-to-noise ratio (S/N) based approaches involving the estimation of the throughput and the noise standard deviation \citep{Mawet14} cannot be used. We replace this definition by an empirical estimation of the contrast corresponding to a predefined detection rate (also called true positive rate) computed at a specific threshold. As it is not possible to reach a 5$\sigma$ confidence level empirically, this threshold corresponds simply to the first detection of a false positive within the entire detection map. The detection rate is computed, for a given angular separation, via the injection of fake companions at different azimuths. The computation of the contrast follows an iterative procedure, where the contrast is increased or decreased depending on the obtained detection rate and the previously tested contrasts \citep[see ][for a detailed presentation of this iterative procedure]{Dahlqvist21}. We selected a detection rate of 95\%, which is the traditional completeness level for the computation of planet detection probability or occurrence rate (see Section \ref{sec:Sensitivity}). This detection rate requires the successive injection of 20 fake companions per considered annulus. We considered nine angular separations ranging from 60 to 1150 mas. From the original 73 ADI sequences forming the SHARDDS survey, we removed 13 ADI sequences because of poor observing conditions, and/or the existence of multiple epochs for several targets. For a few targets, several epochs were kept as they showed a similar level of residual noise. When multiple epochs where available, the lowest contrast was kept for each considered angular separation, to generate a single contrast curve per target. A radial basis multiquadric function (RBF) \citep{Hardy71} was then used to perform the interpolation between the nine angular separations for which a contrast was estimated. Figure \ref{ContrastCurve} provides a consolidated view of the contrast curves, with gray curves showing the individual contrast curves corresponding to each target and the red line providing the median. As can be seen from these curves, the contrast decreases quickly with the angular separation, with a median contrast below $10^{-5}$ at already 300 mas. % However, we observe a relatively high dispersion of the contrasts at close separations, with the contrast ranging from $3\times 10^{-1}$ to $3\times 10^{-4}$ at 100 mas. This high dispersion can be directly linked to the observing conditions. This relationship between the performance in terms of contrasts and the observing conditions will be further investigated in Section~\ref{sec:CCanalysis}. \section{Identification of planetary candidates } \label{sec:Identification} Figure \ref{Target_map} presents the two ADI sequences containing a signal above the previously defined threshold of 0.05, after having applied Auto-RSM on the 16 sequences for which a signal was previously detected. The two ADI sequences contain already known targets, with HD\,206893 B identified in \cite{Milli16b} , and the debris disk from HD\,114082 in \cite{Wahhaj16} which includes also two background stars. In the rest of the section, we propose a new way to extract the photometry and astrometry of point-like sources based on the RSM framework, and apply it to these two datasets. \subsection{RSM NEGFC algorithm} \label{subsec:RSMNEGFC} Like in the negative fake companion (NEGFC) approach \citep{Lagrange10,Marois10,Wertz17}, the astrometry and photometry are determined by injecting a fake companion at the expected position of the planet, with a negative flux providing the photometry. Multiple positions and fluxes are tested and their optimum is defined as the values minimising a loss function defined as the average probability inside an aperture of two FWHM centred on the expected location of the planet. The minimisation relies on a particle swarm optimisation (PSO) framework \citep{Kennedy95}. A series of particles, each defining a set of parameters, travel within the parameter space following an iterative procedure. At each step the velocity of these particles in the parameter space is updated based on the knowledge of the particle's own optimum and the global optimum of the entire swarm. The PSO framework was chosen as it showed, during tests, a higher convergence rate than Bayesian optimisation and allowed multi-core estimation\footnote{Multi-core optimisation is not possible with usual minimisation algorithms such as Nelder-Mead, Newton or Broyden-Fletcher-Goldfarb-Shanno.}, reducing the computation time. More standard minimisation frameworks (Nelder-Mead, Newton, or Broyden-Fletcher-Goldfarb-Shanno) were tested without success because of the non linear behaviour of the selected loss function near the optimum and the presence of multiple local optima. The inertia, the cognitive, and social coefficient parametrising the PSO algorithm help defining the right balance between exploitation of known minima and exploration of the parameter space. Several sets of parameters were tested and the set $[\alpha=0.5,\beta_p=1,\beta_g=1]$ was selected, as it led to a high convergence rate while avoiding local minima. The algorithm is initialised by relying on a detection map generated with the RSM map algorithm using the forward-backward mode, which considers both past and future observations to infer the detection probability. This mode has demonstrated a higher precision in terms of astrometry \citep[see ][]{Dahlqvist21}. Once an initial astrometry has been defined, a range of fluxes is tested to get an initial estimation of the photometry. The PSO framework is then used to minimise the average probability in the two-FWHM aperture centred on the expected position. We relied on ten particles with a maximum number of iterations equal to 20. At the end of the PSO minimisation, the global minimum is kept and a confidence interval is computed based on the computation of the inverted Hessian matrix\footnote{The Hessian matrix is calculated with finite difference derivative approximation.}. We tested additional versions of the planetary signal characterisation algorithm. We tried to subtract a local measure of the noise from the average probability within the two-FWHM aperture. This local noise was computed as the detection probabilities averaged over two sections of the annulus with a width of one FWHM containing the signal, situated at a distance of 1.5 FWHM on either sides of the expected target position. We did not consider the entire annulus, as local features may be observed in the detection map, leading to a potential bias. We also considered replacing the PSO minimisation by a Bayesian optimisation. We tested these different versions along with the NEGFC function provided by the VIP package \citep{Gonzalez17}, which relies on Nelder-Mead minimisation. We based our performance comparison on the ADI sequence obtained on HD\,3003, considering an intermediate angular separation of $4 \lambda/D$. We injected fake companions at eight different azimuths and considered eight different contrasts ranging from $1\times10^{-5}$ to $8\times10^{-5}$. This range goes from a non detection in a traditional S/N map (a detection just above the background with the RSM map) to a very bright planetary signal. This should allow us to investigate the behaviour of the planetary signal characterisation algorithms in two very different regimes. The astrometric error is computed as the root mean squared (rms) position error between the obtained position and the injected fake companion true position, averaged over the eight considered azimuths. The photometric error follows the same approach but comparing in terms of rms the estimated photometry and the true underlying one. Figure \ref{perf_algo} shows the evolution of the astrometric and photometric mean error with the contrast. The upper graph shows a higher performance of the PSO approach without local mean subtraction, except for the highest contrast value. When comparing with the NEGFC algorithm in Figures \ref{perf_algo2}, we see that our approach provides a more accurate estimation of the astromery and photometry for low contrast values, while breaking at high contrast values. This lower performance for very bright companions comes from the fact that a slight shift of the negative injected fake companion compared to the true underlying position, leads to the appearance of bright artefacts near the companion position, and therefore to a high loss function value which prevents its effective minimisation. This is explained by the very high sensitivity of the RSM map algorithm, which is a drawback in this particular case. A way to prevent this behaviour is to apply as an initialisation step the NEGFC algorithm and then use the RSM-based PSO approach. We see from Figures \ref{perf_algo2}, that this approach reduces drastically the error for very bright companions, while unfortunately decreasing the astrometric accuracy when facing faint signals (but increasing the overall photometric accuracy). The optimal solution would be one combining both approaches, relying on the NEGFC approach to initialise the PSO algorithm as from a given brightness threshold. \subsection{Point-source characterisation} \label{subsec:Characterisation} We applied the RSM-based planetary signal characterisation algorithm on the two targets for which signals were detected. The results are presented in Table \ref{Parameters}. Besides the astrometry and photometry, we estimated contrast curves for HD\,206893 at two additional completeness levels, 50\% and 5\%. This could further help us to classify the detected signal between planetary candidates and bright speckle, by considering its relative distance to these contrast curves. In contrast with S/N-based analysis, which relies on Gaussian assumption, there is no linear relationship between companion brightness and completeness level in RSM detection map. The distance between a companion and contrast curves estimated at different completeness levels should therefore give information about the uncertainty associated with the detection. The contrast curves were computed after removing the detected signal via the negative fake companion subtraction technique, using the parameters from Table \ref{Parameters}. Figure \ref{Target_cc} presents the contrast curves along with the detected signal positioned at its estimated contrast and angular separation. No contrast curves were computed for HD\,114082 pertaining to the difficulty of removing the disk via fake companion injections. \begin{table*}[t] \caption{Detected targets photometry and astrometry.} \label{Parameters} \centering \begin{tabular}{lcccc} \hline Target &Radial distance (mas) & Position Angle ($^{\circ}$)& Contrast & Epoch\\ \hline \textbf{Confirmed detections}&&\\ \hline HD\,206893 b & $266.58 \pm 3.25$ &$ 159.76 \pm 0.65$ & $ 4.59 \pm 0.37 \times 10^{-5}$&2015-10-05\\ HD\,114082 BKG (1) & $803.93 \pm 1.06$ & $332.10 \pm 0.08$ & $ 7.49 \pm 0.11 \times 10^{-6}$&2016-02-14\\ HD\,114082 BKG (2) & $1082.67 \pm 0.93$ & $56.75 \pm 0.05$ & $ 1.69 \pm 0.01 \times 10^{-5}$&2016-02-14\\ \end{tabular} \end{table*} We finally computed additional detection maps. We ran the Auto-RSM framework replacing the bottom-up approach by a top-down selection method to define the set of likelihoods cubes used to generate the final RSM detection maps. We also relied on the Auto-SNR framework \citep{Dahlqvist21b} to generate optimised S/N maps. This framework uses the optimised parameters of the Auto-RSM framework for the PSF-subtraction techniques, but relies on a dedicated function to select and combine the optimal set of S/N maps. We eventually computed S/N maps with APCA, NMF, LLSG, and LOCI and simply mean combined them to generate an averaged S/N map. All these detection maps are presented in Figures \ref{Target_map1}, with a yellow circle indicating the position of the detected signals.\\ \noindent\textit{HD\,206893}\\ The first detection of HD\,206893 B dates back to 2015 \citep{Milli16b}, with numerous papers devoted to its characterisation published since \citep[e.g.][]{Grandjean19,Kammerer21}. We see from Figures \ref{Target_cc} and \ref{Target_map1} that HD\,206893 B is a very bright companion, located well above the 95 \% completeness contrast curve, and visible in all detection maps. We estimate a contrast of $ 4.59 \pm 0.37 \times 10^{-5}$, which translates into a mass of $24.76^{+ 0.67}_{-0.62}$ $M_{Jup}$ and $33.22^{+ 0.37}_{-0.34}$ $M_{Jup}$ for respectively the AMES-COND and AMES-DUSTY evolutionary models, using the estimated stellar age of 0.25 Gy taken from Table 1. These estimated masses lie inside or close to the $[5 -30]$ $M_{Jup}$ range defined in\citep[][]{Kammerer21}, while the estimated angular separation of $266.58 \pm 3.25$ mas (10.88 au) is very close to the one determined for the same epoch in \cite{Milli16b}.\\ \noindent\textit{HD\,114082}\\ Although the RSM approach is not designed to unveil large structures, the debris disk around HD\,114082, first detected by \cite{Wahhaj16}, is clearly visible. Two point-like sources are also visible. They are situated at an estimated distance of $803.93 \pm 1.06$ mas and $1082.67 \pm 0.93$ mas from HD\,114082. These signals are visible in all detection maps from Fig.~\ref{Target_map1}. HD\,114082 being in a dense field, we rely on TRILEGAL stellar population model \citep{TRILEGAL} to infer the density of background stars around HD\,114082. This density is then used to estimate the probability of observing two or more background stars at a distance below $1082.67$ mas from HD\,114082, using a spatial Poisson point process. This probability is equal to $63.5$ \%, and increase to $88.5$ \% when considering the probability of observing one or more background stars. Considering these high probabilities and the high inclination of these objects compared to the debris disk, these detections are most likely background stars. A second-epoch follow-up and an astrometric analysis is presented in \citet{Engler2022} and confirmed that those two sources are background sources without proper motion.\\ \section{Contrast curves analysis} \label{sec:CCanalysis} The contrast curves computed in Section \ref{subsec:Contrastcurve} are used throughout this section as a measure of the ADI sequences quality, as well as a metric for the RSM map algorithm performance. \subsection{Influence of clustering} We start by comparing the contrast curves obtained for the cluster centroids and the ones obtained by applying the centroids optimal parameters on the remaining targets of the cluster. The comparison aims to determine if the cluster centroids, for which the optimal parametrisations were computed, do perform better than the other members of the cluster in terms of achievable contrast. This should provide an idea of how far from the optimum we are, the optimum being the case where Auto-RSM is applied on every target. We have estimated the difference between each of the members and their cluster centroid in terms of $\Delta$ mag\footnote{We expressed both contrast curves in terms of magnitude and then subtracted the magnitudes of the members from the one of the cluster center.}, and report in Figure \ref{Perfmembervscenter} the radial evolution of this measure averaged, for each cluster, over their set of members. Looking at the seven curves\footnote{The cluster composed of HD\,14082B, HD\,82943, HD\,107649 was not included in the analysis as two of the three cluster members were rejected, due to the presence of multiple extended speckle-like bright structures in the HD\,107649 detection map and the existence of better epochs for HD\,82943.}, the center seems to perform better for some clusters (see clusters 1-1, 1-3, and 2-3), while for others the cluster members show a higher performance (see clusters 1-2, 2-1, 2-2, and 2-4). Surprisingly, we observe on average a small increase of the performance in terms of contrast for the cluster members at close angular separations. The average performance gain is close to zero at larger separations. This seems to support the use of a reduced number of optimal parameters, as it does not seem to negatively impact the performance within the different clusters. We used the same approach to assess the necessity to rely on multiple optimal parametrisations instead of a single one for the entire survey. This allows us to investigate also the impact of the degree of dissimilarity between optimal parametrisations on the performance, measured in terms of contrast. We considered two sets of clusters, one set of clusters close in terms of parametrisation, cluster 1-1 and 1-2, and one set of clusters presenting a larger level of dissimilarity, cluster 2-3 and 2-4 (see Figure \ref{optiparamcompcluster}). We computed for cluster 1-1 and 2-3, a new set of contrast curves using respectively the optimal parameters of cluster 1-2 and 2-4 (obtained for respectively HD\,3670 and HD\,3003). We then estimated the difference between these new contrast curves and the contrast curves obtained with the optimal parametrisation of their own cluster centroid (respectively HD\,192758 and HD\,181296). These contrast differences, expressed in terms of $\Delta$ mag, are shown in Fig.~\ref{clusterdeltacomp}\footnote{All targets on which the auto-RSM framework had to be applied due to the presence of bright speckles in the first detection maps, were not included in this analysis.}. As can be seen from the mean curves, using the optimal parameters estimated for their own cluster centroid leads on average to a better performance, especially at small angular separation. We see also that the mean distance is larger for the cluster 2-3, which showed a higher degree of dissimilarity in Figure \ref{optiparamcompcluster}. These results highlight the added value, at close separation, of the definition of local optimal parametrisation via Auto-RSM. The reasons for this higher performance are twofold. First, regions with a high level of background residual noise are more difficult to treat and are therefore more sensitive to parametrisation. Secondly, Auto-RSM focuses mainly on close separations to optimise the model parameters, which explain its better performance at these distances compared to other approaches. This confirms the interest of computing several sets of optimal parameters for a large survey to account for dissimilarities in the ADI sequences' characteristics. \subsection{Influence of environmental parameters} We perform a similar correlation analysis as the one made in Appendix \ref{common}, but focusing here on the relationships existing between the parameters characterising the ADI sequences and the performance in terms of achievable contrast. We start by re-expressing every contrast curve in terms of magnitude and average these magnitudes over the set of considered angular distances. We then compute the Pearson correlations between the parameters characterising the ADI sequences and the median contrast, considering the entire SHARDDS dataset. As can be seen from Figure \ref{PerfParam}, the raw contrast at 500 mas, the Strehl and the WDH asymmetry show relatively high correlations and have the expected sign. A higher asymmetry of the WDH is indeed more difficult to treat by the PSF-subtraction techniques, which do not cope well with anisotropy in the speckle field. Despite their lower correlation, the other parameters show also the expected sign. As in Table \ref{paramcor}, the lowest correlation is associated to the autocorrelation measure, indicating that the decay rate of the autocorrelation function is not the best measure of the temporal relationships between the frames. In order to further investigate the relationship between the achievable contrast and parameters characterising the ADI sequences, we propose to rely on linear regression to highlight the parameters contributing the most to the quality of the ADI sequences. Considering the relatively low number of data points with only 60 fully treated observing sequences, and the potential co-linearities existing between the parameters, we rely on a bottom-up approach based on the Akaike information criterion \citep[AIC, ][]{Akaike74} to select one by one the parameters to be included in our model. The AIC provides a measure of the amount of information lost by a model. This measure includes a penalty term increasing with the number of parameters, providing a good trade-off between the model complexity and its goodness of fit. We start by computing the AIC for every parameter and select the parameter having the lowest AIC. We include this parameter to the model and compute again the AIC of this model after adding one at a time each of the remaining parameters. The parameter leading to the highest reduction of the AIC is then included in the model. This procedure is repeated until no more reduction of the AIC is observed. Table \ref{ParametersReg} gives the set of parameters that were selected using this method, along with the parameter values in the linear regression, their standard error, and p-value. We retrieve all three parameters that were already identified as highly correlated to the contrast in Figure \ref{PerfParam}, with in addition the wind speed showing a positive coefficient most probably attributable to the low wind effect. All the selected parameters show a high significance, especially the raw contrast at 500 mas and the WDH asymmetry. This highlights the importance of finding mitigation strategies to tackle the WDH to increase the quality of the ADI sequences \citep[see][]{Cantalloube20a}. With a $R^2$ adjusted for the number of parameters equal to 0.699, this simple model provides already a good indication of the expected contrast, relying on only four parameters that can be quickly computed or are already available in the metadata. \begin{table}[t] \caption{Linear regression coefficients, standard error, and p-value for the five parameters selected via the minimisation of the AIC with as dependent variable the contrast curve median values expressed in $\Delta$ mag. } \label{ParametersReg} \centering \begin{tabular}{lcccc} \hline Parameters &Coefficient&Standard error&p-value\\ \hline Contrast at 500 mas & 0.5863 & 0.082 & 0.000 \\ WDH asymmetry & -0.0468 & 0.013 & 0.001 \\ Strehl & 2.2234 & 1.020 & 0.034 \\ Wind speed & 0.0192 & 0.010 & 0.063 \\ \hline \end{tabular} \tablefoot{ The minimum AIC and the adjusted R$^2$ are respectively equal to 60.04 and 0.699.} \end{table} Following this analysis of the parameters driving the most the quality of the ADI sequences in terms of achievable contrast, we propose to look at existing observation quality ratings. In Figure \ref{Grading}, we report the different ADI sequences of the SHARDDS survey classified in terms of ESO observation quality grading and their respective mean contrast. As can be seen from this graph, apart from a single ADI sequence graded C showing a very low mean contrast, there are no major differences between the contrast distribution among the three grades. The ESO grading system used for this survey was mainly based on the seeing. A more robust multi-factor grading system was introduced in April 2018 \citep{Milli19}. However, a more HCI-oriented grading system based on a multi-factor linear regression, such as the one presented in Table \ref{ParametersReg}, could be an interesting tool to grade HCI observations at the telescope, and/or inform the post-processing of large surveys. \section{Survey sensitivity} \label{sec:Sensitivity} \subsection{Target detection probability} \label{subsec:Detectprob} The median contrast curves provide a good metric for the quality of the ADI sequences of the SHARDDS survey, and its relationship with the observing conditions. However, this analysis did not provide information about the global sensitivity of the SHARDDS survey to planets. In this section, we translate these contrast curves into upper limits on the detectability of planets depending on their semi-major axis and their mass, using respectively an astrodynamic and an evolutionary model. The astrodynamic model relies on Keplerian motion to determine the range of angular separations covered by a planet depending on its orbital elements. The evolutionary model describes how planets cool down over time depending on their mass. Different evolutionary models were developed and refined in the past decades. For the sake of continuity with previous studies, we choose two well-known models, namely the AMES-DUSTY \citep{Chabrier00} and AMES-COND \citep{Baraffe03} models. Both models assume planet formation via direct collapse of part of the disk due to gravitational instabilities. Disk instabilities are assumed to be the main scenario for the formation of giant planets and brown dwarfs at large distance from their host star (>10 au). The tables of cloud-free atmosphere AMES-COND, and dusty atmosphere AMES-DUSTY models for SPHERE were used to convert the contrast curves ($\Delta mag$) into planetary mass curves, knowing the age and the magnitude in H-band of the host star. Having computed the planetary mass sensitivity curves for all targets, we have now to determine the accessible range of angular separations corresponding to a given semi-major axis. This range of angular separations is used alongside the planetary mass curves to compute the detection probabilities for the set of masses and semi-major axis that form the grid points of the planetary detection probability map. We define the range of angular separations for a given semi-major axis, by computing the projected distance between the planet and the host star, as seen from the Earth, for multiple sets of randomly generated orbital elements (eccentricity, inclination, argument of the periapsis, longitude of the ascending node, and mean anomaly). The detailed computation of the projected angular separations is provided in Appendix \ref{Orbit}. For each target of the survey, 150 semi-major axes, ranging from 0.1 to 1 000 au and 100 planetary masses, ranging from 0.1 to 100 $M_{Jup}$ are uniformly distributed in log space to form our grid. For each point in the grid, 5000 sets of orbital elements are defined, using a uniform distribution for the inclination, the argument of the periapsis, the longitude of the ascending node, and the mean anomaly. For the inclination, we rely on a uniform distribution in sine to take into account the higher number of configurations for near edge-on orientations compared to face-on orientations, and ensure isotropy. The eccentricity follows a Beta distribution with parameters $\alpha=0.95$ and $\beta=1.30$, corresponding to the best fit to the full sample of wide substellar companions obtained by \cite{Bowler20}. The planetary detection rate is then computed for each target and each grid point, as the fraction of the 5000 drawn angular distances for which the considered mass lies above the planetary mass sensitivity curves. The obtained values are then averaged over the entire set of targets and multiplied by 0.95 to account for the selected completeness of the contrast curves. Figure \ref{detectprob} shows the resulting planet detection probability maps as a function of companion mass and semi-major axis. We see that higher detection rates are obtained for a semi-major axis range of $[10,100]$ au with masses above $10$ $M_{Jup}$. We have superimposed on this plot, the predicted planets derived from the dynamical constraints presented in \citet{Pearce22}. This study inferred the planet properties (mass, semi-major axis and eccentricity) if the inner edge of the disk is sculpted by one or several planets, and modelled the disk morphology based on ALMA, Herschel or the star spectral energy distribution (SED). We have plotted in Figure \ref{detectprob} the minimum masses and maximum semi-major axes of the planets predicted to be sculpting the inner edges of the disks if one planet is responsible in each of the 21 systems that are common between the SHARDDS sample and that of \citet{Pearce22}. These 21 targets are presented in Appendix \ref{app_disks}. These are the minimum masses and maximum semi-major axes that a single planet would need to sculpt the inner edge of the disk. Alternatively, a more massive planet located further inwards could also have the same effect. The planet masses could also theoretically be lower if multiple planets sculpt each disc, rather than just one planet, or if the inner edge of the disk is smaller than estimated. The disk inner edge was estimated from either a blackbody fit to the Spectral Energy Distribution (SED), or if available, from resolved observation with Herschel or ALMA \cite[see Fig.~9 left in][the data being reproduced here in Appendix \ref{app_disks}]{Pearce22}. Considering the conservative limits we computed for the detection probabilities (95 \% completeness), these planets are relatively close to the detection limit when considering the AMES-COND evolutionary model. Figure \ref{HD38206} shows the contrast curve of HD38206, the most favourable target in terms of mass and semi-major axis, translated into mass curves using the AMES-COND and AMES-DUSTY evolutionary models. We computed the probability distribution of the companion's expected projected separation, using the orbital elements provided in \citep{Pearce22} and assuming a Gaussian distribution for these different orbital elements. As can be seen, the mass curve obtained with AMES-COND is very close to the expected mass of the companion for the region with the highest probability for the projected separation. \subsection{Occurrence rate} \label{subsec:Occurfreq} The definition of planetary detection probabilities allows us to derive statistical constraints on the planet occurrence rate. We consider the statistical approach proposed by \cite{Lafreniere_2007} who build confidence intervals for the planet occurrence rate relying on a Bayesian approach. We start by defining the likelihood of observing a planet characterised by a mass $m\in[m_{min},m_{max}]$ and a semi-major axis $a\in [a_{min},a_{max}]$ around star $i \in [1,N]$\footnote{For the SHARDDS survey $N=53$ as we removed two targets from the initial set of 55 stars because of adverse observing conditions, i.e. Fomalhaut C and HD\,107649} as follows: \begin{eqnarray} \mathcal{L}([d_j]\vert f)=\prod_{i=0}^N (1-fp_i)^{(1-d_i)}(fp_i)^{d_i}\;, \end{eqnarray} where $f$ is the planet occurrence rate we are looking for, $p_i$ the previously derived planet detection probability, and $d_i$ the detections, with $d_i=1$ for the detection of a planet with $m\in[m_{min},m_{max}]$ and $a\in [a_{min},a_{max}]$ around target $i$. The occurrence rates are computed for specific points in the mass-semi-major axis space defined for the estimation of the planet detection probabilities. We replace therefore each of the ranges $m\in[m_{min},m_{max}]$ and $a\in [a_{min},a_{max}]$ by a single mass and semi-major axis point. Following Bayes' theorem, we estimate the posterior probability distribution from the likelihood and the prior probability distribution, which we set to $p(f)=1$, assuming no prior knowledge about the distribution of the occurrence rate. The posterior probability reads: \begin{eqnarray} p(f\vert [d_j])=\frac{\mathcal{L}([d_j]\vert f)p(f)}{\int_0^1 \mathcal{L}([d_j]\vert f)p(f)df}\;, \end{eqnarray} from which we derive the minimum and maximum occurrence rate at a given level of confidence $\alpha$ by solving: \begin{eqnarray} \frac{1-\alpha}{2}=\int_0^{f_{min}} p(f\vert [d_j])df\;,\;\;\;\;\;\; \frac{1-\alpha}{2}=\int^1_{f_{max}} p(f\vert [d_j])df\;. \end{eqnarray} These last expressions simplify for grid points where no detection has been made within the considered set of targets. This is the case for all grid points except the one associated with HD\,206893 B. The simplified expression provides only the maximum occurrence rate, $f_{max}$: \begin{eqnarray} \alpha=\int_0^{f_{max}} p(f\vert [d_j])df\;. \end{eqnarray} For each considered grid point, the occurrence rates are obtained via simplex minimisation using the Nelder-Mead approach imposing a confidence level $\alpha=0.95$. Figure \ref{occfreq} presents the upper limit of the companion occurrence rate obtained for the two considered evolutionary models, as a function of semi-major axis and mass. We see that the occurrence rate is especially low (below 10\%) for companion with masses above $20$ $M_{Jup}$ with a semi major axis ranging between $10$ and $60$ au, because of the high sensitivity of our survey to this region. The lower sensitivity towards the larger semi-major axis, and the sensitivity peak at $30$ au are explained by the stellar distances limited to 100 pc in the SHARDDS survey, as well as the field of view of 1.25 arcsec used in this study. Having considered a completeness level of 95\%, we discarded a large fraction of the cumulative probability distribution of the contrast versus the detection probability. This approach is therefore conservative as it considers the lower bound of the planet detection probability, providing an upper limit of the planet occurrence rates. \section{Conclusion} \label{sec:Conclusion} In this paper, we present an in-depth analysis of the SHARDDS survey in terms of point-source detection, based on the Auto-RSM framework. This framework is an automated optimisation algorithm relying on the RSM algorithm and multiple PSF-subtraction techniques to generate detection maps and unveil potential point sources. Although the SHARDDS survey was mainly designed to image bright debris disks in near-infrared scattered light, the detection of point sources may provide a better understanding of the interaction between planets and debris disks, and give information about the formation and evolution of circumstellar systems. Considering the computational cost of the Auto-RSM framework, as well as the high degree of similarity observed between the optimal parametrisations of different ADI sequences \citep[see ][]{Dahlqvist21b}, we decided to rely on clustering to reduce the number of required optimisations. We divided our dataset into eight clusters using K-means clustering algorithm, based on parameters characterising the ADI sequence itself and the related observing conditions. For each cluster, the most representative ADI sequence was selected and the Auto-RSM framework was applied on it. The generated set of optimal parameters for both the PSF-subtraction techniques and the RSM algorithm was then used to generate detection maps for all the ADI sequences contained in the cluster. The analysis of the obtained detection maps showed the presence of a higher number of bright speckles when reversing the parallactic angles, providing an important reminder that care should be taken when computing detection thresholds based on reversed parallactic angles. Based on the detection maps, we identified high-probability signals in only two ADI sequences: HD\,206893 B which had already been previously detected, and the bright debris disk around HD\,114082. Although these astrophysical objects had already been identified, we proposed a multi-factor detection and characterisation pipeline to confirm the detections and characterise the signals in terms of astrometry and photometry. Following the analysis of the detection maps, we computed for each target a contrast curve at a 95\% completeness level, subtracting the detected signal via the negative fake companion approach when necessary. The median contrast curve demonstrated the high performance of the Auto-RSM framework, reaching a contrast of $10^{-5}$ at 300 mas and $3\times 10^{-6}$ at 600 mas. These contrast curves were then used to assess the performance of the proposed clustering approach. Using the contrast as a performance metric, we found that on average the optimal parametrisation led to slightly higher performance for cluster members compared to cluster centroids. Shifting the optimal parametrisation between clusters led to lower performance in term of contrast, especially at close separation, highlighting the interest of a clustering approach to account for dissimilarities in the ADI sequences characteristics. The quality of an ADI sequence is also shown to be driven by some key observing condition metrics such as the WDH, the Strehl, the wind speed, or the raw contrast, which could allow to one develop a simple and efficient HCI-oriented grading measure. A planet detection probability map was then generated based on these contrast curves and on two different evolutionary models, AMES-COND and AMES-DUSTY. The planet detection probability map showed a high detection probability for a semi-major axis range of $[10,100]$ au with mass above $10$ $M_{Jup}$. We finally computed two planet occurrence rate maps based on the estimated detection probabilities, which showed a very small occurrence rate for companions with masses above 20 $M_{Jup}$ having a semi-major axis between $10$ and $60$ au. The analysis of the SHARDDS survey allowed the development of new tools as well as the improvement of the Auto-RSM framework, allowing it to gain in maturity and become a robust HCI post-processing pipeline, achieving good performance in terms of contrasts. \begin{acknowledgements} We dedicate this work to our friend an colleague Matthew Willson who tragically passed away on 18 January 2022. He will be missed by his family, friends, and colleagues. We thank Tim Pearce for sharing his planet mass and orbital parameters dataset. This work was supported by the Fonds de la Recherche Scientifique - FNRS under Grant n$^{\circ}$ F.4504.18 and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement n$^{\circ}$ 819155). This work has made use of the SPHERE Data Centre, jointly operated by OSUG/IPAG (Grenoble), PYTHEAS/LAM/CeSAM (Marseille), OCA/Lagrange (Nice), Observatoire de Paris/LESIA (Paris), and Observatoire de Lyon/CRAL, and supported by a grant from Labex OSUG@2020 (Investissements d’avenir – ANR10 LABX56). Jonathan P. Marshall acknowledges support from the Ministry of Science and Technology of Taiwan under grant MOST109-2112-M-001-036-MY3. \end{acknowledgements} \bibliographystyle{aa} \bibliography{SHARDDS.bib} \begin{appendix} \section{Parametrisation commonalities and relationship with ADI sequence characteristics} \label{common} Following the computation of the optimal set of parameters for the eight cluster centroids, we propose to investigate the similarities existing between these eight optimal parametrisations. We also consider the relationships existing between the centroids optimal parameters and the set of metrics characterising their ADI sequence. We start by comparing in Figure \ref{optiparamcompcluster}, the obtained optimal set of parameters via a normalised distance for the PSF-subtraction techniques and a dissimilarity index for the RSM algorithm. These measures were computed for each pair of cluster centroids and then averaged over the three possible pairs within each size subset (e.g. for HD\,192758, we have HD\,192758-HD\,3670, HD\,192758-HD\,201219, and HD\,192758-HD\,14082B). The normalised distance was computed considering the 19 parameters required by the ten selected PSF subtraction techniques. For each pair of cluster centroids, we computed the absolute value of the distance between their parameters and normalised them with the mean values of these pairs of parameters\footnote{For centroid A with 20 principal components for APCA and centroid B with 24 principal components, the normalised distance is equal to $4/22=0.18$}. We then averaged the resulting distances over the 19 parameters. The normalisation ensures a proper comparison between the different parameters when consolidating the distances. For the RSM algorithm, a dissimilarity metric replaces the normalised distance as most parameters are non-numerical. This dissimilarity index is simply computed as one minus the percentage of common RSM parameters between a pair of centroids, averaged over the five parameters of the RSM algorithm. Looking at the degree of similarity of the parametrisations within the two size subsets, Figure \ref{optiparamcompcluster} shows an overall higher degree of similarity. We observe a lower degree of dissimilarity for the RSM parametrisation and a lower normalised distance for the PSF subtraction-techniques for the centroids of the subset containing less than 151 observations. For the subset containing more than 151 observations, the slightly higher normalised distance pertain to the high degree of dissimilarity of HD\,181296, which affects strongly the averaged normalised distance. The main drivers of the dissimilarity is the number of segments used for APCA and LLSG\footnote{The number of segments correspond to the number of subdivisions of every annulus during the estimation of the reference PSF when relying on APCA and LLSG.}, the tolerance parameter of LOCI, and the method used to compute the residual speckle noise statistics within the RSM algorithm. These results tend to demonstrate the relatively high stability of the ADI sequence imaged by the VLT/SPHERE instrument and confirm the conclusions drawn in \cite{Dahlqvist21b}. The impact of the dissimilarities in the optimal parametrisations on the performance in terms of achievable contrast is further investigated in Section\ref{sec:CCanalysis}. We now turn to the analysis of the relationship existing between the parameters that we selected in Section \ref{subsec:Clustering} to describe our dataset and the parametrisations of the PSF-subtraction techniques\footnote{Such an analysis is not possible with the parametrisation of the RSM map algorithm as most parameters are non numerical.}. We computed the Pearson correlation between the ten parameters characterising our sample and the PSF-subtraction techniques parameters, considering the eight cluster centroids as data-points. The raw correlations show a significant correlation between these sets of parameters, with overall, around 25\% of the obtained values over 0.5. Table \ref{paramcor} gives the absolute values of the obtained correlations averaged over five classes of parameters, the number of principal components, the FOV rotation threshold, the number of segments, the rank of LLSG, and the tolerance of LOCI. Looking at these consolidated results, the contrast at 500 mas shows the highest average correlation rate, while the exponent of the autocorrelation function has the lowest one. Once averaged over the five considered classes, the percentage of consolidated correlations above 0.5 reach only 16\%, indicating the existence of some discrepancies between the different PSF-subtraction techniques relying on the same parameter. \begin{table*}[h!] \caption{Average absolute Pearson correlations between the PSF-subtraction techniques parameters and the parameters selected to characterised the SHARDDS survey dataset.} \label{paramcor} \centering \footnotesize \begin{tabular}{lcccccccccc} \hline Parameters & \# frames & Contrast & Auto-corr exp & PA & Coherence &Wind speed&Seeing&Strehl&WDH S&WDH A\\ \hline Principal components&0.44&0.41&0.32&0.44&0.36&0.28&0.37&0.39&0.33&0.54\\ FOV rotation threshold&0.32&0.54&0.36&0.16&0.35&0.7&0.37&0.13&0.55&0.16\\ Number of segment&0.41&0.34&0.22&0.42&0.33&0.42&0.29&0.45&0.15&0.24\\ Rank&0.36&0.29&0.14&0.33&0.19&0.14&0.19&0.41&0.17&0.51\\ Tolerance&0.49&0.6&0.17&0.71&0.24&0.34&0.64&0.31&0.21&0.13\\ \hline \end{tabular} \tablefoot{ WDH S and WDH A stand respectively for wind driven halo strength and asymmetry.} \end{table*} \FloatBarrier \onecolumn \section{RSM detection maps} \label{detmap} This section contains the RSM detection maps containing no plausible planetary signals.\\ \FloatBarrier \twocolumn \section{Threshold computation and interpretation} \label{thresh} The radially evolving residual noise measure subtracted from the detection map is estimated by taking, for each annulus, the largest value observed in the detection map generated with reversed parallactic angles. A polynomial fit is then applied on the obtained values to limit the influence of potential outliers (see below) and smooth the curve. This radial threshold is finally subtracted from the original detection maps and any negative value is set to zero. This subtraction reduces the background residual noise and therefore eases the detection of potential planetary candidates. This threshold should however not be considered as a sufficient condition to classify any signal above it as a planetary candidate. As can be seen from Figure \ref{Inverted}, bright structures may appear in the detection map generated with the reversed parallactic angles (right), which explains the use of a polynomial fit when estimating the threshold. Most of the time, the residual noise distributions are similar in the two detection maps, as illustrated with HD\,122652 (2$^{nd}$ epoch). But in some cases, very bright artefacts appear in the detection map with reversed parallactic angles although only a weak level of noise is visible in the original detection map (see HD\,157728). Considering all ADI sequences of the SHARDDS survey, around 20\% of the detection maps computed with the reversed parallactic angles show point-like sources or bright structures above a 0.05 threshold, while this percentage falls to 9\% for the original detection maps. It is therefore preferable to avoid using reversed parallactic angles to define a detection threshold. Detection maps generated with reversed parallactic angles may however be used to reduce the level of residual noise in the original detection maps, as described in this appendix. \FloatBarrier \newpage \section{Detection maps for planetary candidates} This appendix regroups the RSM detection maps obtained with Auto-RSM using either the bottom-up or top-down approaches to select the optimal set of likelihoods cubes (each likelihoods cube corresponding to a PSF-subtraction technique), as well as S/N maps generated via the Auto-S/N approach \citep{Dahlqvist21b} or obtained by averaging the S/N maps generated with APCA, NMF, LLSG, and LOCI, for the two samples containing a potential planetary signal. \FloatBarrier \section{Determination of projected angular separation} \label{Orbit} The projected angular separation is computed based on randomly generated orbital elements and on the predefined semi-major axis, relying on Keplerian motion. We first estimate the true anomaly, which is defined as the angle between the direction of the periapsis and the current position vector of the body in the perifocal plane. Its estimation starts by the definition of the mean anomaly, which provides the fraction of the elliptical orbit that was covered since the periapsis expressed in radian $[0,2\pi]$. The mean anomaly is linked to the eccentric anomaly by the following relationship: \begin{eqnarray} M=\frac{2\pi}{T} t=E-e\sin(E)\;, \end{eqnarray} with $T$ the orbital period and $e$ the eccentricity. This transcendental equation relating time and eccentric anomaly cannot be directly solved. However, there exists a unique solution for every value of the mean anomaly $M$. We rely on the expansion of $E$ in terms of Bessel functions to relate eccentric anomaly and mean anomaly \citep{Curtis}. \begin{eqnarray} E= M + \sum^{ \infty }_{n=1} \frac{2}{n}J_n (ne)\sin(nM)\;, \end{eqnarray} with $J_n(x)$ the Bessel function of the first kind. The sum over $n$ is truncated to $N=100$. The true anomaly $\theta$ is then computed via the following relationship: \begin{eqnarray} \theta= 2\tan^{-1} \left( \sqrt{\frac{1+e}{1-e}}\tan(\frac{E}{2})\right) \;, \end{eqnarray} Once the true anomaly has been estimated, the position vector in the perifocal frame is computed using the elliptic orbit equation: \begin{eqnarray} \bm{r}_p= \frac{h^2}{\mu}\frac{1}{1+e\cos(\theta)}(\cos(\theta)\bm{\hat{p}}+\sin(\theta)\bm{\hat{q}})\;, \end{eqnarray} where the coordinates are normalised, such as $\hat{p}=[1,0,0]$ and $\hat{p}=[0,1,0]$. Using $h=\sqrt{\mu a ( 1-e^2)}$, we get: \begin{eqnarray} \bm{r}_p= \frac{a ( 1-e^2)}{1+e\cos(\theta)}(\cos(\theta)\bm{\hat{p}}+\sin(\theta)\bm{\hat{q}})\;. \end{eqnarray} We project this position vector in the equatorial frame via three Euler rotations: \begin{eqnarray} \bm{r}_e= \left[ \bm{Q} \right] \bm{r}_p\;, \end{eqnarray} with the Euler rotations given by: \begin{eqnarray} \left[ \bm{Q} \right] = \left[ \bm{R}_3(w) \right] \left[ \bm{R}_1(i) \right] \left[ \bm{R}_3(\Omega) \right]\;, \end{eqnarray} where $i$ is the inclination, $w$ the argument of the periapsis, and $\Omega$ the longitude of the ascending node. The normalised distance to the star is then obtained by computing the norm of the position vector in the equatorial frame: \begin{eqnarray} r = \frac{}{} \Vert \bm{r}_e \Vert\;. \end{eqnarray} The angular separation expressed in mas is finally defined as the normalised distance to the star multiplied by the semi-major expressed in mas: \begin{eqnarray} a_{sep} = r a\frac{1000\times 3600 \times 180}{(206265 \pi d) }\;, \end{eqnarray} with $a$ the semi-major axis expressed in au and $d$ the distance from the star expressed in pc.\\ \newpage \onecolumn \section{Disks analysed in \citet{Pearce22}} \label{app_disks} \noindent There are 21 targets in common between the SHARDDS sample analysed in this paper and the sample of \citet{Pearce22}. We do not consider here Fomalhaut C, part of SHARDDS and in \citet{Pearce22} because of the very poor quality of the data. We present these targets in Table \ref{tab_pearce}, with the location of the disk inner radius used in the analysis by \citet{Pearce22} to estimate the planet minimum masses. \begin{table*}[h] \caption{The 21 common targets between SHARDDS and the sample analysed in \citet{Pearce22}. This table is an extract from Table A.1 in \cite[][see references therein]{Pearce22}} \label{tab_pearce} \centering \begin{tabular}{lcc} \hline Target & Disk data & Disk location and extent$^1$ \\ \hline HD203 & SED & $29 \pm 6$ \\ HD377 & SED & $60 \pm 10$ \\ HD3003 & SED & $21 \pm 6$ \\ HD3670 & SED & $100 \pm 20$ \\ HD9672 & ALMA & $62 \pm 4 \rightarrow 210 \pm 4$ \\ HD10472 & SED & $110 \pm 20$ \\ HD13246 & SED & $80 \pm 30$ \\ HD16743 & Herschel 100 $\mu$m & $50 \pm 50 \rightarrow 260 \pm 70$ \\ HD21997 & ALMA & $68 \pm 4 \rightarrow 120 \pm 4$ \\ HD25457 & SED & $45 \pm 8$ \\ HD37484 & SED & $70 \pm 20$ \\ HD38206 & ALMA & $0 \pm 20, 140^{+30} \rightarrow 190 \pm 30, 320^{+50}$ \\ HD69830 & SED & $0.8 \pm 2$ \\ HD107649 & SED & $15 \pm 3$ \\ HD114082 & SED & $29 \pm 6$ \\ HD135599 & SED & $49 \pm 9$ \\ HD172555 & SED & $15 \pm 3$ \\ HD181296 & SED & $81 \pm 10$ \\ HD192758 & Herschel 100 $\mu$m & $40 \pm 40 \rightarrow 180 \pm 50$ \\ HD218340 & SED & $140 \pm 40$ \\ HD221853 & SED & $47 \pm 9$ \\ \end{tabular} \tablefoot{ \tablefoottext{1}{The 'Disc location and extent' column describes the location and shape of the debris disc inner and outer edges: if the disc is resolved and fitted with an asymmetric model (case of HD38206), then the column shows the inner edge pericentre, $q_i$, inner edge apocentre, $Q_i$, outer edge pericentre, $q_o$, and outer edge apocentre, $Q_o$, as '$q_i$, $Q_i$ $\rightarrow$ $q_o$, $Q_o$'. Alternatively, if the disc is resolved and fitted with an axisymmetric model, then the column shows the disc inner edge, $a_i$, and outer edge, $a_o$, as '$a_i$ $\rightarrow$ $a_o$'. Finally, if the disc location is estimated from SED data, then only the corrected blackbody radius is shown.}} \end{table*} \end{appendix}

Title: The Uchuu-SDSS galaxy lightcones: a clustering, RSD and BAO study

Abstract: We present the data release of the Uchuu-SDSS galaxies: a set of 32 high-fidelity galaxy lightcones constructed from the large Uchuu 2.1 trillion particle $N$-body simulation using Planck cosmology. We adopt subhalo abundance matching to populate the Uchuu-box halo catalogues with SDSS galaxy luminosities. These cubic box galaxy catalogues generated at several redshifts are combined to create the set of lightcones with redshift-evolving galaxy properties. The Uchuu-SDSS galaxy lightcones are built to reproduce the footprint and statistical properties of the SDSS main galaxy survey, along with stellar masses and star formation rates. This facilitates direct comparison of the observed SDSS and simulated Uchuu-SDSS data. Our lightcones reproduce a large number of observational results, such as the distribution of galaxy properties, the galaxy clustering, the stellar mass functions, and the halo occupation distributions. Using the simulated and real data we select samples of bright red galaxies at $z_\mathrm{eff}=0.15$ to explore Redshift Space Distortions and Baryon Acoustic Oscillations (BAO) utilizing a full-shape analytical model of the two-point correlation function. We create a set of 5100 galaxy lightcones using GLAM N-body simulations to compute covariance errors. We report a $\sim 30\%$ precision increase on $f\sigma_8$, due to our better estimate of the covariance matrix. From our BAO-inferred $\alpha_{\parallel}$ and $\alpha_{\perp}$ parameters, we obtain the first SDSS measurements of the Hubble and angular diameter distances $D_\mathrm{H}(z=0.15) / r_d = 27.9^{+3.1}_{-2.7}$, $D_\mathrm{M}(z=0.15) / r_d = 5.1^{+0.4}_{-0.4}$. Overall, we conclude that the Planck LCDM cosmology nicely explains the observed large-scale structure statistics of SDSS. All data sets are made publicly available.

https://export.arxiv.org/pdf/2208.00540

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} methods: numerical -- large-scale structure of Universe -- surveys -- galaxies: haloes -- dark matter \end{keywords} \section{Introduction} \label{sec:introduction} Galaxy redshift surveys, measuring the spatial distribution of galaxies throughout cosmic time, have proven to be key observational probes for constraining cosmological models and astrophysical phenomena. For example, the distribution of galaxies on large scales can be used to infer cosmological parameters \citep[e.g.][]{Alam21}, as well as to constrain the properties of dark energy that drives the accelerated expansion of the Universe \citep{Riess98,Perlmutter99}. On smaller scales, the galaxy clustering signal encodes information on how galaxies populate dark matter haloes, on star formation, feedback and other baryonic processes that shape galaxy formation \citep[see][for a review]{Wechsler18}. In order to connect galaxy redshift surveys to theoretical predictions, it is essential to generate high-fidelity galaxy catalogues from cosmological simulations that capture the expected properties and clustering of the observed galaxy sample \citep[e.g.][]{delatorre13,White14,RodriguezTorres16,Lin20}. Hence, they can be used to increase the amount of information that can be extracted from galaxy survey data. Firstly, the cosmological parameters of the simulation are known exactly. This allows us to assess the systematic and statistical errors of cosmological measurements from galaxy surveys, to compute the covariance errors, to test the performance of statistical analyses, and to aid the theoretical interpretation of survey results. Secondly, high-fidelity catalogues based on simulations allow us to assess the systematics arising from such observational effects as selection function and fibre collisions. These effects are a source for incompleteness in the survey sample. They must to be understood in order to minimise their impact in the measured clustering signal \citep[see][and references therein]{Smith19}. In the last two decades, vast observational efforts such as the 2dF Galaxy Redshift Survey (2dFGRS) \citep{Colless01} and the Sloan Digital Sky Survey (SDSS) saga \citep{Alam21}, have driven most of the major discoveries about the Large-Scale Structure (LSS) of our Universe. The relevance of the SDSS surveys is not purely historical -- to date, SDSS continues to be the largest reference galaxy database, offering sky positions, redshifts, spectra and images for millions of galaxies. New technological advances continue to push the depth and volume of galaxy surveys -- the new generation of galaxy surveys, which include the Dark Energy Spectroscopic Instrument (DESI) survey \citep{DESI16}, the Large Synoptic Survey Telescope (LSST) \citep{Ivecic19}, the Subaru Prime Focus Spectrograph (PFS) \citep{Takada14} and the Euclid survey \citep{Laureijs11}, aim to produce unprecedentedly large data sets in an effort to map the Universe to even higher precision. In order to generate high-fidelity simulated galaxy lightcones for these large surveys, cosmological simulations with high-resolution in a large volume are needed. Although it would be desirable to create mocks from hydrodynamical simulations, in which the formation of galaxies is modelled self-consistently by solving the coupled evolution of both baryons and dark matter (DM), such simulations are complex and computationally expensive. The largest to date do not exceed a few hundred Mpc in box size \citep[e.g.][]{Dubois14,Schaye15,Pillepich18,Springel18}. Hydrodynamical simulations are also strongly affected by the modelling uncertainties of complex baryonic processes. Thus, one usually resorts to DM-only simulations. Since galaxy formation physics is not included, dark matter halos must be populated with galaxies in order to produce the desired simulated galaxy catalogue. This can be done using a variety of methods, each comprising different assumptions about the galaxy--halo connection \citep{Wechsler18}.% A common option to generate galaxies from DM halos is to use empirically based methods, such as subhalo abundance matching \citep[SHAM; e.g.][]{Marinoni02,Vale04,Kravtsov04,Conroy06,RodriguezTorres16,Safonova21}. In its simplest form, the main underlying assumption is that every halo and subhalo contains a galaxy, and that there are correlations between halo and galaxy properties. For instance, the most massive and luminous galaxies are generally assumed to reside in the most massive haloes. At face value, one can assign galaxies to haloes by generating a rank-ordered relation between observed galaxy luminosities and simulated halo masses. However, such a one-to-one relation between galaxies and halos is incompatible with observations~\citep[e.g.][]{Trujillo-Gomez11,Shu12} -- an intrinsic scatter must be incorporated in the method to produce realistic catalogues. Furthermore, the clustering of haloes is observed to depend on properties other than halo mass, including the halo formation time, concentration and spin~\citep[e.g.][]{Wechsler01, Gao05, Wechsler06}. Empirical models such as SHAM or the popular HOD method \citep[e.g.][]{Zehavi11} are computationally fast, and are able to workaround the uncertainties in the physics of galaxy formation by constraining the model parameters directly with data. Applications of the SHAM method are able to reproduce observed properties of galaxies in large surveys as the luminosity and stellar mass functions or the luminosity and colour-dependent clustering to high accuracy \citep[e.g.][]{Trujillo-Gomez11,RodriguezTorres16}. From real and simulated survey data, the nature of dark energy can be probed for instance by measuring the growth rate of structure, defined in linear theory as $f(a) = d \mathrm{ln}D(a)/d \mathrm{ln}a$, where $a$ is the scale factor and $D(a)$ is the linear growth function. This parameter can be regarded as a measure of the energy content of the Universe, allowing us to constrain different theories of gravity and dark energy \citep[see e.g.][for a review]{Peebles1980_book,Guzzo2008}. % Galaxy peculiar velocities introduce anisotropies in their observed redshifts. This effect, first described in \citet{Kaiser1987}, is known as Redshift Space Distortions (RSD). Measurements of RSD in galaxy surveys yield $f\sigma_8$, where $\sigma_8(z)$ is the normalization of the power spectrum at redshift $z$ on a scale of $8~\hMpc$. Another source of anisotropy comes from the choice of the fiducial cosmology adopted in the clustering analysis, used for converting redshifts and angles to comoving coordinates. If the fiducial cosmology differs from the observed one, galaxy clusters will appear flattened or elongated% . The study of this so-called Alcock-Paczynski effect can provide an additional source of cosmological information from the data, as well as a means to validate a cosmological model from a simulated galaxy survey. In addition to RSD, one can measure the Baryon Acoustic Oscillation (BAO) feature from the two-point clustering statistics in galaxy surveys% . % This allows us to determine the expansion rate of the Universe \citep[e.g.][]{Eisenstein:2005su,Cole05}. Recently, \citet{Alam21} reported cosmological implications from two decades of SDSS spectroscopic surveys based on clustering measurements from galaxies and quasars in the redshift range $0.1 < z < 3$. % This includes the SDSS DR7 main galaxy sample (MGS), at low redshift $z\sim0.1$, which clustering statistics is compared in this work with the high-fidelity simulated galaxy light-cones built from on our 2.1 trillion $N$-body Uchuu simulation \citep{UchuuDR1}. Our Uchuu-SDSS light-cones are built using the SHAM method to reproduce the basic properties of the SDSS galaxy population, match its sky footprint and selection function, and include the effect of fibre collisions, which facilitates their straightforward comparison with the SDSS MGS data. Furthermore we study and measure RSD and BAO in the real and simulated data to test the Planck base $\Lambda$CDM cosmology model \citep{Planck18}. The Uchuu-SDSS galaxy catalogues include sky positions, redshifts, $ugriz$ apparent and absolute magnitudes stellar masses and star formation rates (SFR), as well as several halo properties. The catalogues are made publicly available at the Skies \& Universes website\footnote{\url{http://www.skiesanduniverses.org/Simulations/Uchuu/}}. The structure of this paper is as follows. In Section~\ref{sec:sdss}, we introduce the SDSS MGS data and define the volume-limited galaxy samples used to validate our simulated galaxy catalogues. In Section~\ref{sec:mock_construction}, we describe the Uchuu simulation and the methodology behind the creation of galaxies from the dark matter halo properties. This includes the construction of light-cones from cubic boxes, the implementation of fibre collisions, and the assignment of additional SDSS galaxy properties. Section~\ref{sec:properties} presents the basic properties of the Uchuu-SDSS light-cones, compared to the SDSS data. In particular, we explore the statistics of several galaxy properties, the galaxy clustering dependence on luminosity, color, and stellar mass, and the galaxy HOD. In Section~\ref{sec:RSD_BAO}, we present our RSD and BAO measurements both in the real and simulated data, and compare them with the fiducial Planck cosmology. Finally, in Section~\ref{sec:conclusions}, we present a summary of our results. \section{SDSS Galaxy Samples} \label{sec:sdss} \subsection{Parent and volume-limited samples} \label{subsec:sdss_samples} \begin{table} \centering \begin{tabular}{cccccc} \hline $^{0.1}M_r^\mathrm{max}$ & $z_\mathrm{max}$ & $N$ & $n_g$ & $V_\mathrm{eff}$ & $f_\mathrm{NN}$ \\ \hline -18.0 & 0.041 & 35359 & 31.95 & 1.11 & 0.043 \\ -18.5 & 0.053 & 49272 & 20.41 & 2.54 & 0.046 \\ -19.0 & 0.064 & 62534 & 14.47 & 4.55 & 0.050 \\ -19.5 & 0.085 & 112652 & 11.09 & 10.68 & 0.058 \\ -20.0 & 0.106 & 119734 & 6.13 & 20.53 & 0.062 \\ -20.5 & 0.132 & 112496 & 3.03 & 39.03 & 0.068 \\ -21.0 & 0.159 & 71795 & 1.13 & 66.95 & 0.079 \\ -21.5 & 0.198 & 33505 & 0.28 & 125.63 & 0.099 \\ -22.0 & 0.245 & 9820 & 0.045 & 218.32 & 0.149 \\ \hline \end{tabular} \caption{For each SDSS volume-limited sample, the columns list (from left to right): the absolute magnitude threshold, the maximum redshift, the number of galaxies, the galaxy number density (in $10^{-3}~h^{3}\mathrm{Mpc}^{-3}$), the effective volume of each sample (in $10^{6}~h^{-3}\mathrm{Mpc}^{3}$), and the fraction of galaxies requiring the nearest neighbour correction. The lower redshift cut for all samples is $z_\mathrm{min}=0.02$. Magnitudes of all galaxies are $k$–corrected and passively evolved to the survey's median redshift of $z = 0.1$ and computed assuming $h=1$.} \label{tab:sdss} \end{table} In this work, we build and compare Uchuu simulated galaxies to observational data from the seventh data release \citep[DR7;][]{Abazajian2009} of the SDSS \citep[][]{York00}. More specifically, we make use of the large scale structure catalog of the SDSS MGS from the NYU Value Added Galaxy Catalogue \citep[NYU-VAGC;][]{Blanton2005}. We restrict our sample to consist only of galaxies in the contiguous northern footprint in regions which have a completeness of $> 90\%$. This parent sample covers an effective area of $\sim6511\ \mathrm{deg}^2$ and contains $\sim497\,000$ galaxies with Petrosian $r$-band magnitudes in the range $14.5<r_\mathrm{pet}<17.6$. In this sample, $\sim6\%$ of targeted galaxies lack a spectroscopically measured redshift due to fibre collisions. We apply a nearest neighbour correction to these galaxies, assigning to them the redshift of the galaxy with which they collided. Apparent magnitudes can be linked to absolute magnitudes via the distance modulus equation, \begin{equation}\label{eq:appmag} r = \magr + 5\log_{10}D_L(z)+ 25 + \kcorrr(z), \end{equation} where $D_L(z)$ is the luminosity distance (in units of $\hMpc$), $\kcorrr(z)$ is the $k$-correction, which takes into account the shift in the bandpass with redshift. Throughout this paper, we denote $k$-corrected unevolved absolute magnitudes in the $r$-band as $\magr \equiv {}^{0.1}M_r - 5\log_{10} h$, where the superscript 0.1 indicates that the rest-frame magnitude has been $k$-corrected to a reference redshift of $z_\mathrm{ref}=0.1$ \citep{Blanton2003b}. A similar notation is used for magnitudes in other bands. We also denote the rest-frame $g-r$ colours as ${}^{0.1}(g-r)$, $k$-corrected to the same reference redshift. Additionally, the absolute magnitudes are corrected for passive evolution using the model of \citet{Blanton2006}. Specifically, the `evolved' magnitude ${}^{0.1}M_r^{h,e}$ is given by \begin{equation} {}^{0.1}M_r^{h,e}=\magr + E(z), \label{eq:e-correction} \end{equation} where the evolution correction $E(z)$ is given by \begin{equation} E(z) = q_0(1+q_1(z-q_{z0}))(z-q_{z0}). \end{equation} This model contains three parameters which we set to values of $q_0=2$, $q_1=-1$, and $q_{z0}=0.1$. From this parent sample of galaxies, we construct nine different volume-limited samples, each of which is complete down to a specified $r$-band magnitude, $^{0.1}M_r^\mathrm{max}$, which is $k$-corrected and corrected for evolution. For our samples, we use the same magnitude thresholds and maximum redshifts as \citet{Guo15}. In all samples, to minimize the impact of peculiar velocities and spureous objects, we keep only galaxies with redshifts $z>0.02$. In Table~\ref{tab:sdss}, for each sample we provide the magnitude threshold, maximum redshift, number of galaxies, number density of galaxies, and effective volume. These quantities can be compared directly to those in Table 1 of \citet{Guo15}. We also show the fraction of galaxies in each sample which require the nearest neighbour correction due to fibre collisions, see Section~\ref{subsec:fibre} for the details. Finally, in order to study in greater detail the BAO signal in the SDSS MGS, we define a BAO sample in which this signal is enhanced, analogously to \citet{Ross15}. This sample, named SDSSbao, is defined by the cuts, % \begin{equation} \begin{array}{l} 0.07 < z < 0.2 \\ \magr<-21.2\\ {}^{0.1}(g-r)>0.8\\ 14.5 < r < 17.6\\ c>0.9, \end{array} \label{eq:hod_cuts} \end{equation} where $c$ is the completeness. \subsection{Stellar masses and star formation rates} \label{sec:stellar_mass_sfr} Uchuu-SDSS galaxy catalogues list stellar masses $M_\ast$ and star formation rates SFRs using two independent sources, which are the MPA/JHU catalogue\footnote{\url{ http://www.mpa-garching.mpg.de/SDSS/DR7/}} (the Max Planck Institute for Astrophysics and the Johns Hopkins University) and the Granada Group catalogue\footnote{\url{https://www.sdss.org/dr17/spectro/galaxy_granada/}. We use the data from stellarMassFSPSGranEarlyDust table.} For the MPA/JHU catalogue, $M_\ast$ was calculated following the methodologies presented in \citet{Kauffmann03} and \citet{Salim07}. The stellar masses in the MPA/JHU catalogue, have been found to be consistent with other estimates \citep[e.g.][]{Taylor11,Chang15,Duarte15,Leslie16}. For galaxies classified as star forming (SF) in the MPA/JHU catalogue, SFRs are calculated using the nebular emission lines within the spectroscopic fibre aperture of $3\arcsec$ as described in \citet{Brinchmann04}. SFRs are calculated using the empirical calibration of $\mathrm{H}_\alpha$~emission lines \citep{Kennicutt98} and corrected from the dust extinction with the Balmer decrement $\mathrm{H}_\alpha/\mathrm{H}_\beta$ \citep{Charlot00}, assuming a Kroupa initial mass function \citep{Kroupa01}. The SFR contribution outside of the fibre is estimated from the galaxy photometry following \citet{Salim07}. Finally, the specific star formation rate (sSFR, defined as SFR/$M_\ast$) were calculated by combining the SFR and stellar mass likelihood distributions as outlined in Appendix A of \citet{Brinchmann04}. % Throughout the paper we use the median values of the resulting probability distribution functions as the sSFR of a galaxy. For the Granada Group catalogue, $M_\ast$ and SFR are calculated using the SDSS spectroscopic redshift and $ugriz$ magnitudes by means of broad-band spectral energy distribution (SED) fitting via Flexible stellar population synthesis technique (FSPS, \citealp{Conroy09}). Our Uchuu-SDSS galaxy catalogues include $M_\ast$ and sSFR obtained from the early formation time and dust attenuation model \citep{Charlot00} with \citet{Kroupa01} initial mass functions. % \section{Constructing the Uchuu-SDSS catalogues} \label{sec:mock_construction} \subsection{The Uchuu Simulation} The Uchuu simulation is a large high-resolution $N$-body cosmological simulation, the largest simulation in the Uchuu suite~\citep{UchuuDR1}. It follows the evolution of $2.1$ trillion ($12\,800^3$) dark matter particles with particle mass resolution of \SI{3.27e8}{\per\h\solarmass} in a (\SI{2.0}{\per\h\giga\parsec})$^3$ comoving periodic box. The simulation adopts the Planck \textLambda CDM cosmological parameters: $\Omega_\mathrm{m} = 0.3089$, $\Omega_\mathrm{b}=0.0486$, $\Omega_\Lambda=0.6911$, $h=0.6774$, $n_\mathrm{s}=0.9667$, and $\sigma_8=0.8159$~\citep{Planck16}. Starting at $z=127$, the subsequent gravitational evolution was solved down to $z=0$ using the TreePM code \textsc{GreeM}~\citep{Ishiyama09,Ishiyama12}, with a gravitational softening length of \SI{4.27}{\per\h\kilo\parsec}. A set of 50 particle snapshots ranging from $z=14$ to $z=0$ were saved, from which bound structures were identified by running the \textsc{RockStar} phase-space halo/subhalo finder~\citep{Behroozi13}. Then merger trees were constructed using the \textsc{Consistent Trees} code~\citep{Behroozi2013b}. All Uchuu data products are publicly available. The characteristics of Uchuu make it ideal for the creation of simulated galaxy catalogues. Its high resolution allows to resolve dark matter haloes down to small halo masses on a large volume. This renders the simulation suitable for the application of SHAM, which requires that subhaloes are identified. The very large volume of Uchuu allows for detailed statistics, as well as the study of large-scale clustering features such as the BAO. \subsection{Box galaxy catalogues} \label{subsubsec:luminosity_SHAM} In order to construct Uchuu galaxy lightcones, we first start with applying SHAM algorithm to assign luminosities to all halos and subhaloes in simulation boxes. Following the basic principle of SHAM, we assign galaxy luminosities to DM haloes by matching the galaxy luminosity function to the cumulative distribution function of a halo property that serves as a proxy of galaxy stellar mass. A possible choice for this halo property is the maximum circular velocity, $V_\mathrm{max}$ \citep[e.g.][]{Conroy06,Trujillo-Gomez11}, defined as the maximum value of the halo circular velocity at the redshift of interest, \begin{equation} V_\mathrm{max}(z)=\max\left(\sqrt{\frac{GM(<r,z)}{r}}\right). \end{equation} Using $V_\mathrm{max}$ generally yields better results compared to the halo mass. Since the maximum velocity is generally reached at smaller scales compared to the halo radius, it characterises both the halo concentration \citep{Campbell18} and the depth of the potential at the typical galactic scales \citep{Chaves-Montero16}. It is also less affected by the tidal stripping suffered by subhaloes upon being accreted by larger haloes \citep{Hayashi03}. Another typical choice, which we adopt in this work, is to use the peak circular velocity, $V_\mathrm{peak}$, defined as the peak value of $V_\mathrm{max}$ over the history of the halo. This makes estimates that reproduce even more closely the properties of observed data \citep[e.g.][]{Reddick13,Chaves-Montero16,Safonova21}. The subhaloes in Uchuu are $90\%$ complete down to $V_\mathrm{peak} \gtrsim \SI{70}{\kilo\meter\per\second}$, while distinct haloes are $90\%$ complete down to $V_\mathrm{peak} \gtrsim \SI{50}{\kilo\meter\per\second}$. This allows us to reach low galaxy luminosities in our Uchuu galaxy catalogues. We match $V_\mathrm{peak}$ with an observationally motivated target luminosity function, $\phi_\mathrm{target}$. We adopt a recipe closely following \citet{Smith17}, which interpolates between the measured luminosity function from SDSS, $\phi_\mathrm{SDSS}$~\citep{Blanton03}, at low redshifts and the luminosity function from the GAMA survey, $\phi_\mathrm{GAMA}$~\citep{Loveday12}, at higher redshifts, as follows \begin{equation} \begin{split} \phi_\mathrm{target}(\magr, z) &= (1-w(z)) \phi_\mathrm{SDSS}(\magr, z) \\ &+w(z)\phi_\mathrm{GAMA}(\magr, z), \label{eq:target_LF} \end{split} \end{equation} where $w(z)$ is a sigmoid function describing this smooth transition at z=0.15, which is given by \begin{equation} w(z) = \left(1+e^{-100(z-0.15)}\right)^{-1}. \end{equation} Both luminosity functions are modelled with an evolving Schechter fit \begin{equation} \phi(M) = 0.4 \ln{10} \, \phi^\ast \left(10^{0.4(M^\ast-M)}\right)^{1+\alpha}\exp{\left(-10^{0.4(M^\ast-M)}\right)}, \end{equation} where the redshift evolution of the Schechter parameters is modelled as \begin{equation} \begin{array}{l} \alpha(z) = \alpha(z_0) \\ M^\ast(z) = M^\ast(z_0) - Q(z-z_0) \\ \phi^\ast = \phi^\ast(0)10^{0.4Pz}, \end{array} \end{equation} where $z_0 = 0.1$~\citep{Blanton03,Loveday12}. The corresponding parameters for the SDSS and GAMA function are shown in Table~\ref{tab:Schechter_params}. Note that this is different to \citet{Smith17}, which used the tabulated measurement from SDSS. \begin{table} \centering \begin{tabular}{c|cc} \hline Parameter & SDSS & GAMA\\ \hline $\alpha$ & $-1.05$ & -1.23 \\ $\phi^\ast / h^3\mathrm{Mpc}^{-3}$ & $1.49 \times 10^{-2}$ & $0.94 \times 10^{-2}$ \\ $M^\ast(z_0)$ & $-20.44$ & $-20.70$ \\ $P$ & $0.18$ & $1.8$ \\ $Q$ & $1.62$ & $0.7$ \\ \hline \end{tabular} \caption{Schechter parameters obtained from a fit to SDSS \citep{Blanton03} and GAMA \citep{Loveday12} data. These two luminosity functions are interpolated in order to obtain our target luminosity function (eq.~\ref{eq:target_LF}).} \label{tab:Schechter_params} \end{table} In our galaxy assignment algorithm, we use the cumulative LF, which gives the number density of galaxies brighter than magnitude threshold.% Starting from the halo $V_\mathrm{peak}$ values, we use a SHAM algorithm to assign galaxy luminosities by matching the $V_\mathrm{peak}$ cumulative number density function to our target galaxy luminosity function, with some added intrinsic scatter. We detail below our algorithm, which is based on the method introduced in \citet{McCullagh17,Safonova21}, \begin{enumerate} \item Sort the haloes by $V_\mathrm{peak}$ in descending order, to compute the $V_\mathrm{peak}$ cumulative number density function, $n_\mathrm{h}(>V_\mathrm{peak})$% . \item Assign an `unscattered' galaxy $r$-band magnitude, $\magr$, to each halo by matching the above $V_\mathrm{peak}$ cumulative halo number density function to the target galaxy luminosity function, preserving the ranking such that large $V_\mathrm{peak}$ haloes host high-luminosity galaxies. \begin{equation} n_\mathrm{g}(<\magr) = n_\mathrm{h}(>V_\mathrm{peak}) \end{equation} \item Define a new `scattered' value of the magnitude, ${}^{0.1}M_r^{h,\mathrm{scat}}$, \begin{equation} {}^{0.1}M_r^{h,\mathrm{scat}} = \mathcal{N}(0, \sigma^2) + \magr, \end{equation} where $\mathcal{N}(0, \sigma^2)$ is a number drawn from a normal distribution with mean $0$ and variance $\sigma^2$. \item Sort the haloes by ${}^{0.1}M_r^{h,\mathrm{scat}}$, and compute the ${}^{0.1}M_r^{h,\mathrm{scat}}$ cumulative distribution, $n_\mathrm{h}(<{}^{0.1}M_r^{h,\mathrm{scat}})$. \item Assign the final $r$-band magnitude by matching the cumulative distribution of ${}^{0.1}M_r^{h,\mathrm{scat}}$ to the target cumulative distribution function, \begin{equation} n_\mathrm{g}(<\magr) = n_\mathrm{h}(<{}^{0.1}M_r^{h,\mathrm{scat}}). \end{equation} In other words, the brightest final luminosities are assigned to the haloes with brightest ${}^{0.1}M_r^{h,\mathrm{scat}}$. % \end{enumerate} All the degrees of freedom in the procedure above are contained in the choice of the scatter parameter $\sigma$, which we can regulate. In our case, we reduce the number of tunable parameters of our model and neglect any possible redshift or luminosity dependence of $\sigma$ by fixing it to a constant value of $0.5 \, \mathrm{mag}$. This choice, despite its simplicity, is able to reproduce the observed galaxy clustering while avoiding unphysicalities in the $(V_\mathrm{peak},\magr)$ galaxy distribution. This is shown in Fig.~\ref{fig:tpcf_box_SDSS}, where we show the monopole of the two-point correlation function of our Uchuu galaxy catalogue at $z=0.093$, the catalogue closest to the median redshift of SDSS, compared to the results from our SDSS samples. Our model is able to recover the SDSS results to good accuracy for a large range of scales and volume-limited samples. The value of $\sigma$ is calibrated to reproduce the observed SDSS galaxy clustering. For more details about the calibration of $\sigma$ and its effect on galaxy clustering, we direct the reader to Appendix~\ref{App:scatter}. We also recover the target LF by construction. The resulting relation between $V_\mathrm{peak}$ and $\magr$ is shown in Fig.~\ref{fig:Vpeak_vs_mag} for galaxies with $^{0.1}M_r <-14.0$ in the $z=0.093$ box. The large volume of Uchuu allows to reach a huge dynamical range in galaxy luminosity $-23 < \magr < -14$. \subsection{Uchuu-SDSS galaxy lightcones} \label{subsec:light_mocks} We use a total of 6 snapshots between $z=0$ and $z=0.5$, which are separated in redshift by approximately 0.1 ($z_\mathrm{snap}=0, 0.093, 0.19, 0.3, 0.43, 0.49$). Lightcones are created from the snapshots by joining them together in spherical shells. An observer is first placed in the box, and the Cartesian galaxy coordinates in each snapshot are converted to equatorial coordinates, and the redshift is calculated, taking into account the line-of-sight velocity. In each simulation snapshot, galaxies in the redshift range $(z_\mathrm{snap-1} + z_\mathrm{snap})/2 < z < (z_\mathrm{snap} + z_\mathrm{snap+1})/2$ are selected, which are then joined together to build the lightcone. If the redshift shell is too big to fit inside a single cubic box, periodic replications are applied. In the final lightcone there are no periodic replications below $z=0.36$. The full-sky lightcone is then cut to the northern contiguous region of the SDSS survey footprint, using a healpix map \citep{Gorski1999, Blanton2005, Swanson2008}. The original healpix map has $N_\mathrm{side}=512$, but we increase the size of the pixels, using $N_\mathrm{side}=128$, and keeping pixels where the completeness in the data is greater than 0.9. This results in a footprint with area $7261~\sqdeg$. The SDSS footprint can be replicated across the full sky to create 4 independent SDSS simulated catalogues. By generating lightcones from eight observer positions (with coordinates at either 0 or $1~\hGpc$, along each of the three dimensions), we are therefore able to create a total of 32 Uchu-SDSS lightcones. The first 8 mocks (constructed with observer at the origin, and at the centre of the box) are independent below $z=0.3$. There is some overlap in the volume between the lightcones at higher redshifts than this, but the fraction of galaxies with $z>0.3$ is very small ($\sim 0.1\%$). We refer to these lightcones as the 8 independent lightcones. The full set of 32 lightcones are independent below $z=0.175$, and we use all 32 to improve the statistics of our RSD and BAO measurements (see Section~\ref{sec:RSD_BAO}). The BAO galaxy sample (eq.~\ref{eq:hod_cuts}) has a maximum redshift of $z=0.2$, so there is some overlap between mocks, and $\sim 25\%$ of galaxies in the sample have $z>0.175$. Combining multiple snapshots in this way to construct a lightcone % has the issue that there are discontinuities at the boundaries between snapshots. It is possible for the same halo to appear twice at either side of the boundary, or to not appear at all, and the duplicated haloes artificially boost the pair counts on very small scales. We investigate this in \citet{Smith2022a}, and find that there is a boost in the real-space clustering on small scales due to this effect. In redshift space, when velocities are included, the effect is greatly reduced. Using snapshots separated by $\sim 0.1$ in redshift is a good compromise which adds evolution to the lightcone, without excessively boosting the clustering below $1~\hMpc$. The clustering measurements on the scales used in a typical RSD analysis ($\gtrsim 20~\hMpc$) are insensitive to the number of snapshots used. \subsubsection{Magnitude evolution} \label{subsubsec:mag_colour_evolution} Each simulation snapshot was constructed to reproduce the target luminosity function at the redshift of the snapshot, $z_\mathrm{snap}$. In the lightcone, this leads to discontinuities in these properties at the boundaries between snapshots. In order to create a simulated lightcone with a smoothly evolving luminosity function, and smooth $n(z)$, we rescale the galaxy absolute magnitudes as a function of redshift. The original magnitude assigned to each galaxy is first mapped to the corresponding number density, using the luminosity function at $z_\mathrm{snap}$. This number density can then be mapped back to an absolute magnitude at the redshift of the galaxy in the lightcone, $z$, using the target luminosity function at the same redshift, $z$. By construction, this reproduces the smooth evolution of the target luminosity function. \subsubsection{Colour assignment} \label{subsubsec:colour_assignment} After galaxy luminosities have been assigned, we add ${}^{0.1}(g-r)$ colours to our simulated galaxy sample. We use an evolving empirical model for the ${}^{0.1}(g-r)$ colours, based on \citep{Skibba2009, Smith17}, but improved to better reproduce the colour-magnitude diagram from the GAMA survey, at a range of redshifts \citep{Smith2022b}. The bi-modality of the colour probability distribution functions is modelled as a sum of two Gaussian distributions. Defining $p({}^{0.1}(g-r); \magr, z) \equiv dN/N/d{}^{0.1}(g-r)$, where $N$ is the number of galaxies, \begin{equation} \begin{split} p({}^{0.1}(g-r); \magr, z) &= f_\mathrm{blue} \, \mathcal{N}_\mathrm{blue}(\mu_\mathrm{blue},\sigma_\mathrm{blue}) \\ &+ (1-f_\mathrm{blue}) \, \mathcal{N}_\mathrm{red}(\mu_\mathrm{red},\sigma_\mathrm{red}), \end{split} \end{equation} where $\mathcal{N}_\mathrm{blue}$ and $\mathcal{N}_\mathrm{red}$ are Gaussian probability distribution functions corresponding to blue and red galaxies, respectively, and $f_\mathrm{blue}$ is the fraction of blue galaxies. The parameters describing this double-Gaussian distribution all depend on $\magr$ and $z$. At a fixed redshift, the mean and rms of each Gaussian, $\mu_\mathrm{red}(\magr)$, $\sigma_\mathrm{red}(\magr)$, $\mu_\mathrm{blue}(\magr)$ and $\sigma_\mathrm{blue}(\magr)$, are modelled as broken linear functions. These functions were fit to the colour-magnitude diagram measured in GAMA, in several redshift bins, which we then interpolate between. Basing the colour distributions on the data from GAMA allows us to create simulated galaxy catalogues to very faint magnitudes, which will be useful e.g. for the DESI Bright Galaxy Survey. However, we find good agreement between these colours and the measurements from the SDSS data. The fraction of blue galaxies at given luminosity, $f_\mathrm{blue}(\magr)$, is modelled differently for central and satellite galaxies. Colours are then drawn randomly from the ${}^{0.1}(g-r)$ colour distributions described above. This method is able to reproduce by construction our target colour distributions. \subsubsection{Apparent magnitudes} The apparent $r$-band magnitude is computed from the absolute magnitude using eq.~\ref{eq:appmag}. In the $r$-band, we use a set of colour-dependent $k$-corrections from the GAMA data \citep[see figure 13 of][]{Smith17}. These are a set of polynomial $k$-corrections, measured in several bins of ${}^{0.1}(g-r)$ colour. These $k$-corrections allow us to calculate apparent magnitudes using the information that was originally available in the simulated catalogue ($r$-band magnitude and $g-r$ colour). In the other bands, we use the $k$-corrections of \citet{Blanton2003b}. We have compared the GAMA and SDSS $r$-band $k$-corrections, and find that the median $k$-corrections at each redshift are in good agreement, differing by no more than $\sim 0.01$. The $1\sigma$ scatter is at a level $< 0.04$. At $z=0.1$, both $k$-corrections agree exactly, since ${}^{0.1}k(z=0.1)=-2.5 \mathrm{log}_{10}(1.1) \approx -0.103$. \subsubsection{Assigning galaxy properties using SDSS data} \label{sec:assign_gal_properties_to_mock} The method we have used to create the lightcone assigns each galaxy a rest-frame $r$-band absolute magnitude, $\magr$, and ${}^{0.1}(g-r)$ colour. In order to assign more properties to the mock, we match galaxies to the SDSS data. We use a k-d tree to find the closest SDSS galaxy, based on $z$, $\magr$ and ${}^{0.1}(g-r)$. Each mock galaxy is then assigned the absolute magnitude in the $u$, $g$, $i$ and $z$-band of the closest-matching galaxy, in addition to its stellar mass and specific star formation rate. The $u$, $g$, $i$ and $z$-band apparent magnitudes are calculated at the redshift of the mock galaxy from the absolute magnitudes, using eq.~\ref{eq:appmag} and the SDSS $k$-corrections. % \subsubsection{Modeling fibre collisions} \label{subsec:fibre} In SDSS, spectroscopic fibres on a single plate cannot be placed closer to each other than the diameter of the fibre plugs. As a result, if two galaxies are in close proximity ($<55''$), a spectrum can only be obtained for one of them. The tiling of SDSS plates slightly alleviates this problem, with $\sim30\%$ of the SDSS footprint covered by multiple plates. These plate `overlap regions' allow for spectroscopic redshifts to be obtained for several galaxies that are within $55''$ of one another. Still, $\sim6\%$ of targeted galaxies in our SDSS sample lack a spectroscopically measured redshift due to their proximity to a neighbouring galaxy. To mimic the fibre collision effect of SDSS, when constructing our mock galaxy catalogues, we employ a procedure adapted from \citet{Szewciw2022}. First, we link together galaxies into friends-of-friends `collision groups'. A galaxy is part of a collision group if its angular distance to any galaxy in that group is $<55''$. Next, we decide whether each galaxy in a group will be `observed' (i.e. receive its spectroscopic redshift) or `unobserved'. When making this choice, we first attempt to maximize the number of galaxies in a collision group that could receive spectroscopic fibres from a single SDSS plate. If there does exist more than one set that maximizes the number of observed galaxies, then we randomly choose one of these sets to be observed. Next, to simulate the effect of SDSS plate overlap regions, we randomly select $\sim30\%$ of the unobserved galaxies to receive their original observed spectroscopic redshifts. The remaining unobserved galaxies are then assigned the redshift of their angular nearest neighbour. Finally, given the new redshifts of these galaxies, we recompute their $k$-corrections \citep[using the colour-dependent $k$-corrections described in][]{Smith17} and absolute magnitudes. It is important to note that this procedure does not fully mimic the role that plate overlap regions play in recovering the redshifts of collided galaxies. In our procedure the collided galaxies whose redshifts are recovered are chosen at random from the full sky. In SDSS, by contrast, recovered galaxies lie in plate overlap regions and thus are spatially correlated. % Furthermore, in SDSS, each overlap region of the sky is covered by a different number of intersecting plates. The number of overlapping plates dictates the number of galaxies whose redshifts can be spectroscopically measured in a given collision group. Our procedure, however, is agnostic with respect to the number of plates required to recover redshifts for any randomly chosen set of collided galaxies. Despite these differences, we apply the procedure described above to our simulated galaxy catalogues. With this relatively simple procedure described above, the global fraction of galaxies affected by fibre collisions in the simulated lightcone ($f_\mathrm{NN} \sim 5.2 \%$) is quite similar to that of SDSS ($f_\mathrm{NN} \sim 5.9 \%$). In Fig.~\ref{fig:fnn}, we show a comparison of $f_\mathrm{NN}$ between SDSS (black points) and each of our eight independent Uchuu-SDSS lightcones (red points) for the different volume-limited samples defined in Table~\ref{tab:sdss}. In both Uchuu and SDSS, we see the same qualitative trend -- $f_\mathrm{NN}$ increases as we move to more luminous samples. This is as expected, since more luminous galaxies tend to be more strongly clustered and thus are more likely to be affected by fibre collisions. There is good agreement for bright volume-limited samples, although for $^{0.1}M_r^\mathrm{max} \leq -19.5$ Uchuu underestimates $f_\mathrm{NN}$ by up to a $2\%$ offset. % After running the fibre assignment algorithm on the 32 Uchuu-SDSS lightcones, we evaluate the fibre assignment completeness in healpix pixels with $N_\mathrm{side}=512$. The lightcones are then cut to pixels where the average completeness $> 0.9$ (averaged over the 32 lightcones). This results in a final area of $6642~\sqdeg$, which is comparable to the footprint of the SDSS data. \vspace{1em} The 32 Uchuu-SDSS lightcone catalogues described in this section, containing \textit{ugriz} magnitudes, stellar masses, star formation rates and fibre assignment information, are made publicly available at Skies \& Universes. A subset of the cubic box catalogues used in the construction of the lightcones, and the companion SDSS data set are also made available. Note that our released box catalogues also include ${}^{0.1}(g-r)$ computed using a similar method as described in Section~\ref{subsubsec:colour_assignment}. % \section{Properties of the Uchuu-SDSS galaxies} \label{sec:properties} \subsection{Galaxy properties} \label{sec:gal_properties} In this section, we illustrate the various galaxy properties stored in the Uchuu-SDSS lightcones, and compare with the galaxy properties in the SDSS data. Fig.~\ref{fig:pie_plot} shows galaxies in a thin slice of the SDSS catalogue and one of the Uchuu-SDSS lightcones, where each galaxy has been coloured based on its stellar mass, illustrating the similarities between the mock and data. The density of galaxies falls with increasing redshift, since faint galaxies with low stellar masses can only be observed close to $z=0$, while the brightest galaxies can be observed over the full redshift range. The density of galaxies appears to be in good agreement between the Uchuu-SDSS and SDSS data. This can been seen quantitatively in Fig.~\ref{fig:dNdz}, which compares the redshift distribution of the 32 Uchuu-SDSS lightcones with SDSS. The Uchuu-SDSS lightcones are in good agreement with SDSS, peaking at $z \sim 0.08$. There is scatter between the 32 lighcones, due to cosmic variance, but the measurement from SDSS is consistent with this scatter. At higher redshifts ($z \sim 0.2$), there is a slight excess of galaxies in the Uchuu-SDSS lightcones compared to the data, due to differences in the luminosity function. While the target luminosity function used to construct the simulated lightcones is in good agreement with the SDSS data at low redshifts, we transition to the luminosity function measured in GAMA at higher redshifts. Using the GAMA luminosity function results in a higher number density of galaxies compared to the SDSS measurements, but the SDSS luminosity function is poorly constrained at this redshift. The absolute magnitudes of galaxies from one of the Uchuu-SDSS lightcones is shown in Fig.~\ref{fig:lightcone_absolute_magnitude}, compared to SDSS. In the $r$-band, magnitudes were assigned to each galaxy to match an evolving target luminosity function from SDSS and GAMA measurements, and we find that this is able to reproduce well the distribution of absolute $r$-band magnitudes in the SDSS data. The magnitudes in the other bands were assigned by matching each simulated galaxy to a galaxy in the data, based on $r$-band magnitude, $g-r$ colour and redshift. By construction, the distribution of these magnitudes is also in good agreement with the SDSS data. The ${}^{0.1}(g-r)$ colour distributions in the Uchuu-SDSS lightcones are shown in the left-hand column of Fig.~\ref{fig:lightcone_g_r_colour}, as a function of $\magr$, in three narrow redshift bins at $z=0.05$, $z=0.1$ and $z=0.15$. In each redshift bin, the colour distribution is bimodal, with a red sequence of galaxies and cloud of blue galaxies. The brightest galaxies are red, while fainter galaxies have a higher blue fraction. The colour distributions show good agreement with the SDSS data, reproducing the same colour evolution with redshift. There is a small discrepancy at low redshifts, where the red sequence is more sloped in Uchuu-SDSS compared to in SDSS, since the colours in the Uchuu-SDSS lightcone were tuned to GAMA measurements. The right-hand column shows the same colour distributions but as a function of the stellar mass. Here we see that the blue galaxies tend to have lower stellar masses than the red galaxies. Again, there is good agreement between the simulated and real data, with a slight discrepancy in the red sequence at low redshifts. The distribution of star formation rates of galaxies in the lightcone is shown in Fig.~\ref{fig:lightcone_ssfr}, in comparison to the SDSS measurements. The upper panel shows stellar mass against sSFR, while ${}^{0.1}(g-r)$ is plotted against sSFR in the lower panel. There is a clear bimodal distribution of quiescent galaxies with low star formation rates on the left, and star-forming galaxies on the right. The quiescent galaxies tend to have higher stellar masses than the star-forming galaxies. Applying a cut of $10^{11}~\mathrm{yr}^{-1}$ in sSFR is able to cleanly cut the galaxy catalogue into these two samples, while we can see in the lower panel that a cut in ${}^{0.1}(g-r)$ would not work as well. \subsection{Luminosity, mass and colour dependence of clustering} In this section, we compare the clustering in our Uchuu-SDSS lightcones with the observational results from the volume-limited SDSS samples. We measure the first non-zero Legendre multipoles ($\ell = 0, 2, 4$) of the redshift-space two-point correlation function (TPCF), defined as, \begin{equation} \xi_\ell(s) = \left( 2\ell+1 \right)\int_{0}^{1}\xi^{s}\left(s,\mu\right)L_\ell(\mu)d\mu, \label{eq:mps} \end{equation} where $s=|\mathbf{s}|$, $\mu=\pi/s$ is the cosine of the angle between the line-of-sight direction and the pair separation vector $\mathbf{s}$. $\xi^{s}\left(s,\mu\right)$ is the two point correlation function and $L_\ell$ is the $\ell^\mathrm{th}$-order Legendre polynomial. These quantities are computed using the publicly available code \texttt{FCFC} \citep[][]{FCFC}. Fig.~\ref{fig:xi_s} compares the monopole, $\xi_0(s)$, quadrupole, $\xi_2(s)$, and hexadecapole, $\xi_4(s)$, of the TPCF of 8 independent Uchuu-SDSS lightcone catalogues against the measurements from the SDSS dataset, for the set of volume-limited samples described in Table~\ref{tab:sdss}. The clustering of our Uhuu-SDSS galaxies is in good agreement with the data for all the volume-limited samples considered, despite the simplicity of our luminosity assignment model -- note that we neglect any potential dependence of our scatter parameter on redshift or luminosity, and consequently our luminosity assignment model has only one tunable parameter. The agreement is poorer for the $\magr<-22$ sample, with the Uchuu-SDSS lightcones underestimating the observed monopole. This suggests that the scatter between our halo mass proxy and galaxy luminosity is likely to decrease for bright galaxies $\magr \lesssim -22$, which is in agreement with previous findings \citep{Stiskalek21}. Similarly, Fig.~\ref{fig:xi_s_Mstar} shows the two-point correlation function for several $M_\ast$ cuts. We find again good agreement between our simulated catalogues and the SDSS data. The poorer agreement for the highest mass cut at $\log_{10}(M_\ast) > 11$ suggests again a decrease in the scatter in the galaxy--halo connection at the high mass end, in agreement with previous studies \citep{Behroozi19}. In order to study the colour-dependence of clustering, we split the sample into two populations of blue and red galaxies. We use a luminosity-dependent colour cut as introduced in \citet{Zehavi05}, \citep[eq.~13 in][]{Zehavi11}. \begin{equation} {}^{0.1}(g-r)_\mathrm{cut} = 0.21 - 0.03~{}^{0.1}M_r^{h,e}. \label{eq:colour_cut} \end{equation} Galaxies above the ${}^{0.1}(g-r)$ cut belong to the red population, while galaxies below the cut belong to the blue population. Fig.~\ref{fig:xi_s_colour} compares the TPCF for blue, red and all galaxies of Uchuu-SDSS with that of the SDSS dataset, for a subset of the volume-limited samples in Table~\ref{tab:sdss}. Uchuu-SDSS is in reasonable agreement with the SDSS data, although the agreement is visibly poorer than for the overall volume-limited samples, specially for the brightest samples. This points to limitations due to the simplicity of our colour-assignment algorithm. As described in Section~\ref{subsubsec:mag_colour_evolution}, colours in the lightcone are randomly drawn from our target colour distribution. % Our results could be improved, by using information about the halo age to assign colours, thus accounting for assembly bias. This would likely require to apply a model involving free parameters, which would need to be finely tuned in order to match the observed colour-dependent clustering. \subsection{Stellar mass function}\label{sec:SMF} In order to further validate our simulated galaxy catalogues, we calculate the stellar mass function (SMF) of the SDSS sample (see Section~\ref{sec:stellar_mass_sfr}) and compare it with that from the set of 8 independent Uchuu-SDSS lightcones (see Section~\ref{sec:assign_gal_properties_to_mock}). The SMF is estimated by the non-parametric $1/V_\mathrm{max}$ method widely used in deriving the galaxy luminosity function. To compute the SMF for SDSS and Uchuu samples, we select all galaxies in a redshift range of $0.02 \leq z \leq 0.2$ with stellar masses $M_\ast \geq 10^9~\hsqMsun$ and $r$-band apparent magnitudes $14.5 \leq r \leq 17.6$. In Fig.~\ref{fig:SMF} (left panel), we present the SMF obtained from the mean of the 8 Uchuu-SDSS lightcones. Results are compared to SMF derived from the SDSS sample. We also compare our results with that obtained by \citet{Moustakas13}. In \citet{Moustakas13}, $M_\ast$ was determined utilizing iSEDfit, a suite of routines used to determine stellar masses, SFRs, and other physical properties of galaxies from the observed broadband SEDs and redshifts \citep[e.g.][]{Kauffmann03,Salim07}. The Uchuu-SDSS is in reasonably good agreement with both MPA-SDSS and SMF obtained by \cite{Moustakas13}. The middle and right panels of Fig.~\ref{fig:SMF} show the SMF of quiescent and star-forming galaxies for each data set. % We adopt $\mathrm{sSFR}=10^{-11}~\mathrm{yr}^{-1}$ as the threshold between quiescent and star-forming galaxies. This value corresponds to the minimum between the two peaks of the bimodal sSFR distribution shown in Fig.~\ref{fig:lightcone_ssfr}. The Uchuu-SDSS SMFs for quiescent and star-forming galaxies galaxies is consistent with those obtained from MPA-SDSS for all masses, although for quiescent galaxies the agreement is slightly poorer compared to the overall sample in the left panel. There is also a good agreement between the Uchuu SMFs and those obtained by \citet{Moustakas13}, with a notable offset at the low- and high-mass ends for quiescent galaxies. One possible reason for this difference is the different methods for the estimation of SFRs used by MPA/JHU and \citet{Moustakas13}, which result in different SFR values. % \subsection{Halo occupation distribution} \label{subsec:hod} Galaxies are known to be biased tracers of the underlying dark matter density field. In order to better understand the connection between galaxies and haloes in Uchuu-SDSS catalogues, we investigate the halo occupation distribution. We compute $\langle N_\mathrm{gal}(>L|M_\mathrm{halo})\rangle$ -- the mean number of galaxies brighter than a given $r$-band luminosity in a halo with virial mass $M_\mathrm{halo}$. The luminosity-dependent HOD is modeled with the functional form described in \citet{Zehavi11}. % We write $\langle N_\mathrm{gal}(>L|M_\mathrm{halo})\rangle$ as a sum of the mean number of central and satellite galaxies. The mean occupation function of the central galaxies is modelled as a step-like function with a cutoff profile softened to account for the scatter between galaxy luminosity and halo mass, and the mean occupation of satellite galaxies is modelled as a power law modulated by a similar cutoff profile, see \citet{Zehavi11} for the details. This HOD model has five free parameters: the mass scale, $M_\mathrm{min}$, and width, $\sigma_{\log{M}}$, of the central galaxy mean occupation, and the cutoff mass scale, $M_{0}$, normalization, $M_{1}'$, and high-mass slope, $\alpha$, of the satellite galaxy mean occupation function. We fit this model to the average HOD obtained from 8 independent Uchuu-SDSS lightcones for the volume-limited samples corresponding to luminosity cuts described in Table~\ref{tab:sdss}. The best fitting HOD parameters are shown in Table~\ref{tab:hod_params}, along with the SDSS estimates from \citet{Zehavi11}. Fig.~\ref{fig:all_hod} shows the mean halo occupation of the Uchuu-SDSS galaxies and the best-fit HOD models. As seen in the figure, the HOD shifts towards more massive haloes as the luminosity threshold increases -- more luminous galaxies occupy more massive haloes. % Our results broadly agree with those of \citet{Zehavi11} over the wide range of brightness thresholds for all volume-limited samples. However, we observe a few discrepancies for some HOD parameters (see Table~\ref{tab:hod_params}). This is likely due to the difference in the % methodologies. While we have computed the halo occupation directly from our values of $\magr$ and $M_\mathrm{halo}$ in the Uchuu-SDSS lightcones, \citet{Zehavi11} obtains it by % fitting the projected correlation function of the observed SDSS data. All our best fit parameters, except $\alpha$, follow an ascending trend as a function of the luminosity threshold. \setlength{\arrayrulewidth}{0.3pt} \begin{table} \centering \begin{tabular}{cccccc} \hline $^{0.1}M_r^\mathrm{max}$ & $\log{M_\mathrm{min}}$ & $\sigma_{\log{M}}$ & $\log{M_{0}}$ & $\log{M_{1}'}$ & $\alpha$ \\ \hline \multirow{2}{*}{-18.0} & 11.34$\pm$0.09 & 0.29$\pm$0.20 & 11.07$\pm$0.08 & 12.60$\pm$0.03 & 0.99$\pm$0.03 \\ & 11.18 & 0.19 & 9.81 & 12.42 & 1.04 \\ [0.75mm] \multirow{2}{*}{-18.5} & 11.48$\pm$0.08 & 0.31$\pm$0.16 & 11.08$\pm$0.07 & 12.73$\pm$0.04 & 1.03$\pm$0.03 \\ & 11.33 & 0.26 & 8.99 & 12.50 & 1.02 \\ [0.75mm] \multirow{2}{*}{-19.0} & 11.66$\pm$0.06 & 0.35$\pm$0.12 & 11.16$\pm$0.07 & 12.83$\pm$0.02 & 0.99$\pm$0.02 \\ & 11.45 & 0.19 & 9.77 & 12.63 & 1.02 \\ [0.75mm] \multirow{2}{*}{-19.5} & 11.85$\pm$0.10 & 0.38$\pm$0.16 & 11.37$\pm$0.09 & 12.95$\pm$0.04 & 0.95$\pm$0.03 \\ & 11.57 & 0.17 & 12.23 & 12.75 & 0.99 \\ [0.75mm] \multirow{2}{*}{-20.0} & 12.12$\pm$0.08 & 0.45$\pm$0.11 & 11.48$\pm$0.09 & 13.17$\pm$0.04 & 0.97$\pm$0.04 \\ & 11.83 & 0.25 & 12.35 & 12.98 & 1.00 \\ [0.75mm] \multirow{2}{*}{-20.5} & 12.43$\pm$0.06 & 0.52$\pm$0.06 & 11.49$\pm$0.11 & 13.47$\pm$0.03 & 1.00$\pm$0.04 \\ & 12.14 & 0.17 & 11.62 & 13.43 & 1.15 \\ [0.75mm] \multirow{2}{*}{-21.0} & 12.86$\pm$0.12 & 0.62$\pm$0.09 & 11.69$\pm$0.12 & 13.83$\pm$0.09 & 1.07$\pm$0.14 \\ & 12.78 & 0.68 & 12.71 & 13.76 & 1.15 \\ [0.75mm] \multirow{2}{*}{-21.5} & 13.32$\pm$0.14 & 0.68$\pm$0.09 & 11.79$\pm$0.14 & 14.25$\pm$0.09 & 1.08$\pm$0.16 \\ & 13.38 & 0.69 & 13.35 & 14.20 & 1.09 \\ [0.75mm] \multirow{2}{*}{-22.0} & 13.98$\pm$0.30 & 0.81$\pm$0.22 & 11.86$\pm$0.22 & 14.76$\pm$0.19 & 1.26$\pm$0.30 \\ & 14.06 & 0.71 & 13.72 & 14.80 & 1.35 \\ [0.75mm] \hline \end{tabular} \caption{Best-fit HOD parameters for our volume-limited samples. For each luminosity cut the top row indicates the best-fit parameters for our Uchuu-SDSS lightcones, while the bottom row shows that for the SDSS data as reported in \citet{Zehavi11}. Error bars on our HOD parameters correspond to standard deviation errors on the parameters. Halo masses are in units of $h^{-1}M_{\odot}$. } \label{tab:hod_params} \end{table} \section{RSD and BAO measurements} \label{sec:RSD_BAO} In this section we study the BAO signal in the SDSS MGS. For this purpose we define a BAO sample (SDSSbao, see Section~\ref{subsec:sdss_samples}) in which this signal is enhanced similar to \citet{Ross15}. We also model the full shape of the TPCF to measure $f\sigma_8$ and obtain the anisotropic BAO distances. While we have the set of 32 Uchuu-SDSS lightcones to compare with the SDSSbao data, in order to further improve our covariance errors on BAO scales, we generate an additional sample of 5100 light-cones, describe below, using a set of lower resolution $N$-body simulations run with the GLAM code \citep{Klypin18}. In Section~\ref{subsec:GLAM_construction} we describe the method used to construct our 5100 GLAM-SDSSbao lightcones, while their clustering properties are explored in Section~\ref{subsec:GLAM_clustering}. Our RSD measurements are presented in Section~\ref{subsec:RSD_measurement}. Finally, we measure the isotropic BAO scale in Section~\ref{subsec:BAO_measurement}. \subsection{Constructing the GLAM lightcones} \label{subsec:GLAM_construction} We generate 1275 GLAM simulations using the same cosmology and linear power spectrum as the Uchuu simulation. The GLAM simulations follow the evolution of $2000^3$ particles of mass $1.07\times 10^{10}\hMsun$ in a cubic box of size $1~\hGpc$ with $N_s=140$ timesteps, and mesh of $N_g=5800$. This numerical set-up yields a spatial resolution of $\Delta x=0.172~\hMpc$. The initial conditions are generated using the Zeldovich approximation starting at $z_\mathrm{ini}=105$. The distinct haloes in GLAM are identified with the Bound Density Maximum halo finder \citep{Klypin97}. Since the GLAM simulations are unable to resolve substructure inside distinct haloes, the SHAM method introduced in Section~\ref{sec:mock_construction} cannot be applied, and we thus resort to a statistical HOD method. First, we compute the HOD of the SDSSbao sample by applying the galaxy selection criteria in eq.~\ref{eq:hod_cuts} to our 8 independent Uchuu-SDSS lightcones (hereafter Uchuu-SDSSbao). We then use the 1275 GLAM halo catalogues available at the mean redshift of the BAO sample ($z \sim 0.1$) to generate a galaxy catalogue for each GLAM box by randomly drawing galaxies from the measured halo occupation statistics for each distinct halo. Fig.~\ref{fig:bao_hod} shows the mean HOD from Uchuu-SDSSbao used to populate with galaxies the GLAM simulations, along with the resulting mean HOD obtained from the 1275 GLAM galaxy catalogues. By construction, the HOD of Uchuu- and GLAM-SDSSbao galaxies are in agreement. It is important to note that the GLAM simulations are only able to resolve haloes larger than $10^{12}~\hMsun$. However, the HOD obtained from the high-fidelity Uchuu-SDSSbao lightcones does not extends to masses below this limit. % The resulting GLAM galaxy catalogues have an average density of $6\times 10^{-4}$ galaxies per unit volume (average $n(z)$ of the Uchuu-SDSSbao sample). Once the GLAM halos for a given box are populated with galaxies, we adjust its number density to match that in the SDSSbao sample. For this step, we use the $n(z)$ presented in \citet{Ross15} as reference. Then we cut it to the northern contiguous region of the SDSS survey footprint. As with the Uchuu lightcones, by replicating the SDSS footprint across the full sky, we can generate a total of 4 independent SDSSbao lightcones from each GLAM box, which allows us to create a total of 5100 GLAM-SDSSbao lightcones. We decided not to apply the fibre collision correction to GLAM-SDSSbao since it effect is negligible on the scales we are going to make use of the lightcones (BAO scales). The GLAM-SDSSbao lightcones mean number density is shown in Fig.~\ref{fig:bao_nz} together with that from Uchuu-SDSSbao and the SDSSbao data. There is a 20\% slight excess of galaxies in Uchuu-SDSSbao as compared to the data and the GLAM-SDSSbao lightcones at high-redshift tail of the distribution. As explained in section \ref{sec:gal_properties}, this is because we use a target luminosity function to build the Uchuu galaxy catalogues that transitions from SDSS to GAMA, since the luminosity function from SDSS is poorly constrained at high redshifts. This difference does not seem to have a significant impact on the performance of our Uchuu-SBSSbao lightcones in reproducing within $1\sigma$ the SDSSbao clustering, however it may explain the small difference seen between Uchuu- and GLAM-SDSSbao (see Fig.~\ref{fig:xi_s_BAO}). % \subsection{Two-point correlation function and covariance matrix} \label{subsec:GLAM_clustering} In this section we explore the clustering on the BAO scales in the three data sets: SDSSbao, Uchuu- and GLAM-SDSSbao. In order to optimise the BAO clustering signal-to-noise ratio, we weight both galaxies and randoms depending on the galaxy number density $n(z)$ using FKP weights \citep{Feldman94,Ross15}, i.e. \begin{equation} w_{\textrm{FKP}}=\frac{1}{1+P_{\textrm{FKP}} n(z)}. \label{eq:fkp_weigths} \end{equation} We set $P_{\textrm{FKP}} = 16\,000~h^{-3}\mathrm{Mpc}^{3}$ , which is close to the measured amplitude at $k = 0.1~\hMpc$. Further, from the 5100 GLAM-SDSSbao lightcones, we infer the covariance matrix, $C$, defined as \begin{equation} C^{\ell_i \ell_j }(s_i,s_j)= \frac{1}{N-1}\sum^N_{n=1} [\xi^{(\ell_i)}(s_i) - \bar{\xi}^{(\ell_i)}(s_i)][\xi^{(\ell_j)}(s_j) - \bar{\xi}^{(\ell_j)}(s_j)], \label{eq:cov_mat} \end{equation} where $N$ is the number of lightcones. Fig.~\ref{fig:xi_s_BAO} shows the monopole, quadrupole and hexadecapole of the two-point correlation function of the SDSSbao data, the mean of the 32 Uchuu-SDSSbao lightcones (described in Section~\ref{subsec:light_mocks}), and the mean of the 5100 GLAM-SDSSbao. We include all bins between 25 and $200~\hMpc$ in $5~\hMpc$ steps. The error bars in the SDSSbao TPCF are calculated from the diagonal elements of the GLAM-SDSSbao covariance matrix. For Uchuu and GLAM, we plot the error in the mean of the correlation functions as shaded regions. % The three data sets show a clear BAO peak at $s \sim \SI{100}{\per\h\mega\parsec}$. Uchuu- and GLAM-SDSSbao generally agree with the SDSSbao observational measurements within $2\sigma$. The GLAM measurements have better statistics than Uchuu, due to the huge number of lightcones in the GLAM suite. Moreover, Uchuu have a larger number density than GLAM, which translates into a lower clustering amplitude. Note that the GLAM lightcones, used to estimate uncertainties from its TPCF covariance matrix, match the SDSSbao number density (see Fig.~\ref{fig:bao_nz}). \subsection{RSD measurements} \label{subsec:RSD_measurement} Galaxy clustering analyses that rely on two-point statistics are not sensitive to the growth rate of structure $f$ directly but instead to $f\sigma_8$, where $\sigma_8$ is the normalization of the linear power spectrum on scale of $8~\hMpc$. We measure the linear growth rate of structure, $f\sigma_8$, and the Alcock-Paczynski parameters, $\alpha_{\perp}$ and $\alpha_{\parallel}$, of the Uchuu-SDSS lightcones and SDSSbao sample described above using the two-point correlation function, $\xi(s,\mu)$. The anisotropic Alcock-Paczynski parameters are defined as \begin{equation}\label{eq:a_par} \alpha_{\parallel} = \frac{D_\mathrm{H}(z)r^{\text{fid}}_{d}}{D_\mathrm{H}^{\text{fid}}(z)r_{d}} \end{equation} and \begin{equation}\label{eq:a_perp} \alpha_{\perp} = \frac{D_\mathrm{M}(z)r^{\text{fid}}_{d}}{D_\mathrm{M}^{\text{fid}}(z)r_{d}}, \end{equation} where $H(z)$ is the Hubble parameter, $D_\mathrm{M}(z)$ is the angular diameter distance, $D_\mathrm{H}(z) = c/H(z)$ is the Hubble distance, and $r_d$ is the sound horizon at the drag epoch. Quantities with a `fid' superscript are calculated for the fiducial cosmology assumed during the analysis, while the quantities without a superscript exist in the true cosmology. The analysis is performed using the Planck fiducial cosmology, as adopted for Uchuu, to convert the redshifts to comoving distances. If the assumed fiducial cosmology does not match the true cosmology, there is a scaling of the BAO peak position parallel and perpendicular to the line-of-sight (as given in eqs.~\ref{eq:a_par},~\ref{eq:a_perp}). Thus, we should recover $\alpha_{\perp}=\alpha_{\parallel}=1$ from the Uchuu- and GLAM-SDSSbao lightcones. % From the covariance matrix $C$ computed in section~\ref{subsec:GLAM_clustering}, we can define the $\chi^2$ statistic as \begin{equation} \chi^2 = \frac{N-p-2}{N-1} (\xi^{\text{Data}}-\xi^{\text{Model}})C^{-1}(\xi^{\text{Data}}-\xi^{\text{Model}})^T, \end{equation} where $p$ is the number of degrees-of-freedom being fitted and $N$ is the number of GLAM-SDSSbao lightcones, and we have included the Hartlap correction \citep{Hartlap_2006}. % Our theoretical model for the TPCF, $\xi^{\text{Model}}$, is based on Lagrangian Perturbation Theory (LPT). To model the distribution of galaxies, one also needs to introduce a bias model that connects the matter density, $\delta_m$, and the galaxy density, $\delta_g$. with two parameters $b_{1}$ and $b_{2}$ named the linear and quadratic Lagrangian bias, respectively. We can then obtain the model power spectrum, and include the effect of peculiar velocities to account for the RSD effect \citep{Seljak2011}. We also model the Fingers-of-God (FOG) effect in Fourier space, using the phenomenological Lorentz model \citep{Taruya_2010}: \begin{equation} P_\mathrm{FOG}(\boldsymbol{k}) = \frac{1}{1+(\boldsymbol{k}\cdot \hat{\boldsymbol{n}} \: \sigma_\mathrm{FOG})^2/2} P(\boldsymbol{k}). \end{equation} where $P(\boldsymbol{k})$ is the non-linear power spectrum without the FOG effect, $\sigma_\mathrm{FOG}$ is the one-dimensional velocity dispersion and $\hat{\boldsymbol{n}}$ is the normalised LOS direction vector. The theoretical power spectrum $P_\mathrm{FOG}$ is obtained using the \texttt{MomentumExpansion} module of \texttt{velocileptors} package \citep[for more details, see][]{Chen_2020,Chen_2021}. Finally, we obtain the correlation function by taking the Fourier transform of $P_\mathrm{FOG}$, \begin{equation} \xi(\boldsymbol{x}) = \int d^3 \boldsymbol{k} e^{i\boldsymbol{k}\cdot \boldsymbol{x}} P_\mathrm{FOG}(\boldsymbol{k}). \end{equation} We fit our RSD model to the correlation function multipoles from three datasets SDSSbao, Uchuu-SDSSbao and GLAM-SDSSbao % in the separation range $[25,145]~\hMpc$, with bins of width $5~\hMpc$. In addition to the cosmological parameters $f\sigma_8$, $\alpha_{\parallel}$ and $\alpha_{\perp}$, we also estimate the Lagrangian biases $b_{1},b_{2}$ and the Fingers-of-God parameter $\sigma_\mathrm{FOG}$. Unphysical values of parameters are avoided by setting the priors to $f\sigma_8>0$, $b_{1}>-1$ and $\sigma_\mathrm{FOG}>0$. The first Lagrangian bias is related to the Eulerian bias by $b_{1,\rm Eulerian}=1 + b_{1}$. We assume the effective redshift of the SDSSbao sample to be $z = 0.15$. The correlation function multipoles corresponding to our best-fit models can be seen in Fig~\ref{fig:xi_s_BAO}. We observe good agreement with Uchuu-SDSSbao and GLAM-SDSSbao measurements. The RSD model predicts position of the BAO peak in the monopole that is too low as compared with the observed position of the peak. However, this disagreement is within the noise limits. The minima of $\chi^2$ is found using \texttt{iminuit} \citep{iminuit}, which achieves convergence near the minimum using the first and approximate second derivatives. Errors are estimated from the region of $\Delta \chi^2 =1$ of the marginalized $\chi^2$ distribution, and they are allowed to be asymmetric. We also run Monte Carlo Markov chains (MCMC) with the \texttt{emcee} package \citep{emcee} in order to compute the likelihood surface of our set of fitted parameters. Their convergence is checked with the Gelman-Rubin convergence test \citep{Gelman1992, Brooks1998}. We first test our RSD pipeline on the GLAM-SDSSbao and Uchuu-SDSSbao light-cones. The best-fit results of the parameters $f\sigma_8$, $\alpha_{\parallel}$ and $\alpha_{\perp}$ are summarised in Fig.~\ref{fig:RSD_Parameter_Spreads}. We confirm that the theoretical model % recovers the fiducial Planck values within 1$\sigma$. % We then apply the pipeline to the SDSSbao sample. % Our results are listed in Table~\ref{tab:RSD_Results}. Both $\chi^2$ minimization and MCMC sampling methods provide consistent results, and the small difference in the errors and parameter values are attributed to a better treatment of the Fingers-of-God effect with MCMC chains. The parameter distributions for $\sigma_\mathrm{FOG}$ is non-Gaussian, as it is restricted to positive values, but the best-fit value is consistent with 0. Additionally, we compare our obtained $f \sigma_8$ and $b_1$ with \citet{Howlett_2015_b}, finding a good agreement, but % we obtain a $\gtrsim 30\%$ increase in precision on $f\sigma_8$ which it can be attributed to our better estimate of the covariance matrix (see Table~\ref{tab:RSD_Results}). \begin{table} \centering \begin{tabular}{cccc} \hline Parameter& $\chi^2$ minimization & MCMC & Reference \\ \hline \\[-1em] $f\sigma_8$ & $0.65^{+0.17}_{-0.16}$ & $0.62^{+0.17}_{-0.17}$ & $0.63 ^{+0.24}_{-0.27}$ \\ \\[-1em] $\alpha_{\parallel}$ & $0.99^{+0.11}_{-0.10}$ & $1.04^{+0.15}_{-0.10}$& N/A \\ \\[-1em] $\alpha_{\perp}$ & $1.18^{+0.10}_{-0.08}$ & $1.17^{+0.07}_{-0.07}$& N/A \\ \\[-1em] $b_{1,\rm Eulerian}$ & $1.59^{+0.19}_{-0.19}$ & $1.62^{+0.22}_{-0.19}$& $1.36^{+0.29}_{-0.26}$ \\ \\[-1em] $b_{2}$ & $-0.3^{+2.1}_{-1.6}$ & $-0.8^{+1.6}_{-1.4}$& N/A \\ \\[-1em] $\sigma_\mathrm{FOG}[\hMpc]$ & $4.5^{+2.8}_{-4.5}$ &$3.6^{+2.5}_{-3.5}$& N/A \\ \\[-1em] \hline \end{tabular} \caption{RSD fitted parameters from the SDSSbao data, obtained by $\chi^2$ minimization and using Bayesian MCMC inference. Only $f\sigma_8$ and $b_{1,\rm Eulerian}$ estimated values from \citet{Howlett_2015_b} are found in the literature.} \label{tab:RSD_Results} \end{table} \begin{table} \centering \begin{tabular}{cccc} \hline Parameter& Uchuu & GLAM & Expected value \\ \hline \\[-1em] $f\sigma_8$ & $0.48^{+0.15}_{-0.15}$ & $0.45^{+0.16}_{-0.16}$ & $0.46$ \\ \\[-1em] $\alpha_{\parallel}$ & $0.94^{+0.11}_{-0.09}$ & $0.94^{+0.16}_{-0.10}$& 1.00 \\ \\[-1em] $\alpha_{\perp}$ & $0.99^{+0.08}_{-0.07}$ & $1.00^{+0.09}_{-0.08}$& 1.00 \\ \\[-1em] $b_{1,\rm Eulerian}$ & $1.42^{+0.17}_{-0.15}$ & $1.40^{+0.19}_{-0.17}$& N/A \\ \\[-1em] $b_{2}$ & $-0.4^{+1.6}_{-1.3}$ & $-0.6^{+1.6}_{-1.3}$& N/A \\ \\[-1em] $\sigma_\mathrm{FOG}[\hMpc]$ & $0^{+9}_{-0}$ &$0^{+22}_{-0}$& N/A \\ \\[-1em] \hline \end{tabular} \caption{RSD fitted cosmological parameters from the means of Uchuu and GLAM correlation functions, obtained using $\chi^2$ minimization in comparison with the values predicted by the fiducial cosmology.} \end{table} The distribution of parameter values and their uncertainties can be seen in Fig. \ref{fig:RSD_Parameter_Spreads}, where the results from the SDSSbao sample together with those from GLAM and Uchuu are shown in blue, orange and red respectively, and the black lines show the expected values of the parameters for the Planck fiducial cosmology. The results from GLAM- and Uchuu-SDSSbao are very consistent within each other and with the SDSSbao data, % meaning that the mocks can be seen as a fair statistical representation of the data. We confirm that the fits we have performed are valid and have an acceptable $\chi^2/\mathrm{dof}$, for GLAM mocks being $\chi^2/\mathrm{dof}= 1.00 \pm 0.18$, for Uchuu mocks being $ \chi^2/\mathrm{dof}= 1.06\pm 0.18 $ and for the SDSSbao sample being $\chi^2/\mathrm{dof} =0.99$. Finally we use our best-fit values of the BAO-inferred $\alpha_{\parallel}$ and $\alpha_{\perp}$ parameters to provide a measurement of the Hubble distance and the (comoving) angular diameter distance at the effective redshift of the SDSSbao sample, i.e. $D_\mathrm{H}(z=0.15) / r_d = 27.9^{+3.1}_{-2.7}$, $D_\mathrm{M}(z=0.15) / r_d = 5.1^{+0.4}_{-0.4}$. These distance results are valuable since at present only the spherically averaged distance $D_\mathrm{V}(z=0.15) / r_d$ has been reported from BAO isotropic measurements from a similar SDSSbao sample by \citet{Ross15}, see below for our own BAO isotropic analysis. \subsection{Isotropic BAO measurements} \label{subsec:BAO_measurement} We also measure the isotropic BAO scale from our Uchuu-SDSSbao and the SDSSbao data at $z=0.15$. The BAO signal can clearly be seen in the correlation function monopole measured from both data set (see Fig.~\ref{fig:xi_s_BAO}). The BAO scale can be extracted by fitting the monopole of the correlation function to a template that includes the dilation parameter, $\alpha$, \begin{equation}\label{eq:alpha} \alpha \equiv \frac{D_\mathrm{V}(z)r^\mathrm{fid}_\mathrm{d}}{D^\mathrm{fid}_\mathrm{V}(z)r_\mathrm{d}}\,, \end{equation} where \begin{equation}\label{eq:DV} D_\mathrm{V}(z) = \left[cz(1+z)^2D^2_\mathrm{A}(z)H^{-1}(z)\right]^{1/3}\,, \end{equation} is the spherically average distance \citep{Eisenstein:2005su}; $\alpha$ is related to the anisotropic Alcock-Paczynski parameters $\alpha_{\perp}$ and $\alpha_{\parallel}$ given in eqs.~\ref{eq:a_par} and \ref{eq:a_perp}, by $\alpha = \alpha_{\parallel}^{1/3}\alpha_{\perp}^{2/3}$. To obtain the best-fit $\alpha$ value, we use Bayesian statistics and maximise the likelihood by adopting the BAO model of the monopole of the redshift-space correlation function as presented in \citet{Ross15}. We perform the fit to the simulations and data monopoles on scales $50 < s < 150~\hMpc$, using separation bins of width $\Delta s = 5\hMpc$. The measured and the best-fit model of the correlation function monopole from the SDSSbao data and the simulated Uchuu and GLAM lightcones are shown in Fig.~\ref{fig:bao_200-5}. We find $\alpha=0.9966\pm0.0067$, $\alpha=1.0036\pm0.0015$ and $\alpha=1.0372\pm0.0390$ from Uchuu, GLAM, and the SDSSbao observations, respectively. Our best-fit $\alpha$ value for the SDSS data and its error is consistent with that reported by \citet{Ross15}, i.e. $\alpha=1.057\pm0.037$. From our dilation parameter estimate we measure a spherically averaged distance $D_\mathrm{V}(z=0.15)/r_\mathrm{d} = 4.43 \pm 0.16$, in agreement with the value $D_\mathrm{V}(z=0.15)/r_\mathrm{d} = 4.47 \pm 0.17$ measured by \citet{Ross15}. The values of $D_\mathrm{V}$ from the Uchuu and GLAM lightcone measurements are $D_\mathrm{V}(z=0.15)/r_\mathrm{d} = 4.257 \pm 0.028$ and $D_\mathrm{V}(z=0.15)/r_\mathrm{d} = 4.304 \pm 0.007$, respectively. \section{Summary} \label{sec:conclusions} The cosmological interpretation of large galaxy surveys requires generation of high-fidelity simulated galaxy data. Here we describe and analyze publicly available Uchuu-SDSS lightcones: a set of simulated SDSS catalogues generated using the Uchuu $N$-body simulation. Uchuu is a large high-resolution cosmological simulation that follows the evolution of the dark matter across cosmic time in the Planck cosmology. The Uchuu-SDSS catalogues are tailored to reproduce the sky footprint and galaxy properties of the SDSS MGS observational sample. This facilitates the direct comparison between our simulated lightcones and the observational data to probe the Planck-$\Lambda$CDM cosmology model using the large-scale clustering signal. The rest-frame $r$-band magnitudes, $\magr$, are assigned to the Uchuu haloes using the SHAM method, with a simple recipe for the scatter in the galaxy-halo connection. By construction, our scheme reproduces the SDSS and GAMA luminosity functions, and it reproduces to good accuracy the SDSS clustering (see Fig.~\ref{fig:tpcf_box_SDSS}), while using only one free parameter -- the scatter. The resulting galaxy catalogues computed at different redshifts are combined in spherical shells and cut to the SDSS sky footprint. By placing observers at different positions in the cubic box, we produce a set of 8 independent Uchuu-SDSS lightcones, with only a small mutual overlap in volume. Additionally, an extended set of 32 lightcones, which overlap for $z>0.175$, are generated to increase the statistics for our SDSS RSD and BAO measurements. Galaxy colours are also produced using a Monte Carlo method that randomly draws colours from the $\magr$ and redshift-dependent $g-r$ distribution obtained from the GAMA survey. Redshifts, $r$-band magnitudes and $g-r$ colours are used to match each simulated galaxy to a galaxy in the SDSS sample. This allows us to assign absolute magnitudes, apparent magnitudes and $k$-corrections in the $u$, $g$, $i$ and $z$-bands in addition to stellar masses and specific star formation rates. Finally, we implement the effect of fibre collisions by applying a nearest neighbour correction to a set of galaxies situated in close angular proximity. Our Uchuu-SDSS lightcones are able to recover the galaxy redshift distribution (see Fig.~\ref{fig:dNdz}) and various galaxy properties from the SDSS survey to very high accuracy (see Figs.~\ref{fig:lightcone_absolute_magnitude},~\ref{fig:lightcone_g_r_colour}~and~\ref{fig:lightcone_ssfr}). We also compute the galaxy correlation functions for several volume-limited samples corresponding to a wide range of luminosities (Fig.~\ref{fig:xi_s}) and stellar mass cuts (Fig.~\ref{fig:xi_s_Mstar}), finding a very good agreement with the SDSS clustering down to the smallest $100\, \hkpc$ scale. The colour- and stellar-mass-dependent galaxy clustering (see Fig.~\ref{fig:xi_s_colour}) is in general agreement with the SDSS results. Similarly, the simulated stellar mass function (Fig.~\ref{fig:SMF}) is in good agreement with that from SDSS data. We also provide the halo occupation distributions and parameters of the Uchuu-SDSS galaxies (Fig.~\ref{fig:all_hod}), which are in agreement with previous SDSS analyses. We explore the RSD and BAO signal in our Uchuu-SDSS lightcones and the SDSS data (see Fig.~\ref{fig:xi_s_BAO}). In order to obtain high-precision covariance matrices for the error estimates of the RSD and BAO measurements, we create a large number of GLAM simulations. We apply HOD method to populate the GLAM halos with galaxies, generating a total of 5100 GLAM-SDSSbao lightcones. We measure $f \sigma_8$ and the anisotropic BAO parameters, $\alpha_{\perp}$ and $\alpha_{\parallel}$, from a full-shape model fit of the TPCF, finding a very good agreement between the SDSSbao data and our GLAM- and high-fidelity Uchuu-SDSSbao lighcones (see Fig.~\ref{fig:RSD_Parameter_Spreads}). Our results are given in Table~\ref{tab:RSD_Results}. We obtain a $\gtrsim 30\%$ increase in precision on $f\sigma_8$, as compared to the previous measurement by \citep{Howlett_2015_b}, which can be attributed to our better estimate of the covariance matrix. We use our best-fit values of the BAO-inferred $\alpha_{\parallel}$ and $\alpha_{\perp}$ parameters to provide a measurement of the Hubble distance and the (comoving) angular diameter distance at the effective redshift of the SDSSbao sample, i.e. $D_\mathrm{H}(z=0.15) / r_d = 27.9^{+3.1}_{-2.7}$, $D_\mathrm{M}(z=0.15) / r_d = 5.1^{+0.4}_{-0.4}$. We highlight that these distance results are valuable since at present only the spherically averaged distance $D_\mathrm{V}(z=0.15) / r_d$ has been reported by \citet{Ross15}. Finally, we measure the isotropic dilation scale, $\alpha$, of the BAO signal by fitting a model template of the BAO peak \citep{Ross15}, obtaining again a good agreement between the simulated and observational SDSS data (Fig.~\ref{fig:bao_200-5}). Based on our results, we conclude that the Planck \textLambda CDM cosmology nicely explains the observed statistics of the large-scale structure of the SDSS main galaxy survey. This work shows the great potential of the Uchuu simulation as a canvas for the creation of simulated galaxies (and quasars) for large surveys. The procedures presented in this paper can be readily applied for the creation of high-fidelity lightcones from Uchuu, and covariance errors from GLAM simulations, tailored for upcoming galaxy surveys such as DESI \citep{DESI16}, Euclid \citep{Laureijs11}, and LSST \citep{LSST2009}. This will aid the evaluation of analysis pipelines, the assessment of observational biases and systematic effects, and enable cosmological models to be probed. \section*{Acknowledgements} We thank Gary Mamon and Marko Shuntov for helpful discussions. We thank Instituto de Astrofisica de Andalucia (IAA-CSIC), Centro de Supercomputacion de Galicia (CESGA) and the Spanish academic and research network (RedIRIS) in Spain for hosting Uchuu DR1 in the Skies \& Universes site for cosmological simulations. The Uchuu simulations were carried out on Aterui II supercomputer at Center for Computational Astrophysics, CfCA, of National Astronomical Observatory of Japan, and the K computer at the RIKEN Advanced Institute for Computational Science. The Uchuu DR1 effort has made use of the \texttt{skun@IAA-RedIRIS} and \texttt{skun6@IAA} computer facilities managed by the IAA-CSIC in Spain (MICINN EU-Feder grant EQC2018-004366-P). This work used the \texttt{DiRAC@Durham} facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure. CAD-P, AS, JE, FP, AK, JR thank the support of the Spanish Ministry of Science and Innovation funding grant PGC2018-101931-B-I00. CAD-P gratefully acknowledges generous funding from the John Simpson Greenwell Memorial Fund. CH-A acknowledges support from the Excellence Cluster ORIGINS which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC-2094 - 390783311. TI has been supported by IAAR Research Support Program in Chiba University Japan, MEXT/JSPS KAKENHI (Grant Number JP19KK0344, JP21F51024, and JP21H01122), MEXT as ``Program for Promoting Researches on the Supercomputer Fugaku'' (JPMXP1020200109), and JICFuS. \section*{Data Availability} The 32 Uchuu-SDSS galaxy lightcones, the 6 Uchuu-box galaxy catalogues at redshifts ${z=\{0, 0.093, 0.19, 0.3, 0.43, 0.49\}}$, the 5100 Uchuu-SDSSbao galaxy lightcones, and the companion SDSS LSS catalogue used in this work are made available at \url{http://www.skiesanduniverses.org/Simulations/Uchuu/}, together With the information on how to read the data. For a list and brief description of the available catalogue columns, please see Appendix~\ref{App:mock_columns}. \bibliographystyle{mnras} \bibliography{references} \appendix \section{Content of the Uchuu-SDSS catalogues} \label{App:mock_columns} Below is a list of the columns of each data set, along with a short description. \subsection{Uchuu-SDSS galaxy lightcones} Each Uchuu-SDSS lightcone has $\sim590\,000$ galaxies in total (excluding the regions of low fibre-collision completeness), with the following columns: \begin{itemize} \item \texttt{galaxy\_type}: indicates whether the galaxy is central or a satellite (0 for centrals, 1 for satellites). \item \texttt{ra}: right ascension (degrees). \item \texttt{dec}: declination (degrees). \item \texttt{z\_cos}: cosmological redshift. \item \texttt{z\_obs}: observed redshift (accounting for peculiar velocities). \item \texttt{z\_obs\_fib}: observed redshift including fibre collisions, i.e. fibre-collided galaxies get a nearest neighbour correction (see Section~\ref{subsec:fibre}). \item \texttt{k\_corr\_r}: $r$-band $k$-correction at a reference redshift $z=0.1$. There are similar columns for the other SDSS photometric bands i.e. \texttt{k\_corr\_u}, \texttt{k\_corr\_g}, \texttt{k\_corr\_i} and \texttt{k\_corr\_z}. \item \texttt{k\_corr\_r\_fib}: $r$-band $k$-correction, recomputed for fibre-collided galaxies using the nearest neighbour-corrected redshift, \texttt{z\_obs\_fib}.% \item \texttt{M\_r}: rest-frame $r$-band absolute magnitude $\magr$, $k$-corrected to $z=0.1$, with no E-correction. Similarly, the absolute magnitudes in the other bands are \texttt{M\_u}, \texttt{M\_g}, \texttt{M\_i} and \texttt{M\_z}. \item \texttt{M\_r\_fib}: $r$-band absolute magnitude accounting for fibre collisions, i.e. recomputed for fibre collided galaxies using the values of \texttt{kcorr\_r\_fib} and \texttt{z\_obs\_fib}. % \item \texttt{m\_r}: apparent $r$-band magnitude, $r$. The apparent magnitudes in the other bands are \texttt{m\_u}, \texttt{m\_g}, \texttt{m\_i} and \texttt{m\_z}. \item \texttt{g\_r}: rest-frame $g-r$ colour $k$-corrected to $z=0.1$, ${}^{0.1}(g-r)$. This is the colour from our colour-assignment algorithm. \item \texttt{g\_r\_obs}: observer-frame $g-r$ colour, from our colour-assignment algorithm. \item \texttt{stellar\_mass\_MPA}: MPA stellar mass ($\hsqMsun$). \item \texttt{stellar\_mass\_granada\_best}: Granada best-fitting stellar mass ($\hsqMsun$). \item \texttt{stellar\_mass\_granada\_median}: Granada median stellar mass ($\hsqMsun$). \item \texttt{ssfr\_MPA}: MPA specific star formation rate ($\log_{10}(1/\mathrm{Gyr})$). \item \texttt{ssfr\_granada\_best}: Granada best-fitting stellar mass ($\log_{10}(1/\mathrm{Gyr})$). \item \texttt{ssfr\_granada\_median}: Granada median stellar mass ($\log_{10}(1/\mathrm{Gyr})$). \item \texttt{idnn}: ID of the nearest neighbour in the original catalogue (-1 if no galaxies are within $55''$; -2 if the galaxy is collided but keeps its true spectroscopic redshift). \item \texttt{is\_collided}: indicates whether a galaxy is fibre-collided (is \texttt{True} if \texttt{idnn} is not either -1 or -2). \item \texttt{completeness}: fibre-collision completeness in the healpix pixel containing the galaxy. \item \texttt{snap}: snapshot number from the original Uchuu-box catalogue. \end{itemize} In addition, the following columns describe the DM host halo properties of the Uchuu-SDSS galaxies, taken from the original values in the Uchuu simulation halo catalogues. \begin{itemize} \item \texttt{First\_Acc\_Scale}: scale factor at which current and former satellites first passed through a larger halo. \item \texttt{Macc}: halo virial mass at accretion, excluding unbound particles ($\hMsun$). \item \texttt{Mvir\_all}: mass enclosed within the virial overdensity, including unbound particles ($\hMsun$).% \item \texttt{Rvir}: virial radius ($\hkpc$). \item \texttt{Vpeak}: peak value of $V_\mathrm{max}$ over the halo history (physical $\si{\kilo\meter\per\second}$). \item \texttt{halo\_id}: halo ID. \item \texttt{halo\_mass}: mass enclosed within an overdensity of $200\rho_\mathrm{crit}$, where $\rho_\mathrm{crit}$ is the critical density at the snapshot redshift ($\hMsun$). \item \texttt{is\_cen}: indicates whether the halo is central (\texttt{True} for centrals, \texttt{False} for satellite). \item \texttt{pid}: ID of the parent central halo for satellite haloes, -1 for central haloes. \item \texttt{pos}: 3D position vector. The coordinate system of the Uchuu cubic box has been shifted so that the observer is at the origin, and there are periodic replications of the box (comoving $\hMpc$). \item \texttt{rs}: scale radius of a fitted NFW profile (comoving $\hkpc$) \item \texttt{vel}: 3D velocity (physical $\kms$). \item \texttt{vrms}: velocity dispersion (physical $\kms$). \end{itemize} Additionally, a set of 10 random catalogues, each containing $10$ times the number of galaxies of an individual catalogue, are provided in order to facilitate the analysis of the simulated catalogues. The randoms are obtained by sampling galaxies randomly from the Uchuu-SDSS catalogues, but reassigning them a random sky position chosen with uniform probability within the footprint of the lightcone. \subsection{Uchuu-box galaxy catalogues} For the $(2~\hGpc)^3$ Uchuu-box galaxy catalogues (at redshifts ${z=\{0, 0.093, 0.19\}}$), there are two galaxy properties: \texttt{M\_r} and \texttt{g\_r}, defined as in the previous subsection. The halo columns are the same as in the Uchuu-SDSS lightcones, with a few exceptions: \texttt{halo\_id} is renamed as \texttt{id}, and \texttt{halo\_mass} is renamed as \texttt{M200c}. The columns \texttt{pos}, \texttt{vel}, and \texttt{is\_cen} are removed, and the following are added: \begin{itemize} \item \texttt{x}: x-position of the halo/galaxy (comoving $\si{\mega\parsec\per\h}$). Similarly for \texttt{y} and \texttt{z}. \item \texttt{vx}: x-velocity of the halo/galaxy (physical $\si{\kilo\meter\per\second}$). Similarly for \texttt{vy} and \texttt{vz}. \end{itemize} \subsection{GLAM-SDSSbao galaxy lightcones} Our 5100 GLAM-SDSSbao lightcones, described in \ref{subsec:GLAM_construction}, have the following columns: \begin{itemize} \item \texttt{galaxy\_id}: indicates whether the galaxy is central or a satellite (1 for centrals, 2 for satellites). \item \texttt{ra}: right ascension (degrees). \item \texttt{dec}: declination (degrees). \item \texttt{z\_cos}: cosmological redshift. \item \texttt{z\_obs}: observed redshift (accounting for peculiar velocities). \end{itemize} In addition, the following columns describe the DM host halo properties of the galaxies, taken from the original values found in the GLAM simulation halo catalogues, \begin{itemize} \item \texttt{Mtotal}: halo virial mass ($\hMsun$). \item \texttt{Rvir}: halo virial radius ($\hkpc$). \item \texttt{rs}: scale radius of a fitted NFW profile (comoving $\hkpc$). \item \texttt{pos}: 3D position vector. The coordinate system of the GLAM cubic box has been shifted so that the observer is at the origin, and there are periodic replications of the box (comoving $\hMpc$). \item \texttt{vel}: 3D velocity (physical $\kms$). \item \texttt{vlos}: velocity vector projected along line-of-sight, with the observer positioned at the origin ($\kms$). \end{itemize} \subsection{SDSS data sample} We also make available the observed SDSS galaxy sample, which consists of $497\,536$ (excluding regions of low fibre-collision completeness).% The columns, which are defined the same way as in the Uchuu-SDSS lightcones (see A1), with the same units, are as follows: \begin{itemize} \item \texttt{indx}: ID of the galaxy. % \item \texttt{ra}% \item \texttt{dec}% \item \texttt{z}: measured redshift, including fibre collisions. \item \texttt{k\_corr\_r}, \texttt{k\_corr\_u}, \texttt{k\_corr\_g}, \texttt{k\_corr\_i}, \texttt{k\_corr\_z}% \item \texttt{M\_r}, \texttt{M\_u}, \texttt{M\_g}, \texttt{M\_i}, \texttt{M\_z}% \item \texttt{m\_r}, \texttt{m\_u}, \texttt{m\_g}, \texttt{m\_i}, \texttt{m\_z}% \item \texttt{g\_r}% \item \texttt{g\_r\_obs}% \item \texttt{mass\_MPA}% \item \texttt{mass\_granada\_best}% \item \texttt{mass\_granada\_median}% \item \texttt{ssfr\_MPA}% \item \texttt{ssfr\_granada\_best}% \item \texttt{ssfr\_granada\_median}% \item \texttt{idnn}% \item \texttt{fgotten}: fibre-collision completeness in the region containing the galaxy. \end{itemize} Additionally, a random file with $67$ times the number of galaxies of each GLAM-SDSSbao catalogue is made available to facilitate the analysis. \section{The effect of scatter in SHAM} \label{App:scatter} As discussed in Section~\ref{subsubsec:luminosity_SHAM}, our SHAM model has only one free parameter $\sigma$, which roughly corresponds to the scatter in $\magr$ at a fixed value of our halo mass proxy, $V\mathrm{peak}$. In practice, the value of $\sigma$ affects mainly the amplitude of the TPCF for a given volume-limited sample, results are shown in Fig.~\ref{fig:tpcf_box_SDSS} for different values of the scatter. Increasing the value of $\sigma$ introduces a larger number of low $V_\mathrm{peak}$ galaxies into the volume-limited samples, which are weakly clustered (and also removes high $V_\mathrm{peak}$ galaxies, which are strongly clustered). This has the effect of decreasing the amplitude of the TPCF. The effect of scatter is consequently larger for strongly clustered volume-limited samples, becoming negligible for samples with magnitude threshold $\magr>-20$. In order to tune the scatter parameter, we optimise the goodness-of-fit of the TPCF monopole for the Uchuu snapshot at $z=0.093$ to the SDSS measurements, finding an optimal value of $\sigma=0.5$. We choose this snapshot since it is the closest to the median SDSS redshift $z=0.1$. This value is in agreement with previous observational estimations of the scatter using SHAM. For example, \citet{Trujillo-Gomez11} estimate the the scatter of $\magr$ at fixed halo circular velocity, by combining an estimate of the intrinsic scatter of the Tully-Fisher relation \citep{Verheijen01} with the scatter resulting from the distribution of dust extinction corrections in SDSS, arriving precisely at a value of $0.5$. \bsp % \label{lastpage}

Title: Towards a comprehensive view of accretion, inner disks, and extinction in classical T Tauri stars: an ODYSSEUS study of the Orion OB1b association

Abstract: The coevolution of T Tauri stars and their surrounding protoplanetary disks dictates the timescales of planet formation. In this paper, we present magnetospheric accretion and inner disk wall model fits to NUV-NIR spectra of nine classical T Tauri stars in Orion OB1b as part of the Outflows and Disks around Young Stars: Synergies for the Exploration of ULLYSES Spectra (ODYSSEUS) Survey. Using NUV-optical spectra from the Hubble UV Legacy Library of Young Stars as Essential Standards (ULLYSES) Director's Discretionary Program and optical-NIR spectra from the PENELLOPE VLT Large Programme, we find that the accretion rates of these targets are relatively high for the region's intermediate age of 5.0 Myr; rates range from $0.5-17.2 \times 10^{-8}$ M$_{\odot}$/yr, with a median value of $1.2\times 10^{-8}$ M$_{\odot}$/yr. The NIR excesses can be fit with 1200-1800 K inner disk walls located at 0.05-0.10 AU from the host stars. We discuss the significance of the choice in extinction law, as the measured accretion rate depends strongly on the adopted extinction value. This analysis will be extended to the complete sample of T Tauri stars being observed through ULLYSES to characterize accretion and inner disks in star-forming regions of different ages and stellar populations.

https://export.arxiv.org/pdf/2208.04986

\title{Towards a comprehensive view of accretion, inner disks, and extinction in classical T Tauri stars: an ODYSSEUS study of the Orion OB1b association} \correspondingauthor{Caeley Pittman} \email{cpittman@bu.edu} \author[0000-0001-9301-6252]{Caeley V. Pittman} \affiliation{Institute for Astrophysical Research, Department of Astronomy, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA} \author[0000-0001-9227-5949]{Catherine C. Espaillat} \affiliation{Institute for Astrophysical Research, Department of Astronomy, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA} \author[0000-0003-1639-510X]{Connor E. Robinson} \affiliation{Department of Physics \& Astronomy, Amherst College, Amherst, MA 01002, USA} \author[0000-0003-4507-1710]{Thanawuth Thanathibodee} \affiliation{Institute for Astrophysical Research, Department of Astronomy, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA} \author[0000-0002-3950-5386]{Nuria Calvet} \affiliation{Department of Astronomy, University of Michigan, 311 West Hall, 1085 S. University Avenue, Ann Arbor, MI 48109, USA} \author[0000-0002-6808-4066]{John Wendeborn} \affiliation{Institute for Astrophysical Research, Department of Astronomy, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA} \author[0000-0001-9797-5661]{Jesus Hern{\'a}ndez} \affil{Instituto de Astronom\'{i}a, Universidad Aut\'{o}noma de M\'{e}xico Ensenada, B.C, M\'{e}xico} \author[0000-0003-3562-262X]{Carlo F. Manara} \affiliation{European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany} \author[0000-0001-7796-1756]{Fred Walter} \affiliation{Department of Physics and Astronomy, Stony Brook University, Stony Brook NY 11794-3800, USA} \author[0000-0001-6015-646X]{P\'eter \'Abrah\'am} \affiliation{Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, E\"otv\"os Lor\'and Research Network, Konkoly-Thege Mikl\'os \'ut 15-17, 1121 Budapest, Hungary} \affiliation{ELTE E\"otv\"os Lor\'and University, Institute of Physics, P\'azm\'any P\'eter S\'et\'any 1/A, 1117 Budapest, Hungary} \affiliation{CSFK, MTA Centre of Excellence, Konkoly-Thege Mikl\'os \'ut 15-17, 1121 Budapest, Hungary} \author[0000-0001-8657-095X]{Juan M. Alcal\'a} \affiliation{Osservatorio Astronomico di Capodimonte, via Moiariello 16, 80131 Napoli, Italy} \author[0000-0002-5171-8376]{S\'ilvia H. P. Alencar} \affiliation{Departamento de Fisica, Universidade Federal de Minas Gerais, Av. Antonio Carlos 6627, 30270-901 Belo Horizonte, MG, Brazil} \author[0000-0003-2631-5265]{Nicole Arulanantham} \affiliation{Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA} \author[0000-0002-1593-3693]{Sylvie Cabrit} \affiliation{Observatoire de Paris, PSL University, Sorbonne University, CNRS, LERMA, 61 Av. de l'Observatoire, 75014 Paris, France} \author[0000-0001-6496-0252]{Jochen Eisl\"offel} \affiliation{Th\"uringer Landessternwarte, Sternwarte 5, D-07778 Tautenburg, Germany} \author[0000-0002-5261-6216]{Eleonora Fiorellino} \affiliation{Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, E\"otv\"os Lor\'and Research Network, Konkoly-Thege Mikl\'os \'ut 15-17, 1121 Budapest, Hungary} \affiliation{CSFK, MTA Centre of Excellence, Konkoly-Thege Mikl\'os \'ut 15-17, 1121 Budapest, Hungary} \affiliation{Osservatorio Astronomico di Capodimonte, via Moiariello 16, 80131 Napoli, Italy} \author[0000-0002-1002-3674]{Kevin France} \affiliation{Laboratory for Atmospheric and Space Physics, University of Colorado Boulder, Boulder, CO 80303, USA} \author[0000-0002-8364-7795]{Manuele Gangi} \affiliation{Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone, Italy} \author[0000-0001-5707-8448]{Konstantin Grankin} \affiliation{Crimean Astrophysical Observatory, Department of Stellar Physics, Nauchny, 298409, Crimea} \author[0000-0002-7154-6065]{Gregory J. Herczeg} \affiliation{Kavli Institute for Astronomy and Astrophysics, Peking University, Yiheyuan 5, Haidian Qu, 100871 Beijing, China} \affiliation{Department of Astronomy, Peking University, Yiheyuan 5, Haidian Qu, 100871 Beijing, China} \author[0000-0001-7157-6275]{\'Agnes K\'osp\'al} \affiliation{Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, E\"otv\"os Lor\'and Research Network, Konkoly-Thege Mikl\'os \'ut 15-17, 1121 Budapest, Hungary} \affiliation{ELTE E\"otv\"os Lor\'and University, Institute of Physics, P\'azm\'any P\'eter S\'et\'any 1/A, 1117 Budapest, Hungary} \affiliation{CSFK, MTA Centre of Excellence, Konkoly-Thege Mikl\'os \'ut 15-17, 1121 Budapest, Hungary} \affiliation{Max Planck Institute for Astronomy, K\"onigstuhl 17, 69117 Heidelberg, Germany} \author[0000-0002-0233-5328]{Ignacio Mendigut\'\i{}a} \affil{Centro de Astrobiolog\'\i{}a (CSIC-INTA), ESA-ESAC Campus, 28692, Villanueva de la Cañada, Madrid, Spain} \author[0000-0001-7351-6540]{Javier Serna} \affil{Instituto de Astronom\'{i}a, Universidad Aut\'{o}noma de M\'{e}xico Ensenada, B.C, M\'{e}xico} \author[0000-0002-4115-0318]{Laura Venuti} \affiliation{SETI Institute, 339 Bernardo Ave, Suite 200, Mountain View, CA 94043, USA} \submitjournal{AJ} \received{15 June 2022} \revised{2 August 2022} \accepted{7 August 2022} \keywords{Stellar accretion disks; Protoplanetary disks; Star formation; Pre-main sequence stars; Reddening law; Interstellar extinction} \section{Introduction} \label{sec:intro} The coevolution of T Tauri stars (TTS) and their surrounding protoplanetary disks is one of the most important subjects in the field of planet formation \citep{manara22}. Accretion of disk material onto the star can be traced by continuum excesses in the near-ultraviolet to optical region of the spectrum \citep{valenti93,gullbring00,hh08}. Emission from the frontally-illuminated inner wall of the disk can be traced by continuum excesses in the near-to mid-infrared region \citep{natta01,tuthill01,dalessio05}. A model that can self-consistently reproduce the spectra of actively-accreting T Tauri stars from the ultraviolet to the infrared is a vital tool for understanding the coevolution of the star and inner disk. To reproduce the NUV-NIR spectra of CTTS, one can combine accretion shock and inner disk wall models to create a consistent description of the physical mechanisms producing the observed spectra, as has been demonstrated in GM Aurigae by \cite{ingleby15}. The magnetospheric accretion model \citep{koenigl91, shu94, hartmann94, cg98,muzerolle98,muzerolle01} has been widely used in explaining observations of UV-excesses and emission lines in CTTS \citep[for a review, see][]{hartmann16}. \cite{bouvier20} and the \cite{gravity20} confirmed that hydrogen \brgamma\ emission associated with the magnetospheric accretion paradigm originates well within TTS corotation radii, reinforcing the hypothesis that the emission comes from accretion columns rather than a wind farther away from the star. When modeling this emission, it is important to fit the optical spectra in addition to the NUV, as accretion estimates that fit excesses only at blue wavelengths underestimate the accretion rates by a factor of $\sim$2 \citep{fischer11}. To account for both NUV and optical continuum excesses in classical T Tauri stars (CTTS), \cite{ingleby13} found that a multi-column accretion shock model should be used \citep[based on the shock model of][]{cg98}. \cite{dalessio05} found that the spectrum of a heated inner wall of a dusty disk located at the dust sublimation radius dominates the emission at NIR-MIR wavelengths. The structure and intensity of emission from the inner wall of protoplanetary disks are determined primarily by the wall's geometry and the mineralogy of the dust \citep[e.g.][]{muzerolle03,mcclure13}. Gas accretion onto the star is likely driven by turbulence in this inner disk, which also necessarily causes diffusion of dust particles that are coupled to the gas; the strength of this diffusion dictates the shape of the inner wall, independent of dust composition, with the wall geometry ranging from short and curved, to tall and vertical \citep{schobert21}. The primary implication of this is that observed emission from a curved wall does not depend heavily on disk inclination $i$, whereas emission from a vertical wall depends strongly on $i$. \cite{mcclure13} best fit the 2--10~\micron\ continuum excesses of four TTS by approximating a curved wall using two vertical walls at different radii, each with a different height, dust size distribution, and composition. The \textit{Hubble UV Legacy Library of Young Stars as Essential Standards} (ULLYSES) DDT program \citep{ullyses20}, combined with the PENELLOPE Very Large Telescope (VLT) Large Program \citep{manara21}, provides an ideal sample for exploring the variation in CTTS system properties across ranges of spectral types, ages, and masses. The {\it Outflows and Disks around Young Stars: Synergies for the Exploration of ULLYSES Spectra} (ODYSSEUS) collaboration \citep{espaillat22} is maximizing the scientific impact of these data by studying accretion, outflows, and disk chemistry in the largest sample of TTS observed with the {\it Hubble Space Telescope} (\hst) to date. This paper focuses on modeling the accretion and inner disks of the first sample of TTS observed by ULLYSES. ULYSSES began observations with Orion OB1, located about 400~pc away \citep{briceno19}. Here, we study nine CTTS in the Orion OB1b star-forming region, which has a mean age of 5.0 Myr and a disk fraction of 12\% \citep{briceno19}. This is intermediate between Cloud B (2.5 Myr, 31\% CTTS) and OB1a (10.8 Myr, 6\% CTTS), indicating an intermediate level of disk evolution in the region. In this work, we model the same Orion OB1b sample of CTTS as \cite{manara21} to demonstrate the significance of using NUV spectra to determine the accretion and extinction properties of these targets. For the first time, we model both accretion and inner disks in a sample of CTTS using contemporaneous spectra from 2000--24000~\AA. Once this analysis is extended to the full ULLYSES sample of $\sim$60 CTTS in nine star-forming regions, we can constrain the relationships between stellar parameters and accretion and inner disk properties to an extent that has not been done before. This paper is organized as follows. In Section~\ref{sec:sample}, we discuss the sample of TTS, multiplicity in the sample, stellar parameters, and observations. In Section~\ref{sec:model}, we describe the accretion shock and disk models used to fit the NUV-NIR continua of the CTTS. Section~\ref{sec:results} presents the modeling results. In Section~\ref{sec:discussion}, we compare our results to those of previous studies and discuss the significance of the choice in extinction law and the wavelength range of data available. Finally, in Section~\ref{sec:summary} we present a summary of our findings and future work that will be possible through ODYSSEUS. \section{Sample and Observations} \label{sec:sample} The ULLYSES program \citep{ullyses20} began observations with a sample of eight CTTS in the Orion OB1b subassociation \citep{briceno01,briceno05,briceno19} and two WTTS (weak-lined TTS, non-accreting sources) in the 25 Ori cluster, which are used as template photospheres in this analysis (see Figure~\ref{fig:17and36} in Appendix~\ref{sec:app_A} for a justification of their classification as WTTS). Each target was observed with both \hst\ and the VLT contemporaneously. One of these targets, CVSO~165, was discovered to be a binary system composed of two CTTS, so 11 TTS were observed in total spanning spectral types M3.5--K6 and masses 0.3--0.9~\msun. \subsection{Objects and stellar parameters} \label{sec:stellar_params} CVSO~17, CVSO~36, and CVSO~109 are known to be physical binaries \citep{tokovinin20}, and CVSO~104 and CVSO~165 have visual companions with which they are not kinematically associated \citep{manara21}. The primary component of CVSO~104 was found to be a spectroscopic binary \citep{kounkel19,manara21,frasca21}, but these components are not resolved in the \hst\ observations. \cite{proffitt21} found that the primary component of CVSO~165 is also itself a binary. The ULLYSES \hst\ observations of this object are able to resolve the two components, which both show signatures of active accretion and are thus both modeled in this analysis. Stellar parameters for all CTTS targets and WTTS photospheric templates used in this work are listed in Table~\ref{tab:stellar_params}. For targets not affected by unresolved binarity in their X-Shooter observations, values for spectral type (SpT), mass (\mstar), distance ($d$), and veiling ($r$) are adopted from \cite{manara21}, and their values for stellar radius (\rstar) and V-band extinction (\av) are included for comparison with the values we derive. Effective temperatures for all targets except the CVSO~165 binary system come from the temperature-spectral type relation given for 5--30 Myr stars in \cite{pm13}. \cite{manara21} derive distances from Gaia Early Data Release 3 parallaxes that have reliable astrometric solutions (typically RUWE $<$ 1.4). For those without reliable solutions, the mean distance to the association is used and an uncertainty of 10\% assumed. \subsubsection{CVSO~165 binary} \label{sec:165} We obtain the effective temperature (\teff) and the visual extinction (\av) for each component of the CVSO~165 system by comparing the HST spectra against the PHOENIX synthetic spectral library \citep{husser13}. We use values of surface gravity adequate for PMS stars (e.g., $3.0<{\rm log(g)}<4.5$) and interpolate the spectral library to obtain theoretical grids with intervals of 50~K in {\teff}. For each theoretical spectrum, we apply values of {\av} from 0 to 5 in steps of 0.01 using the standard interstellar reddening law (R$_V$=3.1) from \citet{cardelli89}. Minimizing $\chi^2$, we obtain the {\teff} and the {\av} from the best match between the observed \hst\ spectra and the reddened synthetic spectra. To avoid contamination from accretion flux, we use the spectral range from 6000--9000 \AA. We also avoid the region around \halpha, fitting a Gaussian function to the line profile and ignoring the spectral range between -5$\sigma$ and +5$\sigma$ from the center of the line. Finally, the uncertainties in the estimated values are obtained using the Monte Carlo (MC) method of error propagation \citep{Anderson1976}, varying the \hst\ spectra fluxes randomly 500 times within their reported uncertainties. The values reported in Table~\ref{tab:stellar_params} are the median and standard deviation of the 500 MC results. We derive V magnitudes for each component of CVSO~165 by multiplying the \hst\ spectrum by the Johnson V-filter transmission curve. We obtained ${\rm V}=13.875\pm0.002$ for CVSO~165A and ${\rm V}=15.775\pm0.004$ for CVSO~165B. The RUWE astrometric parameter is highly sensitive to unresolved binary systems; the high RUWE value of 14.3 for CVSO~165 indicates a poor astrometric solution \citep{Lindegren2020}. Therefore, we assume a distance of $400\pm40$~pc \citep{briceno19} to estimate the stellar luminosity for CVSO~165A and CVSO~165B using the visual magnitude corrected by extinction. Finally, comparing the location on the H-R diagram with the MIST evolutionary model \citep{Dotter2016}, we obtained the stellar masses reported in Table~\ref{tab:stellar_params} for each component. The uncertainties in the masses were obtained using the MC method of error propagation and the uncertainties estimated for the luminosity and \teff. \subsubsection{Veiling} Veiling measurements are necessary to set the flux of the WTTS template photosphere relative to the CTTS spectrum, with the relationship given by $F_{{\rm phot},\lambda}=F_{{\rm CTTS},\lambda}/(1+r_\lambda)$. In our analysis, we scale the WTTS template spectrum to the data using the observed veiling at one wavelength, $\lambda_0$. First, the WTTS spectrum is scaled to the observed CTTS spectrum at $\lambda_0$ with a $\frac{F_{\rm obs}(\lambda _0)}{F_{\rm phot}(\lambda _0)}$ term. Then, it is scaled to the observed veiling with a $\frac{1}{1+r_{\lambda _0}}$ term such that the contribution of the photosphere to the data's continuum flux at $\lambda_0$ is equivalent to the contribution implied by the veiling measurement. Only the absolute flux of the WTTS template spectrum is changed, not the shape of the WTTS spectrum. This scaling process allows us to use the observed veiling to determine the amount of continuum excess emission for which we need to account with the accretion shock and inner disk wall models. Three epochs of veiling measurements at $\lambda_0$=5500~\AA\ (r$_{5500}$) are available for 7 out of 9 of the CTTS from VLT/UVES and VLT/ESPRESSO \citep{manara21}, and we use the value from the epoch closest in time to the \hst/STIS observations for these targets. CVSO~90 has VLT/X-Shooter veiling available at $\lambda_0$=7100~\AA, so this is used in place of r$_{5500}$ for this target. The veiling measured from the unresolved CVSO~165 spectrum is split between the two components according to the ratio of their U-band fluxes. CVSO~104 has not yet been formally modeled at the wavelengths relevant to accretion; however, its r$_{5500}$ has been estimated from modeling the UVES spectrum when the objects are nearly in conjunction (A. Frasca, priv. comm.). The observations used to measure the veiling for each target are shown in Table~\ref{tab:obs_log}, and the veiling values are included in Table~\ref{tab:stellar_params}. \begin{deluxetable}{lcc} \tablecaption{Observation Log \label{tab:obs_log}} \tablehead{ \colhead{Object} & \colhead{Telescope/Instrument} & \colhead{Date (MJD)} } \startdata CVSO~58 & \hst/STIS$^{a}$ & 59184.97 \\ & VLT/X-Shooter$^{b}$ & 59185.25 \\ & VLT/UVES$^{b,c}$ & 59185.09 \\ \midrule CVSO~90 & \hst/STIS$^{a}$ & 59199.60 \\ & VLT/X-Shooter$^{b,c}$ & 59198.06 \\ \midrule CVSO~104 & \hst/STIS$^{a}$ & 59180.21 \\ & VLT/X-Shooter$^{b}$ & 59180.13 \\ & VLT/UVES$^{b,c}$ & 59179.12 \\ \midrule CVSO~107 & \hst/STIS$^{a}$ & 59188.02 \\ & VLT/X-Shooter$^{b}$ & 59187.08 \\ & VLT/UVES$^{b,c}$ & 59188.14 \\ \midrule CVSO~109 & \hst/STIS$^{a}$ & 59181.20 \\ & VLT/X-Shooter$^{b}$ & 59181.15 \\ & VLT/UVES$^{b,c}$ & 59181.17 \\ \midrule CVSO~146 & \hst/STIS$^{a}$ & 59192.85 \\ & VLT/X-Shooter$^{b}$ & 59192.08 \\ & VLT/ESPRESSO$^{b,c}$ & 59193.09 \\ \midrule CVSO~165 & \hst/STIS$^{a}$ & 59197.82 \\ & VLT/X-Shooter$^{b}$ & 59197.08 \\ & VLT/ESPRESSO$^{b,c}$ & 59198.10 \\ \midrule CVSO~176 & \hst/STIS$^{a}$ & 59182.86 \\ & VLT/X-Shooter$^{b}$ & 59185.17 \\ & VLT/UVES$^{b,c}$ & 59183.15 \\ \enddata \tablecomments{$^{a}$Observed through the ULLYSES \hst\ DDT Program \citep{ullyses20}. $^{b}$Observed through the PENELLOPE VLT Large Programme \citep{manara21}. $^{c}$Used for veiling measurement.} \end{deluxetable} \begin{deluxetable*}{lccccccccc} \rotate \tablecaption{Adopted Stellar Properties of Orion OB1 Targets \label{tab:stellar_params}} \tablehead{ \colhead{Object} & \colhead{SpT} & \colhead{Temperature} & \colhead{Luminosity} & \colhead{Radius} & \colhead{Mass} & \colhead{Distance} & \colhead{\av} & \colhead{$r$} & \colhead{Template} \\ & & (K) & (\lsun) & (\rsun) & (\msun) & (pc) & (mag) & & } \startdata \multicolumn{10}{c}{Single CTTS} \\ \cmidrule(lr){2-9} CVSO~58 & K7 & 3970 & 0.32 & 1.19 & 0.81 & 349.00$\pm2.8$ & 0.8 & $0.81\pm0.04$ & HBC 427+{\it TWA 6} \\ CVSO~90 & M0.5 & 3700 & 0.13 & 0.88 & 0.62 & 338.70$^{+3.8}_{-3.7}$ & 0.1 & $1.8\pm0.4^{*}$ & TWA 7+{\it TWA 14} \\ CVSO~107 & M0.5 & 3700 & 0.32 & 1.38 & 0.53 & 330.40$\pm2.5$ & 0.3 & $0.98\pm0.11$ & TWA 7+{\it TWA 14} \\ CVSO~146 & K6 & 4020 & 0.80 & 1.84 & 0.86 & 332.00$\pm1.7$ & 0.6 & $0.44\pm0.10$ & HBC 427+{\it RXJ1543.1-3920} \\ CVSO~176 & M3.5 & 3260 & 0.34 & 1.83 & 0.25 & 302.40$^{+2.9}_{-2.8}$ & 1.0 & $0.34\pm0.16$ & CVSO~17/36+{\it TWA 15A} \\ \midrule \multicolumn{10}{c}{Binary CTTS} \\ \cmidrule(lr){2-9} CVSO~104$^{\dagger}$ & M2 & 3490 & 0.37 & 1.66 & 0.37 & 360.70$^{+3.9}_{-3.8}$ & 0.2 & $0.8^{\star}$ & CVSO~17/36+{\it TWA 2A} \\ CVSO~109A$^{a}$ & M0$\pm0.5$ & 3767.6$\pm81.2$ & $0.59^{+0.17}_{-0.13}$ & $1.81\pm0.25$ & 0.50$_{-0.05}^{+0.07}$ & 400$\pm40$ & 0.06$_{-0.24}^{+0.24}$ & $0.90\pm0.08$ & TWA 7+{\it TWA 14} \\ CVSO~165A & K5.5$\pm1.0$ & 4221$\pm28$ & 0.90$_{-0.15}^{+0.19}$ & 1.78$\pm1.07$ & 0.84$\pm0.05$ & 400$\pm40$ & 0.32$\pm0.05$ & $0.21\pm0.03^{**}$ & RECX 1+{\it RXJ1543.1-3920} \\ CVSO~165B & M1$\pm1.5$ & 3849$\pm7$ & 0.47$_{-0.08}^{+0.10}$ & 1.55$\pm0.93$ & 0.58$\pm0.23$ & 400$\pm40$ & 1.35$\pm0.12$ & $0.15\pm0.02^{**}$ & TWA 7+{\it TWA 14} \\ \midrule \multicolumn{10}{c}{WTTS} \\ \cmidrule(lr){2-9} RECX 1$^{b}$ & K5 & 4140 & 1.0 & 1.8 & 0.9 & 97 & 0.0 & ... & ... \\ {\it RXJ1543.1-3920}$^{c}$ & K6 & 4020 & 0.40 & 1.19 & ... & 150 & $0.1\pm0.1$ & ... & ... \\ HBC 427$^{b}$ & K7 & 3970 & 0.8 & 1.9 & 0.8 & 140 & 0.0 & ... & ... \\ {\it TWA 6}$^{d}$ & K7 & 3970 & 0.11 & 0.67 & 0.66 & 51 & ... & ... & ... \\ {\it TWA 14}$^{d}$ & M0.5 & 3700 & 0.15 & 0.90 & 0.73 & 96 & ... & ... & ... \\ TWA 7$^{b}$ & M1 & 3630 & 0.5 & 1.8 & 0.5 & 50 & 0.0 & ... & ... \\ {\it TWA 2A}$^{d}$ & M2 & 3490 & 0.33 & 1.51 & 0.55 & 47 & ... & ... & ... \\ CVSO~17$^{\dagger}$ & M2 & 3490 & 0.30 & 1.50 & 0.37 & 414.2$^{+9.3}_{-8.9}$ & 0.0 & ... & ... \\ CVSO~36$^{\dagger}$ & M2 & 3490 & 0.22 & 1.28 & 0.39 & 335.5$\pm3.0$ & 0.1 & ... & ... \\ {\it TWA 15A}$^{d}$ & M3.5 & 3260 & 0.11 & 1.00 & 0.30 & 111 & ... & ... & ... \\ \enddata \tablecomments{The following stellar parameters for the CTTS come from \cite{manara21} unless otherwise noted: spectral type, luminosity, mass, distance \citep[derived from GAIA Early Data Release 3 parallaxes when reliable solutions were available, see][]{gaia21}, \av, $r$. All veilings are at 5500~\AA\ unless otherwise noted. The adopted temperature comes from the temperature-spectral type relation for 5-30~Myr stars in \citet[][Table 6]{pm13}, with an average used for intermediate spectral types. The radius is calculated from the luminosity and temperature using the Stefan-Boltzmann relation. Italicized WTTS come from X-Shooter; otherwise, they come from \hst/STIS. The average of CVSO~17 and CVSO~36, weighted by their uncertainties, is used because the individual spectra have low SNR in the NUV. $^{a}$ \cite{espaillat22} $^{b}$ \cite{ingleby13} $^{c}$\citet{manara17} $^{d}$ \cite{manara13} \\ $^{\dagger}$Unresolved spectroscopic binary system \\ $^{*}$Veiling at 7100~\AA\ from X-Shooter \\ $^{**}$Veiling for the primary and secondary components of CVSO~165 attained by scaling the ESPRESSO veiling of $0.36\pm0.05$ for CVSO~165 to the ratio of the components' U-band fluxes \\ $^\star$ Veiling at 5500~\AA\ estimated from modeling the UVES spectrum when the binary components are nearly in conjunction (A. Frasca, priv. comm.)} \end{deluxetable*} \subsection{HST/STIS observations} \label{sec:hst_obs} \hst\ Space Telescope Imaging Spectrograph (STIS) and Cosmic Origins Spectrograph (COS) observations of these targets were taken as part of the ULLYSES DDT program through proposals GO16113, GO16114, and GO16115 (\citealt{2020RNAAS...4..205R}, PI: Julia Roman-Duval). This paper utilizes observations of each target with the following gratings: STIS/G230L (spectral resolution 500-1010, plate scale 0.025''/pixel, and NUV-MAMA pixel size of 25~\micron), STIS/G430L (spectral resolution 530-1040, plate scale 0.051''/pixel, and CCD pixel size of 21~\micron), and STIS/G750L (spectral resolution 530-1040, plate scale 0.051''/pixel, and CCD pixel size of 21~\micron), all using the 52X2 slit. These spectra span a total wavelength range of $1710-10,000$~\AA\ after they are combined and trimmed. See Table~\ref{tab:obs_log} for the times of observations. Values of \lfl\ that are less than $1\times 10^{-15}$~\escm\ are removed. The spectra analyzed here come from ULLYSES Data Release~4, which separates spectra for the resolvable binary systems (CVSO~36, CVSO~104, CVSO~109, and CVSO~165) into their constituent components. The data between 1710 and 3300~\AA\ are dereddened with our derived \av\ using the \cite{whittet04} extinction law based on HD 29647 in Taurus (normalized to the \cite{cardelli89} standard ISM law), as the \cite{whittet04} law removes a potential overcorrection of the 2175~\AA\ bump that is present in standard ISM extinction laws. From 3300~\AA\ onwards, the data are dereddened using the \cite{cardelli89} law to align with the analysis of \cite{manara21}. Both laws assume an interstellar reddening of R$_V=3.1$. \subsection{VLT/X-Shooter observations} \label{sec:xs_obs} Contemporaneous VLT/X-Shooter observations were taken through the ESO PENELLOPE Large Programme \citep[][PI: Carlo Manara; see their Figures F.1 and F.2 for the X-Shooter data overplotted with the \hst\ data]{manara2021data}. See Table~\ref{tab:obs_log} for the times of observations. Though observations were taken in the UVB, VIS, and NIR arms, we use only the NIR spectra (spectral resolution $\sim$11600, plate scale 0.248''/pixel, using the 0.4'' slit) for modeling the inner disk wall because \hst\ data are available out to 1~\micron. As described in \cite{manara21}, these flux-calibrated data are dereddened using the \cite{cardelli89} extinction law and corrected for telluric absorption using {\it molecfit} v3.0.3 \citep{smette15, kausch15}. Again, points with \lfl $<1\times 10^{-15}$ \escm\ are removed. The X-Shooter data for the binary targets CVSO~109 and CVSO~165 are unresolved, so it is necessary to split the total flux between the two components to achieve a better representation of the resolved NIR spectra. As described in \citet{espaillat22}, the X-Shooter spectrum of CVSO~109A is scaled by the J-band fluxes of the two components and their difference found by \cite{tokovinin20}. The J-band flux difference is not available for CVSO~165A and CVSO~165B, so these spectra are simply scaled down to align with the \hst\ continua. This is just an approximation given that the two components have different spectral types. However, our inferred NIR flux ratio of the primary to the secondary of 2.4 is reasonably consistent with our measured V- and R-band flux ratios of 2.6 and 2.7, respectively. The \hst\ and X-Shooter observations of CVSO~176 were separated by about 65 hours, and this seems to have produced a significant discontinuity between the data. This can likely be attributed to variability, as variations on the order of hours to days are expected from magnetosphere-disk interactions \citep{venuti17,sergison20,fischer22}. Both the flux and slope of the \hst\ and X-Shooter continua do not agree. However, if the X-Shooter spectrum is scaled up by a factor of 1.75 (corresponding to a change of 0.6 mag), it aligns with the \hst\ spectrum. This agrees with contemporaneous photometry from AAVSOnet that shows a decrease in Sloan $i$-band flux of 0.6 mag between MJD 9182.67--9189.84.\footnote{\url{http://www.astro.sunysb.edu/fwalter/SMARTS/Odysseus/cvso176.phot.html}} This likely indicates a change in the emission from the inner disk wall. Both the scaled and unscaled X-Shooter data are shown in the model fits to CVSO~109A, CVSO~165A, CVSO~165B, and CVSO~176 presented in Figure~\ref{fig:model_fits}. \subsection{TESS observations} \label{sec:tess} \tess, the Transiting Exoplanet Survey Satellite \citep{ricker14}, has observed these targets on two occasions, in Sector~6 (2018 Dec~11 through 2019 Jan~07) and in Sector~32 (2020 Nov~19 through Dec~17). The latter coincided with the observations reported here, and these data were used to estimate the stellar rotation periods of our targets \citep[expected of order 4--9 days,][]{percy10}. We note that the 27 day viewing window of {\it TESS} samples only a few of the expected periods, so this analysis only reveals gross trends in very complex light curves. We download the full frame image data from the MAST archives using the TESScut software \citep{brasseur19}. {\it TESS} images, while photometrically stable and of continuous cadence, suffer from coarse spatial resolution (21\arcsec\ pixels). {\it TESS} is a single-channel photometer with a 600--1000~nm bandpass. The temporal resolution was 10~minutes in Sector~32 and 30~minutes in Sector~6. We extract the data using aperture photometry with a 1.5 pixel radius. The background is extracted from an annulus between 5 and 10 pixels from the source. Because there are often other sources in the background annulus, we iterate on the background pixels, removing those more than 3$\sigma$ from the median level until we converge on the median background level. We assume that the background is spatially flat in this region. \subsection{WTTS photospheric templates} \label{subsec:templates} The template photospheres constructed for each of the CTTS targets are composed of two WTTS spectra stitched together: the wavelength range between 2000 and $\sim$6,000~\AA\ comes from the \textit{HST}/STIS spectrum closest in spectral type to the CTTS, and the remaining data out to 24,000~\AA\ come from the VLT/X-Shooter spectrum that provided the best photospheric fit in \cite{manara21} (except for CVSO~165B, which has a spectral type different than that assigned to the unresolved CVSO~165 system). All targets have \hst\ WTTS within $\pm1$ spectral subtype except for CVSO~176, which is fit with a WTTS 1.5 subtypes earlier. Each pair of WTTS stitched together as photospheric templates for each CTTS target are listed in the ``Template'' column of Table~\ref{tab:stellar_params}. Note that there are four X-Shooter WTTS for which extinction estimates are unavailable. \av\ for these targets is assumed to be zero because they were chosen from regions of low extinction \citep{manara13}. This should not have a significant effect on the fitted \av\ and \mdot\ values because extinction is most important at wavelengths shorter than these X-Shooter WTTS spectra cover. \section{Accretion and Disk Models} \label{sec:model} The accretion and disk wall models used in this work are computed and fit in order to be consistent with one another. First the accretion shock model is calculated for the specific stellar parameters of each target and fit to the data. Then, the output stellar radius, accretion rate, and shock temperature are used as inputs to the disk wall model, which is then calculated for the given stellar parameters and fit to the data. In Sections~\ref{sec:shockmodel}--\ref{sec:diad}, we describe our implementation of the \cite{cg98} accretion shock model and the D'Alessio Irradiated Accretion Disk radiative transfer model (DIAD, \citealt{dalessio98, dalessio99, dalessio01, dalessio04, dalessio05, dalessio06}). \subsection{Multi-column accretion shock model} \label{sec:shockmodel} We model the NUV and optical \hst\ continua using the \cite{cg98} accretion shock model, updated to include three accretion columns of varying energy fluxes. These approximate a flow with a density gradient, as was recently found in GM Aurigae \citep{espaillat21}. Following \cite{cg98}, we assume a magnetospheric truncation radius (\rin) of 5~\rstar\ for all objects but CVSO~109A.\footnote{Modeling of \halpha\ and \hbeta\ lines of CVSO~109A showed a smaller \rin\ of 2.3~\rstar\ \citep[see][]{espaillat22}, but this analysis is still in preparation for the other objects (Thanathibodee et al., in prep.). We note that changing \rin\ to 2.3~\rstar\ from 5~\rstar\ increases the calculated accretion rate by a factor of 1.4.} \rin, \rstar, and \mstar\ determine the infall velocity of the accreted material, which is assumed to be the freefall velocity. The updated model is solved for individual parameter combinations rather than interpolating over a presolved grid of solutions. We add V-band extinction \av\ as a free parameter in the Markov chain Monte Carlo (MCMC) fitting process, then calculate the stellar radius \rstar\ from the fluxes of the dereddened CTTS spectrum and its associated X-shooter WTTS template scaled by the measured veiling. For details regarding the updates applied to the model, see \cite{re19}. The complete shock model is composed of: a WTTS template photosphere $F_{\rm phot}(\lambda)$ scaled to the veiling measured by \cite{manara21} $r_{\lambda _0}$ (shown in Table~\ref{tab:stellar_params}), plus three accretion columns of low, medium, and high flux densities ($\curf$ = $1\times10^{10}$, $1\times10^{11}$, and $1\times10^{12}$ erg s$^{-1}$ cm$^{-2}$, respectively). The emission from these columns is scaled by the stellar radius \rstar, the distance $d$, and filling factors $f$, which represent the fraction of the stellar surface that is covered by each accretion column. Thus, the total dereddened model flux is given by {\small \begin{equation} \label{eq:full_ftot} F_{\rm tot}(\lambda) = 10^{0.4A_{\lambda _0}}\left[\frac{F_{\rm obs}(\lambda _0)}{F_{\rm phot}(\lambda _0)}\frac{F_{\rm phot}(\lambda)}{1+r_{\lambda _0}}\right] + \left(\frac{R_\star}{d}\right)^2 \sum_{i}^{n}f_i\curf_i(\lambda) \end{equation}} \noindent where $A_{\lambda _0}$ is the extinction as described in Sections~\ref{sec:hst_obs} and \ref{sec:xs_obs}; $F_{\rm phot}(\lambda)$ is the WTTS template photosphere scaled by $F_{\rm obs}(\lambda _0)/F_{\rm phot}(\lambda _0)$ to the observed STIS spectrum at $\lambda _0$; $n$ sums over the three accretion columns of different energy-flux densities; $f_i$ are the filling factors associated with each accretion column; and $\curf_i$ are the accretion shock spectra calculated as the sum of the emission from the heated photosphere and the preshock region of the system for each accretion column. Equation~\ref{eq:full_ftot} is fit to the \hst\ continuum (with emission lines masked out) between 2000-10,000~\AA\ for each object using an MCMC with 2,000 steps, 100 walkers, and a burn-in of 550. \av\ has a tophat prior ranging between 0 and 2 on account of the low extinction reported for Orion OB1 \citep[median $A_V=0.65$ mag, ][]{briceno19}, and the sum of the filling factors is restricted between 0 and 40\% of the stellar surface, as modeling has demonstrated that the footprint of the accretion column can produce detectable emission that covers up to 39\% of the TTS surface \citep{ingleby13,ingleby14,re19}. Once the best-fit model is obtained, the average temperature of the shock ($T_{\rm shock}$) is calculated by fitting blackbody curves to the three columns' spectra, then weighting each column's associated accretion luminosity by its fractional filling factor according to {\small \begin{equation} \label{eq:Tshock} T_{\rm shock} = \left[\frac{f_{1E10}}{f_{\rm tot}}T_{1E10}^4 + \frac{f_{1E11}}{f_{\rm tot}}T_{1E11}^4 + \frac{f_{1E12}}{f_{\rm tot}}T_{1E12}^4 \right]^{\frac{1}{4}}. \end{equation}} \noindent The blackbody given by $T_{\rm shock}$ is an important input to the inner disk wall model, as both the stellar and the accretion luminosities irradiate the wall. \subsection{DIAD} \label{sec:diad} We also model the protoplanetary disk's frontally-illuminated inner dust wall, which is located at the dust sublimation radius, using the D'Alessio Irradiated Accretion Disk radiative transfer models (DIAD, \citealt{dalessio98, dalessio99, dalessio01, dalessio04, dalessio05, dalessio06}). The inner dust wall begins contributing significantly at 1~\micron, after which the total model consists primarily of emission from the photosphere and the disk, with non-zero but less significant emission from the accretion columns at these longer wavelengths. Because our data extend only to 2.4~\micron, our model includes only the inner wall, not the disk behind it. For details regarding the DIAD model used here, see the ODYSSEUS I paper \citep[][Section~6.1.2]{espaillat22}. In short, we model the inner dust wall assuming a fractional abundance of graphite and pyroxine-type silicates of 0.0025 and 0.004, respectively, in accordance with the \citet{draine84} model for the diffuse ISM. The grains are spherical with a size distribution that scales as $a^{-p}$ between grain radii of $a_{\rm min}$ and $a_{\rm max}$ and $p$ of 3.5 \citep{mathis77}. The minimum grain size is held at 0.005~{\micron}. Since the NIR spectra do not extend to 10~{\micron}, we cannot confidently constrain the 10~{\micron} silicate feature. Instead, we assume an $a_{max}$ of 10~{\micron} because \citet{mauco18} fit the 10~{\micron} silicate feature well with this value for the three objects from this analysis that were included in their study (CVSO~104, CVSO~107, and CVSO~109). The wall is illuminated by the stellar luminosity and the accretion luminosity, which is given by the $T_{\rm shock}$ derived for each target from its best-fit accretion shock model. To obtain the best fit to the SED of each target, we adjust the height of the inner wall ($z\rm _{wall}$) between 0.5 and 20 gas scale heights (H), noting that the larger values of $z\rm _{wall}$ are indicative of excess emission likely originating from an optically-thin dust cavity in a pre-transitional disk for which we do not account in this model \citep{mauco18}. We adjust the temperature of the optically thin wall atmosphere ($T\rm _{wall}$) between 1200 and 1800 K. Disk inclination $i$ is estimated as described in Section~\ref{sec:inclinations}. Note that in the case of a vertical wall, the inner disk wall height is degenerate with the inclination of the system. Since these disks are unresolved, we cannot distinguish between a high wall and a highly inclined viewing angle (we receive maximum wall emission from a disk inclined at 60-80$^{\circ}$, see \citealt{dullemond01,calvet05}). The radius in the disk at which the wall is located ($R_{\rm wall}$) is derived using the best-fitting $T_{\rm wall}$ following \begin{equation}R_{\rm wall} \sim \left [{\frac{(L_* + L_{\rm acc})}{16 \pi \sigma_R} } ( 2 + { \frac{\kappa_s} {\kappa_d} }) \right ] ^ {1/2} {1 \over T_{\rm wall}^2 }, \label{eq:rwall} \end{equation} \noindent which assumes that the thickness of the atmosphere is negligible compared to the radius \citep{muzerolle03,dalessio04}, where $\sigma_R$ is the Stefan-Boltzmann constant; $\kappa_s$ and $\kappa_d$ are the mean opacities to the incident and local radiation, respectively; $L_\star$ is the stellar luminosity; and $L_{\rm acc}$ is the luminosity of the stellar accretion shock as given by the output accretion rate (\mdot) and $R_\star$ of the multi-column accretion shock model described in Section~\ref{sec:shockmodel}, with \begin{equation} L_{\rm acc} = (1 - 1/R_{\rm i})(GM_\star\dot{M}/R_\star). \end{equation} \noindent As described in Section~\ref{sec:shockmodel}, \rin\ is taken to be 2.3\rstar\ for CVSO~109A and 5\rstar\ for all other targets. Disk models with $R_{\rm wall}$ located at the dust sublimation radius predict values of $R_{\rm wall}$ between 0.07-0.54 AU, with stronger accretors having larger values of $R_{\rm wall}$ as indicated by Equation~\ref{eq:rwall} \citep{muzerolle03}. \subsection{Inclinations} \label{sec:inclinations} We infer the inclinations of our targets by estimating their stellar rotation periods from their \tess\ light curves and taking measurements of \vsini\ from \cite{manara21} and \cite{kounkel19}. Temporal analysis of the light curves uses the Scargle periodogram analysis \citep{scargle82,horne86} as implemented in IDL\footnote{J\"orn Wilms 2005: \url{http://astro.uni-tuebingen.de/software/idl/aitlib/timing/scargle.html}}. We look for peaks in the power spectral density (PSD) between 1 to 10 days in excess of a 99\% confidence level. When power is found, we fold the data on the periods that show significant power. To minimize long-term secular variability, we construct a running mean of width 1.5 times the period, and subtract that from the light curve prior to folding. We bin the data into 20 phase bins, setting the uncertainty in each phase bin to the variance in that bin and test the binned light curve against the null hypothesis. We also examine the auto-correlation of the light curve. The width of the correlation peak is proportional to the duration of the typical features contributing to the variations. We also consider the positive and negative excursions in the light curve separately: positive excursions may be due to bright patches on the photosphere; negative excursions may be due to occultations of the surface by circumstellar material, or by starspots. To define positive and negative excursions, we de-trend the light curve with a polynomial fit (3$^{\rm rd}$ to 6$^{\rm th}$ order, depending on the number of points) and retain only those points above or below the trend. Brief discussions of the individual targets can be found in Appendix~\ref{sec:obj_comments}. \section{Analysis and Results} \label{sec:results} Figure~\ref{fig:model_fits} shows the best fits of the accretion shock and accretion disk models to the full NUV--NIR spectra of the nine CTTS analyzed here. Tables~\ref{tab:shockmodelparams} and \ref{tab:diskmodelparams} list, respectively, the accretion and disk model parameters associated with each fit. Using our accretion shock model, we derive V-band extinction, stellar radius, accretion rate, and accretion column structure for each target. Our results indicate that accretion remains strong in TTS longer than originally expected. \citet{hartmann16} predict an accretion rate on the order of $3\times 10^{-9}$ \msunyr\ for a 0.7~\msun\ star of 5~Myr (the median age of the Orion OB1b region). However, \citet{ingleby14} measured accretion rates on the order of $1\times 10^{-8}$ \msunyr\ for CVSO~58, CVSO~90, and CVSO~109. Similarly, our derived accretion rates range from 0.5--17.2$\times10^{-8}$ \msunyr, with a median value of 1.2$\times10^{-8}$ \msunyr. These accretion rates are comparable to those of the 1--2~Myr regions Taurus and Chamaeleon I \citep{ingleby13}. These high accretion rates produce associated accretion luminosities that range from 0.07--1.96~$L_\star$, with a median accretion luminosity of 0.25~$L_\star$. Given that the accretion luminosities are comparable to the stellar luminosities for these targets, the accretion rates from the accretion shock model are important inputs to the inner disk wall model. The average accretion shock temperatures range from 4800~K to 10542~K, in agreement with the range predicted by \cite{cg98}. With a mean temperature of 6511~K, these targets have $T_{\rm shock}$ notably lower than the typically-assumed temperature of 8000--10000~K for a single accretion column \citep[e.g.,][]{dalessio98,fischer11}. The average temperatures of the three accretion columns are 4274~K for the low flux-density column; 6695~K for the medium flux-density column; and 10786~K for the high flux-density column. See Table~\ref{tab:Tshock} in the appendix for the best-fit temperatures for each column of each target. We are able to attain satisfactory fits to the NIR excesses of five targets, though there are a number of cases in which the photosphere goes above the data around 1~\micron\ and thus forces the total model fit to be above the data (CVSO~58, CVSO~104, CVSO~107, CVSO~109A, CVSO~165B, and CVSO~176). In the cases of CVSO~107, CVSO~109A, CVSO~165B, and CVSO~176, there is no clear NIR excess above the photospheric and accretion shock emission, so no inner disk wall is fitted to these targets. This may result from variability in the veiling, which dictates the scaling of the WTTS photospheric template. This will be examined further in future work. The lack of NUV spectra of WTTS templates with the same spectral subtype as some of the CTTS may produce scaling that results in mismatches in the NIR. Four out of the five targets that are fit with an inner wall require a wall height in excess of 5 gas scale heights to produce enough emission to account for the excess. These targets are likely pre-transitional disks, which have extra emission from an optically-thin dust component in their inner disk cavities \citep{mauco18}. Our future work will examine whether this additional component produces a better fit to the data. \subsection{Uncertainties} The average accretion shock model uncertainty, quantified by an MCMC nuisance parameter, is 14\%. This statistical goodness-of-fit is a lower limit to the actual uncertainty as it does not take the uncertainties of the input parameters, the photospheric templates, or the extinction curve into account. When accretion rates are calculated considering the uncertainties in \rstar\ and the filling factors, and assuming an uncertainty of 2\rstar\ for the magnetospheric truncation radius \ri, the median percent error increases from 4\% to 32\%. Uncertainties in our derived stellar radii take into account the error in the data fluxes, CTTS distance, veiling, visual extinction, and an assumed 10\% error in the WTTS template fluxes. Our estimates have an average percent error of 8\%. The updated luminosity measurements included in Table~\ref{tab:shockmodelparams} propagate our derived error in \rstar, but no uncertainty is assumed for the stellar temperature. The uncertainties in the average shock temperature are dominated by the errors in the filling factors. Since the errors are often asymmetric, the larger of the upper and lower uncertainties is taken as the error in propagation. A significant source of systematic uncertainty comes from the availability and scaling of the WTTS photospheric templates. There are seven total WTTS templates with NUV spectra available between SpTs K5 and M2, and there are no templates of type K6 or M0. All CTTS studied here are fit with a template within 1.5 subtypes of their own spectral classification, but even this difference in spectral type introduces some amount of error. Fitting each target CTTS with the next closest template on either side, when available, changes the best-fit visual extinction by 0.37 mag on average. Once a template is chosen, it must be scaled to the data by the veiling measurement, which has its own variability and uncertainty. Three measurements of veiling, each separated by a day, were available for all targets except CVSO~90 and CVSO~104. The measured veiling for a given target varied, on average, by a factor of 1.6 across the three epochs. By choosing the epoch closest in time to the \hst\ observations, we minimize the influence of veiling variability, but it cannot be completely removed. Our derivation of inclinations are rough estimates given that the stellar rotation periods in the \tess\ data can easily be obscured by other processes, such as occultation dips and discrete accretion events. Uncertainties have not been determined for the period measurements, so the quoted uncertainties in inclinations come from errors in \vsini\ and \rstar. We note that if the inclination estimates are wrong, the only parameter that would be affected is the height of the inner disk wall $z_{\rm wall}$. \begin{deluxetable}{lccccccccc} \rotate \tablecaption{Best-fit Accretion Shock Model Parameters of Orion OB1 CTTS \label{tab:shockmodelparams}} \tablehead{ \colhead{Object} & \colhead{$A_{V}$} & \colhead{$R_{\star}$} & \colhead{$L_{\star}$} & \colhead{$L_{\rm acc}$} & \colhead{$\dot{M}$} & \colhead{$f_{1E10}$} & \colhead{$f_{1E11}$} & \colhead{$f_{1E12}$} & \colhead{$T_{\rm shock}$} \\ \multicolumn{1}{c}{} & \multicolumn{1}{c}{(mag)} & \multicolumn{1}{c}{(\rsun)} & \multicolumn{1}{c}{(\lsun)} & \multicolumn{1}{c}{(\lsun)} & \multicolumn{1}{c}{(10$^{-8}$ \msunyr)} & \multicolumn{3}{c}{(fraction of stellar surface covered)} & \multicolumn{1}{c}{(K)} } \startdata CVSO~58 & $1.39^{+0.04}_{-0.04}$ & $1.05^{+0.07}_{-0.07}$ & 0.25$\pm$0.03 & $0.202^{+0.011}_{-0.011}$ & $1.03^{+0.05}_{-0.06}$ & $0.231^{+0.010}_{-0.022}$ & $0.00020^{+0.0017}_{-0.00013}$ & $0.0092^{+0.0007}_{-0.0007}$ & 5493$\pm$96 \\ CVSO~90 & $0.92^{+0.03}_{-0.30}$ & $0.84^{+0.09}_{-0.09}$ & 0.12$\pm$0.03 & $0.234^{+0.011}_{-0.08}$ & $1.25^{+0.06}_{-0.4}$ & $0.053^{+0.016}_{-0.03}$ & $0.00031^{+0.024}_{-0.00023}$ & $0.0202^{+0.0011}_{-0.010}$ & 7975$\pm$1105 \\ CVSO~104 & $0.05^{+0.05}_{-0.03}$ & $1.64^{+0.13}_{-0.12}$ & 0.36$\pm$0.06 & $0.0655^{+0.003}_{-0.0021}$ & $1.14^{+0.05}_{-0.04}$ & $0.0304^{+0.0027}_{-0.004}$ & $0.0074^{+0.0005}_{-0.0004}$ & $0.00049^{+0.00006}_{-0.00003}$ & 5239$\pm$105 \\ CVSO~107 & $1.16^{+0.03}_{-0.05}$ & $1.97^{+0.15}_{-0.15}$ & 0.66$\pm$0.10 & $0.331^{+0.012}_{-0.021}$ & $4.84^{+0.17}_{-0.3}$ & $0.076^{+0.004}_{-0.007}$ & $0.00015^{+0.0027}_{-0.00008}$ & $0.00456^{+0.00024}_{-0.0004}$ & 5656$\pm$128 \\ CVSO~109A & $0.83^{+0.04}_{-0.32}$ & $2.55^{+0.20}_{-0.20}$ & 1.18$\pm$0.19 & $0.61^{+0.04}_{-0.23}$ & $17.2^{+1.0}_{-7}$ & $0.0006^{+0.011}_{-0.0005}$ & $0.0073^{+0.0026}_{-0.0021}$ & $0.0051^{+0.0005}_{-0.0026}$ & 8877$\pm$1910 \\ CVSO~146 & $0.28^{+0.03}_{-0.02}$ & $1.25^{+0.09}_{-0.09}$ & 0.37$\pm$0.05 & $0.0926^{+0.003}_{-0.0027}$ & $0.530^{+0.017}_{-0.015}$ & $0.155^{+0.005}_{-0.006}$ & $0.0159^{+0.0011}_{-0.0010}$ & $0.00058^{+0.00005}_{-0.00004}$ & 4975$\pm$27 \\ CVSO 165A & $0.33^{+0.02}_{-0.02}$ & $1.69^{+0.20}_{-0.20}$ & 0.56$\pm$0.13 & $0.0617^{+0.0017}_{-0.0015}$ & $0.708^{+0.020}_{-0.017}$ & $0.1020^{+0.0024}_{-0.003}$ & $0.00287^{+0.0004}_{-0.00028}$ & $0.0000485^{+0.000007}_{-0.0000023}$ & 4800$\pm$14 \\ CVSO 165B & $1.23^{+0.02}_{-0.02}$ & $2.00^{+0.25}_{-0.25}$ & 1.14$\pm$0.29 & $0.176^{+0.005}_{-0.006}$ & $1.65^{+0.05}_{-0.06}$ & $0.00013^{+0.0004}_{-0.00007}$ & $0.00009^{+0.00015}_{-0.00003}$ & $0.00275^{+0.00008}_{-0.00012}$ & 10542$\pm$343 \\ CVSO~176 & $1.44^{+0.03}_{-0.03}$ & $3.26^{+0.31}_{-0.31}$ & 1.08$\pm$0.21 & $0.0815^{+0.0023}_{-0.0020}$ & $4.18^{+0.12}_{-0.10}$ & $0.0108^{+0.0019}_{-0.004}$ & $0.00106^{+0.00026}_{-0.00020}$ & $0.000267^{+0.000017}_{-0.000015}$ & 5043$\pm$254 \\ \enddata \tablecomments{Best-fit accretion shock model parameters for each CTTS. To calculate the models, the stellar mass, distance, temperature, and veiling are adopted as given in Table~\ref{tab:stellar_params}. \rin\ is taken to be 5\rstar\ for all targets except CVSO~109A, which has $R_{\rm i}=2.3$\rstar\ as described in Section~\ref{sec:shockmodel}. For the best-fit temperatures of individual accretion columns, see Table~\ref{tab:Tshock} in the appendix.} \end{deluxetable} \begin{deluxetable}{lcccc} \tabletypesize{\footnotesize} \tablecaption{Periods and Inclinations of Orion OB1 CTTS \label{tab:inclinations}} \tablehead{ \colhead{Object} & \colhead{\vsini} & \colhead{Period} & \colhead{$v$} & \colhead{$i$} \\ \multicolumn{1}{c}{} & \multicolumn{1}{c}{(km/s)} & \multicolumn{1}{c}{(days)} & \multicolumn{1}{c}{(km/s)} & \multicolumn{1}{c}{($^{\circ}$)} } \startdata CVSO~58 & 17.9$\pm$1.3 & 5.7 & 9.3$\pm$0.5 & ... \\ CVSO~90 & 8.3$\pm$1.6 & 5.1 & 8.3$\pm$0.9 & ... \\ CVSO~104 & 7.5$\pm$1.0 & 4.7 & 17.7$\pm$1.3 & 25.1$\pm$4.1 \\ CVSO~107 & 5.9$\pm$0.9 & 6.4 & 15.6$\pm$0.9 & 22.3$\pm$3.9 \\ CVSO~109A & 3.2$\pm$0.9 & 6.5 & 19.8$\pm$1.2 & 9.3$\pm$2.7 \\ CVSO~146 & 5.0$\pm$0.8 & 5.5 & 11.5$\pm$0.7 & 25.8$\pm$4.8 \\ CVSO~165A & 15.4$\pm$0.9 & 4.3 & 19.9$\pm$1.4 & 50.8$\pm$6.5 \\ CVSO~165B & 15.4$\pm$0.9 & 4.3 & 23.5$\pm$1.9 & 40.9$\pm$4.9 \\ CVSO~176 & 18.4$\pm$1.2 & 7.1 & 23.2$\pm$1.9 & 52.4$\pm$7.8 \\ \enddata \tablecomments{Calculated inclinations $i$ for our targets. \vsini\ comes from the VLT modeling that produced the veilings used in this paper \citep{manara21} for all targets except CVSO~90, which takes \vsini\ from \cite{kounkel19}. Stellar rotational velocities $v$ are calculated using the radii we derive here. The derived $v$ for CVSO~58 is smaller than its \vsini, and the calculated $i$ for CVSO~90 has an error of almost $\pm180^\circ$; thus, inclination is assumed to be a standard 60$^\circ$ for these targets, corresponding to cos($i)=0.5$.} \end{deluxetable} \begin{deluxetable}{lccc} \tablecaption{Best-fit Accretion Disk Model Parameters of Orion OB1 CTTS \label{tab:diskmodelparams}} \tablehead{ \colhead{Object} & \colhead{$z_{\rm wall}$} & \colhead{$T_{\rm wall}$} & \colhead{$R_{\rm wall}$} \\ \multicolumn{1}{c}{} & \multicolumn{1}{c}{(H)} & \multicolumn{1}{c}{(K)} & \multicolumn{1}{c}{(AU)} } \startdata CVSO~58 & 5 & 1200 & 0.10 \\ % CVSO~90 & 5.5 & 1600 & 0.06 \\ % CVSO~104 & 5.5 & 1200 & 0.07 \\ % CVSO~146 & 18 & 1800 & 0.05 \\ % CVSO~165A & 8 & 1400 & 0.07 \\ % \enddata \tablecomments{Best-fit parameters for the the accretion disk models given input parameters as shown in Tables~\ref{tab:stellar_params} and \ref{tab:shockmodelparams}. We adopt $a_{\rm min}=0.005$~{\micron}, $a_{\rm max}=10$~{\micron}, and $p=3.5$. Disk inclinations $i$ are taken from Table~\ref{tab:inclinations}.} \end{deluxetable} \section{Discussion} \label{sec:discussion} In Section~\ref{sec:lit_comp}, we place our results in the context of previous studies. In Section~\ref{sec:Av}, we examine the significance of the choice of extinction law and the wavelength range of data available. \subsection{Comparison to previous results} \label{sec:lit_comp} \begin{deluxetable*}{lccccccccccc} \tablecaption{Literature comparison \label{tab:lit_comp}} \tablehead{ \colhead{Object} & \colhead{$\dot{M}_{\rm C05}$} & \colhead{\av$_{\rm ,C05}$} & \colhead{$\dot{M}_{\rm I14}$} & \colhead{\av$_{\rm ,I14}$} & \colhead{$\dot{M}_{\rm M18}$} & \colhead{\av$_{\rm ,M18}$} & \colhead{$\dot{M}_{\rm M21}$} & \colhead{\av$_{\rm ,M21}$} & \colhead{BC$^*$} & \colhead{$\dot{M}_{\rm P22}$} & \colhead{\av$_{\rm ,P22}$} \\ \cmidrule(lr){2-9} \cmidrule(lr){10-12} \colhead{} & \colhead{($\times10^{-8}$)} & \colhead{(mag)} & \colhead{($\times10^{-8}$)} & \colhead{(mag)}& \colhead{($\times10^{-8}$)} & \colhead{(mag)}& \colhead{($\times10^{-8}$)} & \colhead{(mag)} & \colhead{} & \colhead{($\times10^{-8}$)} & \colhead{(mag)} } \startdata CVSO~58 & 0.45 & 0.12 & 1.60 & 0.8$\pm0.5$ & … & … & 0.43 & 0.8 & & $1.03^{+0.06}_{-0.06}$ & $1.39^{+0.04}_{-0.04}$ \\ CVSO~90 & 1.77 & 0.00 & 1.00 & 0.0$\pm0.4$ & … & … & 0.25 & 0.1 & & $1.25^{+0.05}_{-0.5}$ & $0.92^{+0.03}_{-0.30}$ \\ CVSO~104 & 0.75 & 0.00 & … & … & 0.56 & 0.1 & 0.32 & 0.2 & & $1.14^{+0.06}_{-0.04}$ & $0.05^{+0.05}_{-0.03}$ \\ CVSO~107 & 1.09 & 0.32 & 0.25 & 0.7$\pm0.4$ & 0.29 & 0.4 & 5.01 & 0.3 & & $4.83^{+0.19}_{-0.29}$ & $1.16^{+0.03}_{-0.05}$ \\ CVSO~109 & 2.52 & 0.00 & 3.00 & 0.8$\pm0.5$ & 0.67 & 0.0 & 3.24 & 0.1 & A & $17.2^{+1.0}_{-6}$ & $0.83^{+0.04}_{-0.32}$ \\ CVSO~146 & 0.81 & 0.37 & … & … & … & … & 0.27 & 0.6 & & $0.529^{+0.016}_{-0.015}$ & $0.28^{+0.03}_{-0.02}$ \\ CVSO~165 & 0.37 & 0.00 & … & … & … & … & 0.08 & 0.2 & A & $0.708^{+0.021}_{-0.017}$ & $0.33^{+0.02}_{-0.02}$ \\ & … & … & … & … & … & … & … & … & B & $1.65^{+0.05}_{-0.06}$ & $1.23^{+0.02}_{-0.02}$ \\ CVSO~176 & 0.43 & 0.00 & … & … & … & … & 1.45 & 1 & & $4.19^{+0.12}_{-0.11}$ & $1.44^{+0.03}_{-0.03}$ \\ \enddata \tablecomments{Comparison between the accretion rates and visual extinctions derived here and those derived by \cite{calvet05} (C05), \cite{ingleby14} (I14), \cite{mauco18} (M18), and \cite{manara21} (M21). $^*$For the binary targets that are resolved in this study, the BC column indicates which binary component is being referenced.} \end{deluxetable*} \cite{re19} applied the multi-column accretion shock model from \cite{ingleby13} to multi-epoch observations of five CTTS and found that mass accretion rates vary with a spread of a factor of $\sim$2 for a given object. Similarly, \citet{venuti14} found that the accretion rate for a given object has a spread of 0.5 dex (a factor of $\sim$3) from studying about 200 CTTS in NGC 2264. Rather than merely representing this intrinsic variability, the best-fit values for accretion rate and visual extinction presented in this work are systematically higher than those found by previous studies of objects in this sample \citep{calvet05,ingleby14,mauco18,manara21}. See Table~\ref{tab:lit_comp} for the individual values found by each work. \citet{calvet05} modeled all CTTS analyzed here (though their observations of binaries CVSO~109 and CVSO~165 were unresolved), obtaining accretion rates for each using photometric excesses in U--B and U--V. They cite an overall uncertainty of a factor of 3 for each measured accretion rate, with the largest contribution coming from uncertainty in the extinction. They calculate \av\ from the $V-I_C$ color using the \citet{cardelli89} extinction law with $R_V=3.1$. The accretion rates measured here are 0.6-9.7 times that of those presented in \citet{calvet05}, with a median ratio of 2.3. This cannot be attributed to different flux levels, as all of our observed U--V colors are redder than those presented in \citet{calvet05}. Additionally, our measured extinctions are systematically larger than those of \citet{calvet05}, with a median difference of 0.8 mag. Because our accretion rates are systematically higher, it is likely that the different accretion rates result from systematic effects in the different modeling techniques and adopted extinctions rather than solely from true variability. The closest analogue to our modeling technique is that of \citet{ingleby14}, who modeled CVSO~58, CVSO~90, CVSO~107, and the unresolved CVSO~109 system. They used the veiling at V and I to estimate extinction by comparing the observed photospheric V-I$_C$ colors to the standard colors in \citet{kh95}. They note that their inferred V-band extinctions would decrease by 0.2-0.4 mag had they used the colors of \citet{pm13}, and this in turn would decrease their \mdot s by a factor of 1.8–2.4. They then deredden their data using the \citet{whittet04} law towards HD 29647, as we do here. The MagE and MIKE spectra they used for these four targets covered only 3400--9000~\AA, so their fitting does not include the NUV. \citet{ingleby14} thus fit the excess shock emission beyond the Balmer jump using a five-column accretion shock model also based on \citet{cg98} (the three columns used here plus two intermediate columns). However, instead of using the standard accretion shock model, they found that they needed to increase the low-density preshock emission by up to a factor of five to accurately reproduce the bluest regions of the excess emission spectra. This likely explains why the structure of their accretion columns, specifically the area they cover, is notably different from this work. Some of our model fits show a trend of underestimating the flux near the Balmer jump; future work will examine whether increasing this preshock emission produces a better fit to the data, or whether this is merely an effect of Balmer-line crowding that is not accounted for by our continuum model. The ratios of our accretion rates to that of \cite{ingleby14} are 0.7 (CVSO~58), 1.2 (CVSO~90), 19.3 (CVSO~107), and 5.7 (CVSO~109). Note that had they used the same spectral type--color conversion as we did \citep{pm13}, these ratios would have been higher as described above. The differences in our derived accretion rates correspond to the differences in U- and V-band magnitudes between our respective observations. CVSO~58 was dimmer in both bands in our epoch of observations ($\Delta U=0.70$ mag, $\Delta V=0.38$ mag), and we find a lower accretion rate. The other three CTTS were brighter in both bands in our epoch | CVSO~90 ($\Delta U=-0.48$ mag, $\Delta V=-0.53$ mag), CVSO~107, ($\Delta U=-0.61$ mag, $\Delta V=-0.04$ mag), and CVSO~109A ($\Delta U=-1.11$ mag, $\Delta V=-0.09$ mag) | and we found higher accretion rates for these. The greatest increase in U-band belongs to CVSO~109A, which is consistent with the \tess\ light curve that shows that our \hst\ observation occurred near a local maximum in CVSO~109's light curve \citep{espaillat22}. These magnitudes come from observed, rather than dereddened, data, so differences in our treatments of extinction have no effect. Thus, our accretion rates are in broad agreement with those of \cite{ingleby14}. \citet{mauco18} estimated accretion rates for CVSO~104, CVSO~107, and the unresolved CVSO~109 system using the \halpha-\mdot\ relation found for Taurus CTTS by \citet{ingleby13}: log(\mdot)=1.1($\pm0.3$)log(L$_{{\rm H}\alpha})-5.5(\pm0.8$), where the value of the \halpha\ line luminosity was estimated as its equivalent width times the continuum flux. \cite{espaillat22} found that for CVSO~109 this method produces a lower accretion rate than do the accretion shock models. This is what we find here, as the ratios of our accretion rates to theirs are 2.0 (CVSO~104), 16.7 (CVSO~107), and 25.7 (CVSO~109). The discrepancy may result from 1) the small number of sources used to determine this relation (10 CTTS), or 2) the assumption of a uniform distance for all Taurus CTTS (140 pc) in the determination of $L_{{\rm H}\alpha}$. \citet{manara21} analyzed X-Shooter spectra of all of the ULLYSES Orion targets, using a hydrogen slab model to obtain their accretion properties. They fit for \av\ using the \cite{cardelli89} extinction law. X-Shooter spectra are available only beyond 3000~\AA, so the NUV is not included in their analysis. Our multi-column accretion shock model finds higher accretion rates for all targets except CVSO~107, which has the highest accretion rate in the analysis of \cite{manara21}. The median ratio of our accretion rates to theirs is 3.5. Six of our CTTS targets have stellar radii from \cite{manara21} included in Table~\ref{tab:stellar_params}. There are no individual uncertainties available for the \cite{manara21} spectral types and luminosities, but previous work using their method of determining stellar radii showed that objects with spectral types earlier than M4.5 have percent errors less than 25\% \citep{alcala17}. Assuming a percentage error of 25\% for the radii calculated from the \cite{manara21} spectral types and luminosities, we find that our best-fit radii for CVSO~58, CVSO~90, and CVSO~104 are consistent within the errors; those fit for CVSO~107 and CVSO~176 are larger by a factor of 1.4 and 1.8, respectively; and that fit for CVSO~146 is lower by a factor of 1.5. The systematically higher accretion rates presented in this work can in large part be attributed to a) our use of the accretion shock model as opposed to \halpha\ luminosity or an isothermal hydrogen slab model, and b) the larger NUV wavelength coverage of these data, as the NUV is the most important region for constraining the highest energy-flux density accretion column. The latter has the important implication that all ground-based measurements of CTTS accretion rates may be underestimated. Since \hst\ has a finite lifetime, future work should examine whether a correction factor can be determined to account for the systematic underestimation of accretion rates caused by the lack of NUV coverage. \subsection{Extinction law} \label{sec:Av} When modeling NUV observations of CTTS, the choice of extinction law is incredibly important because of the strong attenuation by grains at UV wavelengths and the constraint imposed by the 2175~\AA\ bump. If a law produced for the general ISM is used, such as \citet{cardelli89} or \citet{fitzpatrick19}, the absorption bump feature is strong and will bias results towards lower values of \av. If, by contrast, the \cite{whittet04} law towards HD 29647 (which is embedded in Taurus) is used, the absorption feature is much less pronounced and the fitted \av\ will be higher. No single interstellar extinction law can describe all star-forming regions equally well. This is supported by the finding that Taurus and Ophiuchus exhibit very different ultraviolet extinction functions, which \cite{whittet04} suggest is likely caused by their different populations of massive stars. Ophiuchus's significant population of OB stars produce radiation that maintains the strength of the 2175~\AA\ bump, so the \citet{cardelli89} law is better suited for use there than is the \cite{whittet04} law. Thus, NUV data for specific TTS provide an important constraint on the characteristics of both the individual stars themselves and their surrounding environments. Data from \hst\ are vital for probing the ISM of star-forming regions to distinguish between grain populations that significantly attenuate around 2175~\AA\ and those that do not. For these targets in Orion, the fits produced by the \citet{whittet04} law are better than those produced by the \citet{cardelli89} law, which overcorrects for the 2175~\AA\ bump (see Figure~\ref{fig:Av_comp}). This indicates that Orion OB1b is likely more similar to Taurus than Ophiuchus in terms of its interstellar extinction function. This is supported by an analysis of the ultraviolet interstellar radiation field (ISRF) of the region. The Habing field parameter $G_0$ gives the ratio of the local field enhanced by a neighboring OB star to the typical ISRF ($F_0$) according to \begin{equation} G_0 = \frac{1}{F_0}\frac{L_{\rm FUV}}{4\pi r^2}, \end{equation} \noindent where $F_0$ is assumed to be $1.6\times 10^{-3}$ \escm\ \citep{habing68}, $L_{\rm FUV}$ is approximated as the OB star's luminosity, and $r$ is the true distance between the OB star and the CTTS target of interest \citep{anderson13,mauco16}. \cite{liseau99} found that the $\rho$ Ophiuchi star-forming region has $G_0=20$--140. In contrast, $G_0$ values for our OB1b CTTS targets are much lower. Assuming that the seven most significant OB stars around Orion OB1b ($\zeta$ Ori, $\epsilon$ Ori, $\delta$ Ori, $\eta$ Ori, 22 Ori, 25 Ori, and $\psi^2$ Ori) are all at the median OB1b distance of 400 pc, $G_0$ for our targets has a median value of 0.7 and a maximum value of 67. This, in combination with the low \av\ of the region, demonstrates that though Orion OB1b is an OB region, its interstellar extinction function should align more with that of quiescent Taurus than of hotter, dustier $\rho$ Ophiuchi. This can in part explain the relatively high accretion rates of this region in spite of its intermediate age, as these disks have not been externally photoevaporated by an enhanced ISRF. This is also consistent with the disk models from \cite{calvet05}, which could not reproduce the low long-wavelength infrared fluxes observed in Orion OB1 using small outer disk radii consistent with external photoevaporation ($\sim$30~AU). Instead, they concluded that the disks must be flatter and have larger maximum grain sizes than those in Taurus ($a_{\rm max}$=1~mm). For an analysis of the effect of an enhanced ISRF on circumstellar disk evolution, see \cite{anderson13}. The ULLYSES survey, which will provide over 80 custom-calibrated NUV spectra of CTTS in nine star-forming regions, will allow us to examine the goodness-of-fit of different extinction laws in different environments. Beyond this, it would be ideal to attain extinction curves for all nearby star-forming regions rather than using either a general ISM curve or the curve calculated to the specific environment of Taurus. \section{Summary} \label{sec:summary} \begin{itemize} \item Accretion rates for the 9 CTTS studied here range from $0.5-17.2\times 10^{-8}$ \msunyr, relatively high for the intermediate age of Orion OB1b. \item Our accretion rates and V-band extinctions are systematically higher than those calculated from optical data in previous works, in large part due to our wavelength coverage that extends into the NUV. \item The NIR excesses of the five targets in which an excess is present are fit with 1200--1800~K inner disk walls located at 0.05--0.10~AU from the host stars. \item The choice of extinction law significantly affects the calculated accretion rate and introduces uncertainty that is difficult to quantify. Our analysis indicates that the environment of Orion OB1b is more similar to quiescent Taurus than to hot $\rho$ Ophiuchi. Ideally, extinction curves can be calculated for each star-formation region in the near future. \item This multi-column shock and DIAD analysis will be applied to all nine star-forming regions being observed through ULLYSES, allowing us to extend the analysis across distinct stellar populations and search for correlations between accretion and disk properties in a larger sample. Additionally, it will be applied to the four CTTS being monitored by the ULLYSES program (GM Aur, TW Hya, BP Tau, and RU Lup). \end{itemize} \acknowledgements {Support for this work comes from \hst\ AR-16129, as well as NASA through grant number AR 16129 from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. This work benefited from discussions with the ODYSSEUS team (\url{https://sites.bu.edu/odysseus/}); see \cite{espaillat22} for an overview of the ODYSSEUS survey. J.H. acknowledges support from CONACyT project No. 86372 and the UNAM-DGAPA-PAPIIT project IA102921. P.A., E.F. and \'A.K. acknowledge support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 716155 (SACCRED). C.F.M. acknowledges funding by the European Union under the European Union’s Horizon Europe Research \& Innovation Programme 101039452 (WANDA). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. C.F.M., J.M.A, M.G. and E.F. acknowledge support from the project PRIN-INAF 2019 ``Spectroscopically Tracing the Disk Dispersal Evolution''. S.H.P.A. acknowledges support from CNPq, CAPES and Fapemig. Based on observations obtained with the NASA/ESA Hubble Space Telescope, retrieved from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. This paper utilizes the D’Alessio irradiated accretion disk (DIAD) code. We wish to recognize the work of Paola D’Alessio, who passed away in 2013. Her legacy and pioneering work live on through her substantial contributions to the field.} \bibliography{bibliography}{} \bibliographystyle{aasjournal} \appendix \section{CVSO~17 and CVSO~36} \label{sec:app_A} As shown in Figure~\ref{fig:17and36}, we find that CVSO~17 and CVSO~36 show no significant UV continuum excess in their \hst\ spectra that would indicate active accretion, which aligns with the analysis of their X-Shooter spectra by \cite{manara21} and confirms their photometric characterization as WTTS by \cite{calvet05}. Additionally, an analysis of their far-UV spectra shows no fluorescent H$_2$ emission, which is a clear delineation between CTTS and WTTS \citep{france12,alcala19}. \section{TESS object notes} \label{sec:obj_comments} {\bf CVSO~58:} In Sector~6 there was significant power at 2.87 and 5.31 days. The 2.9 day period folds well, but the likelihood that the amplitude of the period is significant is only 0.12. In Sector~32 significant power exists at 6.45 and 1.41 days, but neither looks periodic after folding. The auto-correlation functions in both sectors show a peak near 5.7 days. This peak is strongest for the dips; during Sector~6 the positive excursions (brightenings) show a less well-defined peak at a period near 6 days. {\bf CVSO~90:} The Sector~6 PSD peaks at 3.15 days, but there is no significant period seen in the folded light curve. The Sector~32 PSD shows a strong and broad peak at 5.11 days. The folded light curve is not significant due to lots of scatter in the half of the period dominated by deep dips. The autocorrelation function shows a peak at a lag of about 4.4 days for the absorption dips, and a peak at a 5.7 day period for the brightenings. {\bf CVSO~104:} This star has a peak in the PSD near 4.7 days in each sector. It is not clear whether the light curve is dominated by bright or dark excursions. We do not attempt to fold the two sectors together because it it impossible to keep track of the phases over this time. {\bf CVSO~107:} The Sector~6 light curve is dominated by two deep dips spaced 12 days apart. The periodogram finds power at 6.3 days, or half that spacing. Power at the same period also dominates in Sector~32. The brightenings show a preferred lag of 6.5 days, while the dips lag at 7.7 days. There are three possible short flares of 2-3 hour duration in Sector~6. {\bf CVSO~109:} The two light curves look similar, with strongest power near 6 days (6.0~days in Sector~6; 6.6 days in Sector~32). After subtracting an 8 day running mean, the folded light curves look sinusoidal, with amplitudes of about 0.07~mag. The autocorrelation functions are broad, consistent with a sinusoidal modulation. {\bf CVSO~146:} This star is partially blended with the brighter A2 star HD~290671 in the $TESS$ images (4.8 pixel separation). The A star has a low amplitude 1.56 day period. There is power at about 9 days in each sector; there is additional power at 5.5~days in Sector~6 and 4.4~days in Sector~32. None of these appear periodic. The autocorrelation power in Sector~6 is strongest for the dips, at a period of 10-12 days; the brightenings show a correlation at 7-8 days; in Sector~32 both the brightenings and fadings show power near 5 days, with comparable peaks near 7 days (brightenings) and 8 days (fadings). {\bf CVSO~165:} $TESS$ cannot resolve this pair. The periodograms at both epochs show strongest power at 4.3 days; in the latter half of Sector~32 there are two prominent (0.1~mag) dips, and possibly 2 others cut off, at this spacing, but that period is not obvious at other times. There is substructure in Sector~6, with the brightenings correlating with lags of 7.7 and 13 days and the fadings correlating at 8.8~days (twice the strongest period) and 11.6 days. In Sector~32 the brightenings and fadings correlate on the 4.3 day period. {\bf CVSO~176:} This was not observed by TESS in Sector~32. In Sector~6 the character is clearly that of a dipper, with fadings up to 0.2~mag. Despite the periodogram placing most of the power at 3.6 days, the autocorrelation shown the strongest peak at 7.3 days, in both brightenings and dips. \section{Accretion column temperatures and mass flux rates} \label{sec:shock_temps} Blackbodies are individually fit to the three accretion column spectra of each target, then a weighted average is taken by scaling the associated accretion luminosities by each column's fractional filling factor according to Equation~\ref{eq:Tshock}. Table~\ref{tab:Tshock} shows the individual temperatures fit to each accretion column. \begin{deluxetable}{lcccc} \tablecaption{Accretion column temperatures \label{tab:Tshock}} \tablehead{ \colhead{Object} & \colhead{T$_{1E10}$} & \colhead{T$_{1E11}$} & \colhead{T$_{1E12}$} & \colhead{T$_{\rm shock}$} \\ \colhead{} & \colhead{(K)} & \colhead{(K)} & \colhead{(K)} & \colhead{(K)} } \startdata CVSO~58 & 4422$\pm$1 & 6880$\pm$3 & 10916$\pm$3 & 5493$\pm$96 \\ CVSO~90 & 4171$\pm$2 & 6816$\pm$3 & 10870$\pm$3 & 7975$\pm$1105 \\ CVSO~104 & 4051$\pm$2 & 6634$\pm$3 & 10698$\pm$3 & 5239$\pm$105 \\ CVSO~107 & 4187$\pm$2 & 6696$\pm$3 & 10679$\pm$3 & 5656$\pm$128 \\ CVSO~109A & 4255$\pm$2 & 6506$\pm$2 & 10714$\pm$2 & 8877$\pm$1910 \\ CVSO~146 & 4459$\pm$1 & 6871$\pm$3 & 10973$\pm$3 & 4975$\pm$27 \\ CVSO~165A & 4675$\pm$1 & 6853$\pm$3 & 10760$\pm$3 & 4800$\pm$14 \\ CVSO~165B & 4309$\pm$1 & 6690$\pm$3 & 10729$\pm$3 & 10542$\pm$343 \\ CVSO~176 & 3938$\pm$2 & 6310$\pm$1 & 10736$\pm$2 & 5043$\pm$254 \\ \enddata \tablecomments{Best-fit temperatures for the three accretion columns and the resultant weighted-average temperature. Uncertainties on T$_{1E10}$, T$_{1E11}$, and T$_{1E12}$ are one standard deviation on the temperature. The error on T$_{\rm shock}$ comes from the propagated accretion column temperature and filling factor uncertainties. Since the filling factor uncertainties are not symmetric, the larger of the two uncertainties is used in the error propagation.} \end{deluxetable} Table~\ref{tab:mass_fluxes} shows the mass flux rates of the individual accretion columns for each target. We find no clear correlations between the column that contributes the most to the total mass flux and either the stellar mass or the total mass accretion rate. Our future analysis of the entire ULLYSES sample will allow us to expand this analysis and look for correlations with stellar age. \begin{deluxetable}{lccccc} \tablecaption{Accretion column mass flux rates \label{tab:mass_fluxes}} \tablehead{ \colhead{Object} & \colhead{M$_\star$/R$_\star$} & \colhead{$\dot{M}_{1E10}$} & \colhead{$\dot{M}_{1E11}$} & \colhead{$\dot{M}_{1E12}$} & \colhead{$\dot{M}_{\rm tot}$} \\ \colhead{} & \colhead{(M$_{\odot}$/R$_{\odot}$)} & \colhead{(10$^{-8}$ \msunyr)} & \colhead{(10$^{-8}$ \msunyr)} & \colhead{(10$^{-8}$ \msunyr)} & \colhead{(10$^{-8}$ \msunyr)} } \startdata CVSO~176 & 0.08 & $0.93^{+0.17}_{-0.4}$ & $0.92^{+0.22}_{-0.18}$ & $2.31^{+0.15}_{-0.13}$ & $4.18^{+0.11}_{-0.12}$ \\ CVSO~104 & 0.23 & $0.227^{+0.021}_{-0.03}$ & $0.55^{+0.04}_{-0.03}$ & $0.360^{+0.05}_{-0.022}$ & $1.14^{+0.06}_{-0.03}$ \\ CVSO~109A & 0.20 & $0.018^{+0.28}_{-0.015}$ & $2.1^{+0.7}_{-0.6}$ & $15.1^{+1.5}_{-5}$ & $17.2^{+1.0}_{-4}$ \\ CVSO~107 & 0.27 & $0.69^{+0.04}_{-0.06}$ & $0.012^{+0.23}_{-0.007}$ & $4.11^{+0.23}_{-0.4}$ & $4.83^{+0.19}_{-0.3}$ \\ CVSO~165A & 0.34 & $0.531^{+0.012}_{-0.017}$ & $0.150^{+0.020}_{-0.014}$ & $0.0253^{+0.004}_{-0.0013}$ & $0.708^{+0.020}_{-0.017}$ \\ CVSO~90 & 0.74 & $0.032^{+0.010}_{-0.019}$ & $0.0018^{+0.16}_{-0.0014}$ & $1.21^{+0.06}_{-0.6}$ & $1.25^{+0.05}_{-0.4}$ \\ CVSO~58 & 0.77 & $0.206^{+0.009}_{-0.019}$ & $0.0018^{+0.016}_{-0.0012}$ & $0.82^{+0.06}_{-0.07}$ & $1.03^{+0.06}_{-0.06}$ \\ CVSO~165B & 0.42 & $0.0007^{+0.0026}_{-0.0004}$ & $0.0053^{+0.010}_{-0.0020}$ & $1.64^{+0.05}_{-0.07}$ & $1.65^{+0.05}_{-0.07}$ \\ CVSO~146 & 0.69 & $0.220^{+0.007}_{-0.008}$ & $0.226^{+0.015}_{-0.014}$ & $0.082^{+0.007}_{-0.006}$ & $0.529^{+0.018}_{-0.016}$ \\ \enddata \tablecomments{Mass flux rates of individual accretion columns with targets listed in order of increasing stellar mass. There is no clear correlation between the column that dominates the mass flux and either the stellar mass or the total accretion rate.} \end{deluxetable}

Title: A JWST/NIRCam Study of Key Contributors to Reionization: The Star-forming and Ionizing Properties of UV-faint $z\sim7-8$ Galaxies

Abstract: Spitzer/IRAC imaging has revealed that the brightest $z\sim7-8$ galaxies often exhibit young ages and strong nebular line emission, hinting at high ionizing efficiency among early galaxies. However, IRAC's limited sensitivity has long hindered efforts to study the fainter, more numerous population often thought largely responsible for reionization. Here we use CEERS JWST/NIRCam data to characterize 118 UV-faint (median M$_{UV}=-19.6$) $z\sim6.5-8$ galaxies. We find that the SEDs are typically dominated by young ($\sim$10-50 Myr), low-mass ($M_\ast\sim10^8\ M_\odot$) stellar populations, with no need for extreme masses ($\sim10^{11} M_\odot$) among our sample in contrast to recent findings in CEERS. Considering previous studies of UV-bright (M$_{UV}\sim-22$) $z\sim7-8$ galaxies, we find evidence for a strong (5-10$\times$) increase in specific star formation rate toward lower luminosities (median sSFR=82 Gyr$^{-1}$ in CEERS). The larger sSFRs imply a more dominant contribution from OB stars in the relatively numerous UV-faint population, perhaps suggesting that these galaxies are very efficient ionizing agents (median $\xi_{ion}=10^{25.7}$ erg$^{-1}$ Hz). In spite of their much larger sSFRs, we find no significant increase in [OIII]$+$H$\beta$ EWs towards fainter M$_{UV}$ (median $\approx$780 $\mathring{A}$). If confirmed, this may indicate that a substantial fraction of our CEERS galaxies possess extremely low metallicities ($\lesssim$3% $Z_\odot$) where [OIII] emission is suppressed. Alternatively, high ionizing photon escape fractions or bursty star formation histories can also weaken the nebular lines in a subset of our CEERS galaxies. While the majority of our objects are very blue (median $\beta=-2.0$), we identify a significant tail of very dusty galaxies ($\beta\sim-1$) at $\approx$0.5$L_{UV}^\ast$ which may contribute significantly to the $z\sim7-8$ star formation rate density.

https://export.arxiv.org/pdf/2208.14999

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} galaxies: high-redshift -- dark ages, reionization, first stars -- galaxies: evolution \end{keywords} \section{Introduction} \label{sec:intro} The epoch of hydrogen reionization reflects when galaxy formation began impacting the large-scale ionization state of the Universe \citep{Dayal2018,Robertson2021}. Current observational constraints imply that nearly all hydrogen atoms in the intergalactic medium were reionized by $z\simeq5.3$ \citep{Bosman2021}, with this process approximately halfway complete by $z\sim7-8$ \citep[e.g.][]{Davies2018,Mason2018_IGMneutralFrac,Jung2020,Planck2020,Wang2020,Yang2020_Poniuaena,Goto2021}. Given the rapidly declining quasar luminosity function at $z>3$, it has long been thought that star-forming galaxies are likely the dominant contributors to reionization \citep[e.g.][]{Madau1999,Ciardi2000}, motivating much effort to characterize these systems in detail. Thanks to a large collection of dedicated surveys, our initial understanding of galaxies present during the reionization era has emerged over the past two decades. In the mid-to-late 2000s, the first small samples of Lyman-break galaxies at $z\gtrsim7$ were identified in deep near-infrared images taken with ground-based facilities as well as the NICMOS camera on the \textit{Hubble Space Telescope} (\HST{}; \citealt{Bouwens2004,Bouwens2008,Kneib2004,Yan2004,Mannucci2007,Stanway2008}). Our census of these early galaxies advanced tremendously in the 2010s following the installation of the Wide Field Camera 3 (WFC3) on \HST{}, affording an $\approx$40$\times$ increased near-IR survey efficiency relative to NICMOS. To date, several deep \HST{} imaging surveys have been conducted in both blind and gravitationally lensed fields \citep{Bouwens2011_HUDF,Grogin2011,Koekemoer2011,Trenti2011,Windhorst2011,Illingworth2013,Lotz2017,Coe2019} resulting in the discovery of approximately 1000 Lyman-break galaxies at $z\sim7-10$ with intrinsic UV continuum luminosities spanning three orders of magnitude ($-22 \lesssim \Muv{} \lesssim -14$; e.g. \citealt{McLure2013,Atek2015a,Bouwens2015_LF,Bouwens2021_LF,Finkelstein2015_LF,Finkelstein2022,McLeod2016,Livermore2017,Ishigaki2018,Oesch2018_z10LF,Salmon2020}). The derived UV continuum luminosity functions at $z\sim7-10$ have revealed a characteristic UV luminosity ($L_\mathrm{UV}^\ast$) equivalent to $\mathrm{M}_\mathrm{UV}^\ast \sim -21$ with a steep faint-end slope ($\alpha \sim -2$) that steadily steepens with increasing redshift \citep[e.g.][]{Finkelstein2015_LF,Ishigaki2018,Bouwens2021_LF}, indicating a highly dominant population of UV-faint ($\Muv{} \gtrsim -20$) galaxies in the early Universe. From the WFC3 data, the UV continuum slopes of $z\sim7-9$ galaxies were typically found to be very blue with $-2.5 \lesssim \beta \lesssim -2$ (where $f_\lambda \propto \lambda^\beta$; e.g. \citealt{McLure2011,Dunlop2012,Finkelstein2012,Rogers2013,Bouwens2014_beta,Bhatawdekar2021}), suggesting little-to-no dust attenuation in most systems. However, the rest-UV regime probed by WFC3 lacked the information necessary to characterize the stellar populations dominating the observed spectral energy distributions (SEDs). Prior to the launch of the \textit{James Webb Space Telescope} (\JWST{}), our primary insights into the ages and stellar masses of $z\gtrsim7$ galaxies have come from the Infrared Array Camera (IRAC) on board the \textit{Spitzer Space Telescope} which could probe the rest-optical emission of reionization-era systems. Early studies frequently found strong photometric excesses between the near-infrared and IRAC 3--5$\mu$m data of faint $z\sim5-7$ galaxies, which was at the time interpreted as extremely strong Balmer breaks indicative of very old ($\sim$300 Myr) and massive ($\Mstar{} \sim 10^{10-11} \Msol{}$) stellar populations \citep[e.g.][]{Egami2005,Eyles2007,Stark2009,Gonzalez2010}. Not only did this imply extremely rapid stellar mass growth at $z>8$, but also little evolution in galaxy specific star formation rates (sSFRs) towards earlier epochs in contrast to expectations \citep[e.g.][]{Weinmann2011,Dave2011,Dave2012,Dayal2012}. By the early 2010s it was realized that high equivalent width (EW) rest-optical nebular emission lines (e.g. H$\alpha$, H$\beta$, and [OIII]$\lambda$4959,5007) can significantly boost the IRAC broadband photometry from high-redshift galaxies \citep[e.g.][]{Schaerer2009,Schaerer2010,Ono2010,Labbe2013,Stark2013_NebEmission,deBarros2014_NebEmission,Gonzalez2014,Smit2014}. This meant that the IRAC photometric excesses seen in many faint $z\gtrsim5$ galaxies could potentially be explained by either a very strong Balmer break, or very high EW lines powered by a much younger ($\lesssim$50 Myr), lower-mass ($\Mstar{} \sim 10^{7-8} \Msol{}$) stellar population. Because higher EW nebular lines are typically associated with more efficient production of hydrogen ionizing photons \citep[e.g.][]{Chevallard2018_z0,Tang2019}, these alternative solutions could have substantial implications for our understanding of how galaxies contributed to reionization. Unfortunately, breaking this degeneracy proved to be very challenging given that IRAC data suffered from inadequate sensitivity ($m\sim26$ at 5$\sigma$ in the deepest fields), strong source confusion due to its broad point spread function (FWHM$\approx$2\arcsec{}), and limited sampling of the rest-optical SED with only two filters at 3--5$\mu$m. The sensitivity limitations of IRAC alone frequently forced studies of faint ($< L_\mathrm{UV}^\ast$) reionization-era galaxy SEDs to resort to stacking, leaving much uncertainty on the demographics of ages, stellar masses, sSFRs, and nebular line strengths among this population. While deep IRAC data in lensing fields enabled studies to overcome some of these issues \citep[e.g.][]{Huang2016_SURFSUP,Strait2020}, the numbers of UV-faint $z\gtrsim7$ galaxies detected at high IRAC S/N remained small, necessitating further work to characterize this population. The successful development, launch, and commissioning of \JWST{} now provides the opportunity to completely transform our understanding of typical ($<\!L_\mathrm{UV}^\ast$; $m\gtrsim26$) reionization-era galaxies. In this work, we specifically focus on utilizing the greatly improved 1--5$\mu$m imaging capabilities of \JWST{}/NIRCam. By reaching 5$\sigma$ depths of $m\sim29$ with just 1 hour integrations, NIRCam is $\sim$30,000$\times$ more sensitive than IRAC at 3--5$\mu$m ($m\approx26$ in 120 hours; e.g. \citealt{Labbe2013,Oesch2013_z9LF}). Such a dramatic gain in sensitivity would alone revolutionize our ability to characterize the rest-optical SEDs of individual faint reionization-era galaxies. Thankfully, \JWST{}/NIRCam also provides two other key improvements over \Spitzer{}/IRAC enabling us to push our understanding of $z\gtrsim7$ systems even further. The large mirror size of \JWST{} results in a $\approx$15-fold improvement in angular resolution over \Spitzer{} at 3--5$\mu$m, greatly mitigating past source confusion issues that would cause large numbers of viable $z\gtrsim7$ candidates to be discarded from rest-optical analyses. Finally, NIRCam provides a large suite of medium- and broad-band filters at $\approx$3--5$\mu$m which not only improves photometric redshift precision (since the strongest nebular lines impact different bands at different redshifts), but also greatly alleviates degeneracies in interpreting photometric excesses as arising from strong Balmer breaks versus very high-EW nebular lines. Consequently, NIRCam data is able to substantially clarify not only the star-forming properties of UV-faint reionization-era galaxies, but their ionizing properties as well. Our primary goal in this work is to begin developing a much clearer picture of the physical properties (e.g. ages, stellar masses, sSFRs, nebular line EWs, and dust optical depths) of typical (sub-$L_\mathrm{UV}^\ast$) reionization-era galaxies by characterizing their rest-UV$+$optical SEDs in detail with \JWST{}/NIRCam imaging data. To do so, we utilize data from the Cosmic Evolution Early Release Science (CEERS\footnote{\url{https://ceers.github.io/}}) survey which provides deep NIRCam imaging in seven bands from 1--5$\mu$m, including four bands at $\approx$3--5$\mu$m. We focus our attention on UV-selected Lyman-break galaxies at $z\sim6.5-8$ where the 3--5$\mu$m filter set of CEERS/NIRCam affords rich insight into both the strength of nebular emission lines as well as the underlying rest-optical continuum. The structure of this paper is as follows. We begin by describing the imaging data, source extraction, photometric calculations, sample selection, and photoionization modelling in \S\ref{sec:sec2}. We then present the variety of NIRCam SEDs measured among our sample of 118 Lyman-break $z\sim6.5-8$ galaxies, discussing implications for demographics in ages, stellar masses, and sSFRs of typical reionization-era systems (\S\ref{sec:SEDs}). Next we discuss two unexpected yet seemingly important classes of sub-$L_\mathrm{UV}^\ast$ $z\sim6.5-8$ galaxies that are newly revealed with the NIRCam data (\S\ref{sec:weakOIII}--\ref{sec:dusty}). We then combine the results from our faint $z\sim6.5-8$ galaxy sample with previous wide-area results on the UV-bright $z\sim7-8$ population (where IRAC data was sufficiently sensitive) to begin investigating any UV luminosity dependence on sSFR or nebular line EWs (\S\ref{sec:sSFRandEW}). Finally, we briefly discuss whether we find evidence of extremely massive ($M_\ast \sim 10^{11}\ M_\odot$) galaxies in our sample, comparing with recent \JWST{}/NIRCam studies (\S\ref{sec:stellarMasses}). Throughout this paper, we quote all magnitudes in the AB system, assume a \citet{Chabrier2003} initial mass function (IMF) with limits of 0.1--300 \Msol{}, and adopt a flat $\Lambda$CDM cosmology with parameters $h=0.7$, $\Omega_\mathrm{M}=0.3$, and $\Omega_\mathrm{\Lambda}=0.7$. We provide a catalog\footnote{\url{https://www.ryan-endsley.com/ceers-z6p5to8-catalog}} listing coordinates, magnitudes, and inferred physical properties among our sample. \section{Imaging Data and Sample Selection} \label{sec:sec2} We begin this section by describing the CEERS \JWST{}/NIRCam imaging data over the Extended Groth Strip (EGS) field, as well as the overlapping optical \HST{}/ACS imaging utilized for our dropout selection (\S\ref{sec:imaging}). Our source extraction and photometric calculations are then detailed in \S\ref{sec:photometry}, wherein we also describe and demonstrate a custom algorithm that removes contaminating flux from nearby objects to help produce reliable photometry. The Lyman-break criteria used to select our final $z\sim6.5-8$ galaxy sample are described in \S\ref{sec:sample} where we also provide a brief description of the photometric properties of the sample. Finally, we describe the star-forming photoionization model SED fitting procedure used to infer the physical properties for each of our galaxies in \S\ref{sec:beagle}. \subsection{\JWST{} and \HST{} Imaging in EGS} \label{sec:imaging} The CEERS \citep{Finkelstein2017_CEERS} \JWST{}/NIRCam imaging utilized in this work were taken in June 2022 and cover $\approx$40 arcmin$^2$ over the EGS field in six broadband filters (F115W, F150W, F200W, F277W, F356W, and F444W) as well as one medium-band filter (F410M). In the redshift window we aim to select galaxies ($z\sim6.5-8$), the rest-optical SED is probed by at least three of these NIRCam filters (F356W, F410M, and F444W) while F277W additionally covers part of the rest-optical regime at $z\lesssim7.3$. The availability of F410M photometry in particular delivers key information on whether any photometric excesses in these bands are likely driven by a strong Balmer break or contamination from high-EW nebular line emission. Before detailing our NIRCam image reduction and photometric calculations, we first briefly describe the optical \HST{} imaging that we employ to identify Lyman-break $z\sim6.5-8$ galaxies across the CEERS NIRCam footprint. The EGS field possesses deep imaging in three optical filters (F435W, F606W, and F814W) taken with the Advanced Camera for Surveys (ACS) as part of the following \HST{} surveys: the All-Wavelength Extended Groth Strip International Survey (AEGIS; \citealt{davis2007}), the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS; \citealt{Grogin2011,Koekemoer2011}), and the Ultraviolet Imaging of the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey Fields (UVCANDELS\footnote{\url{https://archive.stsci.edu/hlsp/uvcandels}}; PI: Teplitz). All ACS data from these surveys were processed into co-added calibrated mosaics using the \textsc{grizli} software \citep{grizli2022} as part of the Complete Hubble Archive for Galaxy Evolution (CHArGE) project (Kokorev in prep.). The CHArGE data products also include mosaics in various \HST{}/WFC3 bands compiled from archival data, which are utilized in companion analyses of the CEERS dataset (Chen et al. in prep). All mosaics from CHArGE are aligned to the \textit{Gaia} astrometric frame with the ACS (WFC3) mosaics set to a pixel scale of 40 mas/pixel (80 mas/pixel). Co-added mosaics for each NIRCam band are produced by first downloading the uncalibrated ({\tt *\_rate.fits}) NIRCam exposures from the MAST Portal.\footnote{\url{https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html}} These data are processed through the \JWST{} Science Calibration Pipeline\footnote{\url{https://jwst-pipeline.readthedocs.io/en/latest/index.html}} where we group exposures by NIRCam filter, pointing, and module combination to achieve precise astrometric alignment (see \citealt{Chen2022}). Before processing the files through the stage 3 step of the pipeline, we revise the photometric calibration specific to each NIRCam detector using the latest updated parameters provided by the NIRCam PI Marcia Rieke (private communication in late August 2022). This updated calibration changes output fluxes by up to $\approx$0.2 mags relative to the default parameters produced by the \JWST{} Science Calibration Pipeline, and we have verified that the total F150W and F200W magnitudes of bright isolated objects in our final mosaics are on average consistent with the WFC3/F160W and WIRCam/$K_s$ magnitudes, respectively, reported in previous EGS catalogs \citep{Skelton2014,Stefanon2017_EGS}. After individual co-added images ({\tt *\_i2d.fits}) are output from the stage 3 step of the pipeline, we align each (one per filter, pointing, and module combination) to the \textit{Gaia} frame. Because the surface density of \textit{Gaia} stars is very low across the CEERS footprint ($\lesssim$3 per module area), direct alignment is not feasible. Instead we utilize \textsc{tweakreg}\footnote{\url{https://drizzlepac.readthedocs.io/en/latest/tweakreg.html}} to align each co-added image to the CHArGE WFC3/F160W mosaic (which is pre-registered to \textit{Gaia}), yielding precise relative alignment (RMS$\approx$6--15 mas). With consistent astrometry across the individual co-adds, we then resample all images for each filter onto a fixed grid with pixel scale of 30 mas/pixel. Background subtraction is then performed on each mosaic using the \textsc{sep} package\footnote{\url{https://sep.readthedocs.io/en/v1.0.x/index.html}}. As a last step to producing our final mosaics, we mitigate $1/f$ noise. To do so, we first run \textsc{Source Extractor} \citep{Bertin1996} on the F200W mosaic. The output source catalog is used to mask objects in the uncalibrated NIRCam exposures before performing a row-by-row background subtraction on these images, largely following the algorithm\footnote{\url{https://www.stsci.edu/jwst/science-planning/proposal-planning-toolbox/simulated-data}} produced by the CEERS team to minimize $1/f$ noise. The background-subtracted uncalibrated files are then re-processed as described in the above paragraph to produce our final mosaics. When performing aperture photometry on the \JWST{} and \HST{} data, we must account for the fact that the width of the point spread function (PSF) varies by a factor of $\approx$4 across the different mosaics (FWHM$\approx$45--180 mas). One option is to convolve all mosaics to the lowest-resolution PSF (WFC3/160W) prior to calculating photometry. However, doing so would significantly weaken the sensitivity in the shortest-wavelength bands which probe the Lyman-alpha break at our redshifts of interest ($z\sim6.5-8$). To mitigate this effect, we choose to convolve all long-wavelength (LW) NIRCam mosaics (F277W, F356W, F410M, F444W) to the PSF of WFC3/F160W while convolving all ACS and short-wavelength (SW) NIRCam mosaics (F115W, F150W, F200W) to the PSF of ACS/F814W. As described below, we introduce a scaling factor to the aperture size used for the LW vs. ACS+SW photometry to account for the differing PSFs. For each mosaic, we generate an empirical PSF by stacking the normalized surface brightness profiles of at least five stars within the mosaic footprint that were visually inspected to be unsaturated and isolated. Convolution kernels are then created using the \textsc{photutils} package \citep{Bradley2022_photutils}, where we verify that the encircled energy distributions after convolution agree with that of the target PSF (WFC3/F160W for LW and ACS/F814W for ACS+SW) to within $\lesssim$2\% accuracy at radii $>$0.05\arcsec{}. \subsection{Source Extraction and Photometry} \label{sec:photometry} Objects within the CEERS footprint are identified by running \textsc{Source Extractor} on an inverse-variance weighted stack of the PSF-convolved F150W and F200W mosaics, yielding a total of $\approx$51,000 sources. We stack the F150W and F200W mosaics for the detection image as we aim to perform a rest-UV selection of $z\sim6.5-8$ in this work. The resulting catalog was also utilized in \citet{Whitler2022_z10} to select galaxies at $z\sim8.5-11$ which appear as F115W dropouts. The photometry of each object identified by \textsc{source extractor} is calculated following procedures adopted in past \HST{}-based analyses of $z\gtrsim6$ galaxies \citep[e.g.][]{Oesch2013_z9LF,Bouwens2015_LF,Bouwens2021_LF,Finkelstein2015_LF,Finkelstein2022,Endsley2021_OIII}. We begin by measuring the flux contained in elliptical apertures with a \citet{Kron1980} factor of $k=1.2$ which is expected to result in approximately maximum signal to noise \citep{Finkelstein2022}. The associated photometric error in each band is calculated as the standard deviation of flux contained in equally-sized Kron apertures randomly distributed in nearby empty regions of the sky as determined from a \textsc{Source Extractor} segmentation map specific to each band. Because we have homogenized the ACS$+$SW and LW mosaics to different PSFs (ACS/F814W and WFC3/F160W, respectively; \S\ref{sec:imaging}), we multiply the aperture size used for the LW bands by a factor of 1.5 which reflects the typical F444W vs. F200W size ratio of twelve UV-bright $z\sim6-8$ galaxy candidates measured from the PSF-homogenized mosaics \citep{Chen2022}. The $k=1.2$ photometry (and their associated errors) are corrected to total values by first multiplying by the ratio of flux measured in $k=2.5$ vs. $k=1.2$ apertures. This correction factor is computed from the PSF-homogenized F200W mosaic for the ACS$+$SW photometry given that our $z\sim6.5-8$ selection requires a $>$5$\sigma$ detection in F200W (\S\ref{sec:sample}). Since we do not explicitly require a detection in any individual LW band, the correction factor for the photometry in these four bands are computed from an inverse variance-weighted stack of their mosaics to maximize S/N. Finally we correct for flux outside the $k=2.5$ apertures using our constructed ACS/F814W and WFC3/F160W mosaics as well as their reported encircled energy distributions at large radii ($>$1\arcsec{}). While our PSF-homogenized mosaics have very high angular resolution (FWHM$<$0.2\arcsec{}), it is still possible for our photometric apertures to be contaminated by considerable flux from neighboring objects. This can introduce significant offsets to not only the total magnitudes of our sources, but also the colors between the SW and LW NIRCam photometry given the different adopted PSF homogenization. To address this concern, we have developed a neighbor subtraction algorithm that is largely based off techniques previously developed for IRAC deconfusion \citep[e.g.][]{Wuyts2007,Labbe2010,McLeod2016,Song2016,Endsley2021_OIII}. In short, this algorithm fits a S{\'e}rsic profile (convolved with the appropriate PSF) to each neighboring object identified in the \textsc{Source Extractor} catalog within a 4$\times$4 arcsec$^2$ region. The fit is performed on the non PSF-homogenized image of each band independently, adopting the source parameters output by \textsc{Source Extractor} as starting values and iterating to find the best-fitting solution via a least-squares optimization. Once the best-fitting surface brightness parameters are determined for each of the non-homogenized images, we then subtract the flux of neighboring objects from the corresponding PSF-homogenized images by convolving the inferred S{\'e}rsic profiles with the F814W or F160W PSF for the ACS$+$SW and LW data, respectively. We demonstrate the performance and benefit of our neighbor subtraction algorithm in Fig. \ref{fig:neighborSubtraction}. In this example, significant flux from a bright neighboring low-redshift object is contaminating the Kron apertures for one of our $z\sim6.5-8$ galaxies, thereby artificially boosting the measured fluxes. Because the LW photometry is performed with a larger aperture size and a broader homogenized PSF, the impact of the contaminating flux is systematically stronger for the LW bands than ACS$+$SW. For this particular example, subtracting the surface brightness profiles of neighboring objects decreases the measured SW fluxes by $\approx$0.1 mag while the LW fluxes decrease by $\approx$0.3--0.4 mags. Not correcting for this contaminating flux would therefore result in seemingly stronger SW-to-LW colors, potentially giving the false impression of a stronger Balmer break and hence higher stellar mass and older age. Given these potential effects, we produce two photometric catalogs -- one that implements the neighbor subtraction algorithm and one without neighbor subtraction -- and perform our $z\sim6.5-8$ galaxy selection on each. This ensures that we identify the maximum number of viable reionization-era galaxies, as some may have biased colors or ACS detections in the original catalog due to contaminating flux from neighbors. \subsection{Selection of Lyman-break \texorpdfstring{$\mathbf{z\sim6.5-8}$}{z ~ 6.5 - 8} Galaxies} \label{sec:sample} We perform a Lyman-break selection to identify galaxies at $z\sim6.5-8$ across the June 2022 CEERS footprint. At $z>6.5$, the intergalactic medium is highly opaque to both Lyman-series ($\lambda_\mathrm{rest} = 912-1216$ \AA{}) and Lyman-continuum (LyC; $\lambda_\mathrm{rest} < 912$ \AA{}) emission \citep[e.g.][]{Inoue2014}. Accordingly, we expect galaxies at $z>6.5$ to show negligible emission in bands that lie blueward of $\approx$9100 \AA{}, resulting in a strong dropout in the \HST{}/ACS photometry relative to that of NIRCam F115W. With this in mind, we develop the follow initial cuts to identify Lyman-break $z\sim6.5-8$ galaxies within CEERS: \begin{enumerate} \item S/N$<$2 in F435W, F606W, and F814W \item F606W$-$F115W$>$1.7 and F814W$-$F115W$>$1.7 \item F115W$-$F200W$<$1.0 \item F814W$-$F115W$>$(F115W$-$F200W)$+$1.5 \end{enumerate} Here, the flux densities in the dropout bands (F606W and F814W) are set to their 1$\sigma$ upper limit in cases of non-detections (S/N$<$1), consistent with the approach of previous works \citep[e.g.][]{Bouwens2015_LF,Endsley2021_OIII}. The first two selection criteria above ensure that the photometry are consistent with expectations of a Lyman-alpha break at $z\gtrsim6.5$. The third and fourth criteria limit our sample to objects that show a much redder F814W$-$F115W color relative to F115W$-$F200W, thereby requiring the presence of a sharp break while still allowing $z\sim6.5-8$ objects with very red rest-UV slopes ($-1.5 \lesssim \beta \lesssim -0.3$) to enter our sample (see \S\ref{sec:dusty}). The overlapping CEERS+ACS area over which we are able to apply our selection is $\approx$31 arcmin$^2$. In addition to the above cuts, we add the following selection criteria to help remove spurious detections and low-redshift interlopers. First, we require that each selected source be detected at S/N$>$5 in F200W, S/N$>$3 in F150W, as well as S/N$>$3 in at least two LW NIRCam bands. Second, we enforce the condition $\chi^2_\mathrm{opt} < 5$ to remove faint low-redshift sources showing $\sim$1--1.5$\sigma$ detections in each ACS band. As defined in \citet{Bouwens2015_LF}, $\chi^2_\mathrm{opt} = \sum \mathrm{abs}(f)/f\, \left(f/\sigma\right)^2$ where $f$ and $\sigma$ represent, respectively, the measured flux density and its error in a given ACS band, while $\mathrm{abs}(f)$ is the absolute value of the flux density. Finally, we limit our sample to sources with F200W$<$28 as fainter $z\sim6.5-8$ galaxies generally cannot be reliably identified given the typical depths of the ACS imaging ($m\approx28.2$ 3$\sigma$ in each). We visually inspect each object satisfying the above selection criteria, removing any impacted by artifacts (e.g. diffraction spikes or `snowballs') or that appear to likely be a spurious detection, often resulting from diffuse emission of nearby bright objects. While it is expected that the surface density of cool brown dwarfs is low in the EGS field ($\sim$0.1 arcmin$^{-2}$; \citealt{Ryan2016_BDsurfaceDensity}), these objects can yield near-infrared colors mimicking a $z\sim7$ Lyman-alpha break \citep[e.g.][]{Stanway2008_contamination}. We search for any bright point source objects with colors consistent with a brown dwarf solution, using both the empirical SPEX 0.8--2.5$\mu$m brown dwarf spectral library \citep{Burgasser2014_SPEX}, as well as the Sonora18\footnote{\url{https://github.com/aburgasser/splat/tree/master/resources/SpectralModels/sonora18}} suite of 0.4--50$\mu$m model brown dwarf spectra (Marley et al. in prep) as references. One likely brown dwarf is identified with this procedure (reasonably consistent with the expected EGS surface density from \citealt{Ryan2016_BDsurfaceDensity}), and we remove this object from our sample. We select $z\sim6.5-8$ galaxies from both our standard photometry catalog, as well as that generated after applying our neighbor subtraction algorithm (\S\ref{sec:photometry}). To ensure that the measured fluxes and colors are reliable, we carefully inspect the postage stamps in every band to ensure that the Kron apertures are not being contaminated by flux from neighboring sources. We only use photometry from the neighbor-subtracted catalog for objects where, upon visual inspection, it is plausible that flux from neighboring objects could be contaminating the apertures. For each such object, we also verify that our neighbor-subtraction algorithm is yielding smooth residuals around the Kron apertures (i.e. that it is not significantly over/under-subtracting flux from neighboring sources within or near the apertures). A small percentage of our remaining candidates fail these checks and are removed from the sample. Due to the powerful sensitivity and angular resolution of the NIRCam mosaics, a subset of our identified $z\sim6.5-8$ galaxy candidates clearly show multiple emission components separated by $\lesssim$0.5\arcsec{} which are identified as separate systems by \textsc{Source Extractor}. We treat any such multi-component galaxy as a single system by computing their photometry in manually constructed elliptical apertures that contain the emission from all individual components satisfying our selection criteria described above. If necessary, we subtract the surface brightness profiles of other neighboring systems if their flux appears to significantly contaminate the manually-constructed apertures. Our final sample consists of 118 $z\sim6.5-8$ Lyman-break candidates across the June 2022 CEERS footprint. Here, we briefly summarize the photometric properties of this sample before using the full multi-band SEDs to infer physical properties in the following sub-section. The F200W magnitudes of the $z\sim6.5-8$ galaxies range between $m=25.3$ to $m=28.0$ (Fig. \ref{fig:photDistns}a), indicating that our sample spans a factor of $\approx$10 in UV luminosity. As expected from the steep faint-end slope of the $z\sim7-8$ UV luminosity function \citep[e.g.][]{Finkelstein2015_LF,Ishigaki2018,Bouwens2021_LF}, the sample is heavily weighted towards fainter objects yielding a median F200W $=$ 27.3. The decline in selection efficiency at $m\gtrsim27.5$ (Fig. \ref{fig:photDistns}a) is expected given the depths of the ACS imaging and our Lyman-break cuts. The sample also shows a wide range of F150W$-$F200W colors ($-0.32$ to 0.45; Fig. \ref{fig:photDistns}b) which reflects the distribution of rest-UV slopes across our 118 galaxies. The median F150W$-$F200W color of our $z\sim6.5-8$ sample is $0.0$, consistent with a flat rest-UV slope ($\beta = -2$ where $F_\lambda \propto \lambda^\beta$) as expected from WFC3 analyses of similarly faint systems \citep[e.g.][]{Bouwens2014_beta,Bhatawdekar2021}. At $z\sim6.5-8$ the F356W, F410M, and F444W bands fully probe the rest-optical and can be contaminated by strong line emission (i.e. [OIII]$\lambda\lambda$4959,5007, H$\beta$ and H$\alpha$; see Fig. \ref{fig:filterContamination}). Previous work with \Spitzer{}/IRAC imaging has demonstrated that a significant fraction of the $z\sim7-8$ galaxy population exhibits very high EW line emission ($\gtrsim$1000 \AA{} rest-frame; \citealt{Smit2014,Smit2015,RobertsBorsani2016,Endsley2021_OIII}), which can result in substantial photometric excesses ($\gtrsim$0.5 mag) in the above NIRCam bands, particularly in F410M given its smaller bandwidth. The impact of such strong line emission is apparent in the F277W$-$F356W vs. F277W$-$F444W color-color space our sample (Fig. \ref{fig:filterContamination}). We identify objects with very red ($\approx$1) F277W$-$F356W colors yet fairly flat ($\approx$0) F277W$-$F444W colors, likely indicating galaxies at $z\lesssim6.75$ where the combined EW of \OIIIHb{} is boosting F356W considerably more than H$\alpha$ in F444W. Conversely, a subset of $z\sim6.5-8$ galaxies show very red F277W$-$F444W colors yet flat F277W$-$F356W which are likely objects at $z\gtrsim7.0$ where \OIIIHb{} redshifts out of F356W and into F444W (Fig. \ref{fig:filterContamination}). There are also objects with red colors in both F277W$-$F356W and F277W$-$F444W which can arise from strong lines at $z\approx6.75-6.90$ where [OIII]$\lambda$5007 falls in both F356W or F444W (Fig. \ref{fig:filterContamination}), though the presence of significant dust attenuation or relatively old stellar populations yielding strong Balmer breaks can lead to such colors as well. Fortunately, the F410M photometry as well as the rest-UV slopes help differentiate between these various effects as we demonstrate below. \subsection{Photoionization Modelling} \label{sec:beagle} We now infer the physical properties of each of the 118 $z\sim6.5-8$ Lyman-break galaxy candidates in our CEERS sample using the SED-fitting code BayEsian Analysis of GaLaxy sEds (\textsc{beagle}; \citealt{Chevallard2016}). \textsc{beagle} fits the input ACS$+$NIRCam photometry to a suite of SED photoionization models, employing the Bayesian \textsc{multinest} algorithm \citep{Feroz2008,Feroz2009} to produce a posterior probability distribution for each free physical parameter. The photoionization models used here \citep{Gutkin2016} include both stellar and nebular emission that were computed by processing the latest \citet{BruzualCharlot2003} stellar population synthesis models through \textsc{cloudy} \citep{Ferland2013}. Consistent with many previous studies at $z\sim6-8$ \citep[e.g.][]{Ono2012,Labbe2013,Stark2013_NebEmission,Gonzalez2014,Smit2014,Salmon2015,Song2016,deBarros2019,Stefanon2022_sSFR}, we assume a constant star formation history (CSFH) in our \textsc{beagle} fits. It has been demonstrated that adopting a non-parametric star formation history (SFH) can yield considerably larger stellar masses ($\sim$0.5--1 dex), particularly for objects with very young CSFH ages ($\lesssim$10 Myr) where light from a recent burst is dominating the observed emission \citep{Carnall2019_SFH,Leja2019,Lower2020,Johnson2021,Tacchella2022_SFHs,Topping2022_REBELS,Whitler2022_z7,Whitler2022_z10}. For these galaxies with very young SEDs, non-parametric SFH fits that disfavor extremely rapid changes in SFR (i.e. the `continuity' prior) tend to add much more star formation at earlier times that is outshined by the recent burst, resulting in stellar masses up to $\approx$2 dex higher than that inferred from CSFH fits. Therefore, non-parametric SFH models with a continuity prior effectively place an upper limit on the stellar mass permitted by the SED (depending on the assumed formation redshift), whereas CSFH fits place an approximate lower limit. While dynamical mass and rest-frame near-infrared SED constraints can help distinguish between these two SFHs \citep[e.g.][]{Tang2022,Topping2022_REBELS}, such information does not yet exist for our sample. Because one of our key goals is to compare the stellar populations dominating the observed rest-UV$+$optical SEDs among our galaxies versus that in brighter $z\sim7-8$ galaxies (\S\ref{sec:sSFRandEW}), we opt to assume a CSFH in our fiducial models. We will consider the impact of non-parametric SFH models for a subset of our objects in \S\ref{sec:dusty} and \S\ref{sec:stellarMasses}, but note that none of our conclusions rely on absolute measures of stellar mass (or sSFR). For our \textsc{beagle} fits, we assume a \citet{Chabrier2003} stellar initial mass function with mass range 0.1--300 \Msol{}, a fixed dust-to-metal mass ratio of $\xi_d = 0.3$, and an SMC dust attenuation curve \citep{Pei1992}. In \textsc{beagle} the dust opacity is parameterized as the V-band optical depth, $\tau_\mathrm{V}$, which is allowed to vary between 0.001 to 5 with a log-uniform prior. For ease of understanding what these dust optical depths imply for UV emission, we often quote the attenuation at 1500 \AA{} rest-frame, $A_{1500}$, which is equivalent to 5.2$\tau_\mathrm{V}$ in the SMC attenuation law. The redshift is allowed to vary in the range $z=5-10$ with a uniform prior, and IGM absorption is incorporated using the empirical model of \citet{Inoue2014}. We adopt log-uniform priors on both stellar mass and ionization parameter, allowing for values in the range $5 \leq \logMstar \leq 12$ and $-4 \leq \mathrm{log}\ U \leq -1$, respectively. Galaxy ages are allowed to take values between 1 Myr and the age of the Universe at the sampled redshift with a log-uniform prior as well. Finally, we adopt a log-uniform prior on metallicity allowing for values of $-2.2 \leq \mathrm{log}(Z/Z_\odot) \leq -0.3$, where we set an upper limit of $\approx$50\% $Z_\odot$ to avoid unreasonable solutions of near-solar metallicity in faint reionization-era galaxies. In these fiducial \textsc{beagle} fits, the stellar and ISM metallicities are equivalent. The resulting fiducial \textsc{beagle} fits are generally able to provide a good match to the measured photometry of our Lyman-break $z\sim6.5-8$ galaxies, with a median best-fitting $\chi^2$ of 6.5 (from 10 fitted photometric points) across the full sample and $\chi^2<20$ for 95\% of the objects. In \S\ref{sec:stellarMasses}, we discuss in detail one object for which our fiducial \textsc{beagle} fits struggle to precisely match the NIRCam photometry and consider other model solutions. When reporting the inferred physical properties for a given galaxy, we adopt the median of the posterior probability distribution as the fiducial value with $\pm$1$\sigma$ errors taken as the inner 68\% credible interval from the posterior. To calculate the rest-UV slopes of our galaxies, we fit $F_\lambda \propto \lambda^\beta$ to the photometry of NIRCam bands determined to fall entirely between the \Lya{} line and the Balmer break (i.e. 1216--3600 \AA{} rest-frame) given the inferred redshift specific to each galaxy. Errors on the rest-UV slope are computed by randomly sampling the redshift from the \textsc{beagle} posterior as well as photometric flux densities given the measured values and their errors. The absolute UV magnitudes, \Muv{}, are defined at 1500 \AA{} rest-frame and are computed from the output redshift and SED posteriors. \section{Results} \label{sec:results} Due to the limitations of previous near-infrared facilities, it has long been very challenging to constrain the SEDs and hence physical properties (e.g. stellar mass, sSFR, age, nebular line EWs) of typical ($<\!L_\mathrm{UV}^\ast$) galaxies in the reionization era. \Spitzer{}/IRAC imaging suffered from inadequate sensitivity, strong source confusion, as well as poor sampling of the rest-optical emission with only two broadband filters at 3--5$\mu$m. While \HST{}/WFC3 was able to deliver relatively sensitive photometry at 1.0--1.6$\mu$m, imaging in moderate-sized ($\sim$100 arcmin$^2$) fields like EGS often reached only $m\sim26.5$ \citep{Grogin2011,Koekemoer2011}, thus limiting insight into the rest-UV emission from typical reionization-era galaxies in these fields as well. With the tremendous advancements in 1--5$\mu$m imaging sensitivity, angular resolution, and filter suite enabled by \JWST{}/NIRCam, we now begin characterizing the rest-UV+optical SEDs of typical reionization-era galaxies in detail. Before describing the results of our full $z\sim6.5-8$ CEERS sample in this section, here we first briefly demonstrate how the NIRCam data from CEERS greatly improve our understanding of $<L_\mathrm{UV}^\ast$ $z\sim7-8$ galaxies in EGS relative to what was previously possible with \HST{} and \Spitzer{} data. In Fig. \ref{fig:HSTSpitzerComparison}, we directly compare the rest UV+optical SED constraints for a typical (F150W$\approx$27) galaxy in our sample when using \HST{} and \Spitzer{} photometry \citep{Stefanon2017_EGS} versus \JWST{}/NIRCam. Even with the deep \HST{} and \Spitzer{} data over the EGS field, this galaxy is undetected in IRAC and only barely detected ($\lesssim$3$\sigma$) in the WFC3 bands, resulting in very little constraining power on the shape of the SED. Consequently, the stellar mass, dust optical depth, sSFR, and \OIIIHb{} EW are all highly uncertain from previous data (see Fig. \ref{fig:HSTSpitzerComparison}). With the CEERS/NIRCam imaging, this scenario changes dramatically. The galaxy is detected at S/N$>$20 in every NIRCam band, revealing a significant F410M excess indicative of high-EW \OIIIHb{} emission at $z\approx7-7.5$, as well as a red rest-UV slope implying strong dust attenuation. From the sensitive 7-band 1--5$\mu$m NIRCam photometry, uncertainties on inferred physical properties decrease to approximately $\pm$0.05--0.1 dex, a vast improvement relative to the $\approx$ $\pm$0.25--1 dex uncertainties from \HST{}+\Spitzer{} data (see Fig. \ref{fig:HSTSpitzerComparison}). In the remainder of this section, we utilize the advanced 1--5$\mu$m imaging capabilities of NIRCam to first characterize the physical properties (e.g. age, stellar mass, dust optical depth) of our large (N=118) $z\sim6.5-8$ CEERS galaxy sample (\S\ref{sec:SEDs}). In each of the following two sub-sections, we discuss an unexpected population of galaxies identified in our sample: sources with young SEDs and weak \OIIIHb{} emission (\S\ref{sec:weakOIII}) as well as very red galaxies that confidently lie at $z\sim7$ given their signatures of strong optical line emission (\S\ref{sec:dusty}). We then combine the results from our CEERS sample with past studies of the UV-bright $z\sim7-8$ galaxy population to assess whether reionization-era galaxy sSFRs or \OIIIHb{} EWs correlate strongly with UV luminosity (\S\ref{sec:sSFRandEW}). Finally, we discuss whether we find any evidence of extremely massive ($M_\ast \sim 10^{11}\ M_\odot$) galaxies at $z\sim7-8$, comparing with previous works (\S\ref{sec:stellarMasses}). \subsection{Demographics of UV-faint \texorpdfstring{$\mathbf{z\sim6.5-8}$}{z ~ 6.5 - 8} Galaxies} \label{sec:SEDs} Before detailing the NIRCam SEDs among our sample of 118 $z\sim6.5-8$ Lyman-break galaxies, we first briefly describe two basic properties of this sample -- their photometric redshifts and absolute UV magnitudes. As expected from our dropout selection criteria (\S\ref{sec:sample}), the photometric redshifts of galaxies in our sample range between $z=6.29$ to $z=8.08$ (Fig. \ref{fig:inferredPropertyHistograms}a). The median redshift of the sample is closer to the low end of the distribution ($z=6.80$), consistent with expectations of a declining UV luminosity function at higher redshifts \citep[e.g.][]{Finkelstein2015_LF,Ishigaki2018,Bouwens2021_LF}. The inferred absolute UV magnitudes of the CEERS $z\sim6.5-8$ galaxies encompass the range $-21.0 \leq \Muv{} \leq -18.9$ with a median $\Muv{} = -19.6$ (Fig. \ref{fig:inferredPropertyHistograms}b), consistent with the F200W magnitude distribution measured among the sample (see Fig. \ref{fig:photDistns}a). Adopting a characteristic UV luminosity of $L_\mathrm{UV}^\ast = -20.5$ at $z\sim6.5-8$ \citep[e.g.][]{Bowler2017,Bowler2020,Harikane2022_LF}, the typical galaxy in our sample has a UV luminosity of 0.4 $L_\mathrm{UV}^{\ast}$ with the full sample spanning 0.2--1.6 $L_\mathrm{UV}^\ast$. The vast majority (94\%) of galaxies in our sample are classified as sub-$L_\mathrm{UV}^\ast$ systems. Now equipped with four deep photometric data points at $\approx$3--5$\mu$m (including in one medium band), we begin investigating the rest-optical SEDs of relatively faint ($\sim$0.4 $L_\mathrm{UV}^{\ast}$) galaxies in the reionization era. The CEERS/NIRCam data clearly reveal a variety of CSFH ages among UV-faint $z\sim6.5-8$ Lyman-break galaxies (Fig. \ref{fig:ageSEDs}). For the bulk of our sample, the measured flux density in either F356W, F410M, or F444W is consistent with extrapolating a power-law SED from the rest-UV photometry (see left two columns of Fig. \ref{fig:ageSEDs}). This implies that young (\ageCSFH{}$\sim$30 Myr) stellar populations dominate their rest-UV+optical SEDs yielding large specific star formation rates (\sSFRCSFH{}$\sim$30 Gyr$^{-1}$). Nonetheless, we do identify a subset of objects with F356W, F410M, and F444W flux densities boosted by $\sim0.5-1$ mag relative to the rest-UV, with approximately equal excesses in each of the these three bands (see third column of Fig. \ref{fig:ageSEDs}). Such photometry imply the presence of a prominent Balmer break, consistent with relatively evolved stellar populations (\ageCSFH{}$\sim$100--500 Myr) and low-to-moderate \sSFRCSFH{} ($\sim$2--10 Gyr$^{-1}$). In addition to sources showing signs of prominent Balmer breaks, we also find a subset of $z\sim6.5-8$ galaxies exhibiting a very strong ($\approx$1--2 mag) photometric excess in F356W, F410M, or F444W, with the extent of the excess differing significantly from band to band (see rightmost column of Fig. \ref{fig:ageSEDs}). The photometric excess patterns of these galaxies are consistent with contamination from exceptionally high-EW nebular lines (\OIIIHb{} EW$\gtrsim$1500 \AA{}), implying very young CSFH ages ($\lesssim$10 Myr) and correspondingly very large \sSFRCSFH{} ($\gtrsim$100 Gyr$^{-1}$). We find a roughly similar fraction of very young (\ageCSFH{}=1--10 Myr) and relatively old (50--550 Myr) galaxies in our sample (29\% and 19\%, respectively), with the remaining 52\% of our sample having moderately young CSFH ages (10--50 Myr; see Fig. \ref{fig:inferredPropertyHistograms}c). NIRCam imaging enables not only much stronger constraints on the stellar population ages dominating the SEDs of sub-$L_\mathrm{UV}^\ast$ $z\sim6.5-8$ galaxies, but also on the stellar masses (see Fig. \ref{fig:HSTSpitzerComparison}). As expected from the typically young CSFH ages ($\sim$30 Myr) and faint UV luminosities ($\Muv{} \sim -19.6$) of our galaxies, we find that our sample is largely composed of low-mass ($M_\ast < 10^9\ M_\odot$) objects in context of our CSFH models. The median CSFH stellar mass of our galaxies is \MstarCSFH{} = $10^{7.9} \Msol{}$, with individual objects having stellar masses ranging over $>$2 dex from $10^{7.1} \Msol{}$ to $10^{9.4} \Msol{}$ (see Fig. \ref{fig:inferredPropertyHistograms}d). We show our CEERS galaxies on the stellar mass versus UV luminosity plane in Fig. \ref{fig:MstarMuv}, where we color-code the data point for each galaxy by its CSFH age. Unsurprisingly, we find that more luminous objects tend to have larger inferred stellar masses. As discussed in \S\ref{sec:beagle}, for galaxies with very young SEDs our CSFH fits are only modeling light from a recent burst resulting in low-mass ($\lesssim10^8\ M_\odot$) solutions, though it is likely that some of these systems have true stellar masses that are substantially ($\gtrsim$3$\times$) larger. In sections \S\ref{sec:dusty} and \S\ref{sec:stellarMasses}, we consider the impact of adopting non-parametric SFH models for a subset of our sample. Because the CEERS/NIRCam data deliver sensitive imaging across 1--5$\mu$m, we can investigate not only the rest-optical SEDs of our galaxies, but their rest-UV slopes as well. As expected from previous studies using \HST{}/WFC3 data (\citealt{McLure2011,Dunlop2012,Finkelstein2012,Rogers2013,Bouwens2014_beta,Bhatawdekar2021}), we find that our relatively faint ($\Muv{} \sim -19.6$) CEERS galaxies tend to have blue UV slopes ($\beta \sim -2$; see Fig. \ref{fig:inferredPropertyHistograms}e). Consequently, our \textsc{beagle} SED fitting results imply little dust attenuation in the majority of our sample (median $\tau_\mathrm{V} = 0.020$ or $A_{1500} = 0.10$ mag; Fig. \ref{fig:inferredPropertyHistograms}f). The $\beta$ vs. \Muv{} relation of our $z\sim6.5-8$ galaxies is presented in \citet{Topping2022_blueSlopes}. Therein, we discussed a subset of extremely blue objects ($\beta \lesssim -3$) exhibiting NIRCam SEDs consistent with not only negligible dust attenuation, low metallicities ($\lesssim$5\% $Z_\odot$), and young CSFH ages ($\lesssim$20 Myr), but also high ionizing photon escape fractions ($\fesc{} \gtrsim 50$\%) which mitigate reddening from nebular continuum emission. In each of the following two sub-sections, we discuss an unexpected population of $z\sim6.5-8$ Lyman-break galaxies identified in our sample that we have not yet detailed. The first population we discuss are objects with NIRCam SEDs suggesting both young CSFH ages yet relatively weak \OIIIHb{} emission. Next, we describe a surprisingly large number of galaxies with very red UV slopes, as well as strong long-wavelength photometric excesses which confidently place each system at $z\sim7$. We argue that the abundance of each of these galaxy classes may have important implications for our understanding of star formation, ionizing photon escape, and chemical evolution in the reionization era. \subsection{The Nature of Weak [OIII]+H\texorpdfstring{$\mathbf{\beta}$}{beta} Emitters at \texorpdfstring{$\mathbf{z\sim6.5-8}$}{z ~ 6.5 - 8}} \label{sec:weakOIII} Even prior to the launch of \JWST{}, it was relatively easy to identify $z\sim7-8$ galaxies with very high \OIIIHb{} EWs ($>$1000 \AA{}) given that these objects would show a substantial ($\gtrsim$1 mag) excess in one of the \Spitzer{}/IRAC bands, yielding a bright (and relatively high S/N) detection \citep[e.g.][]{Smit2014,Smit2015,RobertsBorsani2016,deBarros2019,Strait2020,Endsley2021_OIII}. However, it has remained very difficult to quantify the population of sub-$L_\mathrm{UV}^\ast$) reionization-era galaxies with low \OIIIHb{} EWs given that these sources would generally remain poorly detected in IRAC. With the highly advanced 3--5$\mu$m imaging capabilities of NIRCam, it is now possible to not only easily identify, but also characterize weak \OIIIHb{} emitters among sub-$L_\mathrm{UV}^\ast$ reionization-era galaxies. As expected, we identify a subset of galaxies in our CEERS $z\sim6.5-8$ sample that exhibit NIRCam SEDs consistent with both relatively old stellar populations (\ageCSFH{}$>$300 Myr) and weak \OIIIHb{} emission (EW$<$500 \AA{}; see e.g. IDs 1714 and 20717 in Fig. \ref{fig:ageSEDs}). The comparatively weak nebular lines in these evolved objects are expected given that the rest-optical continuum will be boosted over time by the build-up of A stars, while the nebular line luminosity will remain roughly constant as OB stars undergo short life cycles. However, as detailed below we also find a seemingly significant population of UV-faint $z\sim6.5-8$ Lyman-break galaxies with SEDs implying both young CSFH ages and weak \OIIIHb{}. The CEERS NIRCam data reveal several candidate $z\sim6.5-8$ galaxies with strong Lyman-alpha breaks (F814W$-$F115W=1.7--3.4; median 2.3) as well as SEDs implying young CSFH ages yet also little-to-no photometric excesses in the long-wavelength bands. We show the SEDs of four such objects in our sample in Fig. \ref{fig:youngWeakOIIISEDs} where it can be seen that their F356W, F410M, or F444W photometry are reasonably consistent with a power-law SED extrapolated from the shorter-wavelength NIRCam photometry. The lack of significant Balmer breaks in these systems indicates that their light-weighted ages are young (\ageCSFH{}$<$50 Myr), implying that a substantial portion of their emergent light should be coming from recently-formed OB stars which are highly efficient producers of LyC photons. Accordingly, we might expect to see strong nebular line emission in such seemingly young objects, as indeed found among a sample of UV-faint (median $\Muv{} \approx -19.5$) extreme emission line galaxies (EELGs) at $z\sim2$ where \OIIIHb{} EWs$>$800 \AA{} are nearly ubiquitous at \ageCSFH{}$<$50 Myr (\citealt{Tang2019}; see Fig. \ref{fig:age_EW_metallicity}). However, we find little-to-no significant photometric excesses in the long-wavelength bands for several of the young $z\sim6.5-8$ galaxies in our sample, implying considerably lower \OIIIHb{} EWs at fixed CSFH age relative to $z\sim2$. From our fiducial \textsc{beagle} fits, young (\ageCSFH{}$<$50 Myr), relatively weak \OIIIHb{} emitters (EW$<$600 \AA{}) comprise 23\% (27/118) of our CEERS sample, suggesting that they may be a significant class of sub-$L_\mathrm{UV}^\ast$ reionization-era galaxies that are comparatively rare at $z\sim2$. While the \citet{Tang2019} $z\sim2$ EELG sample mentioned above was selected on [OIII] emission, the adopted selection threshold of [OIII]$\lambda$5007 EW$>$225 \AA{} still allows for the inclusion of relatively weak \OIIIHb{} emitters (EW$\approx$350--600 \AA{}) and such galaxies were indeed found with \ageCSFH{}$\sim$100--500 Myr (Fig. \ref{fig:age_EW_metallicity}). Below, we discuss potential physical origins for the weak \OIIIHb{} and young SEDs that appear present in our $z\sim6.5-8$ sample, though we emphasize that spectroscopic follow-up will be necessary to confirm their redshifts. In the context of our fiducial \textsc{beagle} fits, the combination of young CSFH age ($<$50 Myr) and relatively low \OIIIHb{} EW ($<$600 \AA{}) is often reproduced with extremely low metallicities ($\approx$1--3\% $Z_\odot$; see Fig. \ref{fig:age_EW_metallicity}). In such metal-poor models, [OIII] emission is greatly weakened relative to the more chemically evolved ($\sim$20\% $Z_\odot$) systems with similarly high \sSFRCSFH{} ($\gtrsim$20 Gyr$^{-1}$) found at both lower redshifts ($z\sim2$) and bright UV magnitudes ($\Muv{} \lesssim -21$) at $z\sim7-8$ (e.g. \citealt{Endsley2021_OIII,Tang2021_UVlines}). Even though H$\beta$ EWs increase with decreasing metallicity, the relative contribution of [OIII]$\lambda\lambda$4959,5007 to \OIIIHb{} is generally far more dominant ([OIII]/H$\beta$ $\sim$ 5--10; e.g. \citealt{Steidel2016,Maseda2018,Sanders2018,Tang2019}) such that extremely low metallicities result in much smaller total \OIIIHb{} EWs at fixed \sSFRCSFH{}. While our fiducial \textsc{beagle} models demonstrate that extremely low metallicity solutions can reproduce relatively weak \OIIIHb{} emission at young CSFH ages (see Figs. \ref{fig:youngWeakOIIISEDs} and \ref{fig:age_EW_metallicity}), here we discuss two alternative physical explanations. Because nebular emission lines are byproducts of the interaction between hydrogen ionizing photons and gas in the ISM, these lines can be weakened at high LyC escape fractions (\fesc{}; e.g. \citealt{Zackrisson2013}). If the LyC leakage is driven by density-bounded conditions, we would expect to see weaker [OIII] and H$\beta$ emission at very high \fesc{} \citep{Plat2019}. In Fig. \ref{fig:youngWeakOIIISEDs_otherExplanations}a, we illustrate how high escape fractions ($\fesc{} \sim 0.5$) can also plausibly explain weak \OIIIHb{} in a subset of $z\sim6.5-8$ galaxies with young SEDs (see also \citealt{Topping2022_blueSlopes}). Here, we are using the \textsc{beagle} photoionization models introduced in \citet{Plat2019} which allow for non-zero ionizing photon escape from HII regions within the galaxy (i.e. $f_\mathrm{esc,HII} > 0$) due to density-bounded conditions. While such ionizing photons must also travel through the ISM and CGM to reach the intergalactic medium, sources with large $f_\mathrm{esc,HII}$ would be prime candidates for key contributors to reionization. Another way nebular line emission can be weakened in a seemingly young galaxy is if that object has experienced a sharp decline in star formation rate within the past $\sim$10 Myr, resulting in very few (if any) remaining OB stars. To illustrate how such a star formation history could reproduce young CSFH SEDs with weak \OIIIHb{}, we take our fiducial \textsc{beagle} fitting setup but force the SFR to 0 $M_\odot$/yr over the most recent 15 Myr (with a constant non-zero SFR at $>$15 Myr). Such a recent turnoff in star formation can produce SEDs that provide a satisfactory match the observed NIRCam photometry for a subset of our young weak \OIIIHb{} emitters (see Fig. \ref{fig:youngWeakOIIISEDs_otherExplanations}b). In these solutions, the observed SED is dominated by light from a population of stars that started forming $\sim$50 Myr ago and assembled $\sim$10$^9$ $M_\odot$ before star formation plummeted in the most recent 15 Myr. From an empirical standpoint, such bursty star formation histories are consistent with the presence of extremely high \sSFRCSFH{} ($>$100 Gyr$^{-1}$) systems we identify in our sample. With implied mass-doubling times of $<$10 Myr, these galaxies are caught in a phase of intense star formation activity that likely was proceeded and will be followed by periods with much lower SFRs. A subset of our young, weak \OIIIHb{} emitters may reflect these periods of relatively inactive star formation. Consistent with this picture, recent high-resolution hydrodynamic simulations also predict that $z\sim7-8$ galaxies frequently undergo episodes of extreme star formation between periods of weak-to-moderate star formation \citep[e.g.][]{Ma2018_burstySFH}. It is currently not clear which of these three physical scenarios is responsible for the apparent young SEDs with weak \OIIIHb{}. In principle, low metallicities, high \fesc{}, and bursty star formation histories could contribute to the effect in different galaxies. Fortunately, future \JWST{} surveys will soon allow us to not only confirm the existence of these reionization-era systems, but also better understand the physical origin of their SEDs. For example, direct temperature measurements could test for the presence of extremely metal-poor populations \citep{ArellanoCordova2022,Curti2022,Rhoads2022,Schaerer2022,Taylor2022,Trump2022}. Moreover, with very deep NIRSpec spectra it will be possible to determine if O stars are contributing significantly to the rest-UV continuum emission by searching for P-Cygni profiles of e.g. CIV$\lambda$1540. A complete lack of such O-star features would be consistent with the bursty star formation history scenario discussed above, while we would expect O stars to contribute significantly to the rest-UV continuum in the very high \fesc{} and extremely low metallicity scenarios. \subsection{Very Red UV-faint \texorpdfstring{$\mathbf{z\sim6.5-8}$}{z ~ 6.5-8} Galaxies in Small Areas} \label{sec:dusty} We identify several very red $z\sim6.5-8$ galaxies with strong line emission in our CEERS sample. The NIRCam SEDs of these four objects are shown in Fig. \ref{fig:redEELGseds} where they all exhibit long-wavelength photometric excesses implying \OIIIHb{} EW$>$800 \AA{} as well as high S/N ($\gtrsim20\sigma$) short-wavelength photometry implying red UV slopes ($\beta \geq -1.3$). The combination of very strong Ly$\alpha$ breaks (F814W$-$F115W$\geq$2.3) and clear signatures of strong nebular line contamination in each of these galaxies gives us confidence that they are robust reionization-era systems. The photometric redshifts and absolute UV magnitudes of these galaxies span $z=6.6-7.0$ and $-19.9 \leq \Muv{} \leq -19.7$, respectively, indicating that they all fall under the classification of sub-$L_\mathrm{UV}^\ast$ sources. However, as expected from their very red UV-slopes ($-1.3 \leq \beta \leq -0.6$) all four systems are inferred to be heavily reddened ($A_{1500} = 1.3-2.2$ mag) implying that they would have appeared as UV-bright ($-22.0 \lesssim \Muv{} \lesssim -21.0$) galaxies in the absence of dust. Given the implied large dust reservoirs in these systems, we would perhaps expect them to be fairly massive ($M_\ast\gtrsim$10$^{9}$ $M_\odot$; e.g. \citealt{Dayal2022_REBELS,Ferrara2022_REBELS}). While our CSFH \textsc{beagle} fits suggest that these objects have relatively low stellar mass ($\approx$2--4$\times$10$^{8}$ $M_\odot$), such fits account for only the recent burst suggesting that significantly higher stellar mass solutions may be feasible \citep[e.g.][]{Carnall2019_SFH,Tacchella2022_SFHs,Whitler2022_z7}. Indeed, we find that stellar masses of $\approx$10$^{9}$ $M_\odot$ to $\approx$3$\times$10$^{10}$ $M_\odot$ are possible when adopting \textsc{prospector} non-parametric SFH models utilizing the continuity prior (see \S\ref{sec:beagle}). In recent years, a growing number of $z\sim7-8$ Lyman-break galaxies have been identified with fairly high S/N ($\gtrsim$5$\sigma$) near-IR photometry implying very red UV slopes ($\beta \sim -1$; e.g. \citealt{Bowler2014,Smit2015,Stefanon2019,Endsley2021_OIII}) and a considerable number of these objects are now spectroscopically confirmed via far-infrared follow-up \citep{Smit2018,Bouwens2021_REBELS,Endsley2022_radioConfirmation,Fujimoto2022}. However, all of these previously-known very red $z\sim7-8$ systems appear UV luminous ($\Muv{} < -21$) even in the presence of the substantial dust reddening. This is a direct result of the limited near-IR sensitivity of ground-based cameras as well as \HST{}/WFC3, making it very challenging to confidently identify red $z\sim7-8$ galaxies at more typical UV luminosities ($\Muv{} \gtrsim -20$). NIRCam's greatly improved near-IR sensitivity (and SED sampling) relative to WFC3 changes this scenario dramatically as illustrated in Fig. \ref{fig:HSTSpitzerComparison}. One of our very red CEERS galaxies (ID 8025) is barely detected in the CANDELS EGS WFC3 data (S/N$\lesssim$3) such that it was previously impossible to measure the UV slope with high confidence resulting in the extremely broad dust optical depth posterior in Fig. \ref{fig:HSTSpitzerComparison}. With CEERS/NIRCam data, this object is detected at S/N$>$20 in F115W, F150W, and F200W, clearly revealing a strong increase in flux density over $\approx$1500--2500 \AA{} rest-frame indicating a very red UV slope ($\beta = -0.90 \pm 0.03$) and strong dust attenuation ($A_{1500} \approx 2.2$). The fact that the NIRCam data reveal four very red ($-1.3 \leq \beta \leq -0.6$) $z\sim6.5-8$ galaxies in the CEERS area suggests that there remains a significant tail of heavily dust-reddened galaxies down to at least $\approx$0.5 $L_\mathrm{UV}^\ast$ in the reionization era. This may have important consequences for our understanding of the extent to which obscured star formation contributes to stellar mass assembly in the very early Universe. The dust optical depths inferred by our \textsc{beagle} fits imply that $\approx$70--90\% of the star formation in our four young, very red CEERS galaxies is obscured, resulting in individual obscured CSFH SFRs of 6--91 $M_\odot$ yr$^{-1}$ for each object. Even though these young, very red systems represent only 3\% of our CEERS sample by number, their combined obscured SFR$_\mathrm{CSFH}$ (156 $M_\odot$ yr$^{-1}$) makes up 17\% of all (unobscured$+$obscured) inferred stellar mass growth across our full sample (923 $M_\odot$ yr$^{-1}$). The obscured SFRs of our young, very red galaxies may even be underestimated from the \textsc{beagle} SED fits. ALMA follow-up has shown that the total SFR inferred from the rest-UV+optical SED is often a factor of $\approx$2 lower than the SFR derived by combining UV and far-infrared data, at least among very UV-bright ($\Muv{} \sim -22$) $z\sim7-8$ galaxies \citep{Topping2022_REBELS}. Moreover, at least one very red UV-bright $z\simeq7$ galaxy (COS-87259; $\beta = -0.6 \pm 0.3$) has been found to be extremely bright in the far-infrared with SCUBA-2 and ALMA data, implying an obscured star formation rate of $\approx$1300 $M_\odot$ yr$^{-1}$ (\citealt{Endsley2022_radioConfirmation,Endsley2022_radioAGN}; see also the similar object in GOODS-N described in \citealt{Fujimoto2022}). Deep far-infrared follow-up will be critical to assess the extent of obscured star formation in the relatively UV-faint ($\lesssim$0.5 $L_\mathrm{UV}^\ast$) and very red $z\sim7$ galaxies identified from deep \JWST{} data. \subsection{The \texorpdfstring{M$\mathbf{_\mathrm{UV}}$}{Muv} Dependence of sSFRs and [OIII]+H\texorpdfstring{$\mathbf{\beta}$}{beta} EWs} \label{sec:sSFRandEW} Within the past few years, it has become possible to statistically characterize the sSFRs and \OIIIHb{} EWs of UV-bright ($-23 \lesssim \Muv{} \lesssim -21$) $z\sim7-8$ galaxies thanks to the growing collection of $>$deg$^2$ data sets with deep optical, near-infrared, and IRAC imaging (e.g. \citealt{Mauduit2012,McCracken2012,Jarvis2013,Steinhardt2014,Ashby2018,Aihara2022}). These UV-luminous reionization-era galaxies tend to exhibit SEDs consistent with high \OIIIHb{} EWs ($\approx$700 \AA{}) as well as young CSFH ages and hence large \sSFRCSFH{} (\citealt{Stefanon2019,Endsley2021_OIII,Topping2022_REBELS,Whitler2022_z7}). In the more local Universe ($z\lesssim2$), it has been shown that such galaxy properties are clearly associated with high ionizing photon production efficiency \citep{Chevallard2018_z0,Tang2019}, as well as perhaps higher LyC escape fractions \citep[e.g.][]{Izotov2016b,Flury2022_LyCdiagnostics,Saxena2022}. Unfortunately, our understanding of sSFRs and \OIIIHb{} EWs at fainter magnitudes has been hindered by IRAC's poor sensitivity, leaving much uncertainty on the possible extent to which galaxy ionizing properties correlate with UV luminosity in the reionization era \citep[e.g.][]{Finkelstein2019,Naidu2020}. Now equipped with deep NIRCam imaging, we can begin constraining the demographics of sSFRs and \OIIIHb{} EWs among sub-$L_\mathrm{UV}^\ast$ $z\sim7-8$ galaxies to compare with the more UV-luminous population. We find that the NIRCam SEDs of CEERS $z\sim6.5-8$ galaxies generally imply very large CSFH specific star formation rates, with 70\% of our sample having a fiducial \sSFRCSFH{}$>$30 Gyr$^{-1}$. Following previous studies \citep{Topping2022_REBELS}, we compute the median \sSFRCSFH{} of our sample and its associated uncertainty by bootstrap resampling values from the \textsc{beagle} posteriors. That is, we randomly select 118 galaxies with replacement from our CEERS sample, pulling a \sSFRCSFH{} from the \textsc{beagle} posterior probability distribution for each galaxy, and repeat this process 10000 times. This yields a median and inner 68\% credible interval $\sSFRCSFH{} = 82^{+24}_{-23}$ Gyr$^{-1}$ (Fig. \ref{fig:sSFR}), consistent with a scenario in which the bulk of UV-faint ($\sim$0.4 $L_\mathrm{UV}^\ast$) reionization-era systems have recently experienced a strong upturn in their star formation activity. We compare the \sSFRCSFH{} derived among our relatively faint CEERS sample (median $\Muv{} = -19.6$) with results obtained from very UV-luminous $z\sim7$ galaxies whose rest-optical properties have been characterized with IRAC data. Leveraging ultra-deep optical narrow-band imaging across the 1.5 deg$^2$ COSMOS UltraVISTA field, we previously assembled a large sample of UV-bright ($-22.5 \lesssim \Muv{} \lesssim -21$) Lyman-break galaxies at $z\simeq6.6-6.9$ where the IRAC 3.6$\mu$m band is contaminated by \OIIIHb{} while the 4.5$\mu$m band cleanly probes the rest-optical continuum \citep{Endsley2021_OIII}. After only considering objects with robust IRAC deconfusion, we obtain a final sample of 36 UV-bright (median $\Muv{} = 21.4$) $z\simeq6.6-6.9$ COSMOS galaxies (see \citealt{Whitler2022_z7}). We fit these COSMOS galaxies with \textsc{beagle} adopting the same prior set that we use for the CEERS sample, finding a median inferred $\sSFRCSFH{} = 47^{+38}_{-19}$ Gyr$^{-1}$, potentially hinting at a modest ($\approx$2$\times$) increase in \sSFRCSFH{} between $\sim$2 $L_\mathrm{UV}^\ast$ and $\sim$0.4 $L_\mathrm{UV}^\ast$ at $z\sim7-8$ (see Fig. \ref{fig:sSFR}). To better test for any UV luminosity dependence on \sSFRCSFH{} at $z\sim7-8$, we also consider the large (N=40) sample of extremely UV-bright (median $\Muv{} = -22.0$) galaxies observed with ALMA as part of the Reionization Era Bright Emission Line Survey (REBELS; \citealt{Bouwens2021_REBELS}). From similar \textsc{beagle} rest-UV$+$optical SED fits, the $\sim$4 $L_\mathrm{UV}^\ast$ REBELS sample has an inferred median $\sSFRCSFH{} = 9.5^{+2.4}_{-2.0}$ Gyr$^{-1}$ \citep{Topping2022_REBELS}, which is nearly an order of magnitude lower than that of our CEERS sample (Fig. \ref{fig:sSFR}). Given the availability of ALMA data, the CSFH sSFRs of the REBELS objects were also determined using unobscured$+$obscured SFRs computed from the UV and IR luminosities (and the same CSFH stellar masses). While this approach results in a slightly higher median \sSFRCSFH{} for the REBELS sample ($18^{+7}_{-5}$ Gyr$^{-1}$; \citealt{Topping2022_REBELS}), the data remain consistent with an approximately 5$\times$ higher typical sSFR at $\sim$0.4 $L_\mathrm{UV}^\ast$ relative to $\sim$4 $L_\mathrm{UV}^\ast$ (Fig. \ref{fig:sSFR}). It may be possible that the rest-UV$+$optical SED fits also significantly underestimate the \sSFRCSFH{} for a subset of the CEERS galaxies, particularly those with very red UV slopes ($\beta \sim -1$; \S\ref{sec:dusty}). Such a scenario would only boost the inferred median \sSFRCSFH{} of our CEERS sample, thereby strengthening our conclusions of a strong correlation with \Muv{}. However, the typically bluer UV-slopes of fainter $z\sim7-8$ galaxies suggest that obscured star formation likely plays a less important role in for the CEERS sample as a whole. The strong UV luminosity correlation with \sSFRCSFH{} described above implies that the light emerging from UV-faint $z\sim7-8$ galaxies is generally more heavily dominated by recently-formed OB stars relative to UV-bright systems at the same epoch. This in turn may suggest that fainter, more numerous reionization-era galaxies are often more efficient producers of LyC photons. To explore this, we consider the inferred ionizing photon production efficiencies (\xiion{}) among the CEERS and COSMOS samples, where \xiion{} is defined as the rate at which LyC photons are produced ($\dot{N}_\mathrm{ion}$) divided by the intrinsic stellar UV continuum luminosity at 1500 \AA{} (i.e. that corrected for both dust attenuation and nebular continuum emission). From our fiducial \textsc{beagle} fits, we find a very high median \xiion{} among the CEERS galaxies (\logxiion{} = 25.72$^{+0.03}_{-0.03}$) which is $\approx$0.1 dex higher than that inferred from the brighter COSMOS systems (\logxiion{} = 25.61$^{+0.08}_{-0.02}$). Moreover, we infer that 20\% of the CEERS galaxies possess extremely high \xiion{} ($>10^{25.8}$ erg$^{-1}$ Hz) compared to only 5\% in the COSMOS sample. The implied increase in \xiion{} towards lower luminosities may suggest that UV-faint galaxies contribute more to reionization than predicted in models that assume \xiion{} is independent of \Muv{} \citep{Naidu2020}. Larger \xiion{} values at lower luminosities would also alleviate requirements for a high \fesc{} among the numerous, faint galaxy population in order to reach the LyC photon budget necessary for reionization. Nonetheless, the very blue UV slopes ($\beta \lesssim -2$) of typical sub-$L_\mathrm{UV}^\ast$ reionization-era galaxies suggest these systems may often be efficient leakers of LyC photons \citep[e.g.][]{Chisholm2022}. Having investigated the dependence of CSFH sSFR on UV luminosity at $z\sim7-8$, we now turn our attention to the \OIIIHb{} EWs which we expect to be closely tied to \sSFRCSFH{} \citep[e.g.][]{Tang2019}. For this analysis, we infer the distribution of \OIIIHb{} EWs among our $z\sim6.5-8$ CEERS sample and compare to that of the UV-bright $z\simeq6.6-6.9$ COSMOS sample described above \citep{Endsley2021_OIII,Whitler2022_z7}. Following the approach of \citet{Endsley2021_OIII}, the EW distribution of each sample is assumed to follow a log-normal shape and is constrained by the \textsc{beagle} posterior probability distribution of all input galaxies adopting the Bayesian formalism of \citet{Boyett2022_OIII}. We find that the log-normal \OIIIHb{} EW distribution of our CEERS sample is described by a median EW of $780^{+50}_{-50}$ \AA{} and a standard deviation of $0.29^{+0.02}_{-0.03}$ dex (see Fig. \ref{fig:EWdistns}). This EW distribution is consistent with the first \JWST{} spectroscopic EW measurements of individual sub-$L_\mathrm{UV}^\ast$ $z\sim7-8$ galaxies \citep[e.g.][]{Schaerer2022}. Given the strong decrease in \sSFRCSFH{} towards brighter \Muv{} found above (Fig. \ref{fig:sSFR}), we might expect to see evidence of considerably lower \OIIIHb{} EWs among the COSMOS galaxies. However, the data are consistent with only a slight difference between the COSMOS ($\sim$2 $L_\mathrm{UV}^\ast$) and CEERS ($\sim$0.4 $L_\mathrm{UV}^\ast$) EW distributions, with that of the COSMOS galaxies parametrized by a median EW of 650$^{+110}_{-90}$ \AA{} and standard deviation of 0.24$^{+0.06}_{-0.05}$ dex (Fig. \ref{fig:EWdistns}). This result is in part driven by the considerable subset (23\%) of our $z\sim6.5-8$ CEERS sample showing SEDs consistent with both high \sSFRCSFH{} ($>$20 Gyr$^{-1}$) and relatively low \OIIIHb{} EWs ($<$600 \AA{}; see \S\ref{sec:weakOIII}), as such a population is not evident from IRAC data in COSMOS. Upcoming Cycle 1 \JWST{} surveys will clarify the extent to which \sSFRCSFH{} and \OIIIHb{} EW correlate with UV luminosity in the reionization era by extending to wider areas (e.g. COSMOS-Webb, PRIMER), pushing to deeper depths (e.g. JADES, WDEEP), and delivering spectra (e.g. FRESCO, UNCOVER). Spectroscopic efforts will simultaneously provide more direct constraints on LyC photon production efficiency, helping build our understanding of the relative contribution of UV-bright vs. UV-faint galaxies to cosmic reionization. \defcitealias{Labbe2022}{L22} \subsection{The Stellar Masses of UV-faint \texorpdfstring{$\mathbf{z\sim7-8}$}{z ~ 7 - 8} Galaxies} \label{sec:stellarMasses} Shortly after NIRCam imaging first became available, several $z\sim7-20$ candidates were identified in small ($\leq$40 arcmin$^2$) fields \citep{Adams2022,Atek2022,Castellano2022_GLASS,Donnan2022,Finkelstein2022_z12,Harikane2022_z9to17,Labbe2022,Leethochawalit2022,Morishita2022,Naidu2022_z12,Whitler2022_z10,Yan2022} some of which were reported to show SEDs suggesting extremely large stellar masses ($\approx10^{10-11}\ M_\odot$; \citealt{Naidu2022_z17,Labbe2022}). The large abundance of very early massive galaxies implied by these initial studies poses strong challenges for our understanding of baryon accretion onto the largest halos thought to exist at these epochs in context of $\Lambda$CDM \citep{BoylanKolchin2022,Ferrara2022_massivez10,Lovell2022}. While this potentially points to a failure in our basic understanding of cosmology and/or galaxy formation, it is worth investigating whether considerably lower mass solutions may be viable for some of these reported extremely massive $z\gtrsim7$ candidates. Here we specifically focus on comparing our stellar mass results with the $z\sim7-11$ sample in \citet{Labbe2022}, hereafter \citetalias{Labbe2022}, since they also utilize the CEERS data and their redshift selection window best overlaps with that considered in this work ($z\sim6.5-8$). Our fiducial \textsc{beagle} CSFH fits suggest that the NIRCam SEDs of CEERS $z\sim6.5-8$ galaxies are consistent with low stellar masses. Specifically, we infer a median stellar mass of $M_\ast = 10^{7.9}\ M_\odot$ among our full sample, with only a small fraction (7\%) of objects having $M_\ast > 10^{9}\ M_\odot$ and none with $M_\ast > 10^{10}\ M_\odot$ (see Figs. \ref{fig:inferredPropertyHistograms}d and \ref{fig:MstarMuv}). In contrast to our findings, \citetalias{Labbe2022} report the identification of seven massive ($M_\ast > 10^{10}\ M_\odot$) $z\sim7-11$ galaxies in the CEERS/NIRCam area, including two objects with extremely high mass ($M_\ast \sim 10^{11}\ M_\odot$). After cross-matching the \citetalias{Labbe2022} galaxy coordinates with our $z\sim6.5-8$ catalog, we find one $z\sim7$ object which overlaps between the two samples. This galaxy (ID 47624 in our catalog) is the most massive object in the \citet{Labbe2022} sample with a reported $M_\ast = 10^{11.2}\ M_\odot$. We additionally identify one \citetalias{Labbe2022} galaxy that satisfies the Lyman-break $z\sim8.5-11$ selection criteria presented in \citet[][ID 41858 in our catalog]{Whitler2022_z10}. Even though ID 41858 did not enter the final \citet{Whitler2022_z10} sample because its partial F115W dropout implies a significant probability of $z<8.2$, it remains a highly robust $z\sim8.5$ candidate in our analysis and we therefore consider it for comparison with the \citetalias{Labbe2022} stellar masses below. Upon investigating why the other five \citetalias{Labbe2022} galaxies do not enter our sample, we find that three (including the second most massive object reported in \citetalias{Labbe2022}) fall outside the ACS/F435W footprint and thus were not considered for high-redshift candidate selection in either this work or \citet{Whitler2022_z10}. The final two \citetalias{Labbe2022} galaxies are very faint in F150W and F200W and thus were not identified in our rest-UV detection image by \textsc{source extractor} (\S\ref{sec:photometry}). \citetalias{Labbe2022} utilized a F277W$+$F356W$+$F444W stack for their detection image which more closely resembles a rest-optical selection, though we emphasize that our rest-UV \textsc{source extractor} catalog does contain the two highest-mass ($M_\ast >10^{10.9}\ M_\odot$) galaxies reported in \citetalias{Labbe2022}. The NIRCam SED of the \citetalias{Labbe2022} source that overlaps with our $z\sim6.5-8$ selection (ID 47624) is shown in Fig. \ref{fig:AGN}, revealing an extremely red UV slope ($\beta = -0.57^{+0.03}_{-0.02}$) as well as very strong photometric excesses in F356W, F410M, and F444W relative to F277W. Given the exceptional brightness of this source in the three reddest NIRCam bands ($m=23.85-24.44$), the strength of the photometric excesses is confidently found to vary considerably from band to band. Because of the high S/N pattern of the LW excesses, our CSFH \textsc{beagle} fits strongly prefer a solution where the SED is dominated by a very young ($\approx$4 Myr) stellar population yielding strong nebular line emission (\OIIIHb{} EW = 1240$^{+80}_{-70}$ \AA{}; see Fig. \ref{fig:AGN}a). The nebular line solution allows for more flexibility in the LW colors relative to a solution where the photometric excesses are the result of an approximately power-law rest-optical continuum boosted by an extremely strong Balmer break. As expected from the very young solution, our CSFH fits yield a relatively low stellar mass of $M_\ast \approx 10^{8.6}\ \Msol{}$, which is about 300$\times$ lower than the $M_\ast \approx 10^{11.2}\ \Msol{}$ solution reported by \citetalias{Labbe2022}. We also find that the $z\sim8.5$ galaxy overlapping the \citetalias{Labbe2022} sample and the \citet{Whitler2022_z10} selection (ID 41858) shows strong, high S/N LW photometric excesses that vary from band to band, as well as a flat rest-UV slope ($\beta \approx -2.0$). The implied strong line emission in this object also suggests that its SED is dominated by a very young stellar population (\ageCSFH{}$\sim$5 Myr) with low stellar mass ($M_\ast \approx 10^{7.5}\ M_\odot$). These \textsc{beagle} CSFH fits demonstrate that relatively low-mass ($\sim$10$^8\ M_\odot$) solutions are plausible for at least two $z\sim7-8.5$ objects reported in \citetalias{Labbe2022} which would alleviate tension with models of galaxy formation \citep{BoylanKolchin2022,Lovell2022}. Below we comment on models that yield more massive solutions. It is well known in the literature that CSFH models can significantly underpredict the total stellar mass for galaxies with young SEDs given that these models will only account for stars formed during the recent burst (e.g. \citealt{Carnall2019_SFH,Leja2019,Lower2020}). Adopting non-parametric SFH models that disfavor rapid changes in the star formation rate will necessarily predict considerably more stellar mass assembly before the onset of the recent burst (see \S\ref{sec:beagle}). Such non-parametric SFH models have been employed in a number of recent studies focusing on $z\gtrsim7$ galaxies \citep[e.g.][]{Stefanon2022_IRACz10,Tacchella2022_NIRCamNIRSpec,Tacchella2022_SFHs,Topping2022_REBELS,Whitler2022_z7,Whitler2022_z10}, including \citetalias{Labbe2022} which helps explain the extremely high stellar mass they report for ID 47624 ($M_\ast = 10^{11.2}$ $M_\odot$). As expected, when we fit our photometry for ID 47624 with the non-parametric SFH models described above (following the approach of \citealt{Whitler2022_z7} with a uniform prior formation redshift from $z_\mathrm{form} = 10-30$), we obtain a substantially higher mass solution ($\approx$10$^{10.7}$ $M_\odot$) relative to our \textsc{beagle} CSFH fits (see Fig. \ref{fig:AGN}b). However, we note that the non-parametric SFH fit still requires rapid, strong changes in SFR to match the measured photometry for ID 47624. In the SFH posterior, nearly all of the stellar mass forms between $z_\mathrm{form}\sim20$ and $z\approx11$ (SFR$\approx$200 $M_\odot$ yr$^{-1}$), with very little star formation activity thereafter (SFR$\lesssim$1 $M_\odot$ yr$^{-1}$) until the most recent $\sim$few Myr when star formation ramps back up to power the high S/N excess in F410M. As can be seen in Fig. \ref{fig:AGN}, neither the CSFH nor non-parametric SFH models are able to precisely reproduce the observed LW colors of ID 47624 (the most massive galaxy reported in \citetalias{Labbe2022}), with each resulting in best-fitting $\chi^2 \approx 60$ across the 10 fitted data points. Notably, we clearly identify the characteristic 6-pointed diffraction pattern of JWST in the F356W, F410M, and F444W detections of this source (see Fig. \ref{fig:AGN}), indicating that the light dominating the LW bands is coming from a very compact region ($r < 300-400$ pc). While this is by no means conclusive evidence of an AGN in ID 47624, it at least opens the possibility that the LW excesses are significantly assisted by AGN line emission and we consider whether allowing for such emission (in addition to that from star formation) provides sufficient flexibility in model nebular line strengths and line ratios to better match the data. Utilizing the \textsc{beagle} AGN models in development by Emma Curtis-Lake et al. (to be described in Vidal-Garc{\'i}a et al. in prep), we find that including type II narrow-line AGN emission yields a good fit to the measured photometry (best-fitting $\chi^2 = 7$; Fig. \ref{fig:AGN}c). In this scenario, the substantial nebular line contribution from an AGN alleviates the need for a very recent upturn of star formation activity, thereby allowing for a much older CSFH solution ($\sim$200 Myr) and a corresponding higher mass-to-light ratio. The resulting inferred stellar mass of $\approx10^{10}$ $M_\odot$ with the SF$+$AGN models falls between that predicted from the (star formation only) CSFH and non-parametric SFH models described above. Overall, our results suggest that the photometry of ID 47624 can be reproduced with models that predict relatively moderate stellar mass ($\sim$10$^{9-10}$ $M_\odot$). Nevertheless, because this object shows signatures of a very young SED (\ageCSFH{}$\approx$4 Myr), we do find a wide ($\approx$2 dex) range of possible stellar mass solutions depending on model assumptions. This is indicative of the challenges faced when attempting to infer the absolute stellar masses of galaxies with extremely young SEDs, as was discussed in \citet{Whitler2022_z7}. Fortunately, there are steps that can be taken to help tighten the range of possible stellar masses for $z\gtrsim7$ galaxies with very young SEDs. Dynamical mass estimates from NIRCam WFSS or NIRSpec IFU observations would clarify whether very massive ($>10^{10}\ M_\odot$) solutions are plausible for any of these systems \citep[e.g.][]{Tang2022,Topping2022_REBELS}, while targeted searches for the most luminous $z\gtrsim10$ galaxies would help constrain any rigorous star formation activity at much earlier epochs \citep[e.g.][]{Castellano2022_GLASS,Donnan2022,Harikane2022_HdropsWideArea,Harikane2022_z9to17,Naidu2022_z12}. \section{Summary and Future Directions} \label{sec:summary} In this work, we characterize the star-forming and ionizing properties of 118 UV-faint Lyman-break $z\sim6.5-8$ galaxies in the CEERS ERS data. This task is made possible by the greatly improved sensitivity, angular resolution, and filter suite of \JWST{}/NIRCam relative to \Spitzer{}/IRAC, resulting in far better constraining power on the demographics of rest-optical SEDs among sub-$L_\mathrm{UV}^\ast$ reionization-era galaxies (see Fig. \ref{fig:HSTSpitzerComparison}). We infer the physical properties of each CEERS $z\sim6.5-8$ galaxy by fitting their 0.4--5$\mu$m photometry to a suite of star-forming photoionization model SEDs with \textsc{beagle} \citep{Chevallard2016,Gutkin2016}. Following many previous studies at $z\sim6-8$, we adopt constant star formation history (CSFH) models for our fiducial fits, and we comment on how alternative star formation histories would change our results. To promote comparisons between independent analyses, we provide an online catalog\footnote{\url{https://www.ryan-endsley.com/ceers-z6p5to8-catalog}} listing coordinates, magnitudes, and inferred physical properties among our sample. Our main conclusions are summarized below, where we note how future observations can help clarify outstanding questions raised by our results. \begin{enumerate} \item The photometric redshifts of galaxies in our sample range between $z=6.29$ to $z=8.08$ with a median of $z=6.80$, while the inferred absolute UV magnitudes fall between $-21.0 \leq \Muv{} \leq -18.9$ with a median of $-19.6$ (see Fig. \ref{fig:inferredPropertyHistograms}). Adopting a characteristic UV luminosity corresponding to M$_\mathrm{UV}^\ast = -20.5$ \citep[e.g.][]{Bowler2017,Harikane2022_LF}, this implies that the typical galaxy in our sample is 0.4 $L_\mathrm{UV}^\ast$ and 94\% of our sources are classified as sub-$L_\mathrm{UV}^\ast$ systems. \item The NIRCam data clearly reveal a wide variety of CSFH ages among the sample (Fig. \ref{fig:ageSEDs}). For the bulk of galaxies in our sample, the measured flux density in either F356W, F410M, or F444W is consistent with extrapolating a power-law SED from the rest-UV photometry, implying that their SEDs are dominated by young stellar populations (\ageCSFH{}$\sim$30 Myr). Nonetheless, we do identify a considerable subset of objects ($\approx$20\% of the sample) with significant and nearly equal flux excesses in the three reddest NIRCam bands, implying the presence of a prominent Balmer break consistent with relatively evolved stellar populations (\ageCSFH{}$\sim$100-500 Myr). Another $\approx$30\% of galaxies in our sample show very strong long-wavelength photometric excesses with the extent of the excess varying significantly from band to band, implying contamination from exceptionally high-EW nebular lines (\OIIIHb{} EW$\gtrsim$1500 \AA{}) powered by very young stellar populations (\ageCSFH{}$\lesssim$10 Myr). In context of our fiducial CSFH models, this broad distribution in ages directly implies that sub-$L_\mathrm{UV}^\ast$ reionization-era galaxies have a wide diversity of \sSFRCSFH{} ($\approx$2 Gyr$^{-1}$ to $>$500 Gyr$^{-1}$). \item We find that UV-faint ($\sim$0.4 $L_\mathrm{UV}^\ast$) $z\sim6.5-8$ galaxies typically have very large specific star formation rates (median \sSFRCSFH{}=82$^{+24}_{-23}$ Gyr$^{-1}$; see Fig. \ref{fig:sSFR}), consistent with a scenario in which many of these systems recently experienced a strong upturn in their star formation activity. Combining our results with previous \Spitzer{}/IRAC studies of the UV-bright $z\sim7-8$ galaxy population \citep{Endsley2021_OIII,Topping2022_REBELS,Whitler2022_z7}, we find evidence for a strong ($\approx$5--10$\times$) increase in \sSFRCSFH{} between $\sim$4 $L_\mathrm{UV}^\ast$ and $\sim$0.4 $L_\mathrm{UV}^\ast$ (Fig. \ref{fig:sSFR}). This implies that recently-formed OB stars contribute a larger fraction of the emergent light from fainter $z\sim7-8$ galaxies, resulting in a slight increase in the typical ionizing photon production efficiency in the CEERS sample (\xiion{} $\approx$ 10$^{25.72}$ erg$^{-1}$ Hz in CEERS) relative to the UV-bright systems (\xiion{} $\approx$ 10$^{25.61}$ erg$^{-1}$ Hz). This $\approx$0.1 dex increase in \xiion{} coupled with the steep faint-end slope of the $z\sim7-8$ UV luminosity function could imply that sub-$L_\mathrm{UV}^\ast$ reionization-era galaxies contribute more to reionization than previously thought. Future \JWST{} surveys will help clarify the relative contribution of bright vs. faint galaxies to cosmic reionization by extending to wider areas (e.g. COSMOS-Webb, PRIMER), pushing to deeper depths (e.g. JADES, WDEEP), and delivering spectra (e.g. FRESCO, UNCOVER). \item We infer the \OIIIHb{} EW distribution among UV-faint ($\sim$0.4 $L_\mathrm{UV}^\ast$) $z\sim6.5-8$ galaxies assuming it follows a log-normal function, finding a median EW = 780$^{+50}_{-50}$ \AA{} with a standard deviation of 0.29$^{+0.02}_{-0.03}$ dex (Fig. \ref{fig:EWdistns}). Despite the fact that we find much larger \sSFRCSFH{} among our CEERS galaxies, we find only a slight increase in the \OIIIHb{} EWs relative to more UV-luminous systems ($\sim$2 $L_\mathrm{UV}^\ast$; Fig. \ref{fig:EWdistns}). This is largely due to the substantial fraction ($\approx$23\%) of our CEERS galaxies that show SEDs consistent with high \sSFRCSFH{} ($>$20 Gyr$^{-1}$) yet relatively weak \OIIIHb{} (EW$<$600 \AA{}; see Fig. \ref{fig:youngWeakOIIISEDs}). Such a population is not evident among brighter $z\sim7-8$ galaxies from IRAC data, nor among existing samples at $z\sim2$ covering similar \OIIIHb{} EWs \citep{Tang2019}. In context of our fiducial CSFH \textsc{beagle} models, young SEDs with weak \OIIIHb{} are often reproduced with extremely low metallicities ($\lesssim$3\% $Z_\odot$; see Fig. \ref{fig:age_EW_metallicity}) which greatly suppresses the [OIII] emission. However, we also demonstrate that high ionizing photon escape fractions ($f_\mathrm{esc,HII} \gtrsim 0.5$) or a recent ($\sim$10 Myr) sharp decline in star formation rate can also reproduce the SEDs for a subset of our galaxies (Fig. \ref{fig:youngWeakOIIISEDs_otherExplanations}). Future spectroscopic observations will be necessary to not only confirm the existence of this seemingly important class of faint reionization-era galaxies, but also to better determine the physical origin of their weak \OIIIHb{}. \item As expected from previous studies utilizing deep \HST{}/WFC3 data, we find that our UV-faint CEERS galaxies generally exhibit very blue UV slopes (median $\beta = -2.0$; Fig. \ref{fig:inferredPropertyHistograms}). Nonetheless, we identify four galaxies with high S/N ($\gtrsim$20$\sigma$) short-wavelength NIRCam photometry indicating very red UV slopes ($-1.3\leq\beta\leq-0.6$), as well as long-wavelength photometric excesses implying strong \OIIIHb{} (EW$>$800 \AA{}) at $z\sim7$ (Fig. \ref{fig:redEELGseds}). While all four systems are classified as $\approx$0.5 $L_\mathrm{UV}^\ast$ from the data, their very red UV slopes imply that they would appear as far more luminous galaxies in the absence of dust ($\approx$2--4 $L_\mathrm{UV}^\ast$). The fact that we identify four such objects in CEERS suggests that there remains a significant tail of heavily dust-reddened $z\sim7$ galaxies down to faint M$_\mathrm{UV}$. The \textsc{beagle} fits imply that $\approx$70--90\% of their star formation is obscured which in total contributes 17\% of all (unobscured$+$obscured) stellar mass growth inferred among our CEERS sample, even though these objects make up only 3\% of the sample by number. Deep far-infrared follow-up will be critical to assess the extent of obscured star formation in UV-faint yet very red $z\sim7$ galaxies identified from deep \JWST{} data. \item Our fiducial CSFH \textsc{beagle} fits imply that the NIRCam SEDs of UV-faint ($\Muv{} \sim -19.6$) $z\sim6.5-8$ galaxies are consistent with low stellar masses (median $M_\ast = 10^{8.0}\ M_\odot$), and none of the objects in our sample have \MstarCSFH{}$>$10$^{10}$ $M_\odot$. We investigate one $z\sim7$ object in our sample that was reported to have an extremely large stellar mass ($10^{11.2}$ $M_\odot$) in a previous work \citep{Labbe2022} which challenges models of galaxy formation \citep{BoylanKolchin2022,Ferrara2022_massivez10,Lovell2022}. This galaxy shows an extremely red UV slope ($\beta = -0.6$) with a high S/N LW photometric excess pattern that implies strong nebular line emission and hence a very young (\ageCSFH{}$\approx$4 Myr) SED. Because the recent burst may be outshining an older stellar population, we find a very wide range of plausible stellar mass solutions for this galaxy ($\approx$10$^{8.6}$ $M_\odot$ to $\approx$10$^{10.7}$ $M_\odot$) depending on the assumed star formation history (Fig. \ref{fig:AGN}a,b). Given the extreme LW brightness of this object, we are able to clearly identify the characteristic 6-pointed diffraction pattern of JWST in the F356W, F410M, and F444W detections (see Fig. \ref{fig:AGN}). While this is by no means conclusive evidence of an AGN, it at least opens the possibility that the LW excesses are significantly assisted by AGN line emission and we find that the inclusion of such models yield a good fit to the measured photometry with $M_\ast \approx 10^{10}$ $M_\odot$ (Fig. \ref{fig:AGN}c). Dynamical mass estimates from future spectroscopic observations will help clarify the range of plausible stellar mass solutions for $z\gtrsim7$ galaxies with very young SEDs \citep[e.g.][]{Tang2022,Topping2022_REBELS}. \end{enumerate} \section*{Acknowledgements} RE and DPS acknowledge funding from JWST/NIRCam contract to the University of Arizona, NAS5-02015. DPS acknowledges support from the National Science Foundation through the grant AST-2109066. LW acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No. DGE-2137419. The authors sincerely thank NIRCam PI Marcia Rieke for sending updated photometric calibration information, Mengtao Tang for providing information on their $z\sim2$ EELG sample, Jacopo Chevallard for granting access to the \textsc{beagle} tool used for much of our SED fitting analysis, as well as Gabe Brammer for providing the ACS imaging mosaics of the EGS field as part of CHArGE program. This material is based in part upon High Performance Computing (HPC) resources supported by the University of Arizona TRIF, UITS, and Research, Innovation, and Impact (RII) and maintained by the UArizona Research Technologies department. This research made use of \textsc{astropy}, a community-developed core \textsc{python} package for Astronomy \citep{astropy:2013, astropy:2018}; \textsc{matplotlib} \citep{Hunter2007_matplotlib}; \textsc{numpy} \citep{van2011numpy}; and \textsc{scipy} \citep{jones_scipy_2001}. \section*{Data Availability} The \HST{}/ACS and \JWST{}/NIRCam images used in this work are available through the Mikulski Archive for Space Telescopes (\url{https://mast.stsci.edu/}). We provide an online catalog ({\url{https://www.ryan-endsley.com/ceers-z6p5to8-catalog}}) listing the coordinates, magnitudes, and inferred physical properties of our $z\sim6.5-8$ galaxy sample in CEERS, as well as the NIRCam photometric zeropoint information utilized in our analysis. Additional data products will be made available upon reasonable request to the corresponding author. \bibliographystyle{mnras} \bibliography{paper_ref} % \appendix \bsp % \label{lastpage}

Title: PNV J00444033+4113068: early superhumps with 0.7 mag amplitude and non-red color

Abstract: In the first days of WZ Sge-type dwarf nova (DN) outbursts, the 2:1 resonance induces a spiral arm structure in the accretion disk, which is observed as early superhumps in optical light curves. This paper reports our optical observations of an eclipsing WZ Sge-type DN PNV J00444033+4113068 during its 2021 superoutburst with the 3.8m Seimei telescope and through VSNET collaboration. The eclipse analysis gave its orbital period as 0.055425534(1) d. Our observations confirmed early superhumps with an amplitude of 0.7 mag, the largest amplitude among known WZ Sge-type DNe. More interestingly, its early superhumps became the reddest around their secondary minimum, whereas other WZ Sge-type DNe show the reddest color around the early superhump maximum. The spectrum around the peak of the outburst showed the double-peaked emission lines of He II 4686\AA~ and H$\alpha$ with a peak separation of $\ge 700$ km/s, supporting a very high-inclination system. With the early superhump mapping, the unique profile and color of the early superhump of PNV J00444033+4113068 are successfully reproduced by the accretion disk with vertically extended double arm structure. Therefore, the large amplitude and unique color behavior of the early superhumps in PNV J00444033+4113068 can be explained by the 2:1 resonance model along with other WZ Sge-type DNe.

https://export.arxiv.org/pdf/2208.04251

\SetRunningHead{Y. Tampo et al.}{PNV J00444033+4113068 2021 superoutburst} \title{PNV J00444033+4113068: early superhumps with 0.7 mag amplitude and non-red color} \author{ Yusuke~\textsc{Tampo}\altaffilmark{\ref{affil:Kyoto}*}, Keisuke~\textsc{Isogai}\altaffilmark{\ref{affil:KyotoOkayama}}$^,$\altaffilmark{\ref{affil:Kyoto}}$^,$\altaffilmark{\ref{affil:MuSCAT}}, Naoto~\textsc{Kojiguchi}\altaffilmark{\ref{affil:Kyoto}}, Makoto~\textsc{Uemura}\altaffilmark{\ref{affil:HAO}}, Taichi~\textsc{Kato}\altaffilmark{\ref{affil:Kyoto}}, Tam\'as~\textsc{Tordai}\altaffilmark{\ref{affil:Trt}}, Tonny~\textsc{Vanmunster}\altaffilmark{\ref{affil:Van1}}$^,$\altaffilmark{\ref{affil:Van2}}, Hiroshi~\textsc{Itoh}\altaffilmark{\ref{affil:Ioh}}, Pavol~A.~\textsc{Dubovsky}\altaffilmark{\ref{affil:Vih}}, Tom\'a\v{s}~\textsc{Medulka}\altaffilmark{\ref{affil:Vih}}, % Yasuo~\textsc{Sano}\altaffilmark{\ref{affil:san}}$^,$\altaffilmark{\ref{affil:san2}}$^,$\altaffilmark{\ref{affil:san3}}, Franz-Josef~\textsc{Hambsch}\altaffilmark{\ref{affil:ham1}}$^,$\altaffilmark{\ref{affil:ham2}}$^,$\altaffilmark{\ref{affil:dfs}}, Kenta~\textsc{Taguchi}\altaffilmark{\ref{affil:Kyoto}}, Hiroyuki~\textsc{Maehara}\altaffilmark{\ref{affil:NAOJOkayama}}$^,$\altaffilmark{\ref{affil:KyotoOkayama}}, Junpei~\textsc{Ito}\altaffilmark{\ref{affil:Kyoto}}, and Daisaku~\textsc{Nogami}\altaffilmark{\ref{affil:Kyoto}} } \authorcount{affil:Kyoto}{ Department of Astronomy, Kyoto University, Kyoto 606-8502, Japan} \email{$^*$tampo@kusastro.kyoto-u.ac.jp} \authorcount{affil:KyotoOkayama}{ Okayama Observatory, Kyoto University, 3037-5 Honjo, Kamogatacho, Asakuchi, Okayama 719-0232, Japan} \authorcount{affil:MuSCAT}{ Department of Multi-Disciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan} \authorcount{affil:HAO}{ Hiroshima Astrophysical Science Center, Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima 739-8526, Japan} \authorcount{affil:Trt}{ Polaris Observatory, Hungarian Astronomical Association, Laborc utca 2/c, 1037 Budapest, Hungary} \authorcount{affil:Van1}{ Center for Backyard Astrophycis Belgium, Walhostraat 1a, B-3401 Landen, Belgium} \authorcount{affil:Van2}{ Center for Backyard Astrophycis Extremadura, e-EyE Astronomical Complex, ES-06340 Fregenal de la Sierra, Spain} \authorcount{affil:Ioh}{ Variable Star Observers League in Japan (VSOLJ), 1001-105 Nishiterakata, Hachioji, Tokyo 192-0153, Japan} \authorcount{affil:Vih}{ Vihorlat Observatory, Mierova 4, 06601 Humenne, Slovakia} \authorcount{affil:san}{ Variable Star Observers League in Japan (VSOLJ), Nishi juni-jou minami 3-1-5, Nayoro, Hokkaido, Japan} \authorcount{affil:san2}{ Observation and Data Center for Cosmosciences, Faculty of Science, Hokkaido University, Kita-ku, Sapporo, Hokkaido 060-0810, Japan} \authorcount{affil:san3}{ Nayoro Observatory, 157-1 Nisshin, Nayoro, Hokkaido 096-0066, Japan} \authorcount{affil:ham1}{ Groupe Européen d’Observations Stellaires (GEOS), 23 Parc de Levesville, 28300 Bailleau l’Evêque, France} \authorcount{affil:ham2}{ Bundesdeutsche Arbeitsgemeinschaft für Veränderliche Sterne (BAV), Munsterdamm 90, 12169 Berlin, Germany} \authorcount{affil:dfs}{ Vereniging Voor Sterrenkunde (VVS), Oostmeers 122 C, 8000 Brugge, Belgium} \authorcount{affil:NAOJOkayama}{ Subaru Telescope Okayama Branch Office, National Astronomical Observatory of Japan, National Institutes of Natural Sciences, 3037-5 Honjo, Kamogata, Asakuchi, Okayama 719-0232, Japan} \KeyWords{accretion, accretion disk --- novae, cataclysmic variables --- stars: dwarf novae --- stars :individual (PNV J00444033+4113068)} \section{Introduction} \label{sec:1} Dwarf novae (DNe) are accreting white dwarf (WD) binaries that possess an accretion disk and show recurrent outbursts (see \cite{war95book, hel01book}). It is widely accepted that the mechanism of DN outbursts is explained by the tidal-thermal disk instability model [for a review, see \citet{osa96review}]. WZ Sge-type DNe form a subclass in DNe. Their outbursts are characterized by early superhumps observed for the first 5-10 days of the outburst, which is a double-wave variation with a period almost identical to the orbital period of the system \citep{ish02wzsgeletter,kat15wzsge}. While most of WZ Sge-type DNe show early superhumps with the amplitude less than 0.05 mag \citep{kat15wzsge,kat22WZSgecandle}, high-inclination systems show the amplitude larger than 0.1 mag [e.g., OV Boo \citep{pat08j1507}, V455 And \citep{mat09v455and, Pdot}, ASASSN-18do (vsnet-alert 21921\footnote{http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/21921})]. In addition, the multi-color observations of early superhumps showed the reddest color around the early superhump maxima \citep{mat09v455and, nak13j0120, iso15ezlyn, ima18HVVirJ0120}. These results have suggested that the early superhumps can be explained by the orbital rotational effect of the outer disk with a non-axisymmetric vertical structure \citep{nog97alcom, mat09v455and, ima18HVVirJ0120}. The theoretical understanding of early superhumps is considered to be the occurrence of the 2:1 resonance between the secondary star and the Keplerian accretion disk, resulting in vertical deformation of the accretion disk and the appearance of a double spiral arm pattern \citep{lin79lowqdisk, osa02wzsgehump, kun04SHSPH, kun05earySHSPH}. On the other hand, ordinary superhumps are induced by the eccentric disk through the 3:1 resonance \citep{whi88tidal,osa89suuma, hir90SHexcess}. As the 2:1 resonance suppresses the growth of the 3:1 resonance \citep{lub91SHa}, early superhumps are always observed before ordinary superhumps. Observational evidence of the spiral arm structure in WZ Sge-type DNe was first found in the WZ Sge 2002 superoutburst \citep{bab02wzsgeletter,kuu02wzsge}. Applying Doppler tomography with He II 4686\AA, a double-arm structure in the accretion disk was deduced. Another observational approach of studying a disk structure is to model the profile of early superhumps \citep{mae07bcuma, uem12ESHrecon}. By modeling the multi-color early superhump profiles with self-occultation of the vertically extended disk, called "early superhump mapping", \citet{uem12ESHrecon,nak13j0120} revealed the double arm spiral structure in the accretion disk, highlighting the occurrence of the 2:1 resonance in the accretion disk of WZ Sge-type DNe. However, due to the limited samples, the diversity of the disk structure is still not investigated. PNV J00444033+4113068\footnote{http://tamkin1.eps.harvard.edu/unconf/followups/J00444033+4113068.html} (= AT 2021aaxp, hereafter PNV J0044) was discovered as an M31 classical nova candidate by Koichi Itagaki at 16.5 mag on 2021-10-09.4579. However, double-peaked emission lines of H$\alpha$ and He II 4686\AA~ were detected in the follow-up spectroscopic observation, classifying PNV J0044 as a foreground large-amplitude DN rather than an M31 classical nova \citep{tag21PNVJ0044}. Later photometric observations detected early superhumps with the amplitude of 0.7 mag, confirming PNV J0044 as a WZ Sge-type DN (vsnet-alert 26319 \footnote{http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/26319}). The quiescence counterpart is likely V$=22.278$ mag according to the Revised LGGS UBVRI photometry of the M31 and M33 stars catalog \citep{mas16M31M33stars}. In this paper, we present our optical observations and analyses of PNV J0044 during its 2021 outburst. Section \ref{sec:2} presents the overview of our observations of the superoutburst, and Section \ref{sec:3} shows the results of our analysis. We discuss the properties of PNV J0044 in Section \ref{sec:4} and give the summary of this paper in Section \ref{sec:5}. \section{Observations and Analysis} \label{sec:2} \subsection{photometric observations} Our time-resolved CCD photometric observations of PNV J0044 were carried out by the Variable Star Network (VSNET) collaborations \citep{VSNET}. We also performed the simultaneous $g$-, $r$- and $i$-band photometry with the TriColor CMOS Camera and Spectrograph (TriCCS\footnote{http://www.kusastro.kyoto-u.ac.jp/$ \textasciitilde$kazuya/p-triccs/index.html}) mounted on the 3.8m Seimei telescope \citep{kur20seimei} at Okayama Observatory of Kyoto University. The instruments for our photometric observations and our observation logs are summarized in tables E1 and E2 \footnote{Tables E1 and table E2 are available only on the online edition as Supporting Information. }, respectively. All the observation epochs in this paper are described in the Barycentric Julian Day (BJD). VSNET observations were unfiltered, and the zero point of these data was adjusted to the observations by T. Vanmunster for our period analysis. For the magnitude calibration of TriCCS data, the AAVSO comparison star 000-BNN-553 (= Gaia DR3 369265315130536960) with $g = 16.039(3)$, $r = 15.621(3)$, and $i = 14.473(1)$ at $(\alpha, \delta)_{\rm J2000.0} =$ (\timeform{00h44m44s.96}, $+$\timeform{41D13'12''.9}) (Pan STARRS DR1; \cite{panstarrs1}) was adopted. We also extracted photometric survey data from the Zwicky Transient Facility (ZTF; \cite{ZTF}) alert broker Lasair \citep{lasair} and the Asteroid Terrestrial-impact Last Alert System (ATLAS; \cite{ATLAS}) to examine the global light curve profiles. These survey data were not included in our period analysis. The phase dispersion minimization (PDM; \cite{PDM}) method was applied for period analysis of the superhumps in this paper. The 1$\sigma$ errors for the PDM analysis was determined following \citet{fer89error, pdot2}. Before period analysis, the global trend of the light curve was removed by subtracting a smoothed light curve obtained by locally weighted polynomial regression (LOWESS: \cite{LOWESS}). \subsection{spectroscopic observations} We performed our spectroscopic observation of PNV J0044 on BJD 2459497.21 using the fiber-fed integral field spectrograph (KOOLS-IFU; \cite{mat19koolsifu}) mounted on Seimei telescope \citep{kur20seimei}. We applied VPH-blue as a grism, which has a resolution of $R\sim 500$ and a wavelength coverage of 4,200-8,000\AA. Our data reduction was performed using IRAF\footnote{IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Associations of Univesities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.} in the standard manner (bias subtraction, flat fielding, aperture determination, spectral extraction, wavelength calibration with arc lamps, and flux calibration with a standard star). The preliminary result was already reported by \citet{tag21PNVJ0044}. \section{Results} \label{sec:3} \subsection{Overall light curve during the superoutburst} \label{sec:3.1} Figure \ref{fig:2021lc} shows the global light curve of PNV J0044 during the superoutburst in 2021. Its peak magnitude is $\sim15.3$ mag on BJD 2459497, and hence the outburst amplitude reached $\sim7$ mag. Our time-resolved observations between BJD 2459500 and 2459504 showed clear early superhumps, whereas on BJD 2459507, the variation profile was featureless due to the low S/N of the data. The slope of the outburst decay became gentler around BJD 2459506, which can be attributed to the alternation of early and ordinary superhumps \citep{kat15wzsge}. The outburst probably ceased on BJD 2459520, and the later phase after BJD 2459525 can be a rebrightening. As there are no upper limit observations in Lasair and ATLAS, the rebrightening profile appears to be type-A (plateau rebrightening; \cite{ima06j0137}). We note that lacks of time-resolved observations of the later outburst phase prevent us from drawing any solid conclusions. \subsection{eclipse and orbital period} \label{sec:3.2} In figure \ref{fig:enlargeLC}, the enlarged light curves in the $g$, $r$ and $i$ bands on BJD 2459502 (upper left panel) and 2459515 (upper right panel) observed with TriCCS are presented. In both panels, deep eclipses are recognized. The depth of the eclipses was $\sim$ 0.3 mag on BJD 2459502 and $\sim1.5$ mag on BJD 2459515. We determined the eclipse minima by fitting the Gaussian function, and then the orbital ephemeris of PNV J0044 was obtained as Equation \ref{eq:porb}. \begin{equation} \label{eq:porb} \phi_0 = \rm{BJD~} 2459500.3237(1) + 0.055425534(1) \times E \end{equation} The period obtained is close to the period minimum \citep{kni11CVdonor, kat22updatedSHAmethod}. Therefore, PNV J0044 is one of the best candidates for period bouncing objects, while we cannot constrain its mass ratio due to the lack of further information such as ordinary superhump periods. \subsection{early superhumps} \label{sec:3.3} In the upper left panel of figure \ref{fig:enlargeLC}, the double-peaked early superhumps are recognized on BJD 2459502. The period of early superhumps was calculated through PDM analysis using data outside of the eclipse (lower left panel of figure \ref{fig:enlargeLC}). The yield period is 0.05535(1) d, which is $\sim0.1\%$ shorter than the orbital period obtained from the eclipse analysis (Section \ref{sec:3.2}). This slight difference from the orbital period is also observed in other WZ Sge-type DNe (e.g., \cite{ish02wzsgeletter}). The superhump maxima are summarized in table E3 \footnote{Table E3 is available only on the online edition as Supporting Information.}. The amplitude of early superhumps is $\sim 0.7$ mag including the eclipse or $\sim 0.5$ mag without the eclipse, which is the largest value for the amplitude of early superhumps in known WZ Sge-type DNe (\cite{kat15wzsge, kat22WZSgecandle} and reference there in). As a larger amplitude is observed in a system with higher inclination, PNV J0044 can have the largest inclination angle in WZ Sge-type DNe. In addition, the $g-i$ color of the early superhumps became the reddest around the secondary minimum, and the peaks did not show significant redder color. This behavior is in contradiction with other WZ Sge-type DNe such as V455 And \citep{mat09v455and} and OT J012059.6+325545 \citep{nak13j0120}, which showed the reddest color around the primary maximum of the early superhumps. Since the redder color around the maximum of early superhumps is considered to be responsible for the vertical expansion of the outer disk \citep{mat09v455and, nak13j0120}, the early superhumps in PNV J0044 may not be explained in the same manner. On BJD 2459515, in addition to the eclipse, PNV J0044 showed the variation with the double-peaked profile. Our PDM analysis using data outside the eclipse yielded a period of 0.0566 (2) d for this variation, which is 1.35\% longer than the orbital period. This variation is most likely ordinary superhumps based on the global light-curve profile; however, the superhump profile may not be prominent as this epoch corresponds to the end phase of the outburst. \subsection{spectroscopic observation} \label{sec:3.4} Our spectrum on BJD 2459497.21 observed with KOOLS-IFU on the Seimei telescope is presented in figure \ref{fig:spec}. This epoch corresponds to the orbital phase $\phi \sim $ 0.875 (out of the eclipse). The spectrum showed the blue continuum attributing to the multi-temperature disk black body and the double-peaked emission lines of H$\alpha$ and He II 4686. He II 5411 and C $_{\rm III}$/N $_{\rm III}$ Bowen blend emission lines were detected as well. H$\beta$ line was likely in emission with a deep absorption core. Na D absorption line was detected. The peak separation of H$\alpha$ and He II 4686\AA~ is $\ge$700 km/s. This separation is comparable to WZ Sge at its outburst peak \citep{nog04wzsgespec}, although noting that our spectrum was obtained with a low-resolution grism (R$\sim$500). Such a large peak separation around the optical peak again supports that PNV J0044 is a high-inclination system. The combined equivalent width (EW) of the He II 4686 and Bowen blend (-14.1\AA) was significantly larger than that of H$\alpha$ (-6.3\AA). As in most WZ Sge-type DNe the EW of He II 4686 is weaker than H$\alpha$ (\cite{tam21seimeiCVspec} and reference therein), this result also highlights that PNV J0044 is a high-inclination system, in which the He II emission from the heated arm structure contributes greatly to the line strength \citep{bab02wzsgeletter, mor02DNspectralatlas, tam21seimeiCVspec}. \section{Discussion} \label{sec:4} As described in Section \ref{sec:3}, the early superhumps of PNV J0044 become redder around the secondary minimum rather than around the primary maximum, which is very unique color trend compared with other WZ Sge-type DNe. Moreover, the amplitude of early superhumps is the largest among known WZ Sge-type DNe so far \citep{kat15wzsge, kat22WZSgecandle}. Therefore, it is worth examining whether the above properties of the early superhumps of PNV J0044 can be modeled by the vertically extended spiral arm structure similar to other modeled objects (V455 And; \cite{uem12ESHrecon}, OT J012059.6+325545; \cite{nak13j0120}). We therefore performed the early superhump mapping using the code developed in \citet{uem12ESHrecon}. This model assumes that early superhumps are caused by vertical deformation and orbital rotation effects of the accretion disk. The disk is assumed to radiate as blackbody and the disk temperature $T$ gradient to the disk radius $R$ to be the standard disk model ($T \propto R^{-3/4}$; \cite{sha73alphadisk}). For this analysis, we used the simultaneous observations with TriCCS in the $g$, $r$ and $i$ bands. The system parameters adopted for the early superhump mapping are summarized in table \ref{tab:1}. Since the orbital period $P_{\rm orb}$ of PNV J0044 is close to the period minimum \citep{kni11CVdonor, kat22updatedSHAmethod}, we applied 0.08 as the mass ratio $q~(=M_{\rm Secondary}/M_{\rm WD})$ assuming that PNV J0044 follows the standard evolution path of DNe. The WD mass $M_{\rm WD}$ was adopted as 0.8 $M_\odot$, which is the typical value of the WD mass in DNe (\cite{pal22WDinCVs} and the reference therein). For the outer disk radius $R_{\rm out}$, the 2:1 resonance radius (0.6$a$ where $a$ is the binary separation) was applied \citep{osa02wzsgehump}. The inclination of the system $i$ was assumed to be 85$^\circ$. This is because, as OV Boo with the early superhump amplitude of 0.27 mag is estimated to have the inclination of 83$^\circ$ \citep{pat08j1507}, the inclination of PNV J0044 is expected to be larger than OV Boo. The innermost temperature of the accretion disk $T_{\rm in}$ was set as 150,000K, which gives the most reasonable fit. \begin{table} \caption{System parameters adopted in the early superhump mapping of PNV J0044.} \centering \label{tab:1} \begin{tabular}{cc} \hline Parameter & \\ \hline \hline $P_{\rm orb}$ & 0.055425534 d \\ $q$ & 0.08 \\ $i$ & 85$^\circ$ \\ $M_{\rm WD}$ & 0.8 $M_\odot$ \\ $T_{\rm in}$ & 150,000 K \\ $R_{\rm out}$ & 0.6 $a$\commenta\\ \hline \multicolumn{2}{l}{\commenta $a$ is the binary separation.}\\ \end{tabular} \end{table} Figure \ref{fig:eshmodel} presents the reconstructed height map of the accretion disk of PNV J0044 (middle and right panels). The middle panel shows the height scale normalized by the binary separation, and the right panel shows the ratio of the disk height $h$ over the radius of the disk $r$. The synthesized light curves (solid lines in the left panel of figure \ref{fig:eshmodel}) are presented along with the phase-averaged early superhumps in the $g$ (green circles), $r$ (red squares) and $i$ (pink diamonds) bands. As seen in the left panel, the reconstructed accretion disk well explains the observed profile and color of early superhumps of PNV J0044. In the middle and right panels of figure \ref{fig:eshmodel}, the outermost region of the reconstructed disk shows two flaring parts in the upper left [around $(X,Y)=(-0.3,0.4)$] and lower right [around $(X,Y)=(0.1,-0.5)$] quadrants. In addition to this, the elongated arm structure into the inner disk is recognized around $(X,Y)=(0.2,0.3)$ and $(X,Y)=(-0.2,-0.3)$. The ratio of disk height to radius is less than 0.25 at the arm positions. These structures, such as the phase and height of the double-armed spirals, are consistent with the previously modeled disk height map of other WZ Sge-type DNe (V455 And; \cite{uem12ESHrecon}, OT J012059.6+325545; \cite{nak13j0120}). Therefore, the early superhumps in PNV J0044 can be understood in the same manner as other WZ Sge-type DNe, even though its amplitude was the largest and the color trend was different from other WZ Sge-type DNe. The unique point of PNV J0044 is that the inner arm structure is more evident than the other objects. A closer look reveals that OT J012059.6+325545 does not have the lower-left inner arm structure compared to PNV J0044 and V455 And \citep{nak13j0120}. In the case of V455 And, the height ratio to disk radius at the position of the inner arms is comparable to that of the outer disk \citep{uem12ESHrecon}. However, the inner structure of PNV J0044 has a larger height ratio than its outer spiral arms. As these inner structures are hotter and bluer than the outer disk, this feature enables PNV J0044 not to be the reddest around the early superhump maximum, whereas other WZ Sge-type DNe show the reddest color around the early superhump maximum. Even though this result can indicate that the inner disk in PNV J0044 is truly extended, another interpretation is possible; as our model assumes the temperature $T$ gradient to the disk radius $R$ to be the standard disk model ($T \propto R^{-3/4}$; \cite{sha73alphadisk}), the inner elongated arm structure around $(X,Y)=(-0.3,-0.1)$ can mean a hotter temperature in the outer disk around $(X,Y)=(-0.5,-0.1)$. The same discussion can be applied to the upper right side of the disk. Therefore, our results can also mean that the temperature of the outer disk in PNV J0044 and WZ Sge-type DNe is hotter than that of the standard disk model. An other independent method to test the temperature structure during the early superhump phase would be required to examine the disk temperature and height structure simultaneously. \section{Summary} \label{sec:5} We report optical observations during the outburst of an eclipsing WZ Sge-type dwarf nova PNV J0044 in 2021. Through the analysis of eclipses, its orbital period was determined as 0.055425534(1) d. PNV J0044 showed early superhumps with the amplitude of 0.7 mag, which is the largest among known WZ Sge-type DNe. This result proposes that PNV J0044 can be a WZ Sge-type DN with the highest inclination. Moreover, its early superhumps showed the reddest color around the secondary minimum, whereas those of other well-observed WZ Sge-type DNe become the reddest around the primary maximum. The spectra of PNV J0044 around the optical peak showed the double-peaked emission lines of H$\alpha$ and He II 4686\AA~ with the peak separation of $\ge$ 700 km s$^{-1}$, which supports the idea that PNV J0044 is a high-inclination system. By applying the early superhump mapping to the multi-color and simultaneous observations with TriCCS, our result showed that the accretion disk of PNV J0044 in the early superhump phase accompanies a double-armed spiral structure. This result confirms that the large amplitude and unique color trend of the early superhumps in PNV J0044 originate from the 2:1 resonance as well as other WZ Sge-type DNe. \begin{ack} Y. T. acknowledges support from the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 21J22351. U.M., T.K., and D.N. acknowledge support from the JSPS KAKENHI Grant Number 21K03616. This work was partially supported by the Slovak Research and Development Agency under the contract No. APVV-20-0148. The authors thank the TriCCS developer team (which has been supported by the JSPS KAKENHI grant Nos. JP18H05223, JP20H00174, and JP20H04736, and by NAOJ Joint Development Research). Lasair is supported by the UKRI Science and Technology Facilities Council and is a collaboration between the University of Edinburgh (grant ST/N002512/1) and Queen’s University Belfast (grant ST/N002520/1) within the LSST:UK Science Consortium. ZTF is supported by National Science Foundation grant AST-1440341 and a collaboration including Caltech, IPAC, the Weizmann Institute for Science, the Oskar Klein Center at Stockholm University, the University of Maryland, the University of Washington, Deutsches Elektronen-Synchrotron and Humboldt University, Los Alamos National Laboratories, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW. This research has made use of ``Aladin sky atlas'' developed at CDS, Strasbourg Observatory, France \cite{Aladin2000,Aladinlite}. \end{ack} \section*{Supporting Information} The following Supporting Information is available on the online version of this article: Tables E1, E2 and E3. \bibliographystyle{pasjtest1} \bibliography{main} \appendix

Title: Galactic bar resonances with diffusion: an analytic model with implications for bar-dark matter halo dynamical friction

Abstract: The secular evolution of disk galaxies over cosmic time is largely driven by resonances between the orbits of 'particles' (e.g. stars or dark matter) and the rotation of non-axisymmetric features (e.g. spiral arms or a bar). Such resonances are also often invoked to explain present-day kinematic and photometric features observed in the Milky Way and external galaxies. In simplified cases these resonant interactions are well understood: for instance, the secular dynamics of a test particle trapped near a resonance of a steadily rotating bar is easily analyzed using the angle-action tools pioneered by Binney, Monari and others. However, their treatments don't address the stochasticity and messiness of real galaxies - effects which have, with few exceptions, been previously captured only in complex N-body simulations. In this paper we propose a simple kinetic equation to describe the distribution function of particles near an orbital resonance with a rigidly rotating bar, using the pendulum approximation and allowing for diffusion of the particles' slow actions. We solve this kinetic equation for various values of the diffusion strength $\Delta$. We then apply our theory to the calculation of the dynamical friction torque felt by a bar embedded in a spherical dark matter halo. For $\Delta = 0$ we recover the classic result of Tremaine & Weinberg that the friction vanishes in the time-asymptotic (phase-mixed) limit, whereas for $\Delta > 0$ we find that diffusion suppresses phase mixing, leading to a finite negative torque, suggesting that real galactic bars always decelerate.

https://export.arxiv.org/pdf/2208.03855

command. \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \newcommand{\md}{\mathrm{d}} \newcommand{\me}{\mathrm{e}} \newcommand{\mi}{\mathrm{i}} \newcommand{\nn}{\nonumber} \def\p{\partial} \newcommand{\pder}[2]{\frac{\partial#1}{\partial#2}} \newcommand\cmtla[1]{{\color{blue}[LA: #1]}} \newcommand\cmtch[1]{{\color{red}[CH: #1]}} \newcommand{\btheta}{\bm{\theta}} \newcommand{\bvphi}{\bm{\phi}} \newcommand{\bJ}{\bm{J}} \newcommand{\pattern}{\Omega_\mathrm{p}} \newcommand{\Js}{J_\mathrm{s}} \newcommand{\rp}{r_\mathrm{p}} \newcommand{\ra}{r_\mathrm{a}} \newcommand{\pD}[2]{\frac{\partial #2}{\partial #1}} \newcommand{\pDD}[2]{\frac{\partial^2 #2}{\partial #1^2}} \newcommand{\D}[2]{\frac{{\rm d} #2}{{\rm d} #1}} \newcommand{\DD}[2]{\frac{{\rm d}^2 #2}{{\rm d} #1^2}} \newcommand\bb[1]{\mbox{\boldmath{$#1$}}} \newcommand\grad{\bb{\nabla}} \newcommand\bcdot{\,\bb{\cdot}\,} \newcommand\bdbldot{\,\bb{:}\,} \newcommand\btimes{\,\bb{\times}\,} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mcb}[1]{\bb{\mathcal{#1}}} \newcommand{\msf}[1]{\mathsf{#1}} \newcommand{\msb}[1]{\bb{\mathsf{#1}}} \newcommand\bs[1]{\boldsymbol{#1}} \shorttitle{Galactic bar resonances with diffusion} \shortauthors{Hamilton et al.} \graphicspath{{./}{figures/}} \begin{document} \title{Galactic bar resonances with diffusion: \\ an analytic model with implications for bar-dark matter halo dynamical friction} \correspondingauthor{Chris Hamilton} \email{chamilton@ias.edu} \author[0000-0002-5861-5687]{Chris Hamilton} \affiliation{Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540} \author[0000-0002-2642-064X]{Elizabeth A. Tolman} \affiliation{Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540} \author[0000-0002-5263-9274]{Lev Arzamasskiy} \affiliation{Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540} \author[0000-0001-8096-7518]{VinГ­cius N. Duarte} \affiliation{Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543} \section{Introduction} \label{sec:Introduction} Galaxies are sculpted by resonances. Corotation, Lindblad and ultraharmonic resonances of rotating non-axisymmetries like bars and spirals are likely responsible for disk heating radial migration and mixing in the Milky Way \citep{Binney1988-zy, Sellwood2002-lv, Minchev2012-ml,Sridhar2019-zo}, for the formation of Solar neighborhood moving groups \citep{Dehnen2000-yz,Hunt2019-qo,Kawata2021-pr}, and for producing the rings and dark gaps observed in external galaxies \citep{Buta1986-hq,Buta2017-we,Krishnarao2022-rh}. The stability (or otherwise) of self-gravitating oscillation modes is dictated by the number of stars in the system that are able to resonate with the mode and hence suck energy out of it \citep{Palmer1987-np,Weinberg1994-xq,Sellwood2014-te,Rozier2020-fc}. Orbital resonances between stars in galactic disks, amplified by collective effects, drive rapid relaxation of the stellar distribution function (DF) \citep{Sellwood2012-ro,Fouvry2015-nk}. The infall of heavy satellites and slowing of bars is likely caused by dynamical friction through the resonant trapping of dark matter particles \citep{Lynden-Bell1972-ve,Tremaine1984-wt,Kaur2018-wp,Banik2021-ug,Chiba2021-cc}. Thus the study of secular evolution of galaxies is largely a study of resonant dynamics: to quote from \citet{Weinberg2007-bv}, `\textit{resonances are not the exception but are required for galaxy evolution!}' The existence of resonant structures in galaxies is an inevitable consequence of the quasiperiodicity of most stellar/dark matter orbits in the mean galactic potential. As such, resonant interactions are often best described in angle-action coordinates. \cite{Binney2012-vp,Binney2016-zh,Binney2018orbital,Binney2020-mw,Binney2020-yg} has pioneered the use of these coordinates to construct analytic equilibrium Galaxy models, and to calculate the distribution function (DF) of resonantly trapped stars. In the latter case one assumes that the galactic potential consists of an axisymmetric mean field plus a rigidly-rotating non-axisymmetric perturbation (see also \citealt{Monari2016-wb,Monari2017-tu,Monari2019-er}) --- an alternative way to capture the same physics is to integrate numerically the orbits of an ensemble of test particles in this prescribed potential (e.g. \citealt{Sellwood2019-xd,Hunt2019-qo}). As a result of these efforts, resonances are now often employed in galactic dynamics as diagnostic tools: reliable dynamical features whose imprints tell us something about the underlying galaxy. For instance, several authors have attempted to infer the size and pattern speed of the Milky Way's bar by fitting test particle models of its resonant imprint to Solar neighbourhood kinematic data \citep{Dehnen2000-yz,Antoja2014-vc,Trick2019-qp,Trick2021-pp}. Even more ambitiously, \cite{Chiba2020-th,Chiba2021-cc} have suggested one might retrace the history of the Galactic bar's pattern speed by identifying a `tree ring' structure in the kinematics of stars that have been trapped in the bar at different times during the Milky Way's history. Again, the analytic and numerical tools for this analysis treated the stars as test particles under the influence of a rigidly rotating perturbation. A crucial element missing from these models is the diffusion of the particles in question due to stochasticity in the potential. We will refer to all models which do not include stochastic effects as \textit{collisionless}. Stochasticity is an inevitable feature of real galaxies, arising due to the gravitational influence of passing stars and molecular clouds, transient spiral structure, dark matter substructure, or whatever \citep{Binney2013-cf} (to say nothing of the bar's fluctuating pattern speed and strength, overlap of resonances from other non-axisymmetric structure, and so on --- see \citealt{Minchev2012-ml,Wu2016-og,Fujii2018-tt}). Stochasticity is also inherent to simulated galaxies, e.g. because of the necessity of representing the dynamics of a very large number $N$ of stars/dark matter particles with a much smaller number of simulated particles, resulting in some level of numerical diffusion \citep{Weinberg2007-bv, Weinberg2007-zo, Sellwood2009-jh}. Regardless of the source of diffusion, the implicit justification for using collisionless theory in the past has been that the libration period of particles trapped in a resonance, despite being much longer than the orbital period $t_\mathrm{cross}$, is still very short compared to the relaxation timescale $t_\mathrm{relax} \sim Nt_\mathrm{cross}$. Yet for processes that depend sensitively on resonant features, the important diffusive timescale is not the relaxation timescale $t_\mathrm{relax}$ but the time to diffuse \textit{across the width of the resonance} $t_\mathrm{diff} \ll t_\mathrm{relax}$. When this is comparable to or smaller than the libration time, conclusions drawn from collisionless theory may need to be revised. The purpose of the present paper is to confront this `collisional' reality in the simplest possible model. To do this we develop a reduced kinetic description for an ensemble of particles in the vicinity of a bar resonance, including diffusion. In our kinetic equation, the secular part of the problem --- namely the smooth particle-bar interaction --- is treated using slow-fast angle-action variables and the pendulum approximation \citep{Chirikov1979-fj,Monari2017-tu}, while the stochastic forcing is crudely lumped into a single diffusion coefficient $D$ which acts only upon the slow actions (and which we do not attempt to calculate self-consistently). This reduced problem then turns out to be mathematically \textit{identical} to a problem previously studied in the theory of tokamak fusion plasma devices (\citealt{BerkPPR1997,Duarte2019-mi}; see also \citealt{Pao1988-cs}). In the tokamak context, stars are replaced with energetic ions, the galactic bar is replaced with a long-lived Alfv\'en wave, and the stochasticity stems not from molecular cloud passages or dark matter substructure but from collisions between the minority energetic ion species and the background thermal ion and electron species. Much like galactic dynamicists, tokamak theorists are interested in how the particle DF is heated on timescales much longer than the particles' characteristic orbital period. What tokamaks and galaxies have in common is an underlying integrable mean-field structure that admits quasiperiodic orbits, and resonant wave-particle interactions that slowly modify those orbits. Several results can, therefore, be pulled from the tokamak literature that have not previously been employed in a stellar-dynamical context. Perhaps the paradigmatic example of secular particle-bar interaction is the calculation of the dynamical friction torque on a bar embedded a spherical dark matter halo \citep{Hernquist1992-mo,Debattista2000-dn,Athanassoula2003-xj}. In a classic example of collisionless theory, \cite{Tremaine1984-wt} analyzed this problem and found that in the time-asymptotic limit the DF of resonantly trapped dark matter particles phase mixes completely, and the resulting symmetry leads to a vanishing torque on the bar (see also \citealt{Chiba2022-qt}). As an application of our theory we revisit the problem here in a more general, collisional framework, and show that diffusion always injects some level of asymmetry into the DF even in steady state, suggesting that the slowing rate of the bar never vanishes. The rest of this paper is organised as follows. In \S\ref{sec:Single_particle} we recap the basic angle-action formalism required to describe the secular dynamics of a single test particle near resonance in a rigidly rotating potential. In \S\ref{sec:KT_resonance} we consider an ensemble of such particles, and write down and solve the kinetic equation that includes both this secular effect and a diffusive term. In \S\ref{sec:Dynamical_friction} we apply what we have learned to the calculation of the dynamical friction felt by a galactic bar through coupling with its host dark matter halo. In \S\ref{sec:Discussion} we comment on the limits of our theory, and discuss our results in relation to the existing literature in both stellar dynamics and plasma physics. We summarise in \S\ref{sec:Summary}. \section{Dynamics of a test particle near resonance} \label{sec:Single_particle} In this section we focus on the dynamical evolution of a test particle orbiting in a smooth galactic potential including a rigidly rotating bar perturbation, ignoring any stochastic effects. We show that near a resonance the motion of the particle through phase space can be effectively reduced to that of a pendulum. This is a classic calculation \citep{Chirikov1979-fj} which has been employed numerous times in galactic dynamics \citep{Tremaine1984-wt,Binney2018orbital,Sridhar2019-zo}, but we repeat the key steps here in order to establish notation. We mostly follow the notational choices of \citet{Chiba2022-qt}. Let the gravitational potential of the galaxy be $\Phi$. We suppose that $\Phi$ can be decomposed into a dominant, time-independent spherical background part $\Phi_0$, and a rigidly rotating bar perturbation $\delta \Phi$. Given the spherical symmetry of $\Phi_0$ it is natural to construct a spherical coordinate system $\bm{x} \equiv (r, \vartheta, \varphi)$ which is fixed in the inertial frame with origin at the center of the Galaxy. We let the midplane of the Galactic disk correspond to $\vartheta = \pi/2$, and let the bar be symmetric with respect to this midplane so that it rotates in the $\varphi$ direction only with constant pattern speed $\pattern$. We have chosen $\Phi_0$ to be spherical in order to facilitate comparison with the dynamical friction calculations of e.g. \citet{Tremaine1984-wt,Banik2021-ug,Chiba2022-qt}, where the test particles in question are dark matter particles and $\Phi_0$ represents the dark matter halo potential, but it would be straightforward to modify our results to other contexts such as to stars orbiting in the Galactic disk \citep{Binney2018orbital}, or stars trapped at orbital resonances in triaxial halo potentials \citep{Yavetz2020-hj}. In practice we will always take our background spherical potential to be a Hernquist sphere \begin{equation} \Phi_0 = -\frac{GM}{r_\mathrm{s}+r}, \label{eqn:Hernquist_potential} \end{equation} with total mass $M=1.5\times 10^{12} \, M_\odot$ and scale radius $r_\mathrm{s}=20$ kpc, and take the bar perturbation to be of the form \begin{eqnarray} &&\delta \Phi(\bm{x}, t) = \Phi_\mathrm{b}(r) \sin^2 \vartheta \cos[2(\varphi - \pattern t)], \label{eqn:bar_explicit} \\ \label{eqn:bar_details} && \Phi_\mathrm{b}(r) = -\frac{Av_\mathrm{c}^2}{2}\left(\frac{r}{r_\mathrm{CR}}\right)^2\left( \frac{b+1}{b+r/r_\mathrm{CR}} \right)^5, \end{eqnarray} where $r_\mathrm{CR} = v_\mathrm{c}/\Omega_\mathrm{p}$ and $A = 0.02, \, b = 0.28,\, v_\mathrm{c} = 238 \, \mathrm{km}\,\mathrm{s}^{-1},\, \Omega_\mathrm{p} = 35 \, \mathrm{km} \, \mathrm{s}^{-1} \, \mathrm{kpc}^{-1} = 35.8 \, \mathrm{Gyr}^{-1}$. However, we will not need to use these explicit formulae until \S\ref{sec:numerical_results}. Now we consider an individual test particle orbiting in the combined potential $\Phi$, and aim to describe its dynamics in the inertial frame. Its Hamiltonian is \begin{equation} H = H_0 + \delta \Phi, \end{equation} where \begin{equation} H_0 = \frac{\bm{v}^2}{2} + \Phi_0(\bm{x}), \end{equation} and $\bm{v}$ is the particle's velocity in the inertial frame. The Hamiltonian $H_0$ is globally integrable, i.e. there exists a set of global angle-action coordinates $(\btheta,\bJ)$ such that $H_0$ is a function of $\bm{J}$ only. In practice we take \citep{Binney2008-ou}: \begin{equation} \btheta = (\theta_r, \theta_\psi, \theta_\varphi), \,\,\,\,\,\,\,\,\,\,\, \bJ = (J_r, L, L_z). \end{equation} The angles $\bm{\theta}$ quantify the phase of oscillations in the radial direction, the phase of oscillations in the azimuthal direction within the particle's orbital plane, and the longitude of ascending node of the particle's orbit (which is fixed for spherical $\Phi_0$) respectively. The corresponding actions $\bJ$ quantify the amplitude of radial excursions, the total angular momentum, and the $z$-component of angular momentum of the particle's orbit. In the limit of a vanishingly weak bar perturbation $\delta \Phi\to 0$, the Hamiltonian $H = H_0$ only depends on $\bJ$; in this case the actions $\bJ$ are perfectly conserved while the angles $\btheta$ evolve linearly as $\bm{\theta}(t) = \bm{\theta}(0) + \bm{\Omega}({\bJ})t$ where \begin{equation} \boldsymbol{\Omega}(\bJ) \equiv \frac{\partial H_0}{\partial \bJ} = (\Omega_r, \, \Omega_\psi, \, \Omega_\varphi), \end{equation} is the vector of orbital frequencies of the unperturbed problem. Since $\varphi$ is fixed for a particle orbiting a spherical potential we always have $\Omega_\varphi = 0$. Canonical Hamiltonian perturbation theory allows us to describe the modification of orbits by the finite bar perturbation $\delta \Phi$ \citep{Binney2008-ou}. However, canonical theory breaks down near orbital resonances, namely locations $\bJ_\mathrm{res}$ in action space such that \begin{equation} \bm{N}\bcdot \bm{\Omega}(\bJ_\mathrm{res}) = N_\varphi \pattern, \label{eqn:resonance_condition} \end{equation} for some vector of integers $\boldsymbol{N} = (N_r, N_\psi, N_\varphi)$. To describe orbits in the vicinity of such resonances it is best to make a canonical transformation to a new set of coordinates, which are again angle-action coordinates of the unperturbed problem (\citealt{Lichtenberg2013-pw,Binney2020-mw}). Precisely, we map $(\btheta, \bJ) \to (\btheta',\bJ')$, where $\btheta' = (\btheta_\mathrm{f}, \theta_{\mathrm{s}})$ consists of the `fast' and `slow' angles \begin{eqnarray} \bm{\theta}_{\mathrm{f}} &=& (\theta_{\mathrm{f}1}, \theta_{\mathrm{f}2}) \equiv \left( \theta_r, \theta_\psi \right), \label{eqn:fast_angles} \\ \theta_s &\equiv& \bm{N}\bcdot \boldsymbol{\theta} - N_\varphi \pattern t, \label{eqn:slow_angle} \end{eqnarray} and $\bJ' = (\bJ_{\mathrm{f}}, J_\mathrm{s})$ consists of the corresponding fast and slow actions \begin{align} \bm{J}_{\mathrm{f}} &= (J_{\mathrm{f}1}, J_{\mathrm{f}2}) \equiv \left( J_r-L_z N_r/N_\varphi, \, L- L_z N_\psi/N_\varphi \right), \label{eqn:fast_actions} \\ J_\mathrm{s} &\equiv L_z/N_\varphi. \label{eqn:slow_action} \end{align} Having made this transformation we may rewrite $H$ in terms of the new coordinates. It takes the form \begin{eqnarray} H(\btheta',\bJ') &=& H_0(\bJ') - N_\varphi \pattern J_\mathrm{s} \nn \\ && + \sum_{\bm{k}\neq \bm{0}} \Psi_{\bm{k}}(\bJ') \exp(i\bm{k}\bcdot \bm{\theta}'), \label{eqn:Hamiltonian_slowfastangles} \end{eqnarray} where $\bm{k} = (k_1, k_2, k_\mathrm{s})$ is a vector of integers and we have expanded $\delta \Phi$ as a Fourier series in the new angles $\btheta'$, i.e. written $\delta \Phi(\bm{x}, t) = \sum_{\bm k} \Psi_{\bm k}(\bJ') \exp(i\bm{k}\bcdot \btheta')$. The coefficients $\Psi_{\bm k}$ are easily computed for the simple bar model \eqref{eqn:bar_explicit} --- see \cite{Tremaine1984-wt} and Appendix B of \cite{Chiba2022-qt}. The special thing about the form (\ref{eqn:Hamiltonian_slowfastangles}) of the Hamiltonian is that it has no explicit time dependence (or rather, the time-dependence has been absorbed into the definition of the angle $\theta_\mathrm{s}$; see equation \eqref{eqn:slow_angle}). The fast angles $\btheta_\mathrm{f}$ evolve on the orbital timescale whereas $\theta_\mathrm{s}$ evolves on the much longer timescale $\sim (\bm{N}\bcdot \bm{\Omega} - N_\varphi \pattern)^{-1}$. Thus we may average $H$ over the unimportant fast motion; the result is \begin{eqnarray} \mathcal{H} &\equiv& \frac{1}{(2\pi)^2}\int \md \btheta_\mathrm{f} \, H(\btheta', \bJ') \nn \\ &=& H_0(J_\mathrm{s}) - N_\varphi \pattern \Js + \sum_{k\neq 0} \Psi_{k}(J_\mathrm{s}) \exp(ik {\theta}_\mathrm{s}), \label{eqn:averaged_H} \end{eqnarray} where we used the shorthand $\Psi_{(0,0,k)} = \Psi_k$, and the dependence on fast actions $\bJ_\mathrm{f}$ is now implicit. Hamilton's equations tell us that $\bJ'$ evolves according to $\md \bJ'/\md t = - \partial \mathcal{H}/\partial \btheta' = (-\partial \mathcal{H}/\partial \theta_\mathrm{s}, 0, 0)$. Thus the fast actions $\bJ_\mathrm{f}$ are constants under the time-averaged perturbation, and so at a fixed $\bJ_\mathrm{f}$ we find that the dynamics reduces to motion in the `slow plane' $(\theta_\mathrm{s}, J_\mathrm{s})$. We can simplify $\mathcal{H}$ further by exploiting the fact that we only care about motion in the vicinity of the resonance (indeed, we already made this restriction implicitly by assuming that $\theta_\mathrm{s}$ evolves much more slowly than $\btheta_\mathrm{f}$). To do this, let the resonant action $\bJ_\mathrm{res}$ from equation (\ref{eqn:resonance_condition}) transform to to $\bJ'_\mathrm{res} = (\bJ_\mathrm{f}, J_\mathrm{s, res})$, and define \begin{equation} I \equiv J_\mathrm{s} - J_{\mathrm{s, res}} , \,\,\,\,\,\,\,\,\,\,\, \phi_k = \theta_\mathrm{s} + \frac{\arg \Psi_k}{k}, \label{eqn:I_phi_definitions} \end{equation} and let $\phi_k \in (-\pi, \pi)$. Then Taylor expanding the right hand side of \eqref{eqn:averaged_H} for small $I$ and discarding constants and unimportant higher order terms, we find\footnote{For a discussion of the accuracy of the approximations made in deriving equation \eqref{eqn:averaged_H_expanded}, and the leading-order corrections to it, see \cite{Kaasalainen1994-ks}, \cite{Binney2018orbital, Binney2020-yg}.} \begin{eqnarray} \mathcal{H} &\approx& \frac{1}{2} H_0''(J_\mathrm{s, res})I^2 + 2 \sum_{k > 0} \vert \Psi_k(J_\mathrm{s,res}) \vert \cos (k \phi_k). \label{eqn:averaged_H_expanded} \end{eqnarray} Normally it is the case that the sum over $k$ in equation \eqref{eqn:averaged_H_expanded} is dominated by a single, small value of $k$. Here we will focus on resonances $\bm{N}$ with $N_\varphi = 2$, since these give the dominant contributions to the dynamical friction on galactic bars, although the formalism we develop can be applied to other resonances with minimal adjustments. For $N_\varphi=2$ the only contribution to equation \eqref{eqn:averaged_H_expanded} is from $k=1$ (\citealt{Chiba2022-qt}, Appendix B). The resulting simplified $\mathcal{H}$ therefore takes the pendulum form \citep{Chirikov1979-fj,Lichtenberg2013-pw}: \begin{equation} \mathcal{H} = \frac{1}{2}GI^2 - F\cos \phi, \label{eqn:resonant_Hamiltonian} \end{equation} where $\phi \equiv \phi_1$ and \begin{equation} G \equiv \frac{\partial^2 H_0}{\partial J_\mathrm{s}^2}\Bigg\vert_{J_{\mathrm{s,res}}}, \,\,\,\, \,\,\,\,\,\, \,\,\,\, F = -2\vert \Psi_1(J_{\mathrm{s,res}})\vert . \label{eqn:F_and_G} \end{equation} In the cases of interest to us $G<0$, and of course $F<0$ always. An explicit expression for $-F$ is given in equation (B9) of \citet{Chiba2022-qt}. The variables $(\phi, I)$ are canonical variables for the pendulum Hamiltonian \eqref{eqn:resonant_Hamiltonian}. Their equations of motion are \begin{equation} \frac{\md \phi}{\md t} = \frac{\partial \mathcal{H}}{\partial I} = GI,\,\,\,\,\,\,\,\, \,\,\,\,\, \frac{\md I}{\md t} = -\frac{\partial \mathcal{H}}{\partial \phi} = -F\sin \phi, \end{equation} which are the equations of a simple pendulum. The pendulum moves at constant `energy' $\mathcal{H}$, either on an untrapped `circulating' orbit with $\mathcal{H} > F$ (so that $\phi$ periodically takes all values in $(-\pi, \pi)$), or on a trapped `librating' orbit with $\mathcal{H} < F$ (so that the pendulum oscillates around $(\phi, I) = (0,0)$). The separatrix between the two orbit families has the equation $\mathcal{H} = F$. The timescale for infinitesimally small oscillations around $(\phi, I) = (0,0)$ --- the so-called \textit{libration time} --- is given by \begin{equation} t_\mathrm{lib} \equiv \frac{2\pi}{\sqrt{FG}}. \label{eqn:t_libration} \end{equation} The maximum width in $I$ of the librating `island' is at $\phi = 0$, where it spans $I\in (-I_\mathrm{h}, I_\mathrm{h})$ and $I_\mathrm{h}$ is the \textit{island half-width}: \begin{equation} I_\mathrm{h} \equiv 2\sqrt{\frac{F}{G}}. \label{eqn:half_width} \end{equation} Equations (\ref{eqn:t_libration}) and (\ref{eqn:half_width}) are the fundamental scales involved in the dynamics of test particles governed by the pendulum Hamiltonian (\ref{eqn:resonant_Hamiltonian}). \section{Kinetic theory near a resonance} \label{sec:KT_resonance} In \S\ref{sec:Single_particle} we learned how to describe the dynamical evolution of a single resonant test particle in the smooth, rigidly rotating potential $\Phi_0+\delta \Phi$. In this section we will add a stochastic forcing to the dynamics. The motion of a single particle is no longer deterministic in this case, so we must turn to a statistical description of an ensemble of such test particles. We do this using a kinetic equation. We wish to understand how an ensemble of particles behaves near a particular resonance $\bm{N}$ associated with the slow action $J_{s,\mathrm{res}}$. Let us therefore consider particles that all share the same fast actions $\bJ_\mathrm{f}$, but may differ in their values of fast angles $\btheta_\mathrm{f}$, slow angle $\theta_\mathrm{s}$ and slow action $\Js$. Averaging the ensemble over the fast angles, we can describe the resulting density of particles in the $(\phi, I)$ plane (equation (\ref{eqn:I_phi_definitions})) using a smooth distribution function (DF) which we call $f(\phi, I, t)$. The number of particles in the phase space area element $\md \phi \, \md I$ surrounding $(\phi, I)$ at time $t$ is then proportional to $\md \phi\, \md I f(\phi, I, t)$. The kinetic equation governing $f$ is \begin{equation} \frac{\partial f}{\partial t} + \{ f, \mathcal{H}\} = C[f], \label{eqn:kinetic_equation_general} \end{equation} where $\{\cdot,\cdot\}$ is a Poisson bracket encoding the smooth advection in the $(\phi, I)$ plane: \begin{equation} \{ f,\mathcal{H}\} = \frac{\partial f}{\partial \phi} \frac{\partial \mathcal{H}}{\partial I} - \frac{\partial f}{\partial I} \frac{\partial \mathcal{H}}{\partial \phi}, \label{eqn:Poisson_bracket} \end{equation} and $C[f]$ is a `collision operator' encoding the effect of stochastic fluctuations in the potential. For $\mathcal{H}$ we use the pendulum approximation (\ref{eqn:resonant_Hamiltonian}), while for $C[f]$ we take a very simple diffusive form with constant diffusion coefficient $D$: \begin{equation} C[f] = D \frac{\partial^2 f}{\partial I^2}. \label{eqn:collision_operator} \end{equation} The diffusion coefficient $D$ can be estimated for a given theoretical model of e.g. scattering by passing stars and molecular clouds \citep{Binney1988-zy,Jenkins1990-qs,S_De_Simone2004-gr}, dark matter substructure \citep{Penarrubia2019-su, El-Zant2020-oc, Bar-Or2019-ge}, spurious numerical heating in simulations \citep{Weinberg2007-bv,Ludlow2021-wk}, or can potentially be calibrated from data \citep{Ting2019-la,Frankel2020-vy} --- see \S\ref{sec:Delta}. We emphasize that we make no attempt at self-consistency here, as we simply impose a diffusion coefficient by hand, rather than calculating it from $f$. Similarly, we do not account at any stage for the self-gravity of the perturbed DF \citep{Weinberg1985-en,Dootson2022-cu}. Putting equations \eqref{eqn:resonant_Hamiltonian} and \eqref{eqn:kinetic_equation_general}--\eqref{eqn:collision_operator} together, we have: \begin{equation} \frac{\partial f}{\partial t} + GI\frac{\partial f}{\partial \phi} - F\sin \phi\frac{\partial f}{\partial I}= D\frac{\partial^2 f}{\partial I^2}. \label{eqn:kinetic_equation} \end{equation} In writing down the kinetic equation \eqref{eqn:kinetic_equation} we have made a few key assumptions: \begin{itemize} \item We have ignored any diffusion in $\phi$. \item We have assumed that the diffusion coefficient $D$ is a constant, independent of $\phi$ or $I$. \item We have assumed that it is legitimate to consider an ensemble at fixed fast action $\bJ_\mathrm{f}$, which implicitly ignores any diffusion in $\bJ_\mathrm{f}$. \end{itemize} We can justify the first two assumptions heuristically on the grounds that the resonant island is \textit{narrow}, i.e. the extent of the island in $I$ is usually very small compared to the whole of slow action space, whereas it spans the whole of slow angle $(\phi)$ space.\footnote{This `narrow resonance approximation' is standard in tokamak plasma theory \citep{Su1968-ga,Callen, Duarte2019-mi,Tolman2021-ly}. % } As a result, the features in the DF created by the resonance are going to be much sharper in $I$ than in $\phi$, so a term like $\partial^2 f/\partial \phi^2$ in the collision operator is expected to be sub-dominant. (We explore the addition of angle-scattering terms in the kinetic equation further in Appendix \ref{sec:angle_scattering}.) Also, the narrowness of the island means we are only interested in the dynamics over a small portion of action space, so $D$ ought not to vary too much across the domain of interest and can be estimated locally. The third assumption, namely that of no diffusion in $\bJ_\mathrm{f}$, is difficult to justify in the general case --- there is a priori no reason to assume that stochastic kicks in $\bJ_\mathrm{f}$ will be small compared to those in $J_\mathrm{s}$, and such kicks will couple different $\bJ_\mathrm{f}$ in a proper kinetic equation via terms like $\sim \partial^2 f/\partial J_{\mathrm{f}1}^2$. Nevertheless we will stick with this assumption because it allows a great simplification of the kinetic theory, and the basic results and intuition we gain from solving the simplified theory will be invaluable when attacking more realistic problems. We discuss this assumption further in \S\ref{sec:Discussion}. Before attempting to solve the kinetic equation \eqref{eqn:kinetic_equation}, one more simplification is in order. Naively, equation \eqref{eqn:kinetic_equation} seems to depend on three parameters: $G, F$ and $D$. However, we can reduce this to one effective parameter by introducing certain dimensionless variables. First we note that the typical timescale for a particle to diffuse all the way across the resonance (a distance $\sim 2I_\mathrm{h}$ in action space) is the \textit{diffusion time}, \begin{equation} t_\mathrm{diff} \equiv \frac{(2I_\mathrm{h})^2}{2D}= \frac{8F}{GD}. \label{eqn:diffusion_time} \end{equation} Now let us define the dimensionless variables \begin{eqnarray} \tau &\equiv& \sqrt{GF}t = \frac{2\pi \,t}{t_\mathrm{lib}}, \label{eqn:dimensionless_time} \\ j &\equiv& \sqrt{\frac{G}{F}}I = \frac{2I}{I_\mathrm{h}}, \label{eqn:dimensionless_action} \\ \Delta &\equiv& \sqrt{\frac{G}{F^3}}D = \frac{4\, t_\mathrm{lib}}{\pi\, t_\mathrm{diff}}, \label{eqn:dimensionless_diffusion} \end{eqnarray} where the libration time $t_\mathrm{lib}$ is given in equation \eqref{eqn:t_libration} and the island half-width $I_\mathrm{h}$ is defined in equation \eqref{eqn:half_width}. Clearly, $\tau$ is a dimensionless measure of time normalized by the libration time, $j$ is a dimensionless measure of the slow action variable relative to the resonant island width, and $\Delta$ is the \textit{diffusion strength}, i.e. the ratio of libration timescale to diffusion timescale. Treating $f$ as a function of these variables, i.e. writing\footnote{Note that $f$ still depends parametrically on $\bm{N}$ and $\bJ_\mathrm{f}$, though we are suppressing the explicit dependence here to keep the notation clean.} $f(\phi, j,\tau)$, equation (\ref{eqn:kinetic_equation}) becomes: \begin{equation} \frac{\partial f}{\partial \tau} + j\frac{\partial f}{\partial \phi} - \sin \phi\frac{\partial f}{\partial j}= \Delta \frac{\partial^2 f}{\partial j^2}. \label{eqn:kinetic_equation_dimensionless} \end{equation} We see that in these variables the kinetic equation depends on the single dimensionless parameter $\Delta$, which we discuss next. \subsection{The diffusion strength $\Delta$} \label{sec:Delta} The dimensionless diffusion strength $\Delta \approx t_\mathrm{lib}/t_\mathrm{diff}$ (equation \eqref{eqn:dimensionless_diffusion}) plays a central role in this work, since it measures the importance of diffusion relative to that of the bar perturbation near resonance. The regime $\Delta \gg 1$ corresponds to very strong diffusion, whereas $\Delta \ll 1$ corresponds to very weak diffusion; the `collisionless limit' favored by many classic calculations corresponds to $\Delta =0$ (see \S\ref{sec:Introduction}). A key aim of this paper is to understand how the behavior of a DF near a resonance depends on $\Delta$, and how this might affect galactic phenomena like bar-halo friction. As we will see, even for $\Delta$ values as small as $\sim 0.1$ the behavior predicted by the kinetic equation \eqref{eqn:kinetic_equation_dimensionless} can be significantly different from the collisionless limit. We can estimate $\Delta$ heuristically by noting that $t_\mathrm{diff}$ is related to the relaxation time $t_\mathrm{relax}$, which is the time required for diffusive processes to change a particle's action by order of itself \citep{Binney2008-ou,Fouvry2018-gi}. If we assume the diffusion coefficient $D$ is roughly constant over the whole of action space (which is a gross oversimplification but will suffice for an order-of-magnitude estimate), then from equation \eqref{eqn:diffusion_time} we have: \begin{eqnarray} \frac{t_\mathrm{diff}}{t_\mathrm{relax}} \sim \left(\frac{I_\mathrm{h}}{J_\mathrm{s, res}}\right)^2 \sim \bigg\vert \frac{\Psi_1}{H_0} \bigg\vert \sim A \ll 1. \label{eqn:ratio_diff_relax} \end{eqnarray} Here $A$ is the dimensionless strength of the bar (equation \eqref{eqn:bar_details}); for the Milky Way, $A \sim 0.02$ \citep{Chiba2020-th}. Putting equations (\ref{eqn:dimensionless_diffusion}) and (\ref{eqn:ratio_diff_relax}) together, we estimate \begin{eqnarray} \Delta &\sim& \frac{t_\mathrm{lib}}{A t_\mathrm{relax}} \label{eqn:Delta_approximate} \\ &\sim& 10 \times \left( \frac{A}{0.02} \right)^{-1} \left( \frac{t_\mathrm{lib}}{2\, \mathrm{Gyr}} \right) \left( \frac{t_\mathrm{relax}}{10\, \mathrm{Gyr}} \right)^{-1}. \nonumber \end{eqnarray} Typically, important Galactic bar resonances like the corotation resonance have libration times of $t_\mathrm{lib} \sim 1-2$ Gyr or even longer (see \S\ref{sec:Dynamical_friction} for examples; of course $t_\mathrm{lib}$ also depends on $A$, albeit weakly, as $t_\mathrm{lib} \propto A^{-1/2}$). The relaxation time $t_\mathrm{relax}$ depends on the context in which the kinetic equation is being applied. Let us give some examples. If we consider a virialized, isolated dark matter halo of enclosed mass $M(r)$ made up of particles of mass $m$ and velocity dispersion $\sigma(r)$, where $r$ is the distance to the center of the halo, then the relaxation time for the dark matter particles is \citep{Binney2008-ou}: \begin{equation} t_\mathrm{relax} \sim \frac{\sigma^3 r^3}{G^2 m M} \sim \frac{M}{m} t_\mathrm{cross}, \label{eqn:2body} \end{equation} where we used $\sigma^2 \sim GM/r$ and introduced the typical crossing time $t_\mathrm{cross} \sim \sigma/R$. For standard cold dark matter models $M/m$ is enormous, meaning $t_\mathrm{relax}$ is hugely longer than the age of the Universe which results in a negligible $\Delta \lll 1$, so collisionless theory is valid. However, $t_\mathrm{relax}$ turns out to be much smaller in some alternative dark matter models. For instance in fuzzy dark matter theory, particles with true masses $m \sim 10^{-22}$ eV behave collectively as quasiparticles of size $\sim 1$ kpc and effective mass $m_\mathrm{eff} \gtrsim 10^7 M_\odot$ (see \citealt{Hui2017-rw}, equation (38)). As a result one finds (\citealt{Hui2017-rw}, equation (32)): \begin{eqnarray} t_\mathrm{relax} &\sim& \, 10\, \mathrm{Gyr} \times \left( \frac{\sigma}{100\, \mathrm{km} \, \mathrm{s}^{-1}} \right)^{2} \nn \\ && \times \left( \frac{r}{5\, \mathrm{kpc} }\right)^{4} \left( \frac{m}{10^{-22}\, \mathrm{eV}/c^2} \right)^{2}. \end{eqnarray} Thus even at $r\sim 10$ kpc one can find a $\Delta$ value $\sim 0.1$. Alternatively, one can consider $f$ to be the DF not of a real halo but of a simulated halo; one can then replace $M/m$ in equation (\ref{eqn:2body}) with the number of particles being used to represent that halo in an $N$-body simulation. This gives the effective $\Delta$ resulting from numerical noise alone \citep{Ludlow2021-wk}. As another example, we might let our DF $f$ describe not dark matter particles, but \textit{stars} trapped near bar resonances in the Milky Way disk \citep{Monari2017-tu,Trick2019-qp, Dootson2022-cu}. In this case one can set $t_\mathrm{relax}$ equal to the typical diffusion timescale for the orbital actions of disk stars, as measured from kinematic and chemical data from GAIA and APOGEE (i.e. the `age-velocity dispersion relation', see \citealt{Mackereth2019-cq,Frankel2020-vy}). Such measurements suggest that $t_\mathrm{relax}$ is on the order of the age of the Galaxy; it follows from equation \eqref{eqn:Delta_approximate} that typical $\Delta$ values are $\gtrsim 1$. Of course, such estimates ignore the fact that the bar resonances themselves likely have some influence on the diffusion of stars, particularly when they overlap with other bar resonances and/or resonances associated with spiral waves \citep{Minchev2010-la}, meaning it may be unrealistic to decouple secular and stochastic effects like we have proposed here. We assume throughout this paper that resonances are isolated, so we can make no quantitative statement here about the effect of resonance overlap. We leave this as an avenue for careful future study. \subsection{Solution of the kinetic equation} \label{sec:solution} Unfortunately, the kinetic equation (\ref{eqn:kinetic_equation_dimensionless}) cannot be solved analytically in general. In this section we present time-dependent numerical solutions to (\ref{eqn:kinetic_equation_dimensionless}) for a simple choice of initial DF and for various values of $\Delta$. In particular we find that the solution always tends towards some steady-state form. There are two interesting classes into which the steady-state solutions fall --- for $\Delta \ll 1$ the steady DF is dominated by the resonant island structure, while for $\Delta \gg 1$ it is dominated by diffusion and the resonance leaves little trace. In fact, these limiting cases have already been investigated analytically in the context of wave-particle interactions in plasma \citep{Pao1988-cs,Berk1998-hv,Duarte2019-mi}. Inspired by these works, in \S\ref{sec:weak_scattering} and \S\ref{sec:strong_scattering} we derive some analytical solutions to (\ref{eqn:kinetic_equation_dimensionless}) in the $\Delta \ll 1$ and $\Delta \gg 1$ limits respectively. We will refer to those solutions throughout this section in order to illuminate our numerical results. Let the initial global distribution of particles be a function of unperturbed actions $\bJ$ only, so by Jeans' theorem the initial `background' DF $f_0(\bJ)$ --- normalized so that $\int \md \btheta \md \bJ \, f = 1$ --- is a steady-state solution of the unperturbed collisionless Boltzmann equation governed by $H_0(\bJ)$ \citep{Binney2008-ou}. Since the canonical transformation $(\btheta, \bJ) \to (\btheta',\bJ')$ does not mix up angles and actions (equations \eqref{eqn:slow_angle}--\eqref{eqn:fast_actions}), it follows that when written in the new coordinates $f_0$ will just be a function of $\bJ'$. In the language of the dimensionless variables we used to write down equation \eqref{eqn:kinetic_equation_dimensionless}, at fixed $\bm{J}_\mathrm{f}$ the initial DF in the $(\phi, j)$ plane will depend only on $j$, so $f_0 = f_0(j)$. For simplicity, we will only focus on DFs which are initially \textit{linear} in $j$. This is well-motivated provided the resonance is narrow (\S\ref{sec:KT_resonance}) since any initial DF which depends only on $j$ can be Taylor-expanded around the resonance at $j=0$ to give a locally linear form. Thus we take as our initial condition \begin{equation} f(\phi, j, \tau=0) = f_0(j=0) + \alpha j \equiv f_\mathrm{init}(j), \label{eqn:linear_DF} \end{equation} where \begin{equation} \alpha \equiv \frac{\partial f_0}{\partial j}\bigg\vert_{j=0} % . \end{equation} Of course the linear approximation (\ref{eqn:linear_DF}) should be corrected in detailed modelling \citep{Pao1988-cs} but it is sufficient to elucidate the main physics of equation (\ref{eqn:kinetic_equation_dimensionless}) \citep{White2019-nl}. At $t=0$ we switch on the bar potential, and the resulting evolution is described by equation (\ref{eqn:kinetic_equation_dimensionless}). Of course this is not realistic --- true galactic bars do not just appear out of nowhere but instead grow gradually in strength in the first few Gyr of the galaxy's lifetime. However, the qualitative results one finds by growing the bar slowly are not too different from imposing it instantaneously at $t=0$ \citep{Chiba2022-qt}\footnote{In fact \citet{Chiba2022-qt} show that, if anything, in the collisionless regime the DF exhibits even sharper features when the bar is grown slowly than when it is imposed instantaneously. Since diffusion feeds off sharp features in the DF (equation (\ref{eqn:collision_operator})), diffusive effects may be even more important in this case.}. Hence, for simplicitly we will consider only an instantaneously emerging bar here. We solve \eqref{eqn:kinetic_equation_dimensionless} numerically with the initial condition \eqref{eqn:linear_DF} and the boundary condition that $f \to f_\mathrm{init}$ for $j\to \pm \infty$ at all times. We can most easily examine the nature of the resulting solutions by defining a dimensionless auxiliary DF: \begin{equation} g(\phi, j, \tau) \equiv \frac{f(\phi, j, \tau) - f_0(0)}{\alpha}, \label{eqn:dimensionless_DF} \end{equation} which measures the modification of $f$ compared to its background value on resonance $f_0(0)$; it is obvious from \eqref{eqn:linear_DF} that $g(\phi, j, \tau=0) \equiv g_\mathrm{init}(j) = j$. Figure \ref{fig:dimensionless_DF} shows colored contours of this initial distribution, overlaid by black contours of the Hamiltonian $\mathcal{H}$ (equation \eqref{eqn:resonant_Hamiltonian}). The separatrix between librating and circulating orbit families is shown with a dashed black line. In Figure \ref{fig:slices} we plot contours of the numerical solution $g(\phi, j, \tau)$ for $\Delta = 0.001,\, 0.1, \,1,\, 10$ in rows (a)--(d) respectively. In each row, from left to right we plot the solution at different `times' $\Delta^{1/3} \tau$. First consider row (a), which is in the limit of very weak diffusion ($\Delta =0.001$). Physically, we expect that in this limit the DF will \textit{phase mix} around and within the resonant island. The reason for phase mixing is that in the absence of diffusion ($\Delta = 0$), the trajectories of individual particles trace contours of constant $\mathcal{H}$ in the $(\phi, j)$ plane; and since adjacent contours correspond to slightly different libration/circulation periods, the initial DF gets sheared out along these contours, reaching an approximate steady state after several libration times $t_\mathrm{lib}$. We see that by the third column $g\approx 0$ inside the separatrix and $g$ is smeared almost evenly on contours of constant $\mathcal{H}$ outside of the separatrix. Now consider panel (d), which corresponds to the opposite limit of very strong diffusion, represented here by $\Delta = 10$. In this case the resonance has little effect on $g$ at any time. This again is as expected since the initial linear DF (\ref{eqn:linear_DF}) is annihilated by the collision operator $C[f]$ (equation (\ref{eqn:collision_operator})). In other words, wherever the bar perturbation induces some curvature in the DF, strong diffusion immediately acts to remove it and restore the linear initial condition. The cases $\Delta = 0.1$ and $\Delta =1$ (rows (b) and (c)) are intermediate between these two extremes. In the following subsections we unpick further the different features of these solutions and explain how their behavior depends on $\Delta$. A reader who is satisfied with the basic picture shown in Figure \ref{fig:slices} can skip these details, and go directly to the more astrophysically interesting \S\ref{sec:Dynamical_friction}. \subsubsection{Skew-symmetry} \label{sec:skew_symmetry} In every panel of Figure \ref{fig:slices} we notice the following skew-symmetry property: \begin{equation} g(\phi, j, \tau) = -g(-\phi, -j, \tau). \label{eqn:skew} \end{equation} This follows easily from the symmetry of equation \eqref{eqn:kinetic_equation_dimensionless} under the replacements $\phi \to -\phi$ and $j \to -j$, and the corresponding antisymmetry of the initial condition $g_\mathrm{init}(j) = j$. \subsubsection{Steady state} \label{sec:steady_state} In each row (a)--(d) of Figure \ref{fig:slices} we find that by the third column the solution has approximately reached its steady state, which reach reflects a balance between the secular bar torque (which wants to churn the DF around the island) and diffusion (which wants to restore the linear initial condition). The fact that a steady state is possible when diffusion is constantly injecting energy into the system is due to the boundary condition of a unperturbed DF at $\pm \infty$, which act as a particle source/sink. How long does it take to reach the steady state? For $\Delta \ll 1$, where the bar resonance dominates, the typical steady-state timescale is a few libration periods, say \begin{equation} t_\mathrm{steady} \sim 3 \, t_\mathrm{lib} \sim 3 \Delta \, t_\mathrm{diff}, \,\,\,\,\,\,\,\,\,\,\,\, (\Delta \ll 1) \label{sec:steady_state_timescale_weak} \end{equation} which corresponds to $\Delta^{1/3} \tau_\mathrm{steady} \lesssim 20$ --- see equation (\ref{eqn:dimensionless_time}). Thus in this weakly-diffusive limit we typically have $t_\mathrm{lib} \ll t_\mathrm{steady} \ll t_\mathrm{diff}$. Meanwhile in the diffusion-dominated regime $\Delta \gg 1$, we show in Appendix \ref{sec:strong_scattering} that \begin{equation} t_\mathrm{steady} \sim \frac{t_\mathrm{lib}}{\Delta^{1/3}} \sim \Delta^{2/3} t_\mathrm{diff},\,\,\,\,\,\,\,\,\,\,\,\, (\Delta \gg 1) \label{sec:steady_state_timescale_strong} \end{equation} i.e. $\Delta^{1/3} \tau_\mathrm{steady} \sim 2$. Hence in this case we have the hierarchy $t_\mathrm{diff} \ll t_\mathrm{steady} \ll t_\mathrm{lib}$. In general, for a given bar strength, stronger diffusion always leads to a more rapidly achieved steady state. \subsubsection{Angle-averaged distribution} \label{sec:angle_averaged} It is instructive to average the solution $f(\phi, j, \tau)$ over the slow angle $\phi$ and investigate the resulting DF \begin{equation} \langle f \rangle_\phi \equiv \frac{1}{2\pi} \int_{-\pi}^\pi \md \phi \, f(\phi, j, \tau). \end{equation} In Figure \ref{fig:phi_averaged_g} we plot the corresponding auxiliary DF (see equation (\ref{eqn:dimensionless_DF})): \begin{equation} \langle g - g_\mathrm{init} \rangle_\phi= \frac{\langle f \rangle_\phi - f_0(0) }{\alpha} - j, \label{eqn:phi_averaged_g} \end{equation} as a function of $j$ for the same $\Delta$ values as in Figure \ref{fig:slices}. In panels (a)--(c) we show this quantity for times $t/t_\mathrm{lib} = 0.25, 1, 40$ respectively; in particular, we know from \S\ref{sec:steady_state} that panel (c) always corresponds to the steady state. The vertical dotted lines in these panels are at $j=\pm 2$, which marks the maximum extent of the separatrix in the $(\phi, j)$ plane (Figures \ref{fig:dimensionless_DF}, \ref{fig:slices}). We notice immediately that $\langle g - g_\mathrm{init} \rangle_\phi$ is an odd function of $j$, which follows from the skew-symmetry (\ref{eqn:skew}), and that in the steady-state (panel (c)) it exhibits a single maximum and a single minimum, before decaying to zero at $j\to\pm \infty$. For very small $\Delta$ the amplitude of these extrema actually grows with $\Delta$, peaking around $\Delta \sim 0.01$, before decaying as $\Delta$ is increased further. Furthermore, in the very small $\Delta$ limit the location of the extrema is $\vert j\vert \approx 1.5$, which is inside the maximum extent of the separatrix. Increasing $\Delta$ broadens the distribution and shifts the locations of the extrema to larger $\vert j\vert$. For instance, for $\Delta = 1$ the amplitude of the $\langle g - g_\mathrm{init} \rangle_\phi$ curve is around half of what it was for $\Delta = 0.001$ while the location of its extrema has shifted outside the separatrix to $\vert j \vert \approx 2.5$. In the strong-diffusion regime $\Delta \gg 1$ the distribution becomes very broad and its amplitude decreases dramatically. In Figure \ref{fig:Max_Values}a we plot the maximum value of $\langle g - g_\mathrm{init} \rangle_\phi$ in steady state as a function of $\Delta$. We see that this quantity is roughly constant at small $\Delta \ll 1$, and of order unity in amplitude. For large $\Delta \gg 1$ it decays like $\propto \Delta^{-1}$, which can also be shown analytically (\citealt{Duarte2019-mi}, equation (13)). It makes physical sense that the maximum value of $\langle g - g_\mathrm{init}\rangle_\phi \to 0$ as $\Delta \to \infty$, since for infinitely strong diffusion the DF should never evolve from its linear initial condition. \subsubsection{Asymmetry in slow angle} \label{sec:asymmetry} Finally, when computing the dynamical friction torque on the bar in \S\ref{sec:Dynamical_friction} it will turn out that the key quantity we must extract from our kinetic theory is Im $f_1(j, \tau)$ where $f_1$ is the first Fourier coefficient of the DF: \begin{equation} f_1(j, \tau) \equiv \frac{1}{2\pi}\int_{-\pi}^{\pi} \md \phi \, f(\phi, j, \tau) \exp(-i\phi). \label{eqn:f1} \end{equation} This quantity is obviously a measure of the asymmetry of the (slow-)angular distribution of particles; when there is no such asymmetry, there can be no frictional torque \citep{Tremaine1984-wt}. In Figure \ref{fig:Im_f1} we plot the dimensionless quantity $\mathrm{Im} \, g_1 = \alpha^{-1} \mathrm{Im} \, f_1$ as a function of $j$ in the steady state $\tau\to \infty$, for various $\Delta$. We see from Figure \ref{fig:Im_f1} that $\mathrm{Im} \, g_1$ is always even in $j$, which follows from the skew-symmetry (\ref{eqn:skew}). Its amplitude is very small in the weak-diffusion regime $\Delta \ll 1$, reflecting the (approximate) symmetry in $\phi$ of the (nearly) phase-mixed steady-state --- see Figure \ref{fig:slices}a,b. Indeed we know that in the perfectly collisionless regime \begin{equation} \mathrm{Im}\, g_1(j, \tau \to \infty) =0, \,\,\,\,\,\,\,\,\,\,\,\, (\Delta =0) \label{eqn:Img1_TW84} \end{equation} However, once the diffusion strength $\Delta$ is increased beyond $\sim 0.1$ the peak value of $\mathrm{Im}\,g_1$ grows significantly, then begins to decay with $\Delta$ for $\Delta \gg 1$. The dashed curves we show for $\Delta \geq 1$ in this plot correspond to the approximate analytic solution valid for $\Delta \gg 1$ derived in \S\ref{sec:strong_scattering}, namely \begin{equation} \mathrm{Im}\, g_1(j, \tau \to \infty) \approx \frac{\pi}{2} \mathcal{R}_\Delta(j), \,\,\,\,\,\,\,\,\,\,\,\, (\Delta \gg 1) \label{eqn:Img1} \end{equation} where \begin{equation} \mathcal{R}_\Delta(j) \equiv \frac{1}{\pi \Delta^{1/3}} \int_0^\infty \md y \, \exp\left( -\frac{y^3}{3}\right)\cos \left( \frac{j y}{\Delta^{1/3}}\right). \label{eqn:resonance_function} \end{equation} It is not hard to show that $\mathcal{R}(j)$ has the properties of a collisionally-broadened resonance function \citep{Duarte2019-mi}, namely $\int^\infty_{-\infty} \md j \, \mathcal{R}_\Delta(j) =1 $ and $\lim_{\Delta \to 0} \mathcal{R}_\Delta(j) = \delta(j)$. We see from Figure \ref{fig:Im_f1} that the approximation (\ref{eqn:Img1}) is accurate to within several percent or better for $\Delta \gtrsim 4$. It is also apparent from Figure \ref{fig:Im_f1} that the width of the $\mathrm{Im}\, g_1(j, \tau\to \infty)$ curve increases monotonically with $\Delta$. Indeed, in the $\Delta \gg 1$ limit we know from equations (\ref{eqn:Img1})-(\ref{eqn:resonance_function}) that this width grows like $\propto \Delta^{1/3}$, and that the total area under the $\mathrm{Im}\, g_1(j, \tau \to \infty)$ curve is constant. The reason for this lack of $\Delta$-dependence at large $\Delta \gg 1$ is that strong diffusion essentially renders the bar-halo interaction linear (see \S\ref{sec:implications}). Unlike in the nonlinear phase, the linear dynamics is insensitive to time delay effects that arise from successive integrations of the kinetic equation within a perturbative approach \citep{Berk1996-wp}. Lastly, in Figure \ref{fig:Max_Values}b we extract the steady-state value of $\mathrm{max}( \mathrm{Im}\, g_1)$ and plot it as a function of $\Delta$. For small $\Delta \ll 1$ we have roughly $\mathrm{max}( \mathrm{Im} \, g_1) \propto \Delta^{4/5}$. This is an empirical scaling which is hard to explain mathematically (see \S\ref{sec:weak_scattering}). Basically, stems from the fact that weak diffusion `fills in' the DF near the inner edge of the separatrix, where we would have $g=0$ were there no diffusion at all; The amplitude of $g$ in the filled-in `strip' is $\mathcal{O}(1)$, while the strip's thickness grows monotonically with $\Delta$. For large $\Delta \gg 1$ we see that $\mathrm{max}( \mathrm{Im} \, g_1) \propto \Delta^{-1/3}$, which agrees with (\ref{eqn:Img1})--(\ref{eqn:resonance_function}). The transition between these two regimes occurs around $\Delta \sim 2$, where $\mathrm{max}( \mathrm{Im} \, g_1) \sim 1$. \section{Bar-dark matter halo dynamical friction} \label{sec:Dynamical_friction} As a galactic bar ages, it transfers angular momentum to its host dark matter halo and consequently its rotation rate slows. The mechanism responsible is dynamical friction: the bar produces a perturbation in the dark matter distribution function $f$ and hence in the dark matter density, which then back-reacts to produce a torque $\mathcal{T}$ on the bar, draining its angular momentum. Problems of bar-halo coupling --- and more generally the problem of angular momentum transfer between a massive perturber and a distribution of particles --- have been the focus of many classic studies in galactic dynamics (e.g. \citealt{Lynden-Bell1972-ve,Tremaine1984-wt,Weinberg1985-en,Debattista1998-nv,Athanassoula2003-xj}) and continue to inspire modern research \citep{Kaur2018-wp,Chiba2020-th,Collier2021-nf,Banik2021-ug,Lieb2021-tv,Chiba2022-qt,Kaur2022-tb,Dootson2022-cu}. They are also strongly analogous to wave-particle interaction problems in plasma (see the Discussion). One can write down a rather general formula for the dynamical friction torque $\mathcal{T}$ on the bar as follows. Like we have throughout this paper, let the host dark matter halo be spherical and let the bar rotate around the $z$-axis and have potential $\delta \Phi(\bm{x},t)$. From Hamilton's equation the $z$-component of the specific torque felt by an individual dark matter particle due to the bar is \begin{equation} \frac{\md L_z}{\md t} = -\frac{\partial \delta \Phi}{\partial \theta_\varphi}. \end{equation} Let the DF of dark matter particles be $f$. Then by Newton's third law and using $\md \bm{x} \md \bm{v} = \md \btheta \md \bJ$, the total torque on the bar (divided by the mass of the halo) due to the dark matter particles is equal to \begin{equation} \mathcal{T}(t) = \int \md \btheta \md \bJ \, f(\btheta, \bJ,t) \frac{\partial \delta \Phi(\btheta, \bJ, t)}{\partial \theta_\varphi}. \label{eqn:torque_general} \end{equation} The challenge is to compute $f$ for a given perturbation $\delta \Phi$, and then perform the integral \eqref{eqn:torque_general}. \subsection{Linear theory} \label{sec:Dynamical_friction_linear} \cite{Lynden-Bell1972-ve} (hereafter LBK) and \cite{Weinberg2004-ss} compute $f$ using the linearized collisionless Boltzmann (Vlasov) equation, ignoring the self-gravity of the perturbation to the dark matter distribution. Fourier expanding the potential as $\delta \Phi = \sum_{\bm{N}} \delta \Phi_{\bm{N}}(\bJ,t) \exp(i\bm{N}\bcdot \bm{\theta})$ and similarly for the DF $f$, the linear phase space response induced by the bar is \begin{eqnarray} f_{\bm{N}}(\bJ, t) = i \bm{N} \bcdot \frac{\partial f_0}{\partial \bJ} \int_0^t \md t' \delta\Phi_{\bm{N}} (\bm{J}, t') \me^{-i\bm{N}\bcdot \bm{\Omega}(t-t')}, \nn \\ \end{eqnarray} where $f_0(\bJ)$ is the unperturbed DF. Then putting $\delta\Phi_{\bm{N}} (\bm{J}, t) = \delta\Phi_{\bm{N}} (\bm{J}) \exp(-i N_\varphi \pattern t)$ and performing the integral over $t'$, one finds a `linear torque' $ \mathcal{T}^\mathrm{lin} = \sum_{\bm{N}} \mathcal{T}^\mathrm{lin}_{\bm{N}}$, where the contribution from resonance $\bm{N}$ is: \begin{eqnarray} \mathcal{T}^\mathrm{lin}_{\bm{N}}(t) \equiv (2\pi)^3 N_\varphi \int && \md \bJ \vert \delta \Phi_{\bm{N}} \vert^2 \bm{N}\bcdot \frac{\partial f_0}{\partial \bJ} \nn \\ && \times \frac{\sin[(\bm{N}\bcdot \bm{\Omega} - N_\varphi \pattern)t]}{\bm{N}\bcdot \bm{\Omega} - N_\varphi \pattern}. \label{eqn:Torque_linear} \end{eqnarray} Taking the time-asymptotic limit $t\to \infty$ one arrives at the classic `LBK torque formula': \begin{eqnarray} \mathcal{T}^\mathrm{LBK}_{\bm{N}} \equiv (2\pi)^3 N_\varphi \int && \md \bJ \vert \delta \Phi_{\bm{N}} \vert^2 \bm{N}\bcdot \frac{\partial f_0}{\partial \bJ} \nn \\ & & \times \pi \delta \left(\bm{N}\bcdot \bm{\Omega} - N_\varphi \pattern \right), \label{eqn:LBK} \end{eqnarray} which predicts that angular momentum is transferred to and from the dark matter halo exclusively at resonances. Importantly, for sensible (i.e. decreasing) dark matter DFs $f_0(\bJ)$ the LBK torque is finite and negative, implying a long-term transfer of angular momentum away from the bar and hence a decay in its pattern speed. In practice the time-asymptotic limit may not be valid since the torque takes several Gyr to converge, by which time various properties of the galaxy may have changed \citep{Weinberg2004-ss}. Nevertheless, the LBK formula (\ref{eqn:LBK}) is a good benchmark against which we can compare the magnitude of the torque arising from more sophisticated calculations. \subsection{Nonlinear theory} Since they employ linear theory, LBK and \cite{Weinberg2004-ss} do not account for the inevitable nonlinear particle trapping that occurs sufficiently close to each resonance. Recognizing this shortcoming, \citet{Tremaine1984-wt} --- hereafter TW84 --- and \cite{Chiba2022-qt} calculate $f$ \textit{including} the particle trapping effect. To do this they first convert to slow-fast angle-action variables $(\bm{\theta}',\bm{J}')$ \textit{around each resonance} $\bm{N}$ as in \S\ref{sec:Single_particle}. In these variables the total `nonlinear torque' on the bar $\mathcal{T}^\mathrm{nl}$ can be written as a sum over $\bm{N}$ of contributions \begin{equation} \mathcal{T}^\mathrm{nl}_{\bm N}(t) = \int \md \bm{\theta}' \md \bm{J}' \, f(\bm{\theta}', \bm{J}', t) N_\varphi \frac{\partial \delta \Phi(\btheta', \bJ')}{\partial \theta_\mathrm{s}}. \label{eqn:Torque_slowfast} \end{equation} If we now Fourier expand the potential $\delta \Phi$ in slow angle space as in equation (\ref{eqn:Hamiltonian_slowfastangles}), and also expand the DF as $f = \sum_{\bm{k}} f_{\bm{k}}(\bJ',t) \exp(i \bm{k}\bcdot \bm{\theta}')$, then \eqref{eqn:Torque_slowfast} reads \begin{equation} \mathcal{T}^\mathrm{nl}_{\bm N}(t) = - i(2\pi)^3 N_\varphi \sum_{\bm{k}} k_\mathrm{s} \int \md \bm{J}' \, f_{\bm{k}}(\bm{J}', t) \Psi^*_{\bm{k}}(\bm{J}'), \label{eqn:friction_slowfast} \end{equation} where one must be careful to properly divide phase space into non-overlapping sub-volumes that are dominated by individual resonances (see \S4.5 of \citealt{Chiba2022-qt} for a discussion). Since we always choose our initial $f$ to be independent of angles (i.e. dependent only on $\bJ'$), we know from \S\ref{sec:Single_particle} that there will be no fast-angle dependence in $f$ at later times. This means that $f_{\bm{k}}= \delta_{k_1}^0 \delta_{k_2}^0 f_{(0,0,k_\mathrm{s})}$, so \begin{eqnarray} \mathcal{T}^\mathrm{nl}_{\bm N}(t) = 2(2\pi)^3 N_\varphi \sum_{k>0} k \int \md \bm{J}' \, \mathrm{Im} \left[ f_{k}(\bm{J}', t) \Psi^*_{k}(\bm{J}') \right], \nn \\ \label{eqn:nonlinear_torque} \end{eqnarray} where we have used the shorthand $f_{(0,0,k)} = f_k$, and similarly for $\Psi_k$. At this stage, TW84 and \cite{Chiba2022-qt} calculate the torque $\mathcal{T}^\mathrm{nl}_{\bm N}(t)$ by substituting for $f_k$ the collisionless DF, i.e. the solution to the kinetic equation \eqref{eqn:kinetic_equation_dimensionless} with $\Delta =0$. In the time-asymptotic limit they arrive at the classic result \begin{equation} \mathcal{T}^\mathrm{TW84}_{\bm N} = 0. \label{eqn:TW84} \end{equation} This zero torque result is a consequence of the symmetry of the dark matter density distribution that arises when the DF completely phase mixes within and around the resonant island. Simply put, in the fully phase-mixed state there are the same number of particles `pushing' on the bar as `pulling' on it --- see Figures 4, 7 and 16 of \cite{Chiba2022-qt} for illustration. \\ \\ The linear and nonlinear analytic torque calculations pursued by the above authors (LBK; \citealt{Weinberg2004-ss}; TW84; \citealt{Chiba2022-qt}) and similar efforts by others \citep{Weinberg1989-cj,Banik2021-ug,Banik2022-rj,Kaur2018-wp,Kaur2022-tb} were all \textit{collisionless}: their DF $f$ only responded to the smooth combined potential of the underlying equilibrium galaxy $\Phi_0$ and the rigidly rotating perturbation $\delta \Phi$, in effect setting $\Delta = 0$. The only related (semi-)analytical study we know of to have included the included the effects of diffusion ($\Delta > 0$) in this problem is \cite{Weinberg2007-bv}, but their focus was on the evolution of the bar pattern speed and the associated rate at which resonances sweep through phase space. In the remainder of this section we will investigate how finite $\Delta$ affects the dynamical friction torque on galactic bars, focusing on the corotation resonance $\bm{N} = (0,2,2)$ identified as dominant by \cite{Chiba2022-qt}. \subsection{The corotation torque density} \label{sec:corotation_torque_density} We want to calculate the torque on the bar using the nonlinear torque equation (\ref{eqn:nonlinear_torque}), and substituting for $f$ our solution to the kinetic equation discussed in \S\ref{sec:solution}, for different values of $\Delta$. Following \cite{Chiba2022-qt} we use as our dark matter halo model the Hernquist sphere (\ref{eqn:Hernquist_potential}), and we take our unperturbed dark matter DF $f_0(\bJ)$ to be the isotropic Hernquist DF \citep{Hernquist1990-oe}. In Figure \ref{fig:Resonant_Line} we show colored contours of $f_0(L, J_r)$ at fixed (arbitrary) inclination \begin{equation} \beta \equiv \arccos(L_z/L). \end{equation} For our bar potential we use the model (\ref{eqn:bar_explicit})--(\ref{eqn:bar_details}), with the choices of numerical parameters given below equation (\ref{eqn:bar_details}). We will focus only on the corotation resonance $\bm{N} = (0,2,2)$, so the implicit equation for the resonant line in phase space is \begin{equation} \Omega_\psi(L, J_r) = \Omega_\mathrm{p}, \end{equation} with $\Omega_\mathrm{p} = 35 \, \mathrm{km} \, \mathrm{s}^{-1}\mathrm{kpc}^{-1} = 35.8$ Gyr$^{-1}$. We plot this resonant condition with a solid black line in Figure \ref{fig:Resonant_Line}. We also choose two particular resonant locations, shown with black dots, corresponding to $J_r = 500$ kpc$^2$ Gyr$^{-1}$ and $J_r = 50$ kpc$^2$ Gyr$^{-1}$ respectively, and will refer to these momentarily. Next, given the choice of $\bm{N}$ it follows from (\ref{eqn:fast_actions})--(\ref{eqn:slow_action}) that \begin{eqnarray} &&\bJ_\mathrm{f} = \left(J_r, \, L(1-\cos \beta) \right), \label{eqn:Corotation_fast_actions} \\ && J_\mathrm{s} = \frac{L \, \cos \beta}{2}, \label{eqn:Corotation_slow_action} \end{eqnarray} Thus at fixed inclination, $L$ is (proportional to) the slow action, while $J_r$ is a fast action. Suppose the bar is initially absent and the DF is $f_0(\bJ)$, and then at $t=0$ we suddenly turn on the bar perturbation. Particles that were initially on zero inclination orbits in the background spherical potential ($\beta = 0$) will remain at zero inclination even under the finite bar perturbation, which follows from the conservation of the fast action $J_{\mathrm{f}2}$ (equation (\ref{eqn:Corotation_fast_actions})). As a result, for these particles $L$ is a genuine slow action (up to a factor of $2$), meaning they move in the $(L, J_r)$ phase space only along the horizontal lines of constant $J_r$, for instance along one of the black dotted horizontal lines shown in Figure \ref{fig:Resonant_Line}. Ignoring diffusion for now, particles that are initially sufficiently close to the resonance will be trapped by it and will oscillate back and forth across the solid black resonant line (librating orbits). Particles that are somewhat further away from the resonance will also oscillate in $L$ but will not cross the resonant line (circulating orbits). Before we proceed with our torque calculation, one conceptual hurdle must be overcome. Namely, for a given diffusion coefficient $D$ the corotation torque $\mathcal{T}^\mathrm{nl}_{(0,2,2)}(t)$ involves contributions from a range of different $\Delta$ values. To demonstrate this, in Figure \ref{fig:Resonant_Line} we ilustrate the extent in $j$ of the librating orbits (i.e. the resonance width) for $\beta = 0$ with red bars; in other words these red bars correspond to $j \in (-2, 2)$. We notice that the resonance width depends on the choice of $J_r$. Similarly, with the yellow bars in Figure \ref{fig:Resonant_Line} we show the resonance width for the same $J_r$ values but a different inclination, namely $\beta = \pi/2$, and again the width changes\footnote{There is a minor subtlety here: for orbits that are initially at nonzero inclination, the inclination precesses under the bar perturbation, so the action-space evolution of an initially inclined particle cannot be fully captured in a single $\beta = $ constant diagram. Regardless, the width of the resonance clearly depends on $\beta$.}. Moreover it is easy to show that the libration time $t_\mathrm{lib}$ depends on both $\beta$ and $J_r$. It follows that $\Delta$ (equation (\ref{eqn:dimensionless_diffusion})) is not a constant but rather a function of $(J_r, \beta)$. The corotation torque $\mathcal{T}^\mathrm{nl}_{(0,2,2)}(t)$ involves an integration over $L$, $J_r$ and $\cos\beta$ (see equation (\ref{eqn:nonlinear_torque})), and therefore has contributions from many different $\Delta$. Since our purpose in paper is to understand how the physics of resonances depends on $\Delta$ (rather than to provide the most accurate possible computation of the frictional torque) we choose to isolate a quantity that involves only a single value of $\Delta$. This quantity is the \textit{corotation torque density} $ \mathcal{S}$, defined such that \begin{equation} \mathcal{T}^\mathrm{nl}_{(0,2,2)}(t) = \int \md J_r \, \md \cos\beta \, \mathcal{S}(t \vert J_r, \cos\beta). \end{equation} Thus $\mathcal{S}(t)$ measures the contribution to the total corotation torque from dark matter particles with radial actions $\in (J_r, J_r + \md J_r)$ and inclinations $\in (\cos\beta, \cos\beta + \md \cos \beta)$. Importantly, for a fixed $J_r$ and $\cos \beta$, the torque density $\mathcal{S}$ has contributions from only a single value of $\Delta$, which we can therefore choose by hand. The formula for $\mathcal{S}$ turns out to be \begin{equation} \mathcal{S}(t \vert J_r, \cos\beta) = 2(2\pi)^3 \int \md L \,L \Psi_1\, \mathrm{Im} \, f_1, \label{eqn:corotation_torque_density} \end{equation} where (\citealt{Chiba2022-qt}, Appendix B): \begin{equation} \Psi_1(\bJ) = \frac{(1+\cos\beta)^2}{8\pi} \int_0^\pi \md \theta_r \Phi_\mathrm{b}(r) \cos [2(\theta_\psi-\psi)]. \label{eqn:corotation_potential_perturbation} \end{equation} For a given $\Delta$ value and choice of $(J_r, \cos\beta)$, we can compute $\mathrm{Im}\,f_1$ using the numerical solution to the kinetic equation \eqref{eqn:kinetic_equation_dimensionless} that we described in \S\ref{sec:KT_resonance}, setting the initial condition to be of the form (\ref{eqn:linear_DF}) by linearizing the isotropic Hernquist DF (shown in Figure \ref{fig:Resonant_Line}) around the resonance. We can also calculate $\Psi_1(\bJ)$ (equation (\ref{eqn:corotation_potential_perturbation})) efficiently on a grid in $(L, J_r, \cos\beta)$ space using the standard mappings for $(\bm{x}, \bm{v}) \to (\bm{\theta}, \bm{J})$ for spherical potentials (e.g. \S3.5.2 of \citealt{Binney2008-ou}), and the `angular anomaly' trick from Appendix B of \cite{Rozier2019-ms}. As mentioned below equation (\ref{eqn:friction_slowfast}), when performing the integral in (\ref{eqn:corotation_torque_density}) one has to choose the maximum/minimum $L$ (or equivalently, $j$) values at which to cut it off, and this choice will affect the results, as we will see. \subsection{Results} \label{sec:numerical_results} In Figure \ref{fig:Torque_time_1} we show the bar's dimensionless slowing rate \begin{equation} \mathcal{S}(t) / \mathcal{S}_\mathrm{LBK}, \label{eqn:torque_on_halo} \end{equation} where $\mathcal{S}_\mathrm{LBK}$ is the LBK corotation torque density (see equation (\ref{eqn:LBK})): \begin{eqnarray} \mathcal{S}_\mathrm{LBK}(J_r, \cos\beta) = 4(2\pi)^3 \int && \md L \,L\, \vert \Psi_1 \vert^2 \frac{\partial f_0}{\partial L} \nn \\ && \times \pi \delta [2(\Omega_\psi - \pattern)], \label{eqn:LBK_density} \end{eqnarray} which is the result of computing the torque density using linear theory with $\Delta = 0$. In each panel we fix the inclination $\beta = 0$ and radial action $J_r = 500$ kpc$^2$ Gyr$^{-1}$, so the resonance location corresponds to the upper black dot in Figure \ref{fig:Resonant_Line}. The libration timescale (equation (\ref{eqn:t_libration})) for this resonance location is $t_\mathrm{lib} = 1.9$ Gyr. In each panel of Figure \ref{fig:Torque_time_1} we plot the slowing rate (\ref{eqn:torque_on_halo}) for various values of $\Delta \geq 0$. Note that $\mathcal{S}_\mathrm{LBK}$ is negative, so that a positive value of $ \mathcal{S}(t) / \mathcal{S}_\mathrm{LBK}$ means the bar feels a negative torque (slowing it down). The difference between the panels lies in our choice of integration limits in equation (\ref{eqn:corotation_torque_density}): we choose the limits to be $j=\pm j_\mathrm{cut}$ for $j_\mathrm{cut} = 2$, $4$ and $\infty$ in panels (a)--(c) respectively.\footnote{Except when $-j_\mathrm{cut}$ corresponds to a negative $L$; in this case we set the minimum $j$ to whichever value corresponds to $L=0$.} Note that $\mathcal{S}_\mathrm{LBK}$ (equation (\ref{eqn:LBK_density})) does not depend on $j_\mathrm{cut}$. Let us focus first on panel (a), which is for $j_\mathrm{cut} = 2$, meaning that the edges of our integration domain just graze the maximum extent of the separatrix in the $(\phi, j)$ plane. In the collisionless case ($\Delta = 0$, black dotted line) we see that the slowing rate oscillates on the timescale $\sim t_\mathrm{lib}$. This is the same behavior as found by \cite{Chiba2022-qt} --- indeed, the $\Delta = 0$ results shown here are very similar to the top panel of their Figure 11.\footnote{Note however that they plot the total corotation torque $-\mathcal{T}_{(0,2,2)}$, whereas we only plot the corotation torque density.} In the time-asymptotic limit for $\Delta = 0$ we recover the TW84 result (c.f. equation (\ref{eqn:TW84})): \begin{equation} \mathcal{S}(t\to \infty) = 0, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\Delta =0), \label{eqn:S_TW84} \end{equation} which follows from the symmetry of the fully phase-mixed DF in slow angles, i.e. $\mathrm{Im}\, f_1(j, \tau\to \infty) = 0$ (Figure \ref{fig:Im_f1}). However, as one deviates from $\Delta = 0$ to finite (but small) $\Delta > 0$, the behavior changes. The steady state is reached sooner than it was in the collisionless case, following our expectations from \S\ref{sec:steady_state}. In addition, the slowing rate in the steady state is manifestly positive for all $\Delta > 0$. This is a consequence of the fact that while phase-mixing attempts to abolish the asymmetry of the the slow-angle distribution, diffusion replenishes it (e.g. Figure \ref{fig:slices}b) leading to a finite $\mathrm{Im} \, f_1$ (Figure \ref{fig:Im_f1}). Indeed the slowing rate in Figure \ref{fig:Torque_time_1} grows with $\Delta$ until, for $\Delta \sim 1$, it is actually larger than the LBK value. This growing trend continues up to around $\Delta \sim 10$, when the slowing rate starts to decay with increasing $\Delta$, albeit rather gradually; even by $\Delta = 40$ the steady-state slowing rate is barely smaller than the LBK prediction. This story mostly repeats itself in panels (b) and (c), i.e. for $j_\mathrm{cut} = 4$ and $j_\mathrm{cut} \to \infty$ respectively, with some key differences. First of all, the transient early time behavior of the slowing rate ($t < t_\mathrm{lib}$) is sensitive to the choice of $j_\mathrm{cut}$, especially for small $\Delta$, as might be expected e.g. from the first two panels of Figure \ref{fig:slices}a. Yet for $t \gtrsim t_\mathrm{lib}$ the $\mathcal{S}(t)$ curves for small $\Delta \lesssim 1$ in Figure \ref{fig:Torque_time_1} take a near-universal form independent of $j_\mathrm{cut}$. This is because for $\Delta \lesssim 1$ the steady-state value of $\mathrm{Im}\,f_1(j)$ is negligible outside of the separatrix region $j \in (-2, 2)$ (see Figure \ref{fig:Im_f1}). Physically, under very weak diffusion the disturbances to the initial DF produced by the bar cannot propagate very far beyond the resonant island (as reflected in e.g. Figure \ref{fig:phi_averaged_g}c), so it is not possible to produce angular asymmetries at large $j$. This renders the slowing rate insensitive to $j_\mathrm{cut}$ as long as one chooses a value larger than about $2$. Essentially the same conclusion was drawn by \cite{Chiba2022-qt} when they noted that the great majority of the torque in the collisionless limit came from trapped orbits as opposed to circulating ones. On the contrary, in the strong-diffusion limit $\Delta \gg 1$ the slowing rate \textit{is} sensitive to $j_\mathrm{cut}$, because disturbances are able to propagate much further. Mathematically, for $\Delta \gg 1$ the curve of $\mathrm{Im}\,f_1(j, \tau \to \infty)$ becomes very broad in $j$, with a width $\propto \Delta^{1/3}$ --- see \S\ref{sec:asymmetry}. To make this analysis more quantitative, in Figure \ref{fig:Torque_steady} we plot the steady-state value of the slowing rate as a function of $\Delta$, again for $j_\mathrm{cut} = 2, 4, \infty$. We see that at the low-$\Delta$ end, the steady-state slowing rate follows closely the scaling $\propto \Delta^{4/5}$ irrespective of $j_\mathrm{cut}$. This reflects that fact that for small $\Delta$ the width of the $\mathrm{Im}\, f_1$ curve is approximately independent of $\Delta$ (Figure \ref{fig:Im_f1}) while its amplitude scales like $\propto \Delta^{4/5}$ (Figure \ref{fig:Max_Values}b). Meanwhile at the high-$\Delta$ end in Figure \ref{fig:Torque_steady}, the steady-state slowing rate decays as $\propto \Delta^{-1/3}$ for $j_\mathrm{cut} = 2$ but actually converges towards a constant for $j_\mathrm{cut} \to \infty$. This can be understood as follows. By cutting off the integral in (\ref{eqn:corotation_torque_density}) at $j = \pm 2$ we render the steady-state torque sensitive only to the height, and not the width, of the $\mathrm{Im} \, f_1$ curve, and we know from equations (\ref{eqn:Img1})--(\ref{eqn:resonance_function}) and Figure \ref{fig:Max_Values}b that this height scales like $\Delta^{-1/3}$. But by extending the domain of integration out to infinity we are able to count all contributions to $\mathrm{Im} \, f_1$; and since we know from \S\ref{sec:asymmetry} that $\int_{-\infty}^\infty \md j\, \mathcal{R}_\Delta(j) = 1$ is independent of $\Delta$, it is unsurprising that the torque converges for large $\Delta$. In Figure \ref{fig:Torque_time_2} we repeat the same calculation as in Figure \ref{fig:Torque_time_1}b except this time for a much smaller radial action $J_r = 50$ kpc$^2$ Gyr$^{-1}$, so the resonance location corresponds to the lower black dot in Figure \ref{fig:Resonant_Line}. The libration timescale in this case is shorter, $t_\mathrm{lib} = 1.3$ Gyr. Though we are now considering dark matter particles that are typically on much more circular orbits than in Figure \ref{fig:Torque_time_1}b, the results we find are remarkably similar. The key differences are that in Figure \ref{fig:Torque_time_2} the amplitude of the slowing rate is almost an order of magnitude larger, which reflects the larger value of $\vert L \Psi_1 \vert$ for these more circular orbits, and its oscillation period is shorter owing to the shorter libration time. For small $\Delta$, apart from these rescalings there is almost no difference with the results of Figure \ref{fig:Torque_time_1}b. This again follows from the fact that for weakly-diffusive systems $\mathrm{Im}\,f_1$ tends to be negligible outside of $j\in (-2,2)$. We can show that the factor $\Psi_1$ does not vary much over this range, meaning it can be approximated by a constant in (\ref{eqn:corotation_torque_density}), and the only remaining $\bJ_\mathrm{res}$ dependence in (\ref{eqn:corotation_torque_density}) lies in the overall prefactor and in the scaling of the time coordinate $\tau = 2\pi t/t_\mathrm{lib}(\bJ_\mathrm{res})$. \subsection{Discussion and implications for bar evolution} \label{sec:implications} Which value of $j_\mathrm{cut}$ is the physically relevant one? This is actually a nontrivial question, since in a detailed calculation one would need to consider all important resonances $\bm{N}$, then look carefully at the volume of phase space dominated by the corotation resonance and choose $j_\mathrm{cut}$ so that the domain of integration does not extend beyond this volume \citep{Chiba2022-qt}. Moreover, even if corotation was the only important resonance, we could still not fully justify taking $j_\mathrm{cut} \to \infty$ here because throughout this paper we used the simplification that the background DF $f_0(\bJ)$ could be linearized around the resonance location (equation (\ref{eqn:linear_DF})). It is clear from Figure \ref{fig:Resonant_Line} that as we venture far from the resonance this linear approximation becomes a very poor one. We could of course generalise our results to more realistic initial conditions \citep{Pao1988-cs}, but this is beyond the scope of this paper. Also, far from the resonance the pendulum approximation (\ref{eqn:resonant_Hamiltonian}) eventually breaks down. Thus we do not claim to have computed the torque density with great accuracy here, nor are we able to provide a `correct' value of $j_\mathrm{cut}$. Nevertheless, regardless of the choice of $j_\mathrm{cut}$ we see from Figure \ref{fig:Torque_steady} that the steady state torque density for $J_r = 500$ kpc$^2$ Gyr$^{-1}$ and $\beta = 0$ (the upper resonant location in Figure \ref{fig:Resonant_Line}) is comparable to the LBK value (\ref{eqn:LBK_density}) over the entire range $\Delta \in (1, 100)$, and scales robustly as $\propto \Delta^{4/5}$ for small $\Delta$. Although not shown here, equivalent plots using $J_r = 50$ kpc$^2$ Gyr$^{-1}$ (the lower resonance in Figure \ref{fig:Resonant_Line}), as well as other choices of $J_r$, show very similar results. We also note that performing these calculations again for different values of $\beta \neq 0$ gives qualitatively the same results. (However, as mentioned in \S\ref{sec:corotation_torque_density} the interpretation of such results is more subtle because for $\beta \neq 0$ the orbital inclination precesses, meaning the particles with inclination $\beta$ at time $t$ are not the same particles as were at inclination $\beta$ at time $t - \delta t$.) This suggests the following heuristic formula \begin{equation} \mathcal{S}(t\to\infty) \sim \mathrm{min}(\Delta^{4/5}, 1) \times \mathcal{S}_\mathrm{LBK}. \label{eqn:heuristic} \end{equation} The fact that the LBK torque formula provides a good order-of-magnitude estimate over such a wide range of diffusion strengths $\Delta$ can be understood as follows (see also \citealt{Johnston1971-ir}). Physically, diffusion tends to render the bar-particle interaction problem \textit{linear} in the sense that particles are never truly trapped by the bar --- for instance, they never undergo a full libration orbit around the origin in the $(\phi, j)$ plane (Figure \ref{fig:slices}) before receiving a kick to a new $j$ value. In other words, when diffusion is important, particle trajectories are well-described in the original angle-action variables $(\btheta, \bJ)$, and approximately consist of the unperturbed motion ($\bJ = $ const, $\btheta \propto \bm{\Omega}t$) punctuated by frequent instantaneous jumps in $\bJ$. This means that one can calculate the dynamical friction torque using the same technique as in the collisionless LBK theory (\S\ref{sec:Dynamical_friction_linear}), while accounting for the additional diffusion. The effect of this additional ingredient is basically to broaden the resonance line in action space, so that the $\delta$-function encoding exact resonances in the LBK formula (\ref{eqn:LBK_density}) gets replaced by the function $\mathcal{R}_\Delta$ (equation (\ref{eqn:resonance_function})). But since the `integral under the line' of this broadened resonance function is always unity regardless of $\Delta$, the torque we calculate in the broadened case does not differ much from the LBK result. More quantitatively, using the identity $\int_{-\infty}^\infty \md x \, p(x) \, \delta\left( q(x) \right) = p(x_0) / \vert q'(x_0) \vert $ where $x_0$ is assumed to be the lone zero of $q(x)$, we can rewrite the LBK formula (\ref{eqn:LBK_density}) as \begin{equation} \mathcal{S}_\mathrm{LBK} = (2\pi)^4 \left( L\vert \Psi_1 \vert^2 \frac{\partial f_0}{\partial L} \bigg/ \bigg\vert \frac{\partial \Omega_\psi}{\partial L} \bigg\vert \right)_\mathrm{res}, \label{eqn:LBK_density_2} \end{equation} where the subscript `res' indicates that everything inside the bracket is to be evaluated on resonance. Next let us crudely approximate the integrand of the nonlinear torque (\ref{eqn:corotation_torque_density}) with its resonant value and perform the integral by multiplying this with some resonance width $\delta L$. Comparing the result with (\ref{eqn:LBK_density_2}), using the fact that for our choice of slow actions $(\partial \Omega_\psi / \partial L)_\mathrm{res} = G/4$ and $(\partial f_0/\partial L)_{\mathrm{res}} = \alpha / I_\mathrm{h}$, and employing the definitions (\ref{eqn:F_and_G}) and (\ref{eqn:half_width}), one can show that \begin{equation} \frac{\mathcal{S}(t\to \infty)}{ \mathcal{S}_\mathrm{LBK}} \approx \frac{2}{\pi} \frac{\delta L}{I_\mathrm{h}} \mathrm{Im} \, g_1(0). \end{equation} We know from \S\ref{sec:asymmetry} that for large $\Delta$, the resonance width is $\delta L \sim \Delta^{1/3} I_\mathrm{h}$ and $\mathrm{Im}\,g_1(0) \sim \Delta^{-1/3}$. It follows that unless $\Delta \ll 1$ we always have $ \mathcal{S}(t\to \infty)/ \mathcal{S}_\mathrm{LBK} \sim 1$. We may infer some astrophysical implications from the above results, though we emphasise that this requires an extrapolation, because we have not computed the total torque at all but only a few contributions to the corotation torque density. Moreover, we have neglected time-dependence of the pattern speed and strength of the bar, self-gravity of the perturbed DF, etc. (see the Discussion). Nevertheless, suppose we take the formula (\ref{eqn:heuristic}) to be a broadly correct description not only of the corotation torque density, but of the dynamical friction torque as a whole: that is, we replace $\mathcal{S} \to \mathcal{T}$ and reinterpet the (strictly local, $\bJ_\mathrm{f}$-dependent) diffusion strength $\Delta$ as some characteristic value, perhaps averaged over the important regions of phase space. Then our results suggest that galactic bars will always slow down, since finite diffusion will always replenish some asymmetry in the angular distribution of the particles. They also suggest that if the time-asymptotic limit is valid, then the slow-down timescale predicted by LBK is often a good one to order of magnitude precision, except for cases of very weak diffusion. Further, this time-asymptotic torque grows as $\propto \Delta^{4/5}$ for $\Delta \lesssim 1$, meaning that in some cases a relatively collisional component of a galaxy (e.g. the stellar disk for which $\Delta \sim 1$, see \S\ref{sec:Delta}) might make a significant contribution to the frictional torque even though it carries much less mass than the nearly-collisionless cold dark matter halo (for which $\Delta \to 0$). On the other hand, the time-asymptotic limit can sometimes be irrelevant in practice, particularly if the bar pattern speed changes significantly on a timescale that is not much longer than the libration time, so the resonance sweeps through phase space before the torque has a chance to converge (\citealt{Weinberg2004-ss}). We do not investigate this issue here. \section{Discussion} \label{sec:Discussion} \begin{deluxetable*}{ccccc} \label{table:literature} \tablenum{1} \tablecaption{Guide to some complementary papers in the analytic theory of wave-particle interactions in plasma kinetics (shown in regular text) and galactic dynamics (shown in bold). We split the papers by whether the authors considered the linear or the nonlinear response of the particle DF to the wave perturbation, and whether their calculation did or did not include diffusion. The present paper belongs in the lower-right quadrant.} \tablewidth{0pt} \tablehead{ \nocolhead{\textcolor{red}{plasma} / \textbf{galactic}} & \colhead{\textit{Collisionless}} & & \multicolumn2c{\textit{Collisional}}} \startdata & {\citet{Landau1946-aj}} & & \multicolumn2c{\citet{Auerbach1977-xe}} \\ \textit{Linear theory} & \textbf{\citet{Lynden-Bell1972-ve}} & & \multicolumn2c{\citet{Catto2020-qo}}\\ & \textbf{\citet{Weinberg2004-ss}} & &\\ \hline & \colhead{\fbox{$\Delta = 0$}} & & \colhead{\fbox{$0 < \Delta \ll 1$}} & \colhead{\fbox{$\Delta \gg 1$}} \\ \textit{Nonlinear theory} & \cite{ONeil1965-uy,mazitov1965damping} & & {\citet{Pao1988-cs}} & {\citet{BerkPPR1997}} \\ \textit{(w/particle trapping)} & \textbf{\citet{Tremaine1984-wt}} & & {\citet{Petviachvili1999-mc}} & {\citet{Duarte2019-pl}} \\ & \textbf{\citet{Chiba2022-qt}} & & \multicolumn2c{\textbf{$<$--- this paper (Hamilton et al. 2022) ---$>$}} \\ \enddata \end{deluxetable*} Bar resonances are key drivers of the secular evolution of galaxies. However, analytical studies and test-particle simulations of dynamical interactions between a bar and a population of stars or dark matter particles are often highly idealized in the sense that they are \textit{collisionless}: they only consider the evolution of particle distribution functions (DFs) in smooth prescribed potentials, ignoring the diffusive effects of passing stars, molecular clouds, dark matter substructure, transient spiral waves, etc. Meanwhile, N-body simulations presumably capture (at least some of) this diffusive physics but are difficult to interpret and are rarely linked back quantitatively to the underlying dynamical processes. Moreover, N-body simulations themselves inevitably include some level of \textit{numerical} diffusion, even if they purport to describe a perfectly collisionless system \citep{Weinberg2001-ok,Weinberg2007-bv, Sellwood2013-sr}. In this paper we have taken a step towards reconciling these various approaches by developing a kinetic framework that accounts for both a secular driving of the DF by a rigidly rotating bar perturbation, and a generic diffusion in the associated slow action variable. The kinetic equation we proposed is the simplest possible one that allows us to move beyond the paradigmatic collisionless calculations of \citet{Lynden-Bell1972-ve}, \cite{Tremaine1984-wt}, \cite{Binney2016-zh}, etc. In its dimensionless form (equation (\ref{eqn:kinetic_equation_dimensionless})), the kinetic equation depends on a single dimensionless parameter, the diffusion strength $\Delta$ (equation (\ref{eqn:dimensionless_diffusion})). All past collisionless models have implicitly set $\Delta = 0$, but in stellar disks we can easily have $\Delta \sim 1$, and $\Delta$ can also be significantly different from zero in the dark matter halo depending on the form that the dark matter takes, suggesting that some of the conclusions drawn from collisionless models may need to be revised. Our purpose here has not been to provide a detailed model of the Milky Way's (or any other galaxy's) bar-driven evolution, but to elucidate some theoretical ideas upon which future detailed models may be based. Though we believe we have succeeded in this aim, our model is certainly an unrealistic portrayal of any real galactic bar. For example, we have assumed that the bar's resonances are well-separated in phase space so that there is little resonance overlap. This seems to be a reasonable assumption for dark matter trapping \citep{Chiba2022-qt}, but it is probably not the case for stars in the Galactic disk, where resonance overlap is a significant contributor to transport \citep{Minchev2010-la,Minchev2012-ml}. We have also assumed throughout that the bar is a rigidly rotating structure with constant pattern speed and strength. Even in our dynamical friction calculations we have not accounted for the fact that the friction causes the pattern speed to change, which in turn causes the resonance locations to sweep through action space . Moreover, many simulations show that the bar parameters actually fluctuate with time \citep{Fujii2018-tt}; recently, \cite{Hilmi2020-kb} simulated two Milky-Way-like galaxies and found that both bar pattern speed and strength fluctuate at the level of $10-20$ percent on orbital timescales ($\sim 100$ Myr), mostly due to interactions with spiral arms. In general, weak time-dependence of the pattern speed and/or strength of the bar will cause some broadening of the resonance in action space \citep{Weinberg2007-zo}, just as diffusion did (e.g. Figure \ref{fig:Im_f1}). One might therefore speculate that an inclusion of weakly time-dependent bar parameters in our calculations would amount to little more than an increased effective $\Delta$. Whether this holds true in practice remains to be seen. Another major assumption we have made throughout the paper is that there is no diffusion of particles' fast actions $\bJ_\mathrm{f}$. This assumption is not fully justifiable in the general case --- for example, local isotropically-distributed scattering events should produce diffusion in all three action variables, not just the slow variable $J_\mathrm{s}$. It is worth mentioning that \textit{the same assumption is made implicitly in collisionless models}. In collisionless models, a distinct phase space island exists at every $\bJ_\mathrm{f}$, and the DF phase mixes perfectly within this island similar to Figure \ref{fig:slices}a, with no cross-talk between different $\bJ_\mathrm{f}$ values. This becomes problematic when even a small amount of diffusion is included, because the phase-mixed island structures at each $\bJ_\mathrm{f}$ inevitably produce sharp gradients of $f$ in the $\bJ_\mathrm{f}$ direction, since the location and shape of the resonant island changes as a function of $\bJ_\mathrm{f}$ (for instance $J_\mathrm{s, res}$ itself is a function of $\bJ_\mathrm{f}$ through equation \eqref{eqn:resonance_condition}, and we already argued that the width of the resonance depends on $\bJ_\mathrm{f}$ in \S\ref{sec:corotation_torque_density}). Because of these sharp gradients, a diffusive term like $\sim \partial^2 f/\partial J_{\mathrm{f}1}^2$ in the kinetic equation is liable to become large, and hence $\bJ_\mathrm{f}$ diffusion will start to `fill in' the flattened regions in the DF produced by phase mixing.\footnote{The longitudinal plasma wave calculation of \cite{Pao1988-cs} \textit{did} include three-dimensional diffusion, but he had the advantage of working in velocity space rather than action space, which simplifies the calculations significantly, and it is not clear whether a comparable angle-action calculation could be developed.} This suggests that in reality the effective $\Delta$ is again higher than one would naively estimate based on equation (\ref{eqn:Delta_approximate}). Nevertheless, our study is still useful in that it allows us to quantify, even if only roughly, the impact of stochastic effects upon resonant structures through a single intuitive parameter $\Delta$. If the aforementioned complications do indeed increase the effective $\Delta$, this will only reinforce our main point that in many astrophysically relevant scenarios, diffusive effects cannot be ignored. We also neglected to include any drag term in our collision operator (\ref{eqn:collision_operator}), which is usually valid as long as the scattering agents (molecular clouds, spiral arms, etc.) are sufficiently massive compared to the particles in question \citep{Binney1988-zy}. However drag can be an important ingredient, for instance if $f$ were to describe heavy bodies (e.g. MACHOs) or a population of stars in a tepid disk \citep{Fouvry2015-nk}. In this case resonance lines, in addition to experiencing broadening, are predicted to shift and split \citep{duarte2020shifting}. A note is warranted here on the many correspondences between stellar-dynamical and plasma-kinetic theory. Our kinetic equation (\ref{eqn:kinetic_equation_dimensionless}) turns out to be mathematically identical to an equation employed in a range of plasma kinetic calculations of wave-particle interactions \citep{Pao1988-cs,BerkPPR1997,Duarte2019-mi}. However we have not seen it expressed in this dimensionless single-parameter form before, nor have we seen the solutions examined so closely as in \S\ref{sec:solution}, meaning our results may be of use for future plasma studies. The plasma literature more generally is replete with analyses of nonlinear particle trapping, diffusive resonance broadening, and so on (e.g. \citealt{Dupree1966-dl,Su1968-ga,Ng2006-gu,Black2008-cg,White2019-nl,Catto2020-qo,Catto2021-ed,Tolman2021-ly}). Moreover, the LBK formula \eqref{eqn:LBK} is directly analogous to the classic formula for the Landau damping rate of a Langmuir wave in an electrostatic plasma \citep{Landau1946-aj,Ichimaru1965-zm} or Landau damping in more general geometry \citep{Kaufman1972-ag,Nelson1999-in}. The TW84 calculations that account for nonlinear trapping of particles in resonances are strongly analogous to the `nonlinear Landau damping' calculations by \cite{ONeil1965-uy,mazitov1965damping} (who, unsurprisingly, found that the damping/growth rate of waves goes to zero in the fully phase-mixed limit). In a similar vein, the calculations in the present paper are analogous to those works that have attempted to combine O'Neil's and Mazitov's calculation with a simple model for inter-particle collisions (e.g. \citealt{Zakharov1963-ae,Pao1988-cs,Brodin1997-sj}). The approximate recovery of the LBK torque from the nonlinear torque in the large $\Delta \gg 1$ limit is closely related to the recovery of the Landau damping formula from the O'Neil/Mazitov formula in the limit of strong collisions \citep{Johnston1971-ir,Auerbach1977-xe}. In Table \ref{table:literature} we provide a summary guide to this plasma-kinetic/stellar-dynamical correspondence, and show where the present paper fits into the literature. Finally, in recent years our analytic understanding of dynamical friction in the collisionless ($\Delta=0$) limit has been greatly improved e.g. by the work of \cite{Chiba2022-qt}, \cite{Banik2021-ug,Banik2022-rj} and \citet{Kaur2018-wp,Kaur2022-tb}. To some extent these authors have advocated an `orbit-based' interpretation of dynamical friction. For instance, \cite{Chiba2022-qt} and \cite{Banik2022-rj} calculated the contributions to the friction from individual particles at different points in their orbits, compared the torque coming from trapped orbits to those from untrapped orbits, and so on. Our finite-$\Delta$ calculations show the benefit of a complementary viewpoint, in which dynamical friction is a kinetic process --- something that needs to be understood primarily in terms of distribution functions as opposed to individual particle trajectories. In reality, no particle is permanently trapped or untrapped by a resonance, since for finite $\Delta$ there is a constant flux of particles into and out of the librating region even in steady state. \section{Summary} \label{sec:Summary} In this paper we have investigated the impact of diffusion upon the resonant imprints left by rigidly rotating galactic bars in the distribution function (DF) of stars and dark matter particles, and the subsequent effect this has on bar slowdown through dynamical friction. Our key findings can be summarised as follows. \begin{itemize} \item Using the pendulum approximation to describe the secular forcing by the bar and assuming a simple diffusion in the associated slow action variable, we proposed a kinetic equation for the DF that depends on just one parameter, the diffusion strength $\Delta$, equal to the ratio of the resonant libration period to the diffusion time across the resonance. Though many classic analytic studies took $\Delta = 0$ by default, $\Delta$ can in fact be significantly larger than zero for many cases of astrophysical interest. \item Assuming simple initial/boundary conditions we solved this kinetic equation for a wide range of $\Delta$ values. For $\Delta =0$ and $t\to \infty$ we recovered the classic result in which the DF is spread uniformly (`phase-mixed') along the pendulum Hamiltonian contours in slow angle-action space. This phase-mixed structure was broken for finite $\Delta$, and we provided an analytic understanding of the resulting DF in the $\Delta \ll 1$ and $\Delta \gg 1$ limits. \item We calculated the \textit{corotation torque density} $\mathcal{S}$ felt by a galactic bar due to its frictional interaction with an initially spherical halo of dark matter particles, focusing on the contribution from particles orbiting coplanar with the bar, for various $\Delta$. For $\Delta =0$ we recovered the classic phase-mixed result of \cite{Tremaine1984-wt} that $\mathcal{S} \to 0$ as $t\to \infty$. However, since diffusion replenishes the asymmetry of the DF in the resonant region of phase space, the steady-state torque never vanished for finite $\Delta$. Instead we found $\mathcal{S} \sim \mathrm{min}(\Delta^{4/5}, 1) \times \mathcal{S}_\mathrm{LBK}$, where $\mathcal{S}_\mathrm{LBK}$ is the (negative) linear theory result from \cite{Lynden-Bell1972-ve}, suggesting real bars always slow down. \end{itemize} Our work highlights the fact that resonant phenomena are \textit{delicate}, and that one must therefore be mindful of the effects of diffusion if one is to understand how resonances sculpt galaxies. We have shown this for the particular scenario of bar-dark matter halo coupling, but we suspect that similar conclusions apply to resonant imprints left in Solar neighborhood kinematic data. Further work will be needed to make our rather qualitative claims precise. \begin{acknowledgements} We thank S.~Tremaine, C.~Terquem, R.~Rafikov, R.~Sanderson, M.~Weinberg, K.~Johnston, B.~Kocsis, T.~Yavetz, U.~Banik, E.~Vasiliev, S.~Chakrabarti and members of the CCA Dynamics group for helpful comments and discussions. This work was supported by a grant from the Simons Foundation (816048, CH), by the U.S. Department of Energy under contract DE-AC02-09CH11466 (VND), by the Institute for Advanced Study (LA), and by the W.M.~Keck Foundation Fund at the Institute for Advanced Study (ET). This work was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. \end{acknowledgements} \appendix \section{Analytic results for weak and strong diffusion} In this Appendix we derive some analytic results that allow us to understand the solutions to the kinetic equation (\ref{eqn:kinetic_equation_dimensionless}) in the asymptotic limits of weak diffusion $\Delta \ll 1$ (\S\ref{sec:weak_scattering}) and the strong diffusion $\Delta \gg 1$ (\S\ref{sec:strong_scattering}). \subsection{The limit of weak diffusion, $\Delta \ll 1$} \label{sec:weak_scattering} \subsubsection{Phase-mixed DF in the collisionless limit $\Delta =0$} The (coarse-grained) steady-state phase-mixed DF for $\Delta =0$ is easily calculated by smearing the initial DF evenly over the phase space contours, as described for example by \cite{Binney2016-zh}, \cite{Monari2017-tu} and, in a different context, by \cite{Hamilton2022-gr}. Let us call this phase-mixed DF $f_\mathrm{pm}(\phi, j)$; then we know it will only be a function of $h$, where \begin{equation} h(\phi, j) \equiv \frac{1}{2}j^2 - \cos \phi. \label{eqn:dimensionless_H} \end{equation} is the effective dimensionless Hamiltonian. The initial DF $f_\mathrm{init}$ (equation \eqref{eqn:linear_DF}) written in terms of $(\phi, h)$ is \begin{equation} f_\mathrm{init} = f_0(0) + \alpha j_\pm(\phi, h), \label{eqn:f_init_h} \end{equation} where $j_\pm \equiv \pm \sqrt{2(h+\cos \phi)}$ and we take $+$ or $-$ depending on whether we are applying the equation to positive or negative $j$. Now, for $\Delta = 0$ there are two well-defined orbit families, namely librating ($h < 1$) and circulating ($h>1$) orbits, distinguished by a separatrix ($h=1$). Since the resonant island is symmetric around $j=0$ (e.g. Figure \ref{fig:dimensionless_DF}), it is geometrically obvious that for librating orbits the average phase space density on any given contour is just \begin{equation} f^\mathrm{lib}_{\mathrm{pm}} = f_0(0), \label{eqn:phase_mixed_librating} \end{equation} and hence $g^\mathrm{lib}_{\mathrm{pm}}=0$ (see the final panel of Figure \ref{fig:slices}a). For circulating orbits ($h>1$) we use the fact that the line element in the $(\phi_*, j_*)$ plane at fixed $h$ is \begin{eqnarray} \md \lambda = \md \phi_* \sqrt{1+ \frac{\sin^2\phi_*}{j^2_\pm(\phi_*, h)}}. \end{eqnarray} The phase-mixed DF at fixed $h(\phi, j)$ is then found by averaging \eqref{eqn:f_init_h} over the contour: \begin{eqnarray} f_{\mathrm{pm}}^\mathrm{circ}(h(\phi,j)) = f_0(0) + \alpha \frac{\oint \md \lambda \, j_\pm(\phi_*, h)}{\oint \md \lambda}. \label{eqn:phase_mixed_ratio} \end{eqnarray} Equations (\ref{eqn:phase_mixed_librating})--(\ref{eqn:phase_mixed_ratio}) describe fully the phase-mixed DF $f_\mathrm{pm}$, towards which the solution to (\ref{eqn:kinetic_equation_dimensionless}) tends in the time-asymptotic limit. \subsubsection{Steady-state DF for finite diffusion, $0 < \Delta \ll 1$} When diffusion is finite but weak ($0 < \Delta \ll 1$), the steady-state solution to (\ref{eqn:kinetic_equation_dimensionless}) is modified somewhat from the perfectly phase-mixed solution $f_\mathrm{pm}$ --- see Figure \ref{fig:slices}b and the blue lines in Figures \ref{fig:phi_averaged_g}c and \ref{fig:Im_f1}. We can gain some insight into this modified steady-state DF following the method of \cite{Pao1988-cs} (see also \citealt{Petviachvili1999-mc}). First, we look for a steady-state perturbative solution \begin{equation} f_\mathrm{steady}(\phi, j) = f_\mathrm{pm}(h) + \delta f_\mathrm{steady}(\phi, j), \label{eqn:small_Delta_ansatz} \end{equation} where $\vert \delta f_\mathrm{steady}/f_\mathrm{pm}\vert = \mathcal{O}(\Delta)$. We then have from \eqref{eqn:kinetic_equation_dimensionless} that \begin{equation} j \frac{\partial \delta f_\mathrm{steady}}{\partial \phi} - \sin \phi \frac{\partial \delta f_\mathrm{steady}}{\partial j} \approx \Delta \frac{\partial^2 f_\mathrm{pm}}{\partial j^2}. \label{eqn:steady_state_small_Delta_perturbation} \end{equation} To solve this equation we change variables from $(\phi, j) \to (\phi, h)$, where $h$ is given in (\ref{eqn:dimensionless_H}). Then equation \eqref{eqn:steady_state_small_Delta_perturbation} becomes \begin{equation} \frac{\partial \delta f_\mathrm{steady}}{\partial \phi} = \Delta \frac{\partial }{\partial h} \left[ j_\pm(\phi, h) f_\mathrm{pm}'(h)\right]. \label{eqn:perturbation_new_coords} \end{equation} The solution to \eqref{eqn:perturbation_new_coords} is \begin{eqnarray} \delta f_\mathrm{steady}(\phi, h) = \pm \Delta \frac{\partial}{\partial h} \left[ f_\mathrm{pm}'(h) \int^\phi \md x \sqrt{2(h+\cos x)}\right] = \pm \Delta \frac{\partial}{\partial h} \Bigg[ 2\sqrt{2(1+h)} f_\mathrm{pm}'(h) E\left( \frac{\phi}{2}, \sqrt{\frac{2}{1+h}} \right)\Bigg], \label{eqn:delta_f_weak_scattering_full} \end{eqnarray} where $E(\phi,x)\equiv \int_0^\phi \md y \sqrt{1-x^2 \sin^2 y}$ is an incomplete elliptic integral of the second kind, and we take $+$ or $-$ depending on whether we apply the equation to $j>0$ or $j<0$. For the circulating region of phase space ($h>1$) we can get an explicit form for $f_\mathrm{pm}'(h)$ as follows. We integrate equation \eqref{eqn:perturbation_new_coords} with respect to $\phi$ from $-\pi$ to $\pi$ and get \begin{eqnarray} 0 = \frac{\partial}{\partial h} \left[ 4 {f^\mathrm{circ}_\mathrm{pm}}'(h) \sqrt{h-1} E\left(\sqrt{\frac{-2}{h-1}}\right)\right], \label{eqn:delta_f_small_Delta} \end{eqnarray} where $E(x) \equiv E(\pi/2, x)$ is the complete elliptic integral of the second kind. It follows that the quantity in square brackets is equal to a constant. We can fix this constant by using the fact that very far from the resonance we have $h \approx j^2/2 \gg 1$ and $f_\mathrm{pm}$ is unperturbed, $f_\mathrm{pm} \approx f_\mathrm{init} = f_0(0) \pm \alpha \sqrt{2h}$ (see equation (\ref{eqn:linear_DF})). Then we find that the square bracket is equal to $\pm \pi \sqrt{2} \alpha$, so\footnote{Of course one can also derive an exact expression for ${f^\mathrm{circ}_\mathrm{pm}}'(h)$ by differentiating the phase mixed solution \eqref{eqn:phase_mixed_ratio}, but the approximate expression \eqref{eqn:phase_mixed_derivative} is simpler and sufficiently accurate for our purposes.} \begin{equation} {f^\mathrm{circ}_\mathrm{pm}}'(h) = \frac{\sqrt{2}\pi\alpha}{4\sqrt{h-1}} \left[ E\left(\sqrt{\frac{-2}{h-1}}\right) \right]^{-1}. \label{eqn:phase_mixed_derivative} \end{equation} Combining this with \eqref{eqn:delta_f_weak_scattering_full} we have that for $h>1$, \begin{eqnarray} \delta f^\mathrm{circ}_\mathrm{steady}(\phi, h) = \pm \alpha \Delta \pi \frac{\partial}{\partial h} \Bigg[ \sqrt{\frac{h+1}{h-1}}\frac{E\left( \phi/2,\sqrt{ 2/[1+h] }\right) }{E\left(\sqrt{-2/[h-1]}\right)} \Bigg] . \label{eqn:delta_f_weak_scattering_circulating} \end{eqnarray} Though we do not illustrate it here, equation (\ref{eqn:delta_f_weak_scattering_circulating}) does a fairly good job of reproducing the numerical solution for the region $\vert j\vert > 2$. On the other hand, when we apply the same solution technique to the librating portion of phase space ($h<1$), it fails. The reason is that $f^\mathrm{lib}_\mathrm{pm}(h)=$ constant (see equation \eqref{eqn:phase_mixed_librating}) so that ${f^\mathrm{lib}_\mathrm{pm}}'(h) = 0$, and hence \eqref{eqn:delta_f_weak_scattering_full} would predict $\delta f^\mathrm{lib}_\mathrm{steady} = 0$. This a poor solution, as can be seen by inspecting Figure \ref{fig:slices}b. Of particular importance for dynamical friction calculations (\S\ref{sec:Dynamical_friction}) is the fact that this solution fails to reproduce the major contributions to the $\mathrm{Im}\, g_1$ curve for $\vert j \vert < 2$ shown for small $\Delta$ in Figure \ref{fig:Im_f1}. The reason for the failure of this solution is that for small $\Delta$, sharp gradients in the DF tend to build up near the separatrix, meaning that the term $\Delta \partial^2 \delta f_\mathrm{steady}/\partial j^2$ on the right hand side of equation \eqref{eqn:kinetic_equation_dimensionless} cannot be neglected --- the approximation \eqref{eqn:steady_state_small_Delta_perturbation} therefore breaks down. To remedy this one can treat the region near the separatrix as a boundary layer where $\delta f_\mathrm{steady}$ can be of the same order as $f_\mathrm{pm}$. The reader is referred to \cite{Pao1988-cs} for details of how this calculation works, although in practice it is a cumbersome task and the final expressions are rather unenlightening. The important point is that one gets an additional skew-symmetric contribution to the DF (in the sense of (\ref{eqn:skew})) in a layer on the inner edge of the separatrix, which is of the same order as the phase-mixed solution $f_\mathrm{pm}$. The thickness of the layer grows with $\Delta$ although not in a way that is easily quantified; empirically this leads to the low-$\Delta$ behavior described in \S\ref{sec:solution}, and in particular to the approximate scaling $\mathrm{max}(\mathrm{Im}\,g_1) \propto \Delta^{4/5}$ (Figure \ref{fig:Max_Values}b). \subsection{The limit of strong diffusion, $\Delta \gg 1$} \label{sec:strong_scattering} Let us now consider the opposite limit, in which the timescale for diffusion $t_\mathrm{diff}$ is short compared to the libration time $t_\mathrm{lib}$, i.e. $\Delta \gg 1$. We note that the initial DF (\ref{eqn:linear_DF}) is annihilated by the collision operator on the right hand side of (\ref{eqn:kinetic_equation_dimensionless}) because $\Delta\, \partial^2 f_\mathrm{init}(j)/\partial j^2 = 0$. Thus in the limit of infinitely strong diffusion $\Delta \to \infty$, when this term is the dominant one in the kinetic equation, we expect that the solution $f(\phi, j, \tau)$ will never deviate from $f_\mathrm{init}$. Therefore, without loss of generality let us write \begin{eqnarray} f(\phi, j, \tau) = f_\mathrm{init}(j) + \delta f(\phi, j, \tau). \label{eqn:strong_ansatz} \end{eqnarray} Plugging this into the kinetic equation (\ref{eqn:kinetic_equation_dimensionless}) we find \begin{equation} \frac{\partial \delta f}{\partial \tau} + j\frac{\partial \delta f}{\partial \phi} - \sin \phi \left( \alpha + \frac{\partial \delta f}{\partial j}\right)= \Delta \frac{\partial^2 \delta f}{\partial j^2}. \label{eqn:kinetic_equation_linearized_full} \end{equation} For large but finite diffusion, $\Delta \gg 1$, we anticipate that $\delta f$ will be finite but small (see Figure \ref{fig:slices}d), so that $\epsilon(\phi, j) \equiv \vert (\partial \delta f/\partial j) /\alpha \vert \ll 1$ is small. Our aim in this section will be to compute $\delta f$ to lowest order in this small parameter. Higher order solutions are also possible \citep{Duarte2019-mi} but will not be necessary for our purposes. Let us therefore drop the term $\partial \delta f/\partial j$ from \eqref{eqn:kinetic_equation_linearized_full} and write \begin{equation} \frac{\partial \delta f}{\partial \tau} + j\frac{\partial \delta f}{\partial \phi} - \alpha \sin \phi \approx \Delta \frac{\partial^2 \delta f}{\partial j^2}. \label{eqn:kinetic_equation_linearized} \end{equation} We now take advantage of the periodicity of the coordinate system to expand $\delta f$ as a Fourier series: $ \delta f(\phi, j, \tau) = \sum_{n = -\infty}^\infty \delta f_n(j,\tau) \exp(in\phi)$. Since $f$ is real we must have $\delta f_n^{*}(j,\tau)=\delta f_{-n}(j,\tau)$. Plugging the Fourier expansion in to (\ref{eqn:kinetic_equation_linearized}) and using the orthogonality of Fourier harmonics, we get \begin{equation} \frac{\partial \delta f_n}{\partial \tau} + in j \, \delta f_n + \frac{i \alpha}{2} (\delta_{n-1}^0 - \delta_{n+1}^0) = \Delta \frac{\partial^2 \delta f_n}{\partial j^2}. \label{eqn:kinetic_equation_Fourier} \end{equation} The solution to this equation is $\delta f_n = 0$ for $n\neq \pm 1$, and for $n=\pm 1$: \begin{equation} \delta f_{\pm 1}(j, \tau) = \pm \frac{\alpha}{2i{\Delta^{1/3}}}\int_0^{{\Delta^{1/3}}\tau} \md y \, \exp\left(-\frac{y^3}{3} \mp \frac{ijy}{{\Delta^{1/3}}} \right). \label{eqn:fourier_solution} \end{equation} It follows that the solution to the kinetic equation (\ref{eqn:kinetic_equation_dimensionless}) for $\Delta \gg 1$ is approximately \begin{eqnarray} f(\phi, j, \tau) = f_\mathrm{init}(j) + \frac{ \alpha}{{\Delta^{1/3}} } \int_0^{{\Delta^{1/3}} \tau} \md y \sin\left( \phi - \frac{jy}{{\Delta^{1/3}}}\right)\exp\left(-\frac{y^3}{3}\right). \label{eqn:delta_f_1st_order_solution} \end{eqnarray} The $\exp(-y^3/3)$ factor in the integrand of (\ref{eqn:delta_f_1st_order_solution}) means that the contribution to the integral from $y$ values $\gtrsim 2$ are negligible, meaning the steady state is approximately reached when $\tau \sim \Delta^{-1/3}$. We note also that in this limit we get an explicit expression for the key quantity $\mathrm{Im} f_1$ in steady state: \begin{equation} \mathrm{Im} \, f_{1}(j, \tau \to \infty) = \frac{\alpha}{2{\Delta^{1/3}}}\int_0^{\infty} \md y \, \exp\left(-\frac{y^3}{3}\right) \cos\left(\frac{jy}{{\Delta^{1/3}}} \right), \end{equation} which leads directly to equations (\ref{eqn:Img1})--(\ref{eqn:resonance_function}). Finally, we can use the solution (\ref{eqn:delta_f_1st_order_solution}) to check our assumption that $\epsilon \equiv \vert (\partial \delta f/\partial j) /\alpha \vert \ll 1$. We take the derivative of $f-f_\mathrm{init}$ with respect to $j$ and divide by $\alpha$ to find \begin{eqnarray} \epsilon &\approx& \Delta^{-2/3}\bigg\vert \int_0^{\infty} \md y \, y\sin\left( \phi - \frac{jy}{{\Delta^{1/3}}}\right)\exp\left(-\frac{y^3}{3}\right) \bigg\vert \nn \\ &\leq& \Delta^{-2/3}\bigg\vert \int_0^{\infty} \md y \, y \exp\left(-\frac{y^3}{3}\right) \bigg\vert \approx 0.94\Delta^{-2/3}. \end{eqnarray} Thus $\epsilon$ is small as long as $\Delta \gtrsim$ a few. \section{Diffusion in slow angle} \label{sec:angle_scattering} In the main text we chose a collision operator \eqref{eqn:collision_operator} which produced diffusion only in the slow action but not in any other variable, such as the slow angle. We justified this on the grounds that the narrow resonance mostly produced sharp features in $I$ but not $\phi$. Here we attempt a more quantitative accounting for a possible $\phi$-diffusion. To do this we write down a modified kinetic equation \begin{equation} \frac{\partial f}{\partial t} + GI\frac{\partial f}{\partial \phi} - F\sin \phi\frac{\partial f}{\partial I}= D \frac{\partial^2 f}{\partial I^2} + D_{\phi\phi} \frac{\partial^2 f}{\partial \phi^2} . \label{eqn:kinetic_equation_angle_scattering} \end{equation} We can non-dimensionalize this equation in the same way as in \S\ref{sec:KT_resonance}. We find (c.f. equation \eqref{eqn:kinetic_equation_dimensionless}): \begin{equation} \frac{\partial f}{\partial \tau} + j\frac{\partial f}{\partial \phi} - \sin \phi\frac{\partial f}{\partial j}= \Delta \frac{\partial^2 f}{\partial j^2} + \Delta_{\phi\phi} \frac{\partial^2 f}{\partial \phi^2} , \label{eqn:kinetic_equation_angle_scattering_dimensionless} \end{equation} where (c.f. equation (\ref{eqn:dimensionless_diffusion})): \begin{eqnarray} && \Delta_{\phi\phi} \equiv \frac{D_{\phi\phi}}{\sqrt{FG}}. \end{eqnarray} is approximately the ratio of the libration time to the timescale to scatter by $\mathcal{O}(1)$ in slow angle $\phi$. We may solve equation (\ref{eqn:kinetic_equation_angle_scattering_dimensionless}) perturbatively in the strong diffusion limit $\Delta, \, \Delta_{\phi\phi} \gg 1$. We repeat the analysis of \S\ref{sec:strong_scattering}, replacing equation (\ref{eqn:kinetic_equation_Fourier}) by \begin{equation} \frac{\partial \delta f_{n}}{\partial \tau} + in j \delta f_{n} + \frac{i \alpha}{2} (\delta_{n-1}^0 - \delta_{n+1}^0) = \Delta \frac{\partial^2 \delta f_n}{\partial j^2} - \Delta_{\phi\phi} n^2 \delta f_n. \label{eqn:kinetic_equation_angle_scattering_Fourier} \end{equation} The solution to this equation is $\delta f_n = 0$ for $n\neq \pm 1$ and (c.f. equation \eqref{eqn:fourier_solution}): \begin{equation} \delta f_{\pm 1}(j, \tau) = \pm \frac{\alpha}{2i{\Delta^{1/3}}}\int_0^{{\Delta^{1/3}}\tau} \md y \, \exp\left(-\frac{y^3}{3} - \frac{\Delta_{\phi\phi} y}{\Delta^{1/3}} \mp \frac{ijy}{{\Delta^{1/3}}} \right). \label{eqn:deltaf_pm1_anglediffusion} \end{equation} Thus the scattering in $\phi$ gives rise to an exponential damping of the integrand. However, the this damping will only be the dominant effect (as opposed to $j$-diffusion alone) for $y$ values such that $y^3/3 \ll \Delta_{\phi\phi}y/\Delta^{1/3}$, i.e. $y \ll \Delta_{\phi\phi}^{1/2}/\Delta^{1/6}$. Since an $\mathcal{O}(1)$ diffusion in slow angle would completely change the particle's orbit, we expect the characteristic timescale for this to be on the order of the relaxation time $t_\mathrm{relax}$ --- see \S\ref{sec:Delta}. Hence it is sensible to estimate $\Delta_{\phi\phi}^{1/2}/\Delta^{1/6} \sim (t_\mathrm{lib}/t_\mathrm{relax})^{1/2} \times (t_\mathrm{diff}/t_\mathrm{lib})^{1/6} \sim A^{1/6} (t_\mathrm{lib}/t_\mathrm{relax})^{1/3} < 1$. It follows that $\phi$-diffusion is rarely the dominant effect (except at early times when $\Delta^{1/3}\tau \ll 1$), and so will do little to change the qualitative results and scalings derived in the main text in the $\Delta \gg 1$ limit. This is in line with our intuition --- which also holds for $\Delta \ll 1$ --- that a narrow resonance will attempt to produce much sharper structure in $j$ than in $\phi$, and so the $j$-diffusion is the more important process. \vspace{5mm} \bibliography{sample631}{} \bibliographystyle{aasjournal}

Title: Statistics of Galactic-Scale Quasar Pairs at Cosmic Noon

Abstract: The statistics of galactic-scale quasar pairs can elucidate our understanding of the dynamical evolution of supermassive black hole (SMBH) pairs, the duty cycles of quasar activity in mergers, or even the nature of dark matter, but have been challenging to measure at cosmic noon, the prime epoch of massive galaxy and SMBH formation. Here we measure a double quasar fraction of $\sim 6.2\pm0.5\times 10^{-4}$ integrated over $\sim 0.3-3$ arcsec separations (projected physical separations of $\sim 3-30\,{\rm kpc}$ at $z\sim 2$) in luminous ($L_{\rm bol}>10^{45.8}\,{\rm erg\,s^{-1}}$) unobscured quasars at $1.5<z<3.5$, using Gaia EDR3-resolved pairs around SDSS DR16 quasars. The measurement was based on a sample of 60 Gaia-resolved double quasars (out of 487 Gaia pairs dominated by quasar+star superpositions) at these separations, corrected for pair completeness in Gaia, which we quantify as functions of pair separation, magnitude of the primary, and magnitude contrast. The double quasar fraction increases towards smaller separations by a factor of $\sim 5$ over these scales. The division between physical quasar pairs and lensed quasars in our sample is currently unknown, requiring dedicated follow-up observations (in particular, deep, sub-arcsec-resolution IR imaging for the closest pairs). Intriguingly, at this point the observed pair statistics are in rough agreement with theoretical predictions both for the lensed quasar population in mock catalogs and for dual quasars in cosmological hydrodynamic simulations. Upcoming wide-field imaging/spectroscopic space missions such as Euclid, CSST and Roman, combined with targeted follow-up observations, will conclusively measure the abundances and host galaxy properties of galactic-scale quasar pairs, offset AGNs, and sub-arcsec lensed quasars across cosmic time.

https://export.arxiv.org/pdf/2208.04979

\title{Statistics of Galactic-Scale Quasar Pairs at Cosmic Noon} \author[0000-0003-1659-7035]{Yue Shen} \affiliation{Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA} \affiliation{National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA} \author[0000-0003-4250-4437]{Hsiang-Chih Hwang} \affiliation{School of Natural Sciences, Institute for Advanced Study, Princeton, 1 Einstein Drive, NJ 08540, USA} \author{Masamune Oguri} \affiliation{Center for Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan} \affiliation{Department of Physics, Graduate School of Science, Chiba University, 1-33 Yayoi-Cho, Inage-Ku, Chiba 263-8522, Japan} \author{Nianyi Chen} \affiliation{McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA} \author{Tiziana Di Matteo} \affiliation{McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA} \affiliation{NSF AI Planning Institute for Physics of the Future, Carnegie Mellon University, Pittsburgh, PA 15213, USA} \author{Yueying Ni} \affiliation{McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA} \affiliation{NSF AI Planning Institute for Physics of the Future, Carnegie Mellon University, Pittsburgh, PA 15213, USA} \author{Simeon Bird} \affiliation{Department of Physics \& Astronomy, University of California, Riverside, 900 University Ave., Riverside, CA 92521, USA} \author[0000-0001-6100-6869]{Nadia Zakamska} \affiliation{Department of Physics and Astronomy, Johns Hopkins University} \affiliation{School of Natural Sciences, Institute for Advanced Study, Princeton, 1 Einstein Drive, NJ 08540, USA} \author[0000-0003-0049-5210]{Xin Liu} \affiliation{Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA} \affiliation{National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA} \author{Yu-Ching Chen} \affiliation{Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA} \author[0000-0001-5253-1338]{Kaitlin M. Kratter} \affiliation{University of Arizona, 933 N Cherry Ave, Tucson, AZ 85721, USA} \keywords{black hole physics --- galaxies: active --- quasars: general --- surveys} \section{Introduction}\label{sec:intro} The formation of binary SMBHs ($M_{\rm BH}\gtrsim 10^6\,M_\odot$) is the inevitable consequence of galaxy mergers and the prevalence of SMBHs in galactic nuclei \citep[e.g.,][]{Begelman_etal_1980}. After the merger of two galaxies, the two SMBHs will in-spiral in the merged galaxy due to dynamical friction from tens of kpc to $\sim 10\,$parsec \citep[e.g.,][]{milosavljevic01,Yu_2002,DEGN}. The two SMBHs become a gravitationally bound binary at $\lesssim 10$\,parsec separations, and interactions with stars continue to shrink the binary orbit. The evolution of the binary SMBH below $\sim 1$\,parsec depends on the properties of stellar orbits in the galactic potential and the effects of gas \citep[e.g.,][]{DEGN,DeRosa_etal_2019}. But if the binary orbit can shrink to scales $\ll 1$\,parsec, gravitational wave (GW) radiation will take over in shrinking the binary orbit and eventually lead to the coalescence of the two SMBHs. The GW signals during the final in-spiral and coalescence of the binary SMBH are highly anticipated from ongoing pulsar timing arrays \citep[e.g.,][]{pta20} and future GW facilities such as the Laser Interferometer Space Antenna \citep[e.g.,][]{lisa22}. SMBH pairs at galactic-scale separations (tens of kpc to tens of parsec) represent the best-understood stage in theoretical studies of binary SMBH formation. The abundance of these wide separation pairs sets the initial conditions of binary SMBHs expected from galaxy mergers. Their pair separation statistics constrain the evolutionary timescales of galactic-scale SMBH pairs, which can be compared with analytical calculations or numerical simulations \citep[e.g.,][and references therein]{DEGN,DeRosa_etal_2019}. Dynamical friction dominates the orbital evolution of these pairs before they become bound binaries. Nevertheless, there are still lingering theoretical uncertainties in this regime, and the timescale spent at these galactic-scale separations depends on the galaxy potential, mass ratio of the merging galaxies, properties of the stellar cores surrounding each SMBH, as well as the effects of gas (both dynamical and accretion onto SMBHs) and dark matter halo properties \citep[e.g.,][]{milosavljevic01,Yu_2002,Callegari_etal_2009,Callegari_etal_2011,Khan2013,McWilliams_etal_2014,Kelley_etal_2017,Tremmel2018,Tamfal_etal_2018,ChenN_etal_2022}. Observationally, galactic-scale SMBH pairs can be identified as dual Active Galactic Nuclei (AGNs) or luminous dual quasars (conventionally defined by $L_{\rm bol}\gtrsim 10^{45}\,{\rm erg\,s^{-1}}$), if both SMBHs are active. Inactive SMBH pairs on galactic-scales are difficult to identify at cosmological distances. Pairs with only one active SMBH may appear as an offset AGN \citep[e.g.,][]{Barrows_etal_2016}. But the detection of offset AGNs becomes challenging at $z>1$, requiring deep imaging/spectroscopy and robust measurements of the host galaxy centroid, as well as careful treatments of selection effects \citep[e.g.,][]{Stemo_etal_2021}. These dual AGNs/quasars signpost galactic-scale SMBH pairs, and can be used to constrain the underlying SMBH pair population, if the AGN duty cycle can be reliably inferred from hydrodynamic simulations. With sufficient statistics to explore the diversity of the SMBH pair population, such as host galaxy properties and redshift evolution, observations of these pairs will enable critical comparisons with theoretical models. The pairing and dynamical evolution of SMBHs at $z\sim 2$ is of particular importance. The specific galaxy merger rate is much higher at cosmic noon than at lower redshifts \citep[e.g.,][]{Duncan_etal_2019}, where both luminous quasars and global star formation reached their peak activity around $z\sim 2$ \citep[e.g.,][]{Madau_Dickinson_2014,Richards_etal_2006a}. This is the prime epoch of the growth of massive SMBHs and galaxies, and the onset of formation of the most massive (e.g., $>10^8\,M_\odot$) SMBH binaries, whose eventual coalescence will dominate the GW signal in the pulsar timing array band. The statistics of galactic-scale SMBH pairs at cosmic noon, as traced by dual quasars, provide critical constraints on the dynamical friction timescales, as well as the impact of galaxy mergers on the fueling of SMBHs. The pair statistics down to $\sim 1$\,kpc may even constrain the nature of dark matter. For example, in the fuzzy dark matter model \citep{Hu2000} and neglecting baryonic effects, SMBH pairs would never get much closer than $\sim$1 kpc because fuzzy dark matter fluctuations inhibit the orbital decay and inspiral at $\sim$~kpc scales \citep{Hui2017}, resulting in a ``pile up'' of SMBH pairs at $\sim 1$~kpc. A spike in the dual quasar fraction towards $\sim 1\,$kpc, above the level that can be explained by quasar duty cycle enhancement in mergers, may be the smoking gun signature of fuzzy dark matter. Unfortunately, given the stringent spatial resolution requirement (e.g., sub-arcsec for $\sim$kpc scales) and the apparent rareness of such pairs, the observational inventory of $z>1.5$ dual quasars at $\lesssim$ tens of kpc separations remains scarce. There are only a handful of serendipitously discovered $\sim$kpc-scale dual/offset AGNs known at lower redshifts \citep[e.g.,][]{Komossa_etal_2003,Comerford_etal_2009b,Liu_etal_2010b,Civano_etal_2010,Goulding_etal_2019}. Dedicated wide-area searches of binary quasars\footnote{For historical reasons, these wide-separation pairs are referred to as ``binary quasars'' \citep[e.g.,][]{Djorgovski_1991,Kochanek_etal_1999} as the two SMBH+galaxy systems are bound to each other. } at $z>1.5$ have compiled tens of quasar pairs at projected separations of $10\,{\rm kpc}\lesssim r_p\lesssim 50\,{\rm kpc}$ \citep[e.g.,][]{Hennawi_etal_2006,Hennawi_etal_2010,Myers_etal_2008,Kayo_Oguri_2012,More_etal_2016,Eftekharzadeh_etal_2017}, starting to probe the galactic-scale environment of quasar pairs. But the $r_p<10\,$kpc regime of high-redshift quasar pairs remains largely unexplored \citep[fig.~1 in][]{Chen_etal_2022}, due to the lack of efficient quasar identification for sub-arcsec pairs that are typically unresolved in ground-based data. Assuming no merger-enhanced AGN duty cycles and applying dynamical friction prediction of galactic-inspiral timescales \citep[i.e., the dynamical friction timescale $t_{\rm df}$ is roughly proportional to $r$, the 3D pair separation, e.g.,][]{Yu_2002,Chen_etal_2020b}, we expect a $\sim$kpc-scale dual quasar fraction of $f_{QQ}\sim 5\times 10^{-5}$ among all quasars, extrapolated from the observed quasar pair statistics on tens of kpc scales \citep[e.g., $f_{QQ}\sim 5\times10^{-4}$,][]{Kayo_Oguri_2012}. To test these expectations, we need to search a large parent quasar sample in order to build up the statistics of rare dual quasars. In this work we measure the galactic-scale (i.e., $r_p\lesssim 30\,{\rm kpc}$) quasar pair fraction at $z\sim 2$ using a different approach than earlier studies \cite[e.g.,][]{Hennawi_etal_2006,Hennawi_etal_2010,Myers_etal_2008,Kayo_Oguri_2012,More_etal_2016,Eftekharzadeh_etal_2017,Silverman_etal_2020}, focusing on the $r_p<10$\,kpc regime that has been poorly explored before. Our approach builds on the all-sky Gaia survey Early Data Release 3 \citep{Fabricius_etal_2021}, which provides precise coordinates, magnitudes, and astrometric measurements for all-sky sources to as faint as $G\sim 21$. In particular, Gaia's nominal $\sim 0\farcs2$ resolution enables the identification of close-separation companions around distant quasars, with quantifiable completeness in resolved pairs as a function of angular separation (\S\ref{sec:data}). Importantly, Gaia proper motion measurements enable efficient separation of stars and quasars, a unique advantage that previous quasar pair searches based on photometric color selection did not have. There is no need to update our analysis using the recent Gaia DR3 release \citep{gaiadr3} since the photometric and astrometric content is essentially unchanged from EDR3 to DR3. This paper is organized as follows. In \S\ref{sec:data} we describe the sample and data used in our systematic search of high-redshift small-scale quasar pairs, with an emphasis on quantifying the completeness of Gaia EDR3 resolved pairs. We present our results in \S\ref{sec:result}, where we compare the observed pair statistics with theoretical predictions of lensed quasars and quasar pairs. We discuss the implications of our findings in \S\ref{sec:disc} and summarize in \S\ref{sec:con}. In this work, we focus on luminous unobscured broad-line quasars exclusively, given the survey depth of Gaia. Occasionally we use the term ``dual quasars'' to refer to physical quasar pairs on galactic scales, following the convention for dual AGNs at $z<1$ \citep[e.g.,][]{Comerford_etal_2009b} that have much lower luminosities than our quasars. By default quasar pairs refer to physically associated pairs within the merging galaxies, rather than unrelated, projected quasar pairs at different redshifts. For practical purposes, we use the term ``double quasars'' to collectively refer to quasar pairs and lensed quasars. We adopt a flat $\Lambda$CDM cosmology with $\Omega_\Lambda=0.7$, $\Omega_{M}=0.3$ and $H_0=70\,{\rm km\,s^{-1}\,Mpc^{-1}}$. Pair physical separations are measured in proper units. \section{Data}\label{sec:data} We start from the latest compilation of spectroscopically confirmed quasars in SDSS-DR16 \citep[DR16Q,][]{Lyke_etal_2020}, and restrict our search to $z>1.5$ quasars. This redshift cut is crucial to this study, and ensures negligible emission from the host galaxy within the Gaia bandpass, which would complicate the source detection and astrometry measurements \citep{Hwang_etal_2020}. We then search for Gaia EDR3 sources in a 3\arcsec\ radius circular region around each SDSS quasar. We further require the matched Gaia sources to have $G<20.25$, which balances the needs for pair statistics and high completeness in Gaia detection and astrometric measurements. For example, \citet{Fabricius_etal_2021} demonstrated nearly 100\% completeness of photometric detection at $G=20$ in low stellar density fields with Gaia EDR3, applicable to SDSS quasars. We have tested Gaia's photometric detection completeness for single sources using the DR16Q quasar catalog, and find that the completeness is $\sim 98.12\pm 0.41\%$ even in the faintest bin $G=[20,20.25]$. Our $G<20.25$ flux limit roughly corresponds to bolometric luminosity $L_{\rm bol}>10^{45.8}\,{\rm erg\,s^{-1}}$ at $z>1.5$ \citep{Shen_etal_2011}, or SDSS $i< 20.13$ (we adopt a magnitude conversion of $G=i+0.12$ assuming a fixed quasar power-law continuum $f_\nu\propto \nu^{-0.5}$). The parent sample satisfying these redshift and magnitude cuts and having single Gaia matches includes 134,796 DR16Q quasars. We focus on Gaia resolved double sources at the SDSS quasar position. Multiple systems with more than two Gaia sources brighter than $G=20.25$ within 3\arcsec\ are only $\sim 2\%$ of double systems, hence negligible. A more important issue is that the completeness of these multiples is much lower and much harder to quantify; thus we ignore this higher-order multiple population. Some quasars with only one matched Gaia source may still be a sub-arcsec quasar pair, which can be recovered with other approaches using additional Gaia parameters \citep[e.g.,][]{Hwang_etal_2020,Shen_etal_2019d,Chen_etal_2022,Mannucci_etal_2022,Makarov_Secrest_2022}, but are not covered here; instead, their contribution to the pair statistics is estimated through the completeness analysis (\S\ref{sec:pair_comp}). \subsection{The Pair Sample}\label{sec:sample} Our initial Gaia-resolved pair sample includes 497 SDSS-DR16Q quasars. However, in 10 pairs both components are bona fide quasars listed in DR16Q and thus are counted twice. Removing these 10 duplicated pairs, our final Gaia-resolved pair sample includes 487 unique pairs. For each pair, the closer Gaia match is designated as the corresponding SDSS DR16Q quasar. This is generally the case. However, in very rare cases of pairs separated by $\lesssim 1\arcsec$, the SDSS optical centroid may be dominated by the companion. Nevertheless, this detail does not affect any of our statistical analyses below. We classify the companion as ``star-like'' in 416 pairs where its proper motion is detected by Gaia at $>3\sigma$ significance; for comparison, only $\sim 2\%$ of Gaia singly-matched quasars have $>3\sigma$ proper motion detection, meaning our proper motion cut will only inadvertently exclude a negligible fraction of bona fide double quasars. The remaining 71 resolved pairs are our initial sample of double quasars. Pair separations are computed using Gaia EDR3 coordinates, which can slightly exceed the 3\arcsec\ cross-matching radius between SDSS and Gaia. The full pair catalog of 487 pairs is presented in Table \ref{tab:sample}. Fig.~\ref{fig:rawdist} (left) shows the distributions of Gaia $BP-RP$ color for the DR16Q quasar in the pair, ``star-like'' and ``quasar-like'' companions for the full Gaia pair sample. Because Gaia photometry is measured within a $3\farcs5\times 2\farcs1$ window \citep{Riello2021}, source deblending may be significantly impacted for the closest pairs. Thus we have excluded pairs with separations $<1\arcsec$ in this color distribution plot to avoid crosstalks in their photometric color measurements. Their color distributions suggest that ``star-like'' companions indeed have different colors than the primary quasars or the ``quasar-like'' companions based on Gaia proper motion detection. \begin{table} \caption{Pair Sample Data}\label{tab:sample} \resizebox{\columnwidth}{!}{% \begin{tabular}{llll} \hline\hline Column & Format & Units & Description \\ (1) & (2) & (3) & (4) \\ \hline SDSS\_NAME & STRING & & J2000 hhmmss.ss$\pm$ddmmss.s \\ Z & DOUBLE & & Default redshift from DR16Q \\ PLATE & LONG & & Plate number (SDSS spec) \\ FIBERID & LONG & & FiberID (SDSS spec) \\ MJD & LONG & & MJD (SDSS spec) \\ GAIA\_RA1 & DOUBLE & deg & Gaia RA \\ GAIA\_DEC1 & DOUBLE & deg & Gaia DEC \\ GAIA\_RA2 & DOUBLE &deg & Gaia RA\\ GAIA\_DEC2 & DOUBLE &deg & Gaia DEC \\ G1 & DOUBLE & mag & Gaia G mag\\ G2 & DOUBLE & mag & Gaia G mag \\ BP\_RP1 & DOUBLE & mag & Gaia BP-RP color \\ BP\_RP2 & DOUBLE & mag & Gaia BP-RP color \\ PM\_SIG1 & DOUBLE & & PM significance\\ PM\_SIG2 & DOUBLE & & PM significance\\ PAIR\_SEP & DOUBLE & arcsec & Pair separation\\ TYPE & STRING & & Pair classification \\ KNOWN & STRING & & Literature classification \\ F\_COMP & DOUBLE & & pair completeness (\S\ref{sec:pair_comp})\\ \hline \hline\\ \end{tabular} } {\raggedright Notes. For each pair, index 1 refers to the DR16Q quasar and 2 refers to the companion, regardless of their relative brightness (i.e., the quasar can be fainter than the companion, especially at large pair separations). Gaia measurements are from EDR3 (null values are ``NaN''). The column ``TYPE'' indicates pair classification: ``QQ'' refers to double quasar; ``QS\_PM'' refers to quasar+star pair based on proper motion; ``QS\_PCA'' refers to quasar+star pair based on spectral PCA analysis; one quasar (J0033+2015) is a known quasar+star pair \citep{More_etal_2016} and we set its TYPE=``QS\_KNOWN''. The associated FITS file is available in the online version of this paper. } \end{table} Fig.~\ref{fig:rawdist} (right) shows the distributions of pair separation for double systems with ``star-like'' and ``quasar-like'' companions. The separation distribution for ``star-like'' companions rapidly decline towards smaller separations, as anticipated from the reduction of geometric cross section and the constant sky density of a foreground (star) population, modulo pair-resolving incompleteness towards $\lesssim 1\arcsec$ separations. In contrast, the separation distribution for ``quasar-like'' companion remains more or less constant, suggesting that it is an intrinsic population associated with the primary quasar. Both the color and separation distributions in Fig.~\ref{fig:rawdist} indicate that the classification of star and quasar companions based on proper motion is reasonably good. Of course, it is possible that some detected proper motions are caused by systematics (especially for sub-arcsec pairs where the two sources overlap in photometric/astrometric measurements). Here we opt to exclude these potential double quasars mis-identified as quasar+star pairs due to bad proper motion measurements, in order to maintain a high-purity double quasar sample. Likewise, we expect that there is still residual contamination of star superposition in these close pairs which we classified as ``quasar-like'' companions, especially at $\lesssim 1\arcsec$ separations where the measurement of Gaia proper motion is either unavailable or could be impacted by the close neighbor. We estimate this residual contamination rate using 43 pairs at $<1\farcs5$ separations from the initial sample of 71 double quasars. These pairs are close enough such that the SDSS fiber spectroscopy (with a fiber diameter of 2\arcsec\ or 3\arcsec) encloses most light from both components. We use a spectral Principal Component Analysis (PCA) technique to decompose the SDSS spectrum into potential quasar+star superpositions, using quasar and stellar PCA templates from the SDSS website. Fig.~\ref{fig:pca} shows that such superpositions can be reliably identified from the SDSS spectrum, provided that the companion is not substantially fainter (e.g., by a factor of $\sim 10$ in flux) than the primary quasar (96.6\% of Gaia resolved pairs in our sample have flux contrast ratio $<10$). However, automatic classifications with PCA-decomposed spectra are often unreliable due to degeneracies in the decomposition and noise in the data. Therefore we manually inspect all PCA decomposition results and flag obvious star superpositions. This spectral analysis indicates that there is $\sim 23\%$ (10/43 in the subset of $<1\farcs5$ pairs) contamination of star+quasar superposition in this subset of pairs. These apparent star superpositions have separations between 0\farcs2 and 1\farcs2, with no obvious dependence of the contamination rate on pair separation given the small number statistics. The PCA results for these 10 apparent quasar-star superpositions are shown in Fig.~\ref{fig:pca}. We remove these apparent quasar+star pairs from our double quasar sample. There is no way to remove additional stellar contamination in the $>1\farcs5$ pairs without additional follow-up observations. However, the proper motion measurements are much more reliable for pairs separated by $>1\arcsec$ to remove star superpositions in our initial cut. Thus we expect the residual contamination rate is substantially smaller than $\sim 20\%$ at $>1\farcs5$ separations. The same spectral analysis reveals no obvious, physically unrelated, projected quasar pairs, in which case we would observe different emission line redshifts in the spectrum if the redshift difference is $>2000\,{\rm km\,s^{-1}}$ \citep[the common definition of projected quasar pairs, e.g.,][]{Hennawi_etal_2006,Hennawi_etal_2010}. This is consistent with our expectation from reduced cross section of chance superpositions for our $<3\arcsec$ separation pairs: \citet{Hennawi_etal_2010} estimated $\sim 30\%$ of the double quasars at $<60\arcsec$ separations are projected pairs, which would imply negligible projected pairs at $<3\arcsec$ separations. After removing foreground star superpositions, most of the remaining 61 pairs should either be genuine dual quasars, or gravitationally lensed quasar images. Extended host galaxy emission from old stellar populations at $z>1.5$ would be too faint to be detectable in the Gaia band, and compact UV-emitting star formation regions in the host galaxy is unlikely to be brighter than our flux limit (which implies quasar luminosities). However, the population of lensed quasars cannot be readily removed. Indeed, resolved Gaia pairs have been used to identify candidate gravitationally lensed quasars and confirmed in follow-up observations \citep[e.g.,][]{Lemon_etal_2017,Lemon_etal_2018,Lemon_etal_2022,Krone-Martins_etal_2018}. We cross-match the 61 pairs in our sample with the Gravitationally Lensed Quasar Database\footnote{https://research.ast.cam.ac.uk/lensedquasars/; latest version in 2019.} and the follow-up sample of Gaia DR2-selected candidate lenses and quasar pairs in \citet{Lemon_etal_2022}, as well as additional SDSS quasar lens and pair searches \citep[][]{Hennawi_etal_2006,Hennawi_etal_2010,Oguri_etal_2008,Inada_etal_2012,Myers_etal_2008,Kayo_Oguri_2012,More_etal_2016,Eftekharzadeh_etal_2017}. We find that there are 25 systems that are reported lenses (but only four of them have image separations $<1\arcsec$) in follow-up observations. There are 5 systems reported as a physical quasar pair. These publicly reported cases are indicated in the ``KNOWN'' column in Table~\ref{tab:sample}. It is possible that there are additional sources observed in the literature that are missed from the above resources. Mis-classifications of lenses and pairs among these reported cases are rare but possible (see discussions in \S\ref{sec:disc1}). Finally, three additional pairs among the 61 (J082341.08$+$241805.6, J084129.77$+$482548.4, and J212243.01$-$002653.8) have been observed in our pilot follow-up with HST (optical and IR) and/or VLA. J0823 and J0841 are confirmed double quasars, more likely dual than lensed quasars (Y.~Chen et~al., in prep.). J2122 was reported as a dual/lensed quasar based on resolved 2-band optical HST color \citep[][]{Chen_etal_2022}, pending further confirmation from additional follow-ups. During cross-matching our full SDSS+Gaia pair sample (487 pairs) with the above literature on quasar pairs and lenses (as well as our ongoing follow-up), we found one system (J135306.34+113804.7) classified by us as a quasar-star pair based on Gaia proper motion turns out to be a lensed quasar \citep{Inada_etal_2012}. On the other hand, only one system (J003337.58+201538.1) classified by us as a double quasar (separated by 1\farcs69) turns out to be a quasar-star pair based on spatially resolved optical spectroscopy \citep{More_etal_2016}. We remove J0033+2015 from further analysis, leaving a final cleaned double quasar sample of 60 objects. Unfortunately, the completeness of follow-up observations of candidate quasar pairs or lensed quasars is difficult to quantify and varies across different surveys. Moreover, the constraints on the lensed quasar population in the sub-arcsec regime are essentially absent. For these reasons, we statistically evaluate the contribution of lensed quasars in our pair sample using mock catalogs, as described in \S\ref{sec:result}. Nevertheless, the fact that $\sim$half of our double quasar sample are already confirmed lensed quasars or quasar pairs indicates that our SDSS+Gaia selection is highly effective, and the resulting sample of 60 objects has a high purity of genuine double quasars. Fig.~\ref{fig:z_sep} displays the distribution of these 60 pairs in the redshift-separation space. These double quasars have pair separations between 0\farcs4 and $\sim 3\arcsec$, and form the basis of our subsequent analyses. At the sample median redshift of $z=2$, these pairs probe projected separations of $3\lesssim r_p\lesssim 30\,{\rm kpc}$, i.e., on galactic scales. Individual pairs may still have 3D separations exceeding 30\,kpc, but statistically this population still traces the radial distribution of quasar pairs. Projection effects are properly taken into account when comparing with theoretical predictions in \S\ref{sec:result} and \S\ref{sec:disc}. The fact that close photometric companions within 3\arcsec\ of SDSS quasars are dominated by the foreground (star) population signifies the necessity of additional metrics to remove foreground contamination in quasar pair searches. This high foreground contamination rate is verified in a random offset test. We shuffle the positions of SDSS quasars by 1 arcmin and search for Gaia sources within a 3\arcsec-radius circle. This random offset test maintains the foreground stellar density distribution applicable to the SDSS quasar sample. We find a chance star superposition around 0.68\% of the quasars when limiting to the same $G<20.25$ limit of Gaia sources. This is even higher than the $\sim 0.36\%$ rate for the observed sample above. The main reason for the lower superposition contamination rate in the observed quasar sample is due to SDSS selection. SDSS quasars were targeted by color selection with photometry (with $\sim 1\farcs4$ seeing) and spectroscopically confirmed with 2\arcsec\ or 3\arcsec-diameter fiber spectroscopy. The presence of a star brighter than the quasar itself will both impact the target selection and the spectroscopic classification. In this sense, the SDSS quasar sample is biased against close pairs with bright star companions. If we further require $G$ is no more than 2.5 magnitude brighter than the quasar in the offset test, we find 0.47\% quasars have chance foreground superpositions, roughly consistent with the observed rate. Additional effects, e.g., incompleteness in resolving pairs at $<1\arcsec$, would further reduce the observed foreground contamination rate. \subsection{Pair Completeness}\label{sec:pair_comp} The raw observed pair statistics as a function of separation (Fig.~\ref{fig:rawdist} right) suffer significantly from incompleteness in the sub-arcsec regime, as Gaia can only resolve the pair at $\sim 0\farcs2$ resolution in the along-scan direction (this is somewhat remedied by multiple scans along different directions). Moreover, the presence of a close neighbor decreases the probability of detecting both sources photometrically by Gaia. The pair-resolving completeness as a function of separation has been estimated \citep{Fabricius_etal_2021} using the Washington Double Star Catalog (WDS) catalog \citep{Mason_etal_2001}, demonstrating significant improvements of EDR3 over DR2. Based on the WDS catalog, the pair completeness is $\sim 50\%$ (20\%) at 0\farcs5 (0\farcs3) separations. However, the WDS catalog has a different magnitude distribution (i.e., much brighter) than the parent SDSS quasar sample, and it is reasonable to expect that the completeness of Gaia-resolved pairs depends on both magnitude and magnitude contrast. Therefore, we carry out an independent measurement of the pair completeness in Gaia EDR3, as detailed below. We consider the detectability of close pairs as functions of the magnitude of the brighter primary source, the magnitude difference between the two sources, and their angular separation. There have been previous studies focusing on the Gaia completeness correction as functions of angular separation \citep{Fabricius_etal_2021} and magnitude differences \citep{El-Badry2018b}, but the exact completeness correction depends on the detailed selection criteria of the sample of interest \citep{El-Badry2018b}. We assemble a random pair sample where the pairs are dominated by random stellar pairs, and derive the completeness correction by comparing the observed number of pairs with the expected number of random pairs from a constant sky density of stars, $N ds \propto sds$, where $s$ is the projected angular separation. Following \cite{Hwang2022ecc}, we collect all pairs in the crowded field at $30^\circ<l<55^\circ$ and $5^\circ<b<7^\circ$. This region is chosen such that the Gaia source density is high at low Galactic latitudes, and the region is not strongly affected by dust extinction. We query all Gaia EDR3 sources within this sky region, without any other criteria. Then we collect all pairs with angular separations $<10$\arcsec. To reduce the binary star contribution, which is more prominent at $G<16$ (because brighter stars are closer and thus their binary companions are more likely to be spatially resolved), we further impose a cut on parallaxes $<0.5$\,mas, resulting in 16.7 million unique pairs. We derive the completeness correction as a function of three parameters: magnitude of the brighter primary ($G_{pri}$), magnitude difference ($\Delta G = G_{sec}-G_{pri}$ where $G_{sec}$ is the $G$-band magnitude of the secondary), and angular separation. To this end, we bin these random pairs by $G_{pri}=15-21$ with steps of 1\,mag, $\Delta G=0-3$ with steps of 0.5\,mag, and angular separations $0\farcs 1-4\arcsec$ with steps of 0\farcs2. Then at each point of the 3D parameter grid, we compute the completeness factor $f_{\rm comp}=N_{obs}/N_{model}$, where $N_{obs}$ is the observed number of pairs in the grid and $N_{model}$ is the expected number of pairs from the model. The model is computed as follows. First, for every primary magnitude bin, we start with the first magnitude difference bin (i.e., $0<\Delta G<0.5$) and compute the expected number of pairs along the separation bin based on the observed number of pairs at 5-10\arcsec\ and the expected geometric distribution ($N ds \propto sds$). Next, under the assumption that the sample is dominated by random pairs and therefore the magnitude difference distribution is independent of pair separation, we use the magnitude difference distribution from pairs at 5-10\arcsec\ as the ground truth, and apply this distribution to smaller separations to obtain the expected pair counts as a function of $\Delta G$ at different separations. Our completeness correction uses the expected geometric distribution $Nds\propto sds$, which is applicable when the sample is indeed dominated by random pairs in each 3D bin. While the overall sample is dominated by random star pairs in this crowded region \citep{Hwang2022ecc}, if we naively bin the sample into the 3D grid without the parallax cut, there are some noticeable binary contributions that cause the completeness correction $>1$ at $(G_{pri}<16) \wedge (\Delta G<0.5) \wedge$ (angular separations $<1$\arcsec), where $\wedge$ is the logical AND operator. This binary contribution in the observed data is due to the fact that brighter stars are closer, and thus their binary companions are more likely to be resolved by Gaia, producing an excess of ``twin'' wide binary population with $\Delta G<0.25$ \citep{El-Badry2019,Hwang2022twin}. After we remove nearby stars by the criterion of parallax$<0.5$\,mas, the binary contributions are strongly suppressed and the completeness correction is well-behaved. Fig.~\ref{fig:pair_comp} shows the pair completeness as a function of angular separation. Each panel represents a different range of the primary's G-band magnitude $G_{pri}$, and each colored symbol is for different magnitude difference $\Delta G$. For a handful of bins the completeness correction can slightly exceed unity due to Poisson fluctuations, and we manually set $f_{\rm comp}$ to 1 in these bins (our quasar pairs rarely fall in these bins anyway). The black lines in Fig.~\ref{fig:pair_comp} are the model-fitted completeness of Gaia EDR3 derived from \citet{Fabricius_etal_2021} using the Washington Double Star Catalog \citep{Mason_etal_2001}, which does not take primary magnitudes and magnitude differences into account. The top panels show that our completeness correction for bright $G_{pri}$ bins agrees well with the black line. At the fainter end of $G_{pri}>18$, however, our completeness correction shows that $\Delta G$ plays an important role which is not captured by the black line, emphasizing the importance of deriving the customized completeness correction for our quasar sample. Due to Gaia's detection limit at $\sim21$\,mag, only $\Delta G<1.5$ have completeness correction available in the $20<G_{pri}<21$ panel (bottom right). The pair completeness correction on the 3D grid of ($G_{pri}$, $\Delta G$ and $\Delta\theta$) is available as an electronic FITS table with its content described in Table \ref{tab:comp}. \begin{table} \caption{Binned Pair Completeness}\label{tab:comp} \resizebox{\columnwidth}{!}{% \begin{tabular}{llll} \hline\hline Column & Format & Units & Description \\ (1) & (2) & (3) & (4) \\ \hline GPRI & FLOAT[2] & mag & Boundary of $G_{pri}$ \\ DG & FLOAT[7] & mag & Boundaries of the $\Delta G$ grid \\ DTHETA & FLOAT[20] & arcsec & Boundaries of the $\Delta\theta$ grid \\ FCOMP & FLOAT[19,6] & & Pair completeness \\ \hline \hline\\ \end{tabular} } {\raggedright Notes. For each row of the FITS table, GPRI is the boundary of $G_{pri}$, and DG and DTHETA are the boundaries (not bin center) of the $\Delta G$ and $\Delta\theta$ grids in that $G_{pri}$ bin, respectively. The bin size for the $\Delta G$ grid is 0.5 mag, and the bin size for the $\Delta\theta$ grid is 0\farcs2. The binned completeness FCOMP is set to ``NAN'' if the observed number of star pairs is $< 3$ in that bin. } \end{table} \begin{table} \caption{Binned Quasar Pair Statistics}\label{tab:pair_stat} \resizebox{\columnwidth}{!}{% \begin{tabular}{lcccccc} \hline\hline $\Delta\theta$ (\arcsec) & $N_{\rm QQ}$ & $N_{\rm QQ, corr}$ & $\sigma_{-}$ & $\sigma_{+}$ & $\sigma_{\rm poisson}$ & $N_{\rm QQ, EDR3}$ \\ (1) & (2) & (3) & (4) & (5) & (6) & (7) \\ \hline 0.4 & 1 & 8.0 & 8.0 & 8.0 & 8.0 & 3.7 \\ 0.6 & 9 & 21.0 & 6.7 & 6.7 & 7.0 & 15.8 \\ 0.8 & 7 & 9.2 & 2.8 & 2.9 & 3.5 & 9.0 \\ 1.0 & 4 & 4.7 & 2.4 & 2.4 & 2.4 & 4.6 \\ 1.2 & 6 & 6.9 & 3.4 & 2.3 & 2.8 & 6.5 \\ 1.4 & 6 & 6.5 & 2.2 & 3.1 & 2.7 & 6.3 \\ 1.6 & 1 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\ 1.8 & 5 & 5.0 & 2.0 & 2.0 & 2.2 & 5.2 \\ 2.0 & 3 & 3.0 & 2.0 & 2.0 & 1.7 & 3.1 \\ 2.2 & 6 & 6.1 & 2.1 & 2.1 & 2.5 & 6.1 \\ 2.4 & 4 & 4.0 & 2.0 & 2.0 & 2.0 & 4.1 \\ 2.6 & 2 & 2.0 & 1.0 & 1.0 & 1.4 & 2.0 \\ 2.8 & 3 & 3.0 & 2.0 & 1.0 & 1.7 & 3.1 \\ 3.0 & 3 & 3.0 & 2.0 & 2.0 & 1.7 & 3.1 \\ 0.3--3.1 & 60 & 83.5 & 7.1 & 7.5 & -- & 73.5 \\ \hline \hline\\ \end{tabular} } {\raggedright Notes. Pair statistics are measured in $\Delta\theta$ bins with a linear bin size of $0\farcs2$. Columns (3)--(5) are the pair statistics corrected for completeness ($N_{\rm QQ, corr}$), with the uncertainties ($\sigma_{-}$ and $\sigma{+}$) estimated from bootstrap resampling. Column (6) lists the uncertainties in $N_{\rm QQ, corr}$ estimated from Poisson counting uncertainties from the raw pair counts $N_{\rm QQ}$. Column (7) lists the corrected pair counts using the estimated completeness in \citet{Fabricius_etal_2021}, which remains less than unity even at $\Delta\theta>2\arcsec$. } \end{table} \section{Results}\label{sec:result} To calculate the pair fraction, we define the parent quasar sample as the $\sim 134$\,k SDSS quasars with the same magnitude and redshift cuts as our pair sample, but are unresolved by Gaia. The overall abundance of double quasars with separations over $\sim 0\farcs3-3\arcsec$ is negligible compared to the parent single quasar population (e.g., even after completeness correction, the total pair fraction over these scales is of order $10^{-4}$ to $10^{-3}$). In any case, the parent sample only provides the denominator in the pair fraction calculation and does not affect the relative fraction as a function of pair separation. We show the completeness-corrected double quasar fraction as a function of angular separation in Fig.~\ref{fig:pair_stat}, where the pair fraction is defined as the ratio between the number of pairs in each separation bin and the total number of quasars in the parent sample. In detail, we use the binned completeness estimates $f_{\rm comp}$ over a grid of G magnitude of the primary, magnitude contrast and angular separation as quantified in \S\ref{sec:pair_comp}. Each quasar pair in our sample is weighted up by $1/f_{\rm comp}$, and the correction is significant only in the sub-arcsec regime. We estimate the uncertainties of the corrected pair statistics using bootstrap resampling of the pairs, which are consistent with the Poisson uncertainties estimated from the raw pair counts in each separation bin (see Table~\ref{tab:pair_stat}). The cumulative pair fraction within $0\farcs3-3\farcs1$ is $6.2\pm0.5\times 10^{-4}$ among $z>1.5$ quasars. Dividing these quasar pairs at their median redshift $\langle z \rangle =2$, we measure overall pair fractions over these scales of $6.6\pm1.2\times 10^{-4}$ and $5.9\pm1.0\times 10^{-4}$ in the lower ($\langle z \rangle =1.7$) and higher ($\langle z \rangle =2.4$) redshift bins, respectively, indicating there is no strong evolution in the pair fraction over the redshift range probed by our sample. For completeness, we also present in Table~\ref{tab:pair_stat} the corrected pair statistics using the pair-resolving completeness estimated in \citet{Fabricius_etal_2021} based on the WDS catalog. As demonstrated in \S\ref{sec:pair_comp}, our quasars are substantially fainter than sources in the WDS catalog, and the pair completeness in the sub-arcsec regime is somewhat lower than that in \citet{Fabricius_etal_2021}. Nevertheless, the corrected pair statistics using the completeness in \citet{Fabricius_etal_2021} are consistent with our fiducial estimates within $1\sigma$. The uncertainty in the pair statistics is the largest in the smallest $\Delta\theta$ bin, where there is only one observed quasar pair at 0\farcs4 separation, J0841+4825. This particular pair was first reported in \citet{Shen_etal_2021NatAs} as a genuine double quasar, although their data were insufficient to rule out the lensed quasar scenario. As shown in Fig.~\ref{fig:pair_stat}, the completeness-corrected double quasar fraction (per linear separation bin) gradually rises towards smaller separations (for reference, 1\arcsec\ corresponds to $\sim 8.5$\,kpc at $z\sim 2$). A constant pair fraction with separation is consistent with a quasar auto-correlation function of $\xi(r)\propto (r/r_0)^{-2}$, where $r$ is the 3-dimensional pair distance and $r_0$ is the correlation length. The steepening of the pair fraction towards small separations implies a steepening in small-scale quasar clustering at $\lesssim 30\,$kpc physical scales. Not all pairs in our final cleaned sample of 60 systems are physical quasar pairs, as some of them should be gravitationally lensed quasars. The lensed quasar statistics in the sub-arcsec regime is not well constrained observationally, and therefore we use an updated mock catalog of lensed quasars from \citet{Oguri_Marshall_2010} to estimate the lensed quasar contribution \citep[see also][]{Lemon_etal_2022}. The updated version uses a galaxy velocity dispersion function for all types of galaxies (in contrast to only early-type galaxies considered in the original version of the mock) and imposes no lower limit on the image separation (in contrast to the lower limit of image separation of $0\farcs5$ in the original version of the mock). We include all lensed systems (e.g., doubles, quads, etc.) in the mock catalog with two (and only two) images above the flux limit, with the same redshift cut of $z>1.5$ as for our observed sample. The mock catalog uses SDSS $i$ band magnitude, and we adopt $i<20.2$ for individual resolved images that roughly corresponds to the same $G$-band limit used for the observed pair sample. Varying the flux limit in the mock lensed quasar catalog by one magnitude introduces less than a factor of two in the lensed quasar fractions. Lensed quasars with more than two images above the flux limit would not have been included in our pair sample. The lensed quasar fraction (blue lines in Fig.~\ref{fig:pair_stat}) also shows a gradual increase towards the sub-arcsec regime, mainly due to the increase in the abundance of less massive lens galaxies. In Fig.~\ref{fig:pair_stat_sim}, we compare our double quasar fraction with predictions for dual AGNs at $z\sim 2$ in the cosmological hydrodynamic simulation \texttt{ASTRID} \citep{ChenN_etal_2022c}. \texttt{ASTRID} is a recently developed large-volume, high-resolution (with a gravitational softening of $1.5\,{\rm kpc}/h$ and a dark matter mass resolution of $9.6\times 10^6\,M_\odot$) cosmological hydrodynamic simulation that studies the evolution of galaxies and SMBHs. It utilizes a new version of the \texttt{MP-Gadget} \citep{MPGadget2018} simulation code to solve the gravitational evolution (with an N-body tree-particle-mesh approach), hydrodynamics (with Smoothed Particle Hydrodynamics), and astrophysical processes with a series of subgrid models. With a comoving volume of $(250\,{\rm Mpc}/h)^3$, \texttt{ASTRID} is the largest galaxy formation simulation up to date that covers the epoch of the cosmic noon. The large volume of \texttt{ASTRID} can provide a statistical sample of the rare quasar population, and the high resolution enables detailed studies of the quasar pair statistics and environments down to galactic scales. Details of the \texttt{ASTRID} simulation and the SMBH population overview can be found in \citet{Bird2022,Ni2022,ChenN2022b}, and a comprehensive analysis of the dual AGN population predicted by \texttt{ASTRID} can be found in \cite{ChenN2022b,ChenN_etal_2022c}. Given the simulation volume of \ASTRID, we can only explore dual AGNs with lower luminosities than our quasar sample. For instance, there are 3 dual AGNs at $z\sim 2$ in \ASTRID\ that sample the same luminosity and pair separation ranges as our observed sample, which is not enough for detailed statistical analysis. We therefore use two lower bolometric luminosity cuts, $L_{\rm bol}>10^{44.8}\,{\rm erg\,s^{-1}}$ and $L_{\rm bol}>10^{43}\,{\rm erg\,s^{-1}}$, to select the parent single AGNs and dual AGNs in \ASTRID. We impose a BH mass cut of $M_{\rm BH}>10^7\,M_\odot$ in the simulated AGNs -- this BH mass scale is resolved in \ASTRID. The simulated dual AGNs are restricted to have radial separations $<50\,{\rm kpc}$ and transverse separations $<30\,{\rm kpc}$. The resulting simulated dual AGN sample includes 59 and 1282 pairs for the two luminosity cuts, respectively. As shown in Fig.~\ref{fig:pair_stat_sim}, the dual AGN fraction is generally lower for the higher luminosity cut. Nevertheless, both simulated samples show an enhancement in the pair fraction towards the smallest separations, as seen in the observed sample. If we could increase the luminosity threshold further in the simulated AGN sample as pair statistics allow, we would expect to see further reduced dual fraction. This luminosity trend in the dual AGN fraction can be qualitatively understood as follows: if assuming no merger-enhanced AGN duty cycles $f(L_{\rm min})$, the dual AGN fraction (among all AGNs with the same luminosity threshold $L_{\rm min}$) depends on $L_{\rm min}$ through the duty cycle, i.e., $\propto f(L_{\rm min})$, while the fraction of pairs with a single AGN among all AGNs is constant with the luminosity threshold. As $L_{\rm min}$ increases, the duty cycle $f(L_{\rm min})$ decreases, leading to reduced dual AGN fraction among all AGNs. The average AGN duty cycle at $z\sim 2$ in the \ASTRID\ simulation (for all $M_{\rm BH}>10^7\,M_\odot$ SMBHs) roughly decreases by a factor of 10 from $L_{\rm bol}>10^{43}\,{\rm erg\,s^{-1}}$ to $L_{\rm bol}>10^{44.8}\,{\rm erg\,s^{-1}}$, and by another factor of $\sim 10$ from $L_{\rm bol}>10^{44.8}\,{\rm erg\,s^{-1}}$ to $L_{\rm bol}>10^{45.8}\,{\rm erg\,s^{-1}}$ (N.~Chen et~al., in prep). Thus we expect the simulated dual AGN fraction for $L_{\rm bol}>10^{45.8}\,{\rm erg\,s^{-1}}$ (matching our observed sample) would be a factor $\sim 10$ smaller than the red solid points in Fig.~\ref{fig:pair_stat_sim}, which would match the observed statistics. However, we do expect somewhat enhanced AGN duty cycles in galaxy mergers, which would elevate the simulated dual AGN fraction. The above comparison with simulations should be interpreted with some caution. First of all, currently the simulated sample does not distinguish between obscured and unobscured AGNs, while our quasar sample only contains unobscured objects. Simulations at $z\sim2$ have shown that many luminous dual AGNs are completely obscured in gas-rich mergers \citep{ChenN_etal_2022c}, which would significantly reduce the observable fraction of unobscured dual quasars. Secondly, it might be necessary to further match the SMBH masses in this comparison, i.e., the observed SDSS quasars have BH masses $>$ a few $\times 10^8\,M_\odot$ \citep[e.g.,][]{Shen_etal_2011}. In any case, we conclude that the observed double quasar statistics are roughly consistent with predictions for the dual AGN population from simulations. The intriguing finding that the observed double quasar statistics are consistent with theoretical predictions for both lensing and simulated dual AGNs may indicate that these two populations are comparable in number by coincidence. A complete division between lenses and dual quasars in our sample with follow-up observations will fully address this important issue. We further discuss the implications of our observed pair statistics in \S\ref{sec:disc}. Our definition of the quasar pair fraction is free of a selection bias related to the flux limit and source blending. When selecting a sample of unresolved systems (either single quasars or unresolved pairs), potential pairs or lensed images would boost the combined flux to above the flux limit, and enhance the presence of pairs in the parent sample. In case of gravitational lenses, this is the magnification bias. However, our Gaia sample is a resolved pair sample, and each component of the pair is above the flux limit. In other words, our pair fraction is defined as the fraction of $G<20.25$ quasars that have a resolved quasar companion that is also brighter than $G=20.25$. Pairs with either of the components fainter than the flux limit, even if the other component or the combined flux is above the flux limit, would not have been included in our pair sample to contribute to the numerator. The parent quasar sample has the same flux limit, and could include fainter pairs or lensed images that boost the combined fluxes above the threshold, but such small-scale pairs/lensed images are rare and would only slightly perturb the denominator in our pair fraction calculation. We next examine the flux ratios of the observed quasar pairs. The sample statistics is insufficient to explore the flux ratios as a function of separation in detail, and hence we focus the discussion on the distribution for the full quasar pair sample with $\Delta\theta<3\arcsec$. However, dividing the quasar pair sample into wide-separation ($\Delta\theta>1\arcsec$) and close-separation ($\Delta\theta<1\arcsec$) pairs, there is no noticeable difference in the pair flux ratio distribution. Fig.~\ref{fig:flux_ratio} shows that the observed quasar pair flux ratio distribution peaks near unity. The flux limit ($G<20.25$) in our sample selection reduces the dynamic range in the flux ratio of the observed pairs, biasing the distribution of flux ratios towards more equal-flux values. To illustrate this effect, we consider the wide-separation ($1\arcsec<\Delta\theta\lesssim 3\arcsec$) quasar pairs in our sample, since this subset does not suffer from pair-resolving completeness as much as the close-separation pairs do (\S\ref{sec:pair_comp}). In other words, the main selection effect is due to the flux limit $G<20.25$ for both components of the pair. In Fig.~\ref{fig:flux_ratio_selection} we demonstrate the effect of the flux limit with simple, idealized simulations, with an assumed intrinsic flux ratio ($\Delta G$) distribution (solid lines). The magnitude distribution of the primary (brighter) component follows the observed distribution (Fig.~\ref{fig:pair_prop}). We test two different input $\Delta G$ distribution: (1) a Gaussian distribution with mean $\Delta G=1$ (mag) and a dispersion of 0.3 mag, and (2) a uniform distribution of $\Delta G$ within [0,3]\,mag. When the input $\Delta G$ distribution narrowly peaks at a non-equal ratio value (the Gaussian distribution case), the observed pair flux ratio distribution (dotted lines) is slightly shifted to smaller values, but the peak is more or less preserved. On the other hand, if the input distribution is broad (the uniform distribution case), the resulting observed $\Delta G$ distribution peaks at equal flux ratio. In Fig.~\ref{fig:flux_ratio}, we show the AGN pair flux ratio distribution from the \ASTRID\ simulation at $z\sim 2$ \citep[][]{ChenN2022b}, again restricting to physical AGN pairs with transverse separations $r_p<30\,{\rm kpc}$ and radial separations $<50\,{\rm kpc}$. Because the AGN population in the simulations is limited by the simulation volume, we relax the luminosity threshold for both components to be $L_{\rm bol}> 10^{44.8}\,{\rm erg\,s^{-1}}$. The pair flux ratios from the simulated AGN pairs also peak around equal-flux ratio, although the peak is somewhat less prominent than that of the observed sample. If we lower the luminosity threshold in the simulated sample, more pairs with large luminosity ratios will be included in the sample, further weakening the peak prominence at equal-flux ratio. Similarly, if we increase the luminosity threshold to match our sample ($L_{\rm bol}>10^{45.8}\, {\rm erg\,s^{-1}}$), we expect simulated dual AGNs would produce a prominent peak around unity flux ratio, similar to the observed distribution. The intrinsic pair flux ratio distribution (for the SMBH pair population) from the simulations, however, is much broader if we relax the flux limit on the fainter component. Synchronized growth of the pair of SMBHs that rapidly drives their masses towards equality does not seem to be the case on these $<$ tens of kpc scales. Finally, we show the flux ratio of lensed quasar images in Fig.~\ref{fig:flux_ratio}, using the same mock catalog described above. Coincidently, flux ratios of lensed quasar images also peak around unity flux ratio. This peak is primarily due to selection effects. Double lenses with large magnification factors (which tend to have small magnitude contrasts) from intrinsically fainter quasars are over-represented in the flux-limited sample due to the lensing magnification bias. Quad lenses often have two bright images with similar magnifications near the critical curve and this population preferentially resides in the small-separation regime. The magnification bias would also enhance the presence of equal-flux quad lenses (only the two brightest images) in the flux-limited sample. Therefore, the observed pair flux ratio distribution cannot be used to readily distinguish the lensing and quasar pair scenarios in the statistical sense. \section{Discussion}\label{sec:disc} Since we focus on luminous quasars at $z>1.5$ with bolometric luminosities $L_{\rm bol}>10^{45.8}\,{\rm erg\,s^{-1}}$ and the bulk of the sample are near the flux limit, we make the assumption that the intrinsic quasar pair fraction (as a function of separation) is more or less constant over the luminosity range probed in this work. This simplifies most of the following discussions. The luminosity dependence of pair statistics will be explored in future work with improved sample statistics and more extended dynamic range in quasar luminosity. \subsection{Lensing vs Pairs}\label{sec:disc1} Fig.~\ref{fig:pair_stat} shows that the measured quasar pair fraction in our Gaia sample agrees well with the predicted lensed quasar population at $<1\farcs5$ separations. However, we caution that the lensed quasar fraction is based on mock catalogs and may be different from the actual lensed quasar population at these small separations. In reality, the observed quasar pairs are comparable in number to lensed quasars over few \arcsec\ separations \citep[e.g.,][]{Hennawi_etal_2006,Hennawi_etal_2010,Kayo_Oguri_2012}, but the relative numbers between lensed quasars and pairs are unconstrained at $\lesssim 1\arcsec$. There is also an observational bias that preferentially removes sub-arcsec lensed quasars from the oberved pair sample: lensed quasars are associated with lensing galaxies, which could (if the lensing galaxy is at $z\lesssim 1.5$) change the optical colors of the unresolved system and reduce the probability of selection from ground-based surveys such as the SDSS. Therefore we expect at least some of these double quasars are genuine pairs rather than lenses. This is indeed the case, as there are already several confirmed quasar pairs in our sample from the literature, as discussed in \S\ref{sec:sample}. Follow-up observations of the full sample of 60 double quasar systems will conclusively reveal the division between lensed quasars and physical pairs over the full range of $\sim 0\farcs3-3\arcsec$ separations. It is notoriously difficult to distinguish these two scenarios at high redshift and small separations \citep[e.g.,][]{Shen_etal_2021NatAs,Yue_etal_2021}. Minor spectral dissimilarities between the two components of the pair are insufficient to rule out lensing \citep[e.g.,][]{Shen_etal_2021NatAs}, while spectral similarities are equally insufficient to rule out a quasar pair since different quasars can look similar in their spectral appearances \citep[e.g.,][]{Rochais_etal_2017}, particularly at $z>1.5$ where optical spectroscopy only covers the rest-frame UV broad lines. Spatially-resolved near-IR spectroscopy may be able to reveal the differences in the narrow emission lines, e.g., \OIII, in a quasar pair, but is challenging given the S/N requirement and relatively weak narrow-line emission in high-$z$, high-luminosity quasars \citep[e.g.,][]{Shen_2016}. Multi-wavelength coverage of the two resolved components may help reject the lensing scenario, if the spectral energy distributions are markedly different, e.g., with additional high-resolution radio imaging of the resolved pair. The most decisive and efficient observation to rule out lensing, however, is probably the non-detection of a potential lens galaxy in deep imaging. In the case of $z>1.5$ candidate quasar pairs at sub-arcsec separations, this test requires high spatial resolution and deep IR imaging, ideally from HST or JWST. Indeed, existing optical imaging data (even taken with HST) are too shallow to rule out the lensing hypothesis \citep[e.g.,][]{Shen_etal_2021NatAs,Chen_etal_2022} for $z>1.5$ double quasars, even though statistics may slightly favor the dual quasar scenario over lensing \citep{Shen_etal_2021NatAs}. High-$z$ lens galaxies will be faint in the optical, and the non-detection limit of the lens placed by HST optical imaging is not stringent enough. For larger-separation pairs, deep IR imaging from ground would be sufficient to rule out (or confirm) the lensing scenario based on the non-detection (or detection) of a lens galaxy. Deep IR imaging may also be able to reveal tidal features in the host of the pair, offering additional evidence for physical merging pairs. In what follows, we remain agnostic about the division between lensing and pairs in our sample, and discuss different outcomes if one or the other population dominates our pair sample. \subsection{Dynamical friction, quasar duty cycles, and recoiling SMBHs} The overall double quasar fraction from our sample, $f_{QQ}\sim 6\times 10^{-4}$ ($r_p\sim 3-30$\,kpc) among all $G<20.25$ quasars at $\langle z\rangle\approx 2$, is lower than the dual AGN fraction ($\sim 10^{-2}$) at similar redshifts and separations predicted from recent hydrodynamic simulations \citep[e.g.,][and references therein]{Hirschmann_etal_2014,Dubois_etal_2014,Steinborn2016,Rosas-Guevara_etal_2019,DeRosa_etal_2019,Volonteri_etal_2022}. The main reason for this apparent discrepancy is due to the fact that these simulations do not have sufficient volume to probe the most luminous quasars, and focus on the much less luminous AGN population ($L_{\rm bol}>10^{43}\,{\rm erg\,s^{-1}}$). These low-luminosity AGNs have much higher duty cycles than luminous quasars. In general, the dual AGN fraction among AGNs increases as the luminosity threshold decreases, as seen in the simulations (Fig.~\ref{fig:pair_stat_sim}), as well as the observed high dual AGN fraction ($\gtrsim$ few percent) among low-luminosity AGNs in the nearby Universe \citep{Liu_etal_2011,Koss_etal_2012}. Improvements in both the observed sample (to fainter flux limits) and in the simulation volume over the next few years will enable a better comparison. A substantial fraction of AGNs in these simulations are also optically obscured, and would not be included in our sample. If obscuration occurs more often in merging pairs than in single AGNs, the dual AGN fraction for the unobscured population will be reduced compared with that for all AGNs. Even if the obscured fraction is the same among single AGNs and AGNs in pairs, requiring both AGNs in the pair to be unobscured would also lead to a reduced dual AGN fraction for unobscured AGNs (similar to the duty cycle argument). On the observational side and focusing on quasar luminosities ($L_{\rm bol}\gtrsim 10^{45}\,{\rm erg\,s^{-1}}$), \citet{Kayo_Oguri_2012} reported a dual quasar fraction of $\sim 5\times 10^{-4}$ over $0.6<z<2.2$ and $10\lesssim r_{p}\lesssim 100\,{\rm kpc}$, which is roughly in line with our measured double quasar fraction over smaller separations and higher redshifts. On the other hand, using ground-based optical imaging of resolved pairs around SDSS quasars from the Hyper Suprime-Cam Subaru Strategic Program, \citet{Silverman_etal_2020} reported a double quasar fraction (dual and lensed quasars combined) of $0.26\pm0.18\%$ (requiring a pair flux ratio $>0.1$) over $r_p=3-30\,{\rm kpc}$ with no redshift evolution, which is a factor of $\sim 4$ higher (albeit still within $\sim 1\sigma$) than our pair fraction over the same separations. There is a slight difference in the selection of double quasars between our work and \citet{Silverman_etal_2020}: while the flux limit of the primary SDSS quasar is the same, we require the companion is also brighter than this flux limit, while \citet{Silverman_etal_2020} includes companions that can be ten times fainter than the primary SDSS quasar. Therefore we expect some of the double quasars (candidates) in \citet{Silverman_etal_2020} would not pass our selection. Furthermore, the \citet{Silverman_etal_2020} measurement is based on a ground-based imaging pair sample with loose color selection of quasars, and spectroscopic follow-up is required to remove foreground star contamination in these apparent pairs, as acknowledged by \citet{Silverman_etal_2020}. Our earlier results based on HST imaging and spectroscopic follow-up of high-redshift candidate quasar pairs have shown that such stellar contamination is significant (e.g., $>50\%$) for pure photometric color selection \citep{Chen_etal_2022}. Foreground star contamination would also be a problem in other predominantly imaging samples of dual/offset AGN candidates \citep{Stemo_etal_2021}. In our SDSS+Gaia approach, the additional proper motion information and the rejection of foreground star superpositions with spectral PCA delivered a much cleaner double quasar sample. The relative frequency of quasar pairs as a function of separation in the $r_{p}\sim 3-30\,{\rm kpc}$ regime is determined by the dynamical friction timescale and the duty cycle of quasar activity in mergers, both of which are functions of separation. If the quasar duty cycle remains constant over these separations, simple prediction from dynamical friction implies a roughly constant pair fraction per linear separation bin towards smaller separations \citep[e.g.,][]{Yu_2002,Chen_etal_2020b}. If the pair statistics shown in Fig.~\ref{fig:pair_stat} are dominated by physical quasar pairs, then the rising pair fraction (per linear separation bin) towards small separations indicates that the quasar duty cycle is elevated towards smaller separations, or that the dynamical friction timescale deviates from the scaling predicted in analytical calculations by, e.g., \citet{Chen_etal_2020b}. The observed rising quasar pair fraction towards small separations for our high-redshift sample is consistent with observations at low redshift, where the AGN pair fraction also increases towards small separations at $r_{p}\lesssim 30\,{\rm kpc}$ \citep[e.g.,][]{Ellison_etal_2011,Liu_etal_2012,Stemo_etal_2021}. Such an elevation of SMBH accretion at small pair separations, i.e., late stages of galaxy mergers, is also seen in some hydrodynamic simulations \citep[e.g.,][]{Capelo_etal_2017}. On the other hand, if the pair statistics shown in Fig.~\ref{fig:pair_stat} are dominated by lensed quasars, and the intrinsic physical pair fraction is flat or even decreasing towards smaller separations, it would imply little enhanced (or even reduced) quasar activity towards the $\sim {\rm kpc}$ regime in galaxy mergers, which would be at odds with numerical simulation results and low-redshift observational results. Alternatively, it may imply that at $z\sim 2$, pairs of SMBHs decay more rapidly towards the $\sim {\rm kpc}$ regime than predicted by dynamical friction from stars, for example, accelerated by the presence of gas \citep[e.g.,][]{Callegari_etal_2009} expected in high-redshift gas-rich mergers, or by the build-up of a dense nuclear stellar cusp around one or both SMBHs \citep[e.g.,][]{VanWassenhove_etal_2014}. Either way, our sample of 60 double quasars can be used to address these different scenarios and constrain the dynamical friction evolution of the SMBH pair, as well as the duty cycle of quasar activity in mergers. To that end, we are conducting follow-up observations to differentiate the pairs versus lensing scenarios for our sample, and will present the results in future work. We end this section by pointing out the possibility that a tiny fraction of these quasar pairs might contain an accreting recoiled SMBH from the prior merger of two SMBHs \citep[e.g.,][and references therein]{Blecha_etal_2016}. However, there are still significant theoretical uncertainties on this putative population of recoiling SMBHs and observational challenges to distinguish them from insprialing SMBHs in galaxy mergers. Perhaps host galaxy properties can be useful to identify recoiling SMBHs as offset AGNs, e.g., if these rogue SMBHs predominately reside in early type galaxies long after the merger. \subsection{Fuzzy Dark Matter and a Possible $\sim$\,kpc Pile-up} In the fuzzy dark matter (FDM) model and ignoring baryonic effects, SMBH pairs in galaxy mergers will stall at $\lambda_{\rm FDM}\sim$\,kpc scales due to energy injection from fluctuations of dark matter particles on their de Broglie wavelength $\lambda_{\rm FDM}$ \citep[e.g.,][]{Hui2017}. If the duty cycle of quasar activity is independent of pair evolution, we expect to see a dramatic pile up of quasar pairs near the stall distance, because these pairs spend much longer time there (i.e., $\sim$ Hubble time) compared to their lifetime during previous galactic inspiral. Our quasar pair sample does not yet well probe the $<1\,$kpc regime, and we do not observe any sudden spike in the pair fraction towards $\sim 0\farcs2$ (corresponding to $\sim 1.6\,{\rm kpc}$ at $z\sim 2$). Pair statistics with future data sets (see \S\ref{sec:con}) will probe the sub-kpc regime and constrain the nature of FDM. However, absence of evidence is not evidence of absence. The potential lack of a pile up of quasar pairs below $\sim 1\,$kpc can be explained by baryonic effects, i.e., the pair orbit can further decay regardless of the energy pumping from FDM fluctuations. In addition, in the final stage of pair evolution long after the initial galaxy merger, accretion onto SMBHs may become much less efficient, leading to a diminished fraction of dual quasars among these stalled $\sim$\,kpc SMBH pairs. The most exciting aspect of this test is to potentially reveal that there is indeed a pile up of quasar pairs on $\lesssim$\,kpc scales, which would offer strong support to the FDM model. Compared to other observational tests \citep{Hui2017}, the statistics of $\sim$kpc-scale quasar pairs offer a simple but potentially definitive test (but see below), hinging on the discovery of such a pile-up of SMBH pairs. On the other hand, certain dynamical processes associated with baryonic matter might also lead to the stalling of SMBH pairs at $\sim$kpc scales \citep[][and references therein]{lisa22}. For example, in the case of massive (e.g., $>10^7\,M_\odot$) SMBHs, clumpiness in the host galaxy and inhomogeneous gas and stellar density profiles can lead to inefficient inspiral and potentially stalling of the SMBH pair at $\sim$kpc separations \citep[e.g.,][]{Tamburello_etal_2017,Pfister_etal_2019,Bortolas_etal_2020}. This is still in early theoretical investigations, and observations of host galaxies of high-redshift dual quasars might offer insights on these dynamical processes. \section{Conclusions}\label{sec:con} In this work we have measured the quasar pair statistics over $\sim 0\farcs3-3\arcsec$ separations at $z>1.5$ (median redshift $\langle z \rangle\approx 2$), using a sample of 60 resolved double quasars from Gaia EDR3 \citep{Fabricius_etal_2021}. These pairs are selected by cross-matching the Gaia EDR3 catalog with spectroscopically confirmed quasars from SDSS DR16 \citep{Lyke_etal_2020}. Both members of the pair are flux limited to $G<20.25$, therefore our pair sample corresponds to the luminous quasar population at cosmic noon, with $L_{\rm bol}>10^{45.8}\,{\rm erg\,s^{-1}}$ at $z>1.5$. We efficiently separate quasars and stars in resolved pairs using Gaia proper motion measurements and PCA analysis of SDSS spectra (\S\ref{sec:sample}). We quantify the pair completeness in Gaia EDR3 as functions of pair separation $\Delta\theta$, magnitude of the primary, and magnitude contrast of the pair (\S\ref{sec:pair_comp}). The completeness-corrected pair fraction (per linear separation bin; among all $z>1.5$ quasars at $G<20.25$) increases towards smaller separations, and is elevated by a factor of $\sim 5$ from $\Delta\theta\sim 3\arcsec$ to $\Delta\theta\sim 0\farcs3$. The integrated pair fraction over $\sim 0\farcs3-3\arcsec$ scales (corresponding to projected physical separations of $\sim 3-30\,{\rm kpc}$ at $z\sim 2$) is $\sim 6.2\pm0.5\times 10^{-4}$, with no obvious evolution in the redshift range of our sample. The major caveat of the current analysis is that the division between physical quasar pairs and gravitationally lensed quasars is unknown, especially in the sub-arcsec regime. Previous searches of high-redshift quasar pairs and lensed quasars on $>1\arcsec$ scales have revealed that both populations contribute significantly to the observed double quasars \citep[e.g.,][]{Hennawi_etal_2006,Hennawi_etal_2010,Myers_etal_2008,Kayo_Oguri_2012,More_etal_2016,Eftekharzadeh_etal_2017}. It is then reasonable to expect that there are both bona fide quasar pairs and lensed quasars in the sub-arcsec regime. We are conducting follow-up observations for the complete sample of 60 double quasars presented here, and will refine our constraints on the quasar pair statistics. This work represents a meaningful advance on observational constraints on the formation and evolution of SMBH pairs at high redshift. Granted, the depth of Gaia and SDSS limits such a systematic search to the most luminous quasars, missing the bulk of rapidly growing SMBHs at cosmic noon. The important and more abundant populations of single offset AGNs in mergers and obscured AGNs are also not explored with the Gaia+SDSS sample. Nevertheless, this approach with Gaia+SDSS has delivered some of the first statistical measurements of quasar pair fraction in a redshift-separation regime that has just started to be explored in a systematic fashion \citep[e.g.,][]{Silverman_etal_2020,Stemo_etal_2021,Chen_etal_2022}. With continued Gaia observations (more resolved pairs from different scanning directions), we expect to recover additional luminous quasar pairs at $z>1.5$ to improve the statistics. However, the intrinsic abundance of luminous quasar pairs at cosmic noon is low. In order to significantly improve the pair statistics, to extend to lower AGN luminosities, and to explore the diversity in SMBH and host properties, it is necessary to carry out similar searches with deeper, wide-area surveys at sub-arcsec resolution. Upcoming wide-field space missions, such as Euclid \citep[][to be launched in $\sim 2023$]{Euclid_EWS}, the Chinese Space Station Telescope \citep[CSST,][to be launched in $\sim 2024$]{Zhan_2021}, and the Nancy Grace Roman Space Telescope \citep[Roman,][to be launched before $\sim 2027$]{Spergel_etal_2015}, will provide the perfect combination to perform systematic searches of SMBH pairs across cosmic time. All three missions will carry out a wide-field imaging survey in multiple filters with $\sim 0\farcs05-0\farcs2$ resolution and depths of $\sim 25-28$ AB mag, with additional spectroscopic capabilities. The combined photometric data cover a broad wavelength range across UV-optical-near-infrared. These data can be used to efficiently select candidate quasar pairs based on photometric colors and spectroscopic information, down to the diffraction limit of these space telescopes. Dedicated follow-up observations of these candidates can confirm the nature of these pairs, if needed. In particular, the deep IR imaging from Euclid and Roman will be useful to test the lensing scenario for high-redshift double quasars. In addition, the capability of detecting the host galaxy in deep IR imaging and measuring sub-arcsec offset of point sources within will enable the systematic discovery of single offset AGNs in high-redshift mergers. Host galaxy measurements will also allow a detailed look at the populations of dual and offset AGNs in different types of galaxies, shedding light on AGN fueling and recoiling SMBHs. With combined data sets from these upcoming space-based surveys, we will conclusively measure the abundances of galactic-scale quasar and AGN pairs, offset AGNs, and sub-arcsec lensed quasars across most of the cosmic history, with unprecedented statistics and coverage of the parameter space of SMBHs and host galaxies. \acknowledgments We thank P. Capelo and Joe Hennawi for useful discussions. This work is partially supported by NSF grants AST-2009947 (YS) and AST-2108162 (YS, XL). MO acknowledges support by JSPS KAKENHI Grant Numbers JP22H01260, JP20H05856, JP20H00181. HCH acknowledges the support of the Infosys Membership at the Institute for Advanced Study. NLZ acknowledges support by NASA through grant HST-GO-15900 from the Space Telescope Science Institute and by the Institute for Advanced Study through J. Robbert Oppenheimer Visiting Professorship and the Bershadsky Fund. This research was supported in part by the National Science Foundation under Grant No. PHY-1748958. We are grateful to the hospitality of the Kavli Institute for Theoretical Physics at UC Santa Barbara and the KITP Conference: Building Bridges: Towards a Unified Picture of Stellar and Black Hole Binary Accretion and Evolution (May 2022) during which part of the work was performed. \bibliography{refs}

Title: Unified neutron star EOSs and neutron star structures in RMF models

Abstract: In the framework of Thomas-Fermi approximation, we study systematically the EOSs and microscopic structures of neutron star matter in a vast density range with $n_\mathrm{b}\approx 10^{-10}$-2 $\mathrm{fm}^{-3}$, where various covariant density functionals are adopted, i.e., those with nonlinear self couplings (NL3, PK1, TM1, GM1, MTVTC) and density-dependent couplings (DD-LZ1, DDME-X, PKDD, DD-ME2, DD2, TW99). It is found that the EOSs generally coincide with each other at $n_\mathrm{b}\lesssim 10^{-4}$ fm${}^{-3}$ and 0.1 fm${}^{-3}\lesssim n_\mathrm{b} \lesssim 0.3$ fm${}^{-3}$, while in other density regions they are sensitive to the effective interactions between nucleons. By adopting functionals with larger slope of symmetry energy $L$, the curvature parameter $K_\mathrm{sym}$ and neutron drip density generally increase, while the droplet size, proton number of nucleus, core-crust transition density, and onset density of non-spherical nuclei decrease. All functionals predict neutron stars with maximum masses exceeding the two-solar-mass limit, while those of DD2, DD-LZ1, DD-ME2, and DDME-X predict optimum neutron star radii according to the observational constraints. Nevertheless, the corresponding skewness coefficients $J$ are much lager than expected, while only the functionals MTVTC and TW99 meet the start-of-art constraints on $J$. More accurate measurements on the radius of PSR J0740+6620 and the maximum mass of neutron stars are thus essential to identify the functional that satisfies all constraints from nuclear physics and astrophysical observations. Approximate linear correlations between neutron stars' radii at $M=1.4 M_{\odot}$ and $2 M_{\odot}$, the slope $L$ and curvature parameter $K_\mathrm{sym}$ of symmetry energy are observed as well, which is mainly attributed to the curvature-slope correlations in the functionals adopted here.

https://export.arxiv.org/pdf/2208.12893

\hyphenpenalty=6000 \tolerance=1000 \hyphenation{Eq} \title{Unified neutron star EOSs and neutron star structures in RMF models} \author{Cheng-Jun~Xia$^{1,2,3}$} \email{cjxia@yzu.edu.cn} \author{Toshiki Maruyama$^{3}$} \email{maruyama.toshiki@jaea.go.jp} \author{Ang Li$^{4}$} \email{liang@xmu.edu.cn} \author{Bao Yuan Sun$^{5, 6}$} \email{sunby@lzu.edu.cn} \author{Wen-Hui Long$^{5, 6}$} \email{longwh@lzu.edu.cn} \author{Ying-Xun Zhang$^{7, 8}$} \email{zhyx@ciae.ac.cn} \affiliation{$^{1}${Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China} \\$^{2}${School of Information Science and Engineering, NingboTech University, Ningbo 315100, China} \\$^{3}${Advanced Science Research Center, Japan Atomic Energy Agency, Shirakata 2-4, Tokai, Ibaraki 319-1195, Japan} \\$^{4}${Department of Astronomy, Xiamen University, Xiamen 361005, China} \\$^{5}${School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China} \\$^{6}${Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China} \\$^{7}${China Institute of Atomic Energy, Beijing 102413, People's Republic of China} \\$^{8}${Guangxi Key Laboratory Breeding Base of Nuclear Physics and Technology, Guilin 541004, China} } \date{\today} \section{\label{sec:intro}Introduction} Due to the challenges reside in simulating dense matter with lattice QCD, the state and composition of stellar matter inside compact stars is still unclear and exhibits large ambiguities. In particular, the uncertainties in the equation of state (EOS) and the corresponding microscopic structures are still sizable~\cite{Dutra2012_PRC85_035201, Dutra2014_PRC90-055203, Xia2020_PRD102-023031, LI2020, Hebeler2021_PR890-1}, which was shown to play important roles in the properties and evolutions of compact stars~\cite{Pons1999_ApJ513-780, Horowitz2004_PRC69-045804, Lattimer2012_ARNPS62-485, Janka2012_ARNPS62-407, Bauswein2012_PRD86-063001, Rueda2014_PRC89-035804, Watanabe2017_PRL119-062701, Sotani2019_MNRAS489-3022, Koeppel2019_ApJ872-L16, Baiotti2019_PPNP109-103714, Schuetrumpf2020_PRC101-055804, Bauswein2020_PRL125-141103, Gittins2020_PRD101-103025, Preau2021_MNRAS505-939}. The properties of nuclear matter around the saturation density ($n_0\approx 0.16\ \mathrm{fm}^{-3}$), on the contrary, are well constrained with various terrestrial experiments, astrophysical observations, and nuclear theories, where the binding energy is $B\approx -16$ MeV, the incompressibility $K = 240 \pm 20$ MeV~\cite{Shlomo2006_EPJA30-23}, the symmetry energy $S = 31.7 \pm 3.2$ MeV and its slope $L = 58.7 \pm 28.1$ MeV~\cite{Li2013_PLB727-276, Oertel2017_RMP89-015007}. Note that those quantities can be further constrained by considering the up-to-date astrophysical observations, heavy ion collision data, measurements of the neutron skin thickness for $^{208}$Pb in PREX-II~\cite{PREX2021_PRL126-172502}, as well as predictions of chiral effective field theory, e.g., those in Refs.~\cite{Zhang2020_PRC101-034303, Essick2021_PRL127-192701}. Meanwhile, at vanishing densities, the interaction among nucleons is negligible so that nuclear matter exhibits gas phase with well understood properties~\cite{Yang2019_PRC100-054314, Yang2021_PRC103-014304}. At subsaturation densities, due to the liquid-gas phase transition of nuclear matter, a mixed phase with various nonuniform structures is expected, i.e., nuclear pasta~\cite{Baym1971_ApJ170-299, Negele1973_NPA207-298, Ravenhall1983_PRL50-2066, Hashimoto1984_PTP71-320, Williams1985_NPA435-844}, which is typically found in the crusts of neutron stars and the core region of supernovae at the stage of gravitational collapse. By employing spherical and cylindrical approximations for the Wigner-Seitz (WS) cell~\cite{Pethick1998_PLB427-7, Oyamatsu1993_NPA561-431, Maruyama2005_PRC72-015802, Togashi2017_NPA961-78, Shen2011_ApJ197-20}, five types of geometrical structures for nuclear pasta were obtained aside from the uniform phase, i.e, droplets, rods, slabs, tubes, and bubbles, while further investigations have revealed more complicated structures~\cite{Magierski2002_PRC65-045804, Watanabe2003_PRC68-035806, Newton2009_PRC79-055801, Nakazato2009_PRL103-132501, Okamoto2012_PLB713-284, Schuetrumpf2013_PRC87-055805, Schneider2014_PRC90-055805, Schuetrumpf2015_PRC91-025801, Fattoyev2017_PRC95-055804, Schuetrumpf2019_PRC100-045806, Sagert2016_PRC93-055801, Berry2016_PRC94-055801, Kashiwaba2020_PRC101-045804}. Nevertheless, the investigation on the EOSs and microscopic structures of nuclear pasta is still far from complete due to the uncertainties in the nuclear energy density functional~\cite{Douchin2001_AA380-151, Sharma2015_AA584-A103, Fortin2016_PRC94-035804, Pearson2018_MNRAS481-2994, Vinas2021_Symmetry13-1613, DinhThi2021_AA654-A114, Newton2022_EPJA58-69}, while a unified treatment is preferred so that the uncertainties do not get larger~\cite{Fortin2016_PRC94-035804, DinhThi2021_AA654-A114}. For stellar matter at larger densities, as we are entering the multimessenger era, constraining the EOS with pulsar observations has reached unprecedent accuracy. For example, the observation of two-solar-mass pulsars~\cite{Demorest2010_Nature467-1081, Antoniadis2013_Science340-1233232, Fonseca2016_ApJ832-167, Cromartie2020_NA4-72, Fonseca2021_ApJ915-L12} have excluded various soft EOSs for dense stellar matter. The multi-messenger observations of the binary neutron star merger event {GRB} 170817A-{GW}170817-{AT} 2017gfo have constrained the tidal deformability of $1.4 M_{\odot}$ neutron star with $70\leq \Lambda_{1.4}\leq 580$ and the radii $R=11.9\pm1.4$ km~\cite{LVC2018_PRL121-161101}, indicating a soft EOS at small densities. Additionally, based on pulse-profile modeling with {NICER} and {XMM}-Newton data, the simultaneous measurements of the masses and radii for PSR J0030+0451 and PSR J0740+6620~\cite{Riley2019_ApJ887-L21, Riley2021_ApJ918-L27, Miller2019_ApJ887-L24, Miller2021_ApJ918-L28} suggest that their radii are similar ($\sim$12.4 km) despite the large differences in masses. In such cases, the likelihood of a strong first-order phase transition inside two-solar-mass pulsars may be reduced~\cite{Pang2021_ApJ922-14}. The purpose of our current study is twofold. First, we examine the structures of neutron stars without introducing any new degrees of freedom that leads to first-order phase transitions. Since the radius and crust thickness of a neutron star are sensitive to the EOSs~\cite{Fortin2016_PRC94-035804}, a unified description for neutron star matter is thus necessary~\cite{Fortin2016_PRC94-035804, DinhThi2021_AA654-A114}. This leads to the second purpose of our study, where we have obtained 11 EOSs and the corresponding microscopic structures of neutron star matter in a unified manner adopting the numerical recipe proposed in Ref.~\cite{Xia2022_PRC105-045803}. In particular, as was done in Refs.~\cite{Maruyama2005_PRC72-015802, Avancini2008_PRC78-015802, Avancini2009_PRC79-035804, Gupta2013_PRC87-028801}, the properties of nuclear matter are fixed with relativistic mean field (RMF) models~\cite{Meng2016_RDFNS}, which was very successful in describing finite nuclei~\cite{Reinhard1989_RPP52-439, Ring1996_PPNP37_193-263, Meng2006_PPNP57-470, Paar2007_RPP70-R02, Meng2015_JPG42-093101, Meng2016_RDFNS, Chen2021_SCPMA64-282011, Typel1999_NPA656-331, Vretenar1998_PRC57-R1060, Lu2011_PRC84-014328} and nuclear matter~\cite{Glendenning2000, Ban2004_PRC69-045805, Weber2007_PPNP59-94, Long2012_PRC85-025806, Sun2012_PRC86-014305, Wang2014_PRC90-055801, Fedoseew2015_PRC91-034307, Gao2017_ApJ849-19}. Two types of RMF Lagrangian are considered, i.e., those with nonlinear self couplings (NL3~\cite{Lalazissis1997_PRC55-540}, PK1~\cite{Long2004_PRC69-034319}, TM1~\cite{Sugahara1994_NPA579-557}, GM1~\cite{Glendenning1991_PRL67-2414}, MTVTC~\cite{Maruyama2005_PRC72-015802}) and density-dependent couplings (DD-LZ1~\cite{Wei2020_CPC44-074107}, DDME-X~\cite{Taninah2020_PLB800-135065}, PKDD~\cite{Long2004_PRC69-034319}, DD-ME2~\cite{Lalazissis2005_PRC71-024312}, DD2~\cite{Typel2010_PRC81-015803}, TW99~\cite{Typel1999_NPA656-331}). The paper is organized as follows. In Sec.~\ref{sec:the} we present the theoretical framework for the covariant density functionals adopted here and fixing the microscopic structures of neutron star matter. The obtained EOSs and microscopic structures of neutron star matter are presented in Sec.~\ref{sec:results}, while the corresponding neutron star structures and the possible correlations with the symmetry energy coefficients are investigated. We draw our conclusion in Sec.~\ref{sec:con} \section{\label{sec:the} Theoretical framework} \subsection{\label{sec:the_RMF} RMF models} The Lagrangian density of RMF models for the neutron star matter considered here reads \begin{eqnarray} \mathcal{L} &=& \sum_{i=n,p} \bar{\psi}_i \left[ i \gamma^\mu \partial_\mu - \gamma^0 \left(g_\omega\omega + g_\rho\rho\tau_i + A q_i\right)- m_i^* \right] \psi_i \nonumber \\ &&\mbox{} + \sum_{l=e,\mu} \bar{\psi}_l \left[ i \gamma^\mu \partial_\mu - m_l + e \gamma^0 A \right]\psi_l - \frac{1}{4} A_{\mu\nu}A^{\mu\nu} \nonumber \\ &&\mbox{} + \frac{1}{2}\partial_\mu \sigma \partial^\mu \sigma - \frac{1}{2}m_\sigma^2 \sigma^2 - \frac{1}{4} \omega_{\mu\nu}\omega^{\mu\nu} + \frac{1}{2}m_\omega^2 \omega^2 \nonumber \\ &&\mbox{} - \frac{1}{4} \rho_{\mu\nu}\rho^{\mu\nu} + \frac{1}{2}m_\rho^2 \rho^2 + U(\sigma, \omega), \label{eq:Lagrange} \end{eqnarray} where $\tau_n=-\tau_p=1$ is the 3rd component of isospin, $q_i=e (1-\tau_i)/2$ the charge, and $m_{n,p}^*\equiv m_{n,p} + g_{\sigma} \sigma$ the effective nucleon mass. The boson fields $\sigma$, $\omega$, $\rho$, and $A$ take mean values with only the time components due to time-reversal symmetry. Then the field tensors $\omega_{\mu\nu}$, $\rho_{\mu\nu}$, and $A_{\mu\nu}$ vanish except for \begin{equation} \omega_{i0} = -\omega_{0i} = \partial_i \omega, \rho_{i0} = -\rho_{0i} = \partial_i \rho, A_{i0} = -A_{0i} = \partial_i A.\nonumber \end{equation} The nonlinear self couplings of the mesons are determined by \begin{equation} U(\sigma, \omega) = -\frac{1}{3}g_2\sigma^3 - \frac{1}{4}g_3\sigma^4 + \frac{1}{4}c_3\omega^4, \label{eq:U_NL} \end{equation} which effectively account for the in-medium effects and are essential for the covariant density functionals NL3~\cite{Lalazissis1997_PRC55-540}, PK1~\cite{Long2004_PRC69-034319}, TM1~\cite{Sugahara1994_NPA579-557}, GM1~\cite{Glendenning1991_PRL67-2414}, and MTVTC~\cite{Maruyama2005_PRC72-015802} adopted here. Alternatively, the in-medium effects can be treated with density dependent coupling constants according to the Typel-Wolter ansatz~\cite{Typel1999_NPA656-331}, where \begin{eqnarray} g_{\xi}(n_\mathrm{b}) &=& g_{\xi} a_{\xi} \frac{1+b_{\xi}(n_\mathrm{b}/n_0+d_{\xi})^2} {1+c_{\xi}(n_\mathrm{b}/n_0+d_{\xi})^2}, \label{eq:ddcp_TW} \\ g_{\rho}(n_\mathrm{b}) &=& g_{\rho} \exp{\left[-a_\rho(n_\mathrm{b}/n_0 + b_\rho)\right]}. \label{eq:ddcp_rho} \end{eqnarray} Here $\xi=\sigma$, $\omega$ and the baryon number density $n_\mathrm{b} = n_p+n_n$ with $n_0$ being the saturation density. In addition to the nonlinear ones, we have also adopted the density-dependent covariant density functionals DD-LZ1~\cite{Wei2020_CPC44-074107}, DDME-X~\cite{Taninah2020_PLB800-135065}, PKDD~\cite{Long2004_PRC69-034319}, DD-ME2~\cite{Lalazissis2005_PRC71-024312}, DD2~\cite{Typel2010_PRC81-015803}, and TW99~\cite{Typel1999_NPA656-331}, where the nonlinear self-couplings in Eq.~(\ref{eq:U_NL}) vanish with $g_2=g_3=c_3=0$. For completeness, the parameter sets adopted in this work are listed in Table~\ref{table:param}, where $a_{\sigma, \omega}=1$ and $b_{\sigma, \omega}=c_{\sigma, \omega}=a_\rho=0$ if nonlinear self-couplings are adopted. \begin{table*} \caption{\label{table:param} The adopted parameters for the covariant density functionals with nonlinear self couplings (NL3~\cite{Lalazissis1997_PRC55-540}, PK1~\cite{Long2004_PRC69-034319}, TM1~\cite{Sugahara1994_NPA579-557}, GM1~\cite{Glendenning1991_PRL67-2414}, MTVTC~\cite{Maruyama2005_PRC72-015802}) and density-dependent couplings (DD-LZ1~\cite{Wei2020_CPC44-074107}, DDME-X~\cite{Taninah2020_PLB800-135065}, PKDD~\cite{Long2004_PRC69-034319}, DD-ME2~\cite{Lalazissis2005_PRC71-024312}, DD2~\cite{Typel2010_PRC81-015803}, TW99~\cite{Typel1999_NPA656-331}).} \begin{tabular}{c|cccccccc|ccc} \hline \hline & $m_n$ & $m_p$ & $m_\sigma$& $m_\omega$ & $m_\rho$ & $g_\sigma$ & $g_\omega$ & $g_\rho$ & $g_2$ & $g_3$ & $c_3$ \\ & MeV & MeV & MeV & MeV & MeV & & & & fm${}^{-1}$ & & \\ \hline NL3 & 939 & 939 & 508.1941 & 782.501 & 763 & 10.2169 & 12.8675 & 4.4744 & $-$10.4307 & $-$28.8851& 0 \\ PK1 & 938 & 938 & 511.198 & 783 & 770 & 10.0289 & 12.6139 & 4.6322 & $-$7.2325 & 0.6183 & 71.3075\\ TM1 &939.5731 &938.2796 & 514.0891 & 784.254 & 763 & 10.3222 & 13.0131 & 4.5297 & $-$8.1688 & $-$9.9976 & 55.636 \\ GM1 & 938 & 938 & 510 & 783 & 770 & 8.87443 & 10.60957 & 4.09772 & $-$9.7908 & $-$6.63661& 0 \\ MTVTC & 938 & 938 & 400 & 783 & 769 & 6.3935 & 8.7207 & 4.2696 & $-$10.7572 & $-$4.04529& 0 \\ \hline DD-LZ1 & 938.9 & 938.9 &538.619216 & 783 & 769 & 12.001429 & 14.292525 & 7.575467 & 0 & 0 & 0 \\ DDME-X & 938.5 & 938.5 &547.332728 & 783 & 763 & 10.706722 & 13.338846 & 3.619020 & 0 & 0 & 0 \\ PKDD &939.5731 &938.2796 &555.5112 & 783 & 763 & 10.7385 & 13.1476 & 4.2998 & 0 & 0 & 0 \\ DD-ME2 & 938.5 & 938.5 &550.1238 & 783 & 763 & 10.5396 & 13.0189 & 3.6836 & 0 & 0 & 0 \\ DD2 &939.56536&938.27203&546.212459 & 783 & 763 & 10.686681 & 13.342362 & 3.626940 & 0 & 0 & 0 \\ TW99 & 939 & 939 & 550 & 783 & 763 & 10.7285 & 13.2902 & 3.6610 & 0 & 0 & 0 \\ \hline \end{tabular} \begin{tabular}{c|cccc|cccc|cc} \hline \hline & $a_\sigma$&$b_\sigma$&$c_\sigma$&$d_\sigma$ &$a_\omega$& $b_\omega$ & $c_\omega$ & $d_\omega$ & $a_\rho$ & $b_\rho$ \\ \hline DD-LZ1 &1.062748 &1.763627 &2.308928 & 0.379957 &1.059181 &0.418273 &0.538663 &0.786649 &0.776095 & 0 \\ DDME-X &1.397043 &1.334964 &2.067122 & 0.401565 &1.393601 &1.019082 &1.605966 &0.455586 &0.620220 & $-$1 \\ PKDD &1.327423 &0.435126 &0.691666 & 0.694210 &1.342170 &0.371167 &0.611397 &0.738376 &0.183305 & $-$1 \\ DD-ME2 &1.3881 &1.0943 &1.7057 & 0.4421 &1.3892 &0.9240 & 1.4620 &0.4775 &0.5647 & $-$1 \\ DD2 &1.357630 &0.634442 &1.005358 & 0.575810 &1.369718 &0.496475 &0.817753 &0.638452 &0.983955 & $-$1 \\ TW99 &1.365469 &0.226061 &0.409704 & 0.901995 &1.402488 &0.172577 &0.344293 &0.983955 &0.515000 & $-$1 \\ \hline \end{tabular} \end{table*} Carrying out standard variational procedure, the equations of motion for bosons are fixed by \begin{eqnarray} (-\nabla^2 + m_\sigma^2) \sigma &=& -g_{\sigma} n_\mathrm{s} - g_2\sigma^2 - g_3\sigma^3, \label{eq:KG_sigma} \\ (-\nabla^2 + m_\omega^2) \omega &=& g_{\omega} n_\mathrm{b} + c_3\omega^3, \label{eq:KG_omega}\\ (-\nabla^2 + m_\rho^2) \rho &=& \sum_{i=n,p} g_{\rho}\tau_{i} n_i, \label{eq:KG_rho}\\ -\nabla^2 A &=& e(n_p - n_e - n_\mu). \label{eq:KG_photon} \end{eqnarray} The scalar and vector densities are determined by \begin{eqnarray} n_{s} &=& \sum_{i=n,p} \langle \bar{\psi}_i \psi_i \rangle = \sum_{i=n,p} \frac{{M^*}^3}{2\pi^2} f\left(\frac{\nu_i}{M^*}\right),\\ n_i &=& \langle \bar{\psi}_i\gamma^0 \psi_i \rangle = \frac{\nu_i^3}{3\pi^2}, \end{eqnarray} where $\nu_i$ represents the Fermi momentum and $f(x) = x \sqrt{x^2+1} - \mathrm{arcsh}(x)$. The total energy of the system is then fixed by \begin{equation} E=\int \langle {\cal{T}}_{00} \rangle \mbox{d}^3 r, \label{eq:energy} \end{equation} with the energy momentum tensor \begin{eqnarray} \langle {\cal{T}}_{00} \rangle &=& \sum_{i=n,p,e,\mu} \frac {{m^*_i}^4}{8\pi^{2}} \left[x_i(2x_i^2+1)\sqrt{x_i^2+1}-\mathrm{arcsh}(x_i) \right] \nonumber \\ && + \frac{1}{2}(\nabla \sigma)^2 + \frac{1}{2}m_\sigma^2 \sigma^2 + \frac{1}{2}(\nabla \omega)^2 + \frac{1}{2}m_\omega^2 \omega^2 + c_3\omega^4 \nonumber \\ && + \frac{1}{2}(\nabla \rho)^2 + \frac{1}{2}m_\rho^2 \rho^2 + \frac{1}{2}(\nabla A)^2 - U(\sigma, \omega), \label{eq:ener_dens} \end{eqnarray} where $x_i\equiv \nu_i/m^*_i$ with $m_e^*=m_e = 0.511$ MeV and $m_\mu^*=m_\mu = 105.66$ MeV. In the Thomas-Fermi approximation (TFA), the optimum density distributions $n_i(\vec{r})$ are fixed by minimizing the total energy $E$ at given total particle numbers $N_i=\int n_i \mbox{d}^3 r$, dimension $D$, and WS cell size $R_\mathrm{W}$, which follows the constancy of chemical potentials, i.e., \begin{equation} \mu_i(\vec{r}) = \sqrt{{\nu_i}^2+{m_i^*}^2} + \Sigma^\mathrm{R} + g_{\omega} \omega + g_{\rho}\tau_{i} \rho + q_i A = \rm{constant}. \label{eq:chem_cons} \end{equation} Note that the ``rearrangement" term $\Sigma^\mathrm{R}$ needs to be considered if the density-dependent couplings are adopted in the Lagrangian density~\cite{Lenske1995_PLB345-355}, i.e., \begin{equation} \Sigma^\mathrm{R}= \frac{\mbox{d} g_\sigma}{\mbox{d} n_\mathrm{b}} \sigma n_\mathrm{s}+ \frac{\mbox{d} g_\omega}{\mbox{d} n_\mathrm{b}} \omega n_\mathrm{b}+ \frac{\mbox{d} g_\rho}{\mbox{d} n_\mathrm{b}} \rho \sum_i\tau_i n_i. \label{eq:re_B} \end{equation} \subsection{\label{sec:the_EOS} Microscopic structures of neutron star matter} Neutron star matter at different densities exhibits various microscopic structures. At $n_\mathrm{b}\lesssim 0.0003\ \mathrm{fm}^{-3}$, neutron rich nuclei and electrons form Coulomb lattices, which can be found in the outer crusts of neutron stars and white dwarfs. At larger densities, neutrons start to drip out and form neutron gas, then the neutron star matter is essentially a liquid-gas mixed phase and can by found in the inner crust region of a neutron star. As density increases, the liquid phase will eventually take non-spherical shapes that resembles pasta, which are hence referred to as nuclear pasta~\cite{Baym1971_ApJ170-299, Negele1973_NPA207-298, Ravenhall1983_PRL50-2066, Hashimoto1984_PTP71-320, Williams1985_NPA435-844}. At densities $n_\mathrm{b}\gtrsim 0.08\ \mathrm{fm}^{-3}$, the core-crust transition takes place inside a neutron star, where the uniform phase is energetically more favorable for neutron star matter. To obtain the microscopic structures of neutron star matter, we solve the Klein-Gordon equations and the density distributions iteratively inside a WS cell. Adopting the spherical and cylindrical approximations~\cite{Maruyama2005_PRC72-015802}, the derivatives in the Klein-Gordon equations~(\ref{eq:KG_sigma}-\ref{eq:KG_photon}) are then reduced to one-dimensional, i.e., \begin{eqnarray} \mathrm{1D:}\ \ \ \ && \nabla^2 \phi(\vec{r}) = \frac{\mbox{d}^2\phi(r)}{\mbox{d}r^2}; \label{eq:dif_1D} \\ \mathrm{2D:}\ \ \ \ && \nabla^2 \phi(\vec{r}) = \frac{\mbox{d}^2\phi(r)}{\mbox{d}r^2} + \frac{1}{r} \frac{\mbox{d}\phi(r)}{\mbox{d}r}; \label{eq:dif_2D}\\ \mathrm{3D:}\ \ \ \ && \nabla^2 \phi(\vec{r}) = \frac{\mbox{d}^2\phi(r)}{\mbox{d}r^2} + \frac{2}{r} \frac{\mbox{d}\phi(r)}{\mbox{d}r}, \label{eq:dif_3D} \end{eqnarray} which can be solved via fast cosine transformation fulfilling the reflective boundary conditions at $r=0$ and $r=R_\mathrm{W}$~\cite{Xia2021_PRC103-055812}. The density distributions of fermions are obtained with Eq.~(\ref{eq:chem_cons}) fulfilling the $\beta$-stability condition $\mu_n=\mu_p+\mu_e=\mu_p+\mu_\mu$, where in practice we have adopted the imaginary time step method~\cite{Levit1984_PLB139-147} to obtain the density profiles for the next iteration. Note that at each iteration, the total particle numbers fulfill global charge neutrality condition \begin{equation} \int \left[n_p(\vec{r}) - n_e(\vec{r}) - n_\mu(\vec{r})\right] \mbox{d}^3 r\equiv 0. \end{equation} Different types of microscopic structures can be obtained with Eqs.~(\ref{eq:dif_1D}-\ref{eq:dif_3D}), i.e., droplet, rod, slab, tube, bubble, and uniform. At given average baryon number density $n_\mathrm{b}$, we then search for the energy minimum among six types of nuclear matter structures with optimum cell sizes $R_\mathrm{W}$. Note that the effects of charge screening are included in our calculation with electrons move freely within WS cells, which is expected to affect the microscopic structures of nuclear pasta~\cite{Maruyama2005_PRC72-015802}. With the density profiles fixed by fulfilling the convergency condition, the droplet size $R_\mathrm{d}$ and WS cell size $R_\mathrm{W}$ are then determined by \begin{equation} R_\mathrm{d} = \left\{\begin{array}{l} R_\mathrm{W}\left(\frac{\langle n_p \rangle^2}{\langle n_p^2 \rangle}\right)^{1/D}, \text{\ \ \ \ \ \ \ droplet-like}\\ R_\mathrm{W} \left(1- \frac{\langle n_p \rangle^2}{\langle n_p^2 \rangle}\right)^{1/D}, \text{\ \ bubble-like}\\ \end{array}\right., \label{Eq:Rd} \end{equation} where $\langle n_p^2 \rangle = \int n_p^2(\vec{r}) \mbox{d}^3 r/V$ and $\langle n_p \rangle = \int n_p(\vec{r}) \mbox{d}^3 r/V$ with the WS cell volume \begin{equation} V = \left\{\begin{array}{l} \frac{4}{3}\pi R_\mathrm{W}^3,\ D = 3\\ \pi a R_\mathrm{W}^2 , \ D = 2\\ a^2 R_\mathrm{W}, \ \ D = 1\\ \end{array}\right.. \label{Eq:V} \end{equation} In order for the volume to be finite for the slabs and rods/tubes at $D = 1$ and 2, here we have adopted a finite cell size $a = 30$ fm. Meanwhile, as we decrease the density, it is found that $R_\mathrm{W}$ grows drastically and quickly exceeds the limit for any viable numerical simulations. In such cases, as was done in our previous study~\cite{Xia2022_PRC105-045803}, at densities $n_\mathrm{b}\lesssim 10^{-4}$ fm${}^{-3}$ we divide the WS cell into a core with radius $R_\mathrm{in}=35.84$ fm and a spherical shell with constant densities. \section{\label{sec:results} Results and Discussion} \subsection{\label{sec:pasta_beta} Neutron star matter} \begin{table} \caption{\label{table:NM} Saturation properties of nuclear matter corresponding to the covariant density functionals indicated in Table~\ref{table:param}. } \begin{tabular}{c|ccccccc} \hline \hline & $n_0$ & $B$ & $K$ & $J$ & $S$ & $L$ & $K_\mathrm{sym}$ \\ & fm${}^{-3}$ & MeV & MeV & MeV & MeV & MeV & MeV \\ \hline NL3 & 0.148 & $-$16.25 & 271.7 & 204 & 37.4 & 118.6 & 101 \\ PK1 & 0.148 & $-$16.27 & 282.7 &$-27.8$ & 37.6 & 115.9 & 55 \\ TM1 & 0.145 & $-$16.26 & 281.2 & $-285$ & 36.9 & 110.8 & 34 \\ GM1 & 0.153 & $-$16.33 & 300.5 & $-216$ & 32.5 & 94.0 & 18 \\ MTVTC & 0.153 & $-$16.30 & 239.8 & $-513$ & 32.5 & 89.6 & $-6.5$ \\ \hline DD-LZ1 & 0.158 & $-$16.06 & 230.7 & 1330 & 32.0 & 42.5 & $-20$ \\ DDME-X & 0.152 & $-$16.11 & 267.6 & 874 & 32.3 & 49.7 & $-72$ \\ PKDD & 0.150 & $-$16.27 & 262.2 & $-119$ & 36.8 & 90.2 & $-81$ \\ DD-ME2 & 0.152 & $-$16.13 & 250.8 & 477 & 32.3 & 51.2 & $-87$ \\ DD2 & 0.149 & $-$16.02 & 242.7 & 169 & 31.7 & 55.0 & $-93$ \\ TW99 & 0.153 & $-$16.24 & 240.2 & $-540$ & 32.8 & 55.3 & $-125$ \\ \hline \end{tabular} \end{table} The nuclear matter properties around the saturation density are illustrated in Table~\ref{table:NM} for various covariant density functionals adopted here, which covers a wide range for the incompressibility $K$, the skewness coefficient $J$, the symmetry energy $S$, the slope $L$ and curvature parameter $K_\mathrm{sym}$ of nuclear symmetry energy. Based on those functionals, we then investigate the EOSs and microscopic structures of neutron star matter adopting the numerical recipe introduced in Sec.~\ref{sec:the}. In Fig.~\ref{Fig:EOS-beta} we present the obtained energy per baryon, pressure, and proton fraction for the most favorable nuclear shape with optimum WS cell size $R_\mathrm{W}$. For comparison, the corresponding results for uniform matter are presented in the left panels as well. As we decrease the density, the proton fraction $Y_p$ of the uniform phase decreases and eventually vanishes for most of the functionals. Nevertheless, as indicated in Table~\ref{table:param}, adopting realistic neutron and proton masses for the covariant density functionals TM1, PKDD, and DD2, the proton fraction $Y_p$ of the uniform phase does not vanish but increases to 1 as we decrease the density at $n_\mathrm{b} \lesssim 10^{-4}$ fm${}^{-3}$, which is reasonable as protons are more stable than neutrons. The contribution of electrons are then present in order to reach local charge neutrality condition $n_p=n_e$. Once nonuniform nuclear structures emerge, the proton fraction $Y_p$ deviates significantly from that of the uniform phase, which approaches to $Y_p=0.43$-0.45 at vanishing densities. The energy per baryon are then reduced by up to 8 MeV. Note that the absolute values of the energy per baryon at vanishing densities are sensitive to the adopted nucleon masses, while the obtained binding energy for various functionals coincide with each other. At vanishing densities, the pressure mainly comes from the contributions of electrons and is thus increasing with $Y_p$. Except for those adopting realistic nucleon masses, the obtained pressure for the nonuniform phase is larger than that of the uniform one as predicted by most of the functionals. The neutron drip densities $n_\mathrm{d}$ can be obtained by equating the chemical potential of neutrons with their mass, i.e., $\mu_n(n_\mathrm{d})=m_n$. The obtained values of $n_\mathrm{d}$ for various functionals are then indicated in Table~\ref{table:phase}, where those with the density-dependent couplings generally predict smaller neutron drip densities compared with that of nonlinear ones. Then at $n_\mathrm{b} \lesssim 10^{-4}$ fm${}^{-3}<n_\mathrm{d}$, neutron star matter are comprised of Coulomb lattices of nuclei and electrons, where the similar values for the pressure are obtained with various functionals in this density range. In such cases, the EOSs of neutron star matter at $n_\mathrm{b} \lesssim 10^{-4}$ fm${}^{-3}$ generally coincide with each other except for the slight differences (within 0.1\%) in the energy density due to the variations in the nucleon masses indicated in Table~\ref{table:param}. At larger densities with $n_\mathrm{b} \gtrsim n_\mathrm{d}$, we note that the slope of the energy per baryon, pressure, and proton fraction change suddenly as neutron gas starts to coexist with the liquid phase of nuclear matter, which forms the nuclear pasta typically found in the inner crusts of neutron stars. In contrast to the stellar matter located in the outer crusts of neutron stars, as indicated in the left panel of Fig.~\ref{Fig:EOS-all}, the EOSs of the pasta phase are sensitive to the adopted nuclear energy density functional. It is found that the EOSs obtained with nonlinear couplings become stiffer at the energy density $E/n_\mathrm{b} \gtrsim 20$ MeV fm${}^{-3}$ ($n_\mathrm{b} \gtrsim 0.02$ fm${}^{-3}$), while the EOSs obtained with density-dependent couplings vary more smoothly with density. In general, the relative uncertainty in the EOSs of the pasta phase grows with density and then decreases once reaches the peak at $E/n_\mathrm{b} \approx 20$ MeV fm${}^{-3}$. The corresponding differences in the EOSs at subsaturation densities are expected to affect the radii and crust thickness of neutron stars, which will be illustrated in Fig.~\ref{Fig:MR}. Note that for the functional DD2, the obtained results for nuclear pasta deviate significantly from other functionals. This is mainly because we have employed the single nucleus approximation (SNA) and neglected the contributions of light clusters as initially proposed in Ref.~\cite{Typel2010_PRC81-015803}. For more suitable treatments adopting the extended nuclear statistical equilibrium model, one can refer to Ref.~\cite{Fischer2014_EPJA50-46} with the publicly available EOS HS(DD2), which is more reasonable than the DD2 EOS presented in the left panel of Fig.~\ref{Fig:EOS-all} with a too large proton fraction. \begin{table*} \caption{\label{table:phase} Densities (in fm${}^{-3}$) for shape transitions, which are obtained by varying the density in a step of 0.002 fm${}^{-3}$. The neutron drip densities $n_\mathrm{d}$ obtained with $\mu_n(n_\mathrm{d}) = m_n$ and critical densities $n_\mathrm{DU}$ for the occurrence of DU processes with $Y_p(n_\mathrm{DU})=14.8\%$ are indicated as well.} \begin{tabular}{c|ccccc|cccccc} \hline \hline Transition & NL3 & PK1 & TM1 & GM1 & MTVTC & DD-LZ1 & DDME-X & PKDD & DD-ME2 & DD2 & TW99 \\ \hline $n_\mathrm{d}$ ($10^{-4}$) & 2.4 & 2.7 & 2.3 & 3.1 & 3.1 & 1.9 & 1.9 & 2.3 & 2.0 & 1.7 & 1.8 \\ \hline droplet-rod & - & - & - & - & - & 0.059 & 0.065 & - & 0.063 & 0.041 & 0.063 \\ rod-slab & - & - & - & - & - & 0.065 & 0.073 & - & 0.071 & 0.057 & 0.071 \\ slab-tube & - & - & - & - & - & 0.069 & - & - & 0.073 & 0.087 & 0.075 \\ tube-bubble & - & - & - & - & - & - & - & - & - & 0.101 & - \\ core-crust & 0.057 & 0.061 & 0.061 & 0.067 & 0.061 & 0.071 & 0.077 & 0.065 & 0.075 & 0.111 & 0.077 \\ \hline $n_\mathrm{DU}$ & 0.228 & 0.230 & 0.236 & 0.309 & 0.328 & - & - & 0.325 & - & - & - \\ \hline \end{tabular} \end{table*} If we further increase the density, the uniform phase becomes energetically more favorable once exceeding the core-crust transition densities indicated in Table~\ref{table:phase}, e.g., $n_\mathrm{b} \gtrsim 0.08$ fm${}^{-3}$. The corresponding energy per baryon, pressure, and proton fraction of neutron star matter are indicated in the right panels of Fig.~\ref{Fig:EOS-beta}. In contrast to the cases at smaller densities, the uncertainties in those quantities grow drastically as density increases at $n_\mathrm{b} \gtrsim 0.3$ fm${}^{-3}$, where the less constrained higher order coefficients such as $L$ and $K_\mathrm{sym}$ start to play important roles. If the EOSs do not cross with each other, we note that the stiffness of EOS is directly linked to the maximum mass of neutron stars as indicated in Fig.~\ref{Fig:MR}. Despite their evident differences in the energy per baryon, as indicated in the right panel of Fig.~\ref{Fig:EOS-all}, we note that the EOSs obtained with the functionals DD-LZ1 and DDME-X coincide with each other at $n_\mathrm{b} \gtrsim 0.08$ fm${}^{-3}$. The obtained proton fractions of neutron star matter at $n_\mathrm{b} \gtrsim 0.08$ fm${}^{-3}$ show distinctive trends between the functionals with nonlinear self-couplings and density dependent ones, which is attributed to the differences in the higher order coefficients $L$ and $K_\mathrm{sym}$ of nuclear symmetry energy as indicated in Table~\ref{table:NM}. It is found that $Y_p$ increases with density if nonlinear self-couplings are adopted, while for density dependent ones $Y_p$ approaches to a constant value ($\sim$0.14). It is worth mentioning that if isovector scalar channel ($\delta$ meson) are included in density-dependent covariant density functionals, the proton fraction may deviation from the trend and increase with density~\cite{Wang2014_PRC90-055801}. Meanwhile, we note that a peculiar density-dependent behavior of $Y_p$ (reaching its maximum at $n_\mathrm{b} \approx 0.5$ fm${}^{-3}$) is obtained with the functional PKDD, which is attributed to the large slope $L$ but negative curvature parameter $K_\mathrm{sym}$ of nuclear symmetry energy. In principle, the proton fraction is directly connected to the most efficient cooling mechanism of neutron stars. Once the momentum conservation is fulfilled with $Y_p\gtrsim 14.8\%$, the direct Urca (DU) processes $n\rightarrow p + e^-+\bar{\nu}_e$ and $p + e^-\rightarrow n+\nu_e$ will take place and rapidly cools the neutron star down~\cite{Klaehn2006_PRC74-035802, Page2006_NPA777-497}. As indicated in Fig.~\ref{Fig:EOS-beta}, the critical densities $n_\mathrm{DU}$ for the occurrence of DU processes can be obtained once $Y_p>14.8\%$, where the corresponding values are presented in Table~\ref{table:phase}. It is found that the DU processes only take places if the functionals with nonlinear self-couplings and PKDD are employed. Beside the EOSs, the variation in the microscopic structures has significant implications on the transport and elastic properties of neutron star matter, which would in turn affect various phenomenons observed in neutron stars~\cite{Chamel2008_LRR11-10, Caplan2017_RMP89-041002}. In Fig.~\ref{Fig:Micro-beta} we present the obtained microscopic structures of nonuniform neutron star matter corresponding to the EOSs in Figs.~\ref{Fig:EOS-beta} and \ref{Fig:EOS-all}, where the proton number $Z$, WS cell radius $R_\mathrm{W}$, and droplet size $R_\mathrm{d}$ are indicated. As density increases, the droplet, rod, slab, tube, bubble, and uniform phases appear sequentially for the nuclear pasta in neutron stars. The transition densities between different nuclear shapes are indicated in Table~\ref{table:phase}. We note that for the functionals predicting large slope $L$ of symmetry energy, only the droplet phase emerge for the nuclear pasta in $\beta$-equilibrium, while as indicated in Fig.~\ref{Fig:nt-sym} the core-crust transition densities $n_\mathrm{t}$ are smaller than those predicting smaller $L$ as well. This is consistent with previous studies, where the proton number of nuclei, the core-crust transition density, and the onset density of non-spherical nuclei generally decrease with $L$~\cite{Oyamatsu2007_PRC75-015801, Xu2009_ApJ697-1549, Grill2012_PRC85-055808, Bao2015_PRC91-015807, Shen2020_ApJ891-148, Xia2021_PRC103-055812}. The obtained results with the functional NL3 and DD-ME2 generally coincide with those in Ref.~\cite{Grill2012_PRC85-055808} with slightly larger core-crust transition density, while those of TM1 coincide with Ref.~\cite{Bao2015_PRC91-015807}. Meanwhile, according to Fig.~\ref{Fig:nt-sym}, it is evident that $n_\mathrm{t}$ also decreases with the curvature parameter $K_\mathrm{sym}$ of symmetry energy, which is closely related to the curvature-slope correlations~\cite{Pais2012_PRL109-151101, Li2020_PRC102-045807}. To show the consequences of adopting a functional that does not follow the curvature-slope correlation, in Fig.~\ref{Fig:nt-sym} we present the results predicted by FSUGarnet using the compressible liquid drop model~\cite{Parmar2022_PRD105-043017}, where the $L$-$n_\mathrm{t}$ correlation still holds approximately but not for the $K_\mathrm{sym}$-$n_\mathrm{t}$ correlation. Note that the functional DD2 predicts rather large $n_\mathrm{t}$, which will be reduced if the extended nuclear statistical equilibrium model is adopted including the contributions of light clusters~\cite{Fischer2014_EPJA50-46}. Similar to the EOSs of neutron star matter, as indicated in Fig.~\ref{Fig:Micro-beta}, the microscopic structures vary little with respect to the adopted functionals at $n_\mathrm{b} \lesssim 10^{-4}$ fm${}^{-3}$. For example, slightly different proton numbers and droplet sizes are obtained at vanishing densities (e.g., $n_\mathrm{b} \approx 10^{-10}$ fm) adopting various functionals, which vary within the ranges $Z\approx28$-35 and $R_\mathrm{d}\approx 5$-5.5 fm and increase with density at $n_\mathrm{b}\lesssim 10^{-4}$ fm${}^{-3}$. The obtained WS cell radius $R_\mathrm{W}$ is decreasing with density and is insensitive to the adopted functionals at $n_\mathrm{b} \lesssim 10^{-4}$ fm${}^{-3}$. We note that the differences on the microscopic structures start to grow once $n_\mathrm{b} \gtrsim n_\mathrm{d}$, where the values of $Z$ and $R_\mathrm{d}$ as functions of density may exhibit different trends for various functionals. At larger densities with $n_\mathrm{b} \gtrsim 0.01$ fm${}^{-3}$, the proton number $Z$ and WS cell radius $R_\mathrm{W}$ are generally decreasing, while the droplet size $R_\mathrm{d}$ increases. Throughout the vast density range considered here, in consistent with previous investigations~\cite{Oyamatsu2007_PRC75-015801, Xu2009_ApJ697-1549, Grill2012_PRC85-055808, Bao2015_PRC91-015807, Shen2020_ApJ891-148, Xia2021_PRC103-055812}, the obtained values of $Z$ and $R_\mathrm{d}$ approximately decrease with $L$ if different functionals are adopted, while the values of $R_\mathrm{W}$ are close to each other. Note that rather large values of $Z$, $R_\mathrm{d}$, and $R_\mathrm{W}$ are obtained with the functional DD2, which is mainly due to the SNA adopted here and neglecting light clusters. \subsection{\label{sec:star} Neutron stars} Based on the unified EOSs of neutron star matter presented in Figs.~\ref{Fig:EOS-beta} and \ref{Fig:EOS-all}, the structures of neutron stars are obtained by solving the TOV equation \begin{eqnarray} &&\frac{\mbox{d}P}{\mbox{d}r} = -\frac{G M E}{r^2} \frac{(1+P/E)(1+4\pi r^3 P/M)} {1-2G M/r}, \label{eq:TOV}\\ &&\frac{\mbox{d}M}{\mbox{d}r} = 4\pi E r^2, \label{eq:m_star} \end{eqnarray} where $G=6.707\times 10^{-45}\ \mathrm{MeV}^{-2}$ is the gravity constant. In Fig.~\ref{Fig:MR} we present the mass-radius relations of neutron stars corresponding to the covariant density functionals indicated in Table~\ref{table:param}. Various constraints from pulsar observations are indicated in Fig.~\ref{Fig:MR}, i.e., the binary neutron star merger event GW170817~\cite{LVC2018_PRL121-161101}, the simultaneous measurements of masses and radii for PSR J0030+0451 and PSR J0740+6620~\cite{Riley2019_ApJ887-L21, Riley2021_ApJ918-L27, Miller2019_ApJ887-L24, Miller2021_ApJ918-L28}, and the measured mass of a compact object involved in a compact binary coalescence from the gravitational-wave signal GW190814~\cite{Abbott2020_ApJ896-L44}. The open triangles in Fig.~\ref{Fig:MR} correspond to the critical masses $M_\mathrm{DU}$ for DU processes with the central densities exceeding $n_\mathrm{DU}$. It is expected that the neutrino emissivity is enhanced significantly for neutron stars with $M>M_\mathrm{DU}$~\cite{Spinella2018_Universe4-64}, which cool down too rapidly within just a few years~\cite{Blaschke2004_AA424-979}. It is found that the DU processes only take places if the functionals with nonlinear self-couplings and PKDD are employed ($M_\mathrm{DU}\approx 0.9$-1.3 $M_{\odot}$), which have large slopes of symmetry energy with $L\gtrsim 90$ MeV. We note that all functionals predict neutron stars with maximum masses exceeding 2 $M_{\odot}$~\cite{Fonseca2021_ApJ915-L12}, while the functionals NL3, DD-LZ1, and DDME-X predict even larger maximum masses supporting the possibility that the secondary object observed in GW190814 is a neutron star~\cite{Abbott2020_ApJ896-L44}. Nevertheless, as indicated in Table~\ref{table:NM}, the incompressibility, symmetry energy and its slope for nuclear matter obtained with the functional NL3 exceed the constraints from start-of-art studies~\cite{Shlomo2006_EPJA30-23, Zhang2020_PRC101-034303, Essick2021_PRL127-192701}, leading to neutron stars with too large radii and masses. A combined constraint on the masses and radii of neutron stars suggest that DD2, DD-LZ1, DD-ME2, and DDME-X are the most probable functionals that are consistent with observations. However, to support massive neutron stars, their skewness coefficients $J$ are much larger than expected, which was constrained to be $J=-700\pm 500$ MeV from fits of generalized Skyrme force to breathing-mode energies~\cite{Farine1997_NPA615-135} and $J=-390^{+60}_{-70}$ MeV from empirical pressures in relativistic heavy-ion collisions~\cite{Xie2021_JPG48-025110}. The maximum masses of neutron stars obtained by the two functionals MTVTC and TW99 are close to 2 $M_{\odot}$, while the corresponding radii are slightly small and locate in the lower ends of the PSR J0740+6620 constraints~\cite{Riley2021_ApJ918-L27, Miller2021_ApJ918-L28}. The functionals PKDD, GM1, TM1, PK1, and NL3 predict slightly too large radii according to the constraint derived from the binary neutron star merger event GW170817~\cite{LVC2018_PRL121-161101}, which are attributed to the much larger values for $K$ and/or $L$ as indicated in Table~\ref{table:NM}. Nevertheless, if exotic phases with the emergence of new degrees of freedom such as mesons ($\pi$, $K$, etc.), heavy baryons ($\Delta$, $\Lambda$, $\Sigma$, $\Xi$, $\Omega$, etc.), and deconfinement phase transition into quarks ($u$, $d$, $s$) were to take place, the corresponding EOSs would become softer, which effectively reduce the radii of compact stars and comply with the observational constraints~\cite{Baym2018_RPP81-056902, Sun2019_PRD99-023004, Xia2020_PRD102-023031, LI2020, Dexheimer2021_PRC103-025808, Dexheimer2021_EPJA57-216, Sun2021_PRD103-103003, Tu2022_ApJ925-16, Sun2022}. It was augured that the sudden spin-ups (glitches) of pulsars are due to the angular momentum transfers from the superfluid component of a neutron star's interior to its solid crust~\cite{Anderson1975_Nature256-25}, whose characteristic properties could provide additional constraints on neutron star structures. In particular, the fractional crustal moment of inertia ${I_\mathrm{c}}/{I}$ can be measured with \begin{equation} \frac{I_\mathrm{c}}{I} \gtrsim \frac{2\tau_c}{T}\sum_i\left(\frac{\Delta\Omega_p}{\Omega_p} \right)_i, \label{eq:DI_tau} \end{equation} where $\tau_c$ represents the characteristic age of the pulsar, $T$ the total time span for glitch monitoring, and ${\Delta\Omega_p}/{\Omega_p}$ the fractional frequency jump of glitches. To explain the glitches observed in the Vela pulsar, the fractional crustal moment of inertia was constrained to be ${I_\mathrm{c}/I} \gtrsim 1.4\%$~\cite{Link1999_PRL83-3362}. However, it was argued that the entrainment of superfluid neutrons by the solid crust could lower its mobility and increase the lower limit to ${I_\mathrm{c}/I} \gtrsim 7\%$, causing the ``glitch crisis" where many nuclear EOSs fail to meet the constraint~\cite{Andersson2012_PRL109-241103, Chamel2012_PRC85-035801, Li2016_ApJS223-16}. Nevertheless, it is worth mentioning that the entrainment effect may be suppressed if the pairing gap is of order or greater than the strength of the lattice potential~\cite{Watanabe2017_PRL119-062701}, where the constraint can be reduced to ${I_\mathrm{c}/I} \gtrsim 2.4\pm0.1\%$~\cite{Li2017_IAUS13-360}. For slowly rotating neutron stars, the fraction of crustal moment of inertia can be estimated with~\cite{Link1999_PRL83-3362} \begin{equation} \frac{I_\mathrm{c}}{I} \approx \frac{28\pi P_\mathrm{t} R^3}{3M}\frac{1 - 1.67\beta -0.6\beta^2}{\beta+ \frac{2 P_\mathrm{t}}{n_\mathrm{t}m_n}\left(\frac{1}{\beta} +5-14\beta \right)}, \label{eq:DI} \end{equation} where $P_\mathrm{t}$ is the pressure at core-crust transition density $n_\mathrm{t}$ and $\beta=GM/R$ the compactness of a neutron star. The obtained results are then presented in Fig.~\ref{Fig:DI}, where the crustal moment of inertia is decreasing with mass. It is evident that ${I_\mathrm{c}/I}$ is sensitive to the EOS, and in particular the crust one since it determines the mass and thickness of a neutron star's crust. Therefore a unified treatment for the EOSs of uniform (core) and nonuniform (crust) neutron star matter is essential to obtain accurately the radii, crust properties, core-crust transition density, as well as the corresponding microscopic structures. In order to meet the constraints of Vela pulsar as indicated by the horizontal lines and band, we note that a neutron star should not be more massive than a critical value, which varies with the EOSs and the effectiveness of the entrainment effect. Nevertheless, to distinguish the EOSs from one another, more detailed investigations on pulsar glitches are required in future studies. To examine the possible correlations between the macroscopic neutron star structures and microscopic nuclear matter properties, in Fig.~\ref{Fig:param} we present the radii of neutron stars at $M=1.4 M_{\odot}$ and $2 M_{\odot}$ as well as the slope $L$ and curvature parameter $K_\mathrm{sym}$ of nuclear symmetry energy. It is evident that there are linear $L$-$K_\mathrm{sym}$ correlations in RMF models~\cite{Pais2012_PRL109-151101, Li2020_PRC102-045807}, where $K_\mathrm{sym}$ increases with $L$. For the macroscopic neutron star structures, it is found that $R_2$ generally coincides with $R_{1.4}$ except for the two cases obtained with the functionals TW99 and MTVTC, where the maximum masses are close to $2 M_{\odot}$ with $R_2<R_{1.4}$. In such cases, if the observed radii $R_{1.4}$ are indeed close to $R_2$ as observed in {NICER} and {XMM}-Newton missions~\cite{Riley2019_ApJ887-L21, Riley2021_ApJ918-L27, Miller2019_ApJ887-L24, Miller2021_ApJ918-L28}, then the maximum mass $M_\mathrm{max}$ of neutron stars could easily surpass $2.3 M_{\odot}$ as indicated in the lower-left panel of Fig.~\ref{Fig:param}, which approaches to the upper limit ($\le2.35 M_{\odot}$) according to the numerical simulations of binary neutron star merger event GW170817~\cite{Rezzolla2018_ApJ852-L25, Ruiz2018_PRD97-021501, Shibata2019_PRD100-023015}. Meanwhile, we note that the maximum mass $M_\mathrm{max}$ generally increases with radius, which reaches $2.77 M_{\odot}$ at $R_{1.4}= R_2=14.6$ km for the functional NL3. The linear correlations between neutron stars' radii and $L$ ($K_\mathrm{sym}$) are also observed. In the top-left panel of Fig.~\ref{Fig:param} we find $R_{1.4}$ increases with $L$, which is consistent with previous investigations that the radius and tidal deformability are closely related to $L$~\cite{Zhu2018_ApJ862-98, Tsang2019_PLB795-533, Dexheimer2019_JPG46-034002, Zhang2019_EPJA55-39, Zhang2020_PRC101-034303, Li2020_PRC102-045807}. At the same time, as indicated in the lower-right panel of Fig.~\ref{Fig:param}, the radii of two-solar-mass neutron stars $R_2$ seem to have a better correlation with the higher order coefficient $K_\mathrm{sym}$ instead of $L$, which is attributed to the larger density range covered in those stars. Such kind of correlations provide opportunities to constrain higher order coefficients of nuclear symmetry energy in the absence of strangeness via future radius measurements with both pulse-profile modeling~\cite{Riley2021_ApJ918-L27, Miller2021_ApJ918-L28} and gravitational wave observations~\cite{LVC2018_PRL121-161101}. Nevertheless, it is worth mentioning that the correlations are mainly due to the particular choices of covariant density functionals. If we consider a functional that does not follow the $L$-$K_\mathrm{sym}$ correlation, such as FSUGarnet~\cite{Parmar2022_PRD105-043017} indicated in Fig.~\ref{Fig:param}, the $K_\mathrm{sym}$-$R_{1.4,2}$ correlations become less evident than the $L$-$R_{1.4,2}$ correlations. \section{\label{sec:con}Conclusion} Based on the numerical recipe presented in our previous study~\cite{Xia2022_PRC105-045803}, in this work we investigate systematically the EOSs and microscopic structures of neutron star matter in a vast density range with $n_\mathrm{b}\approx 10^{-10}$-2 $\mathrm{fm}^{-3}$ adopting various covariant density functionals (NL3~\cite{Lalazissis1997_PRC55-540}, PK1~\cite{Long2004_PRC69-034319}, TM1~\cite{Sugahara1994_NPA579-557}, GM1~\cite{Glendenning1991_PRL67-2414}, and MTVTC~\cite{Maruyama2005_PRC72-015802}, DD-LZ1~\cite{Wei2020_CPC44-074107}, DDME-X~\cite{Taninah2020_PLB800-135065}, PKDD~\cite{Long2004_PRC69-034319}, DD-ME2~\cite{Lalazissis2005_PRC71-024312}, DD2~\cite{Typel2010_PRC81-015803}, and TW99~\cite{Typel1999_NPA656-331}). All the results are obtained in a unified manner adopting Thomas-Fermi approximation, where spherical and cylindrical symmetries are assumed for the WS cells. The optimum configurations of neutron star matter in $\beta$-equilibrium are obtained by searching for the energy minimum among six types of nuclear matter structures (droplet, rod, slab, tube, bubble, and uniform) at fixed baryon number density $n_\mathrm{b}$. The effects of charge screening are accounted for with electrons moving freely around the nucleus~\cite{Maruyama2005_PRC72-015802}, where the proton number of nucleus $Z$, droplet size $R_\mathrm{d}$, and WS cell size $R_\mathrm{W}$ become larger compared with the previous investigations neglecting the charge screening effects~\cite{Xia2022_PRC105-045803}. Note that we have adopted the SNA without any light clusters, which is not applicable for the functional DD2 as initially intended~\cite{Typel2010_PRC81-015803}. In such cases, we recommend Ref.~\cite{Fischer2014_EPJA50-46} for a more suitable EOS HS(DD2) obtained with the extended nuclear statistical equilibrium model. The neutron drip densities of neutron star matter are found to be $n_\mathrm{d}\approx 2\text{-}3\times 10^{-4}$ fm${}^{-3}$, where those with the density-dependent couplings generally predict smaller $n_\mathrm{d}$ than that of non-linear ones. At smaller densities, neutron star matter are comprised of Coulomb lattices of nuclei and electrons with pressure mainly comes from electrons, where the EOSs of neutron star matter generally coincide with each other (discrepancy within 0.1\%). At $n_\mathrm{b}>n_\mathrm{d}$, the EOSs are sensitive to the adopted functionals, where the relative difference grows and reaches the peak at $n_\mathrm{b} \approx 0.02$ fm${}^{-3}$. The relative uncertainty of the EOSs decreases and remains small at $n_\mathrm{b} \lesssim 0.3$ fm${}^{-3}$, which however grows drastically at larger densities. For the microscopic structures, it is found that only the droplet (crust) and uniform (core) phases emerge if the covariant density functionals with nonlinear self-couplings are adopted, while non-spherical shapes (rod, slab, tube, and bubble) may appear if density-dependent couplings are employed with generally smaller slope $L$ of symmetry energy. The corresponding core-crust transition densities $n_\mathrm{t}$ decreases with $L$ as well. Meanwhile, the obtained droplet size $R_\mathrm{d}$ and proton number of nucleus $Z$ approximately decrease with $L$, while the values of WS cell size $R_\mathrm{W}$ are close to each other. These observed trends generally coincide with previous investigations~\cite{Oyamatsu2007_PRC75-015801, Xu2009_ApJ697-1549, Grill2012_PRC85-055808, Bao2015_PRC91-015807, Shen2020_ApJ891-148, Xia2021_PRC103-055812}. Additionally, similar correlations with the curvature parameter $K_\mathrm{sym}$ are observed as well, which is closely related to the curvature-slope correlations~\cite{Pais2012_PRL109-151101, Li2020_PRC102-045807}. The neutron star structures are then investigated adopting the unified EOSs. For all functionals considered in this work, the corresponding maximum masses of neutron stars exceed the two-solar-mass limit, while the functionals NL3, DD-LZ1, and DDME-X can even accommodate the mass of the secondary object observed in GW190814~\cite{Abbott2020_ApJ896-L44}. A combined constraint on both the masses and radii from pulsar observations~\cite{LVC2018_PRL121-161101, Riley2019_ApJ887-L21, Riley2021_ApJ918-L27, Miller2019_ApJ887-L24, Miller2021_ApJ918-L28} suggest that DD2, DD-LZ1, DD-ME2, and DDME-X are the most probable functionals for describing neutron star matter, while those of MTVTC and TW99 predict radii close to the lower ends of the PSR J0740+6620 constraints~\cite{Riley2021_ApJ918-L27, Miller2021_ApJ918-L28}. Nevertheless, in order to support massive neutron stars, the skewness coefficients $J$ for DD2, DD-LZ1, DD-ME2, and DDME-X are much larger than expected~\cite{Farine1997_NPA615-135, Xie2021_JPG48-025110}, which could be disentangled if the radius of PSR J0740+6620~\cite{Riley2021_ApJ918-L27, Miller2021_ApJ918-L28} and the maximum mass of neutron stars~\cite{Rezzolla2018_ApJ852-L25, Ruiz2018_PRD97-021501, Shibata2019_PRD100-023015} can be measured with higher accuracy. The functionals PKDD, GM1, TM1, PK1, and NL3 predict slightly too large radii according to the GW170817 constraint~\cite{LVC2018_PRL121-161101}, which can be reduced if exotic phases emerge at the center of neutron stars. Finally, we note there are approximate linear correlations between neutron stars' radii ($R_{1.4}$ at $M=1.4 M_{\odot}$ and $R_2$ at $M=2 M_{\odot}$) and the slope $L$ of nuclear symmetry energy. Since we have adopted covariant density functionals with approximate curvature-slope correlations, the correlations of those quantities with the curvature parameter $K_\mathrm{sym}$ of symmetry energy is observed as well. It was shown that the neutron star structures are sensitive to the EOSs both in the core and crust regions, where a unified description for neutron star matter is required~\cite{Fortin2016_PRC94-035804, DinhThi2021_AA654-A114}. At the same time, the microscopic structures of neutron star matter play important roles in the corresponding transport and elastic properties, which affect various physical processes in neutron stars~\cite{Chamel2008_LRR11-10, Caplan2017_RMP89-041002}. Particularly, we have estimated the critical densities $n_\mathrm{DU}$ and neutron star masses $M_\mathrm{DU}$ at $Y_p=14.8\%$, above which the DU processes will take place and cool the neutron star too rapidly within just a few years~\cite{Klaehn2006_PRC74-035802, Page2006_NPA777-497}. We note that the DU processes only take place if functionals with $L\gtrsim 90$ MeV are adopted. The critical density lies in the range $n_\mathrm{DU}\approx0.23$-0.33 fm${}^{-3}>n_\mathrm{t}$, so that the DU processes is sensitive to the core EOSs. Meanwhile, the crust EOSs are closely connected to the fractional crustal moment of inertia ${I_\mathrm{c}}/{I}$, which can be constrained by the characteristic properties of glitches observed in pulsars. It is shown that ${I_\mathrm{c}}/{I}$ is sensitive to the adopted EOS and in particular the crust one, which provide opportunities to constrain neutron star structures and the corresponding EOS based on glitch monitoring. Further constraints may be obtained if we apply the current results to the investigations of other topics in pulsars such as asteroseismology~\cite{Kouveliotou1998_Nature393-235, Hurley1999_Nature397-41, Hansen1980_ApJ238-740, Schumaker1983_MNRAS203-457, McDermott1988_ApJ325-725, Strohmayer1991_ApJ375-679, Passamonti2012_MNRAS419-638, Gabler2018_MNRAS476-4199, Sotani2012_PRL108-201101, Sotani2016_MNRAS464-3101, Kozhberov2020_MNRAS498-5149}, gravitational waves with respect to the strength of astromaterials~\cite{Horowitz2009_PRL102-191102, Chugunov2010_MNRAS407-L54, Horowitz2010_PRD81-103001, Caplan2018_PRL121-132701, Baiko2018_MNRAS480-5511, Abbott2020_ApJ902-L21}, neutrino-pasta scattering~\cite{Horowitz2016}, and evolution of magnetic field~\cite{Pons2013_NP9-431, Gao2017_ApJ849-19}. In such cases, the EOSs and microscopic structures of neutron star matter obtained in this work should be applicable for the investigations on the structures and evolutions of compact stars in a unified manner. \section*{ACKNOWLEDGMENTS} We would like to thank Prof. Nobutoshi Yasutake and Prof. Toshitaka Tatsumi for fruitful discussions. This work was supported by National SKA Program of China No.~2020SKA0120300, National Natural Science Foundation of China (Grant No.~11875052, No.~11873040, No.~11705163, and No.~11525524), the science research grants from the China Manned Space Project (No. CMS-CSST-2021-B11), the Youth Innovation Fund of Xiamen (No. 3502Z20206061), the Fundamental Research Funds for the Central Universities (Grant No.~lzujbky-2021-sp36), and the National Key R\&D Program of China No.~2018YFA0404402. \newpage

Title: Light dark matter around 100 GeV from the inert doublet model

Abstract: We made global fits of the inert Higgs doublet model (IDM) in the light of collider and dark matter search limits and the requirement for a strongly first-order electroweak phase transition (EWPT). These show that there are still IDM parameter spaces compatible with the observational constraints considered. In particular, the data and theoretical requirements imposed favour the hypothesis for the existence of a scalar dark matter candidate around 100 GeV. This is mostly due to the pull towards lower masses by the EWPT constraint. The impact of electroweak precision measurements, the dark matter direct detection limits, and the condition for obtaining a strongly enough first-order EWPT, all have strong dependence, sometimes in opposing directions, on the mass splittings between the IDM scalars.

https://export.arxiv.org/pdf/2208.13705

\title{Light dark matter around 100 GeV from the inert doublet model} \author{Shehu AbdusSalam} \author{Leila Kalhor} \author{Mohammad Mohammadidoust} \affiliation{Department of Physics, Shahid Beheshti University, Tehran, Islamic Republic of Iran} \tableofcontents \section{Introduction} The observed matter-anti-matter asymmetry and dark matter (DM) constituent of the universe are decisive indications for physics beyond the standard model (SM) of particle physics. The theoretical and model building developments for addressing these include the electroweak baryogenesis scenario~\cite{Sakharov:1967dj, Kuzmin:1985mm} which requires that earlier, the universe undergoes a strong first-order electroweak phase transition. Particles that are stable and weakly, or indirectly, coupled to the SM particles but with acceptable relic densities can be considered for explaining the DM part of the universe~\cite{Scherrer:1985zt}. Within the SM, there is one complex Higgs doublet which led to the prediction of the now observed Higgs boson. Extensions of the SM Higgs sector with additional $SU(2)_L$ n-tuples provide interesting theoretical scenarios that could simultaneously account for baryogenesis via a strong first-order electroweak phase transition(EWPT) in the early universe and the observed DM density. There are many considerations with various DM and EWPT phenomenology perspectives for addressing non-minimal Higgs sectors. For instances, see the non-exhaustive selection~\cite{Silveira:1985rk, Burgess:2000yq, Cirelli:2005uq,Aoki:2009vf, Cheung:2012xb, Morrissey:2012db, Blinov:2015vma, Belyaev:2016lok, Chiang:2018gsn, GAMBIT:2018eea,Chao:2018xwz, Liu:2020dok, Bandyopadhyay:2021ipw, Fan:2022dck, Khan:2015ipa, Datta:2016nfz}, the references therein and their citations. In~\cite{AbdusSalam:2013eya}, various models extending the SM Higgs sector using different Higgs multiplet representations were compared based on the EWPT and DM constraints. The analyses showed that the inert Higgs doublet model(IDM) turned out to be the favoured model. The IDM, an extension of the SM by a second Higgs doublet with no direct couplings to fermions, is one of the simplest scenarios with which a strong first-order EWPT can be realised and at the same time provide a candidate DM particle. With a $\mathbb{Z}_{2}$ symmetry imposed, the lightest $\mathbb{Z}_{2}$-odd particle will be stable and hence a suitable DM candidate~\cite{Deshpande:1977rw} with thermal relics that could explain the observed DM density. Many groups have analysed the IDM in the context of DM, EWPT, and collider phenomenology such as in~\cite{Fromme:2006cm, Cao:2007rm, Miao:2010rg, Honorez:2010re,Ilnicka:2015jba, Chowdhury:2011ga, Borah:2012pu, Gil:2012ya, AbdusSalam:2013eya, Cline:2013bln, Goudelis:2013uca, Gustafsson:2012aj, Swiezewska:2013uya, Dolle:2009ft, Blinov:2015vma, Belyaev:2016lok, Chiang:2018gsn, Dercks:2018wch, Chao:2018xwz, Banerjee:2019luv, Fabian:2020hny, Aoki:2021oez}. In this paper, we are going to make the first statistically convergent Bayesian global fit of the IDM in the light of the requirement that the inert Higgs particle simultaneously lead to a strong first-order EWPT and accounts for the observed cold DM relic density. This should be complementary to the work reported in~\cite{Eiteneuer:2017hoh}, where a frequentist global fit analysis of the IDM were made using constraints including the DM indirect detection limits, which we do not consider here. The requirement for a strong first-order EWPT is central to the analysis presented here but was not addressed in~\cite{Eiteneuer:2017hoh}. Further, for our analysis, we have derived the IDM Lagrangian from the most general renormalisable potential proposed in~\cite{Chao:2018xwz} which differs from previous general potentials used within the literature. Ultimately, we wish to derive the Lagrangians for other representations from this and compare with the IDM to go beyond~\cite{AbdusSalam:2013eya}. Using {\sc LanHEP}~\cite{Semenov:2014rea}, the model Lagrangian was written in the form required by {\sc micrOMEGAs}~\cite{Belanger:2013oya,Belanger:2010pz,Belanger:2008sj, Belanger:2006is} for computing DM properties and another form required by {\sc BSMPT} (Beyond the Standard Model Phase Transitions), a tool for computing beyond the SM (BSM) electroweak phase transitions~\cite{Basler:2018cwe}. We found that the collider, DM, and theoretical constraints applied to the IDM reveal strong support for the existence of an inert Higgs boson around 100 GeV. The most important of the constraints, namely, the oblique parameters from electroweak precision measurements, the dark matter direct detection limits, and the condition for obtaining a strongly enough first-order EWPT, all have a strong dependence on the mass splittings between the IDM scalars, $\Delta m_i$. A deeper study of the correlations with $\Delta m_i$ is an interesting direction beyond the scope of the fits presented here which we hope to address in another project. The layout of this paper is as follows. In section~\ref{idm_and_constraints}, we introduce the inert doublet model as a simple extension of the standard model with one additional Higgs doublet $Q$ and an unbroken $\mathbb{Z}_{2}$ symmetry under which $Q$ is odd while all other fields are even. This discrete symmetry prevents the direct coupling of $Q$ to fermions and, crucial for dark matter, guarantees the stability of the lightest odd particle. In section~\ref{subsec:constraints}, we describe the theoretical conditions that the IDM parameters must satisfy in order to be acceptable. The constraints from collider searches, DM-related limits and the requirement for a strong first-order EWPT are also described in that section. In section~\ref{sec:fits}, we present the result of the global fits and analyses of the IDM parameter space. Our conclusions are presented in the last section. \section{The inert doublet model (IDM)} \label{idm_and_constraints} The $SU(2)_L \times U(1)_Y$ gauge group representations are labelled by isospin and hypercharge, $(J,Y)$. $J$ takes integer and semi-integer values, and $Y$ can have any real value. The electric charge of each component of the multiplet is given by $Q =T_3 + \frac{Y}{2}$. Here, $T_3$ is the third component of $SU(2)_L$ group generators that can take $n = 2J+1$ values $T_3 = J, J-1, ..., -J$ in the {\bf n} representation. In order for one of the components to be a DM candidate, its electric charge must be zero. This constrains the possible values of the hypercharge, $Y$, for each $J$. For even(odd) values of {\bf n}, the value of $Y$ must be an odd (even) integer and it is necessary that $|Y| \leqslant 2J$. For the IDM, ${\bf n} = 2$, there is only one value for hypercharge, $|Y| = 1$. We only consider representations with a positive value of Y. Representations with a negative value of Y are similar to the positive ones. In~\cite{Chao:2018xwz}, the most general renormalisable scalar potential, V, with the SM Higgs doublet, H, and an electroweak multiplet Q of arbitrary $SU(2)_L$ rank and hypercharge, Y, was developed. Imposing a discrete $\mathbb{Z}_{2}$ symmetry, under which $Q$ is odd while all the SM fields are even, prevents the lightest $\mathbb{Z}_{2}$-odd particle from decaying into SM particles. Thus, it could play the role of the DM candidate. Specialising to the IDM case, the scalar potential is given by \begin{align} \label{eqn:Doublet} V = \, & \mu_h^2|H|^2 + \lambda_h|H|^4 + \mu^2_{Q}Q^{\dagger}Q + \lambda_1 [(QQ)_1(\overline{Q}\,\overline{Q})_1]_{0}\, + \alpha(H^{\dagger}H)(Q^{\dagger}Q) + \beta[(\overline{H}H)_{1}(\overline{Q}Q)_{1}]_{0} \nonumber \\ &+ \Big\{\kappa_1[(HH)_{1}(\overline{Q}\,\overline{Q})_{1}]_{0} + H.c.\Big\} \end{align} \begin{equation} \textrm{ with } H \equiv \begin{pmatrix} H^{+} \\ H^{0} \end{pmatrix} =\frac{1}{\sqrt{2}} \begin{pmatrix} x_1+i x_2 \\ h + i x_3 \end{pmatrix} \quad \textrm{ and } \quad Q = \frac{1}{\sqrt{2}} \begin{pmatrix} y_1+i y_2 \\ S + i R \end{pmatrix}. \end{equation} Here $\mu_{Q}, \lambda_1, \alpha, \beta$, and $\kappa_1 \equiv K$ represent the free parameters of the IDM; $h$ is the SM-like neutral Higgs field, with the vacuum expectation value (VEV) $ \left< h \right> \equiv v \approx 246 \, \textrm{GeV}$; and $x_1$, $x_2$, and $x_3$ are the electroweak Goldstone bosons. In unitary gauge, the parameters used here map to the commonly used $\lambda_{1,2,\dots, 5}$ notation~\cite{Belyaev:2016lok} as follows: \begin{equation} \left( \lambda_h, \quad \frac{1}{\sqrt{3}} \, \lambda_1, \quad \alpha, \quad \frac{1}{2\sqrt{3}} \, \beta, \quad \frac{2}{\sqrt{3}} \, \kappa_1\right) \rightarrow \left( \lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5 \right). \end{equation} $\overline{H}$ and $\overline{Q}$ are the similarity transformation-related equivalents of the $SU(2)_L$ representations for $H$ and $Q$ respectively. For the IDM, with $J =\frac{1}{2}$, the similarity transformation matrix V is equal to $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ so that \begin{equation} \overline{H}= V H^{*}=\frac{1}{\sqrt{2}} \begin{pmatrix} h - i x_3\\ -x_1+i x_2 \end{pmatrix} \quad \textrm{ and } \quad \overline{Q}= V Q^{*}=\frac{1}{\sqrt{2}} \begin{pmatrix} S - i R \\ -y_1+i y_2 \end{pmatrix}. \end{equation}\\ This way, $(\overline{Q}_{j=\frac{1}{2}}\,Q_{j=\frac{1}{2}})_J$ represents the combination of two $j=\frac{1}{2}$ doublets with total isospin $J$. The isospin addition rules should be used. For instance \begin{equation} (\overline H H)_1 = \begin{pmatrix} (H^0)^* H^+ \\ \frac{1}{\sqrt{2}}\left[( H^0)^*H^0 - (H^+)^* H^+ \right] \\ -(H^+)^* H^0 \end{pmatrix}. \end{equation} The mass terms for the neutral scalar $S$, the pseudoscalar state, $R$, and for the charged scalar, $Y^{\pm}$, after electroweak symmetry breaking are \begin{equation}\label{eqn:MDM} M_{S}^2 = \mu_Q^2 +(\frac{\alpha}{2}+\frac{\sqrt 3}{3}K - \frac{\sqrt 3}{12}\beta)v^2, \, M_{R}^2 = \mu_Q^2 +(\frac{\alpha}{2}-\frac{\sqrt 3}{3}K - \frac{\sqrt 3}{12}\beta)v^2, \, M_{Y^{\pm}}^2 = \mu_Q^2 +(\frac{ \alpha}{2}+\frac{\sqrt 3}{12}\beta)v^2. \end{equation} For $S$ to be the DM candidate particle, and stable, $M_S < M_R , M_{Y^{\pm}}$ must be satisfied. Accordingly, this choice will imply that $K<0$ and $\frac{\sqrt 3}{3}K+\frac{\sqrt 3}{6}\beta<0$. Using the Higgs portal notations, \begin{equation} \Lambda_1 = \frac{\alpha}{2}+\frac{\sqrt 3}{3}K - \frac{\sqrt 3}{12}\beta, \quad \textrm{ and } \quad \bar\Lambda_1 = \frac{\alpha}{2}-\frac{\sqrt 3}{3}K - \frac{\sqrt 3}{12}\beta \end{equation} are respectively related to the triple and quartic couplings between the SM Higgs $h$ and the DM candidate $S$ or the pseudoscalar $R$. The parameters $\alpha$ and $\beta$, on the other hand, determines the mass term, and describe the $h$ interaction with the charged scalars $Y^{\pm}$. The parameter $\lambda_1$ describes the quartic self- and non-self couplings of extended Higgs sector particles. The vertex factors are summarised in Table~\ref{tab:table1}. The portal parameters can be expressed in terms of the mass parameters and $\lambda_1$ as follows \begin{equation} \alpha = 2 \Lambda_1 - \frac{1}{2v^2}(3M_S^2-2M_{Y^\pm}^2-M_R^2), \, \beta = -\frac{\sqrt 3}{v^2}(M_S^2+M_R^2-2M_{Y^\pm}^2), \, K = \frac{\sqrt 3}{2v^2}(M_S^2-M_R^2). \end{equation} \begin{table} \begin{center} \caption{The vertex factors for the IDM couplings.} \label{tab:table1} \begin{tabular}{|l|l||l|l|} \hline \textbf{vertex} & \textbf{factor} & \textbf{vertex} & \textbf{factor} \\ \hline $SSSS$ & $2\sqrt3\lambda_1$ & $hhY^+Y^-$ & $\alpha+\frac{\sqrt3}{6}\beta$ \\ $RRRR$ & $2\sqrt3\lambda_1$ & $hY^+Y^-$ & $(\alpha+\frac{\sqrt3}{6}\beta)v$ \\ $Y^+Y^-Y^+Y^-$ & $4\frac{\sqrt3}{3}\lambda_1$ & $hhRR$ & $2\bar\Lambda_1$ \\ $SSRR$ & $\frac{2\sqrt3}{3}\lambda_1$ & $hSS$ & $2\Lambda_1 v$ \\ $SSY^+Y^-$ & $\frac{2\sqrt3}{3}\lambda_1$ & $hhSS$ & $2\Lambda_1$ \\ $RRY^+Y^-$ & $\frac{2\sqrt3}{3}\lambda_1$ & {} &\\ \hline \end{tabular} \end{center} \end{table} In all, the IDM has five free parameters which can be chosen to be $M_S, M_R, M_{Y^{\pm}}, \lambda_1$, and $\Lambda_1$. For the global fits, the mass parameters were allowed in the range, 1 to 5 TeV. The parameter $\Lambda_1$ was allowed in [-1, 1] while $\lambda_1$ was fixed at 0.1. In what follows, we are going to explain the set of theoretical and experimental results used for constraining the IDM parameters space. \section{The constraints on the IDM} \label{subsec:constraints} \subsection{Theoretical constraints} \paragraph{Vacuum Stability:} A scalar potential has to be bounded from below for describing a stable physical system. Within the SM, this means that the self-coupling of the Higgs boson, $\lambda_h$, has to be positive. For the IDM, the vacuum of the potential has to be stable in the limit of large values along all possible directions of the field space. This will require that~\cite{Kannike:2012pe} \begin{equation} \lambda_h > 0, \, \lambda_1 > 0; \, \alpha + 2\sqrt{\lambda_1 \lambda_h}>0; \, \textrm{ and } \alpha+\frac{\sqrt 3}{6}(\beta-|K|)+2\sqrt{\lambda_1 \lambda_h}>0. \end{equation} \paragraph{Perturbativity and unitarity:} {For calculations using perturbation theory, the relevant couplings used as expansion parameters should not be too large. This can be imposed by requiring that the absolute values of the coupling parameters be less than $4\pi$~\cite{Ginzburg:2005dt}.} We also require that unitarity should not be violated for all scalar $2 \rightarrow 2$ scattering. The perturbative unitarity conditions~\cite{Ginzburg:2005dt, Belyaev:2016lok} applied to the IDM are $|U_i| \leq 8 \pi$, where $i = 1, \ldots 10$, \begin{eqnarray} U1 &=& \lambda_h + \frac{\sqrt 3}{3}\lambda_1 + \frac{\sqrt {3\lambda_1^2 - 6{\sqrt 3}\lambda_1 \lambda_h+3 \beta^2+9 \lambda_h^2}}{3}, \quad U5 = \alpha - \frac{{\sqrt 3}}{6}\beta - 4 \frac{\sqrt 3}{3}K \nonumber\\ U2 &=& \lambda_h + \frac{\sqrt 3}{3}\lambda_1 - \frac{\sqrt {3\lambda_1^2 - 6{\sqrt 3}\lambda_1 \lambda_h+3 \beta^2+9 \lambda_h^2}}{3}, \quad U6 = -\alpha + \frac{{\sqrt 3}}{6}\beta + 4 \frac{\sqrt 3}{3}K, \nonumber\\ U3 &=& \lambda_h + \frac{\sqrt 3}{3}\lambda_1 + \frac{\sqrt {3\lambda_1^2 - 6{\sqrt 3}\lambda_1 \lambda_h+3 K^2+9 \lambda_h^2}}{3}, \quad U7 = \alpha - \frac{{\sqrt 3}}{6}\beta, \nonumber\\ U4 &=& \lambda_h + \frac{\sqrt 3}{3}\lambda_1 - \frac{\sqrt {3\lambda_1^2 - 6{\sqrt 3}\lambda_1 \lambda_h+3 K^2+9 \lambda_h^2}}{3}, \quad U8 = \alpha + \frac{\sqrt 3}{2} \beta,\nonumber\\ U9 &=& \alpha + \frac{\sqrt 3}{6} \beta + \frac{\sqrt 6}{3}K, \quad \textrm{ and } \quad U10 = \alpha + \frac{\sqrt 3}{6} \beta - \frac{\sqrt 6}{3} K. \end{eqnarray} \subsection{Limits from collider searches} The main approach for the phenomenological exploration of BSMs is the confrontation of the models with limits from experiments. The large electron-positron(LEP), Tevatron and the large hadron collider (LHC) experiments publish exclusion limits based on precision measurements or the non-observation of new particles. For exploring and fitting the IDM parameter space to data, the categories of collider limits used are explained as follows. \paragraph{LEP:} Precision measurement results by LEP exclude the possibility that massive SM gauge bosons decay into inert particles. This requires that~\cite{Dercks:2018wch, Cao:2007rm, Belyaev:2016lok} \begin{equation} M_{R,S} + M_{Y^\pm} \geq M_W, \quad M_R + M_S \geq M_Z, \textrm{ and } 2M_{Y^\pm} \geq M_Z. \end{equation} {The LEP results also} give rise to the exclusion of an intersection of mass ranges which can be evaded by fulfilling all of the following conditions {simultaneously}~\cite{Belyaev:2016lok,Ilnicka:2015jba} \begin{equation} \label{eq:LEPsplit} M_{S} > 80 GeV, \quad M_{R} > 100 GeV, \quad M_{R}-M_{S} < 8 GeV. \end{equation} {There is also the limit $M_{Y^{\pm}} > 70 GeV$ } from searches for charged Higgs pair production. \paragraph{Oblique parameters:} The values of the S, and T (with U=0) oblique parameters for a given point in the IDM parameter space can be computed as~\cite{hep-ph/0603188, Belyaev:2016lok} \begin{equation} \label{STUeq} \begin{split} S & = \frac{1}{72\pi \left( x_{2}^{2} - x_{1}^{2} \right)^{3}} \left[ x_{2}^{6}f_{a}\left( x_{2} \right) - x_{1}^{6} f_{a} \left( x_{1} \right) + 9x_{1}^{2}x_{2}^{2} \left( x_{2}^{2}f_{b} \left( x_{2} \right) - x_{1}^{2}f_{b} \left( x_{1} \right) \right) \right]\\ T & = \frac{1}{32\pi^{2}\alpha v^{2}} \left[ f_{c} \left( M_{Y^{\pm}}^{2}, M_{R}^{2} \right) + f_{c} \left( M_{Y^{\pm}}^{2}, M_{S}^{2} \right) - f_{c} \left( M_{R}^{2}, M_{S}^{2} \right) \right] \\ &\simeq \frac{1}{24\pi^{2}\alpha v^{2}}(M_{Y^\pm}-M_S)(M_{Y^\pm}-M_R)\, = \frac{\Delta m_{Y^\pm} \, ( \Delta m_{Y^\pm} - \Delta m_R)}{24\pi^{2}\alpha v^2}\,. \end{split} \end{equation} Here, $\Delta m_{Y^\pm} = M_{Y^\pm}-M_S$, $\Delta m_R = M_R-M_S$, $x_{1} \equiv \frac{M_{S}}{M_{Y^{\pm}}} $, $ x_{2} \equiv \frac{M_{R}}{M_{Y^{\pm}}} $, $\alpha\approx 1/127$ denotes the fine-structure constant at the scale of the $Z$ boson mass, $f_{a}(x) \equiv -5 + 12\ln x$, $f_{b}(x) \equiv 3-4\ln x$, and \begin{equation} \label{eq:fs} f_{c}(x,y) \equiv \begin{cases} \frac{x+y}{2} - \frac{xy}{x-y}\ln \frac{x}{y} &\mathrm{for} \ x\neq y \\ 0 &\mathrm{for} \ x=y \end{cases}\,. \end{equation} Only model points with S and T oblique parameters within 1-sigma of the PDG~\cite{ParticleDataGroup:2020ssz} average were accepted. There is a strong correlation between these parameters and the limit on other observables used for the IDM analyses. In particular, both the requirement for a strong first-order EWPT $\frac{v_c}{T_c} > 1$ and the upper bound on the inert singlet versus nucleons elastic scattering cross section, $\sigma^{SI}$, are strongly dependent on the IDM mass differences. This is also the case for the oblique parameters as can be seen in Eq.(\ref{STUeq}). The new precision measurements of the top-quark and W boson masses~\cite{CDF:2022hxs, CMS:2022kcl} will change the allowed ranges for the oblique parameters. We check that the IDM fit presented here are compatible with these within 2-sigma range of the allowed interval from an updated global fit~\cite{2204.04204}, which includes these new measurements, of new physics models to electroweak precision data. \paragraph{Limits implemented in HiggsBounds:} The Higgs sector predictions based on the IDM are compared with corresponding cross section limits for various processes studied at LEP, Tevatron, and LHC to determine whether the IDM parameter point has been excluded at 95\% C.L. or not. HiggsBounds~\cite{0811.4169, 1102.1898, 2006.06007} incorporates results from LEP~\cite{hep-ex/0107034, hep-ex/0107032, hep-ex/0107031,hep-ex/0111010, CERN-ALEPH-2002-019, hep-ex/0206022, hep-ex/0401022, hep-ex/0401026, hep-ex/0404012, hep-ex/0501033, hep-ex/0410017, hep-ex/0602042, 0707.0373, 0812.0267, 1301.6065}, the Tevatron~\cite{0809.3930, 0806.0611, 0908.1811, 0907.1269, 0906.1014, 0905.3381, 1011.1931, 1001.4468, 1001.4481, 1003.3363, 1008.3564, 1107.1268, 1106.4555, 1106.4885, 1108.3331, 1203.3774, 1207.6436}, the ATLAS~\cite{1207.7214, 1112.2577, 1109.3357, 1108.5064, 1202.1415, 1202.1414, 1202.1408, 1204.2760, 1402.3051, 1402.3244, 1407.6583, 1409.6064, 1406.7663, 1406.5053, 1509.00389, 1509.05051, 1507.05930, 1503.04233, 1502.04478, 1509.04670, 1606.04833, 1606.08391, 1710.07235, 1710.01123, 1712.06386, 1709.07242, 1707.04147, 1808.02380, 1808.03599, 1804.01126, 1807.00539, 1807.08567, 1807.07915, 1806.07355, 1811.11028, 1809.06682, 1808.00336, 1909.10235, 1904.05105, 1901.08144, 1907.06131, 1906.02025, 1907.02749}, and the CMS~\cite{1202.1997, 1202.3478, 1202.1416, 1202.1488, 1312.5353, 1307.5515, 1310.3687, 1407.0558, 1404.1344, 1504.00936, 1504.04710, 1510.06534, 1506.02301, 1508.07774, 1510.01181, 1506.08329, 1510.04252, 1506.00424, 1503.04114, 1603.02991, 1603.06896, 1707.02909, 1708.04188, 1701.02032, 1707.07283, 1811.08459, 1808.06575, 1805.04865, 1812.06359, 1809.05937, 1805.10191, 1804.01939, 1805.12191, 1811.09689, 1803.06553, 1911.04968, 1907.07235, 1903.04560, 1912.01594, 1911.10267, 1907.03152, 1908.01115, 1911.03781, 1903.00941, 2001.07763} experiments. \paragraph{Limits implemented in {\sc Lilith}:} Should the second CP-even Higgs boson of the IDM be SM-like, with mass between 123 to 128 GeV, then {\sc Lilith}~\cite{1502.04138,1606.03834,1908.03952} is used for gauging its couplings with respect to the Higgs signal strength measurements from ATLAS~\cite{Aaltonen:2013xpo, Aad:2015gba, Aad:2014eha, Aad:2015ona, Aad:2014eva, Aad:2015vsa, Aad:2015iha, Aad:2015gra, Aad:2014xva, Aad:2014iia, Aaboud:2018xdt, Aaboud:2017vzb, Aaboud:2018pen, Aaboud:2017ojs, Aaboud:2018jqu, Aaboud:2018gay, Aaboud:2019rtt, Aad:2019lpq, Aaboud:2017xsd, Aaboud:2017bja, Aaboud:2017jvq, Aaboud:2017rss} and CMS~\cite{Khachatryan:2014jba, Khachatryan:2014ira, Chatrchyan:2013iaa, Chatrchyan:2013mxa, Chatrchyan:2014nva, Chatrchyan:2013zna, Khachatryan:2014qaa, Khachatryan:2015ila, Khachatryan:2015bnx, Chatrchyan:2014tja, Sirunyan:2018koj,Sirunyan:2018cpi, Sirunyan:2018owy}. For each IDM point with an associated signal strength $\mu_i$ , {\sc Lilith} returns a log-likelihood value \be -2L_{lilith}( \theta) = - 2 \sum_i \log L(\mu_i) = \sum_{i} \left(\frac{\mu_i(\theta) - \hat\mu_i}{\Delta \mu_i}\right)^2. \ee Here $i$ runs over the various categories of Higgs boson production and decay modes combinations for a given point, $\theta$, in the model parameter space. $\hat\mu_i \pm \Delta\hat\mu_i$ represents the experimentally determined signal strengths. Theoretically, the signal strength associated to a model point for a given production mode $X$ and decay mode $Y$ is \begin{equation} \label{muimp} \mu = \sum_{X,Y} \epsilon_{X,Y} \frac{ \sigma(X) \, BR(H \rightarrow Y)}{\left[ \sigma(X) \, BR(H \rightarrow Y) \right]^{SM}} \end{equation} where $\epsilon_{X,Y}$ represents experimental efficiencies, $X \in \{ ggH, VH, VBF, ttH\}$ and $Y \in \{ \gamma \gamma, VV^{(*)}, b \bar{b}, \tau \tau, ttH\}$. In general, for the results from LHC, the elements in $X$ represent: the gluon-gluon fusion (ggH), associated production with a boson (VH), vector boson fusion (VBF) or associated production with top quarks (ttH). The elements in $Y$ represent the Higgs diphoton ($\gamma \gamma$), W or Z bosons ($VV$), bottom quarks ($bb$) or tau leptons ($\tau \tau$) decay modes. For computing the signal strengths $\mu$, the input parameters passed to {\sc Lilith} are the reduced couplings~\cite{Heinemeyer:2013tqa} $C_X^2$ and $C_Y^2$ such that \begin{equation} \sigma(X) = C_X^2 \, \sigma(X)^{SM} \quad \textrm{ and } \quad \Gamma(Y) = C_Y^2 \, \Gamma(Y)^{SM}. \end{equation} These, together with the Higgs boson invisible and undetectable decay branching ratios are computed using the {\sc micrOMEGAs} system for \begin{equation} \mu = ( 1 - BR(H \rightarrow \, undetected) - BR(H \rightarrow \, invisible) ) \frac{ \sum_{X,Y} \epsilon_{X,Y} C_X^2 \, C_Y^2} {\sum_Y \, C_Y^2 \, BR(H \rightarrow Y)^{SM}}. \end{equation} This is then in turn compared with the table of likelihood values as a function of $\mu$ within the {\sc Lilith} database of results from experiments for computing the log-likelihood. \subsection{Dark matter related constraints} The IDM predicts the existence of a neutral scalar field, S as DM candidate. The SM Higgs boson may decay into a pair of the DM candidate particles when kinematically allowed, and can therefore contribute to the invisible SM Higgs boson. For the IDM, we require that the branching ratio of the SM Higgs boson decay to the DM candidate particle be less than $0.15$~\cite{ATLAS-CONF-2020-052}. At early universe times, after freezing-out of equilibrium, the relic density of $S$ can account for the observed density of DM relics. The scattering of S onto nucleons should possibly lead to DM direct detection signatures. There are searches for the elastic scattering of DM with nucleons. It is expected that the recoil energy deposited on nuclei in a detector can be measured. In the absence of discovery, then upper limits on the scattering cross section can be determined. The cross sections can be either spin-independent (SI) or spin-dependent (SD) depending on whether the lightest odd particle effective coupling to the nucleons is via scalar or axial-vector interaction. The currently most stringent direct detection limits are those by PandaX-II~\cite{PandaX-II:2017hlx} and the XENON1T~\cite{XENON:2019gfn} experiments. We use the package {\sc micrOMEGAs} for computing the IDM predictions for the DM candidate relic density and its scattering cross section while interacting with nucleons. These are then compared with the corresponding experimentally determined value, $\Omega_{DM}h^2 = 0.1200 \pm 0.0001$~\cite{1807.06209} for the relic density, and direct detection limits set by PandaX-II~\cite{PandaX-II:2017hlx} and the XENON1T~\cite{XENON:2019gfn} experiments. \subsection{Requirement for strong first-order EWPT} For investigating the EWPT, the finite temperature quantum field theory techniques has to be used -- see~\cite{Quiros:1999jp} for a review. The ground state of the potential at $v= 0$ represents the symmetric phase of the model, while $v \ne 0$ represents the broken phase. Starting with the symmetric vacuum in the early universe, the EWPT is defined as the point in the evolution of the effective potential, $V_{eff}$, where a second minimum with non-zero VEV, $v_c$, develops at the critical temperature $T_c$ such that \begin{equation} \label{veff} V_{eff} (v=0, T_c) = V_{eff} (v=v_c, T_c), \quad V_{eff} = V_{tree}+V_{CW}+V_{CT}+V_T. \end{equation} Here $V_{tree}$, $V_{CW}$, $V_{CT}$ and $V_T$ respectively represents the tree-level potential, Eq.(\ref{eqn:Doublet}), the Coleman-Weinberg potential, the counter-term potential and the thermal corrections at finite temperature T. The latter set of the effective potential terms are described in Appendix~\ref{veffective}. Given the IDM tree-level potential, the other terms above were computed such that the strength of the EWPT at each model point can be determined using BSMPT~\cite{1803.02846, 2007.01725}. The description of the IDM implementation into the BSMPT is given in Appendix~\ref{makeBSMPT}. BSMPT can find the global minimum of $V_{eff}$ and hence determine $T_c$ and $v_c$ at the instance when the phase transition takes place. For the model point to be a possible candidate for electroweak baryogenesis, the EWPT must be strongly first-order in order to suppress sphaleron wash-out within the broken phase region; see~\cite{Morrissey:2012db} for a review. The required condition for a strong first-order EWPT is~\cite{Kuzmin:1985mm} \begin{equation} \xi_c \equiv \frac{v_c}{T_c} > 1. \end{equation} \section{Results of the IDM global fits} \label{sec:fits} The sampling and fit of the IDM parameter space, $\theta$, with the SM Higgs boson mass fixed, within an inflated range for accommodating theoretical uncertainties, at $m_h = 125 \pm 3 \, \textrm{GeV}$, is done using {\sc MultiNest}~\cite{Feroz:2007kg,Feroz:2008xx}. Only model points that pass the set of theoretical and experimental constraints, $d$, described in section~\ref{subsec:constraints} and for which the lightest odd particle is the CP-even IDM Higgs boson, $S$, are passed for implementing the nested sampling algorithm. For these IDM parameter points, we model the likelihood $p( d| \theta)$ of the IDM predictions, $O_i$, corresponding to the i$^{th}$ constrain, with experimental central values $\mu_i$ and uncertainties $\sigma_i$, as \begin{equation} \label{likel} p( d| \theta) = \prod_i \, \frac{ \exp\left[- (O_i - \mu_i)^2/2 \sigma_i^2\right]}{\sqrt{2\pi \sigma_i^2}}. \end{equation} For scenarios where $S$ is pseudo-degenerate with the SM Higgs, additional contributions based on the Higgs signal strength measurements at colliders as implemented in {\sc Lilith} were added to~Eq.(\ref{likel}). The global fit indicates that the lightest inert Higgs should be expected around $m_S = 97.83 \pm 11.49 \, \textrm{GeV}$. At Maximum a Posteriori (MAP) and maximal likelihood, $m_S \sim 100 \, \textrm{GeV}$. This result supports the possibility for the IDM inert Higgs boson account for the observed mild but independent excesses at LEP and CMS experiments~\cite{hep-ex/0306033, CMS-PAS-HIG-14-037, CMS-PAS-HIG-17-013, 1811.08459, CMS-PAS-HIG-21-001} in search for light Higgs bosons. In Figure~\ref{1dposteriors}, the 1-dimensional posterior distributions of the IDM parameters are shown. As is the case for the SM, strong first-order EWPT condition, $\frac{v_c}{T_c} > 1$, translates into an upper bound on the lightest inert Higgs mass. This partly explains way a significant part of the prior region, with $m_S \sim 1 \, \textrm{to}\, 5 \, \textrm{TeV}$ will be disfavoured. Should this requirement be uplifted, multi-TeV $m_S$ are possible as can be seen on the second row of Figure~\ref{1dposteriors} plots. Imposing the upper limit~\cite{PandaX-II:2017hlx, XENON:2019gfn} on the inert singlet versus nucleons elastic scattering cross section, $\sigma^{SI}$, on the IDM fit to data leads to small mass differences $\Delta m_{Y^{\pm}} \sim \Delta m_R \sim {\cal O}(10) \, \textrm{GeV}$. Contrary to this, the strength of the first-order EWPT, $\frac{v_c}{T_c}$, is proportional to the mass splittings $\Delta m_{R, \, Y^{\pm}}$. The tendencies with respect to the mass differences can be seen in Figure~\ref{posterior_2d_2}. Therefore, simultaneously requiring both $\frac{v_c}{T_c} > 1$ and the DM direct detection limits on the IDM fit to data is extremely difficult beyond the scope and the computational resources at our disposal. As such, the direct detection limits were not imposed for final fit of the IDM to data since this extremely slowed the sampling of the model parameter space. Instead, the impact of this limit were assessed via post-processing the posterior samples~\footnote{Including the DM direct detection limit for the IDM fits is an interesting direction we hope to pursue in the future using machine-learning techniques.}. Complementary to post-processing, a dedicated fit of the model with DM direct limit imposed but without the strong first-order EWPT requirement could be used for assessing the tension between both observables. The second row of plots Figure~\ref{1dposteriors} were from such an IDM fit with the direct detection 90\% C.L. exclusions limits~\cite{XENON:2018voc, DarkSide:2018bpj,PICO:2019vsc,CRESST:2019jnq} imposed. As can be seen, this yields posteriors with $\frac{v_c}{T_c} <0.1$. The constraints from collider limits disfavours small values of $m_S$ and $m_{Y^{\pm}}$, and also contribute to the control that leads to the allowed region for $\Delta m_{R, \, Y^{\pm}}$. This particularly important for the oblique parameters constraints which favour relatively lower inert Higgs mass differences. In all, % there are IDM points that satisfy collider and dark matter searches limits including relic density generation, and simultaneously allow for a strong first-order EWPT. A selection of benchmarks are presented in Table~\ref{benchmarks}. \begin{table}[h] \centering \begin{tabular}{|l|l|l|l|l|l|l|l|l|} \hline $m_{S}$ & $m_{R}$ & $m_{Y^{\pm}}$ & $\Lambda_1$ & $\log \, (\Omega_{DM} \, h^2)$ & $\log (\sigma^{SI} [pb])$ & $\frac{v_c}{T_c}$ & $\Delta m_{R}$ & $\Delta m_{Y^{\pm}}$ \\ \hline 93.6989 & 176.7693 & 213.3181 & 0.2853 & -2.10594 & -8.6287 & 1.0761 & 83.0704 & 119.619 \\ 115.0309 & 133.2492 & 209.0929 & 0.3971 & -3.64282 & -9.28878 & 1.2038 & 18.2183 & 94.0621 \\ 101.5063 & 157.1630 & 216.7150 & 0.3678 & -3.36139 & -9.72497 & 1.2374 & 55.6567 & 115.209 \\ 122.3877 & 198.1699 & 260.3158 & 0.4478 & -3.26165 & -10.8541 & 1.5127 & 75.7821 & 137.928 \\ 80.5285 & 151.8615 & 193.4262 & 0.28262 & -2.19879 & -12.0729 & 1.0923 & 71.333 & 112.898 \\ \hline \end{tabular} \caption{A selection of benchmark points for the IDM model which passed all the constraints considered. Masses are in GeV units.} \label{benchmarks} \end{table} \section{Conclusion} We have made a detailed exploration of the parameters and the interplay amongst them for the inert Higgs doublet model in light of the limits from collider experiments, the constraints from dark matter searches and the requirement for a strong first-order electroweak phase transition (EWPT). The software packages BSMPT and micrOMEGAs were used for computing the strength of EWPT and dark matter properties respectively. The collider constraints on the IDM parameter space were applied by using HiggsBounds and Lilith packages. Our analyses include the first global fit of the IDM to data using a statistically convergent Bayesian approach implemented in MultiNest package. For these, we have used, also for the first time, a recently presented most general renormalisable potential for the IDM. This will lead to different phase transition dynamics and coupling constants. The global fits show that the IDM spectra can have DM candidates with all constraints satisfied and at the same time able to produce a strongly first-order EWPT. A selection of benchmark points was provided which can be analysed further with respect to ongoing or future collider experiments. The posterior sample~\footnote{The posterior sample can be downloaded at \url{https://doi.org/10.7910/DVN/TCMXDS}.} could be useful for addressing IDM collider and dark matter phenomenology. Given computing resources, the chain of particle physics phenomenology tools developed can be used for comparisons amongst the extended Higgs sector BSMs in the light EWPT and dark matter candidate particle constraints. There are several interesting research directions that could be built on the result presented here. Besides the S and T parameters constrained already applied for the fits, there already are interesting collider results from the LHC which could possibly probe further the IDM parameters space. For instance, at the LHC, the electroweak production of $Y^\pm$ and $S$ (or $R$) can subsequently lead to the inert Higgs decays into a weak boson and the lightest inert scalar. These can provide the same final states and therefore can be constrained by limits from supersymmetric electroweakino searches~\cite{ATLAS:2021moa, ATLAS:2021yqv}. Implementing such limits on the IDM will, however, require dedicated reinterpretation studies. The fits revealed strong correlations between three important constraints ($\sigma^{SI} [pb]$, $\frac{v_c}{T_c}$, and the electroweak precision oblique parameters S and T) used with a strong dependence on the IDM Higgs mass differences albeit with possible pulls along opposing directions. A better delineation of the IDM with respect to these constraints can be achieved by using machine-learning techniques in exploring the compatible regions in parameter space. \paragraph*{Acknowledgements:} We would like to thank Philip Basler for his kind help with the BSMPT package, Alexander Belyaev for advice on the LanHep package and all developers of micrOMEGAs package for their help. MM would like to thank Najimuddin Khan for discussions about perturbative unitarity, Per Osland for the discussions about oblique parameters. This work was partly performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (www.csd3.cam.ac.uk), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/T022159/1), and DiRAC funding from the Science and Technology Facilities Council (www.dirac.ac.uk). \begin{appendix} \section{The IDM finite temperature effective potential} \label{veffective} The 1-loop finite temperature effective potential for the IDM can br written as follows, following the BSMPT~\cite{1803.02846, 2007.01725} notations, in terms static field configuration $\omega$ and temperature T. \begin{equation} V (\omega, T ) = V(\omega) + V_T(\omega, T ) = V_{tree}(\omega) + V_{CW}(\omega) + V_{CT}(\omega) + V_T(\omega, T ) \end{equation} where $V(\omega)$ consists of the tree-level potential $V_{tree}$, Eq.\pref{eqn:Doublet}, the Coleman-Weinberg potential $V_{CW}$ and the counter-term potential $V_{CT}$. The thermal corrections to the potential is given by $V_T(\omega, T)$. In this section we briefly describe each of these terms and then how they are implemented into the BSMPT package in Appendix~\ref{makeBSMPT}. The notation of~\cite{Camargo-Molina:2016moz} is used for casting the effective potential into the form \begin{align} -\mathcal{L}_S &= L^i \Phi_i + \frac{1}{2!} L^{ij} \Phi_i \Phi_j+\frac{1}{3!} L^{ijk} \Phi_i \Phi_j\Phi_k + \frac{1}{4!} L^{ijkl} \Phi_i \Phi_j \Phi_k \Phi_l \label{Eq:LS}\\ -\mathcal{L}_F &= \frac{1}{2} Y^{IJk} \Psi_I \Psi_J \Phi_k + c.c. \label{Eq:LF}\\ \mathcal{L}_{G} &= \frac{1}{4} G^{abij} A_{a\mu}A_b^\mu\Phi_i\Phi_j \;, \label{Eq:LG} \end{align} with summation over repeated indices implied if one is up and the other is down. In this manner, the IDM scalar multiplets are decomposed into $n_{\text{Higgs}}$ real scalar fields $\Phi_i$, with $i= 1,\dots , n_{\text{Higgs}} = 8$. Here $\Psi_I$, with $I=1,\dots , n_{\text{fermion}}$ represents the Weyl fermion multiplets of the model. The four-vectors $A_\mu^a$, where the index $a$ runs over $n_{\text{gauge}}$ gauge bosons in the adjoint representation of the corresponding gauge group, denotes the gauge bosons of the model. $-\mathcal{L}_S$ denotes the extended Higgs potential (including the SM Higgs parts). This consists of the terms $L^i, L^{ij}, L^{ijk}, L^{ijkl}$ and the real scalar fields $\Phi_i$, with $i, j, k, l = 1, \dots , n_{\text{Higgs}}$. $Y^{IJk}$, with $I , J = 1 \dots n_{\text{fermion}}$, are the couplings for the interactions between the scalar and the fermionic fields. $G^{abij}$, with $a,b = 1\dots n_{\text{gauge}}$, are the couplings for the interactions between the scalar and the bosonic fields. After symmetry breaking the scalar fields are expanded around there VEVs, $\omega_i$ as \begin{equation} \Phi_i(x) = \omega_i + \phi_i(x). \label{eq:phiexpand} \end{equation} Putting Eq.~\pref{eq:phiexpand} in Eqs.~\pref{Eq:LS}-\pref{Eq:LG} gives \begin{align} -\mathcal{L}_S &= \Lambda + \Lambda^i_{(S)} \phi_i + \frac{1}{2} -\Lambda_{(S)}^{ij} \phi_i\phi_j + \frac{1}{3!} \Lambda^{ijk}_{(S)} -\phi_i\phi_j\phi_k + \frac{1}{4!} -\Lambda_{(S)}^{ijkl}\phi_i\phi_j\phi_k\phi_l \\ -\mathcal{L}_F &= \frac{1}{2} M^{IJ} \Psi_I\Psi_J + \frac{1}{2} -Y^{IJk}\Psi_I\Psi_J\phi_k + c.c. \\ \mathcal{L}_G &= \frac{1}{2} \Lambda^{ab}_{(G)} A_{a\mu}A_b^{\mu} -+\frac{1}{2} \Lambda^{abi}_{(G)} A_{a\mu}A_b^{\mu}\phi_i + -\frac{1}{4} \Lambda^{abij}_{(G)} A_{a\mu}A_{b}^\mu\phi_i\phi_j \;, \end{align} where \begin{align} \Lambda &= V^{(0)}(\omega_i) = L^i \omega_i + \frac{1}{2!} L^{ij}\omega_i \omega_j + \frac{1}{3!} L^{ijk} \omega_i \omega_j \omega_k + \frac{1}{4!} L^{ijkl} \omega_i \omega_j \omega_k \omega_l \label{Vtreelevel} \\ \Lambda_{(S)}^i &= L^i + L^{ij} \omega_j + \frac{1}{2} L^{ijk}\omega_j \omega_k + \frac{1}{6} L^{ijkl}\omega_j\omega_k\omega_l \, , \quad \Lambda_{(S)}^{ij} = L^{ij} + L^{ijk}\omega_k + \frac{1}{2} L^{ijkl}\omega_k\omega_l \label{eq:scalarten}, \\ \Lambda_{(S)}^{ijk} &= L^{ijk}+L^{ijkl}\omega_l \, , \quad \Lambda_{(S)}^{ijkl} = L^{ijkl} \, , \quad \Lambda_{(G)}^{ab} = \frac{1}{2} G^{abij}\omega_i\omega_j \label{Gabterm} \, , \quad \Lambda_{(G)}^{abi} = G^{abij}\omega_j\\ \Lambda_{(G)}^{abij} &= G^{abij} \label{Gabijterm} \\ \Lambda_{(F)}^{IJ} &= M^{\ast IL} M_{L}^{\; J} = Y^{\ast ILk}Y_L^{\; Jm} \omega_k\omega_m \;, \quad \mbox{with} M^{IJ} = Y^{IJk}\omega_k. \label{Fterm} \\ \end{align} We compute each of the terms $\omega_i, \, L^i, \, L^{ij}, \, L^{ijk}, \, L^{ijkl}, \, G^{abij}$ and $Y^{IJk}$ in {\tt C++} format and then develop the IDM model files needed for BSMPT to work. \paragraph{The Coleman-Weinberg part of the effective potential} Radiative quantum corrections affects the vacuum structure of potentials at the loop levels. This is accounted for using the 1-loop correction known as Coleman-Weinberg potential~\cite{Coleman:1973jx} given by \begin{equation} V_{CW}(\omega) = \frac{1}{4 \, \left(4\pi\right)^2} \sum_{X={S,G,F}}(-1)^{2s_X} (1+2s_X) Tr[(\Lambda^{xy}_{(X)})^2 ( \log(\frac{1}{\mu^2} \Lambda^{xy}_{(X)} ) - k_X)] \label{CWpotential}, \end{equation} where $s_X$ represents the spin of the field $X$. $X=S,G$ and $F$, respectively, represent scalar, gauge and fermionic fields. The indices $xy$ correspond to the scalar indices $ij$, the gauge indices $ab$ and the fermion indices $IJ$ for $X=S,G$ and $F$, respectively. The sum over $X$ is for all degrees of freedom including colour for the quarks. $\Lambda_{(S)}^{ij}$, $\Lambda_{(G)}^{ab}$ and $\Lambda_{(F)}^{IJ}$ are as given in Eqs.~\pref{eq:scalarten},~\pref{Gabterm} and ~\pref{Fterm}. The $\overline{\mbox{MS}}$ renormalisation scheme constants are \beq k_X = \left\{ \begin{array}{ll} \frac{5}{6} \;, & \quad \mbox{for gauge bosons} \\[0.1cm] \frac{3}{2} \;, & \quad \mbox{otherwise} \end{array} \right. \eeq The renormalisation scale $\mu$ is set to the SM Higgs multiplet VEV at $T=0$, $\mu = v(T=0) \approx 246.22 \textrm{ GeV}$. \paragraph{The counter term part of the effective potential} The BSMPT package was designed to use loop-corrected masses and mixing angles as input. As such, the $\overline{\mbox{MS}}$ renormalisation scheme used for the Coleman-Weinberg part of the effective potential has to the modified into the on-shell renormalisation scheme. It is for this reason that the counter term part of the effective potential, $V_{\text{CT}}$, is added. $V_{\text{CT}}$ is obtained by replacing bare parameters $p^{(0)}$ of the tree-level potential $V^{(0)}$ by the renormalised ones, $p$, and their corresponding counter terms $\delta p$ \begin{align} V^{\text{CT}} &= \sum_{i=1}^{n_p} \frac{\partial V^{(0)}}{\partial p_i} \delta p_i + \sum_{k=1}^{n_v} \delta T_k \left(\phi_k + \omega_k \right) \label{vcounterterm}. \end{align} Here $n_p$ is the number of parameters of the potential. $\delta T_k$ represent the counter terms of the tadpoles $T_k$corresponding to the $n_v$ directions in field space with non-zero VEV. \paragraph{The thermal corrections} The temperature dependent part of the effective potential $V^{(T)}$ is given by \cite{Dolan:1973qd,Quiros:1999jp} \begin{align} V^T (\omega,T) &= \sum{X={S,G,F}}{} (-1)^{2 s_X} (1 + 2 s_X)\frac{T^4}{2\pi^2} J_{\pm}\left(\Lambda^{xy}_{(X)}/T^2 \right)\,, \end{align} where $J_{\pm}$ is for bosons or fermions respectively, \begin{align} J_{\pm}\left(\Lambda_{(X)}^{xy}/T^2\right) &= \textrm{Tr}\left[ \int_{0}^{\infty} \,dk\, k^2 \log\left[ 1 \pm \exp\left( -\sqrt{k^2 + \Lambda^{xy}_{(X)}/T^2}\right) \right] \right] \,. \end{align} Taking the finite temperature effect, the daisy corrections~\cite{Carrington:1991hz} to the scalar and gauge boson masses are also implemented in the BSMPT package for the IDM model. \section{Implementation of the IDM model to BSMPT} \label{makeBSMPT} New models can be implemented in BSMPT. For the IDM, the Lagrangian density terms are written in the required format as described in Appendix~\ref{veffective}. Using $\Phi_i = \lbrace h , x_1 , x_2 , x_3 , y_1 , y_2 , S , R \rbrace$, the code needs $\left\lbrace L^i,L^{ij},L^{ijk},L^{ijkl},Y^{IJk},G^{abij}\right\rbrace$ as in Eqs.~\pref{Eq:LS}, \pref{Eq:LF}, and \pref{Eq:LG} specified in {\tt C++} form. For instance $L^{ij} = 0$ unless for $i=j$ for which we have: \begin{align} L^{i,j=x_{3}}=L^{i,j=x_{2}}=L^{i,j=x_{1}}=L^{i,j=h}= -\mu^2_{h} \\ L^{i,j=R}=L^{i,j=S}= L^{i,j=y_{2}}=L^{i,j=y_{1}}= \mu^2_{Q}. \end{align} The same has to be done for the counter terms \begin{eqnarray} V^{CT} = \delta\mu^2_{h}H^{\dagger}H + \delta\mu^2_{Q}Q^{\dagger}Q + \delta\lambda_1 \left(H^{\dagger}H\right)^2 + \delta\lambda_2 [(QQ)_1(\overline{Q}\,\overline{Q})_1]_{0} + \delta\alpha(H^{\dagger}H)(Q^{\dagger}Q) + \nonumber\\ \delta\beta[(\overline{H}H)_{1}(\overline{Q}Q)_{1}]_{0} + \Big\{\delta\kappa_1[(HH)_{1}(\overline{Q}\,\overline{Q})_{1}]_{0} + H.c.\Big\} + \delta T \left(h + \textrm{VEV}\right). \end{eqnarray} In order to add this part of the IDM effective Lagrangian to the BSMPT package, the coefficients $\left\lbrace L^i,L^{ij},L^{ijk},L^{ijkl},Y^{IJk},G^{abij}\right\rbrace $ have to be computed symbolically and then written in {\tt C++} form. Applying on-shell renormalisation leads to the equations \begin{eqnarray} \left.\partial_{\phi_i} V^{\text{CT}}\right|_{\phi =\langle \phi^c\rangle_{T=0}} &=& -\left.\partial_{\phi_i} V^{\text{CW}}\right|_{\phi =\langle \phi^c \rangle_{T=0}} \\ \left.\partial_{\phi_i}\partial_{\phi_j} V^{\text{CT}} \right|_{\phi=\langle \phi^c \rangle_{T=0}} &=& -\left.\partial_{\phi_i} \partial_{\phi_j} V^{\text{CW}} \right|_{\phi=\langle\phi^c \rangle_{T=0}}. \ \end{eqnarray} Solving these equations with respect to the counter terms gives \begin{eqnarray} \delta\mu^2_{h} &=& -\frac{1}{2} H^{\text{CW}}_{h,h} + \frac{3}{2} H^{\text{CW}}_{x_3,x_3} \\ \delta\mu^2_{Q} &=& -\frac{1}{2} H^{\text{CW}}_{y_2,y_2} - \frac{1}{4} H^{\text{CW}}_{S,S} - \frac{1}{4} H^{\text{CW}}_{R,R}\\ \delta\lambda_{1} &=& \frac{1}{2v^2}\left( -H^{\text{CW}}_{h,h} + H^{\text{CW}}_{x_3,x_3}\right) \\ \delta\lambda_{2} &=& 0\\ \delta K &=& \frac{-\sqrt{3}}{2v^2}\left( H^{\text{CW}}_{S,S} + H^{\text{CW}}_{R,R}\right) \\ \delta\alpha &=& 0\\ \delta\beta &=& \frac{\sqrt{3}}{v^2}\left( H^{\text{CW}}_{S,S} + H^{\text{CW}}_{R,R} - 2H^{\text{CW}}_{y_2,y_2}\right)\\ \delta T &=& vH^{\text{CW}}_{x_3,x_3} - N^{\text{CW}}_{h} \end{eqnarray} Finally for models with a different Yukawa and gauge sectors relative to the SM ones, the thermal corrections codes of the BSMPT has to be modified. For the IDM, the gauge sector differs. To account for this, we modified the function {\tt CalculateDebyeGaugeSimplified()} using \begin{equation} \Pi_{(G)}^{ab} = T^2 \frac{2}{3} \left(\frac{\tilde{n}_H}{8} + 5 \right) \frac{1}{\tilde{n}_H} \sum\limits{m=1}{n_{\text{Higgs}}} \Lambda^{aamm}_{(G)} \delta_{ab} \end{equation} where $\Pi_{(G)}^{ab}$ belongs to daisy correction to thermal masses of gauge bosons, $\tilde{n}_H$ represents the number of Higgs bosons coupled to the SM gauge sector. For the IDM, this leads to \begin{align} \Pi^{a,b=W_{0}}_{G}= \Pi^{a,b=W_{1}}_{G}= \Pi^{a,b=W_{2}}_{G}=2g^2 \\ \Pi^{a,b=B_{0}}_{G}= 2{g^{\prime}}^{2} \end{align} where, $g$ and $g^\prime$ are SM $SU(2)_L$ and $U(1)$ gauge couplings. \end{appendix}

Title: The physical and chemical structure of Sagittarius B2 -- VI. UCHII regions in Sgr B2

Abstract: The giant molecular cloud Sagittarius B2 (hereafter SgrB2) is the most massive region with ongoing high-mass star formation in the Galaxy. Two ultra-compact HII (UCHII) regions were identified in SgrB2's central hot cores, SgrB2(M) and SgrB2(N). Our aim is to characterize the properties of the HII regions in the entire SgrB2 cloud. Comparing the HII regions and the dust cores, we aim to depict the evolutionary stages of different parts of SgrB2. We use the Very Large Array in its A, CnB, and D configurations, and in the frequency band C (~6 GHz) to observe the whole SgrB2 complex. Using ancillary VLA data at 22.4 GHz and ALMA data at 96 GHz, we calculated the physical parameters of the UCHII regions and their dense gas environment. We identify 54 UCHII regions in the 6 GHz image, 39 of which are also detected at 22.4 GHz. Eight of the 54 UCHII regions are newly discovered. The UCHII regions have radii between $0.006 {\rm pc}$ and $0.04 {\rm pc}$, and have emission measure between $10^{6} {\rm pc\,cm^{-6}}$ and $10^{9} {\rm pc\,cm^{-6}}$. The UCHII regions are ionized by stars of types from B0.5 to O6. We found a typical gas density of $\sim10^6-10^9 {\rm cm^{-3}}$ around the UCHII regions. The pressure of the UCHII regions and the dense gas surrounding them are comparable. The expansion timescale of these UCHII regions is determined to be $\sim10^4-10^5 {\rm yr}$. The percentage of the dust cores that are associated with HII regions are 33%, 73%, 4%, and 1% for SgrB2(N), SgrB2(M), SgrB2(S), and SgrB2(DS), respectively. Two-thirds of the dust cores in SgrB2(DS) are associated with outflows. The electron densities of the UCHII regions we identified are in agreement with that of typical UCHII regions, while the radii are smaller than those of the typical UCHII regions. The dust cores in SgrB2(N) are more evolved than in SgrB2(DS) but younger than in SgrB2(M).

https://export.arxiv.org/pdf/2208.07796

\title{The physical and chemical structure of Sagittarius\,B2} \titlerunning{UCH{\sc ii} regions in Sgr\,B2} \subtitle{VI. UCH{\sc ii} regions in Sgr\,B2} \author{F.~Meng\inst{1,2}, {\'A}.~S{\'a}nchez-Monge\inst{2}, P.~Schilke\inst{2}, A.~Ginsburg\inst{3}, C.~DePree\inst{4,5}, N.~Budaiev\inst{3}, D.~Jeff\inst{3}, A.~Schmiedeke\inst{6}, A.~Schw{\"o}rer\inst{2}, V.~S.~Veena\inst{7}, \and Th.~M{\"o}ller\inst{2} } \authorrunning{F. Meng et al.} \institute{University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China\\ \email{mengfanyi@ucas.ac.cn, meng@ph1.uni-koeln.de} \and I.\ Physikalisches Institut, Universit\"at zu K\"oln, Z\"ulpicher Str.\ 77, D-50937 K\"oln, Germany \and Department of Astronomy, University of Florida, PO Box 112055, USA \and NRAO, 520 Edgemont Rd, Charlottesville, VA, USA \and Agnes Scott College, 141 E. College Ave., Decatur, GA 30030, USA \and Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse 1, D-85748 Garching, Germany \and Max Planck Institute for Radio Astronomy, Auf dem H\"{u}gel 69, D-53121 Bonn, Germany } \date{Received ; accepted } \abstract{% \LEt{ General notes. A) You show a preference for US English language conventions, and I have edited accordingly throughout. B1) A\&A uses the past tense to describe the specific methods used in a paper and the present tense to describe general methods and recent findings (within the past ten or so years). Please make sure my edits are accurate in this respect throughout the paper. See Sect. 6 of the Language Guide https://www.aanda.org/for-authors/language-editing/6-verb-tenses B2) This means that the specific steps you took for this particular research should be in the past simple: We used, We extrapolated, The observation was performed, After this change we found, etc. Descriptions of methodology, universal truths, constants, or conclusions should be in the present simple: The first step in the process is, Water boils at 100^{\circ}$C, We find, etc. }\\ The giant molecular cloud Sagittarius B2 (hereafter Sgr\,B2) is the most massive region with ongoing high-mass star formation in the Galaxy. Two ultra-compact \hii (UC\hii) regions were identified in Sgr\,B2's central hot cores, Sgr\,B2(M) and Sgr\,B2(N). } {% Our aim is to characterize the properties of the \hii regions in the entire Sgr\,B2 cloud. Comparing the \hii regions and the dust cores, we aim to depict the evolutionary stages of different parts of Sgr\,B2. } {% We use the Very Large Array in its A, CnB, and D configurations, and in the frequency band C ($\sim$6\,GHz) to observe the whole Sgr\,B2 complex. Using ancillary VLA data at 22.4\,GHz and ALMA data at 96\,GHz, we calculated the physical parameters of the UC\hii regions and their dense gas environment. } {% We identify 54 UC\hii regions in the 6\,GHz image, 39 of which are also detected at 22.4\,GHz. Eight of the 54 UC\hii regions are newly discovered. The UC\hii regions have radii between $0.006\,{\rm pc}$ and $0.04\,{\rm pc}$, and have emission measure between $10^{6}\,{\rm pc\,cm^{-6}}$ and $10^{9}\,{\rm pc\,cm^{-6}}$. The UC\hii regions are ionized by \meng{stars of types} from B0.5 to O6. We found a typical gas density of $\sim10^6-10^9\,{\rm cm^{-3}}$ around the UC\hii regions. The pressure of the UC\hii regions and the dense gas surrounding them are comparable. The expansion timescale of these UC\hii regions is determined to be $\sim10^4-10^5\,{\rm yr}$. The percentage of the dust cores that are associated with \hii regions are 33\%, 73\%, 4\%, and 1\% for Sgr\,B2(N), Sgr\,B2(M), Sgr\,B2(S), and Sgr\,B2(DS), respectively. Two-thirds of the dust cores in Sgr\,B2(DS) are associated with outflows. } {% The electron densities of the UC\hii regions we identified are in agreement with that of typical UC\hii regions, while the radii are smaller than those of the typical UC\hii regions. The dust cores in Sgr\,B2(M) are more evolved than in Sgr\,B2(N). The dust cores in Sgr\,B2(DS) are younger than in Sgr\,B2(M) or Sgr\,B2(N). } \keywords{Stars: formation -- Stars: massive -- Radio continuum: ISM -- Radio lines: ISM -- ISM: clouds -- ISM: individual objects: Sgr\,B2 } \section{Introduction} % \label{sec:introduction} The giant molecular cloud Sagittarius B2 (Sgr\,B2) is the most massive ($\sim 10^7\,M_{\odot}$) region with ongoing high-mass star formation in the Galaxy \citep[see e.g.,][]{Goldsmith:1990aa}. Sgr\,B2 has a higher density ($>10^5\rm\,cm^{-3}$) and dust temperature ($\gtrsim$50--70\,K) compared to other star forming regions in the Galactic plane \citep[see e.g.,][]{Ginsburg:2016aa,Schmiedeke:2016uc,Sanchez-Monge:2017vt}. Sgr\,B2 is located at a distance of $8.34\pm0.16$\,pc, and only $\sim$100\,pc in projection from the Galactic center \citep{Reid:2014aa}\footnote{ A new distance to the Galactic center of $8.127 \pm 0.031$~kpc has been measured \citep{Gravity-Collaboration:2018aa}. For consistency with the papers published within the same series of studies of Sgr\,B2, we use the distance reported by \citet{Reid:2014aa}. }. These features make Sgr\,B2 an excellent case to study high-mass star formation in an extreme high-pressure environment. Such an environment resembles nearby starburst galaxies \citep{Leroy:2018to}. Understanding the structure of the Sgr\,B2 molecular cloud complex is necessary to comprehend the most massive star forming region in our Galaxy, which at the same time provides a unique opportunity to study in detail the nearest counterpart of the extreme environments that dominate star formation in the Universe \citep[see, e.g.,][]{Kruijssen:2013vk,Henshaw:2022vl}. This paper continues our series of studies on Sgr\,B2 \citep{Schmiedeke:2016uc,Sanchez-Monge:2017vt,Pols:2018aa,Schworer:2019aa,Meng:2019aa}. In \citet{Meng:2019aa} we presented the observations of Sgr\,B2(DS), which is a part of Sgr\,B2 giant cloud, and analyzed the physical properties of the \meng{non-thermal} emission within it. In this work we study the whole Sgr\,B2 region. In the central $2\,{\rm pc}$ of Sgr\,B2 there are the two well-known and well-studied hot cores Sgr\,B2(N) and Sgr\,B2(M) \citep[see, e.g.,][]{Schmiedeke:2016uc,Sanchez-Monge:2017vt}, which contain at least 70 high-mass stars with spectral types from O5 to B0 \citep[see, e.g.,][]{Gaume:1995aa,De-Pree:1998aa,De-Pree:2014aa}. Surrounding the two hot cores, there is a larger envelope (hereafter \emph{the envelope}) with a radius of 20~pc that contains more than 99\% of the total mass of Sgr\,B2 \citep{Schmiedeke:2016uc}. Along with the active high-mass star forming activity discovered in Sgr\,B2(N) and Sgr\,B2(M), hints of star formation happening in the envelope are also revealed. \citet{Ginsburg:2018wo}, with ALMA at 3~mm, revealed 271 high-mass protostellar cores distributed throughout the entire Sgr\,B2 region, including the envelope. The luminosities of these dust cores suggest that they must contain objects with stellar masses higher than 8\,$M_\odot$. Due to the high extinction in the infrared bands toward Sgr\,B2 \citep[see][]{Meng:2019aa}, there is no direct evidence of the existence of high-mass stars embedded in the dust cores detected by \citet{Ginsburg:2018wo}.\LEt{ ok like so? if you say "missing" it means that you had the information in the past, but now you cannot find it. } However, since high-mass stars ionize the neutral material surrounding them, the presence and properties of the associated \hii regions reflect the evolutionary stages of these dust cores \citep[see, e.g.,][]{Gonzalez-Aviles:2005aa,Breen:2010aa}. Additionally, since the free-free emission from \hii regions may extend from centimeter to millimeter wavelengths in the spectral domain \citep[see, e.g.,][]{Sanchez-Monge:2013ab}, measuring the luminosities of the associated \hii regions can help us better constrain the luminosities of the dust cores. Therefore, to further characterize the evolutionary stages and physical properties of these dust cores, we investigate the possible \hii regions associated with them. The \hii regions in Sgr\,B2 were targeted by several previous studies. \citet{Mehringer:1993aa} observed the entire Sgr\,B2 with VLA in the 20, 6, and 3.6\,cm bands and identified 15 \hii regions. The resolutions range from $\sim$20\arcsec\ to $\sim$3\arcsec\ when the wavelength changes from 20 to 3.6\,cm, which correspond to the range 0.8--0.12\,pc. The 15 \hii regions, except two unresolved cases, all have sizes $>2$\arcsec. Since the 271 dust cores may contain newly formed high-mass stars \citep{Ginsburg:2018wo}, the associated \hii regions may be ultra-compact \hii (UCH{\sc ii}) and hyper-compact \hii (HCH{\sc ii}) regions, which typically have sizes from $\sim 0.03$\,pc to $\sim 0.1$\, pc \citep[see, e.g.,][]{Kurtz:2002aa,Gonzalez-Aviles:2005aa,Kurtz:2005aa,Breen:2010aa}. Thus, the resolution of \citet{Mehringer:1993aa} is not sufficient to resolve the UCH{\sc ii} and HCH{\sc ii} regions. \citet{Gaume:1990aa,Gaume:1995aa} \LEt{ In the main text (i.e., not in parentheses) use "and" between two references, not ";". Three or more references must be separated by commas and the last reference set off with "and": ref1, ref2, and ref3. Please check for this throughout. }observed Sgr\,B2(N) and Sgr\,B2(M) at 7\,mm and 1.3\,cm and achieved resolutions of 0.065\arcsec\ and 0.25\arcsec, respectively. \citet{Rolffs:2011ty} observed Sgr\,B2(N) and Sgr\,B2(M) in 40\,GHz with a resolution of 0.1\arcsec. Unfortunately, these high-resolution observations do not cover the entire envelope. For example, Sgr\,B2(DS), where $\sim$80 of dust cores reside \citep[see, e.g.,][]{Ginsburg:2018wo,Meng:2019aa}, is left out of these observations. \citet{LaRosa:2000wr,Law:2008tl,Law:2008vw} also observed the entire Sgr\,B2 at centimeter wavelengths, but the resolutions were not high enough to study the UCH{\sc ii} and HCH{\sc ii} regions. In this paper we present Very Large Array (VLA) observations of the entire Sgr\,B2 cloud in the frequency regime 4--8~GHz, with configurations A, BnC, and D. The high resolution ($\lesssim 0.01$\, pc) and large spatial coverage ($\sim 20$\,pc) of our data sets make a systemic and complete study of the UC\hii and HC\hii regions in Sgr\,B2 possible. We also include analysis of the 3\,mm image \citep{Ginsburg:2018wo} as well as the newly acquired \ce{SiO}\,(5--4) data, both of which were observed with the Atacama Large Millimeter/submillimeter Array (ALMA). Thus, we can disentangle the contributions of ionized gas and dust at millimeter wavelengths and better constrain the evolutionary stages of the dust cores. This paper is organized as follows. In Sect.~2 we describe the observations and the data reduction process. In Sect.~3 we present the results. In Sect.~4 we discuss the results. Finally, we summarize this paper in Sect.~6. \section{Observations and data reduction} % \label{sec:observations_and_data_reduction} We \moda{used} the VLA in \moda{its} A, CnB, and D configurations to observe the entire Sgr\,B2 complex in frequency band C (4--8~GHz). In the following we call this band 6\,GHz. The observations with the CnB and D configurations were described in \citet{Meng:2019aa}. The observations with the A configuration were conducted from October 1 to 12, 2016 (project 16B-031, PI: F.\ Meng). We used 64 spectral windows with a bandwidth of 128~MHz each. Mosaic mode was used, with ten pointings for C band. The primary beam of each pointing is 7.5\arcmin. Quasar 3C286 was used as the flux and bandpass calibrator, the SED of which is $S_\nu = 5.059\pm0.021\ {\rm Jy} \times (S/8.435\ {\rm GHz})^{-0.46}$ from 0.5 to 50~GHz \citep{Perley:2013aa}. Quasar J1820-2528, whose flux is 1.3~Jy in the C Band, was used as phase calibrator. The calibration was done using the standard VLA pipelines provided by the NRAO\footnote{The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.}. Calibration and imaging were done in Common Astronomy Software Applications \citep[CASA 4.7.2][]{McMullin:2007aa}. The details of the data processing from the CnB and D configurations are described in \citet{Meng:2019aa}. The A configuration data were originally taken every 2\,s. To shorten the time of processing, we applied \texttt{timebin} in CASA to the measurement sets, which averaged the data taken within 10\,s into one data point. All the pointings of the mosaic in each band were primary beam corrected and the mosaic was imaged using the CASA task \texttt{tclean}. With a robust factor of 0, the image of C band has a synthesized beam of 0.62\arcsec$\times$0.28\arcsec, with a position angle ${\rm (PA)}$ of $7.27^\circ$. The PA is defined positive north to east.\LEt{ The PA is defined as positive from north to east. ? } To mitigate the spatial filtering effect of the A configuration image, we applied the \texttt{feather} algorithm to combine the images from A configuration and from CnB and D configurations. The combined image of the three configurations is shown in Fig.\,\ref{f:coresoverallvla}. The combined image has a resolution identical to that of the A configuration images, while being sensitive to spatial scales up to $\sim240$\arcsec. The sensitivities of the observations are described in Sect.~\ref{sec:results}. Ancillary data include the 22.4 GHz data, the 3\,mm continuum, and \ce{SiO}\,(5--4) data. The observations of the 22.4 GHz data are described in \citet{Gaume:1995aa}, with a resolution of 0.27\arcsec$\times$0.23\arcsec\ (${\rm PA} = 70^\circ$). The root mean square (RMS) noise is 0.38 mJy/beam. The spatial coverage of the 22.4\,GHz data is shown as the blue dashed box in Fig.\,\ref{f:coresoverallvla}. The 96\,GHz continuum data covers the frequency range from 89.5 to 103.3\,GHz. The image in 96\,GHz has a resolution of 0.54\arcsec$\times$0.46\arcsec\ (${\rm PA}=68.31^\circ$), the observational details of which are described in \citet{Ginsburg:2018wo}. Unlike the 22.4\,GHz image, the 96\,GHz image covers the entire area shown in Fig.\,\ref{f:coresoverallvla}. The 22.4\,GHz observations were performed 27 years prior to the 6\,GHz and 96\,GHz observations. If we assume that Sgr\,B2 has a proper motion of $\sim 50$\kms \citep{Henshaw:2016ug}, the corresponding position shift in 27 years is 0.04\arcsec, which is only $\sim$10\% of the beam sizes of our images. In addition, we did not find any visible shift in the positions of the compact sources positions between the 6 and 22.4\,GHz images. Thus, we do not take the astrometry difference into account when analyzing the 6 and 22.4\,GHz data in the following sections. The SiO~(5--4) emission was observed with ALMA (Project 2017.1.00114.S, P.I. A. Ginsburg) and has a resolution of 0.35\arcsec $\times$ 0.24\arcsec, with \meng{PA}\LEt{ is this the same as PA, which you introduced above? Please make sure the abbreviations are consistent throughout. } of $-80^{\circ}$, and spectral resolution of 1.35~km~s$^{-1}$. For the details of the observation and data reduction, see Jeff et al. (in prep.). The typical RMS of the SiO image is 0.9~mJy/beam (0.3\,K). The observation covers Sgr\,B2(S) and the eastern part of Sgr\,B2(DS). \section{Results} % \label{sec:results} In this section we present the image of Sgr\,B2 at 6\,GHz and the UCH{\sc ii} regions identified in it. For the \hii regions, we calculated their actual sizes and physical properties using the observations at 6\,GHz and at 22.4\,GHz. With the 96\,GHz data we characterized the properties of the dense gas that the UC\hii regions reside in. \subsection{Observed parameters of the 6 GHz sources} % \label{sub:observed_parameters_of_the_6_ghz_cores} Figure\,\ref{f:coresoverallvla} displays the image of Sgr\,B2 at 6\,GHz, where the known large-scale \hii regions are denoted N, M, S, AA, DS, and V, following the nomenclature of \citet{Mehringer:1992aa,Mehringer:1993aa,Ginsburg:2018wo,Meng:2019aa}. The maps at 22.4\,GHz and 96\,GHz are presented by \cite{Gaume:1995aa} and \cite{Ginsburg:2018wo}, respectively. Our aim is to study UC\hii regions and since the 22.4\,GHz image has non-complete spatial coverage, we performed compact source identification at the 6\,GHz image only. We used the automatic source extracting algorithm SExtractor \citep{Bertin:1996aa}, which allows us to identify the location of bright compact sources throughout the map. We note that this algorithm also includes a large fraction of artificial sources due to some large-scale artifacts visible in the final image (see the zoomed-in images in Fig.~\ref{f:coresoverallvla}). The variable noise (see Fig.\,\ref{f:rmsmaps}) also has an effect on the exclusion of certain sources with the automatic algorithms. After cross-checking the automatically produced lists of sources at 6\,GHz and 22.4\,GHz, we made the final catalog of continuum sources at 6\,GHz by excluding or adding sources by visual inspection (in a similar way to the approach followed in \citealt{Ginsburg:2018wo} for the ALMA 96\,GHz data).\LEt{ The slash "/" is used for ratios and some instrument pairings. Please check my changes throughout: you can substitute "and", "or", "and/or", or a double hyphen (which can be used to indicate dual nature: Hertzprung--Russell diagram).} Since we focus on compact sources, extended emission larger than 5\arcsec\ or 0.2\,pc (see contours in Fig.\,\ref{f:coresoverallvla}) are excluded in the catalog and the following analysis. In total, 54 compact sources are identified throughout the entire Sgr\,B2 cloud (see Table\,\ref{t:coreparams}), 8 of which are newly identified. Of these 54 cores, 8 are identified in Sgr\,B2(N), 40 in Sgr\,B2(M), 2 in Sgr\,B2(S), and 1 in Sgr\,B2(DS). All 54 compact sources are covered by the 22.4\,GHz image, except core 1. For each of the 54 compact sources we define a minimal circle that can include as much as the total flux density of it at 6\,GHz.\footnote{ \LEt{ Numerous discursive footnotes, especially those that have a direct bearing on the information in the paper, should be avoided. Please move the footnotes to the main text whenever possible. I think this should be in the main text; I have made the necessary changes. }This is done by expanding the circle as long as the flux $S_{\rm total}$ within the circle increases monotonically. When $S_{\rm total}$ remains constant or (due to the ``negative bowl'' in the artifacts) decreases, the corresponding circle is called the minimum circle. } \LEt{ perhaps like so: ... we find the smallest circle that includes the greatest amount of total flux density at 6 GHz }The observed radius of the core and the flux density within the core are denoted $r_{\rm obs6}$ and $S_{\rm 6}$, respectively. For these 54 sources we followed the same photometry procedure at the 22.4\,GHz and 96\,GHz bands, and obtained $r_{\rm obs22}$, $S_{\rm 22}$, $r_{\rm obs96}$, and $S_{\rm 96}$ that correspond to the observed radii and flux densities. The observed radius and flux density in three bands of all the 54 compact sources are listed in Table\,\ref{t:coreparams}. To match the sources across the three images (6, 22.4, and 96\,GHz), we defined that if two sources in two images have a distance (distance between their centers) shorter than either of their radii, they are associated with each other.\LEt{ ... we define two sources in two images as associated with each other if the distance between their centers is smaller than either of their radii. } Afterward, we manually adjust few of the matched sources in the crowed region in Sgr\,B2(M). For all the compact sources at 6\,GHz, emission in the 96\,GHz band is also detected. Among the 15 sources that have no $S_{\rm 22}$, 14 are without reliable 22.4\,GHz detection ($S_{22}<3{\rm RMS}$), and one (\#1 in Tab.\,\ref{t:coreparams}) is not spatially covered by the 22.4\,GHz image. All the 22.4\,GHz sources\footnote{ In \citet{Gaume:1995aa}, extended \hii regions are also included in the catalog. We only compare the compact sources ($r<5$\arcsec) in their catalog with ours.} reported by \citet{Gaume:1995aa} are associated with the 6\,GHz compact sources. We compared the spatial association of the 54 sources with the 271 dust cores identified at 96\,GHz by \citet{Ginsburg:2018wo}. Although there is no one-to-one correspondence between the 271 dust cores and the associated dust emission of the 54 compact sources, we can still conclude that at least 217 (80\%) dust cores are not associated with compact radio sources that are detectable with our sensitivity. Most of the compact sources have $r_{\rm obs6}$ from 0.5\arcsec\ to 1\arcsec, which is comparable to the beam size at 6\,GHz. Even though the 22.4\,GHz image has higher resolution ($\sim 0.25^{\prime \prime}$), the measured $r_{\rm obs22}$ are still not significantly larger than the beam size at 22.4\,GHz. Therefore, most of the compact sources are not well resolved; in other words, $r_{\rm obs6}$ or $r_{\rm obs22}$ cannot accurately represent the actual radii of most of the compact sources. The 96\,GHz compact sources, as indicated by \citet{Ginsburg:2018wo}, are also not resolved. The flux densities of the compact sources are distributed in a wide range; $S_{\rm 6}$ ranges from $\sim 3$\,mJy to $\sim 300$\,mJy at 6\,GHz, as shown in Fig.\,\ref{f:s_cband_distribution}. The mean and median of $S_{\rm 6}$ are 37.3\,mJy and 16.2\,mJy, respectively. To test whether the nature of the emission at 6 and 22.4\,GHz is free-free emission, we interpolated $S_{\rm 6}$ and $S_{\rm 22}$ to derive a spectral index $\alpha_{\rm 6-22}$ for each of the 39 sources that have 22.4\,GHz detection, see the last column of Table\,\ref{t:coreparams}. All 39 sources have $\alpha_{\rm 6-22}>-0.1$, which suggest that their emission at centimeter wavelengths is very probably dominated by thermal free-free emission from ionized \meng{gas} \LEt{ gas? } \citep[see, e.g.,][]{Sanchez-Monge:2013ab}. For the 15 sources without 22.4\,GHz detections, we find that their $S_{\rm 6}$ are not higher than the values of the other sources. If we treat their emission at 6\,GHz as free-free as well \footnote{For single-dish images, the RMS around a certain null detection spot could be used to derive an upper limit of the possible signal, which in this paper can be translated into an upper limit of $\alpha_{\rm 6-22}$; however, because the artifacts in the 22.4\,GHz image contain negative bowls of interferometric images, we do not trust the upper limits derived from the 22.4\,GHz RMS map. }, the derived physical parameters are also within the same ranges as the other sources (see Section\,\ref{sub:cores_in_vla}). Thus, we treat all 54 sources as thermal free-free sources in the following analysis. \subsection{Physical parameters of the H{\sc ii} regions} % \label{sub:cores_in_vla} As we show in Sect.\,\ref{sub:observed_parameters_of_the_6_ghz_cores}, most of the compact sources at 6 and 22.4\,GHz are not well resolved. Therefore, we first need to determine their actual sizes. The observed flux density ($S_{\rm 6}$ and $S_{\rm 22}$) of an \hii region is related to its actual size ($r_{\rm calc}$, which is called calculated radius in this paper), its electron temperature ($T_{\rm e}$), and its emission measure (EM) as \citep[see, e.g.,][]{wilson2013tools} \begin{equation} \label{eq:s-tau} \frac{S_\nu}{\rm mJy}= 8.183\times10^{-4} \left(\frac{r_{\rm calc}}{\rm arcsec}\right)^{2} \left(\frac{\nu}{\rm GHz}\right)^{2} \left(\frac{T_{\rm e}}{\rm K}\right) (1-e^{-\tau_{\nu}}), \end{equation} where the optical depth $\tau_\nu$ is \begin{equation} \label{eq:tau-em} \tau_\nu = 8.235\times10^{-2} \left(\frac{\nu}{\rm GHz}\right)^{-2.1} \left(\frac{T_{\rm e}}{\rm K}\right)^{-1.35} \left(\frac{\rm EM}{\rm pc\,cm^{-6}}\right). \end{equation} In our case the frequency $\nu$ is 6\,GHz or 22.4\,GHz. We assume that $T_{\rm e}$ is $10^4$\,K, following the values given by \cite{Mehringer:1993aa} for Sgr\,B2(N) and Sgr\,B2(M), considering that most of the compact sources are in these two regions. Thus, with the two independent flux density measurements, $S_{6}$ and $S_{22}$, we solve Eq.~\ref{eq:s-tau} and Eq.~\ref{eq:tau-em} to obtain $r_{\rm calc}$ and EM simultaneously for the 39 sources that are detected at both 6 and 22.4\,GHz. For the 15 compact sources that are only detected at 6\,GHz, we deconvolved the beam size from $r_{\rm obs6}$ to estimate $r_{\rm calc}$, as $r_{\rm calc}^2 + r_{\rm beam}^2 = r_{\rm obs6}^2$, and $r_{\rm beam}^2 = 0.62^{\prime\prime}\times0.28^{\prime\prime}$ for 6\,GHz. Then we calculated EM using the estimated $r_{\rm calc}$. The $r_{\rm calc}$ and EM of all the compact sources are listed in Table\,\ref{t:coreparams_derived}; those derived from the deconvolved $r_{\rm calc}$ are flagged with and asterisk (*) in Table\,\ref{t:coreparams_derived}. For all the compact sources that are associated with the 22.4\,GHz detection, we plot the $r_{\rm obs6}-r_{\rm calc}$ diagram (see Fig.\,\ref{f:r_calc_r_obs}). We can see that in general $r_{\rm calc}$ follows the trend of deconvolution (i.e., $r_{\rm calc}^2 = r_{\rm obs6}^2 - 0.62^{\prime\prime}\times0.28^{\prime\prime}$), which suggests that the method we used to calculate $r_{\rm calc}$, although without measuring the compact source size in the image, is in general consistent with the size we observed. The deviation between the $r_{\rm calc}$ from the dashed curve in Fig.\,\ref{f:r_calc_r_obs} is likely due to some uncertainties in the observations and flux measurements. For example, the determination of $r_{\rm obs6}$ is made by including as much of the flux as possible, which might be larger than the actual observed size. In addition, due to the artifacts in the image, we cannot ideally define the boundaries of the compact sources, but only approximate the compact source as a circle with $r = r_{\rm obs}$. The probability distribution of EM of all the 54 compact sources is shown in Fig.\,\ref{f:em_dist}. The EM of these compact sources ranges from $\sim 2\times10^6\,{\rm pc\,cm^{-6}}$ to $\sim 3\times10^9\,{\rm pc\,cm^{-6}}$. Only eight compact sources have ${\rm EM}<10^7\,{\rm pc\,cm^{-6}}$, which means that most of the compact sources are consistent with the parameters derived for UC\hii regions \citep[see][]{Kurtz:2002aa}. From EM we estimated the electron density $n_{\rm e}$ as $n_{\rm e} = [{\rm EM}/(2r_{\rm calc})]^{1/2}$. The distribution of $n_{\rm e}$ is shown in Fig.\,\ref{f:ne_dist}. The $n_{\rm e}$ of the 54 \hii regions are from $\sim4\times10^3\,{\rm cm^{-3}}$ to $\sim5\times10^5\,{\rm cm^{-3}}$ with a mean value of $\sim10^5\,{\rm cm^{-3}}$. Knowing EM, we calculated the flux of Lyman continuum photons ($\dot{N}_{\rm Ly}$) that are needed to ionize these UC\hii regions following Eq.\,C.18 in \citet{Schmiedeke:2016uc}. The derived $\dot{N}_{\rm Ly}$ ranges from $10^{46}\,{\rm s^{-1}}$ to $10^{49}\,{\rm s^{-1}}$ (see Fig.\,\ref{f:nly_dist}). If we assume that these cores are ionized by single stars, these UC\hii regions are ionized by stars from spectral types B0.5 to O6 \citep{Panagia:1973aa}. In this section we describe a method that is independent of the observed source size to derive the radius ($r_{\rm calc}$) and emission measure (EM) of 39 sources detected at 6 and 22.4\,GHz. To verify this a method, we compared the $r_{\rm calc}$, EM, and $\dot{N}_{\rm Ly}$ values of these 39 sources with the values presented by \citet{Gaume:1995aa}, namely $r_{\rm G95}$, $\rm EM_{G95}$, and $\dot{N}_{\rm Ly\,G95}$. Of the 39 sources, 33 are coincident with the UC\hii regions reported by \citet{Gaume:1995aa}. Twenty of the 33 sources have $r_{\rm calc}>r_{\rm G95}$ (see left panel of Fig.\,\ref{f:compare_remnly}). The order of magnitude of EM of the UC\hii regions is consistent in these two studies. Of the 33 sources, 23 have EM larger than $\rm EM_{G95}$ (see middle panel of Fig.\,\ref{f:compare_remnly}). Most of the sources have $\dot{N}_{\rm Ly}$ similar to $\dot{N}_{\rm Ly\,G95}$ (see right panel of Fig.\,\ref{f:compare_remnly}), except source 44. The difference between $\dot{N}_{\rm Ly}$ and $\dot{N}_{\rm Ly\,G95}$ of source 44 is possibly caused by the time-domain flickering of the radio emission in SgrB2 (see the last paragraph of this section). In Fig.\,\ref{f:d_ne} we plot the diagram of $r_{\rm calc}$ and $n_{\rm e}$. It is worth noting that all of the 39 \hii regions have typical $n_{\rm e}$ of UC\hii regions (i.e., $\sim 10^4\,{\rm cm^{-3}}$ to $\sim 10^6\,{\rm cm^{-3}}$; \citealt{Kurtz:2005aa}). However, 16 of these have $r$ smaller than that of typical UC\hii regions ($\sim0.015$ to $\sim 0.05$\,pc). Although some of the sources have sizes similar to HC\hii regions, considering the low density of our \hii regions compared to typical HC\hii regions, and following the previous nomenclature of most of the sources \citep[e.g.,][]{Gaume:1995aa}, we still call all the \hii regions in this study UC\hii regions. Recently, \citet{2020ApJ...899...94R} identified similar \hii regions in W\,51 \LEt{ in the W51 complex? }that have $n_{\rm e}\sim10^4-10^5\,{\rm cm^{-3}}$ and $2\times r\sim 10^{-3}-10^{-2}\,{\rm pc}$. They categorized these sources as HC\hii regions, but they are similar to smaller UC\hii regions and are ionized by early B-type stars, which is in agreement with the spectral type of the ionizing stars for most of our UC\hii regions (see Fig.\,\ref{f:nly_dist}). Unlike the HC\hii regions in W51, our UC\hii regions do not exactly follow the $r-n_{\rm e}$ relationships of UC\hii regions proposed by \citet{Garay:1999ta} and \citet{Kim:2001vy}, and have higher $n_{\rm e}$ (by a factor of $\sim 2$) than the predicted values (see Fig.\,\ref{f:d_ne}). Such a discrepancy might be due to the neutral gas surrounding our UC\hii regions that is denser ($n_{\rm H_2}\gtrsim 10^6\,{\rm cm^{-3}}$) than typical molecular cores ($n_{\rm H_2}\sim 10^4-10^5\,{\rm cm^{-3}}$; see, e.g., \citealt{Bergin:2007aa}). In Sect.\,\ref{sub:dustdensity} a detailed analysis of the gas density surrounding the UC\hii regions is presented. Since we cannot resolve the detailed morphologies of the UC\hii regions, we neglect the possible inhomogeneity of the \hii regions. The various morphologies of UC\hii regions result in modified SEDs other than that described by Eq.~\ref{eq:s-tau} and Eq.~\ref{eq:tau-em}, \citep[see, e.g.,][]{Keto:2003ud,Keto:2008wu}. Additionally, if there are accretion flows to the \hii regions, flickering of the flux on a timescale of $\sim 100$\,yr may occur, which is observed by \citet{De-Pree:2014aa} and modeled by \citet{Peters:2010ug,Peters:2010ta}. Since the 6\,GHz observation (2013) was performed 23 years after the 22.4\,GHz observations (1989), the effects of flickering could be present in some of these cores. Hence, the simultaneous use of 6\,GHz and 22.4\,GHz fluxes may not be appropriate for the analysis of some sources. However, \citet{De-Pree:2014aa} reported the flickering of 4 out of 41 sources in Sgr\,B2 within a similar time range (1989--2014). Such a rarity of cases (10\%) suggests that the flickering may not significantly alter our statistical results. \subsection{Dense gas environment} % \label{sub:dustdensity} We extrapolated the \hii regions' SED (Eq.\,\ref{eq:s-tau}) to subtract the free-free contribution from $S_{96}$ to get the emission purely from dust, which is denoted $S_{\rm dust}$. Twelve UC\hii regions have $S_{\rm dust}$ below $3\times$rms, while the other 42 UC\hii regions have detectable dust emission after subtraction of the free-free emission. Using $S_{\rm dust}$, we evaluated the dust properties in the vicinity of the 42 UC\hii regions. Following \citet{Ossenkopf:1994aa}, we calculated the dust column density $N_{\rm d}$ as \begin{equation} \label{eq:dust_sed} S_{\rm dust} = \frac{2h\nu^3}{c^2} \frac{1}{e^{\frac{h\nu}{k_{\rm B}T_{\rm d}}}-1} \left( 1-e^{ -\kappa_{0}\left(\frac{\nu}{\nu_0}\right)^{\beta}N_{\rm d} } \right) \frac{\pi r^2}{D^2}, \end{equation} in which $T_{\rm d}$ is the dust temperature, $D$ is the distance of Sgr\,B2, and $r$ is the radius of the dust core. Since most of the dust cores are not resolved \citep{Ginsburg:2018wo}, here we use $r = (r_{\rm obs96}^2-r_{\rm beam}^2)^{1/2}$, where $r_{\rm beam} = (0.54^{\prime\prime}\times0.46^{\prime\prime})^{1/2}$ is the effective radius of the 96\,GHz beam. The dust parameters $\kappa_{0}$ and $\beta$ depend on the dust grain properties. Due to the general high temperature and high density of neutral gas revealed by previous studies \citep[see, e.g.,][]{Huttemeister:1993tr,Schmiedeke:2016uc}, we assume that $T_{\rm d} = 100$, $\kappa_{0} = 2.631$, and $\beta = 1.05$ for $\nu_0 = 100\,{\rm GHz}$. The assumption of $\kappa_{0}$ and $\beta$ corresponds to dust grains without ice mantles and with a volume density of $10^8\,{\rm cm^{-3}}$. We assume a gas-to-dust mass ratio of 100 \citep{Ott:2014wk,Giannetti:2017vb} to estimate the gaseous mass $N_{\rm H_2}$, $N_{\rm H_2} = 100N_{\rm d}$. From the column density of molecular gas, $N_{\rm H_2}$, we obtain the volume density of molecular gas (\ce{H2}), \begin{equation} n_{\rm H_2} = \frac{3N_{\rm H_2}}{4r}. \end{equation} We list the $n_{\rm H_2}$ values at the position of the 42 UC\hii regions associated with dust emission in Table\,\ref{t:coreparams_derived}. The distribution of $n_{\rm H_2}$ is shown in Fig.\,\ref{f:nh2_dist}. The $n_{\rm H_2}$ ranges from $\sim 10^6$ to $\sim 10^8\,{\rm cm^{-3}}$. Due to the artifacts and possible small size of the dust core, even a dust core with enough volume density might not be detected by the sensitivity of our data set. We used the method in \citet{Ginsburg:2018wo} to estimate the detection limit of the dust emission. Compared to the typical $n_{\rm H_2}$ of the molecular clouds surrounding UC\hii regions ($10^5\,{\rm cm^{-3}}$) \citep[e.g.,][]{Wood:1989aa}, our UC\hii regions reside in denser neutral gas. The median value obtained in this work is similar to the value ($2\times10^7\,{\rm cm^{-3}}$) measured in the central region of Sgr B2(M) \citep{de-Pree:1995ve,Huettemeister:1995aa,Sanchez-Monge:2017vt}, although our sources are also distributed in the outskirts of Sgr B2(M). \section{Analysis and discussion} % \label{sec:analysis_and_discussion} \subsection{Expansion and equilibrium} % \label{sub:expansion_time} Based on the physical properties of the \hii regions (see Sect.\,\ref{sub:cores_in_vla}) and the properties of their surrounding environment (see Sect.\,\ref{sub:dustdensity}), we now investigate the expansion time for these extremely compact and dense \hii regions. The density of the molecular cloud that UC\hii regions expand into significantly affects the expansion rate \citep[e.g.,][]{Wood:1989aa,de-Pree:1995ve,De-Pree:1998aa}. For the simple case of a spherical UC\hii region ionized by the Lyman continuum flux of $\dot{N}_{\rm Ly}$ and expanding in molecular gas with volume density ${n}_{\rm H_2}$ and electron temperature of $10^4\,{\rm K}$, following \citet{de-Pree:1995ve,De-Pree:1998aa} we calculate the initial Str\"{o}mgren radius $r_{\rm i}$ as \begin{equation} \frac{r_{\rm i}}{\rm pc} = 1.99\times10^{-2} \left(\frac{\dot{N}_{\rm Ly}}{\rm 10^{49}\,s^{-1}}\right)^{1/3} \left(\frac{{n}_{\rm H_2}}{\rm 10^{5}\,cm^{-3}}\right)^{-2/3}. \end{equation} For the 12 UC\hii regions without detected associated dust emission ($S_{\rm dust}$, which is $S_{96}$ subtracting the free-free component), we assumed a uniform dust density of $2\times10^7\,{\rm cm^{-3}}$ following \citet{De-Pree:2015ve}. This value is in agreement with the median of $n_{\rm H_2}$ of the 42 sources that have physical $S_{\rm dust}$. Since we assumed a relatively high gas density compared to typical dust cores \citep[see, e.g.,][]{Bergin:2007aa}, the nondetection of $S_{\rm dust}$ for 12 sources under such an assumption is due to small $r$ but not low $n_{\rm H_2}$. Then we calculated the expansion timescales of all 54 cores applying the expansion equation by \cite{Spitzer:1968wl,Dyson:1980tp}, \begin{equation}{} \label{eq:uchiiexpansion} r_{\rm calc} = r_{\rm i} \left(1 + \frac{7c_{\rm i}t}{4r_{\rm i}}\right)^{4/7}, \end{equation} where $c_{\rm i}$ is the sound speed ($\sim 10\, {\rm km\,s^{-1}}$). The expansion times, $t$, of the UC\hii regions are listed in Table\,\ref{t:coreparams_derived}. In Fig.\,\ref{f:cores_time} the cores are color-coded according to their $t$. In ideal cases most of the cores have an expansion time between $10^4$\,yr and $10^5$\,yr. The expansion times in this section were dervied under the assumption of ideal conditions \citep{Spitzer:1968wl}. Such derived expansion times should be treated as lower limits and the actual expansion time might be longer for to at least three reasons. First. the expansion of UC\hii regions can be halted due to the pressure equilibrium between the ionized gas and the surrounding dense molecular gas \citep[e.g.,][]{De-Pree:1998aa}. We assume that $T_{\rm e}$ of the ionized gas is $10^4\,{\rm K}$, and the molecular temperature $T_{\rm H_2}$ is from $50\,{\rm K}$ to $100\,{\rm K}$. The pressure equilibrium condition ($2n_{\rm e}T_{\rm e}=n_{\rm H_2}T_{\rm H_2}$) can be expressed as $2n_{\rm e} = 5\times10^{-3}n_{\rm H_2}$ and $2n_{\rm e} = 10^{-2}n_{\rm H_2}$ for $T_{\rm H_2} = 50\,{\rm K}$ and $100\,{\rm K}$, respectively. In dust cores with $n_{\rm H_2} \gtrsim \times10^7\,{\rm cm^{-3}}$, the UC\hii regions are mostly in equilibrium with the neutral gas, evident from Fig.\,\ref{f:pressure_eq}, whereas for the lower $n_{\rm H_2}$ regime, the UC\hii regions are in expansion phase owing to ionized gas pressure exceeding the neutral gas pressure. Second, in reality, dust absorption may reduce $r_{\rm i}$, and therefore slows down the expansion \citep[see, e.g.,][]{Wood:1989aa,De-Pree:1998aa}. Third, accretion flow on to the central star will disrupt the expansion process and cause a sudden decrease in the flux and size of the UC\hii region. \citep[see, e.g., Fig.\,10 in][]{Peters:2010tv}. With the current data, we cannot quantify the effect of either of these mechanisms. Future observations, for example high-resolution radio recombination line observations, may provide constraints on the possible effect of accretion on the expansion of the \hii regions. \subsection{Evolutionary stages} % \label{sub:evolutionary_stages} \begin{table} \caption[Associated objects of the dust cores.]{\meng{Number of objects associated with dust cores}\LEt{ Number of objects associated with the dust cores}} \label{t:small.assoc} \begin{tabular}{l r r r r r r} \toprule \hline \noalign{\smallskip} Region & N & M & S & DS & Rest$^{\rm a}$ & Total \\ \hline \noalign{\smallskip} Dust core & 24 & 55 & 46 & 46 & 100 & 271 \\ \hii region & 8 & 40 & 2 & 1 & 3 & 54 \\ Outflow & --$^{\rm b}$ & --$^{\rm b}$ & --$^{\rm b}$ & 31 & 18$^{\rm b}$& 49$^{\rm b}$\\ \ce{CH3OH} Maser & 2 & 2 & 0 & 0 & 5 & 9 \\ \bottomrule \end{tabular} {\\\smallskip\textbf{a}: Rest of the envelope.\\ \textbf{b}: Not fully covered by the \ce{SiO} map.} \end{table} The type of objects that are associated with dust cores suggests the evolutionary stages of the dust cores in star formation activity. \meng{In Table~\ref{t:small.assoc} we summarize the type and number of objects associated with the dust cores in the subregions Sgr\,B2(N), Sgr\,B2(M), Sgr\,B2(S), and Sgr\,B2(DS), and in the rest of the envelope of Sgr\,B2.} Of all the 54 UC\hii regions, 8 are in Sgr\,B2(N), 40 are in Sgr\,B2(M), 2 are in Sgr\,B2(S), 1 is in Sgr\,B2(DS) and 3 are in the rest of the envelope. Although the dust cores are distributed all over Sgr\,B2 \citep{Ginsburg:2018wo} and more than 80 of them are associated with the large \hii region in Sgr\,B2(DS) \citep{Meng:2019aa}, the dust cores are rarely associated with any UC\hii regions outside of Sgr\,B2(N) and Sgr\,B2(M). The percentage of the dust cores that are associated with UC\hii regions are 33\%, 73\%, 4\%, and 1\% for Sgr\,B2(N), Sgr\,B2(M), Sgr\,B2(S), and Sgr\,B2(DS), respectively, while for the remaining part of the envelope the percentage is 3\%. For a dust core, association with the \hii region is a sign of that the core is more evolved \citep[see, e.g.,][]{Breen:2010aa}. Therefore, the cores in Sgr\,B2(M) are the most evolved, while the cores in Sgr\,B2(S) and Sgr\,B2(DS) are the least evolved. The evolutionary stages of the cores in Sgr\,B2(N) are between those of Sgr\,B2(M) and Sgr\,B2(DS) or Sgr\,B2(S). All the dust cores in Sgr\,B2(DS) appear to be pure dust\footnote{ One \hii region is found associated with a core reported by \citet{Ginsburg:2018wo}, but the core emission is pure free-free, which suggests that the core is a dust-free \hii region. Therefore, no dust cores in Sgr\,B2(DS) are associated with \hii regions.}. With the sensitivity of our observation ($\sim 1$\,mJy, for unresolved sources), we constrain the age of the possibly undetectable embedded \hii regions in the dust cores. We consider the $\dot{N}_{\rm Ly}$ from the central star are of $10^{46}$, $10^{47}$, $10^{48}$\,${\rm s^{-1}}$, and dust densities are $10^6$ and $10^8$\,${\rm cm^{-3}}$, which are based on the typical values of the detected UC\hii regions. The evolution of $S_6$ of the \hii regions, based on the expansion model in Sect.\,\ref{sub:expansion_time} and Eq.~8 in \citep{De-Pree:1998aa}, are displayed in Fig.\,\ref{f:evolution}. The temporal evolution of $r$ and EM leads to increase in $S_{\rm 6}$ (see Eq.\,\ref{eq:s-tau}). For the dust cores that have $n_{\rm H_2} = 10^6$\,${\rm cm^{-3}}$, the embedded \hii regions with $\dot{N}_{\rm Ly} \geq 10^{47}$\,${\rm s^{-1}}$ will be observable $\sim100$\,yr after their birth, which contradicts the nondetection of \hii regions in Sgr\,B2(DS). For the dust cores that have $n_{\rm H_2} = 10^8$\,${\rm cm^{-3}}$, the embedded \hii regions with $\dot{N}_{\rm Ly} \geq 10^{47}$\,${\rm s^{-1}}$ will be observable $\sim10^3$\,yr after their birth, which suggests that it is possible that the \hii regions are already formed but still too dim to be detected. The \hii regions with $\dot{N}_{\rm Ly} \leq 10^{46}$\,${\rm s^{-1}}$ will be always undetectable with the current sensitivity. The sizes of the embedded \hii regions will reach the typical size of the UC\hii regions in this study ($\sim 0.01\,{\rm pc}$) in $10^4$ years \citep[see Fig.\,1 in]{de-Pree:1995ve}. Thus, it is possible that stars later than B1 have already formed and ionized the interiors of the dust cores. Collimated outflows are the footprints of the very early stages of star formation activity \citep[see, e.g.,][]{Beuther:2002aa}. Outflows can be traced by SiO emission \citep[e.g.,][]{Schilke:1997aa}. In a recent project, the SiO~(5--4) emission was observed with ALMA (P.I. A. Ginsburg). For the details of the observation and data reduction, see Jeff et. al. (in prep.). The resolution is 0.35\arcsec\ $\times$ 0.24\arcsec, with P.A. of $-80^{\circ}$. The spectral resolution is 1.35~km~s$^{-1}$. The typical RMS of the image is 0.9~mJy/beam. The observation covers Sgr\,B2(S) and the eastern part of Sgr\,B2(DS). The peak intensity map of SiO~(5--4) is shown in Figure~\ref{f:small.sio_peak}. Due to the artifacts around Sgr\,B2(S), we only analyzed the part of the image with declination $<-$28:24:00. We generated the moment one map by masking out all the pixels below $3\sigma$ (see Fig.~\ref{f:small.sio_mom1}). The average velocity difference between the blueshifted and redshifted lobes is found to be $\sim 10$\kms. We visually matched the positions of the dust cores identified by \citet{Ginsburg:2018wo} and the SiO outflows. The dust cores that are 1) spatially associated with an outflow, 2) covered by the SiO image not associated with an outflow, and 3) not covered by the SiO image are indicated in Table~\ref{t:g18_outflow}. Spatially, most of the outflows are associated with dust cores. On the other hand, of all the 120 dust cores covered by the SiO image, 49 are identified as associated with outflows. Particularly, among all the cores in Sgr\,B2(DS), two-thirds are associated with outflows. Such a high fraction confirms that the cores in Sgr\,B2(DS) are at their very early evolutionary stages. \ce{CH3OH} masers serve as additional probes to trace the star formation activity \citep[see, e.g.,][]{Breen:2010aa}. We cross-matched the \ce{CH3OH} masers presented by \citet{Caswell:2010aa} with our dust cores\footnote{We searched for associated masers in a circle with a radius of 1 arcsec, which is double the beam size of the 96\,GHz image.}. In Table~\ref{t:small.assoc} the number of masers that are associated with the dust cores that are in each region are listed. The cores in Sgr\,B2(M) and in Sgr\,B2(N) are associated with two \ce{CH3OH} masers. The cores in Sgr\,B2(DS) have no associated masers. This lack of associated masers also suggests that the cores in Sgr\,B2(DS) are less evolved than those in Sgr\,B2(N) and Sgr\,B2(M). It is worth noting that the dust cores in Sgr\,B2(DS) are distributed around the large-scale \hii region in Sgr\,B2(DS) \citep[See Fig.\,2 in][]{Meng:2019aa}. The large-scale \hii region was ionized by a central O7 star and is still in the expansion phase \citep{Meng:2019aa}. The expansion time of this \hii region is $\sim 10^5\,{\rm yr}$, which is estimated based on the size of the \hii region (0.72\,pc) and the sound speed of the ionized gas, $10\,{\rm km\,s^{-1}}$. This timescale, compared to the expansion time of the \hii regions in Sgr\,B2(M) and Sgr\,B2(N) suggest that the central O7 star possibly formed before the stars in Sgr\,B2(M) and Sgr\,B2(N). Another possible scenario is that the O7 star was ejected from a nearby star forming site (e.g., Sgr\,B1) earlier than $10^5\,{\rm yr}$ ago, and Sgr\,B2 had no star forming activity at that time. The newly formed dust cores in Sgr\,B2(DS), on the other hand, are signs of star forming activity that possibly was triggered by the expansion of the large-scale \hii region in Sgr\,B2(DS). \section{Summary} % \LEt{ please include a few lines before and after the bullet list (which acts as a "frame" for the list. A list should never be left "hanging" alone). Something like this: In this paper we wanted to show ... Our main conclusions can be summarized as follows: } \label{sec:summary} \meng{We observed the Sgr\,B2 complex with VLA A, BnC, and D array at 6\,GHz and identified 54 compact radios sources. We found that 39 of the 54 sources are also detected in the 22.4\,GHz band \citep{Gaume:1995aa}. Our main results are summarized as follows:} \begin{itemize} \item Using the 6\,GHz data of all 54 sources, as well as the 22.4\,GHz data of 39 sources, we calculated the EM, radius, electron densities, and the spectral type of the ionizing stars of all the 54 UC\hii regions. The UC\hii regions have radius between $6\times10^{-3}\,{\rm pc}$ and $4\times10^{-2}\,{\rm pc}$, and have EM between $10^{6}\,{\rm pc\,cm^{-6}}$ and $10^{9}\,{\rm pc\,cm^{-6}}$. We found that the electron densities of these UC\hii regions are in agreement with the values of typical UC\hii regions, while the radii are smaller than for the typical UC\hii regions. We identified that the UC\hii regions are ionized by stars with spectral types ranging between B0.5 to O6. \item Using the 96\,GHz ALMA data, we characterized the dense gas environment where the UC\hii regions are located. We found a typical dense gas density of $\sim10^6-10^9\,{\rm cm^{-3}}$ around the UC\hii regions. Using \citet{Spitzer:1968wl}, we estimated the expansion timescale of the UC\hii regions as $\sim10^4-10^5\,{\rm yr}$. More than half of the UC\hii regions are close to equilibrium with the neutral gas; this means that the pressure of most UC\hii regions and the dense gas surrounding them are comparable. Due to the high pressure of the neutral gas, some natal \hii regions might be optically thick and cannot be detected with the sensitivity of this study. Instead, for the lower $n_{\rm H_2}$ regime ($n_{\rm H_2} \lesssim \times10^7\,{\rm cm^{-3}}$) the ionized gas pressure exceeds that of the neutral gas and the UC\hii regions are still expanding. \item The percentage of the dust cores that are associated with \hii regions are 33\%, 73\%, 4\%, and 1\% for Sgr\,B2(N), Sgr\,B2(M), Sgr\,B2(S), and Sgr\,B2(DS), respectively. Among all the dust cores in Sgr\,B2(DS), two-thirds are associated with outflows that are traced by SiO(5--4). The dust cores in both of Sgr\,B2(M) and Sgr\,B2(N) are associated with two 6.7\,GHz \ce{CH3OH} masers, while the dust cores in Sgr\,B2(DS) have no associated maser. Based on these findings, we suggest that the dust cores in Sgr\,B2(M) are more evolved than those in Sgr\,B2(N). The dust cores in Sgr\,B2(DS) are younger than those in Sgr\,B2(M) or Sgr\,B2(N). \end{itemize} \meng{In this work, we calculated the physical properties of the UC\hii regions and their surrounding neutral gas in Sgr\,B2. We found that the pressure of the UC\hii regions and the dense gas surrounding them are comparable. We also characterized the evolutionary stages of these UC\hii regions and obtained their minimum expansion timescales. } \begin{acknowledgements} FM, ASM, PS, ASchw research is carried out within the Collaborative Research Centre 956, sub-projects A6 and C3, funded by the Deutsche Forschungsgemeinschaft (DFG) - project ID 184018867. AG acknowledges support from the NSF via AST 2008101 and CAREER 2142300. This paper makes use of the following ALMA data: 2013.1.00269.S and 2017.1.00114.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This research made use of Astropy,\footnote{http://www.astropy.org} a community-developed core Python package for Astronomy \citep{Astropy-Collaboration:2013aa,Astropy-Collaboration:2018aa}. \end{acknowledgements} \bibliographystyle{aa} % \bibliography{ref} \begin{appendix} \section{RMS maps} % \label{sec:rms_maps} The rms maps were generated using SExtractor \citep{Bertin:1996aa}. Although the core identification was performed by eye, SExtractor produces rms maps as side-products of automated core identification. The rms maps are shown in Fig.\,\ref{f:rmsmaps}. \section{Tables} % \begin{table*} \begin{small} \caption{Observed parameters} \label{t:coreparams} \centering{ \begin{tabular}{lrrrrrrrrr} \toprule \hline \noalign{\smallskip} \# & RA & DEC & $r_{\rm obs6}$ & $S_{6}$ & $r_{\rm obs22}$ & $S_{22}$ & $r_{\rm obs96}$ & $S_{96}$ & $\alpha_{6-22}$\\ & 17:47:- - & $-$28:- - & arcsec& mJy & arcsec& mJy & arcsec& mJy & \\ \hline \noalign{\smallskip} 1$^{* }$&23.355&25:33.95&0.70&$2.4 \pm 0.1$&...&...&0.35&$0.3 \pm 0.2$ & ... \\ 2$^{a }$&19.485&24:39.75&1.45&$64.9 \pm 0.3$&...&...&1.20&$60.9 \pm 0.2$ & ... \\ 3$^{* }$&20.235&23:44.95&0.60&$7.0 \pm 0.1$&0.60&$14.7 \pm 0.7$&0.60&$7.9 \pm 0.2$ & $0.56 \pm 0.26$ \\ 4$^{a }$&20.428&23:44.25&2.10&$363.1 \pm 0.8$&1.15&$820.5 \pm 1.6$&1.45&$1107.6 \pm 0.5$ & $0.62 \pm 0.27$ \\ 5$^{a }$&20.042&23:18.00&1.65&$123.8 \pm 0.2$&1.10&$160.4 \pm 0.9$&1.05&$135.5 \pm 0.2$ & $0.20 \pm 0.09$ \\ 6$^{a }$&20.049&23:12.60&1.20&$68.1 \pm 0.2$&0.65&$126.1 \pm 0.7$&0.80&$124.8 \pm 0.2$ & $0.47 \pm 0.21$ \\ 7$^{b }$&19.765&23:09.90&0.75&$15.6 \pm 0.1$&0.50&$22.8 \pm 0.7$&0.40&$18.3 \pm 0.2$ & $0.29 \pm 0.13$ \\ 8$^{* }$&20.015&23:08.85&0.55&$28.3 \pm 0.3$&0.55&$64.2 \pm 0.7$&0.55&$52.5 \pm 0.6$ & $0.62 \pm 0.27$ \\ 9$^{a }$&20.106&23:08.45&0.90&$96.9 \pm 0.5$&0.70&$274.6 \pm 0.7$&0.75&$266.3 \pm 1.1$ & $0.79 \pm 0.35$ \\ 10$^{b }$&20.315&23:08.00&0.40&$9.2 \pm 0.6$&0.30&$25.5 \pm 0.7$&0.45&$37.3 \pm 0.9$ & $0.77 \pm 0.35$ \\ 11$^{b }$&19.799&23:06.70&0.80&$31.3 \pm 0.3$&0.60&$63.3 \pm 0.7$&0.55&$39.0 \pm 0.3$ & $0.53 \pm 0.24$ \\ 12$^{b }$&20.148&23:06.60&0.35&$7.0 \pm 0.4$&0.30&$14.1 \pm 0.7$&0.35&$44.0 \pm 1.1$ & $0.53 \pm 0.25$ \\ 13$^{b }$&20.193&23:06.55&0.30&$4.2 \pm 0.4$&0.30&$20.3 \pm 0.7$&0.40&$132.9 \pm 1.6$ & $1.20 \pm 0.54$ \\ 14$^{b }$&20.246&23:06.10&0.65&$30.9 \pm 1.0$&0.45&$28.4 \pm 1.1$&0.45&$46.4 \pm 3.0$ & $-0.06 \pm 0.07$ \\ 15$^{b }$&20.133&23:06.05&0.35&$6.9 \pm 0.4$&0.25&$12.7 \pm 0.7$&0.30&$59.0 \pm 1.0$ & $0.46 \pm 0.23$ \\ 16$^{b }$&20.174&23:05.80&0.40&$11.8 \pm 0.6$&0.30&$45.9 \pm 0.8$&0.50&$267.1 \pm 1.9$ & $1.03 \pm 0.46$ \\ 17$^{b }$&19.693&23:05.65&0.65&$15.9 \pm 0.3$&0.40&$20.2 \pm 0.7$&0.40&$13.6 \pm 0.2$ & $0.18 \pm 0.10$ \\ 18$^{b }$&20.072&23:05.10&0.45&$12.7 \pm 0.4$&0.30&$22.0 \pm 0.7$&0.40&$45.5 \pm 1.5$ & $0.42 \pm 0.19$ \\ 19$^{ab}$&20.159&23:04.70&0.65&$114.6 \pm 0.8$&0.60&$932.9 \pm 1.3$&0.70&$3194.7 \pm 2.8$ & $1.59 \pm 0.70$ \\ 20$^{ab}$&20.303&23:04.40&0.55&$27.5 \pm 1.0$&...&...&0.55&$52.3 \pm 2.7$ & ... \\ 21$^{ab}$&20.231&23:04.20&0.55&$58.4 \pm 0.8$&0.40&$245.2 \pm 0.9$&0.40&$528.6 \pm 1.7$ & $1.09 \pm 0.48$ \\ 22$^{a }$&20.121&23:03.90&0.35&$41.5 \pm 0.3$&0.45&$487.2 \pm 0.9$&0.50&$1353.7 \pm 2.0$ & $1.87 \pm 0.82$ \\ 23$^{a }$&20.174&23:03.50&0.45&$47.1 \pm 0.4$&0.40&$225.3 \pm 0.8$&0.40&$517.8 \pm 1.5$ & $1.19 \pm 0.52$ \\ 24$^{ab}$&20.106&23:03.40&0.30&$15.5 \pm 0.3$&...&...&0.40&$450.9 \pm 1.6$ & ... \\ 25$^{ab}$&20.246&23:03.25&0.25&$4.3 \pm 0.3$&0.25&$12.9 \pm 0.7$&0.30&$80.9 \pm 1.1$ & $0.83 \pm 0.38$ \\ 26$^{b }$&20.106&23:02.90&0.30&$8.1 \pm 0.2$&0.30&$10.3 \pm 0.7$&0.35&$47.2 \pm 1.2$ & $0.18 \pm 0.12$ \\ 27$^{ab}$&20.277&23:02.85&0.45&$24.0 \pm 0.4$&0.45&$198.2 \pm 0.8$&0.50&$387.9 \pm 1.5$ & $1.60 \pm 0.70$ \\ 28$^{ab}$&19.902&23:02.80&0.85&$100.7 \pm 0.3$&0.60&$323.9 \pm 0.8$&0.85&$360.2 \pm 0.6$ & $0.89 \pm 0.39$ \\ 29$^{b }$&19.992&23:02.65&0.45&$13.1 \pm 0.2$&0.40&$17.0 \pm 0.7$&0.40&$19.8 \pm 0.7$ & $0.20 \pm 0.10$ \\ 30$^{b }$&20.269&23:02.05&0.45&$13.0 \pm 0.3$&0.30&$16.6 \pm 0.7$&0.45&$44.4 \pm 1.0$ & $0.19 \pm 0.10$ \\ 31$^{ab}$&20.125&23:02.00&0.95&$56.3 \pm 0.6$&0.55&$62.1 \pm 0.9$&0.55&$73.3 \pm 1.3$ & $0.07 \pm 0.04$ \\ 32$^{b }$&19.898&23:01.90&0.35&$11.3 \pm 0.1$&0.30&$18.5 \pm 0.7$&0.45&$40.5 \pm 0.2$ & $0.37 \pm 0.17$ \\ 33$^{b }$&19.860&23:01.35&0.45&$10.4 \pm 0.1$&0.30&$19.8 \pm 0.7$&0.45&$24.1 \pm 0.2$ & $0.49 \pm 0.22$ \\ 34$^{b }$&20.125&22:60.00&0.70&$22.7 \pm 0.3$&...&...&0.75&$41.6 \pm 0.5$ & ... \\ 35$^{* }$&20.087&22:57.05&0.40&$1.6 \pm 0.1$&...&...&0.50&$5.3 \pm 0.2$ & ... \\ 36$^{a }$&19.598&22:56.25&0.75&$30.9 \pm 0.2$&0.70&$38.7 \pm 0.7$&0.55&$26.7 \pm 0.2$ & $0.17 \pm 0.08$ \\ 37$^{a }$&20.186&22:56.25&0.40&$3.1 \pm 0.1$&...&...&0.40&$3.9 \pm 0.2$ & ... \\ 38$^{a }$&19.515&22:56.15&0.60&$30.3 \pm 0.2$&0.40&$44.2 \pm 0.7$&0.55&$49.6 \pm 0.2$ & $0.29 \pm 0.13$ \\ 39$^{* }$&19.939&22:55.50&0.50&$1.4 \pm 0.1$&...&...&0.50&$3.1 \pm 0.2$ & ... \\ 40$^{a }$&19.526&22:55.30&0.65&$36.6 \pm 0.1$&0.40&$43.0 \pm 0.7$&0.55&$45.6 \pm 0.2$ & $0.12 \pm 0.06$ \\ 41$^{a }$&18.629&22:54.30&1.25&$73.9 \pm 0.5$&...&...&1.15&$84.5 \pm 0.3$ & ... \\ 42$^{* }$&20.000&22:47.45&0.70&$5.4 \pm 0.1$&...&...&0.70&$6.0 \pm 0.2$ & ... \\ 43$^{* }$&19.886&22:47.40&0.60&$3.1 \pm 0.1$&...&...&0.60&$5.4 \pm 0.2$ & ... \\ 44$^{b }$&20.030&22:41.15&0.50&$5.0 \pm 0.1$&0.35&$32.2 \pm 0.7$&0.55&$79.0 \pm 0.2$ & $1.41 \pm 0.62$ \\ 45$^{b }$&19.489&22:26.40&0.65&$6.1 \pm 0.1$&...&...&0.80&$11.7 \pm 0.2$ & ... \\ 46$^{a }$&19.803&22:20.70&1.10&$79.1 \pm 0.1$&0.65&$227.5 \pm 0.8$&0.50&$141.0 \pm 0.4$ & $0.80 \pm 0.35$ \\ 47$^{* }$&19.417&22:19.75&0.60&$3.0 \pm 0.1$&...&...&0.60&$7.3 \pm 0.2$ & ... \\ 48$^{a }$&19.871&22:18.30&0.45&$7.8 \pm 0.1$&0.30&$43.5 \pm 0.7$&0.95&$1401.0 \pm 5.4$ & $1.30 \pm 0.57$ \\ 49$^{a }$&19.898&22:17.00&0.65&$28.2 \pm 0.4$&0.45&$184.2 \pm 0.7$&0.50&$290.9 \pm 3.1$ & $1.42 \pm 0.63$ \\ 50$^{b }$&18.102&22:06.85&0.50&$6.4 \pm 0.1$&0.25&$8.9 \pm 0.7$&0.50&$9.7 \pm 0.2$ & $0.25 \pm 0.15$ \\ 51$^{a }$&19.996&22:04.65&1.20&$162.6 \pm 0.9$&0.80&$411.3 \pm 1.5$&1.00&$271.4 \pm 0.6$ & $0.70 \pm 0.31$ \\ 52$^{a }$&17.337&22:03.60&0.75&$16.5 \pm 0.1$&0.45&$29.8 \pm 1.1$&0.65&$19.8 \pm 0.2$ & $0.45 \pm 0.20$ \\ 53$^{b }$&23.049&21:55.55&0.85&$23.6 \pm 0.1$&...&...&0.90&$30.3 \pm 0.2$ & ... \\ 54$^{a }$&24.830&21:44.34&1.95&$20.6 \pm 0.1$&...&...&0.45&$4.1 \pm 0.2$ & ... \\ \bottomrule \end{tabular} {\\ ${a}$: Associated with the \hii regions identified by \cite{Benson:1984wr}.\\ ${b}$: Associated with the \hii regions identified by \cite{Gaume:1995aa}.\\ ${*}$: Newly identified in this study. } } \end{small} \end{table*} \begin{table*} \begin{small} \caption{Derived parameters} \label{t:coreparams_derived} \centering{ \begin{tabular}{rrrrrrrr} \toprule \hline \noalign{\smallskip} \# & $r_{\rm calc}$ & ${\rm EM}_{\rm calc}$ & $\log_{10}(\dot{N}_{\rm Ly}/{\rm s^{-1}})$ & $n_{\rm e}$ & $n_{\rm H_2}$ & $r_i$ & $t$\\ & $\times 10^{-3}$\,pc & $\times 10^7\,{\rm pc\,cm^{-6}}$ & & $\times 10^4\,{\rm cm^{-3}}$& $\times 10^6\,{\rm cm^{-3}}$& $\times 10^{-3}$\,pc & $\times 10^4\,{\rm yrs}$\\ \hline \noalign{\smallskip} 1&$^*$26.73&$^*$0.24&$^*$46.17&$^*$0.94&...&$^{\dagger}$$^*$0.106&$^{\dagger}$$^*$9.3\\ 2&$^*$57.40&$^*$1.49&$^*$47.64&$^*$1.61&0.22&$^*$0.324&$^*$15.2\\ 3&6.51&30.34&47.05&21.59&...&$^{\dagger}$0.207&$^{\dagger}$0.5\\ 4&46.19&33.91&48.80&8.57&4.69&0.794&5.3\\ 5&33.59&11.90&48.07&5.95&...&$^{\dagger}$0.453&$^{\dagger}$4.6\\ 6&20.84&25.06&47.98&10.97&0.77&0.422&2.1\\ 7&10.95&16.11&47.23&12.13&...&$^{\dagger}$0.237&$^{\dagger}$1.1\\ 8&12.90&33.98&47.70&16.23&...&$^{\dagger}$0.339&$^{\dagger}$1.1\\ 9&23.30&45.84&48.34&14.03&0.33&0.556&2.1\\ 10&7.21&44.38&47.30&24.82&5.94&0.251&0.5\\ 11&13.85&28.75&47.68&14.41&...&$^{\dagger}$0.337&$^{\dagger}$1.2\\ 12&6.54&28.68&47.03&20.95&30.27&0.097&0.8\\ 13&4.79&88.33&47.25&42.92&72.90&0.064&0.7\\ 14&40.83&1.39&47.31&1.85&9.97&0.253&10.1\\ 15&6.66&24.78&46.98&19.29&73.46&0.052&1.4\\ 16&8.02&68.02&47.58&29.12&73.96&0.082&1.4\\ 17&12.22&11.28&47.17&9.61&...&$^{\dagger}$0.227&$^{\dagger}$1.3\\ 18&9.17&22.45&47.22&15.65&16.66&0.168&1.0\\ 19&24.94&183.33&48.99&27.11&244.93&0.109&8.0\\ 20&$^*$20.36&$^*$5.88&$^*$47.33&$^*$5.37&6.66&$^*$0.257&$^*$3.0\\ 21&17.85&74.51&48.32&20.43&182.20&0.079&5.7\\ 22&15.02&384.53&48.86&50.60&157.61&0.132&2.9\\ 23&16.00&87.68&48.29&23.41&184.11&0.077&4.8\\ 24&$^*$8.63&$^*$402.88&$^*$48.39&$^*$68.31&95.39&$^*$0.129&$^*$1.1\\ 25&4.90&48.99&47.01&31.62&106.05&0.042&1.0\\ 26&8.70&11.39&46.88&11.44&37.11&0.076&1.7\\ 27&11.41&188.10&48.32&40.61&44.63&0.202&1.3\\ 28&23.58&53.78&48.42&15.10&3.03&0.591&2.0\\ 29&10.91&11.93&47.10&10.45&3.11&0.214&1.1\\ 30&11.05&11.35&47.09&10.14&13.64&0.173&1.4\\ 31&27.61&6.74&47.66&4.94&4.72&0.329&4.2\\ 32&8.87&20.04&47.14&15.03&10.94&0.209&0.8\\ 33&8.06&26.49&47.18&18.13&2.72&0.228&0.6\\ 34&$^*$26.73&$^*$2.49&$^*$47.20&$^*$3.05&2.25&$^*$0.231&$^*$5.1\\ 35&$^*$13.66&$^*$0.64&$^*$46.02&$^*$2.16&1.35&$^*$0.094&$^*$3.1\\ 36&17.30&10.80&47.45&7.90&...&$^{\dagger}$0.282&$^{\dagger}$2.1\\ 37&$^*$13.66&$^*$1.25&$^*$46.31&$^*$3.02&0.92&$^*$0.117&$^*$2.6\\ 38&15.30&15.95&47.52&10.21&2.51&0.296&1.6\\ 39&$^*$18.18&$^*$0.31&$^*$45.95&$^*$1.30&0.67&$^*$0.089&$^*$5.4\\ 40&20.22&8.74&47.50&6.58&1.93&0.292&2.7\\ 41&$^*$49.30&$^*$2.37&$^*$47.71&$^*$2.19&0.64&$^*$0.342&$^*$11.2\\ 42&$^*$26.73&$^*$0.56&$^*$46.54&$^*$1.44&0.21&$^*$0.140&$^*$7.5\\ 43&$^*$22.51&$^*$0.44&$^*$46.30&$^*$1.41&0.58&$^*$0.116&$^*$6.4\\ 44&18.18&1.12&46.51&2.48&18.89&0.090&5.3\\ 45&$^*$24.63&$^*$0.74&$^*$46.60&$^*$1.73&0.56&$^*$0.146&$^*$6.3\\ 46&21.03&46.73&48.26&14.91&...&$^{\dagger}$0.522&$^{\dagger}$1.8\\ 47&$^*$22.51&$^*$0.43&$^*$46.28&$^*$1.38&0.97&$^*$0.115&$^*$6.4\\ 48&6.51&106.80&47.59&40.51&66.06&0.089&0.9\\ 49&12.39&131.78&48.24&32.62&26.56&0.269&1.2\\ 50&7.28&14.09&46.82&13.92&0.61&0.173&0.7\\ 51&30.47&39.57&48.51&11.39&...&$^{\dagger}$0.633&$^{\dagger}$3.0\\ 52&10.35&23.94&47.35&15.21&...&$^{\dagger}$0.261&$^{\dagger}$0.9\\ 53&$^*$32.96&$^*$1.65&$^*$47.20&$^*$2.23&0.65&$^*$0.232&$^*$7.4\\ 54&$^*$77.55&$^*$0.25&$^*$47.12&$^*$0.56&...&$^{\dagger}$$^*$0.218&$^{\dagger}$$^*$34.7\\ \bottomrule \end{tabular} } \tablefoot{$*$: Sources that are without detection in 22.4\,GHz. Parameters are derived from 6\,GHz flux and the deconvolved core size. $\dagger$: No detection in 96\,GHz. Parameters are derived assuming $n_{\rm H_2} = 2\times10^{7}\,{\rm cm^{-3}}$ } \end{small} \end{table*} \begin{table*} \begin{small} \caption{Which cores in \citet{Ginsburg:2018wo} are associates with outflows.} \label{t:g18_outflow} \centering{ \begin{tabular}{rl|rl|rl|rl|rl|rl} \toprule \hline \noalign{\smallskip} ID & Outflow & ID & Outflow & ID & Outflow & ID & Outflow & ID & Outflow & ID & Outflow \\ \hline \noalign{\smallskip} 1& Y & 47 & -- & 97 & -- & 143 & N & 202 & -- & 258 & Y \\ 2& Y & 48 & -- & 98 & -- & 144 & N & 203 & -- & 259 & Y \\ 3& Y & 49 & Y & 99 & -- & 145 & -- & 204 & -- & 260 & Y \\ 4& Y & 50 & Y & 100 & -- & 146 & -- & 205 & -- & 261 & N \\ 5& Y & 51 & Y & 102 & -- & 147 & N & 206 & -- & 262 & N \\ 6& N & 52 & -- & 103 & -- & 148 & -- & 207 & Y & 263 & N \\ 7& Y & 53 & -- & 104 & -- & 149 & -- & 208 & N & 266 & N \\ 8& Y & 54 & Y & 105 & -- & 150 & -- & 209 & N & 267 & N \\ 9& N & 55 & -- & 106 & -- & 153 & -- & 210 & Y & 268 & N \\ 10& N & 56 & -- & 107 & -- & 154 & -- & 211 & Y & 269 & N \\ 11& N & 57 & -- & 108 & -- & 155 & -- & 212 & Y & 270 & N \\ 12& Y & 58 & -- & 109 & -- & 156 & -- & 213 & -- & 271 & N \\ 13& -- & 59 & -- & 110 & -- & 157 & -- & 214 & -- & 101\_X & -- \\ 14& N & 60 & -- & 111 & -- & 158 & Y & 215 & -- & 175\_G & -- \\ 15& -- & 61 & -- & 112 & -- & 159 & Y & 216 & -- & 177\_B & -- \\ 16& Y & 62 & -- & 113 & -- & 160 & Y & 217 & -- & 180\_E & -- \\ 17& Y & 63 & -- & 114 & -- & 161 & Y & 218 & -- & 181\_D & -- \\ 18& Y & 64 & -- & 115 & -- & 162 & Y & 219 & -- & 182\_Y & -- \\ 19& Y & 65 & -- & 116 & N & 163 & Y & 220 & -- & 244\_C & -- \\ 20& Y & 66 & -- & 117 & N & 164 & -- & 221 & -- & 265\_H & N \\ 21& N & 67 & -- & 118 & N & 165 & -- & 222 & -- & 171\_K3 & -- \\ 22& Y & 68 & -- & 119 & -- & 166 & -- & 223 & -- & 172\_K2 & -- \\ 23& Y & 69 & -- & 120 & Y & 167 & -- & 224 & -- & 173\_K1 & -- \\ 24& Y & 70 & N & 121 & N & 168 & -- & 226 & -- & 174\_f3 & -- \\ 25& Y & 71 & N & 122 & N & 169 & -- & 227 & -- & 176\_f1 & -- \\ 26& -- & 72 & N & 123 & N & 170 & -- & 228 & -- & 234\_f4 & -- \\ 27& -- & 73 & N & 124 & N & 183 & N & 229 & -- & 235\_f2 & -- \\ 28& -- & 74 & N & 125 & N & 184 & N & 230 & -- & 245\_A2 & -- \\ 29& -- & 75 & N & 126 & N & 185 & N & 231 & -- & 264\_k4 & -- \\ 30& -- & 76 & N & 127 & N & 186 & N & 232 & -- & 86\_B9.96 & -- \\ 31& -- & 77 & N & 128 & N & 187 & N & 238 & -- & 87\_B9.99 & -- \\ 32& -- & 78 & N & 129 & N & 188 & N & 239 & -- & 90\_B9.89 & -- \\ 33& -- & 79 & N & 130 & -- & 189 & N & 243 & -- & 92\_I10.52 & -- \\ 34& Y & 80 & N & 131 & -- & 190 & N & 246 & -- & 96\_Z10.24 & -- \\ 35& Y & 81 & N & 132 & -- & 191 & N & 247 & -- & 151\_B10.06 & -- \\ 36& Y & 82 & N & 133 & -- & 192 & N & 248 & N & 152\_f10.32 & -- \\ 37& N & 83 & N & 134 & -- & 193 & N & 249 & N & 178\_f10.37 & -- \\ 38& Y & 84 & -- & 135 & -- & 194 & N & 250 & -- & 179\_f10.38 & -- \\ 39& -- & 85 & -- & 136 & -- & 195 & Y & 251 & -- & 237\_G10.44 & -- \\ 40& -- & 88 & -- & 137 & -- & 196 & Y & 252 & -- & 241\_f10.30 & -- \\ 41& -- & 89 & -- & 138 & -- & 197 & Y & 253 & -- & 225\_f10.33b & -- \\ 42& -- & 91 & -- & 139 & -- & 198 & Y & 254 & N & 233\_f10.27b & -- \\ 43& -- & 93 & -- & 140 & Y & 199 & -- & 255 & N & 236\_f10.303 & -- \\ 44& -- & 94 & -- & 141 & Y & 200 & -- & 256 & Y & 240\_f10.44b & -- \\ 45& Y & 95 & -- & 142 & -- & 201 & -- & 257 & Y & 242\_f10.318 & -- \\ 46& Y & & & & & & & & & & \\ \bottomrule \end{tabular} } \tablefoot{The IDs of the dust cores are as in Table~{3} in \citet{Ginsburg:2018wo}. Cores that are not covered by the SiO image (Fig.~\ref{f:small.sio_mom1}) are marked ``--''. Cores that are covered by the SiO image and associated or not associated with outflows are flagged Y or N, respectively. } \end{small} \end{table*} \section{SiO Images} \begin{sidewaysfigure*} \begin{center} \includegraphics[width=0.75\textwidth]{./small_outflow_peak.pdf} \caption[Peak intensity of SiO (5-4)]{Peak intensity map of SiO (5-4) line. The zoomed-in plots are the regions in Sgr\,B2(DS) with abundant outflows. The resolution of the image is 0.35\arcsec $\times$ 0.24\arcsec, with P.A. of $-80^{\circ}$. The dust cores identified by \citet{Ginsburg:2018wo} are shown as green circles, whose size is not scaled to the size of the cores.} \label{f:small.sio_peak} \end{center} \end{sidewaysfigure*} \begin{sidewaysfigure*} \begin{center} \includegraphics[width=0.75\textwidth]{./small_outflow_mom1.pdf} \caption[Moment 1 of SiO (5-4)]{Moment 1 map of SiO (5-4) line. The zoomed-in plots are the regions in Sgr\,B2(DS) with abundant outflows. The resolution of the image is 0.35\arcsec $\times$ 0.24\arcsec, with P.A. of $-80^{\circ}$. The dust cores identified by \citet{Ginsburg:2018wo} are shown as green circles, whose size is not scaled to the size of the cores.} \label{f:small.sio_mom1} \end{center} \end{sidewaysfigure*} \end{appendix}

Title: Observations of Thomson scattering from a loop-prominence system

Abstract: We describe observations of the white-light structures in the low corona following the X8.2 flare SOL2017-09-10, as observed in full Stokes parameters by the Helioseismic and Magnetic Imager (HMI) of the Solar Dynamics Observatory. These data show both bright loops and a diffuse emission region above them. We interpret the loops as the white-light counterpart of a classical loop-prominence system, intermediate between the hot X-ray loops and coronal rain. The diffuse emission external to the loops is linearly polarized and has a natural interpretation in terms of Thomson scattering from the hot plasma seen prior to its cooling and recombination. The polarimetric data from HMI enable us to distinguish this contribution of scattering from the HMI pseudo-continuum measurement, and to make a direct estimation of the coronal mass in the polarized source. For a snapshot at 16:19~UT, we estimate a mass $8 \times 10^{14}$~g. We further conclude that the volumetric filling factor of this source is near unity.

https://export.arxiv.org/pdf/2208.06007

\title{Observations of Thomson scattering from a loop-prominence system} \correspondingauthor{Juan Carlos Mart\'inez Oliveros} \email{oliveros@ssl.berkeley.edu} \author[0000-0002-2587-1342]{Juan Carlos Mart\'inez Oliveros} \affil{Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA} \author[0000-0001-6854-2779]{Juan Camilo Guevara G\'omez} \affiliation{Rosseland Centre for Solar Physics, University of Oslo, Postboks 1029 Blindern, 0315 Oslo, Norway} \affiliation{Institute of Theoretical Astrophysics, University of Oslo, Postboks 1029 Blindern, 0315 Oslo, Norway} \author[0000-0002-8283-4556]{Pascal Saint-Hilaire} \affil{Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA} \author[0000-0001-5685-1283]{Hugh Hudson} \affil{Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA} \affil{SUPA Department of Physics and Astronomy, University of Glasgow, G12 8QQ, UK} \author[0000-0002-2002-9180]{S\"am Krucker} \affil{Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA} \affiliation{Institute for Data Science, School of Engineering, University of Applied Sciences and Arts Northwestern Switzerland, 5210 Windisch, Switzerland} \keywords{Sun: flares, Sun: corona, Sun: X-rays} \section{Introduction} Classical loop-prominence systems { associated with powerful solar flares} have been observed since the 1930's \citep[e.g.][]{1964ApJ...140..746B,1968IAUS...35..287S}. These are commonly observed in H$\alpha$ and other chromospheric lines in the form of loop arcades. Since their initial observations, loop prominence systems (LPS) were a topic of particular interest for solar physicists as they seemed to have a clear relationship with solar flares. Moreover, LPS display filamentary structures with episodes of plasma condensation \citep{1963ZA.....56..291W}, leading to the somewhat misleading term ``sporadic coronal condensation'' for the flare-associated LPS. \citet{1968IAUS...35..287S} found that nearly 70\% of all LPS are observed in active regions, an in particular active regions that also hosted solar flares and were often associated with the coronal yellow line of Ca~{\sc xv}, implying higher than normal active-region temperatures. The early observations further suggested that the LPS are highly correlated with active regions, and in particular with active regions that can produce highly energetic flares (e.g. proton flares) \citep{1963ZA.....56..291W}. Since their discovery loop-prominence systems have been identified differently with ``loop-prominences", ``flare arcades", ``eruptive loops", ``post-flare loops", or ``post-eruptive arcades'' \citep[for an in-depth discussion see][]{2007SoPh..246..393S}. Although similar in nature, loop-prominence systems should not be confused with ordinary prominences. According to \citet{2007SoPh..246..393S} the key difference between these two phenomena is their evolution. Prominences grow by expanding the individual loops that compose them, while loop-prominence systems grow by the illumination of new higher loops while the lower fade. The time evolution of loop prominence systems is related to the thermodynamic and radiative properties of the chromospheric-temperature plasma. From observations \citep[e.g., ][]{1964ApJ...140..746B} it is clear that these systems do not grow from lower to higher altitudes \citep[see Figure 2 in ][]{1963ZA.....56..291W}, instead their generation is the product of higher loops getting more intense due to cooling to low temperatures, while the lower ones decay. This evolutionary process can take several tens of minutes to several hours \citep{2007SoPh..246..393S}. We now understand this in terms of the cooling of arcade loop systems from X-ray temperatures, down through Ca~{\sc xv} and then abrupt condensation to produce emission in cool lines like H$\alpha$. Although LPS commonly occur in flares near the limb, their observation in white light or pseudo-white light above the solar limb is extremely rare, even more so without the usage of coronagraphs since the signatures are faint relative to the intensity of the emission from the disk. Only a few cases exist in the literature, to our knowledge, describing visual observations of such events: SOL1980-06-21 (X2.6), an event visually observed by J. Harvey and T. Duvall \citep{1983SoPh...86..123H,1990ApJS...73..213C}, SOL1989-08-14 \citep[$\sim$X20; ][]{1992PASJ...44...55H}, and SOL2003-11-04 \citep[$>$X17;][]{2004AAS...204.0213L} may comprise the whole list of such visual observations. Recently, though, \cite{2014ApJ...780L..28M} and \cite{2014ApJ...786L..19S} found two excellent examples in ordinary optical imaging, but from space: SOL2013-05-13T02:17 and SOL2013-05-13T16:00. These observations made use of the Helioseismic and Magnetic Imager \citep[HMI; ][]{2012SoPh..275..229S,2012SoPh..275..207S} of the Solar Dynamics Observatory \citep[SDO; ][]{2012SoPh..275....3P}. \cite{2018ApJ...867..134J} have now also analyzed SOL2017-09-10 using HMI data, and in this paper we analyze its polarization properties. Our study of SOL2017-09-10 focuses on the development of the thermal sources in the flare. Many earlier papers describe this fascinating event in great detail, but mainly bypassing the basic thermal emissions in favor of studying more complicated aspects of the flare development, such as current-sheet formation \citep[e.g.][]{2018ApJ...854..122W} and magnetic reconnection, especially via the new capabilities of microwave imaging spectroscopy \citep[e.g.][]{2018ApJ...863...83G,2020ApJ...900...17Y,2021ApJ...908L..55C}. The HMI observations describe off-limb emission at the solar limb and relatively low in altitude (below 1.03 R\textsubscript{\(\odot\)} elongation). Both of the 2013 flares, the first reported from the HMI database, also showed white-light (WL) footpoint sources at the level of the photosphere. The gradual coronal emissions can be identified as visual counterparts of the classical loop-prominence system, but were brighter than expected and have some ambiguity in their emission mechanisms; a continuum could come from scattering or from the free-bound extension of the hot plasma seen in soft X-rays, or from direct emission (lines and continua), depending mainly upon the density \citep{2018ApJ...867..134J}. In their interpretation, the coronal sources detected by HMI in these flares represent flare loops, initially heated to X-ray temperatures, and detected in the process of cooling. The authors found the HMI flux to exceed the long-wavelength extrapolation of the bremsstrahlung of the flare soft X-ray sources by at least one order of magnitude, implying the contribution of cooler material that could produce free–bound continua and possibly line emission detectable by HMI. Their analysis suggested electron densities as high as 10$^{13}$~cm$^{-3}$, and under these conditions optically-thin Paschen continuum is not important. \citet{2014ApJ...786L..19S} reported the detection of linearly polarized scattered light from an LPS (SOL2013-05-13T02:17 and SOL2013-05-13T16:00) via the HMI white-light polarimetric data. This revealed Thomson scattering and therefore enabled a direct mass estimate and inferences about the source density, using the techniques described in \citet{1930ZA......1..209M}. \citet{2014ApJ...786L..19S} concluded that only a fraction of white-light emission in LPS was due to Thomson scattering and inferred a lower limit of the free electron density of about 3.5$\times$10$^{11}$ cm$^{-3}$, assuming a line-of-sight depth of 2.2~Mm. In this article, we describe similar polarization measurements and analysis of the well-studied SOL2017-09-10 limb flare. The HMI white-light intensity (i.e., Stokes I) of this event were reported by \citet{2018ApJ...867..134J}, whose study took into account all continuum emission processes in the flare loops, as well as optical-depth effects; see also \cite{1992PASJ...44...55H}. The LPS appear in an image annulus extending about 44$''$ above the limb, evolving for several hours in a close relationship with the EUV structures observed by AIA \citep[e.g.][]{2018ApJ...854..122W,2020ApJ...896L..35H}. For reference, we show the spatial relationships of these sources in Figure~\ref{fig:smapaia_94_sswl}. The diffuse HMI source above the LPS was bright enough, in this case, for us to make a first estimate of the density/filling factor for the hot plasma routinely seen as recombination radiation in the soft X-ray and microwave bands in major flares. We note that \cite{2019ApJ...887L..34F,2020ApJ...900..192F}, \cite{2021ApJ...921L..26Z} and \cite{2018ApJ...866...64C} have also analyzed this event with the complementary Mauna Loa K-Coronameter observations. \section{Observations and Instrumentation} On 2017 September 10, a loop prominence system developed during the gradual phase of an X8.2 flare (SOL2017-09-10). This LPS was well observed by the SDO/AIA \citep[e.g.][]{2018ApJ...854..122W} and various other aspects of this classical eruptive flare event have been widely reported in the literature. In particular, \cite{2018ApJ...867..134J} have previously reported on the HMI observations, but not using its polarimetric capability. The HMI instrument contains two cameras (front and side) designed for on-disk observations of the whole Sun, primarily for helioseismology and the characterization of the photospheric magnetic field, and for these purposes it makes high-resolution filtergrams of the photospheric Fe {\sc i} absorption line at 6173.3~\AA. HMI observes the line via six passbands spanning a range of about 345~m\AA~around the target line. It also obtains full Stokes profiles \citep{2012SoPh..275..229S}. The ``front camera" standard data are created from a combination of sequences of 12 distinct images (six filters and two polarizations each called a filtergram), while the ``side camera" data are reconstructed from observations with the same filters and the following polarizations: I+/--U, I+/--Q, and I+/--V. The individual wavelengths (and polarization settings) for these line profiles are not observed simultaneously, but in a programmed sequence extending over 45~s and 135~s frames for the HMI front and side cameras, respectively, and therefore have limitations when rapid transients occur \citep[e.g.,][]{2011SoPh..269..269M,2014SoPh..289..809M}. The resulting images are cropped and later during the process, truncated at about 65$''$ above the limb to generate the standard HMI observables (intensity, velocity and magnetic field). The observations we report here, though unambiguous photometrically, thus occur in a parameter space not optimized by the design of the telescope. Figure~\ref{HMI-AIA} shows the evolution of the LPS at two different and characteristic times, for HMI/STOKES I (top row) and AIA 94/193/211~\AA~composite images (bottom row). The RHESSI 10\%, 30\%, 50\%, 70\%, 90\% intensity contours are shown in red and plotted for the same times. The RHESSI images were reconstructed using the CLEAN algorithm with a period of integration that encompasses the HMI image cadence. Figure~\ref{HMI-AIA} (bottom row) shows the HMI/STOKES I contours (blue) for each evolutionary time at 75\%, 85\%, 95\% over the composite AIA images. On the top row three regions (white boxes) are shown and designated as A, B and C. Regions A and B denote the areas where the HMI/Stokes Q, U and V were analyzed. Region C is our control region. The data provided by HMI team through the Joint Science Operations Center (JSOC) has a cadence of 90 seconds and a roll angle or rotation angle close to 180$^\circ$. The analysis to be described was made before applying the roll angle rotation in order to avoid mathematical artifacts introduced by the interpolation. The data also have a small jitter in every frame, so, the first operation was to recenter all frames in order to have all the images properly aligned in time and later the six filters were average for each Stokes component. In addition, to increase the signal, images were summed over four consecutive frames to produce a single image every 6 minutes, which gives us sufficient temporal resolution for the purpose of this research. Finally, Q and U were rotated to increase the signal to noise ratio for the Stokes parameters throughout by aligning the plane of reference of the measurement with the location of the LPS \citep[see ][]{2004JGRD..109.9205S}. These new Stokes are named Q$'$ and U$'$ on the expectation that most of the polarized signal, in the case of Thomson scattering above a uniform photosphere, will appear in Q$'$ and little in U$'$ \citep{2014ApJ...786L..19S,2021ApJ...923..276S}. Figure~\ref{IQUV_tdist} shows the measured SDO/HMI Stokes parameters in the three regions of interest (A, B, C). The top panel shows the intensity values as a function of time. The GOES 1-8~\AA~X-ray flux is shown as the gray shadow area. It is clear that the highest value is reached in region A which during the event contains the visible part of the LPS. Region B is not as intense as region A and its peak occurred several tens minutes later than the peak in region A. Our control region C shows no appreciable changes in intensity that can be attribute to the LPS. The second panel from the top shows the Stokes parameter Q$'$, U$'$ and V. Regions A and B show a change in the Stokes parameter (Q$'$) during the event of about 20\% with a maximum close to 44\%. The control region C shows no apparent change. The U$'$ and V Stokes parameters show values close to zero. The third panel shows the $\Delta Q’$/$\Delta I$ values for regions A and B, region C is not shown since the values are undetermined in the time interval. Here we note a 19\% increase of this ratio in region A and about 21\% in region B in the time intervals selected to match the RHESSI observations. The bottom panel shows the spatial-temporal evolution of the LPS intensity within a 10$''$ slit centered at the LPS brightest feature. We calculated the average propagation velocity of the LPS during two stages of its evolution. The blue line indicates the apparent motion of the LPS with an average velocity of $7.88~\text{km\,s}^{-1}$ whereas the green line indicates the final phase of the event with an average velocity of $1.75~\text{km\,s}^{-1}$, these values are in agreement with those reported for this type of structures \citep[e.g.][]{1964ApJ...140..746B,2002SoPh..210..341G}. Table~\ref{tab:obs} summarizes the measurements of the excess fluxes for Boxes A, B and C at the two reference times indicated in Figure~\ref{HMI-AIA}, and interpreted as successive loops cooling and draining sequentially. \begin{table}[h] \centering \caption{Observed Fluxes} \medskip \begin{tabular}{ l l l l l l } \hline Box & Time (UT) & Position & I (DN/s) & Q$'$ (DN/s) & Area (arcsec$^2$)\\ \hline A & 16:19 & [982,-140] & 48.7$\pm1.5$ & $\sim$ -0.44$\pm$0.05 & 196 \\ B & 16:19 & [996,-142] & 39.9$\pm1.5$ & $\sim$ -0.27$\pm$0.03 & 196 \\ C & 16:19 & [996,-170] & 37.6$\pm1.5$ & $\sim$ -0.08$\pm$0.009 & 196 \\ \hline A & 16:43 & [982,-140] & 56.1$\pm1.4$ & $\sim$ -0.38$\pm$0.04 & 196 \\ B & 16:43 & [996,-142] & 39.9$\pm1.4$ & $\sim$ -0.24$\pm$0.03 & 196 \\ C & 16:43 & [996,-170] & 37.5$\pm1.4$ & $\sim$ -0.03$\pm$0.004 & 196 \\ \hline \end{tabular} \label{tab:obs} \end{table} \section{Mass, density and filling factor from Thomson scattering}\label{sec:cd} In the following sections we discuss the region \textit{above} the bright loops. This faint and apparently diffuse volume corresponds to the hot plasma of the soft X-ray source, an almost definitive flare component and one that is not well imaged by EUV observations such as those of AIA. \subsection{Mass} The K-component of the white-light continuum observed from coronal structures consists of Thomson-scattered photospheric continuum, with the resulting linear polarization \citep[e.g.][]{1966gtsc.book.....B}. This component of coronal brightness scales directly with $N_e$ (the column electon density) and hence the electron density $n_e$, while collisional atomic emissions (collisional bremstrahlung) scale as $n_e n_i$. The total free electron number ${\mathcal N}_e$ then determines the mass of the scattering material, if the abundances are known and if the location of the source justifies the ``plane-of-the-sky'' approximation. At high densities the atomic emissions processes may dominate and even produce optically-thick continuum \citep{1992PASJ...44...55H, 2018ApJ...867..134J}. This may obscure a part of the polarization source; accordingly the mass estimate represents a lower limit. Such an effect is likely to be a small one because of the efficiency of the thermal collapse \citep{1965ApJ...142..531F}. The dust-scattered F-corona is not important in the lower corona. Thus where Thomson scattering dominates, its brightness directly determines the mass of the scattering material, if the photospheric radiation field is known \citep{2021ApJ...923..276S}: \begin{equation}\label{eq:scat_mass} M = \mu m_p \mathcal{N}_e = \frac{F_s}{F_\lambda} \cdot \frac{\mu m_p (\pi R_\odot^2) }{\sigma_T W} \ \ , \end{equation} where $\mu$ is the mean molecular weight, $m_p$ the mass of the proton, $\mathcal{N}_e$ the number of scattering electrons, $F_s/F_\lambda$ the ratio of scattered irradiance to the total solar spectral irradiance at the HMI wavelength, $W$ a geometrical factor as estimated by \cite{2014ApJ...786L..19S}, and $\mathrm \sigma_T = 6.65\times10^{-25}~\text{cm}^{-2}$ the Thomson cross section. This estimate assumes fully ionized plasma and does not depend upon the source density or filling factor $\eta$, which we discuss in the following sections. Note that we use $\mathcal{N}_e$ for the total number of scattering electrons, $N_e$ for the column density (cm$^{-2}$), and $n_e$ for the number density (cm$^{-3}$) here and below. \subsection{Density}\label{sec:density} With further assumptions we can estimate the density of the scattering source and its filling factor. The basic idea of this technique is to derive the Thomson scattering contribution of the measured flux using the ratio between the observed linear polarization and the total intensity data. Following \citet{1994A&A...290..553R} and \citet{2009SoPh..254...89J} the density $n_e$ of an off-limb source emitting by Thomson scattering is: \begin{equation} n_e = \frac{j(\nu)}{I_0(\nu)}\frac{1}{W}\frac{1}{\sigma_T} = \frac{\varepsilon_I}{I_0}\frac{1}{\sigma_T}\frac{1}{G_I} \, \text{\hspace{3mm} (eq. 14 in \citealt{2021ApJ...923..276S})} \label{density_1} \end{equation} where $j(\nu)$ is the volume emissivity at a given prominence location, I$_0(\nu)$ is the emitted intensity from the solar disk center, $W = G_I$ is a geometrical dilution factor dependent on the height over the limb which represents our knowledge of the source structure. \citet{2014ApJ...786L..19S} described how to use the theoretical values from \citet{1930ZA......1..209M} in order to rewrite Equation~\ref{density_1} in terms of the effective Thomson scattering cross section at a given height $H$ over the solar limb $\sigma_T(H) = \sigma_T \cdot G_I(H)$\footnote{$G_I(H)$ is $\approx$~0.3 for the observed source heights and the known limb darkening at 6173\AA}, the line-of-sight thickness $D$ and the ratio between the scattered intensity from the source at the given height $\Delta I(H)$, understood as the intensity excess value above the baseline, and the intensity from the solar disk center such that: \begin{equation}\label{density_2} n_e(H) = \frac{\Delta I(H)}{I_0}\frac{1}{\sigma_T(H)}\frac{1}{D}\ \ . \end{equation} This allows us to estimate the electron density of a source, with $\Delta I$ being the intensity produced by only the scattered-light component. The observed $\Delta I$, however, also may have a component due to optically-thick sources seen in emission. As these latter sources are unpolarized and the expected degree of polarization due to Thomson scattering is known as a function of height \cite[e.g.][]{2014ApJ...786L..19S}, the two components can be separated using the actual measured degree of polarization $P_m$ and the expected one $P_T$. Taking this factor into account in Equation~\ref{density_2}, it is possible to calculate the density of the material responsible for the Thomson-scattered signal: \begin{equation}\label{density_3} n_e = \frac{\Delta I}{I_0}\frac{1}{\sigma_T(H)}\frac{P_m}{P_T}\frac{1}{D}\ \ . \end{equation} The column density $N$~(electrons cm$^{-2}$) is implicit in Equation~\ref{density_3}; it is the integral of the electron density along the line of sight through the emitting volume. Here we estimate the column depth by taking the width of the 70\% image contour of the RHESSI source. This is shown as the box dimension in Figure~\ref{HMI-AIA}. Thus, the column density, with the polarization-factor correction, becomes: \begin{equation}\label{density_4} N_e = \frac{\Delta I}{I_0}\frac{1}{\sigma_T(H)}\frac{P_m}{P_T}\ \ . \end{equation} The upper panels of Figure~\ref{fig:Ne_ne} show the column densities for regions A and B respectively, as calculated using Equation~\ref{density_4}. If the depolarization factor is not applied, i.e., if the flux originated entirely from Thomson scattering, the column densities would follow the dashed curves. The correction factor in each box increases with time, as expected for sources cooling and recombining. For region A, which is the closest to the photosphere and therefore is reached first by the white-light structure seen in HMI/Stokes~$I$ (see Figure~\ref{HMI-AIA}), the uncorrected column density is as large as $10^{21}~\text{cm}^{-2}$. However, when the column density is corrected by the polarization, gray curve in top left panel, it drops by almost an order of magnitude as this represents the actual Thomson scattered component only. For region B, there is no significant variation between the uncorrected and corrected column densities initially, meaning that the source is mainly dominated by Thomson scattering until the observed structure reaches it at a later stage. Finally, the bottom panels shows the expected electron densities derived from the corrected column densities using Equation~\ref{density_3} and assuming the line-of-sight thickness $D$ between $2.9~\text{Mm}$ and $40.7~\text{Mm}$ for regions~A and~B. Although the electron densities for the two regions are quite similar during the observation, region A shows a slightly higher density, which is expected as it is closest to the photosphere and consequently influenced by the material just above the bright loops. We performed an error propagation analysis to determine the uncertainty in our calculation of the column density and hence the electron density, finding this relative error to be about 11\%, and to be dominated by the observational error of Q'. An additional and ill-defined uncertainty is due to the assumption of a value for the column depth. This geometrical uncertainty does not affect the mass $M$ estimated via Equation~\ref{eq:scat_mass}, using the excess flux $F_S$ observed from each source region. From the first entry in Table~\ref{tab:obs}, we find that $F_s/F_\odot = 6.7 \times 10^{-9}$ for Box A at 16:19~UT. This corresponds to an electron number $4 \times 10^{38}$ and a (total) mass $8 \times 10^{14}$~g. This mass estimate is interesting as it is a simple to measure the gravitational potential energy of the scattering source, here roughly $5 \times 10^{28}$~erg. For further discussion see Section~\ref{sec:interp}. \subsection{X-ray and microwave observations} The polarized continuum observed by HMI relates directly to the free-free and free-bound continua observed in the microwave and X-ray bands. The same electrons contribute both to the emission measure ($\propto n_e^2$) and to the polarized flux ($\propto n_e$). The X-ray continuum also has significant free-bound and bound-bound contributions, whereas the microwaves do not. Each can serve to estimate the value of $n_e n_i \eta V$, and all three wavelength ranges should agree for a hot isothermal source according to Equation \ref{eq:free-free} \citep[e.g.][]{1972SoPh...23..155H}: \begin{equation}\label{eq:free-free} f_\nu \propto (n_e n_i V) \Lambda(n_e, T) \cdot \frac{e^{-h\nu/kT}}{\sqrt{T}}\ \ , \end{equation} where $f_\nu$ is the spectral flux in erg~(cm$^2$~s~Hz)$^{-1}$, including the slowly varying Coulomb logarithm $\Lambda(n_e, T)$. Note that the microwave flux has only a weak dependence upon the source temperature, but that it may have low-frequency cutoffs due to opacity or the Razin-Tsytovich efffect \citep[e.g.][]{2013A&ARv..21...58K}; furthermore the relatively weak thermal microwave continuum also may have to compete with other emission mechanisms such as gyrosynchrotron or plasma radiations. Figure~\ref{fig:SOL2017-09-10_RSTN} shows single-frequency data from the San Vito site of the Radio Solar Telescope Network (RSTN). Based on Equation~\ref{eq:free-free}, we expect to see a long-wavelength extension of the X-ray and white light continua through the IR, mm-wave, and longer radio wavelengths. This extension should have only the weak frequency dependence of the Gaunt factor, and so $f_\nu \approx {\mathrm const.}$ -- a flat spectrum. What we see in the Figure is different from this expectation, mainly because of gyrosynchrotron emission. The major time-series peak prior to 16:20~UT is likely to be from this mechanism, and the emission after about 16:25~UT may also come from a competing source. We interpret the minimum at around 16:20-25~UT in terms of the thermal signature of the hot plasma detected by RHESSI, but only as an upper limit. The spectral character of the minimum-flux epoch is consistent with optically thin free-free emission (Equation~\ref{eq:free-free}, which could reflect the long-wavelength extension of the RHESSI hot plasma and the polarized white-light continuum. The microwave flux does indeed have the right magnitude for the free-free interpretation, but this is just a consistency check. The two boxes A and B introduced in Figure~\ref{HMI-AIA} have been chosen to coincide with the main X-ray sources seen by RHESSI. We defer the details of the RHESSI measurements to Section~\ref{sec:rhessi} but report the result for the thermal emission measure of the source in Box A here. From the thermal component of the spectral fits we can derive a volumetric emission measure $n_e n_i \eta V$~cm$^{-3}$ for the hot component in terms of the volumetric filling factor $\eta$. We compare this with the same quantity derived from the whole-Sun fluxes observed by GOES/XRS. Finally we can also interpret the RSTN\footnote{\url{http://www.ngdc.noaa.gov/stp/space-weather/solar-data/solar-features/solar-radio/rstn-1-second/}} whole-Sun microwave observations for this parameter, noting that the lack of temperature senstivity for the microwave thermal component makes it possible to check for low-temperature flare sources not visible in soft X-rays \citep[e.g.,][]{2016ApJ...819L..30P}. Table~\ref{tab:em} lists all three of these estimates. \begin{table}[h] \smallskip \caption{Volumetric Emission Measures\label{tab:em}} \begin{center} \begin{tabular}{ l r} \hline\noalign{\smallskip} Source & Value (cgs) \\ \hline\noalign{\smallskip} RHESSI & $(3.0 \pm 0.3) \times 10^{50} $ \\ GOES-15/XRS & $(3.0 \pm 0.1) \times 10^{50}$ \\ RSTN/San Vito & $<(3.3 \pm 0.5) \times 10^{50} $ \\ \hline\noalign{\smallskip} \end{tabular} \end{center} \end{table} This agreement between microwave and X-ray emission measures confirms the identification of the sources and precludes the presence of substantial flare emission at low temperatures, according to Equation~\ref{eq:free-free}, this would destroy the agreement between microwave and X-ray emission-measure estimates displayed in the ``loci plot'' of Figure~\ref{fig:loci_plot}. The agreement of the HMI estimate of column density $N_e$ with the radio and X-ray values also allows us, uniquely for this flare, to estimate the line-of-sight thickness of the source $\eta D$ in terms of the observed volumetric filling factor EM and the total electron number ${\mathcal N}$ as derived in Section~\ref{sec:cd}. The emission measure implies an electron content ${\mathcal N}' = n_e' V' = \sqrt{EM \cdot \eta D A}$, which we can combine with the true electron number to give the estimate for the source thickness \begin{equation}\label{eq:thickness} \eta D = \frac{{\mathcal N}^2}{A\cdot EM}\ ,\end{equation} based on simple estimates of the source area $A$ and depending on the volumetric filling factor $\eta$. \subsection{RHESSI imaging}\label{sec:rhessi} The final information needed for characterizing the polarization sources comes from the RHESSI imaging, from which we can obtain estimates of the projected area of the joint X-ray/Thomson-scattering/thermal-microwave source identified by the agreement in Table~\ref{tab:em}. At the time of the observations of the September 10 flare, the RHESSI spectrometer had been operating in space for more than 16 years and only 4 of the 9 detectors were functioning. Even for these four detectors (detectors 1, 3, 6, and 8), radiation damage had substantially affected the spectral resolution and effective area. Because radiation damage affects each detector differently, we performed spectral fitting for each detector separately and then compare the derived parameters to judge their consistency. The RHESSI observations ended at about 16:17:40~UT when the RHESSI spacecraft entered the South Atlantic Anomaly. We selected the interval from 16:16:30 to 16:17:20~UT for our comparison with the HMI observations taken at 16:19~UT. The orbit-night background from 15:49 to 15:52~UT has been subtracted before the standard RHESSI spectral fitting of a thermal component plus a non-thermal component was done. The thermal fits dominate the total count spectrum for energies below 25~keV, so we take the 12-18 keV images shown in Figure~1 to represent the thermal sources. The derived temperatures from the four different detectors range from 18.2 to 19.3~MK in good agreement. The corresponding emission measures vary from 2.6 to 3.5 $\times 10^{50}$ cm$^{-3}$, with the average value of 3.3$\pm$0.5$\times 10^{50}$ cm$^{-3}$ as quoted above in Table~\ref{tab:em}. The temperature and emission measure derived from GOES-15 are 19.1~MK and 3.0$\pm$0.1$\times 10^{50}$ cm$^{-3}$, respectively, in surprisingly good agreement with RHESSI, considering the effects of radiation damage to the effective area of the RHESSI detectors. The RHESSI parameters derived above refer to the total counting rates. We can also use imaging spectroscopy to estimate temperature and emission measure for the RHESSI source contained within Box~A \citep[e.g.][]{2002SoPh..210..229K}. The fluxes from within Box~A are derived from CLEAN images at 1~keV resolution, which were reconstructed in the same way as the images shown in Figure 1. Using the energy range 13--18~keV gave parameters [21.0$\pm$1.1 MK, 1.2$\pm$0.4$\times 10^{50}$ cm$^{-3}$]. Here the errors have been derived by adding Gaussian random noise at 10\% rms in a Monte Carlo procedure; this gave parameters [19.2--22.6~MK, and 0.6--2.0 $\times 10^{50}$ cm$^{-3}$]. The derived temperature of the main source is thus slightly hotter than the flare-averaged value. Considering the uncertainties the difference is of minor significance. The emission measure of the bright source is about a factor of 3 smaller, indicating that the fainter sources outside Box A cannot be neglected and indeed affect the flare-averaged fit parameters. In the final step of the RHESSI analysis, we estimate the source area of the thermal hard X-ray source. We use CLEAN images to estimate the source size of the main compact RHESSI source (source A). From the RHESSI clean image shown in Figure~1 (top), we derive a FWHM source size of the compact part of the source of 15.8$''$. The image is reconstructed with a CLEAN beam of 12.2$''$. Hence, we get a deconvolved source size of 10.0$''$. \subsection{The Box~B interval} RHESSI exited the South Atlantic Anomaly (SAA) around 16:39~UT; the second interval (Box~B) was fully observed by RHESSI with both attenuators still inserted. The same analysis is applied to the later interval as has been done for the earlier interval using the orbit-night background from 16:55 to 16:58~UT. The non-thermal fit is marginal at this time, with very steep spectral slopes ($\approx 8$). The temperature ranges from 16.0 to 17.7 MK with an average value of 17.0$\pm$0.6 MK; and the emission measures vary between 0.7 to 1.3 $\times10^{50}$cm$^{-3}$, with an average value of $1.0\pm0.2\times10^{50}$cm$^{-3}$. GOES at this time gives [T, EM] = [15.5~MK, 1.3$ \times 10^{50}$cm$^{-3}$], again in good agreement with RHESSI. Imaging-spectroscopy fits for the main compact source give higher temperatures (22.8$\pm$1.1 MK). As the temperature of the main source is higher than the flare-averaged temperature, a significantly lower emission measure of $0.10\pm0.04\times10^{50}$cm$^{-3}$ of the compact source already produces a good fit with the observed hard X-ray flux in the box B. As extreme values for the emission measure we found $0.04\times10^{50}$cm$^{-3}$ and $0.3\times10^{50}$cm$^{-3}$. The CLEAN image shown in Figure~1 has an area with the 50\% contour level of 18.5$''$ $\times$18.5$''$, giving a deconvolved source diameter of 14.0$''$. Compared to the earlier interval around 16:19~UT, the source area apparently had roughly doubled in area. \section{Results}\label{sec:results} With these data we can now incorporate the polarization observations into a description of the RHESSI soft X-ray source. The essential augmentation or our knowledge is twofold: we can directly estimate the number of electrons, and hence the mass, of the scattering source; in addition that we can determine the product $\eta D$ (Equation~\ref{eq:thickness}). \subsection{Electron number, mass, and energy}\label{sec:numbers} As described above, the measurement of scattered flux directly determines the total number of free electrons, with the plausible assumptions of plane-of-the-sky location and nominal limb-darkened solar radiance \citep{2021ApJ...923..276S}. We find $4\ \times\ 10^{38}$ for the total electron number. This in turn determines the coronal mass at the time of observation, plus its gravitational potential energy. The values we have derived for the Box~A snapshot are $8 \times 10^{14}$~g and $5 \times 10^{28}$~erg, respectively. \subsection{Constraints on density and filling factor} The right panel of Figure~\ref{fig:loci_plot} summarizes the constraints imposed by the observed value of $\eta D = 4 \times 10^{9}$~cm. To this we have added constraints from the Razin-Tsytovich effect and from free-free opacity \citep{1996AIPC..374..416B}. These limits are based on the assumption that the RSTN three-point spectrum at the Box~A epoch (Figure~\ref{fig:SOL2017-09-10_RSTN}) does not show a spectral break. The free-free limit definitely restricts the density estimate for unity filling factor, shown in the Figure as the thick solid line (the depth-dependence of mean density for $\eta$~=~1). For some part of the range of possible source thicknesses, we can exclude this limit. We would expect a filling factor below unity in any case from the fine structure in the developing arcade, but there is no real constraint from the microwave or X-ray imaging. From the diagram we conclude that the probable value for the mean source density does not exceed $2 \times 10^{10}$~cm$^{-3}$ for line-of-sight depths comparable to the dimension of the integration box. \subsection{Interpretation}\label{sec:interp} This remarkable solar flare (SOL2017-09-10) has allowed us to consider in detail the information about hot coronal emission sources in a major flare. These sources consist of hot plasma, and we have shown for the first time that observations of linear polarization in the optical continuum match the extrapolated soft X-ray thermal emission spectrum right down to the microwave range. This convincingly establishes that flares do not require the injection into the corona of cool material (below about 1~MK), which would readily be detectable in the Paschen continuum and in the microwave free-free emission in the Rayleigh-Jeans spectrum \citep[e.g.][]{2016ApJ...819L..30P}. The absence of few-MK temperatures is also consistent with the AIA-derived emission measure distributions for this event, as reported by \cite{2018ApJ...854..122W}. The polarization brightness is directly proportional to the mass of the hot coronal cloud; this information cannot be obtained otherwise except in model-dependent ways. The coronal mass, $8 \times 10^{14}$~g, estimated some 30 minutes after the onset of the flare's impulsive phase, suggest that the magnetically trapped mass can rival that of the most massive CME, often quoted as about 10$^{16}$~g. The potential energy estimate, however, is less than a percent of total energy estimates for such an event, either flare or CME \citep[e.g.][]{2012ApJ...759...71E}. The standard picture of the development of a major eruptive flare envisions a ``magnetic explosion'' \citep[eg][]{2001ApJ...552..833M} creating a CME, followed by an extended period of magnetic reconnection that forms the loop prominence system finally seen in chromospheric emission lines. The plasma that fills these loops, according to this well-accepted picture, originated via chromospheric ``evaporation'' and appear as the hot cloud our observations show and as depicted in many cartoon descriptions. This hot mass has been driven into the corona and heated, although the details of how this happens remain unclear. Our data are consistent with the idea that the heating can be identified with particle acceleration leading to plasma $\beta$ of order unity \citep{2010ApJ...714.1108K}. The standard picture, dating back at least to \cite{1964ApJ...140..746B}, involves hot mass in larger and larger loops cooling sequentially, implying that the instantaneous mass we infer at the epoch of Box~A is a lower limit to the total mass for the entire flare process. After cooling, the ionized plasma recombines and may become optically thick, radiating strongly via collisional processes \citep{2018ApJ...867..134J} at densities much higher than we infer for the hot plasma. \section{Summary} The successful measurement of linear polarization in the optical continuum has allowed us to make direct estimates of the electron content of the flare's coronal plasma, and hence its mass. The information provided by this new tool is in agreement with previous observations of classical loop prominence systems \citep{1964ApJ...140..746B,2007SoPh..246..393S,1992PASJ...44...55H}. Essentially we have extended the familiar methods of K-corona studies to the bottom of the corona \citep{2021ApJ...923..276S}. The work reported here made serendipitous use of HMI data, which were never intended for studying off-limb sources and are limited in performance, for this purpose, by design factors in the optics and the spectral bandwidth. In particular, we have not attempted to use these data to analyze either the image structure of the polarization or its development in time. Both of these would contain novel information about the evolution of the excess coronal mass created by the flare. Our simple mass estimate is of special interest now because of the remarkable EOVSA microwave observations of this flare (SOL2017-09-10), which reveal important processes outside the regions of the EUV loop/cusp/current-sheet geometry \citep{2020ApJ...895L..50C,2022Natur.606..674F}. Our multi-wavelength analysis does suggest that the polarization source can be identified with its counterparts in the soft X-rays and microwaves at high electron temperatures, and can disambiguate these observations in terms of electron density and filling factor. This just gives us a taste of what may be learned by better-optimized observations in the future. Better imaging polarimetry would allow for a far deeper investigation of the coronal structure of a suitable flare and of its evolution \citep[\textit{e.g.} DKIST,][]{2021SoPh..296...70R}. \acknowledgments J.C.G.G. wish to acknowledge the SolarALMA project, which has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 682462), and by the Research Council of Norway through its Centres of Excellence scheme, project number 262622. H.S.H. thanks the University of Glasgow for hospitality during the completion of this work. \bibliography{biblio} \bibliographystyle{aasjournal}

Title: Tests of photometry: the case of the NGC 3370 ACS field

Abstract: A critical analysis and comparison of different methods for obtaining point spread function (PSF) photometry are carried out. Deep ACS observations of NGC3370 were reduced using four distinct approaches. These reductions explore a number of methodological differences: software packages (DAOPHOT and DOLPHOT), input images (individual and stacked frames), PSF models (synthetic and empirical), and aperture correction methods (automatic and manual). A comparison of the photometry leads to the following results: 1) Photometric incompleteness between individual reductions shows only a minimal difference (<10%). 2) Statistical errors are 20% to 30% smaller for DAOPHOT runs on stacked frames than DOLPHOT runs on individual frames. 3) Statistical errors assigned directly by the photometry codes are 25% to 50% smaller than the errors measured from artificial star tests. 4) Systematic errors are magnitude dependent and become larger at the faint end, at the level of $\sigma_s\sim0.1$ mag. 5) The automatic aperture correction routines in DOLPHOT result in a significant systematic error ($\sigma_s \sim 0.05$ mag). 6) Individual reductions agree well at the 0.02 mag level when the systematic errors are properly corrected through artificial star tests. The reasonable agreement between the reductions leads to important implications that i) the reduction dependent errors can be reduced to a 1% level in the luminosity distance scale, and ii) the stacked frame photometry can be a good means to study non-variable stars in external galaxies.

https://export.arxiv.org/pdf/2208.02824

\title{Tests of photometry: the case of the NGC\,3370 ACS field} \author{In Sung Jang} \affil{Department of Astronomy \& Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637}\email{isjang@astro.uchicago.edu} \keywords{cosmology: distance scale -- cosmology: observations -- galaxies: individual (NGC\,3370) -- galaxies: stellar content -- stars: RGB and AGB} \section{Introduction} Measuring accurate fluxes of resolved stellar objects is one of the most fundamental steps in observational astronomy. This is due to the nature of astronomy---the information content of stellar objects comes primarily from their electromagnetic radiation. Measuring this radiation is therefore the first task in deriving stellar physical properties (e.g. temperature) that can be used to constrain stellar evolution models. When stars are observed in nearby resolved galaxies this flux can then be combined with stellar distance indicators (i.e., standard candles) to determine their accurate luminosity distances, which can then form the basis for determining cosmological parameters, such as the Hubble constant ($H_0$). There have been increasing efforts to determine the local value of the Hubble constant using stellar distance indicators. And, remarkable progress has been made in the past few years: \setlist[2]{noitemsep} \setenumerate{noitemsep} \begin{enumerate}[noitemsep] \item Geometric distances to the local calibrator galaxies have increased in precision (LMC -- \citet{pie19}, SMC -- \citet{gra20}, NGC\,4258 -- \citet{rei19}, Milky Way calibrators -- \citet{gai21}); \item The Tip of the Red Giant Branch (TRGB) method has been used to determine the distances to Type Ia supernova (SN Ia) host galaxies \citep[e.g.,][]{fre19, fre20}, providing a measurement of $H_0$ that is independent of the Cepheid distance scale. \item A larger number of local SN~Ia calibrators and host galaxies have been observed (19 SNe Ia -- \citet{fre21} and 42 SNe Ia -- \citet{rie21}). \end{enumerate} The quoted accuracy of the Hubble constant measured from stellar distance indicators is now nearing $\sim$1\% precision, corresponding to only $\sim$0.02 mag in the luminosity scale. The improved accuracy of the local distance scale is encouraging. However, it should be noted that these measurements rest on an uncertain interface, the photometry. For example, the mean photometric error for a single TRGB star at $\sim$20~Mpc is $\sigma_{F814W} \sim0.15$~mag in $HST$ imaging \citep[e.g.,][]{jan18}. For Cepheid variables, the dispersion in the NIR period-luminosity relation is on average $\sim$0.3~mag for galaxies in the same distance range \citep[e.g.,][]{rie16}. These individual errors are much larger than the final error of the Hubble constant. While statistical errors can be reduced by analyzing a larger number of stars, there may be systematic issues that can not be reduced by increasing the sample size and are more difficult to identify and constrain. Stellar photometry is an art that requires stringent control of errors \citep[e.g., ][]{ste87}. The theory of stellar photometry is straightforward: one must determine the local sky background and then measure the flux excess using a fixed aperture (aperture photometry) or a pre-calculated Point-Spread-Function model (PSF fitting photometry). However, from the very first step, the practical implementation is complex, and many choices must be made. For example, there is no single sky annulus that is optimal for all degrees of stellar crowding. Even using a given sky annulus, a number of statistical schemes can be considered to find a representative value of the sky background: $mean$, $median$, and $mode$\footnote{$mode$ is not necessarily unique to the discrete distribution so its approximation can also be considered: $mode \approx 3\times median - 2\times mean$ (e.g., DAOPHOT).}, and one may further consider different values for the sigma clipping of outlier pixels. In a crowded field, unresolved sources contribute to the sky background, making a gradient within the sky annuli. These issues are all possible sources of error. Artificial star tests have been the preferred method to assess the robustness of crowded-field photometry \citep[e.g., ][]{ste88}. These tests are performed by injecting a large number of artificial stars into the images and then recovering them in the same manner as real stars. This process is computationally expensive, but it does deliver robust estimates of photometric performance (e.g., completeness, random errors, and systematic errors). However, it does not fully encapsulate the all potential sources of error. One example is the error associated with the use of incomplete PSF models. Model PSFs do not exactly match the intrinsic image PSFs, so there is an uncertainty in the measured flux of real sources due to the difference in PSFs. It is not possible to measure this bias from artificial star tests alone, since the artificial stars are injected using the model PSF, which differs from the real sources. A complementary and alternative approach is to undertake and compare independent reductions, from different input parameter sets, or ideally from different software packages \citep{fre01}. Indeed, such tests have been carried out in the literature\footnote{For $HST$ images, we identified the following studies: \citet{kel96, sil96, phe98, kel99, sil99, dol00, rej05, mon10, hat17, jan17a, jan18, fil20, jan21}. }. These studies often found small, but non-negligible magnitude-dependent offsets, such that the faint stars show larger discrepancy ($\Delta mag \lesssim 0.05$). The faint side of photometry is a sensitive test area for systematic effects. The observed discrepancy could be due to the different photometric processing or the systematic bias associated with the stellar crowding and sky subtraction. The latter can be quantified through artificial star tests. Therefore, it would be ideal to combine both approaches---artificial star tests and independent reductions---to address systematic issues in photometry whenever practical. In this paper, we now present detailed photometry tests applied to deep observations of NGC\,3370 taken with ACS/WFC on board the Hubble Space Telescope ($HST$). NGC\,3370 is a moderately inclined late-type disk galaxy (Figure \ref{fig1}) and has been host to a Type Ia Supernova, SN\,1994ae. The ACS observations for this galaxy provide a unique opportunity for this study since two compelling stellar distance indicators have been applied to the $same$ $dataset$ and resulted in accurate distances: Cepheids \citep[][$(m-M)_0 = 32.29\pm0.06$]{rie05} and the Tip of the Red Giant Branch (TRGB) \citep[][$(m-M)_0 = 32.25\pm0.04$]{jan17b}. We reduced this ACS dataset in a number of independent ways, thereby allowing us to compare photometric performance and to test possible reduction-dependent uncertainties. This paper is organized as follows. Section~2 describes the ACS imaging data and data reduction procedures. We provide details of how we obtained PSF photometry from the individual reductions, and how we carried out aperture corrections and artificial star tests. Section~3 presents a detailed comparison of photometric performance (e.g., completeness, precision, and accuracy) measured from artificial stars. Furthermore, we examine magnitudes of real stars to see if there are any reduction-dependent systematics. The primary results are summarized in Section~4. \section{Data and Data processing} \subsection{Archival Image Data} The images for NGC\,3370 used in this study were downloaded from the MAST archive. Observations were taken with the ACS/WFC instrument on $HST$ in three filters: $F435W$ (9,600s), $F555W$ (61,240s), and $F814W$ (24,000s). The $F435W$ data\footnote{Obtained as part of the Hubble Heritage Program (PID: 9696, PI: K. Noll).} were only used to make the color composite image shown in Figure \ref{fig1}, and not used in the subsequent analysis. The $F555W$ and $F814W$ data were originally aimed at detecting Cepheid variables (PID: 9351 and 10802, PI: A. Riess), and here we use them to carry out photometry tests. All of the observations, except for some short $F555W$ exposures (3,640s), were taken in 2003. This means that the dataset is one of early applications of ACS (installed in 2002), when the instrument had better sensitivity and lower CTE losses. Individual frames were processed before extracting the photometry, following the methods described in \citet{jan21}. To briefly summarize, we carried out initial PSF photometry on the individual \texttt{\_flc} images, selecting bright stellar objects. These objects were then used as the alignment sources for Tweakreg in DrizzlePac 2.0 \citep{gon12, avi15}. This process found that some images were not properly aligned with a measurable offset of $\sim$1~pixel with respect to the reference frame. The aligned \texttt{\_flc} images were then passed into Astrodrizzle to generate stacked \texttt{\_drc} images. We prepared one drizzled image from the individual $F814W$ images to serve as the reference frame for the DOLPHOT reductions (Phot\,A and B, see next section for details) using the default pixel scale of $0\farcs05$. Additionally, we prepared a drizzled image in each $F555W$ and $F814W$ to perform drizzled frame photometry in DAOPHOT (Phot\,C and D). These images were generated using a slightly finer pixel scale of 0\farcs03, which mitigates the undersampling of the ACS point spread function (PSF). All drizzled images were generated using a \texttt{final\_pixfrac} of 1 and \texttt{final\_kernel} of `\texttt{square}' (default). \subsection{Point Spread Function (PSF) Photometry} \begin{deluxetable*}{cccccc} % \tabletypesize{\small} \setlength{\tabcolsep}{0.05in} \tablecaption{Summary of photometric reduction methods \label{tab1}} \tablewidth{0pt} \tablehead{\colhead{Processing ID} & \colhead{Photometry tool} & \colhead{Input image} & \colhead{PSF model} & \colhead{Aperture correction} &\colhead{Sky estimation}} \startdata Phot\,A & DOLPHOT & Individual frame (\texttt{\_flc}) & TinyTim & Auto (\texttt{ApCor=1}) & \texttt{Fitsky=2}\\ Phot\,B & DOLPHOT & Individual frame (\texttt{\_flc}) & TinyTim & Manual & \texttt{Fitsky=2}\\ Phot\,C & DAOPHOT & Drizzled frame (\texttt{\_drc}) & Empirical & Manual & ...\\ Phot\,D & DAOPHOT & Drizzled frame (\texttt{\_drc}) & TinyTim & Manual & ...\\ \enddata \end{deluxetable*} % \begin{deluxetable}{lll} % \tabletypesize{\footnotesize} \setlength{\tabcolsep}{0.05in} \tablecaption{DOLPHOT Processing Parameters \label{tab2}} \tablewidth{0pt} \tablehead{\colhead{Processing ID} & \colhead{Parameter} & \colhead{Value} } \startdata All & \texttt{PSFPhot} & 1 \\ All & \texttt{PSFPhotIt} & 2 \\ All & \texttt{Force1} & 0 \\ All & \texttt{SkipSky} & 2 \\ All & \texttt{SkySig} & 2.25 \\ All & \texttt{SecondPass} & 5 \\ All & \texttt{SearchMode} & 1 \\ All & \texttt{SigFind} & 2.5 \\ All & \texttt{SigFindMult} & 0.85 \\ All & \texttt{SigFinal} & 2.5 \\ All & \texttt{MaxIT} & 25 \\ All & \texttt{NoiseMult} & 0.10 \\ All & \texttt{FSat} & 0.999 \\ All & \texttt{RCentroid} & 2 \\ All & \texttt{PosStep} & 0.25 \\ All & \texttt{dPosMax} & 3.0 \\ All & \texttt{RCombine} & 1.5 \\ All & \texttt{SigPSF} & 3.0 \\ All & \texttt{PSFres} & 1 \\ All & \texttt{useWCS} & 1 \\ All & \texttt{Align} & 2 \\ All & \texttt{Rotate} & 1 \\ All & \texttt{FlagMask} & 4 \\ All & \texttt{InterpPSFlib}& 1 \\ All & \texttt{ACSpsfType} & 0 \\ All & \texttt{ACSuseCTE} & 0 \\ All & \texttt{FitSky} & 2 \\ All & \texttt{img\_RAper} & 3 \\ All & \texttt{img\_RChi} & 2 \\ All & \texttt{img\_RPSF} & 15 \\ All & \texttt{img\_RSky} & 15 35 \\ All & \texttt{img\_RSky2} & 4 10 \\ All & \texttt{img\_apsky} & 15 25 \\ \hline Phot\,A & \texttt{ApCor} & 1 \\ \hline Phot\,B & \texttt{ApCor} & 0 \\ \enddata \end{deluxetable} % There are two possible approaches to extracting photometry from multiple images of the same field: 1) Use individual images and photometer them simultaneously. The mean magnitude is then determined from the individual flux measurements. 2) Make a co-added (stacked) frame in each band and perform the photometric measurements on the stack. The first approach allows one to generate time-series photometry of variable stars. Because input images are only passed through the basic processing steps (e.g., bias correction, flat fielding), the intrinsic image PSF is preserved. This helps in achieving high photometric accuracy when the image PSF is well characterized by synthetic models, as is the case with $HST$ data. DOLPHOT is a modified version of HSTphot \citep{dol00} and has a number of useful routines to simultaneously photometer individual $HST$ images using synthetic PSFs (e.g., TinyTim PSFs). It has been widely used in recent studies \citep[e.g., ][]{tul09, dal09, rad11, ana21} and it was also tested in this study. The second approach has been considered to be sub-optimal for accurate photometry because the pixel counts are resampled during the stacking process. However, improved image processing techniques in recent years (e.g., DrizzlePac) have made stacked frame photometry more robust. This approach requires significantly reduced computing resources than photometering the individual images simultaneously. In general, empirical PSFs, which can be constructed from bright stars in the stacked frame, are used to carry out PSF photometry (but see Phot\,D, as described below and in Table \ref{tab1}.). Stacked frame photometry has been used to reduce $HST$ data in many studies\footnote{For ACS/WFC, the following studies can be found: \citet{pir05, rej05, bro06, cal06, alo07, wil07, sav08, bro09, bir10, kal12, ann13, bro14, geh15, lee16b, jan17b, tik19, fil20}} and it is more common in ground-based surveys (e.g., Dark Energy Survey \citep{bur18}, Hyper Suprime-Cam Subaru Strategic Program \citep{aih18}). In this study, we tested both approaches with a number of parameter changes. The details are listed below and also summarized in Table \ref{tab1}.\\ \begin{itemize} \item Phot\,A: uses DOLPHOT with individual frame \texttt{\_flc} images. The input parameters are listed in Table \ref{tab2}. They are consistent with those in the DOLPHOT/ACS User’s Guide, except for \texttt{SigFinal=2.5}. This is slightly lower than the default value (\texttt{3.5}) and makes sure that all of the detected stars (\texttt{SigFind=2.5}) are listed in the final catalog. The PSF to $0\farcm5$ aperture correction is measured using the DOLPHOT implemented routines (\texttt{ApCor=1}). The local sky background is measured with \texttt{Fitsky=2}. This is relevant to most of the DOLPHOT processing in the literature. \\ \item Phot\,B: same as the Phot\,A processing, but with manual aperture corrections (see Section~2.3 for details).\\ \item Phot\,C: uses drizzled stacked frames (\texttt{\_drc}) to measure stellar flux. A single pass of DAOPHOT PSF photometry (\texttt{FIND - PHOT - ALLSTAR - ALLFRAME}) is carried out \citep{ste87, ste94}. We used a readout noise of 4.98e$^-$ and a gain of 1.0e$^-$/ADU, as listed in the header of the drizzled images. PSF images were constructed empirically using $\sim$20 isolated, bright stars in the drizzled frames. Source coordinates are determined from the montaged frame with a detection threshold of $\sim$3$\sigma$, where the sky fluctuation is measured in the outskirts of the galaxy ($SMA\sim3\arcmin$).\\ \item Phot\,D: same as the Phot\,C processing, except that the TinyTim PSFs were used. We used the "\texttt{addstars}" task in DOLPHOT to inject TinyTim PSFs on individual \texttt{\_flc} images and then drizzled them in the same way as the original images. We note that the injected PSFs are corrected for the difference between the image PSFs and the models, measured from the prior DOLPHOT run (i.e., central pixel adjustment). This increases the photometric accuracy.\\ \end{itemize} There are three local sky fitting options generally used in DOLPHOT: \texttt{Fitsky=1}, \texttt{2}, and \texttt{3}. The \texttt{Fitsky=1} option measures the sky background prior to each photometry measurement with an annulus given by \texttt{img\_RSky} (\texttt{=15} and \texttt{35} pixels). Sky measurements in DOLPHOT are sigma-clipped $means$, with \texttt{SkySig} (\texttt{=2.25}) being the sigma used for the clipping. The \texttt{Fitsky=2} option fits the sky first, and the star second (i.e., two single-parameter fits). The sky annulus is given by \texttt{img\_RSky2} (\texttt{=4} and \texttt{10} pixels), typically smaller than \texttt{img\_RSky} used for \texttt{Fitsky=1} and \texttt{3}. The \texttt{Fitsky=3} option makes a single fit for the sky and star simultaneously (i.e., one two-parameter fit). In theory, \texttt{Fitsky=3} with \texttt{img\_RAper=10} should be very similar to \texttt{Fitsky=2} with \texttt{img\_RAper=3} and \texttt{img\_RSky2=4 10}, but \texttt{Fitsky=2} is significantly more robust in extremely crowded fields (A. Dolphin 2021, private communication). % In this study, we chose the \texttt{Fitsky=2} option to simultaneously process the inner crowded and outer sparse regions of NGC\,3370. DAOPHOT measures the local sky background from the $mode$ of pixel counts after sigma clipping. Here the $mode$ is an approximate estimate with an empirical relationship: $mode = 3\,\times median - 2\, \times mean$. \begin{deluxetable*}{ccc|cccccccc|cc} % \tabletypesize{\small} \setlength{\tabcolsep}{0.05in} \tablecaption{A List of the Photometric Standards \label{tab3}} \tablewidth{0pt} \tablehead{\colhead{ID} & \colhead{R.A.} & \colhead{Decl.} & \multicolumn{8}{c}{FLC ($r=0\farcs25$)} & \multicolumn{2}{c}{DRC ($r=0\farcs25$)} \\ & & & \colhead{$F814W$} & \colhead{Err} &\colhead{N} &\colhead{Stdev} & \colhead{$F555W$} & \colhead{Err} &\colhead{N} &\colhead{Stdev} & \colhead{$F814W$} & \colhead{$F555W$}} \startdata 1 & 161.740256 & 17.261232 & 22.340 & 0.009 & 8 & 0.040 & 23.103 & 0.014 & 21 & 0.013 & 22.334 & 23.096 \\ 2 & 161.751915 & 17.281692 & ... & ... & ... & ... & 21.748 & 0.006 & 36 & 0.011 & ... & 21.757 \\ 3 & 161.753383 & 17.270955 & 22.013 & 0.008 & 16 & 0.012 & 23.555 & 0.017 & 32 & 0.015 & 22.013 & 23.543 \\ 4 & 161.755169 & 17.263363 & 23.174 & 0.016 & 13 & 0.027 & ... & ... & ... & ... & 23.153 & ... \\ 5 & 161.767836 & 17.294145 & 22.887 & 0.013 & 13 & 0.013 & ... & ... & ... & ... & 22.889 & ... \\ 6 & 161.775137 & 17.291700 & 21.929 & 0.007 & 12 & 0.016 & 22.703 & 0.010 & 40 & 0.017 & 21.932 & 22.708 \\ 7 & 161.780718 & 17.303599 & 21.854 & 0.007 & 14 & 0.013 & 22.592 & 0.009 & 31 & 0.016 & 21.859 & 22.570 \\ 8 & 161.782637 & 17.286926 & ... & ... & ... & ... & 24.368 & 0.055 & 7 & 0.048 & ... & 24.404 \\ 9 & 161.783071 & 17.286043 & 22.745 & 0.012 & 6 & 0.051 & 24.088 & 0.030 & 15 & 0.032 & 22.741 & 24.092 \\ 10 & 161.789391 & 17.250907 & 20.392 & 0.003 & 16 & 0.004 & 22.736 & 0.010 & 31 & 0.015 & 20.387 & 22.729 \\ 11 & 161.791513 & 17.259049 & ... & ... & ... & ... & 23.460 & 0.016 & 35 & 0.017 & ... & 23.475 \\ 12 & 161.792929 & 17.263690 & 20.481 & 0.003 & 13 & 0.006 & 21.784 & 0.005 & 36 & 0.018 & 20.475 & 21.782 \\ 13 & 161.794224 & 17.267171 & 20.063 & 0.003 & 16 & 0.005 & 20.927 & 0.004 & 47 & 0.007 & 20.060 & 20.926 \\ 14 & 161.798045 & 17.265244 & 21.258 & 0.005 & 15 & 0.005 & ... & ... & ... & ... & 21.252 & ... \\ 15 & 161.801675 & 17.260026 & 22.420 & 0.009 & 14 & 0.020 & 23.307 & 0.014 & 39 & 0.023 & 22.416 & 23.306 \\ 16 & 161.802506 & 17.289145 & ... & ... & ... & ... & 22.894 & 0.011 & 34 & 0.016 & ... & 22.904 \\ 17 & 161.802663 & 17.283882 & 23.017 & 0.014 & 15 & 0.016 & ... & ... & ... & ... & 23.013 & ... \\ 18 & 161.803457 & 17.263566 & 22.845 & 0.012 & 15 & 0.013 & ... & ... & ... & ... & 22.841 & ... \\ 19 & 161.806155 & 17.291105 & 22.029 & 0.007 & 8 & 0.011 & 22.761 & 0.010 & 36 & 0.016 & 22.014 & 22.758 \\ \enddata \end{deluxetable*} % \subsection{Aperture correction} The finite sizes of PSF models and sky annuli used for photometry require an additional flux correction which is called the ``aperture correction". For the photometry of $HST$ data, the correction procedure is divided into two steps: 1) a correction from the PSF fit magnitude to an aperture magnitude at finite radius, and 2) a correction from the finite aperture magnitude to infinity. The second step can be made with the encircled energy values provided by STScI, so this section focuses on the first step, an empirical determination of the aperture correction out to a finite radius. The general procedure of the aperture correction is to select bright and isolated stars (hereafter, photometric standards) and then compare their aperture magnitudes with the PSF fit magnitudes. The mean magnitude difference in each band (or in each exposure) is identified as the aperture correction. DOLPHOT provides an automated routine to do the necessary correction (\texttt{ApCor=1}), which is used in Phot\,A. For the other reductions (Phot\,B -- D), we applied the aperture correction determined manually, following the method described in \citet{jan21}. Figure \ref{fig2} displays the selection of bright point sources we used for the aperture correction. The concentration index, $C$, is the magnitude difference between the small and large aperture radii. This simple parameter is very efficient at distinguishing between point sources and marginally resolved sources \citep[e.g.,][]{whi99}, as shown in the study of globular clusters in the Coma cluster \citep{lee16a}. We derived the concentration index from the photometry of individual \texttt{\_flc} images with aperture radii of $r=0\farcs04$ and $0\farcs125$. The histogram of the detected sources (red line) shows a prominent peak at $C\sim1.05$, which we identified as point sources. We selected 15 and 14 bright point sources in the $F814W$ and $F555W$ bands respectively, after a careful visual inspection of images and the light growth curves. Figure \ref{fig3} shows the selected bright stars in the drizzled frame. Their $r=0\farcs25$ aperture magnitudes are listed in Table \ref{tab3}. Photometry was carried out on both individual (\texttt{\_flc}) and drizzled (\texttt{\_drc}) frames. We found that there is no significant difference between the two image sets. The mean (median) difference in $F814W$ is 0.0045 (0.0040) mag with a standard deviation of 0.0063 mag. The same calculation in $F555W$ gives a mean (median) difference of --0.0017 (0.0015) mag with a standard deviation of 0.0132 mag. This good agreement in aperture photometry indicates that the stellar flux is well preserved after resampling images with DrizzlePac. With the $r=0\farcs25$ aperture magnitudes in hand, we calibrated the PSF fit magnitudes. Here we separated the image types for the correction: PSF fit magnitudes from the individual frames (Phot\,A and B) or drizzled frames (Phot\,C and D) were combined with aperture magnitudes from the same image frames. The corrections from the finite aperture magnitudes ($r=0\farcs25$) to infinity were made with those from \citet{boh16} : --0.1726~mag for $F814W$ and --0.1537~mag for $F555W$. We adopt photometric zero-points from the webtool\footnote{https://acszeropoints.stsci.edu/} provided by STScI: 25.525 for $F814W$ and 25.725 for $F555W$. \subsection{Artificial star tests} Extensive artificial star tests have been carried out to estimate errors and completeness of our photometry. We made a list of $5\times10^5$ artificial stars that has a uniform distribution in the color range of $0.5 < F555W - F814W < 2$ and the magnitude range of $23.0 < F814W < 30.5$~mag. The spatial distribution of the artificial stars was set to mimic the real stars: we selected real stars with $F814W \leq 29$~mag and used their coordinates with random shifts of $|\Delta| \leq 5\arcsec$ along the $X$ and $Y$ directions. We used the same list of the artificial stars for Phot\,A -- D. The artificial stars were injected into images and recovered in the same way as was done for the real stars. For the DAOPHOT-based processing (Phot\,C and D), we injected a small number of artificial stars ($N \sim 10,000$) at a time to preserve the degree of stellar crowding. Instead, we repeated the test for 50 times to sample the full list of the artificial stars ($N = 5\times10^5$). In the case of the DOLPHOT-based processing (Phot\,A and B), such an issue is not present, because the code performs the test one star at a time. Figure \ref{fig4} shows the results of the artificial star tests for Phot\,A. We used sources that passed the point source selection criteria (details are given in the next section). We found that the photometry is moderately deep with a 50\% recovery rate at $F814W \sim 28.1$~mag (top panel). The mean differences in $F814W$ magnitude and $(F555W-F814W)$ color between the injected and recovered stars are measured to be small ($<0.05$~mag) for $F814W < 29$~mag (middle and bottom panels). In Section~3 we explore how the photometric errors and completeness vary depending on the spatial selection, color selection, and the data reduction methods (Phot\,A -- D). \subsection{Point source selection} There are several types of sources in the raw photometry catalogs of the NGC\,3370 field: stars, spatially-resolved star clusters, blended sources, background galaxies, and false stellar detections. Selecting reliable point sources ($\approx$ stars) is one of the main steps before comparing PSF photometry. We inspected photometric diagnostic parameters returned from DOLPHOT and DAOPHOT and chose the sharpness parameter as our criterion for the point source selection. Figure \ref{fig5} displays an illustration of our point source selection criterion applied to Phot\,A. We used the sharpness distribution of artificial stars as a proxy for the genuine point sources (top panel). Red dots indicate the median sharpness values in each magnitude bin. Their standard deviations are marked by error bars. We determine the point source selection criteria by fitting the upper and lower ends of the error bars (yellow lines) with a constant + exponential function as follows: \begin{equation} {\rm Sharpness}_{F814W} = \alpha + \beta\times{\rm exp}(F814W - \gamma). \end{equation} \noindent Here we fit the upper and lower boundaries separately, because the sharpness distributions are not always symmetric. For the upper boundary, we fixed $\alpha = 0.03$, and found $\beta = 0.00023$ and $\gamma = 22.08$. The same scheme for the lower boundary gives $\alpha = -0.03$, $\beta = -0.00032$ and $\gamma = 22.15$. We then applied these selection criteria to the real stars (bottom panel). Sources between the two boundary lines were considered as point sources. We applied the same methodology to other reductions (Phot\,A--D). The derived values are listed in Table~\ref{tab4}. In addition to the sharpness-based selection above, we applied \texttt{type} = 1 (good star) for Phot\,A and B. This additional cut effectively eliminates spurious detections in the DOLPHOT processing. DAOPHOT does not provide the \texttt{type} parameter, so we applied the sharpness-based cut only (Phot\,C and D). \begin{deluxetable}{ccccccc} % \tabletypesize{\small} \setlength{\tabcolsep}{0.05in} \tablecaption{Point source selection \label{tab4}} \tablewidth{0pt} \tablehead{\colhead{ID} & \colhead{Upper boundary} & \colhead{Lower boundary} \\ & ($\alpha$, $\beta$, $\gamma$) & ($\alpha$, $\beta$, $\gamma$) } \startdata Phot\,A & 0.03, 0.00023, 22.08 & --0.03, --0.00032, 22.15 \\ Phot\,B & 0.03, 0.00024, 22.15 & --0.03, --0.00028, 22.04 \\ Phot\,C & 0.10, 0.00039, 21.36 & --0.10, --0.00072, 22.11 \\ Phot\,D & 0.10, 0.00036, 21.11 & --0.10, --0.00107, 22.23 \\ \enddata \end{deluxetable} % It is worth mentioning that there is no single setting of point source selection criteria that is optimal for all local environments and all scientific goals. The selection criteria adopted in this study are designed to sample stars in both the disk and halo simultaneously, and to apply a homogeneous selection (i.e., all sharpness based) to independent reductions as much as possible. Our approach is rather simple and straightforward compared with the selection schemes in the literature that are often based on several diagnostic parameters (e.g., error, chi-square, roundness, and crowding). We found, however, that our selection is good enough for our purposes of comparing photometry between individual reductions, as discussed in the next section. \section{Comparison of photometry} In this section, we used real and artificial star data to assess photometric performance of individual reductions. We start by analyzing the artificial star data in terms of recovery rates (Section 3.1), statistical errors (Section 3.2), and systematic errors (Section 3.3), which are indicators of completeness, precision and accuracy, respectively. We then show a star-by-star comparison of real stars and their CMDs (Section 3.4). \subsection{Photometric completeness} In Figure \ref{fig6}, we present completeness of our photometry for the inner (left) and outer (right) regions of the NGC\,3370 field. The recovery rates taken from individual reductions (Phot\,A -- D) are indicated as a function of input $F814W$ magnitudes. Error bars represent the $1\sigma$ statistical uncertainty in each magnitude bin. They are mostly smaller than the symbol size. It is obvious (and expected) that the inner region ($1\arcmin < SMA \leq 2\arcmin$) has lower recovery rates than the outer region ($SMA > 2\arcmin$) regardless of the reduction method. This is most likely due to the high stellar crowding at the disk region, where the surface brightness is several magnitudes brighter than the outer halo field \citep[e.g.,][]{can09}. The 80\% recovery rates are measured at $F814W \approx 26.5$ and $\approx28$~mag in the inner and outer regions, respectively. Individual reductions show notable features in their respective star recovery rates. First, Phot\,A and B show almost identical completeness curves in both inner and outer regions (solid lines). This is explained by the similarity in the two photometry methods: both Phot\,A and B are based on the DOLPHOT processing with individual frame images. The only difference is that Phot\,A has aperture correction determined automatically from the code itself ($\tt ApCor=1$), and Phot\,B is based on the manual aperture correction. The aperture correction typically amounts to a few hundreds of magnitudes, so Phot\,A and B should have similar photometric performance, in recovery rates, statistical and systematic errors. Phot\,C and D are in the same vein. They were reduced using the same code (DAOPHOT) and the same images (drizzled stacked frames). A small difference is the PSF models: empirical PSFs for Phot\,C and synthetic TinyTim PSFs for Phot\,D. The good agreement between their completeness curves (dashed lines) indicates that the choice of PSF models does not meaningfully change the recovery rates. Second, there is a systematic difference between the two groups of photometry: DOLPHOT runs on individual frames (Phot\,A and B) and DAOPHOT runs on stacked frames (Phot\,C and D). Photometry at the bright side of $F814W \lesssim 25$~mag is almost complete in all the cases. The systematic difference becomes apparent thereafter. In the magnitude range of $25.5$ $\lesssim F814W \lesssim 28.5$~mag, Phot\,C and D are more complete, showing $\sim$5\% (up to $\sim$10\%) higher recovery rates than Phot\,A and B. This trend is seen in both inner and outer regions. The completeness curves are crossed over at the faint side ($F814W \gtrsim 29$~mag) showing higher recovery rates for Phot\,A and B than Phot\,C and D. We note, however, that photometry near the detection limit is easily contaminated by false-positive detections (e.g., sky fluctuations) so that the recovery rates in this magnitude range are less reliable. In summary, we found a reasonable agreement in photometric completeness between independent reductions. While we detected a small systematic difference in some cases, the difference is not observed across the entire magnitude range and doesn't significantly change the general shape of the completeness curves. \subsection{Photometric precision} The statistical (random) error of individual reductions were derived from the artificial star data. We used the standard deviation of the magnitude difference between the injected and recovered stars (e.g., error bars in Figure \ref{fig4}) as a proxy for the statistical errors. This approach is straightforward and provides robust estimates of photometric errors taking into account both the Poisson photon statistics and stellar crowding \citep[e.g.,][]{ste88, gal96, dol02, mak06}. Figure \ref{fig7} displays the statistical errors for the inner (left) and outer (right) regions of NGC\,3370. It is not surprising that the inner crowded region shows larger errors than the outer region. The difference is about 60\%, which means that the errors of the inner region are on average 1.6 times larger than those of the outer region in the same magnitude bin. We also found that there is a systematic difference between individual reductions such that Phot\,A and B have larger statistical errors than Phot\,C and D. To better show the difference, we plot the error ratios between Phot\,A and Phot\,D in the bottom panels. It is clearly seen that Phot\,A has about 20\% (for the inner region) -- 35\% (for the outer region) larger errors than Phot\,D. Similar error ratios can be achieved with Phot\,B and C, as their errors are almost the same as those of Phot\,A and D, respectively. This result indicates that DAOPHOT runs on stacked frames could deliver higher photometric precision (i.e., smaller statistical errors) than DOLPHOT runs on individual frames. We next explore the statistical errors computed internally from the photometry codes. Figure \ref{fig8} shows a comparison of errors derived from two independent approaches: artificial star tests (filled circles with lines) and internal routines in the photometry codes (gray dots and open circles). We chose two representative reductions, Phot\,A (top) and Phot\,B (bottom), to see the errors in the inner (left) and outer (right) region of NGC\,3370. We found that the internally computed errors are smaller than the errors determined from the artificial star experiments in all cases. The average ratio between the two error estimates in the outer region is 1.25 for Phot\,A and 1.38 for Phot\,D. The inner crowded region shows a more significant difference with larger error ratios: 1.94 for Phot\,A and 1.99 for Phot\,D. This result indicates that artificial star experiments are critical to properly estimate photometric errors; using the errors computed directly from photometry codes underestimates the true errors, especially in crowded fields. \subsection{Photometric accuracy} We estimate the photometric accuracy (systematic error) by examining the magnitude offset between the injected and recovered artificial stars (i.e., red circles in Figure \ref{fig4}). The results from the inner (left) and outer (right) regions of NGC\,3370 are shown in Figure \ref{fig9}. A few distinguishable features are evident. First, the systematic errors for Phot\,A are almost identical to those of Phot\,B in both inner and outer regions. Similarly, a good match is seen between the errors of Phot\,C and~D. This, again, is due to the similarity in data processing, as discussed in previous sections. Second, there is a stark difference between the two groups of photometry: DOLPHOT-based reductions with individual frames (Phot\,A and B), and DAOPHOT-based reductions with stacked frames (Phot\,C and D). At the faint magnitude end ($F814W \gtrsim 27$~mag), the offsets have opposite signs: negative offsets for Phot\,A and B (i.e., fainter recovered magnitudes), and positive offsets for Phot\,C and D (i.e., brighter recovered magnitudes). The brighter recovered magnitudes is naturally expected at the faint side as unresolved stars below the detection limit will contribute to source's magnitudes. The origin of the opposite trend for Phot\,A and B is not clear, but we infer that this is likely due to the sky estimation in individual frames, where sources have much lower signal to noise ratio than the stacked frames. Third, the systematic errors at the bright end of the magnitude range ($F814W \lesssim 27$~mag) are almost negligible in both the inner and outer regions. The errors become larger at the faint end, but do not exceed $|\Delta F814W| = 0.15$~mag down to $F814W = 29$~mag. The 50\% recovery rates are measured at $F814W \sim 28.2$~mag in the inner region (see Figure \ref{fig6}). At this level, the inner region gives systematic errors of $\sim$0.05~mag for Phot\,A and B, and $\sim$0.03~mag for Phot\,C and D. The same completeness level for the outer region is measured at $F814W \sim 28.6$~mag, and there we found systematic errors of $\sim$0.11~mag for Phot\,A and B, and $\sim$0.07~mag for Phot\,C and D. It is noted that these systematic errors are well within the statistical errors ($0.2\sim0.3$~mag, see Figure \ref{fig7}). In summary, we detected measurable systematic errors at the faint magnitude range of photometry. The sign of the systematic effects is not the same in all the cases. The photometric accuracy is not much different between individual reductions, though we found slightly smaller systematic errors for Phot\,C and D than Phot\,A and B in the outer region. \begin{deluxetable*}{c|ccc|cccc} % \tabletypesize{\small} \setlength{\tabcolsep}{0.05in} \tablecaption{Summary of magnitude differences between Phot\,D and other reductions \label{tab5}} \tablewidth{0pt} \tablehead{\colhead{Magnitude range} & \multicolumn{3}{c}{Inner region ($1'< SMA \leq 2'$)} & \multicolumn{3}{c}{Outer region ($SMA > 2'$)}\\ ($F814W$) & \colhead{Phot\,A} & \colhead{Phot\,B} & \colhead{Phot\,C} & \colhead{Phot\,A} & \colhead{Phot\,B} & \colhead{Phot\,C} } \startdata $\leq26$ & 0.050 & --0.015 & --0.029 & 0.051 & --0.008 & --0.010 \\ $26.0 - 26.5$ & 0.043 (0.051) & --0.021 (--0.013)& --0.036 (--0.036)& 0.057 (0.061)& --0.010 (--0.004)& --0.025 (--0.025) \\ $26.5 - 27.0$ & 0.040 (0.049) & --0.025 (--0.016)& --0.036 (--0.036)& 0.048 (0.056)& --0.003 (0.005)& --0.030 (--0.031) \\ $27.0 - 27.5$ & 0.029 (0.041) & --0.037 (--0.026)& --0.037 (--0.037)& 0.026 (0.038)& --0.034 (--0.024)& --0.024 (--0.024) \\ $27.5 - 28.0$ & 0.007 (0.043) & --0.060 (--0.030)& --0.038 (--0.039)& 0.015 (0.041)& --0.045 (--0.019)& --0.026 (--0.027) \\ $28.0 - 28.5$ & --0.050 (0.034) & --0.117 (--0.032)& --0.040 (--0.036)& --0.052 (0.017)& --0.113 (--0.038)& --0.026 (--0.027) \\ $28.5 - 29.0$ & --0.111 (0.102) & --0.177 (0.040)& --0.039 (--0.018)& --0.120 (0.080)& --0.173 (0.040)& --0.025 (--0.017) \\ \enddata \tablenotetext{a}{Note: values in parentheses are the offsets corrected for the photometric bias measured from the artificial star experiments.} \end{deluxetable*} % \subsection{Comparison of real stars} With the extensive artificial star data and their photometric properties in hand, we compared real-star photometry. Figure \ref{fig10} displays the magnitude difference between individual reductions for the inner (left) and outer (right) regions of NGC\,3370. We used Phot\,D, which is a stacked-frame DAOPHOT photometry with TinyTim PSFs, as a reference reduction to see residual differences ($\Delta F814W$ = Phot\,D -- others). The red circles indicates the observed median offset in each magnitude bin. These offsets can be corrected for the systematic bias measured from the artificial star experiments, as marked by green circles. The brightest stars exhibit a small scatter in $\Delta F814W$ so systematic offsets can be easily identified. We found that Phot\,A shows larger offsets than Phot\,B and C in both inner and outer regions, as listed in Table \ref{tab5}. The median offsets for stars brighter than $F814W = 26$~mag in Phot\,A are $\Delta F814W$ = 0.050 and 0.051~mag for the inner and outer regions, respectively. These values are much larger than the offsets from the other reductions: $\Delta F814W$ = --0.015 and --0.008~mag (inner and outer regions) for Phot\,B and --0.029 and --0.010~mag (inner and outer regions) for Phot\,C. The larger offsets for Phot\,A are most likely due to the different methodology adopted for the aperture correction: Phot\,A uses the automated routines implemented in DOLPHOT ({\tt{ApCor=1}}) and the other reductions are based on the manual determinations with visual inspections of images (see Section 2.3). According to the DOLPHOT manual, the default aperture correction could have the potential for errors. We infer that the DOLPHOT processing of the NGC\,3370 field (based on Phot\,A) results in a significant error in the aperture correction. This result supports the general conclusion that a careful inspection of the aperture correction is an essential component of obtaining accurate photometry. Looking at the faintest stars, we found that Phot\,A and B show clear magnitude-dependent offsets. The observed offsets (red circles) decrease with increasing magnitude and they reach $\Delta F814W \simeq -0.15$~mag in the faintest magnitude bin. This is explained by the systematic bias of individual reductions. Through the artificial star experiments, we confirmed that Phot\,A and B have a negative systematic bias at the faint end (i.e., recovered magnitudes are fainter than the input magnitudes), and the other two reductions (Phot\,C and D) have an opposite trend (i.e., recovered magnitudes are brighter than the input magnitudes). Therefore, a direct photometry comparison of Phot\,D with Phot\,A and B should show very apparent offsets. The observed offsets in the photometry comparison can be corrected using the measure of photometric bias, as marked by green circles in Figure \ref{fig10}. It is clear that the green circles show reduced magnitude dependent offsets, except for the last point at $F814W = 28.75$~mag. The larger offset for the last point (open circle) is likely due to the nature of the bias correction; the photometric bias is measured as a function of the input magnitudes (Figure\,\ref{fig9}), and the bias correction is applied to the observed magnitudes (Figure\,\ref{fig10}). Therefore, the bias correction is expected to work well when the degree of the correction is not very significant ($\Delta F814W_{corr} \lesssim 0.1$~mag). Acknowledging this limitation, we found stable and approximately constant offsets over the entire magnitude range (Table \ref{tab5}). It is also noted that Phot\,C shows measurable offsets in both the inner and outer regions of NGC~3370. Phot\,C is fainter than Phot\,D. These two sets of photometry were analyzed using almost the same processing method, except for PSF images: empirical PSFs for Phot\,C and Tiny Tim PSFs for Phot\,D. Therefore, the measurable offsets come entirely from the choice of PSF images. \subsection{Color-Magnitude Diagrams} Figure \ref{fig11} presents color-magnitude diagrams (CMDs) of resolved point sources in the inner (top) and outer (bottom) regions of NGC~3370 taken from individual reductions. We used signal-to-noise ratios returned from DOLPHOT to mark an approximate boundary of $S/N = 3$ in $F814W$ and $S/N = 1$ in $F555W$, as shown by a dashed line. Sources below this line are less reliable, as they have large (random and systematic) errors with low recovery rates. CMDs of the inner region show stellar populations with distinct ages: young main sequence stars (a vertical feature at $F555W - F814W \approx 0$), red helium-burning stars (a slanted feature reaching to $F814W = 24.5$~mag at $F555W - F814W \approx 2$), old red giant branch (RGB) stars (a feature below $F814W \approx 28$~mag containing the largest number of stars), and asymptotic giant branch (AGB) stars (a feature between $F814W \approx 27$ and 28~mag having the reddest color). These features are seen in all four CMDs from individual reductions. We note, however, that the presence of the old RGB population is more evident in Phot\,C and D than Phot\,A and B. We infer that this is due to the differences in photometric performance as follows: 1) Phot\,C and D have higher ($\sim$5\%) recovery rates in the RGB magnitude range; 2) Phot\,C and D have smaller ($20\% \sim 30\%$) photometric errors, as marked by error bars in the figure; and 3) Phot\,C and D have a positive systematic bias at the faint side (i.e., brighter recovered magnitudes). For Phot\,A and B, however, the recovered magnitudes are fainter, so the features in CMDs are more elongated. Photometry of the outer region is deeper and more precise than the inner region, owing to its lower stellar density (i.e., less crowding). Indeed, CMDs of the outer region in Figure~\ref{fig11} exhibit a dominant old RGB population in all four panels. Small photometric errors for Phot\,C and D make them useful to investigate fine structures at the fainter magnitudes in the CMD, such as the tip of the RGB. \section{Summary and Discussion} The distance measurements of nearby galaxies have improved significantly in recent years. Given the great importance of the local distance scale in the determination of cosmological parameters, a more stringent control of errors is to be desired. This paper has investigated various systematic issues associated with the data reduction methods, together with presenting a quantitative comparison of photometric performance. We selected deep observations for NGC\,3370 taken with ACS/WFC on board $HST$ (Figure\,\ref{fig1}). The ACS data were reduced using four methods divided into two groups: DOLPHOT runs on individual frames (Phot\,A and B) and DAOPHOT runs on stacked frames (Phot\,C and D) (Table\,\ref{tab1}). These reductions include different aperture correction methods (automatic vs. manual) and PSF models (synthetic vs. empirical). Photometric performance (recovery rates, statistical and systematic errors) of individual reduction methods have been evaluated using artificial star tests. We have found that photometric incompleteness is comparable in all four reductions (Figure\,\ref{fig6}). The systematic difference is smaller than 10\% in recovery rates and it does not change the general shape of the completeness curves. The overall agreement indicates that the DOLPHOT processing does a good job of detecting faint sources, comparable with the stacked frame photometry with DAOPHOT. Photometric precision is measured to be 20\% -- 30\% higher for Phot\,C and D (DAOPHOT runs on stacked frames) than Phot\,A and B (DOLPHOT runs on individual frames). Such a trend is seen in both inner and outer regions of NGC~3370 (Figure\,\ref{fig7}). We also found that statistical errors assigned to each detection by the photometry codes are significantly smaller than the errors estimated from the artificial star tests (Figure\,\ref{fig8}). Therefore, artificial star tests are crucial to properly estimate errors of photometry and their impact on the error budget for subsequent measurements, such as stellar distances and cosmological parameters based upon them (e.g., a local value of $H_0$). Photometric bias is also detected (Figure\,\ref{fig9}). The bias is almost negligible in the bright magnitude range ($F814W \lesssim 27$~mag) but becomes larger at fainter magnitudes, making the recovered magnitudes brighter (for Phot\,C and D) or fainter (for Phot\,A and B). Due to the photometric bias, a direct comparison of real stars shows large magnitude dependent offsets (Figure\,\ref{fig10}). These offsets are reduced after applying the bias correction. Small, but non-negligible systematic offsets ($0.01 \sim 0.03$~mag) are detected in comparisons between Phot\,B, C, and D. This implies that there could be reduction-dependent uncertainties remaining after the cited photometric bias correction has been applied. Similar results can be found in \citet{jan21}. They presented photometry of the NGC~4258 halo field with a number of sky fitting options ({\tt Fitsky = 1}, {\tt 2}, and {\tt 3}) and PSF models (TinyTim and Anderson's PSFs) available in DOLPHOT. A direct photometry comparison showed offsets of $\sim$0.02~mag at the faint side and they too were not entirely eliminated with their bias correction (see their Figures 8 and 9 and A1). Studies of NGC~3370 and NGC~4258 showed that the reduction dependent uncertainties can be reduced to a 0.02~mag, corresponding to 1\% of the luminosity distance. We expect that a systematic survey of nearby galaxies will deliver a more robust estimate of the reduction-dependent uncertainties in the stellar distance scale and the measurement of the Hubble constant. The automatic aperture correction option ({\tt ApCor=1}) in DOLPHOT has also been tested. This option is widely used in DOLPHOT processing, but appears to be subject to larger errors, as documented in the manual. We have found that there is a large discrepancy ($\sim$0.05~mag) between the automatic and manual aperture corrections (Figure\,\ref{fig10}), indicating that the automatic correction routines applied to the NGC~3370 dataset introduces a sizable systematic error. Therefore, the automatic correction option should be used with care and a manual examination is necessary to achieve high accuracy photometry in the DOLPHOT processing. CMDs of independent reductions are comparable in showing distinct stellar populations in NGC~3370 (Figure\,\ref{fig11}). One noticeable difference is that Phot\,C and~D CMDs provide a noticeably clearer delineation of the old RGB population in both the inner and outer regions. This implies that the stacked frame photometry is helpful for studying non-variable stars in external galaxies. \acknowledgments I am grateful to Andrew Dolphin for patiently answering my questions over the past few years. I appreciate Wendy Freedman, Barry Madore, and Myung Gyoon Lee for their careful reviews of the paper. I thank Kayla Owens for improving the original manuscript. I thank the SH0ES team for taking deep images of nearby galaxies used in this and my previous studies. I thank Lucas Macri and Wenlong Yuan for helpful discussions on data reduction. I appreciate Sungsoon Lim for teaching me how to use DAOPHOT and DOLPHOT in my early career years. My thanks to Michele Cantiello for useful discussions regarding the stellar structure of NGC~3370. I would like to thank the members of the CCHP, and GHOSTS teams who have graciously shared their knowledge and insights into resolved stellar populations. My spatial thank to Alisa Brewer, Denija Crnojevic, and Dmitry Makarov. This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

Title: Dark matter substructures affect dark matter-electron scattering in direct detection experiments

Abstract: Recent sky surveys have discovered a large number of stellar substructures. It is highly likely that there are dark matter (DM) counterparts to these stellar substructures. We examine the implications of DM substructures for electron recoil (ER) direct detection (DD) rates in dual phase xenon experiments. We have utilized the results of the LAMOST survey and considered a few benchmark substructures in our analysis. Assuming that these substructures constitute $\sim 10\%$ of the local DM density, we study the discovery limits of DM-electron scattering cross sections considering one kg-year exposure and 1, 2, and 3 electron thresholds. With this exposure and threshold, it is possible to observe the effect of the considered DM substructure for the currently allowed parameter space. We also explore the sensitivity of these experiments in resolving the DM substructure fraction. For all the considered cases, we observe that DM having mass $\mathcal{O}(10)\,$MeV has a better prospect in resolving substructure fraction as compared to $\mathcal{O}(100)\,$MeV scale DM. We also find that within the currently allowed DM-electron scattering cross-section; these experiments can resolve the substructure fraction (provided it has a non-negligible contribution to the local DM density) with good accuracy for $\mathcal{O}(10)\,$MeV DM mass with one electron threshold.

https://export.arxiv.org/pdf/2208.14471

\section{Introduction} \label{sec:intro} It is important to leave no stone unturned in the search for the DM identity. Numerous astrophysical and cosmological observations infer the irrefutable evidence of DM \cite{Bertone:2004pz,Lin:2019uvt,Slatyer:2021qgc,Planck:2018vyg}. Despite these insurmountable evidences of the gravitational interaction of DM, we do not yet know if the DM candidate interacts via other forces. Numerous experiments have been performed to discover the non-gravitational signature of DM, but none of them have revealed a positive result. The DD experiments have been playing a pivotal role in their quest for the DM identity. The typical nuclear recoil (NR) DD experiments, searching for weak-scale DM, have made extraordinary progress\,\cite{SuperCDMS:2015eex, Akerib:2016lao, Cui:2017nnn, DarkSide:2018bpj, XMASS:2018bid, Aprile:2018dbl, EDELWEISS:2019vjv, Amare:2019jul, CRESST:2019jnq, CDEX:2019hzn, Adhikari:2018ljm, PandaX-4T:2021bab, DEAPCollaboration:2021raj, Schumann:2019eaa, DelNobile:2021wmp, Cooley:2021rws, Aalbers:2022dzr}. Typical NR DD experiments lose their sensitivity due to kinematic mismatch for an incident non-relativistic ambient sub-GeV DM (see for instance \cite{Battaglieri:2017aum, Kahn:2021ttr, Mitridate:2022tnv, Essig:2022dfa}).\footnote{Alternatively, one can boost non-relativistic light DM through scattering with energetic particles to overcome the threshold barrier, see for e.g., \cite{Bringmann:2018cvk, Ema:2018bih, Cappiello:2019qsw, An:2017ojc, Wang:2021jic, Granelli:2022ysi, Li:2022jxo, Calabrese:2022rfa, Calabrese:2021src} or by utilizing the Migdal effect\,\cite{Ibe:2017yqa, Dolan:2017xbu, Bell:2019egg, XENON:2019zpr, Essig:2019xkx, Dey:2020sai, Knapen:2020aky, Bell:2021zkr, Bell:2021ihi, Chatterjee:2022gbo, DarkSide:2022dhx}.} In order to fully characterize particle DM properties, it is important to probe DM-electron coupling too. A promising strategy to search for such DM interactions is to consider its scattering with electrons of the target materials\,\cite{Dedes:2009bk, Kopp:2009et, Essig:2011nj, Graham:2012su, Essig:2012yx, Lee:2015qva, Essig:2015cda, Roberts:2016xfw, Essig:2017kqs, Emken:2019tni, Catena:2019gfa, Bloch:2020uzh, Bose:2021cou}. In contrast with nuclear scattering, the maximum sensitivity to DM-electron interaction is typically achieved at a lower DM mass. For e.g., assuming a xenon target and momentum independent scattering cross-section, the maximum sensitivity is achieved at $\sim$ 30 GeV for DM-nuclear scattering and $\sim$ 200 MeV for DM-electron scattering. An ambient DM of mass $\mathcal{O}(10)$ MeV will have a kinetic energy of the $\mathcal{O}(10)$ eV, which is in the ball-park of the atomic ionization energy or the band gap energy of semiconductor. This indicates that a sub-GeV DM can ionize an electron from an atomic shell or facilitate an electron's transition from the valance band to the conduction band. Many experiments like XENON \cite{XENON:2019gfn}, SuperCDMS \cite{SuperCDMS:2018mne}, DarkSide-50 \cite{DarkSide:2018ppu, DarkSide-50:2022hin}, DAMIC \cite{DAMIC:2019dcn}, EDELWEISS \cite{EDELWEISS:2020fxc}, SENSEI \cite{Crisler:2018gci,SENSEI:2020dpa}, PandaX-II \cite{PandaX-II:2021nsg} etc.\,\,are searching for the signatures of such a phenomenon. The boundedness of electrons in the target material makes DM-electron scattering events inelastic. The DM velocity required to have a measurable recoil is rather high, which can be found near the tail of the DM velocity distribution (assuming that it has a Maxwell-Boltzmann form). These tails are quite sensitive to the choice of the DM velocity distribution \cite{Hryczuk:2020trm, Buch:2020xyt, Radick:2020qip, Maity:2020wic}. The present DM velocity distribution depends on the galactic structure formation history. In the well-known paradigm of $\Lambda$CDM (Lambda Cold Dark Matter), bottom-up hierarchical structure formation is a generic feature \cite{10.1093/mnras/183.3.341,Freeman:2002wq, Vogelsberger:2014kha, Springel:2017tpz, Feldmann:2022qvd, Somerville_2015, Vogelsberger:2019ynw}. Larger galaxies are formed from the merger of smaller galaxies (although the merger of similar mass galaxies may also lead to a bigger galaxy \cite{Belokurov_2018, Helmi_2018}). The gravitational field of the Milky Way (MW) is non-uniform, and this non-uniformity gives rise to strong tidal forces. When smaller galaxies accrete into the MW galaxy, the gravitational force disrupts these galaxies resulting in tidal stripping of various components (including DM) of these infalling galaxies. For an ancient merger, the DM component will have time to virialize within the MW, which may lead to an isotropic, isothermal DM halo. This scenario is often referred to as the Standard Halo Model (SHM), with the Maxwell-Boltzmann distribution representing the DM distribution. However, for relatively recent mergers, there will not be sufficient time for virialization, resulting in plenty of substructures both in the stellar and in the DM component\,\cite{Ibata:1994fv, Helmi:1999ks, Ibata:2000ys, Belokurov:2006kc, Lisanti:2011as, Myeong:2017skt, myeong2018shards, Necib:2018iwb, Necib:2019zka, Yuan_2020, 2022arXiv220102404S, 2022arXiv220102405R, 2022arXiv220611248D}. The presence of such additional stellar substructures (beyond the MW stars) have been detected by different sky-surveys like Gaia \cite{Ahn_2012,Myeong:2017skt,Belokurov_2018,2018, 2021ApJ...912L..30Z, 2022arXiv220611248D}, SDSS \cite{Myeong:2017skt}, LAMOST\,\cite{2018ApJS..238...16L, Yan:2022arj}, etc.,\,and have also been predicted in various N-body simulations\,\cite{Diemand:2008in, Vogelsberger:2008qb, Kuhlen:2012fz, Kuhlen:2012ft, Necib:2018igl, Simpson_2019, Helmi_2020, https://doi.org/10.48550/arxiv.2208.08443, https://doi.org/10.48550/arxiv.2208.11135}. Since these stellar substructures arise from merged galaxies, a DM counterpart must be associated with them too (because the DM is also present in the accreted galaxies before their merger). Whether DM would follow stellar distribution or not is a matter of debate. For example, the celestial part of the Sagittarius stream might not substantially overlap with the Solar neighborhood. However, the extended DM counterpart may overlap with our local position \cite{Purcell_2012}. The similarities between DM and stellar distributions in debris flow have been pointed out in Refs.\,\cite{Lisanti:2011as, Lisanti:2014dva}. The dwarf spheroidals, which give rise to the S2-stream, are believed to have similar DM and stellar shape \cite{OHare:2019qxc} before they merged with MW. Therefore the resemblance between stellar and DM substructures is not settled yet; more dedicated studies are needed to understand this. However, the presence of this DM might manifest in the local DM density and velocity distribution: this will result in a difference of the velocity distribution from the normal MB distribution with cut off at the galactic escape velocity\,\cite{Goodman:1984dc, Drukier:1986tm}. DM DD rate is strongly dependent on the local velocity distribution of DM\,\cite{Vergados:2002hc, Green:2003yh, Ling:2009eh, McCabe:2010zh, Fox:2010bz, Fox:2010bu, Catena:2011kv, Peter:2011eu, Frandsen:2011gi, Green:2011bv, Gondolo:2012rs, DelNobile:2013cta, Mao:2013nda, Bozorgnia:2013pua, Fox:2014kua, Feldstein:2014gza, Bozorgnia:2016ogo, Gelmini:2016pei, Laha:2016iom, Benito:2016kyp, Gelmini:2017aqe, Ibarra:2017mzt, Wu:2019nhd, Bozorgnia:2017brl, Fowlie:2017ufs, Ibarra:2018yxq, Herrero-Garcia:2019ntx, Bozorgnia:2019mjk, Poole-McKenzie:2020dbo, Lawrence:2022niq}, and a different DM velocity distribution can result in a large change in our theoretical expectations. The effects of these substructures have been extensively studied in the literature in the context of typical NR DD experiments\,\cite{Gelmini:2000dm, Stiff:2001dq, Freese:2003tt, Freese:2003na, Bernabei:2006ya, Savage:2006qr, Peter:2013aha, OHare:2017rag, OHare:2018trr, Evans:2018bqy, Buckley:2019skk, Ibarra:2019jac, OHare:2019qxc, Buch:2019aiw, DEAP:2020iwi}. This paper aims to study the effect of these DM substructures in the ER DM DD experiments assuming xenon-based detectors. Such a study has been conducted for semiconductor target material in Ref.\,\cite{Buch:2020xyt}. It was shown in Ref.\,\cite{Maity:2020wic} that the effect of such astrophysical uncertainties is quite prominent for xenon targets. Further, in large regions of the DM parameter space, the sensitivity of xenon targets is a few orders of magnitude stronger than those from semiconductor-based experiments\,\cite{XENON:2019gfn, Crisler:2018gci, SENSEI:2020dpa, PandaX-II:2021nsg} implying that xenon detectors will probably play a big role in discovering DM-electron scattering. These facts motivate our detailed study in this manuscript, where we highlight the importance of considering DM substructures while searching for DM-electron scattering. It has been argued in Refs.\,\cite{Ahn_2012, Myeong:2017skt,Belokurov_2018,2018,Necib:2018iwb, Necib:2019zka, Yuan_2020, Ou:2022wvr} that there are plenty of stellar substructures in the local halo. We utilize the results of the LAMOST survey \cite{2018ApJS..238...16L} to present the effect of the DM substructure \cite{Yuan_2020} in DM ER experiments. Without a loss of generality, we demonstrate our results by choosing a few benchmark substructures. We expect broadly similar results for other relevant substructures. In addition, our formalism will be useful for future analysis of DM ER experiments for xenon-based targets. Due to the lack of current understanding of how much of these substructures contributes to the local DM density, we adopt two approaches: an aggressive and a conservative approach where the DM substructure constitutes $100\%$ and $10\%$ of the local DM density, respectively. Our choices are motivated by Ref.\,\cite{2022arXiv220102405R} which states that stellar substructures near the Sun may constitute $\gtrsim 20\%$ of the stellar halo. We also consider the forecast of xenon targets in resolving the fraction of DM substructures components for a few benchmark choices of the DM parameter space. The rest of the paper is organized as follows. In Sec.\,\ref{sec:DMe}, we briefly review the DM-electron scattering in xenon-based detectors. In Sec.\,\ref{sec:DMSS}, we describe DM substructures that we have considered in our analysis. In Sec.\,\ref{sec:DMeSS}, we present our results along with the statistical methodology, and conclude in Sec.\,\ref{sec:conclusion}. \section{DM-electron scattering at xenon} \label{sec:DMe} If the ambient DM particle scatters off an electron of xenon, DM may transfer its kinetic energy to the electrons, leading to free electrons. For example, a non-relativistically moving ambient DM of mass $\sim 100 $ MeV will have kinetic energy $\sim 50$ eV (in the Solar system), which is in the ballpark of the electron ionization energy of xenon. In a two-phase xenon time projection chamber, DM particles interact with the liquid Xe target material, and depending on interaction type (electronic or nuclear), the signal topologies are different. For DM-nuclear interaction, the deposited DM energy produces excited atoms, electron-ion pairs, and some non-observable heat. Some free electrons recombine with ionized atoms to generate more excited atoms. Essentially both the direct and excited states produced by electron-ion recombination make a characteristic scintillation light. This prompt scintillation light, known as S1, is detected in photomultiplier tubes (PMTs) immersed in the liquid Xe at the bottom. Due to an external electric field, the remaining electrons drift through liquid xenon and cross the liquid and gaseous interface, producing proportional scintillation in the upper PMTs. This signal is known as S2. For the ER interactions, almost all the ionized electrons are collected at the upper PMTs through scintillation, producing a dominant S2 signal with a subdominant S1 signal. Hence ER interactions manifest through a large S2/S1 ratio compared to the NR case\,\cite{DiGangion:2021thw}. Let us consider a DM particle of mass $m_{\chi}$ and velocity $v$ scattering off an electron in the xenon atom. Energy conservation implies\,\cite{Bloch:2020uzh} \begin{equation} \label{eq:vmin} v_{\rm min}=\frac{q}{2 m_{\chi}}+\frac{\Delta E_e}{q}, \end{equation} where $v_{\rm min}$ is the minimum DM velocity required to get an ER of $\Delta E_e$, and $q$ is the momentum transfer to the electron. Note that $\Delta E_e$ must be greater than the ionization energy of the corresponding shell $E_{n,l}$ to have an observable recoil $E_e$, i.e., $\Delta E_e = E_{n,l} + E_e$. The differential DM-electron scattering event rate can be written as \cite{Essig:2017kqs} \begin{equation} \label{eq:rateXe} \frac{dR}{d\,{\rm ln}\, E_e}=N_T\frac{\rho_{\chi}}{m_{\chi}}\,\sum_{nl} \frac{\bar{\sigma}_e}{8\mu_{\chi e}^2} \int q dq \,F_{\rm DM}(q)^2\, |f_{\rm ion}^{n,l}(k^{\prime},q)|^2 \,\eta\left(v_{\rm min}(k^{\prime},q),t\right), \end{equation} where $N_T$ is the number of electrons in the target, $\rho_{\chi}$ denotes the local DM density, and DM-electron reduced mass is represented by $\mu_{\chi e}$. DM-electron scattering cross section for a reference momentum transfer, namely $q=\alpha m_e$, is indicated by $\bar{\sigma}_e$. The DM form factor, $F_{\rm DM}(q)$, takes care of the momentum dependency in the cross-section. The ionization form factor is represented by $f_{\rm ion}^{n,l}$ with $n$ and $l$ being the principal and angular momentum quantum number, respectively. The recoil momentum is denoted by $k^{\prime}=\sqrt{2 m_e E_e}$. The time dependency of the recoil signal is described through $t$. The quantity $\eta$, also called the mean inverse speed, depends on the $i^{\rm th}$ DM velocity distribution as \begin{equation} \label{eq:eta} \eta^i(v_{\rm min},t)=\int_{v_{\rm min}}^{\infty} \frac{f_{\rm lab}^i(\mathbf{v},t)}{v} d^3v, \end{equation} where $f_{\rm lab}^i$ is the DM velocity distribution at the the detector's rest frame in the location of the Earth for the $i^{\rm th}$ DM component (which contributes to the DM velocity distribution). The latter can be obtained by boosting the galactic rest frame DM velocity distribution ($f_{\rm gal}$) \begin{equation} \label{eq:galtolab} f_{\rm lab}^i(\mathbf{v},t) = f_{\rm gal}^i(\mathbf{v+v}_{\rm E}(t)), \end{equation} where $\mathbf{v}_{\rm E}$ is the Earth's velocity in the galactic rest frame: \begin{equation} \mathbf{v}_{\rm E}(t)=\mathbf{v}_{\rm LSR}+\mathbf{v}_{\rm pec}+\mathbf{u}_{\rm E}(t). \end{equation} Here $\mathbf{v}_{\rm LSR}$ is the velocity of the local standard of rest (LSR), $\mathbf{v}_{\rm pec}$ is the peculiar velocity of the Sun with respect to the LSR. Conventionally these are expressed in galactic rectangular co-ordinate and expressed as $\mathbf{v}_{\rm LSR}=(0,v_0,0)$, $\mathbf{v}_{\rm pec}=(11.1 \pm 1.5, 12.2 \pm 2, 7.3 \pm 1)$ km/s \cite{Sch_nrich_2010}. Following Refs.\,\cite{Evans:2018bqy, Maity:2020wic}, throughout the paper we fix $v_0=233$ km/s. The uncertainties associated with $v_0$ and other astrophysical parameters have been studied in Refs. \cite{Hryczuk:2020trm, Radick:2020qip, Maity:2020wic} in the context of ER. The time-dependent Earth's velocity is represented by $\mathbf{u}_{\rm E}(t)$ which leads to the well-known annual modulation of the signal. The expression for $\mathbf{u}_{\rm E}(t)$ can be found in \cite{McCabe:2013kea}. The differential event rate given in Eq.\,\eqref{eq:rateXe} can be divided into three parts. The particle physics input is indicated by $\bar{\sigma}_{e}$ and $F_{\rm DM}$. Throughout our analysis, we will do a model-independent analysis with two choices of $F_{\rm DM}$: 1 and $1/q^2$, which appears in large classes of particle physics model \cite{ Holdom:1985ag, Borodatchenkova:2005ct, Chu:2011be, Lin:2011gj, Izaguirre:2015yja, Alexander:2016aln, Boehm:2020wbt, 10.21468/SciPostPhysLectNotes.43}. We will present the results of $F_{\rm DM}= 1$ in the main text and that of $F_{\rm DM}= 1/q^2$ in the appendix. The atomic physics part symbolized by $f_{\rm ion}^{\rm n,l}$ signify ionization probability. The numerical values of the $f_{\rm ion}^{n,l}$ is adopted from {\tt QEdark} \cite{Essig:2015cda, Essig:2017kqs, QEdark}. The local DM density and $\eta$ constitute the astrophysical inputs. The galactic DM velocity distribution is traditionally assumed to be a Maxwell-Boltzmann (MB) distribution truncated at the galactic escape velocity ($v_{\rm esc}$) \begin{equation} f_{\rm gal}^{\rm MB}(\mathbf{v})= \frac{1}{(2 \pi \sigma_v^2)^{3/2} N_{\rm esc}^{\rm MB}}\exp{\left(-\frac{|\mathbf{v}|^{2}}{2 \sigma_{v}^{2}}\right)} \Theta(v_{\rm esc}-|\mathbf{v}|) \,. \label{eq:fvSHM} \end{equation} The isotropic velocity dispersion $\sigma_{v}$ is related to $v_0$: $v_0=\sqrt{2} \sigma_v$. The normalization constant $N_{\rm esc}^{\rm MB}={\rm erf}(z)- 2\pi^{-1/2}z e^{-z^2}$ with $z=v_{\rm esc}/v_0$ and erf is the error function. Throughout the discussion the galactic escape velocity ($v_{\rm esc}$) has been fixed to $528$\,km/s \cite{Evans:2018bqy, Deason_2019}. While the MB distribution may describe the DM velocity distribution which is in equilibrium (hydrodynamical simulations indicate that MB distributions may not adequately describe the velocity distribution of the smooth DM halo component), the equilibration condition will not be met for relatively recent mergers of the MW with other galaxies. These recent mergers will have unique signatures, both in velocity and position space, called substructures. The existence of these substructures is also observed in various N-body simulations. When a galaxy accretes into the Milky Way, the stellar component of the accreted galaxy carries several tell-tale signatures: stellar streams, stellar shards, and stellar debris flow\,\cite{Ibata:1994fv, Helmi:1999ks, Ibata:2000ys, Belokurov:2006kc, Lisanti:2011as, Myeong:2017skt, myeong2018shards, Necib:2018iwb, Necib:2019zka, Yuan_2020, 2022arXiv220102405R, 2022arXiv220102404S, 2022arXiv220611248D}. The recent results of various surveys like Gaia, SDSS, and LAMOST indeed indicate the presence of these stellar substructures. Combining the effect of the substructure with the SHM, we get total average inverse speed as \begin{equation} \eta(v_{\rm min},t)= \int_{v_{\rm min}}^{\infty} \frac{1}{v} \left[ (1-\delta) f_{\rm lab}^{\rm MB}(\mathbf{v},t) + \delta f_{\rm lab}^{\zeta_i}(\mathbf{v},t)\right] d^3v, \label{eq:etacombine} \end{equation} where $f_{\rm lab}^{\zeta_i}(\mathbf{v},t)$ refers to the substructure velocity distribution (discussed in Sec.\,\ref{sec:DMSS}) and $\delta$ represents the fractional contribution that the corresponding component constitutes to the local density of DM.\footnote{If each of the substructures contributes different fractions then instead of one $\delta$ there will be a set of such $\delta$'s. For simplicity, we have ignored the effect of multiple substructures.} In what follows, we will consider the effect of these substructures in DM velocity distribution and the ER DD rate in liquid xenon experiments. \section{DM substructures} \label{sec:DMSS} This section discusses the benchmark DM substructures that we have studied in this work. We have utilized the results of Ref. \cite{Yuan_2020} where the stellar substructure is obtained using the star catalog of LAMOST DR3 \cite{2018ApJS..238...16L}. We choose a few representative substructures to present our results. For clarity, we also mention the name of the associated dynamically tagged groups (DTG) with the relevant substructures \cite{Yuan_2020}. The details of these substructures are summarised in Table \ref{tab:subs}. We emphasize that the chosen substructures are for illustrative purposes only. Further research is required in order to understand the DM content of various substructures and whether the substructure DM profile coincides with the Solar circle. Whether the corresponding DM substructure will follow the same velocity distribution as the stellar substructure or not is currently not understood. Using Via Lactea II high-resolution N -body simulation, it has been shown that DM debris flows closely follows their stellar counterpart \cite{Lisanti:2011as, Lisanti:2014dva}. However, the same is not valid for Sagittarius stream \cite{Purcell_2012}. Nevertheless, we will assume that the velocity distributions of the substructures follow that of the corresponding stellar components. This assumption can be confirmed or refuted by future research. However, the broad conclusion of this study will hold. We note that the substructures we have considered in this paper have similarities with previous considerations \cite{OHare:2019qxc, Buch:2020xyt}. For instance, the Helmi substructure is analogous to S2-substructure \cite{Helmi_2020}. The velocity properties of the Nyx substructure are somewhat similar to the prograde (Pg) stream and are expected to arise from the same Splashed Disk event \cite{Yuan_2020}.\footnote{Ref.\,\cite{2021ApJ...912L..30Z} has argued that Nyx is a part of thick disk.} Some of the considered substructures are also found in Gaia DR3 data at the Solar neighborhood \cite{2022arXiv220102404S, 2022arXiv220102405R, Ou:2022wvr}. \begin{table}[] \centering \begin{tabular}{|c|c|c|c|c|c|c|} \hline \multicolumn{1}{|l|}{\multirow{2}{*}{Substructure}} & \multicolumn{3}{c|}{Mean velocity (km/s)} & \multicolumn{3}{c|}{Velocity dispersion (km/s)} \\ \cline{2-7} \multicolumn{1}{|l|}{} & $\mu_R$ & $\mu_{\phi}$ & $\mu_z$ & $\sigma_R$ & $\sigma_{\phi}$ & $\sigma_z$ \\ \hline HelmiDTG1 & 4.5 & 197.2 & 244.3 & 146.0 & 62.6 & 42.4 \\ \hline HelmiDTG3 & 26.2 & 157.1 & -241.3 & 78.9 & 28.8 & 27.2 \\ \hline PolarDTG11 & -47.9 & 21.8 & 229.2 & 75.4 & 19.2 & 21.5 \\ \hline PgDTG2 & 221.2 & 155.7 & 139.7 & 26.2 & 33.8 & 52.3 \\ \hline Sausage & 2.1 & -0.3 & -8.7 & 136.6 & 35.0 & 72.3 \\ \hline RgDTG28 & -4.0 & -106.1 & -143.2 & 115.8 & 29.3 & 30.3 \\ \hline Sequoia & -36.9 & -273.9 & -87.0 & 138.2 & 36.7 & 65.0 \\ \hline \end{tabular}% \caption{The details of the substructures are used in this paper. The numerical values of the mean velocities and diagonal values of the velocity dispersions are adapted from tables 2 and 3 of \cite{Yuan_2020}. The DTG from which substructures are identified has also been specified.} \label{tab:subs} \end{table} The mean stellar velocities and the diagonal values of the stellar velocity dispersions are given in Table \ref{tab:subs}. In general, DM substructures will have a different velocity distribution than the virialized component (SHM), which will dramatically impact the ER distribution. The galactic velocity distribution for each of the substructures (referred to by $\zeta_i$) can be written as \cite{OHare:2019qxc, Buch:2020xyt} \begin{equation} f_{\rm gal}^{\zeta_i}(\mathbf{v})=\frac{1}{(8\pi^3 \, {\rm det}\, \sigma^{\zeta_i})^{1/2} N_{\rm esc}^{\zeta_i} } {\rm exp} \left(-(\mathbf{v}-\boldsymbol{\mu}^{\zeta_i})^T\frac{1}{2 (\sigma^{\zeta_i})^2 } (\mathbf{v}-\boldsymbol{\mu}^{\zeta_i})\right) \Theta(v_{\rm esc}-|\mathbf{v}|), \label{eq:fvsubs} \end{equation} where $\sigma^{\zeta_i}$ is the velocity dispersion matrix, assumed to be diagonal with the values given in Table \ref{tab:subs} and ${\rm det}\, \sigma^{\zeta_i}$ is the determinant of the the dispersion matrix. The mean velocities of the substructures in the galactic frame are expressed by $\boldsymbol{\mu}^{\zeta_i}$ which are non-zero in contrast to the SHM case, as indicated in Table \ref{tab:subs}. The normalization constant $N_{\rm esc}^{\zeta_i}$ is calculated numerically. The step function represents the cut-off at the galactic escape velocity, although the substructures' velocity distributions are likely to peak at smaller velocities. Therefore this cut-off will have a numerically insignificant effect. The index $\zeta_i$ refers only to the substructure, whereas $i$ includes both the substructures and SHM. Assuming Eq.\,\eqref{eq:fvsubs} as the galactic velocity distributions for the DM substructures, we display the corresponding lab frame speed distributions, $f_{\rm lab}^i(v) = v^2 \int d \Omega f_{\rm lab}^i(\mathbf{v})$, using Eq.\,\eqref{eq:galtolab} in Fig.\,\ref{fig:VelDist}. Expect for the modulation signature (discussed in Sec.\,\ref{subsec:res}), we fix the Earth's velocity to $\mathbf{v_E}\,=\,(39.7, 243.2, 16.4)$\,km/s. The general trend we observe is that the substructures which peak at larger values of $v$ have negative $\mu_{\phi}$. Since the Earth moves with high positive rotational velocity $\sim 250$ km/s, substructures with negative $\mu_{\phi}, $ will hit the Solar system with larger velocities. On the other hand, substructures having large positive $\mu_{\phi}$ co-rotate with the Earth, leading to $f_{\rm lab}^i(v)$ peaking at smaller velocities. This has been displayed in Fig.\,\ref{fig:VelDist}, where the Helmi streams having larger values of $\mu_{\phi}$ peak at relatively smaller velocities, whereas Sequoia having a negative $\mu_{\phi}$ peaks at the higher velocity. We also display the velocity distribution of SHM by the solid black line. For reference we show the required $v_{\rm min} = 428.7$ km/s to obtain a recoil of $20$\,eV with momentum transfer $25$\,keV and $5p^6$ shell for DM mass $100$\,MeV by the vertical black dashed line. Given these velocity distributions, we turn to the discussion of the mean inverse speed $\eta^i(v_{\rm min})$ (using Eq.\,\eqref{eq:eta}) of each of the astrophysical components. The values of $\eta^i(v_{\rm min})$ as a function of $v_{\rm min}$ are depicted in Fig.\,\ref{fig:eta}. Expectedly, $\eta^i(v_{\rm min})$ are monotonically decreasing function of $v_{\rm min}$, which can be understood from the integration over velocity starting from $v_{\rm min}$. The maximum values of $\eta^i(v_{\rm min})$, i.e., $\eta^i(0)$ is larger for the distributions which peak at lower velocities because the mean inverse speed is inversely proportional to the most probable speed (the speed at which velocity distribution attains maximum value) of the distribution. Hence in Fig.\,\ref{fig:eta}, we observe maximum and minimum $\eta^i(0)$ for HelmiDTG3 and Sequoia respectively. For the other distributions, $\eta^i(0)$ lie within the same of HelmiDTG3 and Sequoia. The flatness of $\eta^i(v_{\rm min})$ for Sequoia up to a large value of $v_{\rm min}$ as compared to other distributions is also a manifestation of the higher most probable speed of Sequoia. This indicates the extent to which $v_{\rm min}$ is supported by the distribution. It should also be noted that the flatness of $\eta^i(v_{\rm min})$ is also sensitive to the choice of the velocity dispersion. \section{DM-electron scattering at xenon: effect of substructure} \label{sec:DMeSS} In this section, we discuss the effect of the substructures on the DM-electron scattering rate for liquid xenon experiments. For $F_{\rm DM}(q)=1$, the constraint on the DM-electron scattering cross-section from the xenon detectors dominate when DM mass is $\gtrsim$ 50 MeV. Xenon experiments may have a better prospect of discovering DM-electron scattering, and it is essential that we study this prospect thoroughly. Our work outlines the theory effort toward answering this important question. Following Ref.\,\cite{Essig:2017kqs}, we convert the ER energy ($E_e$) to number of electrons ($n_e$). DM-electron scattering would produce $n_e$ number of observable electrons, unobservable photons, and heat. Some primary electrons would recombine with secondary ions with probability $f_R$. Further, each recoiling electron of energy $E_e$ will give rise to additional secondary $n_e^{(1)}={\rm Floor}[E_e/W]$ quanta (photon or electron). The average energy required to create a single quanta is $W$. Moreover, the scattering process can also lead to the ionization of electrons from the inner shell, which would de-excite by releasing a photon. These photons may also create secondary quanta, $n_e^{(2)}={\rm Floor}[\Delta E_{i,j}/W]$, $\Delta E_{i,j}$ is the difference between binding energies between the relevant inner and outer shells. The number of secondary electrons produced is calculated using a binomial distribution with $n_e^{(1)}+n_e^{(2)}$ trials, having success probability $f_e$. We have chosen fiducial values (i.e., $W=13.6$\,eV, $f_e=0.83$, $f_R=0$) of the relevant parameters to convolute Eq.\,\eqref{eq:rateXe} which will give the differential event rate as a function of number of produced electrons. Our paper does not consider uncertainties associated with $W,\,f_e,$ and $f_R$. In Fig.\,\ref{fig:EventRate}, we show the differential event rate as a function of $n_e$ for $m_{\chi}=100$\,MeV, $\bar{\sigma}_e=10^{-41} \, {\rm cm}^2$, and 1 kg-year exposure. For each event rate, we have assumed that the corresponding astrophysical component (SHM or substructures) constitutes $100\%$ of the local DM density. For $m_{\chi}=100$\,MeV with typical momentum transfer of $\mathcal{O}(10)$\,keV, to obtain a measurable recoil the required minimum DM velocity should be around $500$\,km/s. Hence, the tail of $\eta^i(v_{\rm min})$ dominantly contributes to the recoil rate. Evidently the substructures having the largest value of $\eta^i(v_{\rm min})$ near $v_{\rm min}\sim 500$\,km/s give rise to a larger event rate. \subsection{Neutrino background} \label{subsec:nubag} The scattering of neutrinos with electron/\,nucleon may also give rise to ionization signals in low-threshold DD experiments. Other background sources like radioactive background, Cherenkov radiation, etc.\,which can potentially mimic a DM signal \cite{Du:2020ldo}. The experimental collaborations confront and beat these non-neutrino backgrounds using various experimental techniques to isolate a potential DM signal. However, the neutrinos are an irreducible background that can not be removed by using shielding, purified detector material, and other experimental techniques. Because of this, we have taken neutrinos as the only source of background in our analysis. If other non-neutrino backgrounds are found in the data-set, then our results will degrade proportionally. It has been argued in Refs.\,\cite{Essig:2018tss, Wyenberg:2018eyv} that Solar neutrinos are the main source of background for sub-GeV DM-electron scattering.\footnote{See Refs. \cite{Essig:2018tss, Schwemberger:2022fjl} or discussion related to the prospect of these detectors in probing beyond SM interactions of neutrino.} Neutrino-electron elastic scattering is the dominant contribution of background events for rather large recoil energy ($\sim 10^5$\,eV). Instead, coherent neutrino-nucleon scattering may produce small ionization, which would be the dominant source of background in our consideration. The neutrino-nucleon scattering event rate is \cite{Billard:2013qya, Essig:2018tss} \begin{equation} \frac{dR}{dE_{\rm NR}}=N_T M T \int_{E_{\nu}^{\rm min}} \frac{d\sigma}{dE_{\rm NR}} \frac{d\phi_{\nu}}{dE_{\nu}} dE_{\nu}, \label{eq:nurate} \end{equation} where $N_T,\, M$, and $T$ are the number of target nuclei per unit mass, total mass, and time respectively. The minimum neutrino energy to produce a nuclear recoil of energy $E_{\rm NR}$ is expressed by $E_{\nu}^{\rm min}=\sqrt{m_N E_R/2}$. The differential coherent neutrino nucleon cross section and the differential neutrino flux are denoted by $d\sigma/dE_{\rm NR}$ and $d\phi_{\nu}/dE_{\nu}$ respectively \cite{Essig:2018tss,OHare:2016pjy}. We have utilized low, fiducial, and high ionization models given in Ref.\,\cite{Essig:2018tss} to obtain number of electron $n_e$ for a particular nuclear recoil energy. The corresponding neutrino-induced event rate for fiducial model is displayed in Fig.\,\ref{fig:EventRate} by the grey dashed lines.\footnote{We note that there is a factor $\sim 3$ difference in the event rate between our result and Ref.\,\cite{Essig:2018tss}.} The grey shaded regions represent variation in the event rate for high and low ionization models of $n_e$\,\cite{Essig:2018tss}. Since there is a difference between three ionization models in the low $n_e$/energy bins, hence we observe a large change in the differential event rates at those bins. The discovery limits for low and high ionization models is given in appendix\,\ref{app:neUn}. For one electron threshold, the impact of the ionization model uncertainty leads to less than a factor of $3$ change in the discovery limits. \subsection{Statistical methodology} \label{subsec:stameth} In this section, we discuss the statistical procedure to obtain the discovery limit for DM-electron scattering in the presence of substructures for liquid xenon experiments. We have employed the profile likelihood ratio test \cite{Cowan:2010js} with $\bar{\sigma}_{e}$ and substructure fraction ($\delta$) as the signal parameters of interest. In the following, we briefly discuss this procedure. The binned likelihood for the background and signal model ($\mathcal{M}$), is given by \begin{equation} \mathcal{L}(m_{\chi},\bar{\sigma}_e,\delta,\Phi|\mathcal{M})=\prod_{i=1}^{N_{\rm bins}}\left( \mathcal{P}(N_{\rm obs}^i|N_{\chi}^i+\sum_{j=1}^{n_{\nu}}n_{\nu}^i(\Phi^j)) \right)\prod_{j=1}^{n_{\nu}}\mathcal{G}(\Phi^j) \label{eq:llhood} \end{equation} Here the $N_{\rm bins}$ are the number of energy bins. The Poisson probability ($\mathcal{P}$) at the $i$-th bin is calculated using observed $N_{\rm obs}^i$ and the expected number of events. The expected number of events is the addition of DM events ($N_{\chi}^i$) and the sum of neutrino events ($n_{\nu}^i$) for all the neutrino components ($n_{\nu}$). The Gaussian function ($\mathcal{G}(\Phi^j)$) takes care of the uncertainty in the neutrino fluxes ($\Phi^j$) with mean values and standard deviation given in \cite{Essig:2018tss, OHare:2016pjy}. Depending on the choice of the analysis, we vary one of the signal parameters (either $\bar{\sigma}_{e}$ or $\delta$), treating the other one as a nuisance parameter. We treat $\bar{\sigma}_{e}$ as the signal parameter for the discovery reach. Therefore, the profile likelihood ratio test statistic, which compares the background-only hypothesis ($\mathcal{M}_0$) with the background and signal model ($\mathcal{M}$), is given by \cite{Cowan:2010js, OHare:2020lva, Buch:2020xyt} \begin{equation} q_0 = -2 \, {\rm ln}\left(\frac{\mathcal{L}(\bar{\sigma}_{e}=0,\boldsymbol{\lambda}|\mathcal{M}_0)}{\mathcal{L}\left(\bar{\sigma}_{e},\boldsymbol{\lambda}|\mathcal{M} \right)} \right) \sim \chi_1^2, \label{eq:q0Dis} \end{equation} where $\boldsymbol{\lambda}$ contains the nuisance parameters, i.e., $\delta$ and $\Phi^j$ in this case. The ratio in Eq.\,\eqref{eq:q0Dis} follows a $\chi^2$ distribution with one degree of freedom \cite{Cowan:2010js}. Thus, the significance of rejecting the background-only hypothesis is given by $\sqrt{q_0}$-$\sigma$. In this paper, we present all the discovery limits at the $90\%$ confidence level (CL). We consider $\delta$ as the signal parameter for the prospective detection of DM substructure fraction. The corresponding profile likelihood ratio test to distinguish two neighboring points $\delta_1$ and $\delta_2$ can be written as \cite{Buch:2020xyt} \begin{equation} q_0 = -2\, {\rm ln}\left(\frac{\mathcal{L}(\delta_2,\boldsymbol{\lambda}|\mathcal{M}_{\delta_1})}{\mathcal{L}\left(\delta_2,\boldsymbol{\lambda}|\mathcal{M}_{\delta_2} \right)} \right) \sim \chi_1^2. \label{eq:q0focast} \end{equation} This profile likelihood ratio is employed to reject the null hypothesis, which is that two neighboring points $\delta_1$ and $\delta_2$ are indistinguishable at $68\%$ CL. Both for Eqns.\,\eqref{eq:q0Dis} and \eqref{eq:q0focast} we utilized Asimov data set \cite{Cowan:2010js} to obtain the likelihood ratio test. In this scenario, artificial data is generated using the model's parameters (in our case $\mathcal{M}$). Then the expectation is that the number of observed events ($N_{\rm obs}$) should be equal to the number of the expected event ($N_{\rm exp}$). For a sufficiently large number of observations, the value of the profile likelihood ratio test approaches the median value. Compared to the Monte Carlo simulation, the Asimov data set scenario is computationally more economical while acquiring accurate results. For the $68\%$ and $90\%$ CL limit the required $q_0$'s are $0.99$ and $2.71$ respectively. For a fixed $m_{\chi}$ and $\delta$, the $90\%$ CL discovery limit is obtained by changing $\bar{\sigma}_{e}$ in Eq.\,\eqref{eq:q0Dis} until the required $q_0$ ($2.71$) is achieved. The $68\%$ CL contours in resolving substructure fraction are estimated using Eq.\,\eqref{eq:q0focast}. In this case for a fixed values of $m_{\chi}$, $\sigma_e$, and $\delta_1$, we iterate over $\delta_2$ until the required $q_0$ (= $0.99$) is attained. \subsection{Results} \label{subsec:res} Here we will present the results using the statistical analysis discussed in the previous subsection. The three parameters of interest are DM mass ($m_\chi$), DM-electron cross section ($\bar{\sigma}_e$), and the DM substructure fraction ($\delta$). Given that DM has to be massive, we present our results through two possible choices, keeping one of the other two parameters to a fixed value. In the first part, the results are presented through the discovery limit, which is depicted in DM mass and DM-electron cross-section plane keeping a fixed DM substructure fraction. In the other case, considering a fixed DM-electron cross-section, we present the forecast of the xenon experiments to resolve the substructure fraction for a few benchmark choices of DM particle masses. In Fig.\,\ref{fig:Exclusion}, we present the sensitivity to DM-electron cross-sections for each of the substructures considered in this paper, assuming that the corresponding substructure constitutes $100\%$ of the local DM density. In Fig.\,\ref{fig:Exclusion}, each line represents the minimum DM-electron cross section required to observe the effect of the corresponding substructure in a liquid xenon detector with $1$ kg-year exposure and one electron threshold. The discovery limits for two and three electron thresholds are given in appendix\,\ref{app:ne2and3}. The different discovery limits for different substructures are the implication of non-identical most probable speed. The tail of the DM velocity distribution will be more populous for the substructure having a relatively larger most probable speed. Therefore a sizable number of DM particles will be available to interact with the target electrons. This leads to a larger event rate, as has been depicted in Fig.\,\ref{fig:EventRate}, where for a fixed DM-electron cross section among the considered DM substructures, we obtain the minimum and the maximum number of events for HelmiDTG3 (lowest most probable speed, see Fig.\,\ref{sf:lowfv}) and Sequoia (highest most probable speed, see Fig.\,\ref{sf:highfv}) respectively. Owing to this, the DM-electron cross-section that can be probed for HelmiDTG3 is the largest, whereas the same for Sequoia is the lowest. The event rates and subsequently the discovery limits lie between HelmiDTG3 and Sequoia for the other considered substructures. The light grey shaded region demonstrates the constraint from the ionization signals in the XENON1T experiment \cite{XENON:2019gfn}, which is the most stringent current DD constraint for the parameter space shown in the plot. For reference, we have also shown the discovery limit for the SHM with the solid black line. In reality, these substructures would not contribute $100\%$ to the local DM density. Therefore, we choose two benchmark values of $\delta$, namely $\delta=0.1$ and $\delta=0.2$ (shown in Fig.\,\ref{fig:Exclusiondel}). Further, as mentioned above, we have only considered two substructures, HelmiDTG3 and Sequoia, which lie at two extreme ends. SHM constitutes the rest of the local DM density for the combined DM distribution. If the discovery limit for a particular substructure (with $\delta=1$) is larger compared to SHM, then the same for the combined DM distribution will lie above the SHM limit. This effect would be more pronounced upon increasing $\delta$. In Fig.\,\ref{fig:Exclusiondel}, the combined discovery limit for HelmiDTG3 and Sequoia is displayed by brown and purple lines, respectively. Notably, brown and purple lines lie above and below the SHM scenario. Upon increasing the $\delta$, we observe more deviation from SHM. Importantly, it is still possible to see the effect of these substructures in liquid xenon experiments with this kind of realistic choice of $\delta$. Next, we turn into the discussion of resolving substructure fractions in liquid xenon experiments. Again, we have restricted ourselves to HelmiDTG3 and Sequoia among the considered substructures as these two reside in the extreme ends. The sensitivity in resolving DM substructure at $68\%$ CL is displayed in Fig.\,\ref{fig:contour} for 1 kg-year exposure, one electron threshold, and $\bar{\sigma}_e=10^{-40}\, {\rm cm}^2$, with a few benchmark points. Generically, we observe a better resolution for low DM mass. Comparing Figs.\,\ref{sf:Helmicontour} and \ref{sf:Sequoiacontour} one can see that we will determine the substructure fraction more accurately for Sequoia compared to HelmiDTG3. This is due to Sequoia's large most probable velocity, which leads to a substantial number of DM-electron scattering events. Generically, it is possible to measure the substructure fraction more accurately, which is moving with a higher most probable speed. For $\delta=0.1$, with the considered exposure, threshold, and $\bar{\sigma}_e$, it is difficult to conclude whether the substructure is contributing to local DM density. Interestingly, for DM mass $\sim 50$ MeV, and $\delta = 0.4$, xenon target electron scattering experiments can resolve the substructure fraction with $\sim 50\%$ accuracy. Moreover, the structures of the contours can be understood from Eq.\,\eqref{eq:vmin} and from Fig.\,\ref{fig:eta}. Both for the lower and higher DM masses, the inclination of the contours is reversed as we compare HelmiDTG3 with Sequoia. For low DM masses, $v_{\rm min}$ is larger (for fixed $q$ and $E_e$ from Eq.\,\eqref{eq:vmin}), therefore it is the tail of the distribution which is contributing to $\eta^i$. Thus for HelmiDTG3 with low mass DM, an increment in $\delta$ will reduce the combined value of $\eta$. This reduction could be compensated by increasing DM mass for a fixed number of observed events. This results in a slightly tilted contour towards a higher DM mass. Whereas for higher DM mass (thus smaller $v_{\rm min}$), the maximum value of $\eta$ determines the orientation of contours. For Sequoia, the maximum value of $\eta^i$ is less than that of HelmiDTG3. Hence increasing $\delta$ for the former will reduce combined $\eta$, which can be elevated by reducing $v_{\rm min}$, i.e., by increasing DM mass. We have not discussed a distinctive feature of the DM DD signal: annual modulation\,\cite{Lee:2015qva}, where the signal event rates vary with the time of the year in a specified manner. Due to the rotation of the Sun around the MW, there will be DM wind in the Solar rest frame. Due to the Earth's rotational motion around the Sun, the event rate will vary with time. The event rate will be larger (smaller) when the Sun and the Earth travel in opposite (same) directions, respectively. Due to this distinctive feature, which the background cannot mimic, annual modulation events are expected to be less dependent on the background reductions and identifications. The main task for the modulation discovery limit would be to evaluate the event rate against both time and energy. The corresponding likelihood can be obtained by taking the difference of their individual Poisson distributions, referred to as the Skellam distribution\,\cite{Skellam1946TheFD}. Subsequently, one can estimate the discovery limit using the test statistic defined in Eq.\,\eqref{eq:q0Dis}. Following the prescription in \cite{Buch:2020xyt}, we find that the modulation discovery limit is weaker than the non-modulation counterpart. For example, with SHM or Sequoia, we observed that the modulation discovery reaches are weaker by a factor $\sim 10-100.$ \section{Conclusions} \label{sec:conclusion} The presence of DM in the Universe is well established. Many attempts have been made to discover the connection between DM and SM states. Among them, DD experiments look for the scattering signatures of DM and visible states. There has been a growing interest in the search for light DM (masses $\lesssim 1$\,GeV) through DD. Ambient non-relativistic DM having mass in the sub-GeV range can not impart sufficient energy to produce a measurable recoil in the typical nuclear recoil DD experiments. Electron, being a light particle, can be an excellent target in detecting such light DM. Many target materials have been considered to identify electronic excitation by the scattering of ambient DM. DM velocity distribution is an integral part of calculating the event rate or the exclusion limit of the DD experiments. DM is also an intrinsic part of structure formation; the history of galaxy formation influences its velocity distribution. While it is difficult to track the velocity distribution of DM, however, it may be manifested through stellar distribution. Surveys like Gaia, SDSS, LAMOST, etc., have made unprecedented progress mapping these stellar distributions. These data reveal the presence of stellar clumps and substructures. It is highly likely that there is a DM counterpart to these stellar substructures, called DM substructure. This paper investigates the prospects of detecting these substructures in low threshold DM DD experiments through elastic DM-electron scattering. Specifically, we have explored the prospect of xenon targets experiments in deciphering this. Note that compared to semiconductor targets experiments (like SENSEI), the xenon targets experiments have better sensitivity in the DM mass range of $\mathcal{O}(100)\,{\rm MeV}$. We utilize the results of the LAMOST survey and choose a few benchmark DM substructures. We emphasize that there is no definite proof of the existence of the DM counterpart to the detected stellar substructures. However, it is likely that they exist. If these DM substructures overlap with the Earth's position, then we can observe the imprint of the same in xenon targets experiments through DM-electron scattering. We find that if the substructure constitutes $\gtrsim 10\%$ of the local DM density, then there is a possibility to observe the effect of the substructures in xenon target experiments with the currently allowed DM particle properties. We have also explored the forecast of xenon experiments in resolving the DM substructure fraction. We find that the uncertainty in resolving DM substructure fraction is considerable for higher DM mass compared to lower DM mass. For example, with $m_{\chi}=50\,$MeV, $\bar{\sigma}_{e}=10^{-40}{\,\rm cm}^2$, and one electron threshold in xenon experiments, we can resolve the substructure fraction to $\sim 50\%$ accuracy provided $\delta \sim 0.4.$ The discovery limit and resolving DM substructure fraction are mainly regulated by the most probable velocity of the corresponding velocity distribution. Given this correlation between DD rates and DM velocity distributions, a more detailed understanding of DM substructure is required. High-resolution cosmological simulations and near-future observations will play a crucial role in understanding this. We encourage the experimentalists to continue their excellent work in improving their detector sensitivity so that we are sensitive to such a signal. Our work shows that by pursuing this technique, we will be able to know more about the particle physics and astrophysics of DM and maybe even discover it. \paragraph*{Acknowledgements\,:} We thank Jatan Buch, Ciaran A.\,J.\,O’Hare, Mukul Sholapurkar, and Tien-Tien Yu for useful correspondence. We thank John F.\,Beacom, Ciaran A.\,J.\,O’Hare, and Tien-Tien Yu for comments on the manuscript. TNM thanks IOE-IISc fellowship program for financial assistance. RL acknowledges financial support from the Infosys foundation (Bangalore), institute start-up funds, and Department of Science and Technology (Govt. of India) for the grant SRG/2022/001125. \appendix \section{Discovery limits for two and three electron threshold} \label{app:ne2and3} Throughout the main text, we have considered the reach of the xenon experiments for one kg-year exposure and one electron threshold with $F_{\rm DM}=1$. Here we present the discovery limit with two and three electron thresholds for $\delta=0.1$. The results are depicted in Figs.\,\ref{sf:ne20p1} and \ref{sf:ne30p1}. With higher thresholds, the expected event numbers decrease; thus, the required cross-section to see the possible effect of the substructure increases. Further, the lowest possible DM mass that can be probed also increases. \section{Variation in the discovery limits} \label{app:neUn} As discussed in Sec.\,\ref{subsec:nubag}, background event rate from neutrino may change depending on the ionization model. In this appendix, we present the discovery limit for high and low ionization efficiencies models for $n_e$\,\cite{Essig:2018tss}. We display the result in Fig.\,\ref{fig:neUn}. For each of the substructures, solid lines represent discovery limits for fiducial ionization model and shaded bands show the corresponding uncertainties associated with the ionization models. \section{Momentum dependent DM-electron scattering} \label{app:FDMqm2} In this appendix, we present the discovery limits of the momentum-dependent DM-electron scattering, namely $F_{\rm DM}=\alpha^2 m_e^2/q^2$ for the considered DM substructures. In this case, we also observe a similar tendency, except that the minimum required DM-electron cross section for the discovery of the substructures is larger than the same of $F_{\rm DM}=1$. This is displayed in Fig.\,\ref{fig:FDmqm2}. \bibliographystyle{JHEP} \bibliography{ref.bib}

Title: The structure of the ultrarelativistic prompt emission phase and the properties of the black hole in GRB 180720B

Abstract: In analogy with GRB 190114C, we here analyze the ultrarelativistic prompt emission (UPE) of GRB 180720B observed in the rest-frame time interval $t_{\rm rf}=4.84$--$10.89$~s by Fermi-GBM. We reveal the UPE hierarchical structure from the time-resolved spectral analysis performed in time sub-intervals: the spectrum in each shorter time interval is always fitted by a composite blackbody plus cutoff power-law model. We explain this structure with the \textit{inner engine} of binary-driven hypernova (BdHN) model operating in a quantum electrodynamics (QED) regime. In this regime, the electric field induced by the gravitomagnetic interaction of the newborn Kerr BH with the surrounding magnetic field is overcritical, i.e., $|{\bf E}|\geq E_c$, where $E_c=m_e^2 c^3/(e\hbar)$. The overcritical field polarizes the vacuum leading to an $e^+~e^-$ pair plasma that loads baryons from the surroundings during its expansion. We calculate the dynamics of the self-acceleration of the pair-electromagnetic-baryon (PEMB) pulses to their point of transparency. We characterize the quantum vacuum polarization process in the sequences of decreasing time bins of the UPE by determining the radiation timescale, Lorentz factors, and transparency radius of the PEMB pulses. We also estimate the strength of the surrounding magnetic field $\sim 10^{14}$ G, and obtain a lower limit to the BH mass, $M=2.4~M_\odot$, and correspondingly an upper limit to the spin, $\alpha = 0.6$, from the conditions that the UPE is powered by the Kerr BH extractable energy and its mass is bound from below by the NS critical mass.

https://export.arxiv.org/pdf/2208.14177

\sloppy \title{The structure of the ultrarelativistic prompt emission phase and the properties of the black hole in GRB 180720B}% \author{F.~Rastegarnia\thanksref{e1,addr1,addr2,addr3,addr4} \and R.~Moradi\thanksref{e2,addr1,addr2,addr5} \and J.~A.~Rueda\thanksref{e3,addr1,addr2,addr3,addr4,addr6} \and R.~Ruffini\thanksref{e4,addr1,addr2,addr7} \and Liang~Li\thanksref{e5,addr1,addr2,addr7} \and S.~Eslamzadeh\thanksref{addr1,addr2,addr3,addr4} \and Y.~Wang\thanksref{e7,addr1,addr2,addr5} \and S.~S.~Xue\thanksref{addr1,addr2} } \thankstext{e1}{e-mail: fatemeh.rastegarnia@edu.unife.it} \thankstext{e2}{e-mail: rahim.moradi@inaf.it} \thankstext{e3}{e-mail: jorge.rueda@icra.it} \thankstext{e4}{e-mail: ruffini@icra.it} \thankstext{e5}{e-mail: liang.li@icranet.org} \thankstext{e7}{e-mail: yu.wang@uniroma1.it} \institute{ICRA, Dipartimento di Fisica, Sapienza Universit\`a di Roma, Piazzale Aldo Moro 5, I--00185 Roma, Italy\label{addr1} \and International Center for Relativistic Astrophysics Network, Piazza della Repubblica 10, I--65122 Pescara, Italy\label{addr2} \and ICRANet-Ferrara, Dipartimento di Fisica e Scienze della Terra, Universit\`a degli Studi di Ferrara, Via Saragat 1, I--44122 Ferrara, Italy\label{addr3} \and Dipartimento di Fisica e Scienze della Terra, Universit\`a degli Studi di Ferrara, Via Saragat 1, I--44122 Ferrara, Italy\label{addr4} \and INAF -- Osservatorio Astronomico d'Abruzzo,Via M. Maggini snc, I-64100, Teramo, Italy\label{addr5} \and INAF, Istituto di Astrofisica e Planetologia Spaziali, Via Fosso del Cavaliere 100, 00133 Rome, Italy\label{addr6} \and INAF, Viale del Parco Mellini 84, 00136 Rome, Italy\label{addr7} } \date{Received: date / Accepted: date} \section{Introduction}\label{sec:1} \textcolor{black}{The binary-driven hypernova (BdHN) model has been proposed for the description of long-duration gamma-ray bursts (GRBs), following the induced gravitational collapse (IGC) scenario \cite{2012ApJ...758L...7R}. The progenitor is a binary system composed of a carbon-oxygen core (CO$_{\rm core}$) and a neutron star (NS) companion. These CO$_{\rm core}$-NS binaries are expected to form in the late stages of the binary evolution of massive binaries, e.g., $\sim 15+12 M_\odot$ zero-age main-sequence (ZAMS) stars \cite{2020ApJ...893..148R}. The more massive star undergoes core-collapse supernova (SN) and creates a NS when its thermonuclear evolution is over. After multiple common-envelope phases and binary interactions during the X-ray binary phase of the system (see, \cite{2014ApJ...793L..36F, 2015PhRvL.115w1102F}, and references therein), the hydrogen and helium envelopes of the less massive main-sequence star are stripped, leading to the formation of the CO$_{\rm core}$ (see \cite{2015PhRvL.115w1102F}, and Fig.~1 and section 2 in \cite{2020ApJ...893..148R}). The system at this point is a CO$_{\rm core}$-NS binary in tight orbit, which is taken as the initial configuration of the BdHN model for long GRBs \cite{2014ApJ...793L..36F, 2015ApJ...812..100B,2016ApJ...833..107B,2019ApJ...871...14B,2020ApJ...893..148R}.} In the last decade, theoretical progress in the analysis of BdHNe, including three-dimensional numerical simulations, has identified their sequence of physical events \cite{2012ApJ...758L...7R, 2012A&A...548L...5I, 2014ApJ...793L..36F, 2015PhRvL.115w1102F, 2015ApJ...812..100B, 2016ApJ...833..107B, 2019ApJ...871...14B}. The gravitational core-collapse event of the CO$_{\rm core}$ forms a newborn NS ($\nu$NS) at its center and powers a \textcolor{black}{type Ic} SN explosion. The SN ejecta is partially accreted by the $\nu$NS because of matter fallback and partially by the NS companion at hypercritical (i.e., highly super-Eddington) rates due to a powerful neutrino emission process occurring on top the NS surface \cite{2016ApJ...833..107B, 2018ApJ...852..120B}. \textcolor{black}{The fallback accretion onto the $\nu$NS contribute to the energy of prompt emission and spins up the $\nu$NS (Becerra et al., submitted; see also \cite{2019ApJ...871...14B} and Yu et al., in press). The $\nu$NS rotational energy powers the synchrotron emission leading to the afterglow \cite{2012ApJ...758L...7R, 2014ApJ...793L..36F, 2019ApJ...871...14B, 2021MNRAS.504.5301R, 2021IJMPD..3030007R}.} For orbital periods of a few minutes, the NS companion reaches the critical mass for gravitational collapse leading to the formation of a rotating (Kerr) black hole (BH). These systems have been called BdHN I, \textcolor{black}{which explain the subclass of energetic long GRBs with $E_{\rm iso} \gtrsim 10^{52}$ erg. Up to now, 380 long GRBs have been interpreted as BdHNe I \cite{2021MNRAS.504.5301R}}. \textcolor{black}{Therefore, three pillars of BdHN I, responsible for different episodes of this subclass of long GRBs are: 1) SN, 2) BH, and 3) $\nu$NS. The interplay between these three components leads to different episodes of BdHN I.} For longer orbital periods, the NS companion does not reach the critical mass and hold stable as a more massive, fast rotating NS. These systems have been called BdHN II, \textcolor{black}{which explain the subclass of energetic long GRBs with $E_{\rm iso} \lesssim 10^{52}$ erg}. \textcolor{black}{The emergence of the optical SN naturally expected in the BdHN scenario (see e.g., \cite{2019GCN.25657....1P, 2019GCN.23715....1R, 2019GCN.25677....1D, 2013GCN.14526....1R}) completes the BdHN approach.} The experimental verification of the entire sequence of Episodes in a BdHN I has been recently achieved in GRB 190114C \citep{2021PhRvD.104f3043M} \textcolor{black}{and GRB 180720B \citep{2021arXiv210309158M} }. Thanks to the high quality of the data and the brightness of \textcolor{black}{these sources}, we have identified through a detailed time-resolved analysis the following Episodes of a BdHN I \cite{2021arXiv210309158M, 2021PhRvD.104f3043M}: the emission from the $\nu$NS (the $\nu$NS-rise); the BH formation, \textcolor{black}{known as BH-rise,} originating the ultrarelativistic prompt emission (UPE) phase; the formation of the \textit{cavity} around the newborn BH, \textcolor{black}{formed in the gravitational collapse of the companion NS to the BH, and further depleted by the UPE phase} \cite{2019ApJ...883..191R}; \textcolor{black}{the soft and hard X-ray flares (SXF and HXF) originating from the interaction of the UPE phase with the expanding SN ejecta \citep{2018ApJ...869..151R}}; the X-ray afterglow powered by the rapidly rotating $\nu$NS \cite{2018ApJ...869..101R, 2019ApJ...874...39W, 2020ApJ...893..148R}, and the gigaelectronvolt (GeV) emission from the BH following the UPE \cite{2019ApJ...886...82R, 2021A&A...649A..75M}. \textcolor{black}{We discuss in section~\ref{sec:observation} the observational identification of the above episodes in the case of GRB 180720B.} In this article, \textcolor{black}{we perform a time-resolved spectral analysis of the UPE phase of GRB 180720B and interpret it in the context of the BH formation in a BdHN I. A most relevant result of this kind of analysis has been the discovery of the hierarchical structure of the UPE phase of GRB 190114C.} The spectrum on rebinned shorter time intervals shows always a similar blackbody plus cutoff power-law (BB + CPL) model during the entire UPE phase \cite{2021PhRvD.104f3043M}. The explanation of such a hierarchical structure of the UPE phase has been found in the sequence of microphysical elementary events, in the quantum electrodynamics (QED) regime of vacuum polarization, that occurs in the formation of the BH in the inner engine of the GRB \cite{2019ApJ...886...82R}. The inner engine is the system composed of the newborn rotating BH surrounded by a uniform magnetic field, aligned with the BH rotation axis, and the low-density ($\sim 10^{-14}$ g cm$^{-3}$) matter of the SN ejecta in the cavity. The physical process, which combines the pure general relativistic effect of \textit{gravitomagnetism} and QED works as follows. The gravitomagnetic interaction of the Kerr BH with the magnetic field induces an electric field, and the structure of such an electromagnetic field has been modeled with the Papapetrou-Wald solution \cite{1974PhRvD..10.1680W}. The intensity of the induced electric field is proportional to the BH spin parameter and the magnetic field strength. \textcolor{black}{The newborn BH is not charged. The interaction of the gravitomagnetic field of the Kerr BH with the surrounding magnetic field, $B_0$, induces an electric field around the BH. This electric field is nearly radial, and despite its quadrupolar nature, decreases with distance roughly as $1/r^2$. Hence, it is possible to define an ``effective charge'', $Q_{\rm eff}$, as the proportionality constant of such a field, i.e., $E \approx Q_{\rm eff}/r^2$, where \citep{2019ApJ...886...82R, 2020EPJC...80..300R, 2021A&A...649A..75M} \begin{eqnarray}\label{eq:EFCH} Q_{\rm eff}=\frac{G}{c^3}2 B_0 J. \end{eqnarray}} \textcolor{black}{It can be shown that the Papapetrou-Wald solution, due to theorems by \citet{PhysRevLett.27.529}, \citet{Waldthesis} and \citet{DeWitt:1973uma}, produces the unique solution which, at perturbative level, represents the transformation of the Kerr BH into a charged rotating Kerr-Newman BH, with effective charge given by the above equation, i.e., Eq.~(\ref{eq:EFCH}) of the paper. It can be \textcolor{black}{indeed} checked that for relatively low values of the spin parameter $\alpha = c J/(G M^2) \lesssim 0.6$, the Papapetrou-Wald solution can be approximated by the Kerr-Newman solution \citep{2021A&A...649A..75M}. We take advantage of the above property to estimate the energy and the QED effect with the Kerr-Newman geometry for which an analytic expression for the energy of the dyadoregion has been derived in \citet{2009PhRvD..79l4002C}. In fact, the difficulty of the origin of a charged BH is overcome by the idea of the effective charge.} The inner engine in the UPE phase operates in an overcritical QED regime in which the induced electric field is larger than the critical field for vacuum polarization, i.e., $E > E_c$, where \begin{equation}\label{eq:Ec} E_c = \frac{m_e^2 c^3}{e\hbar} \approx 1.32\times 10^{16}\,\,\rm V/cm. \end{equation} The MeV radiation of the UPE and its associated hierarchical structure is explained by the inner engine in this overcritical regime, and is powered by the rotational energy of the Kerr BH. In this article, we focus on the UPE phase of GRB 180720B. The overcritical electric field \textcolor{black}{of the inner engine} generates an initially optically thick $e^+e^-$ plasma that self-accelerates under its own internal pressure while engulf ambient baryons. The first numerical simulations of the expanding optically thick pair electromagnetic-baryon plasma, called \textit{PEMB} pulse, were presented in \cite{1999A&A...350..334R}. \textcolor{black}{For instance, for a BH mass $10 M_\odot$ and effective charge to mass ratio of $\sim 0.1$, adopted for GRB 991216 \citep{2004IJMPD..13..843R}, the produced $e^+e^-$ plasma pairs lie between the radii $r_1=6 \times 10^6$ cm and $r_2=2.3 \times 10^8$ cm, with total energy of $E_{\rm tot}= 4.8 \times 10^{53}$ erg, and the total number of pairs is $N_{e^+e^-}=2 \times 10^{58}$. This leads to the pair number density of $10^{32} \lesssim \overline{n}_{e^+e^-} \lesssim 10^{37}$ cm$^{-3}$ and the optical depth of $\tau \sim \sigma_T~ \overline{n}_{e^+e^-} \times [r_2-r_1] \sim 10^{16}$--$10^{21} \gg 1$, being $\rm \sigma_T=6.6 \times 10^{-25}$~cm$^{-2}$ the Thomson cross-section. Such an optically thick PEMB pulse self-accelerates outward reaching ultrarelativistic velocities \citep{1999A&A...350..334R, RSWX2} up to Lorentz factors of $\Gamma \sim 300$ at transparency and emits MeV photons. The observation of such thermal photons signs the first evidence of the Kerr BH formation, i.e., the BH-rise. } The high Lorentz factor guarantees the avoidance of the so-called GRB compactness problem \cite{1975NYASA.262..164R,2004RvMP...76.1143P}. \textcolor{black}{In \citet{2020ApJ...893..148R} (see Section 7 therein), it has been discussed that numerical simulations of BdHN I point to the possible presence of a torus-like distribution of matter with higher density on the equatorial plane that can serve to anchor the magnetic field. The physical process leading to the UPE phase requires the presence of low-density ionized matter on the polar regions, i.e., above and below the BH. Therefore, the presence of matter with higher density on the equatorial plane, providing it is not as massive as to change the assumed spacetime geometry, does not interfere with the production of the pair plasma around the BH.} The emitted energy \textcolor{black}{in the UPE phase} is paid by the rotational energy of the BH, hence it reduces its angular momentum, and consequently the intensity of the induced electric field. This process continues until the electric field reaches the value $E_c$ and no more pairs can be created via vacuum breakdown. We here analyze all the above process occurring during the UPE phase in the case of GRB 180720B, which is another BdHN I and its data quality allows us to perform a detailed time-resolved spectral analysis analogous to the one applied successfully to GRB 190114C in \cite{2021PhRvD.104f3043M}. We confirm in this paper the presence in the $10$ keV--$10$ MeV energy band the very same hierarchical structure of the UPE phase in GRB 180720B already found in GRB 190114C. We simulate the above physical process of the inner engine that explains the UPE energetics, luminosity and spectrum and infer from it the magnetic field strength, the initial mass and spin of the BH, and their time evolution. The electro-vacuum Papapetrou-Wald solution used in the inner engine differs from other models of the electromagnetic field structure around astrophysical BHs (see, e.g., \cite{2005MNRAS.359..801K, 2019PhRvL.122c5101P}). A detailed theoretical review of such models is presented by \citet{2005MNRAS.359..801K}. In those models, the field structure and parameters enforce the condition $\mathbf{E} \cdot \mathbf{B} \neq 0$, while in the Papapetrou-Wald solution naturally exist regions where such a condition is naturally satisfied in the Kerr BH surroundings. In the BdHN I, the BH is surrounded by a very-low dense plasma in which the screening of the electric field is unlikely to occur, guaranteeing the stability of the Papapetrou-Wald electromagnetic field structure. This differs from the environment envisaged for describing the mechanism of generating powerful relativistic jets from a black hole system in AGNs and x-ray binary systems, e.g., in \citet{2005MNRAS.359..801K} and \citet{2019PhRvL.122c5101P}, where the surrounding plasma has a much larger density and may cause the screening of the electric field. Therefore, the inner engine parameters and physical processes are different with respect to these models and have been guided by the GRB analysis. Specifically, our approach gives a theoretical explanation to the time-resolved spectral analysis of the UPE phase, and the MeV luminosity observed by Fermi-GBM. {The article is organized as follows.} In Sec.~\ref{sec:observation}, we present the observations of GRB 180720B by different satellites and then introduce the 6 Episodes of this GRB obtained from the observations. In Sec.~\ref{sec:heir}, we perform the time-resolved spectral analysis during the UPE phase of GRB 180720B {thanks to} the recent progress in the understanding of the UPE phase of GRB 190114C \cite{PhysRevD.104.063043}, and the high signal-to-noise (S/N) ratio of the Fermi-GBM data during the UPE phase of GRB 180720B. In Sec.~\ref{sec:5}, {we introduce the structure of the electromagnetic field used to study} the properties of inner engine of GRB 180720B. {We also discuss the physical differences of this electromagnetic field and the operation of the inner engine to extract the rotational energy of the BH with the existing literature on the subject.} In Sec.~\ref{sec:massupe}, we determine the mass and spin of the BH in the inner engine of GRB 180720B during the UPE phase {and in the subsequent evolution}. In Sec.~\ref{sec:vac}, {we describe the vacuum polarization process in the inner engine and how it originates the UPE phase.} In Sec.~\ref{sec:12}, {we analyze} the compactness problem and the general formulation of transparency condition during the UPE phase. In Sec.~\ref{sec:magnetic}, we obtain the magnetic field and BH parameters at transparency point during the UPE phase. We follow the quantum vacuum polarization process down to a timescale of $\tau \sim 10^{-9}$~s, marking the hierarchical structure of the UPE imposed by the observed luminosity and the electromagnetic configuration of the inner engine during the UPE phase. We compute the value of the magnetic field, $B_0$, the Lorentz factors, the baryon loads, the energy emitted and radii at the transparency of each PEMB pulse. {We also discuss relevant differences between our approach and different models in the literature, e.g. by \citet{2005MNRAS.359..801K} and \citet{2019PhRvL.122c5101P}.} In Sec.~\ref{sec:conc}, we {summarize the conclusions} of this work. \section{Observational data of GRB 180720B} % \label{sec:observation} On July 20, 2018, GRB 180720B triggered the Fermi-GBM at 14:21:39.65 UT \citep{GCN22981}, the CALET Gamma-ray Burst Monitor at 14:21:40.95 UT \citep{GCN23042}, the Swift-BAT at 14:21:44 UT \citep{GCN22973}, the Fermi-LAT at 14:21:44.55 UT \citep{GCN22980}, and the Konus-Wind at 14:21:45.26 UT \citep{GCN23011}. The X-ray telescope (XRT) on board the Neil Gehrels Swift Observatory (hereafter Swift), began observing $91$ s after the Fermi-GBM trigger \citep{GCN22973}, MAXI/GSC started at $296$ s \citep{GCN22993} and NuStar covered later times from $243$ ks to $318$ ks \citep{GCN23041}. Just $78$ s after the Fermi-GBM trigger, the $1.5$-m Kanata telescope performed optical and NIR imaging polarimetry of the source field and found a bright optical R-band counterpart within the the Swift-XRT error circle, observed by HOWPol and HONIR attached to the $1.5$-m Kanata telescope \citep{GCN22977}. Additional observations followed by optical, infrared and radio telescopes \citep{GCN22976,2018GCN.22977....1S,GCN22983,GCN22985,GCN22988,GCN23017,GCN23020,GCN23021,GCN23023,GCN23024,GCN23033,GCN23037,GCN23040,2019Natur.575..464A}. Following the optical observation, redshift z = 0.654 was identified from the Fe II and Ni II lines by the VLT/X-shooter telescope \citep{GCN22996}. This allows to determine the cosmological rest frame time $t_{\rm rf}=t_{\rm obs}/(1+z)$, being $t_{\rm obs}$ the observed time, as well as the isotropic energy ($E_{\rm iso}$) and the isotropic luminosity ($L_{\rm iso}$) of this source. GRB 180720B has isotropic energy of $E_{\rm iso}=5.92 \times 10^{53}$~erg during the $T_{90}$ of the Fermi-GBM. The sub-TeV ($100$--$440$~GeV) observation by the High Energy Stereoscopic System (H.E.S.S.) has been also reported for this GRB \citep{2019Natur.575..464A}. The diversity and the statistical significance of the observed data have made this GRB one of the proper candidates to test the GRB models. The luminosity light-curve in radio, optical, and gamma-rays of the GRB 180720B is shown in Fig.~\ref{fig:data}. \subsection{The Episodes of GRB 180720B} Six different episodes relating to six different astrophysical processes have been recently identified in the time domain analysis of GRB 180720B \citep{2021arXiv210309158M}: \textcolor{black}{I) Episode 1 (UPE I): the BH formation caused by hypercritical accretion onto the companion NS in BdHN I, and its subsequent UPE phase originated from vacuum polarization and expanding PEMB pulses with their characteristic Lorentz factor $\Gamma \sim 100$ \citep{1999A&A...350..334R,RSWX2}. This episode pinpoints the first manifestation of the BH (BH-rise) which is the subject of the current paper. The UPE I of GRB 180720B occurs from $~t_{\rm rf}=4.84$~s to $~t_{\rm rf}=6.05$~s. Its measured isotropic energy is $E^{\rm MeV}_{\rm UPE I}=(6.37\pm0.48) \times 10^{52}$~erg, and its spectrum is best fitted by a CPL+BB model (index $\alpha=-1.13$, cutoff energy $E_{\rm c}=2220.569$~keV, and blackbody (BB) temperature $k T = 50.31$~keV in the observer's frame); see Fig.~\ref{fig:lightcurve+spectra}. } \textcolor{black}{II) Episode 2 ($\nu$NS-rise): the fallback of the SN ejecta onto the $\nu$NS spins it up (\cite{2019ApJ...871...14B} and Becerra et al., submitted). The first evidence of this interplay in GRB 180720B, referred to as the \textit{$\nu$NS-rise}, spans from $~t_{\rm rf}=6,05$~s to $~t_{\rm rf}=9.07$~s. The isotropic energy of this phase $E^{\rm MeV}_{\rm \nu Ns}=(1.13\pm0.04) \times 10^{53}$~erg, and its spectrum is best fitted by a CPL model ($\alpha=-0.98$, and $E_{\rm c}=737$~keV, in the observer's frame); see Fig.~\ref{fig:lightcurve+spectra}. } \textcolor{black}{III) Episode 3 (UPE II): it is evidenced by the the first significant observed GeV photon at $~t_{\rm rf}=7.06$~s. This phase also includes the continuation of the UPE phase (UPE II) from $~t_{\rm rf}=9.07$~s to $~t_{\rm rf}=10.89$~s, with an isotropic energy of $E_{\rm UPE II}^{\rm MeV}=(1.6 \pm 0.95) \times 10^{53}$~erg. A CPL+BB model with model parameters of $\alpha= -1.06^{+0.01}_{-0.01}$, $E_{\rm c}=1502.5^{+88.6}_{-87.5}$~keV and $kT= 39.8^{+1.6}_{-1.6}$~keV is the best fit for the spectrum of this phase; see Fig.~\ref{fig:lightcurve+spectra}. } \textcolor{black}{IV) Episode 4 (Cavity): the gravitational collapse of the NS and the consequent BH formation creates a cavity, which becomes further depleted by the expanding PEMB pulses \cite{2019ApJ...883..191R}. The collision and partial reflection of the ultra-relativistic PEMB pulses from the cavity's wall results in radiation with a CPL spectrum that has an energy of $\sim 10^{52}$ erg and a luminosity of $\sim 10^{51}$ erg s$^{-1}$. These values are comparable to the UPE and $\nu$NS-rise energetics \cite{2019ApJ...883..191R}. For GRB 180720B, this emission extends from $t_{\rm rf}=16.94$~s to $~t_{\rm rf}=19.96$~s, with an isotropic energy of $E_{\rm CV}^{\rm MeV}=(4.32 \pm 0.19) \times 10^{52}$~erg, characterized by a CPL spectrum ($\alpha=-1.16$, $E_{\rm c} = 607.96$~keV). Its luminosity and spectrum is given in Fig.~\ref{fig:lightcurve+spectra}.} \textcolor{black}{V) Episode 5 soft X-ray flare (SXF), and VI) Episode 6 hard X-ray flare (HXF): HXF and SXF emissions result from the interaction of the PEMB pulses with the SN ejecta occurring at $r= 10^{12}$ cm from the BH \cite{2018ApJ...869..151R}. The HXF of GRB 180720B extends from $t_{\rm rf}= 28.95$~s to $t_{\rm rf}= 34.98$~s, with $L_{\rm HXF,iso}^{\rm MeV}=(7.8 \pm 0.07) \times 10^{51}$~erg~s$^{-1}$. Its spectrum is best fitted by a CPL model with $E_{\rm c}=(5.5_{-0.7}^{+0.8}) \times 10^2$~keV, $\alpha = -1.198 \pm 0.031$. The SXF occurs from $t_{\rm rf}= 55$~s to $t_{\rm rf}= 75$~s, with $L_{\rm SXF,iso}^{\rm X}=1.45\times 10^{50}$~erg s$^{-1}$. Its spectrum is best fitted by a PL+BB model with $\alpha = -1.79 \pm 0.23$, and $k T=0.99 \pm 0.13$~keV; see Fig.~\ref{fig:lightcurve+spectra}.} \textcolor{black}{The cavity, SXF, and HXF have energetics similar to the UPE phase because they are also created by the interaction of expanding PEMB pulses with SN ejecta; (see, \cite{2021MNRAS.504.5301R}, and references therein).} \textcolor{black}{ One-dimensional \cite{2014ApJ...793L..36F, 2015PhRvL.115w1102F}, two-dimensional \cite{2015ApJ...812..100B}, and three-dimensional \cite{2016ApJ...833..107B, 2019ApJ...871...14B} simulations of BdHN model clearly show that the accretion of the SN ejecta onto the $\nu$NS and NS companion transfers both mass and angular momentum to them. According to numerical simulations of the early evolution phase of BdHN I, the NS companion can reach its critical mass for BH formation before the second peak of fallback accretion onto the $nu$NS (Becerra et al., submitted; see also \cite{2019ApJ...871...14B}). In some cases, this phenomenon allows the $\nu$NS-rise emission to superpose to the UPE. In GRB 180720B, the energy released by the $\nu$NS-rise dominates the UPE phase for about three seconds, resulting in split UPEs I and II. After that, the $\nu$NS-rise emission fades and the UPE becomes visible again. } The detailed explanation of Episodes 4 to 6 of GRB 180720B is presented in \cite{2021arXiv210309158M}. \textcolor{black}{This work is devoted to the UPE I and UPE II phases of GRB 180720B. } Following the explanation of the UPE phase in GRB 190114C \cite{PhysRevD.104.063043}, we first present the detailed spectral analysis of the UPE phase of GRB 180720B and then its astrophysical mechanism based on the inner engine of GRBs \cite{2019ApJ...886...82R} \textcolor{black}{and expanding PEMB pulses \citep{2021PhRvD.104f3043M}}. \section{The time-resolved spectral analysis of the UPE phase}\label{sec:heir} {Due to the high signal-to-noise (S/N) ratio of the Fermi-GBM data acquired during the UPE phase, a refined spectral analysis is performed in the $[4.84$--$6.05]$ time interval in three iterations, and in the $[9.07$--$10.89]$ time interval in five iterations on decreasing time bins, while maintaining reliable statistical significance. The time intervals between iterations are halved.} {For the final iteration of the UPE I, i.e., the third iteration, the UPE I is divided into four time intervals of $\Delta t_{\rm rf} \approx 0.3$~s: [$4.840$s--$5.142$s],[$5.142$s--$5.445$s],[$5.445$s--$5.747$s], [$5.747$s--$6.050$s].} For the last iteration of the the UPE II where reliable statistical significance is still fulfilled, i.e., the fifth iteration, the UPE II is divided into $16$ time intervals of $\Delta t_{\rm rf} \approx 0.11$~s: [$9.07$s--$9.19$s],[$9.19$s--$9.30$s],[$9.30$s--$9.41$s], [$9.41$s--$9.53$s], [$9.53$s--$9.64$s], [$9.64$s--$9.75$s], [$9.75$s--$9.87$s], [$9.87$s--$9.98$s], [$9.98$s--$10.10$s], [$10.10$s--$10.21$s], [$10.21$s--$10.32$s], [$10.32$s--$10.44$s], [$10.44$s--$10.55$s], [$10.55$s--$10.66$s], [$10.66$s--$10.78$s] and [$10.78$s--$10.89$s]. The spectral analysis is performed over each time interval. The presence of a cut-off power-law plus black body (CPL+BB) as the best spectral fit is confirmed in each time interval and for each iterative process. The time intervals both in rest-frame and observer frame, the significance ($S$) for each time interval, the power-law index, cut-off energy, temperature, $\Delta$DIC, BB flux, total flux, the BB to total flux ratio, $F_{\rm BB}/F_{\rm tot}$ and finally the isotropic energy of entire the {UPE phase} and its sub-intervals are shown in {Table~\ref{tab:UPEI} and} Table~\ref{tab:180720B}; see also {Fig.~\ref{fig:UPEI} and} Fig.~\ref{alltogether}. The evolution of the temperature and the luminosity during the UPE phase, as obtained by the time-resolved spectral analysis, are shown in Fig.~\ref{fig:lumupe}. \begin{table*} \small\addtolength{\tabcolsep}{-1pt} \caption{{The time-resolved spectral fit parameters for GRB 180720B (CPL+BB model) during the UPE I phase ($t_{\rm rf}=4.84$ s to $t_{\rm rf}=6.05$ s). This table summarizes the time intervals in the rest and observer frames, their significance ($S$), the power-law index, cut-off energy, temperature, $\Delta$DIC, BB flux, total flux, BB to total flux ratio, $F_{\rm BB}/F_{\rm tot}$, and finally the isotropic energy. The first block (first row) in the table contains the spectral best-fit parameters for UPE I. The second block (second, third, fourth, and fifth rows) contains the time-resolved spectral best-fit parameters for $Delta t_{\rm rf}=0.3$ s.} To select the best model from two different given models, we adopt the deviance information criterion (DIC), defined as DIC=-2log[$p$(data$\mid\hat{\theta}$)]+2$p_{\rm DIC}$, where $\hat{\theta}$ is the posterior mean of the parameters, and $p_{\rm DIC}$ is the effective number of parameters. The preferred model is the model with the lowest DIC score. Here we define $\Delta$DIC=(CPL+BB)-CPL, if $\Delta$DIC is negative, indicating the CPL+BB is better. After comparing the DIC, we find the CPL+BB model is the preferred model over the CPL and other models. } \label{tab:UPEI} \begin{tabular}{c|c|c|c|c|c|c|c|c|c|c} \hline\hline $t_{1}$$\sim$$t_{2}$&$t_{rf,1}$$\sim$$t_{rf,2}$&{$S$}&$\alpha$&$E_{\rm c}$&$kT$&$\Delta$DIC&$F_{\rm BB}$&$F_{\rm tot}$&$F_{\rm ratio}$&$E_{\rm tot}$\\ \hline (s)&(s)&&&(keV)&(keV)&&(10$^{-6}$)&(10$^{-6}$)&&($10^{52}$~erg)\\ Obs&Rest-frame&&&&&&(erg~cm$^{-2}$~s$^{-1}$)&(erg~cm$^{-2}$~s$^{-1}$)&\\ \hline 8.000$\sim$10.000&4.840$\sim$6.050&142.74&-1.13$^{+0.01}_{-0.01}$&2221$^{+184}_{-183}$&50.3$^{+2.8}_{-2.9}$&-108&1.39$^{+0.53}_{-0.35}$&27.14$^{+2.20}_{-2.04}$&0.05&9.53\\ \hline \hline 8.000$\sim$8.500&4.840$\sim$5.142&82.61&-1.06$^{+0.02}_{-0.02}$&2965$^{+316}_{-313}$&64.4$^{+6.1}_{-6.0}$&-84&2.44$^{+1.43}_{-0.92}$&43.61$^{+4.78}_{-4.80}$&0.06&3.83\\ 8.500$\sim$9.000&5.142$\sim$5.445&89.78&-1.11$^{+0.03}_{-0.03}$&1898$^{+266}_{-267}$&56.2$^{+5.0}_{-5.0}$&-51&1.97$^{+1.15}_{-0.75}$&31.47$^{+4.46}_{-4.26}$&0.06&2.76\\ 9.000$\sim$9.500&5.445$\sim$5.747&72.53&-1.07$^{+0.06}_{-0.06}$&953$^{+265}_{-267}$&34.2$^{+10.3}_{-13.9}$&-23&0.37$^{+1.53}_{-0.34}$&15.77$^{+7.32}_{-3.93}$&0.02&1.38\\ 9.500$\sim$10.000&5.747$\sim$6.050&60.82&-1.19$^{+0.05}_{-0.05}$&1788$^{+571}_{-582}$&38.1$^{+4.9}_{-4.9}$&-32&0.76$^{+0.67}_{-0.39}$&15.42$^{+4.80}_{-4.00}$&0.05&1.35\\ \hline \end{tabular} \end{table*} \begin{table*} \small\addtolength{\tabcolsep}{-2pt} \caption{The parameters of the time-resolved spectral best fits of GRB 180720B (CPL+BB model) from the $t_{\rm rf}=9.07$~s to $t_{\rm rf}=10.89$~s. This table reports: the time intervals both in rest-frame and observer frame, the significance ($S$) for each time interval, the power-law index, cut-off energy, temperature, $\Delta$DIC, BB flux, total flux, the BB to total flux ratio, $F_{\rm BB}/F_{\rm tot}$ and finally the isotropic energy. The $\Delta$DICs are reported in column 6. \label{tab:180720B}} \centering \begin{tabular}{c|c|c|c|c|c|c|c|c|c|c} \hline\hline $t_{1}$$\sim$$t_{2}$&$t_{rf,1}$$\sim$$t_{rf,2}$&{$S$}&$\alpha$&$E_{\rm c}$&$kT$&$\Delta$DIC&$F_{\rm BB}$&$F_{\rm tot}$&$F_{\rm ratio}$&$E_{\rm tot}$\\ \hline (s)&(s)&&&(keV)&(keV)&&(10$^{-6}$)&(10$^{-6}$)&&($10^{52}$~erg)\\ Obs&Rest-frame&&&&&&(erg~cm$^{-2}$~s$^{-1}$)&(erg~cm$^{-2}$~s$^{-1}$)&\\ \hline 15.00$\sim$18.00&9.07$\sim$10.89&274.60&-1.06$^{+0.01}_{-0.01}$&1502.5$^{+88.6}_{-87.5}$&39.8$^{+1.6}_{-1.6}$&-226.4&1.99$^{+0.43}_{-0.34}$&45.55$^{+3.11}_{-2.70}$&0.04$^{+0.01}_{-0.01}$&16.0$^{+1.1}_{-0.952}$\\ \hline 15.00$\sim$16.50&9.07$\sim$9.98&190.63&-1.04$^{+0.01}_{-0.01}$&1750.5$^{+112.7}_{-111.1}$&40.5$^{+2.0}_{-2.0}$&-176.6&2.08$^{+0.58}_{-0.46}$&48.03$^{+3.28}_{-3.09}$&0.04$^{+0.01}_{-0.01}$&8.46$^{+0.577}_{-0.543}$\\ 16.50$\sim$18.00&9.98$\sim$10.89&215.76&-1.05$^{+0.02}_{-0.02}$&1151.3$^{+117.3}_{-119.6}$&37.1$^{+2.8}_{-2.8}$&-78.7&1.63$^{+0.69}_{-0.54}$&41.83$^{+4.61}_{-4.04}$&0.04$^{+0.02}_{-0.01}$&7.37$^{+0.812}_{-0.712}$\\ \hline 15.00$\sim$15.75&9.07$\sim$9.53&105.93&-1.07$^{+0.03}_{-0.03}$&1198.0$^{+211.1}_{-217.8}$&31.4$^{+3.3}_{-3.3}$&-41.5&0.94$^{+0.70}_{-0.42}$&23.84$^{+4.65}_{-3.86}$&0.04$^{+0.03}_{-0.02}$&2.1$^{+0.41}_{-0.34}$\\ 15.75$\sim$16.50&9.53$\sim$9.98&163.07&-1.02$^{+0.01}_{-0.01}$&1949.4$^{+126.1}_{-127.9}$&46.2$^{+2.68}_{-2.67}$&-15.4&3.43$^{+1.23}_{-0.84}$&72.68$^{+5.48}_{-5.31}$&0.04$^{+0.01}_{-0.01}$&5.16$^{+0.478}_{-0.423}$\\ 16.50$\sim$17.25&9.98$\sim$10.44&155.67&-1.15$^{+0.02}_{-0.02}$&2382.3$^{+217.5}_{-221.3}$&45.3$^{+2.7}_{-2.7}$&-125.6&2.85$^{+1.00}_{-0.76}$&53.96$^{+4.55}_{-4.28}$&0.05$^{+0.02}_{-0.01}$&4.75$^{+0.401}_{-0.377}$\\ 17.25$\sim$18.00&10.44$\sim$10.89&159.05&-0.93$^{+0.02}_{-0.02}$&684.7$^{+49.7}_{-49.2}$&23.9$^{+3.8}_{-4.0}$&-30.8&0.63$^{+0.93}_{-0.37}$&35.74$^{+3.28}_{-3.21}$&0.02$^{+0.03}_{-0.01}$&3.15$^{+0.289}_{-0.283}$\\ \hline 15.00$\sim$15.38&9.07$\sim$9.30&69.11&-1.06$^{+0.07}_{-0.08}$&711.2$^{+209.5}_{-215.5}$&28.9$^{+5.7}_{-5.6}$&-30.2&0.78$^{+1.14}_{-0.55}$&14.27$^{+6.80}_{-3.54}$&0.05$^{+0.08}_{-0.04}$&0.628$^{+0.299}_{-0.156}$\\ 15.38$\sim$15.75&9.30$\sim$9.53&83.03&-1.01$^{+0.03}_{-0.03}$&1319.4$^{+210.9}_{-208.7}$&31.0$^{+5.2}_{-5.2}$&-28.9&0.83$^{+1.14}_{-0.48}$&32.18$^{+6.45}_{-5.45}$&0.03$^{+0.04}_{-0.02}$&1.42$^{+0.284}_{-0.24}$\\ 15.75$\sim$16.12&9.53$\sim$9.75&109.59&-1.02$^{+0.02}_{-0.02}$&1967.9$^{+193.8}_{-194.9}$&43.6$^{+4.0}_{-4.0}$&-72.6&2.63$^{+1.51}_{-0.96}$&62.61$^{+6.83}_{-6.58}$&0.04$^{+0.02}_{-0.02}$&2.76$^{+0.301}_{-0.29}$\\ 16.12$\sim$16.50&9.75$\sim$9.98&133.10&-1.01$^{+0.02}_{-0.02}$&1919.4$^{+162.1}_{-168.5}$&47.9$^{+3.5}_{-3.5}$&-107.5&4.31$^{+1.60}_{-1.38}$&82.08$^{+8.46}_{-7.17}$&0.05$^{+0.02}_{-0.02}$&3.61$^{+0.372}_{-0.316}$\\ 16.50$\sim$16.88&9.98$\sim$10.21&133.12&-1.09$^{+0.02}_{-0.02}$&2574.3$^{+264.0}_{-267.2}$&55.7$^{+3.8}_{-3.7}$&-117.9&5.16$^{+2.03}_{-1.44}$&83.97$^{+8.79}_{-7.60}$&0.06$^{+0.02}_{-0.02}$&3.7$^{+0.387}_{-0.335}$\\ 16.88$\sim$17.25&10.21$\sim$10.44&89.16&-1.24$^{+0.05}_{-0.05}$&1537.9$^{+522.7}_{-558.0}$&31.9$^{+3.4}_{-3.4}$&-27.8&1.38$^{+0.94}_{-0.57}$&24.25$^{+7.37}_{-6.29}$&0.06$^{+0.04}_{-0.03}$&1.07$^{+0.325}_{-0.277}$\\ 17.25$\sim$17.62&10.44$\sim$10.66&125.76&-0.86$^{+0.03}_{-0.03}$&696.1$^{+59.2}_{-57.7}$&22.5$^{+3.8}_{-3.7}$&-27.3&0.83$^{+1.39}_{-0.48}$&45.89$^{+5.21}_{-4.69}$&0.02$^{+0.03}_{-0.01}$&2.02$^{+0.23}_{-0.206}$\\ 17.62$\sim$18.00&10.66$\sim$10.89&102.97&-1.02$^{+0.04}_{-0.04}$&622.4$^{+77.4}_{-80.6}$&25.7$^{+8.4}_{-9.5}$&-25.5&0.39$^{+1.32}_{-0.34}$&25.51$^{+4.95}_{-3.40}$&0.02$^{+0.05}_{-0.01}$&1.12$^{+0.218}_{-0.15}$\\ \hline 15.00$\sim$15.19&9.07$\sim$9.19&51.57&-1.01$^{+0.14}_{-0.15}$&805.3$^{+449.1}_{-380.0}$&33.0$^{+13.1}_{-18.2}$&-288.8&0.80$^{+5.09}_{-0.77}$&19.23$^{+23.49}_{-7.86}$&0.04$^{+0.27}_{-0.04}$&0.423$^{+0.517}_{-0.173}$\\ 15.19$\sim$15.38&9.19$\sim$9.30&42.03&-1.19$^{+0.09}_{-0.09}$&1201.3$^{+667.6}_{-595.4}$&27.5$^{+4.3}_{-4.2}$&-27.1&0.97$^{+1.06}_{-0.55}$&12.89$^{+8.98}_{-4.04}$&0.08$^{+0.1}_{-0.05}$&0.284$^{+0.198}_{-0.0889}$\\ 15.38$\sim$15.56&9.30$\sim$9.41&53.84&-1.00$^{+0.04}_{-0.04}$&1158.5$^{+201.4}_{-200.2}$&23.4$^{+8.3}_{-8.3}$&-27.1&0.29$^{+1.66}_{-0.26}$&27.59$^{+7.34}_{-4.93}$&0.01$^{+0.06}_{-0.01}$&0.608$^{+0.162}_{-0.109}$\\ 15.56$\sim$15.75&9.41$\sim$9.53&63.61&-1.06$^{+0.05}_{-0.05}$&1839.8$^{+434.0}_{-420.6}$&39.4$^{+7.6}_{-7.0}$&-32.2&1.74$^{+2.60}_{-1.11}$&40.95$^{+11.15}_{-9.24}$&0.04$^{+0.06}_{-0.03}$&0.902$^{+0.246}_{-0.203}$\\ 15.75$\sim$15.94&9.53$\sim$9.64&72.54&-1.04$^{+0.04}_{-0.04}$&1896.8$^{+350.9}_{-351.5}$&40.8$^{+4.7}_{-4.7}$&-30.3&2.78$^{+2.13}_{-1.19}$&51.44$^{+12.23}_{-9.91}$&0.05$^{+0.04}_{-0.03}$&1.13$^{+0.269}_{-0.218}$\\ 15.94$\sim$16.12&9.64$\sim$9.75&83.99&-0.99$^{+0.03}_{-0.03}$&1950.2$^{+231.8}_{-232.1}$&47.5$^{+7.6}_{-7.6}$&-34.3&2.34$^{+3.12}_{-1.29}$&74.72$^{+11.53}_{-9.35}$&0.03$^{+0.04}_{-0.02}$&1.65$^{+0.254}_{-0.206}$\\ 16.12$\sim$16.31&9.75$\sim$9.87&85.09&-0.95$^{+0.04}_{-0.04}$&1379.2$^{+207.4}_{-203.8}$&32.7$^{+5.4}_{-5.3}$&-39.2&1.84$^{+2.29}_{-1.02}$&63.06$^{+12.29}_{-10.56}$&0.03$^{+0.04}_{-0.02}$&1.39$^{+0.271}_{-0.233}$\\ 16.31$\sim$16.50&9.87$\sim$9.98&104.94&-1.05$^{+0.02}_{-0.02}$&2304.7$^{+260.1}_{-261.8}$&62.1$^{+2.8}_{-2.8}$&-85.4&6.72$^{+1.63}_{-1.29}$&97.87$^{+12.08}_{-9.75}$&0.07$^{+0.02}_{-0.01}$&2.15$^{+0.266}_{-0.215}$\\ 16.50$\sim$16.69&9.98$\sim$10.10&107.18&-1.04$^{+0.03}_{-0.03}$&2737.1$^{+346.9}_{-340.9}$&58.4$^{+5.6}_{-5.6}$&-86.1&6.57$^{+3.89}_{-2.56}$&119.20$^{+16.65}_{-14.38}$&0.06$^{+0.03}_{-0.02}$&2.62$^{+0.367}_{-0.317}$\\ 16.69$\sim$16.88&10.10$\sim$10.21&82.58&-1.13$^{+0.13}_{-0.08}$&1910.0$^{+709.1}_{-1074.0}$&58.6$^{+8.6}_{-9.2}$&-86.9&3.67$^{+4.06}_{-3.43}$&53.29$^{+28.29}_{-22.24}$&0.07$^{+0.08}_{-0.07}$&1.17$^{+0.623}_{-0.49}$\\ 16.88$\sim$17.06&10.21$\sim$10.32&64.96&-1.24$^{+0.03}_{-0.03}$&2412.4$^{+580.9}_{-576.0}$&34.7$^{+4.0}_{-4.0}$&-28.1&1.52$^{+1.46}_{-0.72}$&32.97$^{+6.96}_{-5.49}$&0.05$^{+0.05}_{-0.02}$&0.726$^{+0.153}_{-0.121}$\\ 17.06$\sim$17.25&10.32$\sim$10.44&62.39&-1.06$^{+0.08}_{-0.08}$&480.3$^{+112.6}_{-114.6}$&21.1$^{+8.8}_{-8.9}$&-125.2&0.39$^{+3.01}_{-0.35}$&15.20$^{+8.60}_{-3.47}$&0.03$^{+0.2}_{-0.02}$&0.335$^{+0.189}_{-0.0764}$\\ 17.25$\sim$17.44&10.44$\sim$10.55&81.92&-0.89$^{+0.05}_{-0.05}$&720.6$^{+93.9}_{-92.3}$&19.1$^{+3.9}_{-3.8}$&-23.5&0.82$^{+1.62}_{-0.55}$&38.20$^{+8.11}_{-5.42}$&0.02$^{+0.04}_{-0.01}$&0.841$^{+0.179}_{-0.119}$\\ 17.44$\sim$17.62&10.55$\sim$10.66&97.68&-0.84$^{+0.05}_{-0.05}$&713.4$^{+96.8}_{-97.0}$&32.3$^{+11.9}_{-10.7}$&-38.1&1.05$^{+5.66}_{-0.87}$&55.49$^{+13.70}_{-10.34}$&0.02$^{+0.1}_{-0.02}$&1.22$^{+0.302}_{-0.228}$\\ 17.62$\sim$17.81&10.66$\sim$10.78&82.29&-0.95$^{+0.05}_{-0.05}$&628.7$^{+86.6}_{-86.2}$&19.5$^{+9.9}_{-7.8}$&-66.8&0.33$^{+4.15}_{-0.30}$&33.47$^{+9.11}_{-5.06}$&0.01$^{+0.12}_{-0.01}$&0.737$^{+0.201}_{-0.111}$\\ 17.81$\sim$18.00&10.78$\sim$10.89&64.36&-1.08$^{+0.06}_{-0.06}$&565.9$^{+123.9}_{-118.5}$&30.2$^{+7.8}_{-10.3}$&-15.3&0.36$^{+1.63}_{-0.33}$&17.96$^{+6.32}_{-3.42}$&0.02$^{+0.09}_{-0.02}$&0.395$^{+0.139}_{-0.0752}$\\ \hline \end{tabular} \end{table*} The time-resolved spectral analysis over each iteration, reveals a common spectral feature for each time interval characterized by the CPL$+$BB best-fit model with a rest-frame temperature of $kT= 20\sim 60$~keV and the ratio of blackbody flux ($F_{\rm BB}$) to the total flux ($F_{\rm tot}$) of \begin{equation}\label{eq:ratio} 0.01\lesssim \frac{F_{\rm BB}}{F_{\rm tot}} \lesssim 0.07. \end{equation} {In essence, the UPE II is a continuation of the UPE I, and there is no distinction between the two. The observed discontinuity in the UPE phase is caused by the simultaneous occurrence of the UPE phase and the $\nu$NS—rise in this GRB. The temporal coincidence of these two emissions in a BdHN depends on binary parameters, more relevant the orbital period (Rueda et al.; to be submitted). As a result, we assume in this paper that the UPE phase extends from $t_{\rm rf}=4.84$ to $t_{\rm rf} =10.89$~s.} The existence of the BB components in the spectrum of the UPE phase has been identified as the characteristic signature of $e^+~e^-$ pair creation in presence of baryons (the PEMB pulse) originating from the vacuum polarization process \citep{1999A&AS..138..511R,1999A&A...350..334R, 2000A&A...359..855R,2010PhR...487....1R,PhysRevD.104.063043}. This subject will be addressed in Secs.~\ref{sec:vac} and \ref{sec:12}. \section{The properties of inner engine}\label{sec:5} The physics of inner engine was first described in \cite{2019ApJ...886...82R,2020EPJC...80..300R} for GRB 130427A and \cite{2021A&A...649A..75M} for GRB 190114C. The Papapetrou-Wald solution \cite{1966AIHPA...4...83P,1974PhRvD..10.1680W} is used to describe the newborn Kerr BH in the BdHNe I surrounded by a magnetic field and by the low-density plasma in the cavity \citep{2019ApJ...883..191R}. The gravitomagnetic interaction of the newborn Kerr BH with the magnetic field induces a strong electric field in the BH vicinity \cite{2022ApJ...929...56R}. In \cite{PhysRevD.104.063043}, it was shown that the UPE phase of GRB 190114C can originate by the QED process of vacuum polarization in an overcritical field, i.e., $|{\bf E}|>E_c$, where $E_c$ is the critical field for spontaneous $e^+e^-$ pair creation in vacuum and is given by Eq. (\ref{eq:Ec}). Following this framework, we apply in this work the inner engine in the QED overcritical regime to explain the UPE phase of GRB 180720B observed by Fermi-GBM. We give the details of the physical process in Section \ref{sec:vac}. In this section, we focus on the structure of the electromagnetic field around the BH to investigate the conditions under which the overcritical field regime can develop. The components of the electric and magnetic field (in the Carter's orthonormal tetrad) in the approximation of small polar angles can be written as \citep{2019ApJ...886...82R, 2020EPJC...80..300R} \begin{eqnarray} E_{\hat{r}} &=& -\frac{2 B_0 J\,G}{c^3} \frac{ \left(r^2-\hat{a}^2 \right)}{\left(r^2+\hat{a}^2 \right)^2} \label{eq:ER} \\ E_{\hat{\theta}}&=&0 \\ B_{\hat{r}}&=&\frac{B_0 \left(-\frac{4\,G\, J^2 r}{M\left(r^2+\hat{a}^2 \right) }+a^2+r^2\right)}{\left(r^2+\hat{a}^2 \right)}\\ B_{\hat{\theta}}&=& 0. \end{eqnarray} where $\Sigma=r^2+\hat{a}^2\cos^2\theta$, $\Delta=r^2-2 \hat{M} r+\hat{a}^2$, $\hat{M}= G M/c^2$, $\hat{a}=a/c=J/(M\,c)$, being $M$ and $J$ the mass and angular momentum of the Kerr BH. The (outer) event horizon is located at $r_+=(\hat{M}+\sqrt{\hat{M}^2-\hat{a}^2})$. We can now introduce the effective charge \cite{2019ApJ...886...82R} \begin{equation}\label{eq:Qeff} Q_{\rm eff}=\frac{G}{c^3}2 B_0 J, \end{equation} which when is replaced in the charge of the Kerr-Newman solution, it leads to a radial electric field equal to the one of the Papapetrou-Wald solution given by Eq.~(\ref{eq:ER}) \cite{PhysRevD.104.063043}. Therefore, up to linear order in $\theta$ and in the dimensionless BH spin parameter $\alpha \equiv \hat{a}/(G M/c^2)$, the electric field can be written as \begin{equation}\label{eq:ER2} E_{\hat{r}} = -\frac{2 B_0 J\,G}{c^3} \frac{ \left(r^2-\hat{a}^2 \right)}{\left(r^2+\hat{a}^2 \right)^2}\approx -\frac{1}{2}\alpha B_0\frac{r_+^2}{r^2}. \end{equation} The specific value of the mass, spin parameter, and magnetic field in the inner engine will be determined as a function of operative astrophysical processes, which are presented in the next sections. We now discuss how the gravitational collapse of the NS in a BdHN I can lead to an engine with an electromagnetic field structure that can be approximately described by the Papapetrou-Wald solution. To the best of our knowledge, there are no numerical simulations dedicated to demonstrating the formation of this specific configuration. Nevertheless, we refer to Sec. $7$ of \cite{2020ApJ...893..148R} which discusses the nature of the magnetic field around the newborn BH in a BdHN I and its support from numerical simulations of gravitational collapse. Numerical simulations in \cite{2019ApJ...871...14B} show that the magnetized NS companion of the CO$_{\rm core}$, in the accretion process, gains not only a considerable amount of mass but also angular momentum. Therefore, we conclude that the BH forms from the collapse of a magnetized ($10^{12}$--$10^{13}$ G), fast rotating (millisecond period) NS once it reaches the critical mass. The same simulations show that SN material remains bound around the nascent BH in a torus-like structure. This matter is essential to anchor the magnetic field outside the newborn BH. The matter density in the off-equatorial directions is low, as shown by numerical simulations in \cite{2019ApJ...883..191R}. The above picture naturally leads to the inner engine: a rotating BH surrounded by a magnetic field and very low density ionized matter. The numerical simulations of the magneto-rotational collapse of NS starting from the seminal simulations of J. Wilson in \cite{1975NYASA.262..123W, 1978pans.proc..644W}. These early works already showed the amplification of the magnetic field in the gravitational collapse. This result is confirmed by simulations of NS binary mergers leading to a BH surrounded by an accretion disk, whose post-merger system show similarities with the inner engine picture and support the present scenario. Some relevant works on this subject are \cite{2006PhRvL..96c1101D, 2006PhRvL..96c1102S, 2006PhRvD..73j4015D, 2007CQGra..24S.207S, 2008PhRvD..77d4001S, 2011ApJ...732L...6R}. Equation (\ref{eq:ER2}) tells us that if the above processes occur and amplify the magnetic field strength to values $B_0 \gtrsim (2/\alpha) B_c$ near the BH horizon, where $B_c = E_c = 4.41\times 10^{13}$ G, an overcritical electric field will develop and lead to the QED process of vacuum polarization. \section{Mass and spin of BH}\label{sec:massupe} The energy condition is obtained from the mass-energy formula of the Kerr BH \cite{1970PhRvL..25.1596C,1971PhRvD...4.3552C,1971PhRvL..26.1344H} \begin{equation} \label{aone} M^2 = \frac{c^2 J^2}{4 G^2 M^2_{\rm irr}}+M_{\rm irr}^2. \end{equation} The extractable energy of a Kerr BH $E_{\rm ext}$ {is given by the subtracting the irreducible mass, $M_{irr}$, from the total mass of the BH, $M$}: \begin{equation} \label{Eextr} E_{\rm ext}=(M-M_{\rm irr}) c^2=\left(1-\sqrt{\frac{1+\sqrt{1-\alpha^2}}{2}}\right)M c^2. \end{equation} which we use to obtain $M$ as a function of $\alpha$, $M(\alpha)$, by requesting the condition that observed UPE emission originates from BH extractable energy, i.e. \begin{equation} \label{latextr} E_{\rm UPE} = E_{\rm ext}. \end{equation} The goal is to show that the Kerr BH extractable energy can explain the energetics of the UPE phase and, in turn, it leads to estimate the mass and spin of BH. Equation (\ref{Eextr}) has two parameters, $M$ and $\alpha$, hence we must supply another equation to determine them. In BdHNe I the BH originates from the hypercritical accretion of SN ejecta onto the NS, i.e., the BH forms when the NS reaches its critical mass. Therefore, the mass of the BH must satisfy the constraint \begin{equation} \label{meq} M\geq M_{\rm crit}(\alpha), \end{equation} where $M_{\rm crit}(\alpha)$ is the critical mass of a rotating NS with Kerr spin parameter $\alpha$. The NS critical mass value depends on the nuclear equation of state (EOS). In \citet{2015PhRvD..92b3007C}, it was shown that, for instance, for the NL3, GM1 and TM1 EOS, the critical mass for rigidly rotating NS is fitted with a maximum error of $0.45\%$ by the expression \begin{equation}\label{eq:Mcrit} M_{\rm crit}(j)=M_{\rm crit}^{J=0}(1 + k j^p), \end{equation} where $k$, $p$, and $M_{\rm crit}^{J=0}$ are parameters that depend upon the nuclear EOS, being the latter the critical mass in the non-rotating case, and $j \equiv cJ/(G M_\odot)^2$. With the relation between $j$ and $\alpha$, i.e., $j = \alpha (M/M_\odot)^2$, Eq. (\ref{eq:Mcrit}) becomes an implicit non-linear algebraic equation for the NS critical mass as a function of $\alpha$. For instance, we show in Fig.~\ref{fig:Mcrit} the numerical solution of Eq. (\ref{eq:Mcrit}) for the NL3 ($k=0.006$, $p=1.68$) and TM1 ($k=0.017$, $p=1.61$) EOS. We limit the value of the spin parameter to $\alpha_{\rm max}\approx 0.7$, which has been found to be the maximum value attainable by rigidly rotating NS independent on the nuclear EOS (see \cite{2015PhRvD..92b3007C} for details). We now proceed to estimate the mass and spin parameter of the BH at the beginning of the UPE phase, namely at $t_{\rm rf}=4.84$~s. For this task, we solve Eq. (\ref{latextr}) using $E_{\rm UPE}=E_{\rm UPEI}+E_{\rm UPEI}=2.24 \times 10^{53}$ erg, together with the inequality (\ref{meq}). We use the minimum possible value in the latter so to set a lower limit to the BH mass, and correspondingly an upper limit to the spin parameter. For the NS critical mass, we use Eq.~(\ref{eq:Mcrit}) for the TM1 EOS. We obtain the lower limit to the BH mass, $M=2.40 M_\odot$, and the upper limit to the spin, $\alpha = 0.60$. The corresponding irreducible mass of the BH which is assumed to be constant during the radiation process is $M_{\rm irr}=2.28~M_\odot$. Since the MeV emission during the UPE phase is powered by the extractable rotational energy of the Kerr BH [see Eq. (\ref{latextr})], the time derivative of Eq.~(\ref{Eextr}) gives the luminosity \begin{equation} \label{sdown1} L_{\rm UPE}=-\frac{dE_{\rm ext}}{dt}=-\frac{dM}{dt}. \end{equation} The rest-frame $10$~keV--$10$~MeV luminosity light-curve of GRB 180720B during UPE phase is fitted by a power-law with slope of $\alpha_{\rm UPE}=0.36\pm 0.23$ and, amplitude of $(1.21\pm 0.70)\times 10^{53}$~erg~s$^{-1}$. From this luminosity (see Fig.~\ref{fig:GeV+MeV}), and using as initial BH mass and spin at $t_{\rm rf}=4.84$~s the values estimated above, Eq.~(\ref{sdown1}) can be integrated to obtain the time evolution of the BH mass and spin during the UPE phase which is shown in Fig.~\ref{massspinupe}. \section{Vacuum polarization and its role in the formation of UPE phase} \label{sec:vac} The UPE phase has been explained by introducing the concept of the the dyadosphere around the Reissner-Nordström BH \citep{1998astro.ph.11232R} and dyadotorous around the Kerr-Newman geometry \cite{2009PhRvD..79l4002C}. The dyadoregion is the region around the BHs, characterised by the overcritical electric field $|{\bf E}|> E_c$, filled by the highly dense $e^+~e^-$ pairs produced by vacuum polarization process (see \cite{2010PhR...487....1R} for more details). Following Ref. \cite{PhysRevD.104.063043}, we use the dyadoregion framework in the Kerr-Newman geometry taking take advantage of the effective charge defined by Eq. (\ref{eq:Qeff}). For the calculation of the transparency properties of the $e^+e^-$ pair plasma formed from the vacuum polarization process (analyzed in Section \ref{sec:12}), we need the initial energy of the pairs, $E_{e^+e^-}$, and the radius of the dyadoregion, $r_d$. The dyadoregion extends from the BH horizon to the distance $r_d$ at which the electric field has the critical value $E_c$. Applying this condition to the Kerr-Newman electric field one obtains \cite{2009PhRvD..79l4002C} \begin{equation} \label{dyadosurf} \left(\frac{r_d}{\hat{M}}\right)^2=\frac12\frac{\lambda}{\mu\epsilon} -\alpha^2+\left(\frac14\frac{\lambda^2}{\mu^2\epsilon^2} -2\frac{\lambda}{\mu\epsilon}\alpha^2\right)^{1/2}\, \end{equation} with $\epsilon= E_c M_\odot G^{3/2}/c^4\approx 1.873 \times10^{-6}$, and \begin{equation} \label{eq:lamda} \lambda = \frac{Q_{\rm eff}}{\sqrt{G} M} = \frac{2 B_0 J G/c^3}{\sqrt{G} M}, \end{equation} is the effective charge-to-mass ratio. Therefore, the width of the dyadoregion is \begin{equation} \label{eq:width} \Delta_{\rm d}(t)= r_d(t)-r_+(t). \end{equation} The energy of $e^+~e^-$ pairs generated (at a given time) by the inner engine is estimated as the electromagnetic energy stored in the dyadoregion (see \cite{2009PhRvD..79l4002C} for details), i.e., $E_{e^+e^-} = E_{(r_+,r_d)}$, where \begin{eqnarray} \label{Eemxi} E_{(r_+,r_d)} &=&\frac{(2 B_0 J G/c^3)^2}{4r_+}\left(1-\frac{r_+}{r_{\rm d}}\right)+\frac{(2 B_0 J G/c^3)^2}{4\hat{a}}\nonumber\\ &\times&\left[\left(1+\frac{\hat{a}^2}{r_+^2}\right)\arctan\left(\frac{\hat{a}}{r_+}\right)\right.\nonumber\\ &-&\left.\left(1+\frac{\hat{a}^2}{r_d^2}\right)\arctan\left(\frac{\hat{a}}{r_d}\right)\right]. \end{eqnarray} \section{General formulation of transparency condition of the UPE phase}\label{sec:12} We follow the treatment of the transparency introduced in \cite{PhysRevD.104.063043}. The existence of the overcritical electric field around the BH leads to the following sequence of events: 1) The formation of an optically thick dyadoregion around BH dominated by the high density and pressure of the neutral $e^+e^-\gamma$ plasma \citep{2010PhR...487....1R}, formed in a timescale $\sim \hbar/(m_e c^2) \approx 10^{-21}$ s, with total energy $E_{e^+~e^-}^{\mathrm{tot}}=E_{\rm iso}$. This plasma is endowed with a baryonic mass $M_B$, with baryon load parameter ${\cal B}=M_B c^2/E_{\rm iso}$. This optically thick pair electromagnetic-baryon pulse is known as the PEMB pulse first introduced by \cite{1999A&A...350..334R}. 2) The self-acceleration and expansion of the PEMB pulses due to their high internal pressure achieved by pair-plasma thermalization in a very short timescale ($\sim 10^{-13}$~s). They reach ultra-relativistic velocities of up to $\Gamma \sim 100$ in the case of long GRBs \cite{1999A&A...350..334R,2007PhRvL..99l5003A,2009PhRvD..79d3008A}). 3) Emission of thermal radiation. When the PEMB pulses expand with ultra-relativistic velocities, the $e^+~e^-~\gamma$ plasma becomes optically thin \cite{1999A&A...350..334R,2000A&A...359..855R}. The condition of transparency is \begin{eqnarray} \label{eq:transp1} \tau &=& \sigma_T (n_{e^+e^-}+\bar{Z} n_B)\Delta_{d} \approx \sigma_T (\bar{Z} n_B) \Delta_{d},\nonumber \\ & = & \sigma_T \frac{\bar{Z} M_B }{m_N 4 \pi R^2 \Delta_{d}} \Delta_{d} = 1, \end{eqnarray} where $\Delta_{d}$ is the thickness of the PEMB pulses, $\sigma_T$ is the Thomson cross-section, $\bar{Z}$ is the average atomic number of baryons ($\bar{Z}= 1$ for Hydrogen atom and $\bar{Z}= 1/2$ for general baryonic matter), $m_N$ is nucleon mass and $M_B$ is the baryon mass. For the values of ${\cal B}$ considered in the present work, i.e., ${\cal B}=10^{-3}$--$10^{-2}$, we can safely assume $n_{e^+e^-} \ll n_B$. Therefore, from Eq. (\ref{eq:transp1}) the lower bound of transparency radius is \begin{equation} R^{\rm tr} = \left(\frac{\sigma_T}{8 \pi} \frac{M_B }{m_N}\right)^{1/2}=\left(\frac{\sigma_T}{8\pi}\frac{{\cal B} E_{\rm iso}}{m_Nc^2}\right)^{1/2}. \label{eq:transp2} \end{equation} This emission at transparency, previously known as P-GRB, is characterised by a thermal component observed in the spectral analysis of prompt emission of GRBs. The energy of this blackbody component that signs the occurrence of the UPE phase is \begin{eqnarray} \label{eq:energy3} E^{obs}_{\rm P-GRB} = a T^4_{obs} \Gamma^4(1-v/c)^3 4\pi R_{\rm tr}^2 \Delta_{d}, \end{eqnarray} where $a=4\sigma/c$, being $\sigma$ the Stefan-Boltzmann constant. \textcolor{black}{The most efficient process to create the $e^+e^-$ plasma around BH is the vacuum polarization, which proceeds on a quantum timescale of the order of the Compton time, $\hbar/(m_e c^2) \approx 10^{-21}$~s \citep{RSWX2}. The electric field screening time is given by the time it takes to charged particles to induce a field that opposites to the original field. This timescale, of the order of $r_+/c \approx 10^{-5}$~s ($r_+$ is the BH horizon radius), is $16$ orders of magnitudes larger than quantum time scale. } This guarantees that the formation of the $e^+e^-$ pair plasma and its self-expansion by internal pressure starts before any screening process of the electric field could be at work. The dynamics of the expanding plasma from the vicinity of the BH up to the transparency point depends upon the plasma energy, $E_{e^+e^-}^{\mathrm{tot}}$, and the baryon load parameter, ${\cal B}$ \cite{1999A&A...350..334R,2000A&A...359..855R}. \textcolor{black}{As discussed in Sec. XI in \citep{2021PhRvD.104f3043M} and in Secs. \ref{sec:magnetic} and \ref{sec:conc} of this paper, the BH extractable energy powers the energy for the creation of the $e^+e^-$ plasma around the BH, which is then used in the kinetic energy of expansion of the PEMB pulse and in the radiation released at transparency. Therefore, in each of these processes, the Kerr BH loses a fraction of its mass-energy and angular momentum. This implies that the BH mass and angular momentum, at the time $t_0+\Delta t$, are $M = M_0- \Delta M$ and $J = J_0 - \Delta J$, where $\Delta M$ and $\Delta J$ are the BH mass-energy and angular momentum extracted by the PEMB pulse expansion and emission process. We estimate that each process extracts $\Delta M/M \sim \Delta J/J\sim 10^{-9}$. Since the induced electric field depends linearly on $J$, see Eq. (\ref{eq:ER}), the new value of the induced electric field, $E$, is lower than the previous value, $E_0$, fulfilling $E = E_0 (1-\Delta J/J)$. As a result, the system begins a new process in presence of the same magnetic field $B_0$, which is kept constant, and a new, lower effective charge $Q_{\rm eff}= Q_{{\rm eff},0}-\Delta Q_{\rm eff}$, where $\Delta Q_{\rm eff} =2 B_0 \Delta J$. Therefore, we assume that the spacetime evolves from one stationary axially symmetric metric to the next, and at each step the electromagnetic field structure of the inner engine is given by the Papapetrou-Wald solution, and the latter can be approximated by the Kerr-Newman metric of charge $Q_{\rm eff}$. Once the plasma is formed, it self-accelerates, expanding to the point of transparency. We recall that the dynamics of the plasma depends only on the initial conditions of energy and baryon load, which in turn depend only on $M$, $J$ and $Q_{\rm eff}$. This means that the plasma dynamics at times $t>t_0$, being $t_0$ the time of its formation and beginning of the expansion, depends only on the values at $t_0$ of $M$, $J$ and $B_0$. Thus, the QED process of $e^+e^-$ formation and its dynamics leading to transparency of the PEMB pulses efficiently extracts the BH energy without being affected by any screening process of the electric field. Therefore, the decrease of the electric field with time is driven by the BH energy extraction which lowers the BH angular momentum, not because of an electric field screening. This ensures that the above process can repeat over time until the electric field reaches the critical value. For GRB 180720B, this occurs at $t_{\rm rf}=10.89$~s. After this time, the vacuum polarization process does not occur any longer.} The corresponding value of the Lorentz factor at the instant of transparency, $\Gamma$, and the baryon load parameter can be inferred from UPE observables as follows. The calculation involves the following quantities: (a) the isotropic energy of PEMB pulses, $E_{\rm iso}$; (b) the ratio of the blackbody energy of the P-GRB to the isotropic energy, $E^{\rm obs}_{\rm P-GRB}/E_{\rm iso}$; (c) the observed value of the blackbody temperature of the P-GRB, $T_{\rm obs}$; (d) the width of the dyadoregion at decoupling, $\Delta_{\rm d}$, obtained from Eq.~(\ref{eq:width}). The properties of the plasma at transparency are obtained from the solution of the following equations simultaneously. The first equation is obtained by substituting Eq.~(\ref{eq:transp2}) into Eq.~(\ref{eq:energy3}), and dividing it by $E_{\rm iso}$ \begin{equation} \frac{E^{obs}_{\rm P-GRB}}{E_{\rm iso}} = \frac{a T^4_{obs}}{16\Gamma^2}\sigma_T \frac{{\cal B}}{m_N c^2} \Delta_{d}, \end{equation} and the second equation is obtained from the energy conservation \begin{equation} 1 = \frac{E^{obs}_{ \rm P-GRB}}{E_{\rm iso}} + \frac{E_{\rm Kinetic}}{E_{\rm iso}}\, \label{eq:energy6} \end{equation} where $E_{\rm Kinetic}$ is the kinetic energy of the baryonic PEMB pulses \begin{equation} E_{\rm Kinetic} = (\Gamma -1 ) M_B c^2\, . \label{eq:energy7a} \end{equation} By substituting Eq.~(\ref{eq:energy7a}) in Eq.~(\ref{eq:energy6}) we have \begin{equation}\label{eq:energy7} {\cal B} = \frac{1}{\Gamma -1}\left(1-\frac{E^{obs}_{\rm P-GRB}}{E_{\rm iso}}\right). \end{equation} \section{Magnetic field and transparency condition and timescale of radiation during the UPE phase } \label{sec:magnetic} The time evolution of the mass and spin of BH during the UPE was discussed in in Sec~\ref{sec:massupe}; see Fig.~\ref{massspinupe}. In order to calculate the magnetic field during the UPE phase, i.e., in the time interval {$4.84<t_{\rm rf}< 10.89$~s,} we assume that the electric field therein is overcritical, which guarantees the occurrence of the UPE phase. Therefore, we infer a magnetic field strength $B_0=2.14 \times 10^{14}$~G such that the electric field given by Eq.~(\ref{eq:ER2}), at the end of the UPE phase at $t_{\rm rf}= 10.89$~s, fulfills $|E_{r_+}|= E_c$. For this magnetic field, the dyadoregion energy at {$t_{\rm rf}=4.84$ is $5 \times 10^{43}$~erg} obtained from Eq.~(\ref{Eemxi}). Figures~\ref{fig:eduringupe}[a] and \ref{fig:eduringupe}[b] show the evolution of magnitude of electric field and the dyadoregion energy during the UPE phase. The total isotropic energy of the UPE phase is $E^{\rm UPE}_{\rm iso}=2.23 \times 10^{53}$~erg, consequently, there exist $\sim 10^{9}$ PEMB pulses during UPE phase. For the first PEMB pulse, assuming $B_0=1.87 \times 10^{14}$~G, the width of the dyadoregion at transparency point is {$\Delta_{\rm d}= 4.1 \times 10^{4}$~cm;} obtained from Eq.~(\ref{eq:width}). From the hierarchical structure of UPE phase in this GRB presented by Eq.~\ref{eq:ratio}, we have $E^{\rm obs}_{\rm P-GRB}/E_{\rm iso}\sim 0.03$ and the temperature $k T_{\rm obs}\sim 50$~keV; see Table.~\ref{tab:180720B}. With these and following the previous section the transparency radius {$ R^{\rm tr} = 4.5 \times 10^9~\rm cm,$the baryon load parameter ${\cal B} = 3.1 \times 10^{-2},$} and finally the Lorentz factor $\Gamma = 38$, are obtained. After the first PEMB pulse, whose energetics ($\Delta E$) is paid by the rotational energy of the BH (by reducing the $\Delta J$ from the angular momentum of the BH), the system starts over with the new value of the \textcolor{black}{BH mass, angular momentum and} effective charge\textcolor{black}{, as explained above}. \begin{table} \small\addtolength{\tabcolsep}{12pt} \caption{ The parameters of the inner engine and the transparency point, obtained from the starting time of the UPE phase for GRB 180720B {($t_{\rm rf}=4.84$~s)} and GRB 190114C ($t_{\rm rf}=1.9$~s). \label{tab:comparision}} \centering \begin{tabular}{|c||c|c|c} \hline ~&GRB180720B &GRB190114C\\ ~& &\\ \hline \hline $ R^{\rm tr}$ (cm)& $4.5\times 10^9$ & $9.4\times 10^9$ \\ \hline ${\cal B}$& $3.1 \times 10^{-2}$ & $5.1 \times 10^{-3}$ \\ \hline $\Gamma $& 30 & 139 \\ \hline $\tau_{\rm q} $ (s)& $5 \times 10^{-10}$ & $3.1 \times 10^{-9}$ \\ \hline $\Delta_d $ (cm) & $4.1 \times 10^{4}$ & $1.8 \times 10^{5}$\\ \hline $B_0 $ (G) & $1.87 \times 10^{14}$ & $2.3 \times 10^{14}$\\ \hline $|\mathbf{E}|/E_c $ & $1.11$ & $1.25$\\ \hline \end{tabular} \end{table} We infer from the MeV luminosity, the evolution of the radiation timescale $\tau_q(t)$ of the PEMB pulses by requiring it to explain the MeV emission energetics, i.e.: \begin{equation}\label{tauinner} \tau_q(t)=\frac{E_{(r_+(t),r_d(t))}}{L_{\rm MeV}}, \end{equation} where the $E_{(r_+(t),r_d(t))}$ is the energy of dyadoregion from Eq.~(\ref{Eemxi}), determined from the new values of $J$ and $M$ for each PEMB pulse, and $L_{\rm MeV}$ is the MeV luminosity obtained from the best fit in Sec.~\ref{sec:massupe}. These parameters obtained from the starting time of the UPE phase, are similar to those of GRB 190114C; see Table~\ref{tab:comparision}. The evolution of the PEMB pulse timescale, the Lorentz $\Gamma$ factor, transparency radius, are shown in Fig.~\ref{fig:eduringupe} [d], Fig.~\ref{fig:eduringupe}[e] and Fig.~\ref{fig:eduringupe}[f], respectively. { We recall that the inner engine model has been first motivated to explain the GeV emission of GRBs as powered by an electrodynamical process that extracts the rotational energy of the newborn Kerr BH \cite{2019ApJ...883..191R, 2021A&A...649A..75M, 2022ApJ...929...56R}, and in \cite{2021PhRvD.104f3043M} for GRB 190114C and here for GRB 180720B it has been extended to explain the UPE phase. Following \cite{2021A&A...649A..75M, 2021PhRvD.104f3043M}, we summarize some key takeaways of our approach with respect to existing literature on this subject, in particular from numerical simulations. } { There is a vast literature about magnetic fields around BHs and how they may act in a mechanism that could extract the mass-energy of a Kerr BH. \citet{1975PhRvD..12.2959R} made an early attempt using a matter-dominated magnetized plasma accreting in a disk around a pre-existing Kerr BH. They used the infinite conductivity condition, $F_{\alpha \beta} u^\beta = 0$, where $F_{\alpha \beta}$ is the electromagnetic field tensor and $u^\beta$ is the plasma four-velocity, leading to $\mathbf{E}\cdot \mathbf{B}=0$. Under these conditions, the acceleration of particles and processes of energy extraction were not possible. This work was further developed by \citet{1977MNRAS.179..433B}, who introduced the concept of gaps and spontaneous $e^+e^-$ pair creation in the context of a BH, closely following the pulsar theory by \citet{1971ApJ...164..529S} and \citet{1975ApJ...196...51R}, to have regions in the magnetosphere where $\mathbf{E}\cdot \mathbf{B} \neq 0$. They imposed a force-free condition, $F_{\alpha \beta} J^\beta = 0$, where $J^\beta$ is the current density. Their aim was to produce an ultrarelativistic matter-dominated plasma whose bulk kinetic energy could be used to explain the energetics of a jet at large distances from the BH. The alternative view of \citet{1982MNRAS.198..339T} extended the work of \citet{1977MNRAS.179..433B} and analyzed the problem of matter-dominated accretion in a magnetic field anchored to a rotating surrounding disk. The physical system, however, remained the same of \citet{1977MNRAS.179..433B}. } { More recently, numerical simulations based on different models with respect to the one used in this article have been developed with the premise that the background electric field of a electro-vacuum solution (like the Papapetrou-Wald solution) might be screened from the surrounding plasma in the magnetosphere (see e.g. \citet{2005MNRAS.359..801K} and \citet{2019PhRvL.122c5101P}). These simulations have mainly addressed the physics of relativistic jets of plasma emerging from active galactic nuclei and x-ray binary systems and a especially detailed treatment and review of their theoretical models is presented by \citet{2005MNRAS.359..801K}. The choice of parameters and physical processes are different from the ones we have used for the GRB analysis. In our approach, we have been guided by the theoretical explanation of the following crucial observations of GRBs: (1) the time-resolved spectral analysis of the UPE phase; and (2) the MeV luminosity observed by Fermi-GBM. From this, we have identified the physical processes and parameters that have to be fulfilled in order to fit the vast amount of high-quality observational data. Their parameters enforce the condition $\mathbf{E} \cdot \mathbf{B} \neq 0$, while we use the Papapetrou-Wald solution which naturally possesses regions fulfilling such a condition in the BH vicinity. } {In our model, the magnetic field inherited from the collapsed NS is rooted in the surrounding material, and the electric field is created by the interaction of the gravitomagnetic field of the rotating BH with the external magnetic field. Since the electric field is assumed to be overcritical at the beginning, in a very short timescale of the order of the Compton time, $\hbar/(m_e c^2) \sim 10^{-21}$ s, which is much shorter than any electromagnetic process, it is originated a region dominated by the high density and high pressure of the neutral PEMB pulse. The PEMB pulse self-accelerates to the ultrarelativistic regime and finally reaches transparency at a radius $\sim 10^{10}$ cm. } { As soon as the BH forms, the first and the most efficient process in action to produce the $e^+e^-$ plasma and, consequently decreasing the rotational energy of BH, occurs through the Schwinger critical field pair production. Since an overwhelming amount of pair plasma is created in quantum timescales, the plasma expansion by its internal pressure starts well before any electric field screening. This process takes a fraction of angular momentum of the Kerr BH. The BH then is left with a slightly smaller angular momentum $J^*= J - \Delta J$, with $\Delta J/J \sim 10^{-9}$, being $\Delta J$ the angular momentum extracted to the BH and the same magnetic field. This process leads to a new, lower value of the induced electric field. This process continues up to the moment when the electric field becomes undercritical. } { The expanding $e^+e^-$ photon plasma sweeps matter in the cavity reducing the density of the latter to values as low as $\sim 10^{-14}$ g cm$^{-3}$, as shown by numerical simulations in \citet{2019ApJ...883..191R}. This low-density ionized plasma is needed to fulfill an acceleration of charged particles leading to the electrodynamical process around a newborn BH. This density is much lower the Goldreich-Julian density, for instance $\rho_{\rm GJ}\sim 10^{-11}$ g cm$^{-3}$, obtained for the present inner engine parameters. Moreover, the matter energy density inside the cavity is negligible comparing to the electromagnetic energy density, namely $\rho_M/(B^2 - E^2)\sim 10^{-14}$, while in \citet{2005MNRAS.359..801K} (see also \cite{2019PhRvL.122c5101P}), this ratio is $0.05$ or higher. } \section{Discussion and Conclusions}\label{sec:conc} Following a new paradigm opened by the theoretical understanding and data analysis of GRB 190114C \cite{2019arXiv190404162R,PhysRevD.104.063043}, we have analyzed in this paper the UPE phase of GRB 180720B. We have here shown that also in GRB 180720B, a time-resolved spectral analysis conducted on shorter and shorter time intervals reveals the hierarchical structure of the UPE. Namely, the spectrum of the UPE phase, obtained in multiply rebinned time intervals, holds its features and is always fitted by a BB+CPL model (see Fig. \ref{alltogether}). We have shown the statistical significance of such a structure down to a time resolution of $0.11$ s. We have then linked the above hierarchical structure of the UPE phase to a sequence of microphysical elementary events in the QED regime of the inner engine, occurring on a timescale of $\tau_q \sim 10^{-9}$~s. The understanding of the underlying quantum nature is not possible without the discovery of the observed hierarchical structure of the UPE phase. The inner engine is composed of a Kerr BH rotating in a uniform magnetic field $B_0$, aligned with the BH rotation axis, described by the Papapetrou-Wald solution, immersed in a rarefied plasma. The gravitomagnetic interaction of the rotating BH with the magnetic field induces an electric field. The process that originates the $10$~keV--$10$~MeV radiation is triggered by the vacuum polarization that occurs when the induced electric field in the inner engine is overcritical, i.e., $|{\bf E}|>E_c$. This process forms around the BH an optically thick pair $e^+~e^-~\gamma$ plasma whose high internal pressure drives its self-accelerating expansion. During the expansion, the plasma is loaded with baryons forming the PEMB pulse that reaches ultrarelativistic regime with $\Gamma\sim 30$ and the transparency point where the radiation becomes observable \cite{2010PhR...487....1R}. We assume that the magnetic field $B_0 \sim10^{14}$~G is constant during the UPE phase. In the radiation timescale of the PEMB pulses, $\tau_q \sim 10^{-9}$~s, the above process extracts $\Delta J\sim 10^{-9}~J$ of angular momentum of the Kerr BH, leaving it with a new, lower angular momentum $J^* = J- \Delta J$. Since the magnetic field is assumed constant during the UPE phase, the new value of the induced electric field is lower. Then, the system starts a new vacuum polarization process in the presence of the same magnetic field $B_0$, and a new effective charge of $Q^*_{\rm eff}= Q_{\rm eff}-\Delta Q_{\rm eff}$, where $\Delta Q_{\rm eff} =2 B_0 \Delta J$. It leads to the production of approximately $10^9$ PEMB pulses, which one after another reach the transparency point and their radiations form the UPE phase. This process continues till the electric field lowers to $|{\bf E}|<E_c$. The magnetic field in this scenario is inherited from the NS and is amplified in the gravitational collapse to a BH. Consequently, the electric field and consequent effective charge, $Q_{\rm eff}=2 B_0 J G/c^3$, are induced by the gravitomagnetic interaction of the rotating BH with the external magnetic field \citep{2020ApJ...893..148R, 2022ApJ...929...56R}. The electric field is overcritical during the UPE phase. In a quantum timescale, $\hbar/(m_e c^2) \approx 10^{-21}$ s, the dyadoregion characterized by the high density and high pressure of the $e^+e^-\gamma$ plasma develops and dominates over any other electromagnetic process \cite{2010PhR...487....1R}. \begin{acknowledgements} \textcolor{black}{We thank the Referee for the deep remarks and comments which have improved the presentation of the results and readability of the paper.} \end{acknowledgements}

Title: TOI-4562 b: A highly eccentric temperate Jupiter analog orbiting a young field star

Abstract: We report the discovery of TOI-4562 b (TIC-349576261), a Jovian planet orbiting a young F7V-type star, younger than the Praesepe/Hyades clusters ($\sim$ 300-400 Myr). This planet stands out because of its unusually long orbital period for transiting planets with known masses ($P_{\mathrm{orb}}$ = $225.11757^{+0.00027}_{-0.00025}$ days), and because it has a substantial eccentricity ($e$ = $0.81^{+0.05}_{-0.05}$). The location of TOI-4562 near the southern continuous viewing zone of TESS allowed observations throughout 25 sectors, enabling an unambiguous period measurement from TESS alone. Alongside the four available TESS transits, we performed follow-up photometry using the South African Astronomical Observatory node of the Las Cumbres Observatory, and spectroscopy with the CHIRON spectrograph on the 1.5 m SMARTS telescope. We measure a radius of $1.072_{-0.043}^{+0.044}$ R$_{\mathrm{Jup}}$, and a mass of $3.29^{+1.88}_{-0.82}$ M$_{\mathrm{Jup}}$,for TOI-4562 b. The radius of the planet is consistent with contraction models describing the early evolution of the size of giant planets. We detect tentative transit timing variations at the $\sim$ 20 min level from five transit events, favouring the presence of a companion that could explain the dynamical history of this system if confirmed by future follow-up observations. With its current orbital configuration, tidal timescales are too long for TOI-4562 b to become a hot-Jupiter via high eccentricity migration, though it is not excluded that interactions with the possible companion could modify TOI-4562 b's eccentricity and trigger circularization. The characterisation of more such young systems is essential to set constraints on models describing giant planet evolution.

https://export.arxiv.org/pdf/2208.10854

\title{\toib: A highly eccentric temperate Jupiter analog orbiting a young field star.} \correspondingauthor{Alexis Heitzmann} \email{alexis.heitzmann@usq.edu.au} \author[0000-0002-8091-7526]{Alexis Heitzmann} \affiliation{\USQ} \author[0000-0002-4891-3517]{George Zhou} % \affiliation{\USQ} \author[0000-0002-8964-8377]{Samuel N.~Quinn} % \affiliation{\CfA} \author[0000-0003-0918-7484]{Chelsea X. Huang} % \affil{\USQ} \author[0000-0002-3610-6953]{Jiayin Dong} % \altaffiliation{Flatiron Research Fellow} \affiliation{\CCA} \affiliation{\PennStateAA} \affiliation{\PennStateCEHW} \author[0000-0002-0514-5538]{L. G. Bouma} % \altaffiliation{51 Pegasi b Fellow} \affiliation{\Caltech} \author[0000-0001-9677-1296]{Rebekah I. Dawson} % \affiliation{\PennStateAA} \affiliation{\PennStateCEHW} \author[0000-0001-5522-8887]{Stephen C.~Marsden} % \affiliation{\USQ} \author[0000-0001-7294-5386]{Duncan Wright} % \affiliation{\USQ} \author[0000-0001-7624-9222]{Pascal Petit} % \affiliation{\IRAP} \author[0000-0001-6588-9574]{Karen A.\ Collins} % \affiliation{\CfA} \author[0000-0003-1464-9276]{Khalid Barkaoui} % \affiliation{\LiegeUni} \affiliation{\MITAtmo} \affiliation{\IACSpain} \author[0000-0001-9957-9304]{Robert A.~Wittenmyer} % \affil{\USQ} \author[0000-0003-2851-3070]{Edward Gillen} % \altaffiliation{Winton Fellow} \affiliation{\QMU} \affiliation{\CAM} \author[0000-0002-9158-7315]{Rafael Brahm} % \affiliation{\AIUniChile} \affiliation{\MIAChile} \author[0000-0002-5945-7975]{Melissa Hobson} % \affiliation{\MaxPlank} \affiliation{\MIAChile} \author[0000-0002-3439-1439]{Coel Hellier} % \affiliation{\KeeleUni} \author[0000-0002-0619-7639 ]{Carl Ziegler} \affiliation{Department of Physics, Engineering and Astronomy, Stephen F. Austin State University, 1936 North St, Nacogdoches, TX 75962, USA} \author[0000-0001-7124-4094]{C\'{e}sar Brice\~{n}o} \affiliation{Cerro Tololo Inter-American Observatory, Casilla 603, La Serena, Chile} \author{Nicholas Law} \affiliation{Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3255, USA} \author[0000-0003-3654-1602]{Andrew W. Mann} \affiliation{Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3255, USA} \author[0000-0002-2532-2853]{Steve B. Howell} \affil{NASA Ames Research Center, Moffett Field, CA 94035, USA} \author[0000-0003-2519-6161]{Crystal~L.~Gnilka} \affil{NASA Ames Research Center, Moffett Field, CA 94035, USA} \author[0000-0001-7746-5795]{Colin Littlefield} \affiliation{Bay Area Environmental Research Institute, Moffett Field, CA 94035, USA} \affiliation{NASA Ames Research Center, Moffett Field, CA 94035, USA} \author[0000-0001-9911-7388]{David W.~Latham} % \affiliation{\CfA} \author[0000-0001-6513-1659]{Jack J. Lissauer} % \affiliation{Space Science \& Astrobiology Division , MS 245-3, NASA Ames Research Center, Moffett Field, CA 94035, USA} \author[0000-0003-4150-841X]{Elisabeth R.~Newton} % \affiliation{Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA} \author[0000-0001-9626-0613]{Daniel M.~Krolikowski} % \affiliation{\UTAustin} \author[0000-0002-6549-9792]{Ronan Kerr} % \affiliation{\UTAustin} \author[0000-0001-7337-5936]{Rayna Rampalli} % \affiliation{Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA} \author[0000-0001-7371-2832]{Stephanie T. Douglas} % \affiliation{Department of Physics, Lafayette College, 730 High St., Easton, PA 18042, USA} \author[0000-0002-9138-9028]{Nora L.~Eisner} % \affiliation{Sub-department of Astrophysics, University of Oxford, Keble Rd, Oxford, United Kingdom} \author{Nathalie Guedj}% \affiliation{\plahunters} \author{Guoyou Sun} % \affiliation{\plahunters} \author{Martin Smit} % \affiliation{\plahunters} \author{Marc Huten} % \affiliation{\plahunters} \author{Thorsten Eschweiler} % \affiliation{\plahunters} \author{Lyu Abe} \affiliation{Universit\'e C\^ote d’Azur, Observatoire de la C\^ote d’Azur, CNRS, Laboratoire Lagrange, CS 34229, F-06304 Nice Cedex 4, France} \author[0000-0002-7188-8428]{Tristan Guillot} \affiliation{Universit\'e C\^ote d’Azur, Observatoire de la C\^ote d’Azur, CNRS, Laboratoire Lagrange, CS 34229, F-06304 Nice Cedex 4, France} \author[0000-0003-2058-6662]{George Ricker} \affiliation{Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA} \author[0000-0001-6763-6562]{Roland Vanderspek} \affiliation{Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA} \author[0000-0002-6892-6948]{Sara Seager} \affil{Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA} \affil{Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA} \affil{Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA} \author[0000-0002-4715-9460]{Jon M. Jenkins} \affiliation{NASA Ames Research Center, Moffett Field, CA 94035, USA} \author[0000-0002-8219-9505]{Eric B. Ting} % \affiliation{NASA Ames Research Center, Moffett Field, CA 94035, USA} \author[0000-0002-4265-047X]{Joshua N.~Winn} \affiliation{Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA} \author[0000-0002-5741-3047]{David R.~Ciardi} % \affiliation{\NESI} \author[0000-0001-7246-5438]{Andrew M.~Vanderburg} % \affiliation{\kavlimit} \affiliation{\WisconsinMadisonUni} \author[0000-0002-7754-9486]{Christopher~J.~Burke} \affiliation{Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA} \author[0000-0003-1286-5231]{David~R.~Rodriguez} \affiliation{Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA} \author[0000-0002-6939-9211]{Tansu~Daylan} \affiliation{Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA} \affiliation{LSSTC Catalyst Fellow} \section{Introduction} \label{sec:intro} Planetary systems evolve rapidly within the first hundreds of millions of years of formation. The architectures of the systems evolve before settling into their eventual orbital configuration. Planets with extensive gaseous envelopes are expected to undergo contraction and cooling and experience observable changes in radius within this time frame. Observations of planets around young stars help anchor our understanding of this era of rapid change and help define models of planet formation and evolution. In particular, Jovian planets in distant orbits are less affected by stellar irradiation than close-in hot Jupiters. Transiting cold Jupiters around young stars can therefore provide constraints for cooling and contraction of giant planet evolution models. The orbital properties of these planets can also help to narrow down the timescales of dynamical evolution experienced by many other giant planets discovered to date. Numerous mechanisms are responsible for the formation and evolution of close-in Jovian planets. These mechanisms vary by the distribution of planets that they produce and by the timescales at which they operate. We can best assess the prevalence of these multiple formation channels via a census of the gas giant population as a function of time (see \citealt{2018ARA&A..56..175D}). Such a temporal survey of planetary systems can unveil the roles that in-situ formation (review in~\citealt{2014prpl.conf..619C}), disk migration (review in~\citealt{2014prpl.conf..667B}) and high eccentricity migration (review in~\citealt{2018ARA&A..56..175D}) played in shaping our current gas giant population. For example, planets can gravitationally interact with their depleting gas disks, resulting in moderately eccentric final orbits within a few million years \citep[e.g.,][]{2003ApJ...586.1374N, 2015ApJ...812...94D,2021MNRAS.500.1621D}. On the other extreme, excitation via stellar fly-bys can occur on the hundreds of millions of years timescale \citep[e.g.,][]{2016ApJ...816...59S}. Gas giants also undergo significant contraction in the first hundred million years post formation. In models, the rate of contraction is strongly dependent on the initial conditions of the planet post formation, such as their envelope-core mass ratio and initial luminosities \citep[e.g.,][]{2007ApJ...659.1661F,2019A&A...623A..85L}. It is clear, however, that the radius distribution of close-in Jovian planets is shaped by external factors that retard their contraction \citep[e.g.,][]{2002A&A...385..156G,2003A&A...402..701B,2010ApJ...714L.238B}. Young planets in distant orbits provide simpler key tests for gas giant evolution. Missions like Kepler, K2, and the Transiting Exoplanet Survey Satellite (\TESS; \citealt{2015JATIS...1a4003R}) have brought forth a growing number of planetary systems about young stars \citep[e.g.,][]{2019ApJ...880L..17N,2020AJ....160..179M,2020Natur.582..497P, 2022AJ....163..121B,2022arXiv220501112B, 2022AJ....163..289Z}. However, true young Jovian analogues are rare. Interestingly, \citet{2021NatAs...6..232S} measured the masses of the giant planets in the 22 Myr old V1298 Tau system \citep{2019ApJ...885L..12D,2019AJ....158...79D}, finding that the two Jovian planets have already settled to their expected final radii, a process that is predicted to take hundreds of millions of years by contraction models. Other close-in Jovian-sized planets have also been found around young stars \citep{2020AJ....160...33R,2020AJ....160..239B,2021arXiv211009531M}, but strong stellar activity has prevented their mass from being measured. We report the discovery of a young transiting Jovian planet in a distant orbit around a $\sim 300$ Myr old star. \toi hosts a temperate-Jupiter in a 225\,day period orbit near the \TESS continuous viewing zone. Along with additional observations from our ground-based photometric follow-up campaign, five total transits of the planet were obtained, unambiguously identifying the period of the system. Radial velocity monitoring over the following two years provided a mass and eccentricity measurement for the young planet. In addition, data from FEROS helped to constrain the stellar parameters, and high resolution images from Gemini-South and SOAR helped to rule out false positive scenarios, confirming the transit candidate as a true planet. We also constrained the age of \toi via gyrochronology and lithium. Finally we detect a transit timing variations (TTV) signature, indicative of a perturbing companion in the system. \toib is one of the longest period transiting temperate Jupiters discovered by \TESS, and the youngest amongst such planets. Missions like \TESS and \emph{PLATO}~\citep{2014ExA....38..249R} have the potential to uncover this special population that critically constrains cooling models and migration pathways for Jovian planets. \section{Observations} \label{sec:obs} \subsection{\TESS: Photometry} \label{subsec:tess} The transiting planet candidate around \toi was first identified from observations by \TESS. \toi lies in the Southern Continuous Viewing Zone of \TESS, and therefore received near-uninterrupted photometric monitoring during years 1 and 3 of operations. The target received observations at 30\,minute cadence during Sectors 1-8 (2018-07-25 to 2019-02-28) and 10-13 (2019-03-26 to 2019-07-18), and 2\,minute target-pixel-stamp observations during sectors 27-39 (2020-07-04 to 2021-06-24). The transit signature of TOI-4562b was detected by the TESS Science Processing Operations Center (SPOC;~\cite{2016SPIE.9913E..3EJ}) at NASA Ames Research Center during a transit search of sectors 27 through 39 with an adaptive, noise-compensating matched filter~\citep{2002ApJ...575..493J,2010SPIE.7740E..0DJ,2020TPSkdph}. The transit signature passed all the diagnostic tests in the Data Validation report~\citep{Twicken:DVdiagnostics2018} and was fitted with an initial limb-darkened transit model~\citep{Li:DVmodelFit2019}. In particular, the transit signal passed the difference image centroiding test, which localized the source of the transits to within $1.0 \pm 2.5$ arcsec. The TESS Science Office reviewed the diagnostic information and released an alert to the community for TOI-4562b on 28 October 2021 \citep{guerrero:TOIs2021ApJS}. We make use of the MIT Quicklook pipeline \citep{2020RNAAS...4..204H} photometric extraction from the Full Frame Image observations. In addition, where available, we make use of the 2\,minute cadence target pixel file observations from the crowding and flux fraction corrected Simple Aperture Photometry (CROWDSAP) light curves \citep{twicken:PA2010SPIE,morris2020} made available by SPOC. Because of the large stellar variability seen in the light curve, we used the SAP light curves rather than the Pre-search Data Conditioning SAP (PDCSAP) flux and performed the detrending using a high order spline interpolation \citep{2014PASP..126..948V} The full \TESS light curve covering all sectors of observations is presented in Figure~\ref{fig:full_lc}. During the two (non-consecutive) years of near-continuous observations a total of 4 transits were captured by \TESS. Figure~\ref{fig:transits} shows the zoomed in region around each of these transits. \toi was first identified as a potential young star due to its strong rotational modulation \citep{2021AJ....161....2Z}, as part of our program to survey for planets around young field stars. We performed a search for transiting signals around \toi via a Box-least-squares period search \citep{2002A&A...391..369K} after removal of the stellar modulation signal with the splines. This detrending was not the one used for the transit modeling, described in section~\ref{subsec:transits}. \subsection{Follow-up photometry} \label{subsec:LCO} We obtained follow-up photometric confirmation of the planetary transit via the Las Cumbres Observatory Global Network \citep[LCOGT;][]{2013PASP..125.1031B}. Transit opportunities for a 225\,day period planet are rare from the ground (see Table~\ref{tab:upcoming_transits}). We captured the full transit of \toib on 2022-01-03 UTC from the South African Astronomical Observatory (SAAO) node of LCOGT via two 1\,m telescopes. The observations were obtained with the \emph{Sinistro} 4K$\times$4K cameras in the Sloan $i'$ filter. The observations were calibrated via the \textit{BANZAI} pipeline \citep{McCully:2018}, and light curves were extracted via the \textit{AstroImageJ} package \citep[AIJ;][]{2017AJ....153...77C} using circular apertures with radius $4\farcs7$, which exclude flux from all known nearby Gaia EDR3 and TESS Input Catalog stars. The combined light curves (after removing systematics) and best fit model are shown in Figure~\ref{fig:transits}. In addition, a transit on 2022-08-16 was attempted from the SAAO node of LCOGT via one 1\,m telescope, as well as the Antarctica Search for Transiting ExoPlanets (ASTEP) facility \citep{2015AN....336..638G,2016MNRAS.463...45M}, located at the East Antarctic plateau. A 25\,minute segment was captured out of transit, but no portions of a transit event was recorded, and the dataset not included in the modeling presented below. \subsection{CHIRON/SMARTS: Spectroscopy} \label{subsec:chiron} To characterize the radial velocity orbit of \toib and constrain the properties of the host star, we obtained 84 spectroscopic observations of \toi using the CHIRON facility. To capture the long orbital period of \toib, the velocities spanned two observing seasons, from 2020-12-09 to 2022-01-23; the resulting radial velocities are given in Table~\ref{tab:RVs}. CHIRON is a fiber-fed high resolution echelle spectrograph on the 1.5\,m SMARTS telescope at Cerro Tololo Inter-American Observatory, Chile \citep{2013PASP..125.1336T}. Due to the faintness of the host star, spectral observations were obtained in the `fiber' mode of CHIRON, yielding a resolving power of $R\sim28,000$ over the wavelength range of 4100 to $8700$\,\AA{}, and an average signal-to-noise of $\sim 100$ per resolution element at the Mg b line wavelength region. We make use of the extracted spectra from the standard CHIRON pipeline described in \citet{2021AJ....162..176P}. Radial velocities were derived from the observations via a least-squares deconvolution against a non-rotating ATLAS9 spectral template \citep{Castelli:2004}. The resulting broadening profile is fitted via a kernel describing the effects of radial velocity shift, rotational, macroturbulent, and instrumental broadening. The derived velocities are presented in Table~\ref{tab:RVs} and shown in Figure~\ref{fig:RV}. To estimate the spectroscopic properties of the host star, we matched each spectrum against an observed library of $\sim 10,000$ spectra pre-classified by the Spectroscopic Classification Pipeline \citep{2012Natur.486..375B}. The matching was performed by first training the pre-classified library via a gradient boosting classifier using \textsc{scikit-learn}, and then classifying the observed spectrum. We found that \toi has an effective temperature of \teff = \teffCHIRON~K, a surface gravity of log\,$g$\,= \loggCHIRON~dex, and bulk metallicity of [$\mathrm{Fe}$/H] = \fehCHIRON~dex. Since the CHIRON dataset overwhelms the other datasets we obtained for TOI-4562 in quantity, we adopt these parameters as Gaussian priors in the global analysis of the system described in Section~\ref{sec:analysis}. We note a general consensus between the spectral parameters from CHIRON and those presented below in Section~\ref{subsec:FEROSGALAH}. We also check for the possibility that the velocity variations we observe are due to a spectroscopically blended companion rather than the host star. We compare the broadening measured from the line profiles against the velocities and find no correlation. If a blended companion is causing the radial velocity offset, then the line profiles should be broadest at the orbital quadratures, and narrowest at conjunctions. We therefore find no evidence that the velocity variations originate from a blended companion. \subsection{FEROS \& GALAH: Spectroscopy} \label{subsec:FEROSGALAH} The FEROS spectrograph, attached to the MPG 2.2 m \citep{1999Msngr..95....8K} telescope at La Silla Observatory, gathered 11 spectra of \toi. Spectra are co-added, with a signal to noise ratio per spectra ranging between 52 and 82, and atmospheric parameters are derived using \textsc{ZASPE} \citep{2017MNRAS.467..971B}. We find \teff\,=\,6280\,$\pm$\,100 K, log $g$\,= $4.49 \pm 0.10$, [$\mathrm{Fe}$/H]\,= $0.24 \pm 0.05$~dex and \vsini\,= $15.67 \pm 0.5$~\kms. We chose not to include the FEROS data in the RV modelling. All points fall near phases (-0.4, 0.025 and 0.35) where the RV signal is close to 0 and therefore don't meaningfully contribute, while adding one instrument and the associated extra parameters. Using the \textsc{ceres} pipeline \citep{2017PASP..129c4002B}, we also recover chromospheric emission indices, tracers of stellar activity. The core emission of the $\mathrm{H_{\alpha}}$ line at 6562.808 \r{A} is $\mathrm{H_{\alpha}}$\,=\,0.160\,$\pm$\,0.005 (following \citet{2009A&A...495..959B}). Using regions defined by \citet{1991ApJS...76..383D} and calibrations from \citet{1984ApJ...279..763N} we measure the core emission of the Ca II H and K lines around 3933 \r{A} and 3968 \r{A} to be log\,$R^{\prime}_{HK}$\,=\,-4.503\,$\pm$\,0.044. This value is consistent with a young active star \citep{2008ApJ...687.1264M}.\par Finally, legacy spectra from the GALAH survey \citep{2021MNRAS.506..150B} found \teff\,=\,6034\,$\pm$\,77 K, log\,$g$\,=\,4.36\,$\pm$\,0.18, [$\mathrm{Fe}$/H]\,=\,0.08\,$\pm$\,0.06 and \vsini\,=\,15.6\,$\pm$\,2.2\,\kms. \subsection{Gemini-South and SOAR: High resolution direct Imaging} \label{subsec:direct-imaging} A first high resolution image of \toi was obtained on 2022-03-17 with the Zorro Speckle camera on the 8.1 m Gemini-South telescope~\citep{2022FrASS...9.1163H} and is shown in the top of Figure~\ref{fig:highres}. Simultaneous observations were obtained at 562 and 832 nm respectively. Contrast curves were retrieved following \cite{2011AJ....142...19H} for both wavelengths and neither shows sign of a companion in the vicinity of \toib. A difference in magnitude $\Delta$m of 5 is achieved at a separation of $\sim0.1$". This allows us to rule out the presence of bright stellar objects in the same TESS pixel as \toi that would meaningfully impact the transit light curve to a projected distance of $\sim$ \sepvalue~au (given \toi's distance of \distvalue~pc). On 2022-04-19, another high resolution image was acquired with the HRCam instrument on the 4.1 m Southern Astrophysical Research (SOAR) telescope. \toi was observed as part of the SOAR TESS survey~\citep{2020AJ....159...19Z,2021AJ....162..192Z}, and the data was reduced following~\cite{2018PASP..130c5002T}. The image shown in the bottom panel of Figure~\ref{fig:highres} and shows a contrast in the $I$-band of 5 mag within 1" with no sign of a companion, in agreement with the Gemini-South observation. \section{Age of \toi} \label{subsec:age} \toi does not appear in the extensive list of stars with known age and/or belonging to associations and moving groups compiled from the literature in \cite{Bouma_2022}. Similiarly, we do not identify a co-eval population when applying the \textsc{comove} package~\citep{2021AJ....161..171T} that uses Gaia DR3 astrometric parameters to find whether a given possible young star candidate is co-moving with its visual neighbours. This lack of evidence of \toi belonging to any known moving group or open cluster means its age estimation is challenging. The variability seen in both photometry and radial velocity are indicative of the presence of rotationally modulated surface brightness features, likely due to the presence of dark spots and bright plages/faculae. Combined with a fast rotation period (\Prot~= \Protvalue~days), this strongly suggests that \toi is a young and active star. Determining the age of a field star is notoriously difficult \citep{2010ARA&A..48..581S}. In the following paragraphs, we make use of the rotation and lithium abundance of \toi to qualitatively assess its youth. We note that though \toi exhibits signatures of activity and youth indicative of being younger than 1 Gyr, pin-pointing its age will remain difficult without placing it within co-moving populations. With increasingly more sophisticated clustering with updated \emph{Gaia} datasets, we hope that kinematics studies such as \citet{2017AJ....153..257O}, \citet{2018ApJ...856...23G}, \citet{2019AJ....158..122K} and \citet{2020AJ....159..166U} can provide improved census of young associations and groups. \input{Tables/Stellar_parameters} \subsection{Stellar rotation and Gyrochronology} \label{subsec:gyro} Young stars on the zero-age main-sequence spin rapidly. Over the course of a few billion years, mass loss from stellar winds spin-down Sun-like stars. The rotation period of Sun-like stars can be a tracer for their age. Rotation--color--age relationships such as those from \citet{2007ApJ...669.1167B} and \citet{2008ApJ...687.1264M} are calibrated against co-eval clusters and associations, and can provide useful metrics to estimate stellar ages. Recent theoretically-motivated models, which are based in wind braking models and can incorporate core-envelope coupling, also provide such relationships \citep[e.g.][]{2020A&A...636A..76S}. The 25 sectors of observations gathered by \TESS provide the means for a good estimation of the rotation period of \toi. As shown in Figure~\ref{fig:per_TESS}, we ran a Lomb-Scargle period analysis (\citealt{Lomb1976,Scargle1982}) on the entire dataset and measured a rotation period of \Prot~= \Protvalue~days. We note the clear second periodogram peak on Figure~\ref{fig:per_TESS}, close to \Prot. This could be showing differential rotation (i.e., the variation of \Prot~as a function of stellar latitude). This has been largely observed in Kepler stars~\citep{2013A&A...560A...4R}. We could suppose that the rotational modulation of two distinct clumps of surface stellar spots evolving at a different latitude would be at origin of the double peak \citep{1993A&A...269..351L}. In addition, \toi received 4 years of monitoring with the Wide Angle Search for Planets (WASP) Consortium \citep{2006PASP..118.1407P} Southern SuperWASP facility from 2008-2012. WASP-South is located at SAAO, and consists of an array of eight commonly mounted 200\,mm f/1.8 Canon telephoto lenses, each with a $2\mathrm{K} \times 2\mathrm{K}$ detector. A period analysis of the WASP-South light curves reveals a 3.84 day periodicity, in agreement with the \TESS light curves. The long term stability of the signal helps to confirm it as the correct alias of the rotational modulation signal. Finally, we run periodograms on the available light curves from the All-Sky Automated Survey for Supernovae (ASAS-SN, \cite{2014AAS...22323603S,2019MNRAS.485..961J}). Sloan g-band data spanning from October 2017 to April 2022 shows very strong peaks in the periodogram around 3.85 days, agreeing with the other photometric datasets. Johnson V-band data was obtained between October 2016 and September 2018. Despite being less extensive and less densely sampled than the g-band photometry, a moderate peak (FAP $\sim$\,0.2\%) is found at 3.64 days, close to \Prot. Using the age-rotation relationship from \cite{2008ApJ...687.1264M}, we found \toib to be 110-490 (3$\sigma$) Myr old. We note that age estimates from this relationship assumes that the star lies on the slow-sequence of the age-rotation relationship. Stars are often found to be more rapidly rotating than such sequences for a given age, which has been attributed to binarity in cluster populations \citep[e.g.,][]{2016ApJ...822...47D,2020MNRAS.492.1008G}. Though there is no evidence for \toi being part of a binary system, caveats still apply for gyrochronology-based age estimates. For a 1.2 \Msun~star with \Prot = \Protvalue, the model from~\cite{2020A&A...636A..76S} gives a consistent age estimate of 300-400 Myr. The top plot of Figure~\ref{fig:Liew} shows the rotation period of \toi compared with stars of known nearby clusters and associations. \toi's \Prot\,is consistent with that of stars belonging to Group X~\citep{2022arXiv220606254N,2022A&A...657L...3M}, with an estimated age of 300 Myr. \subsection{Lithium} \label{subsec:Li} The convective envelope of low-mass stars (\Mstar\,$<$\,1.5 \Msun) allows efficient transport of lithium to deeper and hotter regions in a star's interior, where it gets destroyed by proton capture. Calibrated with stars in clusters and associations, this lithium depletion can be used as a proxy for stellar age. Using CHIRON spectra (see section \ref{subsec:chiron}), we measured the equivalent width of the lithium doublet at 6707.76 and 6707.91 \r{A}. We fit two Gaussian line profiles of the same depth at the respective wavelengths of the lithium doublet and one auxiliary with a different depth to account for the nearby Fe I line at 6707.43 \r{A}, usually blended with the Li doublet. All profiles share the same width as per the rotational broadening of the star. We measure a lithium equivalent width of 0.084 $\pm$ 0.007 \r{A} from a median combined spectrum of all our CHIRON observations. These data are displayed in Figure~\ref{fig:Liew}. On the same figure, we show the lithium equivalent width as function of effective temperature for stars belonging to clusters with well constrained ages, the Pleiades ($\sim$\,125 Myr), Group X ($\sim$\,300 Myr) and Praesepe ($\sim$\,670 Myr). \toi exhibits a Li equivalent width shallower than most of the Pleiades stars and of comparable strength to stars from the Praesepe cluster, at an effective temperature of 6000 K. Combined with the Gyrochronology analysis, \toi's age is consistent with a star younger than the Praesepe/Hyades clusters (i.e., $\lesssim$ 700 Myr). \section{Analysis and Results} \label{sec:analysis} To best determine the system properties of \toi, we perform a joint modeling of all available photometric and spectroscopic datasets, including stellar isochrone models that constrain the properties of the host star. The paragraphs below detail individual components of this model. \subsection{Transit modeling} \label{subsec:transits} Despite the 225 day orbital period of \toib, the extensive observations of \toi by \TESS allowed four transits to be observed. Spot modulated variability at the $\sim$3\% level is seen on the \TESS light curve due to the active nature of \toi, as expected given its young age. For the purposes of the transit modeling, we detrend the region around each transit epoch with a fourth-order polynomial. The polynomial is fitted using the out-of-transit regions of the light curve within 0.5\,days of the transit center. We model the transits as per \citet{2002ApJ...580L.171M} via the \textsc{batman} package \citep{2015PASP..127.1161K}. Free parameters that describe the transit model include the transit centre \tc\ at each transit epoch, radius ratio $R_{\mathrm{p}}$/$R_{\mathrm{\star}}$, line of sight inclination of the transit $i$, and the eccentricity parameters $\sqrt{e}\cos\omega$ and $\sqrt{e}\sin \omega$. A quadratic model was used to account for Limb Darkening using coefficients $\mu_{1TESS}$ and $\mu_{2TESS}$ fixed to those interpolated from \cite{2017AA...600A..30C} at the atmospheric parameters of \toi for the \TESS transits. We note that $a$/$R_{\mathrm{\star}}$ was not directly sampled but rather computed from the free parameters \Porb, \Mstar, \Rstar\,and planet mass \Mp. For the two (same epoch, different telescopes) SAAO LCOGT transits, the Limb Darkening coefficients $\mu_{1_{LCO}}$ and $\mu_{2_{LCO}}$ are computed for the SDSS i' band from \citet{2011AA...529A..75C}, using the interpolation routine from \cite{2013PASP..125...83E} with \teff\,= 6000\,K, log $g$\,= 4.5 and \met\,= 0.1, computed with the least square method (LSM). For the SAAO LCOGT data, we also incorporate the effects of instrumental systematic variations that are common to ground-based photometric observations via a simultaneous detrending of the light curve against parameters describing the observation airmass to which we add a linear trend with respect to time. All detrended light curves and the best transit model fits are shown in Figure~\ref{fig:transits}.\par \subsection{Radial velocity modeling} \label{subsec:rv} The radial velocities obtained over the 2 consecutive orbits of \toib were modeled using a Keplerian orbit. Some fitted parameters are shared with the transits and stellar isochrone fitting, such as \tc, \Porb, \ars, \Rstar, \Mstar, $i$, \secosw~and \sesinw. To model the velocities, we add the planet mass \Mp, a radial velocity offset \offsetchiron, and a white noise term for each year of data, $\sigma_{Y1}$ and $\sigma_{Y2}$ to account for the stellar noise being noticeably different from the first year to the next. The semi-amplitude of the planetary signature \Kamp~was computed from the above parameters. The orbital solution and the associated likelihood from the fit to the data are computed from \Kamp, \tc, \Porb, \secosw~and \sesinw~via the \textsc{radvel} package \citep{2018PASP..130d4504F}.\par We also try to add a Gaussian Process using a Quasi-Periodic kernel, implemented through \textsc{radvel} to model the stellar noise apparent in the data. The resulting parameter values do not yield a significant difference, therefore not justifying the necessity to use a correlated noise model to account for the stellar intrinsic variability seen in the radial velocities. With one datapoint a day at most, the sampling is too sparse for the Gaussian Process to correctly grasp the $\sim$ 4 days stellar period. Crudely assuming a spot covering 0.6-1.2\% ($\delta_{spot}$) of the stellar surface, we can approximate an activity induced radial velocity semi-amplitude $K_{act}$ of $v$ sin $i\times \delta_{spot} \sim 100-200$ \ms, comparable to the jitter level seen in Figure~\ref{fig:RV}. We attempted to fit a second longer period circular planet to the radial velocities. We used uniform priors for the period ($\mathcal{U}[300:2000]$ days), planet mass ($\mathcal{U} [0.002:0.1]$~\Msun) and $t_0$ ($\mathcal{U}[1398:3398]$ TBJD). The posterior distribution are not clearly converging, favouring larger periods and smaller masses. With a \Kamp~of $\sim$ 70 \ms, the best solution is clearly below the activity level and therefore not trustworthy. Long term data is needed to attempt to constrain a longer period companion. \subsection{Spectral energy distribution model} \label{subsec:sed} To constrain the host star parameters \Rstar, \Mstar, [$\mathrm{Fe}$/H] and \teff\,we also model the spectral energy distribution of \toi simultaneously to the transit and radial velocity models. The stellar parameters are modeled using the MESA Isochrones \& Stellar Tracks \citep{2011ApJS..192....3P,2013ApJS..208....4P,2015ApJS..220...15P,2016ApJ...823..102C}. We interpolate evolution tracks using the \textsc{minimint} package \citep{sergey_koposov_2021_5610692} against \Mstar, age, [$\mathrm{Fe}$/H] and the photometric bands $B$, $V$, \emph{Gaia} $G$, $Bp$, $Rp$, 2MASS bands $J$, $H$, and $K$. \Rstar\,is derived from the isochrone predicted values for log\,$g$ and \Mstar. To account for uncertainties in the stellar evolution models, we adopt a 4\% uncertainty floor in stellar radius, and 5\% floor in stellar mass, where appropriate \citep{2022ApJ...927...31T}. For the effective temperature \teff, we apply a Gaussian prior such that the predicted \teff\,interpolated from the isochrone is compared against that measured from the CHIRON spectra as an additional likelihood term. Predicted fluxes from the SED model are corrected for interstellar reddening with the \textsc{PyAstronomy} \textsc{unred} package, that uses the parameterization from \cite{1999PASP..111...63F}. Extinction is a free parameter, with a maximum value of $E(B-V) =$\,0.1542 mag, as estimated from the \citet{2011ApJ...737..103S} maps over a 5 arcmin radius\footnote{Obtained from the \href{https://irsa.ipac.caltech.edu/applications/DUST/}{NASA/IPAC Infrared Science Archive}} around \toi. We also incorporate a Gaussian prior on the distance modulus via the observed \emph{Gaia} parallax to \toi. We offset Gaia DR3's parallax value by -0.023861 mas, the parallax zero-point offset estimated using the routine from~\cite{2021A&A...649A...4L}\footnote{\url{https://gitlab.com/icc-ub/public/gaiadr3_zeropoint}} and function of ecliptic latitude, magnitude and colour. At each MCMC jump step, the observed spectral energy distribution is compared against the interpolated MIST model predictions for a given tested stellar parameter. \subsection{Global model} \label{subsec:globalfit} The global model includes simultaneous fits of the \TESS and ground-based photometric datasets (\ref{subsec:transits}), the CHIRON velocities (\ref{subsec:rv}), and stellar isochrone model (\ref{subsec:sed}), as shown in Figure~\ref{fig:transits},~\ref{fig:RV} and ~\ref{fig:SED} respectively. We explore the best fit parameters and the posterior distribution via the Affine Invariant Markov chain Monte Carlo Ensemble sampler \textit{emcee} \citep{2013PASP..125..306F}. The resulting parameters for \toib are given in Table~\ref{tab:planet}. \par The availability of the radial velocities not only allows us to recover \toib's mass, but also helps to break the degeneracy between $e$ and $\omega$. Figure~\ref{fig:corner_e_w} illustrates this by showing the resulting posterior distributions for $e$ and $\omega$ from our global model versus a model excluding the radial velocities. \input{Tables/Planet_parameters} \section{Discussions and Conclusions} \label{sec:conclusion} We report the discovery of \toib, a temperate gas giant on a highly eccentric orbit around a young Sun-like star. The planet has a mass of~\mpjupvalue~\MJup~and a radius of \rpjupvalue~\RJup. With an orbital period of \pervalue~days, it is to date the second longest period planet in the \TESS sample (after TOI-2180b, \citealt{2022AJ....163...61D}). \toib' resides in a highly elliptic orbit ($e$\,=\eccvalue), and has, based on~\citep{2020A&A...636A..76S}, an age younger than the Praesepe and Hyades clusters. A representation of its orbit alongside the inner Solar System planets is shown in Figure~\ref{fig:orbits}. \subsection{Radius evolution} \label{sec:contraction} At the end of their accretion phase, newly formed gas giants are expected to have radii larger than 1 \RJup. As the planet core radiates its primordial internal heat, Jovian mass planets will cool down via Kelvin-Helmholtz contraction to $\sim 1$~\RJup. Only Hot Jupiters, orbiting extremely close to their parent star are expected to remain inflated due to their increased irradiation. According to cooling models \citep{2003A&A...402..701B,2007ApJ...659.1661F,2008A&A...482..315B,2019A&A...623A..85L}, shown on Figure~\ref{fig:massradius}, the most drastic changes in radius occur at the earliest ages. Measuring radii of young gas giants like \toib is therefore essential to set constraints on these such models, as emphasized in \cite{2007ApJ...659.1661F}.\par The current picture is unclear as the recently measured mass of V1298 Tau\,b\,\&\,e\, \citep{2021NatAs...6..232S} yield much denser planets than predicted at 20 Myr old and require dramatic heavy element enrichment to somewhat reconcile with cooling models (see Figure~\ref{fig:massradius}). Conversely, \toib's radius is as expected for its age. At the closest approach to its host star ($\sim0.18$\,AU), it receives stellar irradiation of $\sim$\,9.3\,$\times$\,10\,$^4$\,W m$^{-2}$, or $\sim$\,68 times that of Earth. Although above the $\sim$ \,1.6\,$\times$ 10\,$^4$\,W m$^{-2}$ threshold to trigger inflation, given by \cite{2018A&A...616A..76S} for planets more massive than 2.5~\MJup, \toib's orbital eccentricity means this level of irradiation affects the planet for a very short fraction of the orbit, not sufficient to trigger radius inflation. \input{Tables/Upcoming_transits} \subsection{Dynamical history of TOI-4562 b and benefits of additional follow-up} \label{subsec:evolution} In its current observed state, \toib's semi major axis and eccentricity (see Figure~\ref{fig:ecc_sma}) are not in favour of a high eccentricity migration scenario as a circularization of its orbit would take orders of magnitudes longer than the age of the universe ($\tau_{circ} \sim$\,1\,$\times10^{7}$\,Gyr, \citealt{1966Icar....5..375G}). It is possible, however, that the planet is experiencing ongoing eccentricity cycles and we happen to be observing it at a lower eccentricity. Reduction of the star-planet distance at periastron at the eccentricity peak of such cycles might allow the circularization process to be triggered as described in \cite{2014ApJ...781L...5D}. Disk-planet interactions can in principle excite the eccentricity of the orbit \citep{2015ApJ...812...94D} but this is restricted to low ($e \lesssim$\,0.2) values, as shown with the red area on Figure~\ref{fig:ecc_sma}. \cite{2021MNRAS.500.1621D} proposed that migration inside wide gaps carved in protoplanetary disks could result in gas giants with eccentricities up to 0.4. This is still insufficient to explain the very high eccentricity from \toib's orbit. \par Another possible scenario to account for \toib's very high eccentricity is in-situ formation (or alternatively, smooth disk migration), followed by excitation from a companion. This can occur via secular interactions, or slow angular momentum exchanges with another body located further out, either periodically through e.g.,\,von Zeipel-Lidov-Kozai cycles \citep{1910AN....183..345V,1962P&SS....9..719L,1962AJ.....67..591K,2016ARA&A..54..441N,2008ApJ...678..498N} or chaotically in secular chaos \citep{2011ApJ...735..109W,Hamers2017}. High eccentricity can also be triggered sporadically in planet-planet scattering \citep{1996Natur.384..619W,1996Sci...274..954R,2006ApJ...638L..45F,2008ApJ...686..580C}, or stellar fly-bys \citep{2016ApJ...816...59S,2021ApJ...913..104R}. Planet-planet scattering could have happened quickly and potentially early if triggered by the dissipation of the gas disk or if the planets were initially closely spaced. Constraints on an outer companion (if not ejected as a result of scattering) could provide crucial insights on dynamical evolution timescales give the young age of the system.\par The five transits of \toib show modest deviation from a linear ephemeris fit on the 5~--~20 min level (see Figure~\ref{fig:ttv}). This potential detection of a transit timing variation signal suggests the presence of a companion in the system, to which \toib probably owes its high eccentricity. The existing data are not sufficient to set meaningful constraints on the companion and most configurations for period (i.e., inner or outer companion), eccentricity and mutual inclination remain possible. \toib will be observed by \TESS again in its second extended mission in 2023. In Table~\ref{tab:upcoming_transits}, we show future opportunities to continue monitoring transits of \toib in the years to come. Combining these with long-term radial velocity follow-up might enable us to unravel the 3-D architecture and dynamical history of this system, as has been successfully performed for Kepler-419\,b\,\&\,c\,\citep{2012ApJ...761..163D,2014ApJ...791...89D}. \par The orbital astrometric motion of an outer companion could be retrieved from \textit{Gaia} in the upcoming release of astrometric solutions for $\sim$~1.3 billion stars \citep{2021A&A...649A...2L}. When archival Hipparchos and \emph{Gaia} observations have been analysed jointly for previous brighter systems \citep[e.g.,][]{2021AJ....162...12V}, astrometric accelerations have often yielded constraints for outer stellar massed companions to key exoplanet systems. Additional \emph{Gaia} observations over the next $\sim 10$ years will allow us to achieve similar constraints for \toi. Combined with the diffraction limited adaptive optics observations estimated to reach $\sim$ 35 au (see section~\ref{subsec:direct-imaging}), these constraints can inform the presence of exterior stellar companions and provide means to distinguish between evolution scenarios. Another candidate tracer for dynamical history is the angle between the star's rotation axis and the planet's orbital axis, or (sky projected) obliquity. From \Prot, \Rstar~and \vsini, we estimate the stellar inclination with respect to the line of sight to have a $3\sigma$ lower bound of $70^\circ$ as per \cite{2020AJ....159...81M}, consistent with being well aligned. Similarly to other planetary characteristics, the young ($<$\,1 Gyr) end of the obliquity distribution is under sampled. Recent measurements resulting from \TESS discoveries reveal a remarkable systematic alignment of young systems, including the Jupiter-sized planet HIP 67522 b \citep{2020AJ....160...33R,2021ApJ...922L...1H}, as well as a number of smaller planets (e.g., AU Mic b \& c; \cite{2020Natur.582..497P,2020A&A...643A..25P,2020A&A...641L...1M,2020ApJ...899L..13H, 2021AJ....162..137A}, DS Tuc Ab; \cite{2019ApJ...880L..17N,2020ApJ...892L..21Z,2020AJ....159..112M}, TOI 942 b \& c \cite{2021ApJ...917L..34W}, and TOI 251 \cite{2021AJ....161....2Z}). The estimated amplitude of the Rossiter McLaughlin effect \citep{1924ApJ....60...15R,1924ApJ....60...22M} for \toib is $\Delta$V\,$\sim$\,70--150\,\ms. Given the $\sim 4$ hours transit duration, combined with a brightness of \Vmag = 12.098 and a rotational broadening of $v$\,sin\,$i$ = 17.5\,\kms, this is well within the grasp of a 4m-class telescope and such an eccentric system would provide a precious addition to the age-obliquity distribution. It is important to note that the long orbital period remains a major obstacle to transit spectroscopy for ground-based facilities. In the coming years, we aim to conduct extensive follow-ups of the \toi system to unravel the full architecture of the system and potentially provide insights into the processes shaping the current gas giant planet distribution. Such follow-up will include radial velocities, ground and space based photometry, astrometry and transit spectroscopy for obliquity measurements and/or atmospheric characterisation. \input{Tables/RVs} We respectfully acknowledge the traditional custodians of the lands on which we conducted this research and throughout Australia. We recognize their continued cultural and spiritual connection to the land, waterways, cosmos and community. We pay our deepest respects to all Elders, present and emerging people of the Giabal, Jarowair and Kambuwal nations, upon whose lands the MINERVA-Australis facility at Mount Kent is located. This research has been supported by an Australian Government Research Training Program Scholarship. GZ thanks the support of the ARC DECRA program DE210101893. GZ, SQ thank the support of the \TESS Guest Investigator Program G03007. CH thanks the support of the ARC DECRA program DE200101840. EG gratefully acknowledges support from the David and Claudia Harding Foundation in the form of a Winton Exoplanet Fellowship. This work was supported by an LSSTC Catalyst Fellowship awarded by LSST Corporation to TD with funding from the John Templeton Foundation grant ID \# 62192. This research has used data from the CTIO/SMARTS 1.5m telescope, which is operated as part of the SMARTS Consortium by \href{http://secure-web.cisco.com/1TL5nionOJJUGi7T0X_YvX7RLRwbVQl20QG7s4LKeK1vpFY8M3UHYMuONVvV2D2hxli_pMi4YkHdTYel4ogZ3sJWN4axM8-5IsyCIPeIj7BfVIBOvp9a8iRKv2IM-wTBpjGA3xxZcH5lT4FNKBIoEstyJEEyUYzEKbDQyL4T1LQSiukl5eTarVlkS9YJbHf_HrjiuXV1gM1uXr7gdIdCbZg4CfJa_N8Qw38oz0KhpJ74RZ0rIcyg3XKCc6-HCDjlBrMtX3cpMKa1Kcya1SxY0UxXY0WkwM0zGeXYUYfbkp1Ce6jIBY8Evcz-YcyODRE4QWMlPqSDV66bKv5F1R3-RrkcH91Y7INyFOP6qJfGJKLRFJT-KNphpqmNc4Pf7zLVOIBjCEKsANmt1XTtzQN5AIPwKf-F1qd4b6KCZrqjHZIA/http\%3A\%2F\%2Fwww.recons.org}{RECONS} This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement. This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is funded by NSF. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. Funding for the \TESS mission is provided by NASA's Science Mission directorate. We acknowledge the use of public \TESS Alert data from pipelines at the \TESS Science Office and at the \TESS Science Processing Operations Center. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This paper includes data collected by the \TESS mission, which are publicly available from the Mikulski Archive for Space Telescopes (MAST). Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. The Center for Exoplanets and Habitable Worlds is supported by the Pennsylvania State University and the Eberly College of Science. RB and MH acknowledge support from ANID – Millennium Science Initiative – ICN12\_009. NE thanks everyone who takes part in the Planet Hunters TESS citizen science project, which contributes to finding new and exciting planetary systems. \facility{TESS, Exoplanet Archive, CTIO 1.5\,m, LCOGT, Gemini:Zorro, CTIO SOAR, ESO 2.2\,m} \software{\textsc{AstroImageJ}~\citep{2017AJ....153...77C}, \textsc{astropy}~\citep{2013A&A...558A..33A,2018AJ....156..123A}, \textsc{batman}~\citep{2015PASP..127.1161K}, \textsc{emcee}~\citep{2013PASP..125..306F}, \textsc{pyastronomy}~\citep{pya}, \textsc{comove} (\url{https://github.com/adamkraus/Comove}), \textsc{pyphot} (\url{https://mfouesneau.github.io/pyphot/}), \textsc{radvel}~\citep{2018PASP..130d4504F}, \textsc{scikit-learn}~\citep{scikit-learn}, \textsc{minimint} (\url{https://zenodo.org/record/4900576}), \textsc{numpy}~\citep{harris2020array}, \textsc{matplotlib}~\citep{Hunter:2007}, \textsc{astropy}~\citep{astropy:2013,astropy:2018}, \textsc{unred} (\url{https://github.com/pbrus/unredden-stars}), \textsc{pandas}~\citep{reback2020pandas}, \textsc{corner}~\citep{corner}} \bibliography{main}{} \bibliographystyle{aasjournal}

Title: A New Polar Ring Galaxy Discovered in the COSMOS Field

Abstract: In order to understand the formation and evolution of galaxies fully, it is important to study their three-dimensional gravitational potential for a large sample of galaxies. Since polar-ring galaxies (PRGs) provide useful laboratories for this investigation, we have started our detailed study of a sample of known PRGs by using the data set obtained by the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP). During the course of this study, we have discovered a new PRG, identified as SDSS J095351.58+012036.1. Its photometric redshift is estimated as z ~ 0.2. The polar ring structure in this PRG appears to be almost perpendicular to the disk of its host galaxy without any disturbed features. Therefore, this PRG will provide us with useful information on the formation of such an undisturbed polar structure. We discuss its photometric properties in detail.

https://export.arxiv.org/pdf/2208.12388

command. \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \shorttitle{} \shortauthors{} \graphicspath{{./}{figures/}} \begin{document} \title{A New Polar Ring Galaxy Discovered in the COSMOS Field} \author[0000-0003-1966-5742]{Minoru Nishimura} \affiliation{The Open University of Japan \\ 2-11 Wakaba, Mihama-ku \\ Chiba, Chiba, Japan} \author[0000-0001-6473-5100]{Kazuya Matsubayashi} \affiliation{Institute of Astronomy, The University of Tokyo \\ 2-21-1 Osawa \\ Mitaka, Tokyo, Japan} \author[0000-0001-5211-7807]{Takashi Murayama} \affiliation{Astronomical Institute, Graduate School of Science, Tohoku University\\ Aramaki, Aoba\\ Sendai, Miyagi, Japan} \author[0000-0003-2247-3741]{Yoshiaki Taniguchi} \affiliation{The Open University of Japan \\ 2-11 Wakaba, Mihama-ku \\ Chiba, Chiba, Japan} \keywords{Galaxy colors(586) --- Galaxy photometry(611) --- Galaxy structure(622)} \section{Introduction} Polar ring galaxies (PRGs) are galaxies with a polar structure (either a ring or a disk of gas, dust, and stars) rotating in a plane almost perpendicular to the major axis of its host galaxy. The archetypical PRG is NGC 2685 (or Arp 336). \citet{sandage_hubble_1961} gives the following note on this galaxy; ^^ ^^ There are two axes of symmetry for the projected image; most galaxies have only one." Its host galaxy is an S0 galaxy, with the polar structure rotating through the minor axis of the host galaxy \citep{schechter_ngc_1978}. From a theoretical point of view, it is considered that the stable maintenance of the polar structure is due to the precessional motion of the polar structure \citep{steiman-cameron_stable_1982}. So far, more than 400 candidates of PRGs have been discovered to date (e.g., \cite{whitmore_cnew_1990}, \cite{moiseev_cnew_2011}). However, among them, only dozens have been confirmed as real PRGs by spectroscopic observations (e.g., \cite{egorov_metallicity_2019}). Since most galaxies reside in large-scale structures, interactions or mergers among galaxies are not rare events for them. In order to understand the dynamical and morphological evolution of galaxies, it is important to study how such interactions or mergers affect the evolution of galaxies by using large unbiased samples of PRGs. Indeed, observational properties of PRGs allow us to investigate a wide range of issues related to their galaxy formation and evolution: e.g., the baryonic matter accretion (e.g., \cite{egorov_metallicity_2019, smirnov_active_2020}), the rate of galaxy interactions or mergers (e.g., \cite{reshetnikov_polar-ring_2011, reshetnikov_galaxies_2019}), and the 3D distribution of mass in the dark halo (e.g., \cite{khoperskov_be_2014, zasov_dark_2017}). In typical PRGs, their host galaxies are mostly early-type galaxies (E/S0), while their polar structures are generally young, blue, and gas rich \citep{reshetnikov_new_2019}. In addition, the observed polar structures show a wide variety in their morphology; e.g., a narrow ring or a wide annulus (\cite{whitmore_few_1991}), a spindle-, a Saturn-, or a worm-like structure \citep{faundez-abans_morphology_1998} , inner polar structure \citep{moiseev_inner_2012}. As for the formation of PRGs, the following three mechanisms have been theoretically proposed; galaxy mergers (e.g., \cite{bekki_formation_1997, bekki_formation_1998, bournaud_formation_2003}), accretion of matter from an approaching galaxy (e.g., \cite{reshetnikov_global_1997, bournaud_formation_2003}), and cold accretion from filaments in intergalactic space (e.g., \cite{maccio_origin_2006, brook_formation_2008, snaith_halo_2012}). However, it is still uncertain which mechanism is important for the formation of PRGs. In order to improve the understanding of the formation and evolution of PRGs, we have started our systematic search for PRG candidates using the data set obtained by the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP; \cite{aihara_hyper_2018}). During the course of this search, we have discovered a new PRG candidate SDSS J095351.58+012036.1 (hereafter J0953) . This galaxy is located at the edge of the Cosmic Evolution Survey (COSMOS; \cite{scoville_cosmic_2007}) field. This survey covers a 2 square degree field. It is designed to probe the galaxy formation and evolution as functions of both cosmic time (redshift) and the local galaxy environment. It is noted that only one PRG candidate has been found in the COSMOS Field; SPRC 093 \citep{moiseev_cnew_2011}. In this paper, we discuss the observational properties of our new PRG candidate J0953. Throughout this paper, we use the following cosmological parameters; $H_0$ = 70 km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\rm m}$ = 0.3, and $\Omega_{\Lambda}$ = 0.7. All magnitudes given in this paper are in the AB magnitude system. \section{SDSS J095351.58+012036.1} The new PRG candidate in our study, J0953, is found at the eastern edge of the COSMOS field in Wide-layer images from the second Public Data Release (PDR2; \cite{aihara_second_2019}) of HSC-SSP. The HSC-SSP survey consists of the following three layers: Wide, Deep, and UltraDeep. The Wide layer covers about 300 deg$^2$ in all the five broad-band filters ($grizy$) to the nominal survey exposure (10~min in $gr$ and 20~min in $izy$). The HSC-SSP PDR2 data have already been processed with the data reduction pipeline hscPipe version 6 (\cite{bosch_hyper_2018}, \cite{aihara_second_2019}), including the following procedures; the point spread function (PSF) modeling, object detection, and photometry. In figure \ref{fig:5image} (a), we show the $g$-, $r$-, and $i$-band composite image of J0953 taken by hscMap tool\footnote{hscMap: $\langle $https://hsc-release.mtk.nao.ac.jp/hscMap-pdr2/app/$\rangle $} (\cite{koike_hscmap_2019}) with SDSS TRUE COLOR mixer. Apparently, a polar structure or ring can be seen in an NE-SW direction almost perpendicular to the disk of the host galaxy. J0953 is identified as a galaxy by the Sloan Digital Sky Survey (SDSS; \cite{york_sloan_2000}); SDSS J095351.58+012036.1. Its photometric properties are reported as follows; $u^{\prime}=21.49\pm 0.25$ mag, $g^{\prime}=19.73\pm 0.03$ mag, $r^{\prime}=18.83\pm 0.02$ mag, $i^{\prime}=18.34\pm 0.025$ mag, and $z^{\prime}=18.02\pm 0.05$ mag. These data give a photometric redshift of $0.146\pm 0.036$ \citep{beck_photometric_2016}. Note that there is no spectroscopic observation of J0953 and thus no spectroscopic redshift is available. In table \ref{tab:data} we present the photometric properties of J0953 taken from the HSC-SSP database by CAS search tool\footnote{CAS search: $\langle $https://hsc-release.mtk.nao.ac.jp/datasearch/$\rangle $}. It is noted that the magnitudes given in this table are slightly brighter than the SDSS magnitudes given above. We consider that this slight difference is probably due to the deeper imaging of the HSC-SSP Wide layer. Using these photometric data, the HSC-SSP group estimates photometric redshifts of J0953 using the two codes. One is the DEmP code (\cite{hsieh_estimating_2014}), and the other is the MIZUKI code (\cite{tanaka_photometric_2015}). These two codes give $z = 0.19$ (DEmP) and $z = 0.20$ (MIZUKI), respectively. Since these are consistent within their errors, we adopt the photometric redshift of J0953, $z = 0.20$ in this paper. This gives the luminosity distance of J0953, 980 Mpc. \begin{deluxetable*}{cc} \tablenum{1} \tablecaption{Properties of J0953 from the HSC-SSP Database\label{tab:data}} \tablewidth{0pt} \tablehead{ \colhead{Property Name} & \colhead{Data} } \startdata RA (J2000.0) & 09h53m51s.59 \\ Dec (J2000.0) & +01D20'36".3 \\ $g$ & $19.43$ \\ $r$ & $18.58$ \\ $i$ & $18.12$ \\ $z$ & $17.75$ \\ $y$ & $17.65$ \\ Photometric redshift (MIZUKI) & $0.20^{+0.09}_{-0.06}$ \\ Photometric redshift (DEmP) & $0.19^{+0.05}_{-0.05}$ \\ Stellar mass (MIZUKI) & $3.85^{+3.10}_{-2.06} \times 10^{10}$ $M_\odot$ \\ Star formation rate (MIZUKI) & $2.66^{+7.77}_{-2.32}$ $M_{\odot}$ y$^{-1}$ \\ Interstellar absorption: $A_V$ (MIZUKI) & $0.65^{+0.22}_{-0.33}$ \\ \enddata \tablecomments{We note that the photometric errors in all the five bands are less than 0.01 mag.} \end{deluxetable*} We note that there are neither companion galaxies nor satellite galaxies around J0953. In figure \ref{fig:5image}, however, there appear two faint objects about 5.7 arcsec in both east and west of J0953. Based on their photometric properties, they are background galaxies with photometric redshifts of $z \sim 1.3$, respectively. \section{Results and Discussion} We discuss the possibility that J0953 is the PRG. To do so, we need to examine whether at least one of the two components that appear to be vertically intersecting the host galaxy or polar structure is the disk or the ring. First, PSFs of the $g, r, i, z,$ and $y$ bands of J0953 are obtained by using the PSF picker\footnote{PSF picker: $\langle $https://hsc-release.mtk.nao.ac.jp/psf/pdr2/$\rangle $}. Next, the FWHM values of these PSFs are obtained by using the IRAF\footnote{IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.} task IMEXAMINE. The PSF of each band in arcsec is as follows; 0.75 ($g$), 0.85 ($r$), 0.63 ($i$), 0.78 ($z$), and 0.64 ($y$). Since the PSF size is the largest in the $r$ band, the other band images are convolved with Gaussian kernels to fit to the $r$-band one by using the IRAF task GAUSS. In the subsequent analyses, we use these smoothed images. In order to investigate structural properties of both the polar ring and its host galaxy, we have made the $g$-, $r$-, $i$-combined image of J0953 to improve the signal-to-noise ratio. Then, we have carried out the two-component fitting for both the polar ring and the host galaxy, each of which has a S\'{e}rsic component. In this analysis, we use the GaLight package\footnote{GaLight is a Python-based open-source package for 2D model fitting of galaxy images in cooperation with lenstronomy (\cite{birrer_lenstronomy_2018}; \cite{birrer_lenstronomy_2021}.) } (version 0.1.6: \cite{ding_mass_2020}; \cite{ding_galaxy_2021}). The fitting results are summarized in table \ref{tab:parameters}. The obtained ellipses with the half-light radius are shown in figure \ref{fig:5image} (b). The S\'{e}rsic index for the host galaxy is 2.94, suggesting that the host galaxy has an elliptical galaxy-like structure rather than an exponential disk. However, \citet{vika_megamorph_2015} reports that the average of the S\'{e}rsic index for the Sab-Sb galaxies is 2.9. Therefore, it is also possible that the host galaxy is a disk galaxy. On the other hand, the polar structure has the flatter light profile than the exponential disk; its S\'{e}rsic index is 0.47, being much less than 1. This small S\'{e}rsic index is consistent with that the polar component has a ring-like structure. Further imaging observation with a higher spatial resolution is necessary to confirm if the polar structure is a ring or not. \begin{deluxetable*}{lcc} \tablenum{2} \tablecaption{Structural parameters \tablenotemark{a} \label{tab:parameters}} \tablewidth{0pt} \tablehead{ \colhead{Parameter Name} & \colhead{Host galaxy} & \colhead{Polar structure} } \startdata effective radius [$\arcsec$] & $0.89$ & $2.12^{+0.01}_{-0.01}$ \\ S\'{e}rsic index & $2.94^{+0.02}_{-0.02}$ & $0.47$ \\ position angle [$\arcdeg$] & $102.2^{+0.5}_{-0.5}$ & $31.4^{+0.1}_{-0.1}$ \\ axis ratio\tablenotemark{b} & $0.71$ & $0.28$ \\ \enddata \tablenotetext{a}{The structural parameters were estimated on the combined image of the $g$, $r$, and $i$ images by GaLight.} \tablenotetext{b}{The axis ratio is the semi-minor to semi-major axis ratio.} \tablecomments{No error means that the error is less than 0.01 in each case.} \end{deluxetable*} Next, with the structure parameters fixed as those in table \ref{tab:parameters}, we have performed model decomposition by using GaLight in each of the $g$, $r$, $i$, $z$, and $y$ images of J0953 to measure apparent magnitudes of each component. Table \ref{tab:magnitude} lists the apparent magnitudes and the absolute magnitudes of the host galaxy and the polar structure. \begin{deluxetable*}{lccccc} \tablenum{3} \tablecaption{Estimated magnitudes.\label{tab:magnitude}} \tablewidth{0pt} \tablehead{ \colhead{} & \colhead{$g$} & \colhead{$r$} & \colhead{$i$} & \colhead{$z$} & \colhead{$y$} } \startdata $m_{\rm HG}$ & $20.07$ & $19.14$ & $18.67$ & $18.29$ & $18.17$ \\ $m_{\rm PR}$ & $21.02$ & $20.41$ & $20.00$ & $20.01$ & $19.76$ \\ $M_{\rm HG}$ & $-20.30$ & $-20.92$ & $-21.34$ & $-21.73$ & $-21.82$ \\ $M_{\rm PR}$ & $-19.09$ & $-19.52$ & $-19.85$ & $-19.88$ & $-20.22$ \\ \enddata \tablecomments{ \hangindent6pt\noindent Note that $m$ and $M$ are the apparent and absolute magnitudes, respectively. \hangindent6pt\noindent HG and PR are the host galaxy and the polar structure, respectively. \hangindent6pt\noindent We note that the photometric errors in all the five bands are less than 0.01 mag. \hangindent6pt\noindent The absolute magnitudes are corrected for the Galactic extinction using the values listed in the HSC-SSP PDR2 data, which are estimated from the dust maps given in \citet{schlegel_maps_1998} ; see also \cite{aihara_second_2019}. \hangindent6pt\noindent The $k$-correction is applied for all the absolute magnitudes (\cite{chilingarian_analytical_2010}). } \end{deluxetable*} The spectral energy distribution (SED) of the galaxy is a composite of the spectral emissions from all the stars contained therein. Therefore, we can estimate the stellar mass of both the host galaxy and the polar structure. Here, we use the absolute magnitudes in $i$-band and the color $g - i$ following the method outlined by \citet{taylor_galaxy_2011}, and obtain the stellar mass of the host galaxy and the polar structure are $26.18 \times 10^9 M_\odot$ and $4.23 \times 10^9 M_\odot$, respectively. The characteristic absolute magnitude $M_*$ corresponding to the characteristic luminosity $L_*$ of the galaxies are suggested from \citet{blanton_luminosity_2001} as follows; $u^{\prime}=-18.34\pm 0.08$ mag, $g^{\prime}=-20.04\pm 0.04$ mag, $r^{\prime}=-20.83\pm 0.03$ mag, $i^{\prime}=-21.26\pm 0.04$ mag, and $z^{\prime}=-21.55\pm 0.04$ mag; see table 1 of \citet{blanton_luminosity_2001} for details on the sample galaxies. Using the color conversion formulae from the SDSS system to the HSC system from \citet{komiyama_stellar_2018}, these $M_*$ are converted to $g=-20.12$ mag and $i=-21.31$ mag. The $g-$ and $i-$ band absolute magnitudes of the J0953 host galaxy are -20.30 mag and -21.34 mag, respectively. Therefore, the host galaxy of J0953 is possibly brighter and heavier than the galaxies of the characteristic luminosity $L_*$ corresponding to the characteristic absolute magnitude $M_*$. Figure \ref{fig:color} is the color-color diagrams to estimate one of the information, and the colors of the host galaxy and the polar structure of this galaxy at $z \sim 0.20$ are plotted. Figure \ref{fig:color} is the color-color ($r -i$ vs.~$g - r$) diagram of the host galaxy and the polar structure compared with that of each Hubble type of typical galaxies at $z \sim 0.2$ (\cite{fukugita_galaxy_1995}). The colors of the host galaxy suggest a Sab-Sb galaxy while host galaxies of typical PRGs are mostly early-type galaxies (E/S0) (\cite{reshetnikov_new_2019}). The polar structure is similar in colors to Scd galaxies. This is consistent with that the polar structure is blue and probably younger than the host galaxy. In order to confirm if this galaxy is truly a PRG, it is necessary to make spectroscopic observations, to investigate the kinematical properties of both the host galaxy and the polar structure. \section*{Acknowledgements} We would like to thank an anonymous referee for his/her useful comments. The HSC collaboration includes the astronomical communities of Japan and Taiwan, and Princeton University. The HSC instrumentation and software were developed by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was contributed by the FIRST program from Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS), Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University. This paper makes use of software developed for the Large Synoptic Survey Telescope. We thank the LSST Project for making their code available as free software at $\langle $http://dm.lsst.org$\rangle $. The Pan-STARRS1 Surveys (PS1) have been made possible through contributions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation under Grant No. AST-1238877, the University of Maryland, and Eotvos Lorand University (ELTE) and the Los Alamos National Laboratory. This paper is based on data collected at the Subaru Telescope and retrieved from the HSC data archive system, which is operated by Subaru Telescope and Astronomy Data Center (ADC) at NAOJ. Data analysis was in part carried out on the Multi-wavelength Data Analysis System operated by ADC. \bibliography{PASP_220824}{} \bibliographystyle{aasjournal}

Title: Optical throughput and sensitivity of JWST NIRSpec

Abstract: To achieve its ambitious scientific goals, the Near-Infrared Spectrograph, NIRSpec, on board the Webb Space Telescope, needs to meet very demanding throughput requirements, here quantified in terms of photon-conversion efficiency (PCE). During the calibration activities performed for the instrument commissioning, we have obtained the first in-flight measurements of its PCE and also updated the modeling of the light losses occurring in the NIRSpec slit devices. The measured PCE of NIRSpec fixed-slit and multi-object spectroscopy modes overall meets or exceeds the pre-launch model predictions. The results are more contrasted for the integral-field spectroscopy mode, where the differences with the model can reach -20%, above 4 micron, and exceed +30%, below 2 micron. Additionally, thanks to the high quality of the JWST point-spread function, our slit-losses, at the shorter wavelength, are significantly decreased with respect to the pre-flight modeling. These results, combined with the confirmed low noise performance of the detectors, make of NIRSpec an exceptionally sensitive spectrograph.

https://export.arxiv.org/pdf/2208.04876

\keywords{JWST, NIRSpec, near-infrared spectrograph} \section{INTRODUCTION} \label{sec:intro} % The Near-Infrared Spectrograph (NIRSpec) is one the four focal plane instruments on the Webb Space Telescope; its main scientific goal is to enable multi-object spectroscopy of faint high redshift galaxies and thus advance our understanding of the early stage of galaxy formation at the dawn of our universe. Nevertheless its innovative design allows sensitive observations of a wide range of targets to be performed, including detailed spectroscopic studies of the atmospheres of exoplanets, to choose an example at the opposite end of the Universe distance scale[\citenum{pfa+2022}]. To support these studies, NIRSpec is capable of carrying out low, medium and high resolution spectroscopy, both in single- object mode employing any one of five fixed slits (FS), or a 3.1$\times$3.2 arcsec$^2$ integral field unit (IFU) for Integral Field Spectroscopy observations (IFS), or in multi-object (MOS) mode employing a novel programmable micro-shutter array (MSA) covering a 3.6$\times$3.4 arcmin$^2$ field of view[\citenum{pfa+2022}]. NIRSpec is equipped with a double pass-prism for low-resolution spectroscopy over the entire wavelength-range, 0.6 $-$ 5.3 $\mu$m, and two sets of three diffraction gratings providing spectral resolutions of $R{\simeq}1000$ and $R{\simeq}2700$, respectively. The two sets of gratings are formally specified to cover the 1.0-5.0\!~$\mu$m wavelength region in each of three overlapping bands: 1.0-1.8\!~$\mu$m (Band~I), 1.7-3.0\!~$\mu$m (Band~II) and 2.9-5.0\!~$\mu$m (Band~III). The nominal instrument configurations, and corresponding combination of dispersive elements and filters, are listed in \autoref{tab:NIRS_modes}. The shorter wavelength F070LP long-pass filter listed is included to enable observations below 1\!~$\mu$m with the G140M and G140H gratings, which are free of contaminating second-order light only up to a wavelength of 1.27\!~$\mu$m. \begin{table}[ht] \caption{Nominal NIRSpec science configurations} \label{tab:NIRS_modes} \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline\hline Band & Disperser element & Resolution $\lambda / \Delta\lambda$ & Filter & Spectral range / $\mu$m\\ \hline \rule[-1ex]{0pt}{3.5ex} 0 & G140M, G140H & 1000, 2700 & F070LP & 0.7--1.2 \\ \hline \rule[-1ex]{0pt}{3.5ex} I & G140M, G140H & 1000, 2700 & F100LP & 1.0--1.8 \\ \hline \rule[-1ex]{0pt}{3.5ex} II & G235M, G235H & 1000, 2700 & F170LP & 1.7--3.1 \\ \hline \rule[-1ex]{0pt}{3.5ex} III & G395M, G395H & 1000, 2700 & F290LP & 2.9--5.2 \\ \hline \rule[-1ex]{0pt}{3.5ex} n/a & PRISM & ~100 & CLEAR & 0.6--5.3 \\ \hline \end{tabular} \end{center} \end{table} \subsection{Photon-conversion efficiency} A key performance parameter in every optical instrument, either a telescope or microscope, is its efficiency: how many photons will make it through the optical chain the instrument is comprised of. Barring systematics, the throughput and the noise performance of the detectors are the parameters driving the photometric sensitivity of an instrument. NIRSpec is a complex instrument that supports many different modes of observations. Its optical train is reflective throughout, save for the order-separation filters and the low resolution dispersive prism. The three primary optical modules of NIRSpec are each implemented in the form of three-mirror anastigmats employing high-order aspherical surfaces[\citenum{geyl11}]. Counting the five plane fold mirrors and the disperser, the light entering NIRSpec undergoes a total of 15 reflections (in the FS/MOS mode) before reaching the detector array. Coupled with the telescope element of JWST, the total number of reflections a photon goes thorough in its path from the primary mirror to the detectors at the NIRSpec focal plane is 19! The IFU mode adds a further 8 gold-coated reflections to the light path. Ultimately, though, we do not count photons but the electrons generated in the semi-conductor substrate of the detectors, with their own performance in terms of quantum efficiency (QE). For this reason, we describe the throughput of our instrument in terms of Photon Conversion Efficiency, PCE. To achieve its ambitious science goals, NIRSpec was designed with demanding requirements in terms of optical efficiency across its entire wavelength range - and therefore particular attention was placed in the design and manufacturing of its reflective elements, making sure that all the optical surfaces received the highest quality optical coatings and were kept extremely clean during instrument assembly. With the aim of achieving high optical throughput at the blue end, all mirrors are coated with protected silver. Additionally, the two Teledyne H2RG focal plane detectors have excellent QE, overall exceeding 70\% , as illustrated by Rauscher et al.\,[\citenum{bernie2014}]. Preliminary measurements of the NIRSpec PCE were obtained during the NIRSpec ground-testing campaign that took place in 2013. The measurements, though, were affected by significant uncertainties (due to limitations of the ground-testing equipment), and, in addition, in 2015 the detector arrays were replaced with new devices. Therefore, before flight, the PCE performance of NIRSpec was based on a radiometric model of the instrument obtained by combining the measured absolute QE of the replaced arrays with individual-component measurements made on each optical element or appropriate witness samples, together with the nominal reflectivity of the telescope optics[\citenum{light16}]. The PCE values predicted before flight were very high, reaching in excess of 50\% in prism configuration, and at peak blaze in all six gratings - see Jakobsen et al.[\citenum{pfa+2022}]. The moment of truth finally came during the commissioning of the instrument, with the first observations of a spectro-photometric standard star. \section{OBSERVATIONS AND PROCESSING} JWST was launched on the 25th of December of 2021. After deployment and reaching the target halo-orbit around the Sun-Earth L2 point, the telescope and instrumentation underwent a six months period of commissioning activities to prepare the observatory for its scientific observations. During this phase, NIRSpec commissioning team commanded NIRSpec to perform internal calibration exposures and on-sky observations, acquiring the data necessary to assess the instrument's in-flight performance and generate calibration reference data[\citenum{tspie+2022}]. For the absolute spectro-photometric calibration of the instrument, we observed the standard stars 1808347 (TYC 4433-1800-1), an A3 star selected from STScI Calibration database CALSPEC, as well as two other reference stars during one of the first instrument check-out exposures on sky: wd1057+719 (a white dwarf) and p177d (G0), using all the dispersers, for a total of 9 filter-disperser combinations\footnote{Proposal ID (PID) 1128}. To asses the PCE for the MOS and FS mode, the star was observed through the square 1.6 arcsec-wide slit (S1600A1), which has minimal geometrical loss, with typical values of 3--9\% (see Sect.\,\ref{sec:losses}). To enable the subtraction of the background emission a two-point 1-arcsec dither in the spatial direction was executed within the slit, in each configuration, while, in the case of the IFS mode, a 4-point nod was executed across the IFU aperture field-of-view. The exposures were processed with our NIRSpec ramp-to-slope pipeline that performs the following basic data reduction steps: bias subtraction, reference pixel subtraction, linearity correction, dark subtraction and finally count-rate estimation, including jump detection and cosmic-ray rejection -- see Birkmann et al.[\citenum{st_spie+2022}] for more details on this last step. From the count-rate images the wavelength calibrated spectra were obtained using the Stage 2 of the NIRSpec pipeline to perform the following operations for the FS: subtract background (combining the exposures from the two-nods); extract sub-image containing the spectral trace and assign wavelength and spatial coordinates to each pixel therein; generate a rectified spectrum re-sampled on regular 2D-grid; compute the 1D-spectrum obtained by spatial integration. Figure \ref{fig:example} shows the spectra for configurations F070LP-G140H and F290-G395M, as examples. For the IFS exposures, after background subtractions and rectification of the 30 IFU traces, Stage 2 of NIRSpec pipeline builds the three-dimensional data cube from where the 1D-spectrum is obtained by spatial integration of a circular area. The PCE of the instrument through the given aperture is given simply by dividing the observed number of electron rate per wavelength element by the value of photons flux collected by the primary mirror (per wavelength element). \section{RESULTS} The PCE for the MSA/FS mode and the IFS mode are shown in Fig.\,\ref{fig:prismpce} for the low-resolution configuration, together with predictions from the model of the instrument response that we developed and used throughout to estimate the sensitivity of NIRSpec before flight. In the figure, the throughput response of the different components entering the model are also shown. The model does not include the light losses due to the finite size of the aperture, which however do affect the data presented here - in the case of the square-aperture these are expected to be relatively small. The PCE values, in MSA/FS mode, for the medium and high resolution configurations are shown in Fig.\,\ref{fig:gpce} - also compared to the model. For the IFU, the PCE values for the grating configurations available are shown in Fig.\,\ref{fig:pce_ifu}. Although not presented here, we note that for the configurations for which observations of more than one standard star was available we obtained results completely consistent with each other. In general the efficiency of NIRSpec meets or exceeds the predictions of our pre-flight model. Significantly higher than predicted performances are achieved in particular for the high-resolution configurations, for the MOS/FS mode, and for all configurations below $\sim2.6~\mu$m, for the IFS mode. We notice a dip in efficiency with regard to predictions above $\sim 4~\mu$m -- in particular for the IFS mode, where we see 10--20\% less throughput than predicted. In the case of the FS/MOS, this difference is (at least partly) explained by the effect of the slit losses and the fact that they become more significant at these wavelength, due to the geometrical truncation of the larger point-spread function (PSF) - see next section. In the case of IFS mode, the differences between data and predictions are likely due, partly, to diffraction losses at the level of the IFU 100 mas-slice, which affect the observations (prevalently in the red) but are not included in the model and, partly, to slight inaccurate assumptions in the gold-reflectivity as a function of wavelength, at cryogenic temperatures (recall that the IFU adds 8 gold-coated reflections to the light-path). \subsection{Path-losses} \label{sec:losses} Ultimately, in the case of NIRSpec, the total amount of light registered when observing an object is also affected by the light losses occurring in the slit device employed. These depend on a variety of factors: the nature and shape of the target, the slit employed, the positioning of the object within the aperture, and the optical quality of the image formed within the slit. For simplicity we will limit our discussion here to the case of a centered point source. In this case, when employing the relatively wide square aperture (S1600A1), the losses are small, but for the narrower slits and the micro-shutters these are substantial. In the IFS mode, where the flux of the scientific target in the image plane can be recovered by summing over multiple slices, the path losses are limited to their diffraction component and can be in excess of 10\% at the longest wavelengths. Before flight, we computed the expected path losses for a point source, in the various apertures, using a Fourier-optics approach based on the available measured wavefront error maps of all surfaces in the light path[\citenum{pfa+2022}], including predictions for the telescope. Indeed, because the major source of losses is due to the truncation of the PSF in the silt plane by the geometry of the aperture, the quality of the image is of fundamental importance. Fig.\,\ref{fig:stransm} shows the predicted transmission coefficient as a function of wavelength for slit S200A1/A2, for the pre-flight computation and the updated values using a representative map of the wavefront error obtained during commissioning -- see [\citenum{otespie2022}]. Since the PSF achieved in flight is of such a high quality, in particular with a higher Strehl ratio, the transmission from our narrowest slit is predicted to improve significantly up to $\sim 4$~$\mu$m, where the truncation of the PSF core starts to be the dominant factor in the loss of flux. The computed transmission coefficients for all the fixed slits as a function of wavelength based on the in-flight PSF are also shown in Fig.\,\ref{fig:stransm} - right panel; the transmission coefficients shown here include both the geometrical losses due to the finite size of the aperture, the dominant term, and the diffraction losses in the collimator pupil, contributing by a few percent[\citenum{pfa+2022}]. While the relative improvement of the transmission, with respect to the pre-flight prediction, is most noticeable in the blue, the absolute level of losses are higher on the red-side of NIRSpec wavelength range where the PSF full-width half-maximum is larger: indeed, this is partly the reason for an apparent drop in performance in the red when comparing the observed PCE with the model predictions (which does not include path losses), for the FS/MOS mode. Using the transmission coefficients for slit S1600A1, to correct for this effect for the case of the F290LP filter, both M and H gratings, brings the PCE values right in line with the predictions also for these configurations, as shown in Fig.\,\ref{fig:corr_pce_g395m}. We have currently not yet updated our estimates of path losses in the case of the IFS mode. \section{DISCUSSION AND CONCLUSION} For many years during the development of NIRSpec, we had to rely on a model of the instrument PCE, anchored to measurements obtained during the instrument level testing that took place in 2013 - when the instrument was not yet in its final flight configurations. For the MOS/FS mode, the anticipated PCE, excluding slit losses, was very high, reaching in excess of 50\% for the prism, and at peak blaze in all six gratings [\citenum{pfa+2022}]. For the IFS the PCE performance was predicted to be approximately 50\% to 10\% lower, going from the blue- to the red-end of NIRSpec operating wavelength range[\citenum{bal+2022}], since the IFS introduces eight further reflections. The observations performed during commissioning demonstrates that NIRSpec is equipped with extraordinarily efficient optics and detectors. In the MOS/FS mode, the observed PCE overall meets or exceeds the predictions; for the IFS mode, the performance in the blue are significantly higher than anticipated ($+$30\% below 2~$\mu$m) and lower than expected above 4~$\mu$m (up to $-$20\%). In this case, the reasons for the discrepancy between the model and the measurements could be a combination of unaccounted for diffraction losses at the level of the IFU 100 mas-slices and inaccurate assumptions in terms of the gold-reflectivity as a function of wavelength, at cryogenic temperatures. Beside the optical efficiency, the other fundamental parameter driving the photometric sensitivity of an instrument such as NIRSpec is the noise performance of the detector. NIRSpec is equipped with extremely low-noise Teledyne H2RG arrays and, as illustrated by S. Birkmann et al.[\citenum{st_spie+2022}] at this conference, the behaviour of the two sensors is excellent, with noise figures that meet the anticipated in-flight performances. Hence, NIRSpec achieves the sensitivity projected before launch by Ferruit et al. [\citenum{fjg+2022}], for the FS and MOS mode, and B{\"o}ker et al. [\citenum{bal+2022}], for the IFS mode. JWST/NIRSpec is confirmed to be the most sensitive near-IR spectrograph currently available for astronomical studies ready to deliver exciting new observations. \bibliography{nirspec-pce} % \bibliographystyle{spiebib} %

Title: Looking for the Signals of the Missing Baryons in the Extragalactic Background Light

Abstract: The missing baryons in the universe are assumed to be hidden in the whole space as a warm-hot intergalactic medium (WHIM). Finding them is one of the important subjects in modern cosmology. In this paper, we point out that the very high energy electron/positron rays may light up the WHIM due to the anomalous bremsstrahlung according to the improved Bethe-Heitler formula. The resulting excess of the extragalactic background light (EBL) can be observed by the direct measurement method. A possible explanation on the difference between the direct and indirect measurements of EBL is also proposed. Thus, we open a new window to probe the WHIM properties via the EBL.

https://export.arxiv.org/pdf/2208.10363

\hoffset = -1truecm \voffset = -2truecm \baselineskip = 10 mm \title{\bf Looking for the Signals of the Missing Baryons in the Extragalactic Background Light} \author{ Wei Zhu$^1$\footnote{Corresponding author, E-mail: wzhu@phy.ecnu.edu.cn}~ and Rong Wang$^2$\footnote{E-mail: rwang@impcas.ac.cn} \\ \normalsize $^1$ Department of Physics, East China Normal University, Shanghai 200241, China \\ \normalsize $^2$ Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China \\ } \date{} \newpage \vskip 3truecm {\bf keywords}: Missing Baryons; Improved Bethe-Heitler Formula; Extragalactic Background Light \newpage \section{Introduction} The standard cosmological model (SCM) considers that most $(95\%)$ of the universe is composed of dark energy $(70\%)$ and dark matter $(25\%)$, while the universe's remaining $5\%$ is the visible matter partly in the form of stars, planets, interstellar dust... However these observed matter so far can only account for half of the visible matter in the SCM predictions [e.g. 1,2]. It implies that at least $1/2$ of the baryons are now ``missing'' [3,4]. Cosmological simulations show that these ``missing'' baryons in the intergalactic medium may condense into a filamentary web and undergo shocks that heat the baryons up to the temperatures $T\sim 10^5-10^7K$ [5,6]. Finding this warm-hot intergalactic medium (WHIM), which contains the missing baryons, is crucial to validate the SCM. The WHIM is hot and highly ionized, so that it could absorb or emit far-ultraviolet and soft X-ray photons. However, these characteristic lines are formed in the atomic bound state. This spectral method is invalid if the WHIM is too hot to show up in the Lyman-absorption, or too diffused to emit the detectable X-rays [e.g. 7]. Recently, the dispersion of $X$-ray afterglows from fast radio bursts (FRBs) is used to prove the existence of the WHIM [8,9]. Although the preliminary results from these studies tend to support the WHIM model, the conclusion has a large uncertainty since the FRB sources were produced occasionally and their positions in the universe are undetermined. Still, all mentioned measurements above are not enough to give a complete physical picture of the WHIM. The purpose of this article is to try to find the possible signals of the missing baryons in the visible light band, which relates to the extragalactic background light (EBL). The EBL is the diffuse background radiation accumulated over the cosmic history from ultraviolet (UV) to infrared (IR) wavelengths. A peak in the EBL spectrum around $1~\mu m$ is formed by the direct emission from stars. Measurements of both the EBL spectral intensity and its evolution are important to study both the star formation and the galaxy evolution processes [10]. We find that the cosmic rays may light up the WHIM and lead to the observed excess of the EBL. Direct measurements of the optical EBL are made by observing the total sky brightness and with the measurements or modeling of all the foreground components [e.g.11,12]. In this method, these foreground components must be very accurately known. However, the bright zodiacal light (ZL), which is created by the scattered sunlight off the interplanetary dust particles, may arise a large uncertainty for the weak EBL intensity. To get rid of this problem, the indirect techniques were developed: the counting galaxies and the gamma-ray attenuation measurements [e.g.13]. These indirect measurements are complementary for the direct measurements and they are important to constrain the large uncertainty of EBL. At the same time, the direct surface photometry measurements have been improved. For example, the detectors are set at the vantage points, where the earth's atmosphere and the light from interplanetary dust are not the appreciable components of the diffuse sky brightness. These researches show an excess brightness above the indirectly measured EBL intensity. In particular, the NASA's New Horizons spacecraft is presently over 50 AU away from the sun, where the ZL almost disappears. A recent report from New Horizons shows a new component of unknown origin of EBL of $8.06\pm1.92~nW m^{-2}sr^{-1}$ [14]. This discrepancy is interesting because it might point to the presence of a truly diffuse emission component in the universe that is not resolvable into point sources. Let us imagine a given small range in the sky, where very high energy (VHE) cosmic electrons and positrons interact with the Coulomb field in the WHIM and emit a lot of soft bremsstrahlung photons. According to quantum electrodynamics (QED), the contribution of these soft photons to the EBL intensity is only $n_{\gamma}\sim 10^{-20}~nWm^{-2}sr^{-1}$ (see section 3), which is much smaller than the observed $n_{\gamma}\sim 10~nWm^{-2}sr^{-1}$. Therefore, the contributions of the normal bremsstrahlung photons can be neglected. However, the above conclusion should be reconsidered if we use an improved Bethe-Heitler formula [15]. The reason is as follows. The electromagnetic shower is a common phenomenon in astronomy, which comprise the bremsstrahlung and the pair production in the nuclear Coulomb field (figures 1b and 1c). The quantum theory of the electromagnetic shower is based on the Bethe-Heitler formula [16], which has been widely applied in many branches in physics including astrophysics. However, the recent work [15] pointed out that the Bethe-Heitler formula should be modified to a new form in the dilute ionized gas, where the target recoil can be neglected for the incident VHE cosmic rays. In this case both the bremsstrahlung and the pair production cross sections have an unexpected big increment, as $\sigma_{brem}\sim 1/m^2_e$ in the Bethe-Heitler formula Eq. (2.1) is replaced by $\sigma_{brem}^{MD}\sim 1/\mu^2$ in the modified version Eq. (2.2) and the screening length (or the Debye length) is $\mu^{-1}\gg m_e^{-1}$. This leads to a big screening-length effect, which was neglected for a long time. The ionized gas of extreme low density is an ideal place for testing the anomalous bremsstrahlung and pair production since the ionized atoms have a large screening length. The nuclear Coulomb potential may spread into such a broad space, where the bremsstrahlung or pair production induce almost no recoil of the target since the incident high energy electrons or photons are far away from the nucleus. This result can explain the strange difference among the precise measurements of cosmic electron-positron fluxes at the GeV-TeV band by Alpha Magnetic Spectrometer (AMS), Fermi Large Area Telescope (Fermi-LAT), DArk Matter Particle Explorer (DAMPE) and Calorimetric Electron Telescope (CALET) [17]. We find that the anomalous cross section has about eight-orders of magnitude higher than the normal cross section in this example. We will discuss the corrections from the anomalous bremsstrahlung to the directly measured EBL intensity in section 3. We find that the distribution of the EBL intensity will be moved upward if adding the bremsstrahlung contributions. The computed result is close to the observation data. Then we discuss the anomalous pair production in the indirect measurement method using gamma-ray attenuation with the same theoretical framework in section 4. We noticed that the data of gamma-ray attenuation coincide with that of the counting galaxies. However, the processes in Figures 1b and 1c do not contribute to the counting galaxies since the WHIM is diffusely distributed in the space. Therefore, we consider that the screening length $\mu^{-1}$ takes a smaller value in the gamma-ray attenuation method. We will give an explanation for that in section 5. We find a strong dependence of the directly measured EBL intensity on the screening length in the electronic structure of the WHIM. This result may open a new window to expose the electronic properties of the WHIM during the universe evolution. \section{Improved Bethe-Heitler formula} We sketch and review some results of the improved Bethe-Heitler formula from the work [15]. When charged particles scatter off the electric field of proton or nucleus, they can emit real photons in such interactions. This is the famous bremsstrahlung (braking radiation). A quantum-mechanical description of bremsstrahlung emission at the Born approximation is the Bethe-Heitler formula. This formula for the high-energy incident electrons and at the soft photon limit reads $$\sigma_{brem}(E,\omega)d\omega=\frac{4\alpha_s^3}{m^2_e}\ln\frac{2E}{\mu}\frac{d\omega}{\omega}, \eqno(2.1)$$ where $\mu^{-1}$ is the screening length. Equation (2.1) presents a strong reduced bremsstrahlung cross section since $1/m^2_e$ is much smaller than the geometric cross section $1/\mu^2$ of atom, where $\mu^{-1}\geq 10^{-8}~cm$. The work [15] proved that if the target recoil can be neglected for a VHE incident electron, the Bethe-Heitler formula should be modified as, $$\sigma^{MD}_{brem}(E,\omega)d\omega=\frac{4\alpha_s^3}{\mu^2}\ln\frac{2E}{\mu}\frac{d\omega}{\omega}. \eqno(2.2)$$ It predicts a big enhanced bremsstrahlung crosse section. This is called the anomalous bremsstrahlung effect. The deeply ionized gas in the cosmic space is an ideal place to see the anomalous bremsstrahlung effect since the dilute ionized atoms have a large screening length, where the nuclear Coulomb potential may spread into such a broader space, and the recoil of the bremsstrahlung event can be neglected since the incident high energy electrons are far away from the nuclei. Bremsstrahlung under this environment exhibits a big cross section. On the other hand, the bremsstrahlung probability also decreases with the reduced density. The above two opposite effects lead to a limited kinematic window, where the bremsstrahlung cross section becomes anomalous. According to the analysis of the work [17], the TeV-band of electron energy is such a possible window. The similar argument also satisfies for the electron-positron pair creation from the real high energy photon. We consider a high energy photon traversing the atomic Coulomb field. This photon has a probability of transforming itself into a pair of electron-positron. The corresponding cross section in the leading approximation is given by, $$d\sigma_{pair}=\frac{\alpha^3}{m_e^2}\ln\frac{4\omega^2}{\mu^2}(1-z)[(1-z)^2+z^2]dz, \eqno(2.3)$$ where $z=E_e/\omega$. In a situation where the target recoil can be neglected, we have the anomalous pair production cross section, $$d\sigma_{pair}^{MD}=\frac{\alpha^3}{\mu^2}\ln\frac{4\omega^2}{\mu^2}(1-z)[(1-z)^2+z^2]dz \eqno(2.4)$$ The above improved Bethe-Heitler formula has been used to explain the difference of VHE electron-positron fluxes in the ionosphere about $400\sim 500~km$ height above the earth ground, where the oxygen atoms are not only strongly ionized but also of extremely dilute density. The WHIM in the cosmology scale may provide a new platform to test the improved Bethe-Heitler formula. We will discuss about this in the following sections. \section{Anomalous bremsstrahlung in the WHIM} The direct measurement of the EBL intensity was carried out by the special camera in the space. After the subtraction of the contributions from the ZL and other foreground components, the data show an obvious excess of EBL photons compared to the indirectly measured EBL intensity. This indicates the presence of an unknown diffuse emission component. However, the conclusion remains uncertain since the estimations of the ZL contributions have large uncertainties. The NASA's New Horizons spacecraft is an excellent platform for the EBL observation since it is presently over 50 AU away from the sun, where the zodiacal light is almost disappeared. A recent report from the New Horizons confirms an EBL component of unknown origin of $n_{\gamma}=8.06\pm1.92~nW m^{-2}sr^{-1}$ [14]. Although some of the authors have argued that the EBL includes a substantial component of light from stars tidally removed from galaxies, or a population of faint sources in extended halos. None of these hypotheses may be correct [14]. We try to explain this excess of the EBL photons and to point out that it could be arisen from the contributions of the anomalous bremsstrahlung when the VHE cosmic electron/positron fluxes pass through the WHIM. We consider a small range in the deep space, where the incident VHE cosmic electron-positron rays interact with the Coulomb field of the WHIM and emit soft bremsstrahlung photons. The screening length in equation (2.2) depends on the ionization state and spatial distribution of the missing baryons in the WHIM. If the probing range is small enough, the resolution of camera can not distinguish the WHIM structure. Thus, we take an average screening length $\mu^{-1}$ in equation (2.2) to describe the probed WHIM. The contributions of the anomalous bremsstrahlung to the EBL intensity is $$n_{\gamma}(\omega)d\omega =\int_{E_{min}}^{E_{max}} n_{\gamma}(E,\omega)dEd\omega =\int_{E_{min}}^{E_{max}} J_e(E)P(E,\omega)dEd\omega. \eqno(3.1)$$ where $J_e(E)$ is the electron-positron flux at energy $E$. We take $E_{min}=0.1~TeV$ and $E_{max}=10~TeV$ according to the Ref. [17]. $P(E,\omega)d\omega/\omega$ is the photon number emitted by the electrons in the range $\omega\rightarrow\omega+d\omega$. The relation between the emission power spectrum $P(E,\omega)d\omega$ and the bremsstrahlung cross section is written as [17], $$\sigma_{brem}^{MD}(E,\omega)d\omega=\frac{P(E,\omega)d\omega}{n_pv_e\omega}, \eqno(3.2)$$ where $n_p\sim 1/m^3$ is the average density of the missing baryons in the WHIM and $v_e\simeq c$. We use the parameterized distribution function $$J_e(E)=10^{-2}\left(\frac{E}{100GeV}\right)^{-2}[m^{-2}s^{-1}sr^{-1}], \eqno(3.3)$$ to model the electron-positron flux in the energy range $100~GeV<E<1~TeV$ [17] and assume that this estimation is available in the intergalactic space. Note that the azimuth angles in both $n_{\gamma}(\omega)$ and $J_e(E)$ are almost coincide, since the soft photons are concentrated in the moving direction of the high energy electrons. Thus, we have, $$n_{\gamma}(\omega)=\int_{E_{min}}^{E_{max}}J_e(E)\frac{4\alpha_s^3}{\mu^2}\ln\frac{2E}{\mu}n_pv_edE.\eqno(3.4)$$ This result shows a flat distribution and its height sensitively depends on the value of $\mu$. In Figure 2 we show our resulting $n_{\gamma}(\omega)\sim \lambda$ (dashed curve) with $\mu=10^{-15}~GeV$ or $\mu^{-1}=10~cm$. Dotted curve is a global fitting of the EBL intensity data from the indirect method [19]. The sum of the above two components (solid curve) is consistent with the data from New Horizons spacecraft (solid point) [14]. We also present some other directly measured EBL intensities (all other points) [20-22], where the ZL contributions are deducted by using the special models rather than the measurements in the observations. Although our estimation is still preliminary, one can find that the contribution from the anomalous bremsstrahlung to the EBL density seems possible. On the other hand, the large excess around $\lambda\sim 1~\mu m$ was suspected to be from the highly redshifted ultraviolet radiation of the first generation of stars. However, the evolution models based on the observations of high-redshift galaxies indicate that the brightness caused by first stars are much lower than the observed excesses [23]. The increment of the EBL due to the anomalous bremsstrahlung may compensate this deficiency. Note that the solution of equation (3.4) can not be applied without restrictions. The applications of the anomalous bremsstrahlung requests very low photon energy so that neglecting the recoil effect is safe. On the other hand, the radiation corrections are needed when the photon energy $\rightarrow 0$ in equation (3.4) [24]. The infrared cutoff should be used to stop the infinite increasing of $\sigma^{MD}\propto 1/\omega$ due to the negative corrections of the virtual processes. The radiation corrections depend on the experimental conditions (energy and angular resolutions). Thus, we assume that the intensity $n_{\gamma}\propto \omega \sigma^{MD}$ is suppressed when the photon energy $\omega$ is below the resolution of the detector. Therefore, we restrict the solution of equation (3.4) to be in the range of $\lambda \approx 0.1-3~\mu m$, which is shown in figure 2. Finally, according to the Bethe-Heitler formula (2.1), equation (3.4) should be replaced by $$n_{\gamma}(\omega)=\int_{E_{min}}^{E_{max}}J_e(E)\frac{4\alpha_s^3}{m_e^2}\ln\frac{2E}{\mu}n_pv_edE,\eqno(3.5)$$ and we have $n_{\gamma}\sim 10^{-20}~nWm^{-2}sr^{-1}\ll 10~nWm^{-2}sr^{-1}$. This is why the contributions of the normal bremsstrahlung photons are neglected. \section{Anomalous pair production in the WHIM } An indirect measurement of the EBL intensity is based on the absorption of gamma rays via the pair production off real EBL photons. This effect leads to the distance-dependent suppression of the gamma-ray flux from active galactic nuclei (AGNs)[10,13]. AGNs are the extragalactic objects characterized with extremely luminous electromagnetic radiations. The missing baryons in the WHIM may deform the gamma-ray distributions due to the anomalous pair production process described by eqution (2.4). We begin from a case without the WHIM. The EBL may modify the propagation of VHE gamma-rays traveling through intergalactic space due to pair production in the channel $\gamma_{VHE}+\gamma_{EBL}\rightarrow e^-+e^+$ (figure 1a). The observed VHE gamma-ray flux on the earth is related to the initial flux at the source as, $$\Phi_{\gamma}^{ob}(E_{\gamma})=\Phi_{\gamma}^{in}(E_{\gamma})e^{-\tau_{\gamma\gamma}},\eqno(4.1)$$ where the optical depth $\tau_{\gamma\gamma}$ relates to the pair production cross section of two photons $\sigma_{\gamma\gamma}$ and the density $n_{\gamma}$ of the soft photon in the EBL as, $$\tau_{\gamma\gamma}(E_{\gamma}) =\int_0^R\int_{\lambda_{min}}^{\lambda_{max}}\sigma_{\gamma\gamma}(E_{\gamma},\lambda)n_{\gamma}( \lambda, r)d\lambda dr,$$ $$~~~~around~E_{\gamma}\sim 1~TeV. \eqno(4.2)$$ This is the foundation of the indirect method of the EBL measurement. If we considering the WHIM, the contributions of pair production in the nuclear Coulomb field of the WHIM (figure 1c) will be added to the observed optical-depth data in equation (4.2). Therefore, the measured optical depth based on the pair production model should include the contribution $\gamma+p\rightarrow e^++e^-$ due to the WHIM, i.e., $\tau_{\gamma\gamma}\rightarrow \tau_{\gamma\gamma}+\tau_{\gamma p}$. The optical depth in the WHIM is evaluated with the anomalous pair production as, $$\tau_{\gamma p}(E_{\gamma})=\int_0^R\sigma_{pair}^{MD}(E_{\gamma},r)n_p(E_{\gamma},r) dr,\eqno(4.3)$$ where $n_p$ is the proton density in the WHIM. We have $\sigma_{\gamma p}\sim 0.1\sigma_{\gamma\gamma}$ around $E\sim 1~TeV$ [24], with $n_p/n_{\gamma}\sim 10^{-9}-10^{-10}$ and $\sigma_{pair}^{MD}\sim (m_e/m_{\mu})^2\sigma_{Pair}$. To have the similar magnitude of gamma-ray attenuation from the WHIM as that from the EBL, $\tau_{\gamma p}\sim \tau_{\gamma\gamma}$, the screening length of the WHIM is required to be, $$\left(\frac{m_e}{\mu}\right)^2=10^{10}-10^{11}. \eqno(4.4)$$ Note that the counting-galaxy method is irrelevant to the WHIM structure since the virtual photons in the Coulomb field are not counted in this method. The EBL data deduced from the counting galaxies and the gamma-ray attenuation method coincide with each other [13]. Therefore, the correction to the optical depth from the anomalous pair production on the WHIM should be neglected. This implies that $(m_e/\mu)^2\ll 10^{10}-10^{11}$, or $\mu^{-1}\ll 10^{-5}~cm$ in equation (4.3). We will discuss these results in the next section. \section{Discussions} Both the EBL and the WHIM are the products of the universe evolution, and they convey some important information of the cosmic formation history. Many works studied the evolution of the optical depth $\tau_{\gamma\gamma}$ [e.g. 12,13]. In this work, we focus on the evolution of the properties of the WHIM with the redshift, based on the optical-depth correction from the WHIM with the improved Bethe-Heitler formula. The screening length $\mu^{-1}$ is sensitively dependent on the ionization status of the missing baryons in the WHIM. We take two extreme examples to illustrate it. (i) If the missing baryons exist in the form of neutral atoms, we have $\mu^{-1}\sim 10^{-8}~cm$ or $n_{\gamma}\sim 10^{-15}~nW m^{-2}sr^{-1}$ calculated with equation (3.4); (ii) The missing baryons and electrons are uniformly distributed in the space as the plasma form, where the maximum of the average screening length is $\mu^{-1}=(1/n_p)^{1/3}\sim 10^2~cm$, and we have $n_{\gamma}\sim 10^5~ nWm^{-2}sr^{-1}$. These two examples show that as we use the average screening length to describe the ionization property of the atom gas in the WHIM, its value has a broadly reasonable range from the beginning of re-ionization (i.e., $\mu^{-1}>10^{-8}~cm$) to a maximally ionized state of the baryon-electron plasma. The standard cosmological model predicts that the missing baryons in the universe evolution are concentrated in the WHIM due to continuously shock-heated process [1,2]. Cosmological simulations find that the WHIM contains a large fraction of the baryons at redshift $z\simeq 0$ in the form of highly-ionized gas [25]. Theoretically, an atom which is bounded in an infinite heavy web can completely absorb the recoil effect. However, this bounded atom can not avoid the obvious recoil corrections due to the intense collisions at a short impact parameter. Therefore, these bounded atoms in the WHIM do not have the anomalous bremsstrahlung or the anomalous pair production. We consider the atomic gas inside the WHIM at high redshift. The direct EBL intensity is measured by using a long time exposition of the camera at low redshift $z\sim 0$. We have extracted a large screening length $\mu^{-1}\simeq 10~cm$ at $z\simeq0$ in section 3. It implies that a lot of atoms in the WHIM at $z\rightarrow 0$ are almost re-ionized in the complete baryon-electron plasma state. This result is consistent with the observations, which gives the evidence that a significant portion of the baryons are located at $z\simeq 0$ in the WHIM [26]. On the other hand, the gamma rays in the indirect measurement of the EBL intensity come from the AGNs at large redshift. The contributions to the optical depth from highly-ionized gas in the WHIM at low redshift only account for a small part of the total optical depth in equation (4.3). Therefore, we have the average screening length $\mu^{-1}\ll 10^{-5}~cm$ for the anomalous pair production discussed in section 4, which leads to the result $\tau_{\gamma p}\ll \tau_{\gamma \gamma}$. Thus, we explained why we use two different values of the parameter $\mu^{-1}$ in sections 3 and 4. Usually, the absorptions or emissions of far-ultraviolet and soft X-ray photons [27], or the hyperfine transition of neutral hydrogen (called 21-cm line) [28] are used to look for the signals of the missing baryons in the WHIM. However, the WHIM is extremely difficult to be observed, for its tremendously low density and high temperature are believed to elude the detections of the absorption and emission signals. Therefore, the properties of the WHIM at $z=0$ is still elusive with the traditional detecting methods. Comparing with the characteristic spectra of the WHIM, the sensitive dependence of the improved Bethe-Heitler cross section on the average screening length provides a new method to complement the study of the WHIM structure. In summary, we try to establish a connection between the WHIM and the EBL. The found relation answers the following cosmological questions: Where are the missing baryons? Is the sky absolutely dark if all star lights are removed? According to an improved Bethe-Heitler formula, we find that the WHIM may be lighted up by the VHE cosmic electron/positron fluxes, and this effect can be observed by the directly measured EBL intensity. Using a sensitive relation between the EBL intensity and the WHIM electronic structure, the improved Bethe-Heitler formula may open a new window to study the WHIM structure. \noindent {\bf ACKNOWLEDGMENTS:} This work is supported by the National Natural Science Foundation of China (No.11851303). R.W. acknowledges the support from the National Natural Science Foundation of China (No.12005266). \newpage \newpage

Title: Properties of shocked dust grains in supernova remnants

Abstract: Shockwaves driven by supernovae both destroy dust and reprocess the surviving grains, greatly affecting the resulting dust properties of the interstellar medium (ISM). While these processes have been extensively studied theoretically, observational constraints are limited. We use physically-motivated models of dust emission to fit the infrared (IR) spectral energy distributions of seven Galactic supernova remnants, allowing us to determine the distribution of dust mass between diffuse and dense gas phases, and between large and small grain sizes. We find that the dense ($\sim 10^3 \,{\rm cm}^{-3}$), relatively cool ($\sim 10^3 \, {\rm K}$) gas phase contains $>90\%$ of the dust mass, making the warm dust located in the X-ray emitting plasma ($\sim 1 \,{\rm cm}^{-3}$/$10^6 \, {\rm K}$) a negligible fraction of the total, despite dominating the mid-IR emission. The ratio of small ($\lesssim 10 \, {\rm nm}$) to large ($\gtrsim 0.1 \, {\rm \mu m}$) grains in the cold component is consistent with that in the ISM, and possibly even higher, whereas the hot phase is almost entirely devoid of small grains. This suggests that grain shattering, which processes large grains into smaller ones, is ineffective in the low-density gas, contrary to model predictions. Single-phase models of dust destruction in the ISM, which do not account for the existence of the cold swept-up material containing most of the dust mass, are likely to greatly overestimate the rate of dust destruction by supernovae.

https://export.arxiv.org/pdf/2208.11137

\label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \begin{keywords} dust, extinction -- ISM: supernova remnants -- ISM: evolution \end{keywords} \section{Introduction} Core-collapse supernovae (CCSNe) both form \citep{dunne2009,matsuura2011,gomez2012,delooze2017,delooze2019,chawner2019,niculescu2021} and destroy \citep{jones1996,slavin2015} significant quantities of dust. The balance between these two processes determines whether CCSNe are net dust sources or sinks, which has important consequences for the evolution of the interstellar medium (ISM) \citep{morgan2003,delooze2020,galliano2021}. With extensive observational evidence for efficient dust formation by CCSNe, the key uncertainties in models of ISM evolution are now those related to dust destruction {in the shockwaves driven by these same objects.} {Although theoretical predictions for the destruction of newly-formed dust in CCSN ejecta} span nearly the entire range from complete destruction to complete survival (\citealt{kirchschlager2019} and references therein), models of dust destruction in the ISM have settled on a typical gas mass `cleared' of dust of order $\sim 10^3 \msun$ {per SN} (\citealt{jones1996,slavin2015}, although \citealt{kirchschlager2022} report larger values). {This corresponds to} $\sim 10 \msun$ of dust destroyed per SN for a typical Galactic dust-to-gas (DTG) ratio of $\sim 0.01$, {much larger than the observed dust masses of $\lesssim 1 \msun$ found in supernova remnants (SNRs), suggesting that CCSNe are net destroyers of dust under present-day ISM conditions.} These models {generally assume that} shocks propagate into a spherically symmetric, uniform density ISM, {with properties appropriate for the warm neutral medium ($\nh \sim 1 \pcc$, $T \sim 10^4 \kel$). The real ISM is multi-phase, with most of the mass concentrated in colder, denser gas ($\sim 30 \pcc$/$100 \kel$; \citealt{mckee1977}).} More realistic ISM structures can significantly reduce the quantity of dust destroyed \citep{hu2019,martinez2019}, as denser regions of the ISM experience less violent shock interactions. Models also typically assume the standard \citet{mathis1977} (MRN) grain size distribution for the ISM dust, but this may vary depending on the ISM phase \citep{hirashita2009} and can be altered by the SN itself \citep{hoang2019}, with potentially dramatic effects on the resulting mass of dust destroyed \citep{kirchschlager2019}. Observationally, infrared (IR) studies of SNRs often find dust temperatures lower than those predicted by models of grains in the high-temperature ($\gtrsim 10^6 \kel$) gas {produced by SN shocks in low-density material \citep{seok2015,koo2016,chawner2020}. Using data extending into the far-IR, \citet{chawner2020b} showed that the observed spectral energy distribution (SED) of the Tornado SNR cannot be explained by dust grains in the hot, X-ray emitting shocked material swept up by the expanding SNR. The far-IR emission requires the presence of colder grains, located within the SNR but not exposed to the high-temperature gas. This cold dust component has a mass at least an order of magnitude larger than that of the warmer ($\sim 100 \kel$) grains located in the hot gas. The same phenomenon was found for the three SNRs investigated in \citet{priestley2021}, suggesting that it is not uncommon. Most observational studies of dust in SNRs \citep[e.g.][]{borkowski2006,williams2006,temim2012} have focused on the mid-IR emission, which is mainly produced by the warm grains, and thus may represent a very small fraction of the total dust mass within the SNR. The dust properties derived from the mid-IR data, and the resulting destruction efficiencies, are potentially unrepresentative of most of the swept-up material.} {A significant issue in interpreting far-IR observations of SNRs is that the dust properties are poorly constrained; in \citet{chawner2020b} and \citet{priestley2021}, we were forced to assume standard ISM properties for both the cold dust, and for the radiation field presumably responsible for heating it. The impact of the SNR blastwave can be expected to significantly alter both of these properties. In this paper, we develop a comprehensive model for the IR emission of shocked dust grains in SNRs, allowing us to derive the properties of these grains from observational data, and thus provide empirical constraints on models of dust destruction.} \begin{table*} \centering \caption{Distance, radius, {estimated age}, volume, the hot component density and temperature, the {initial} ISM density and shock velocity {reproducing these hot component properties}, and the scaling factor to reproduce observed X-ray luminosities, for each SNR in our sample. Note that for G11, G27 and G29, `volume' refers to {that of} the shell of shocked material, rather than the spherical volume. References: (1) \citet{priestley2021}; (2) \citet{chawner2020b}; (3) \citet{green2004}; (4) \citet{ranasinghe2018}; (5) \citet{verbiest2012}; (6) \citet{sawada2011}; (7) \citet{frail1996}; (8) \citet{kilpatrick2016}; (9) \citet{chawner2020}; (10) \citet{kothes2007}; (11) \citet{tian2014}; (12) \citet{keohane2007}; (13) \citet{park2010}; (14) \citet{leahy2020}; (15) \citet{koo2016}; (16) \citet{chawner2019}.} \begin{tabular}{ccccccccccc} \hline SNR & $D$/kpc & $r$/pc & Age/$\kyr$ & $V$/pc$^3$ & $\nh$/cm$^{-3}$ & $T$/$10^6 \kel$ & $n_{\rm ISM}$/cm$^{-3}$ & $v_{\rm sh}$/km s$^{-1}$ & $f$ & Ref. \\ \hline G11 & $4.4$ & $2.8$ & $1.4-2.4$ & $33.7$ & $6.8$ & $8.2$ & $1.7$ & $770$ & $0.028$ & 1, 3, 15, 16 \\ G27 & $5.8$ & $2.9$ & $0.8-2.1$ & $37.1$ & $6.5$ & $9.1$ & $1.6$ & $810$ & $0.019$ & 1, 4, 15, 16 \\ G29 & $5.8$ & $3.2$ & $<0.8$ & $59.4$ & $1.6$ & $26.0$ & $0.4$ & $1370$ & $0.010$ & 1, 5, 15, 16 \\ Tornado & $11.8$ & $4.5$ & $2-8$ & $382$ & $0.5$ & $8.5$ & $0.13$ & $780$ & $0.091$ & 2, 6, 7 \\ G43 & $11.3$ & $8.2$ & $1-4$ & $2310$ & $1.0$ & $18.6$ & $0.25$ & $1170$ & $0.121$ & 8, 9, 12, 15, 16 \\ G340 & $15.0$ & $13.6$ & $2.6$ & $10530$ & $1.0$ & $11.6$ & $0.25$ & $920$ & $0.121$ & 9, 10, 13, 16 \\ G349 & $11.5$ & $4.5$ & $1.8$ & $382$ & $3.4$ & $7.0$ & $0.85$ & $710$ & $0.091$ & 9, 11, 14, 15, 16 \\ \hline \end{tabular} \label{tab:snrprop} \end{table*} \begin{table*} \centering \caption{Median SNR dust masses returned by the MCMC for each component, with the 16th and 84th percentiles as uncertainties, {and the estimated initial swept-up dust mass $M_0$, assuming an initial density $n_{\rm ISM}$ and volume $V$ from Table \ref{tab:snrprop}, and a DTG mass ratio of $0.01$.}} \begin{tabular}{cccccc} \hline & \multicolumn{4}{c}{Dust mass / $\msun$} & \\ SNR & Hot/large & Hot/small & Cold/large & Cold/small & $M_0/\msun$ \\ \hline \multicolumn{6}{c}{Carbon} \\ \hline G11 & $3.0^{+0.7}_{-0.8} \times 10^{-3}$ & $< 10^{-4}$ & $<0.2$ & $4.1^{+0.7}_{-0.8}$ & $0.020$ \\ G27 & $4.6^{+0.1}_{-0.9} \times 10^{-3}$ & $< 10^{-4}$ & $9.6^{+1.3}_{-1.7}$ & $<0.5$ & $0.021$ \\ G29 & $0.017^{+0.003}_{-0.003}$ & $< 2 \times 10^{-4}$ & $0.004^{+0.396}_{-0.003}$ & $<0.3$ & $0.008$ \\ Tornado & $0.86^{+0.05}_{-0.04}$ & $< 10^{-4}$ & $<0.2$ & $12.5^{+0.6}_{-0.9}$ & $0.017$ \\ G43 & $1.61^{+0.07}_{-0.09}$ & $< 5 \times 10^{-3}$ & $<2$ & $<1$ & $0.203$ \\ G340 & $<0.7$ & $< 4 \times 10^{-3}$ & $167^{+98}_{-31}$ & $123^{+14}_{-99}$ & $0.928$ \\ G349 & $0.30^{+0.03}_{-0.02}$ & $< 10^{-4}$ & $<0.5$ & $38.2^{+1.2}_{-1.8}$ & $0.114$ \\ \hline \multicolumn{6}{c}{Silicate} \\ \hline G11 & $5.3^{+1.3}_{-5.0} \times 10^{-3}$ & $< 10^{-3}$ & $<4$ & $6.6^{+1.0}_{-1.2}$ & $0.020$ \\ G27 & $6.4^{+0.1}_{-2.5} \times 10^{-3}$ & $< 6 \times 10^{-4}$ & $40.5^{+5.0}_{-6.4}$ & $<0.6$ & $0.021$ \\ G29 & $0.014^{+0.003}_{-0.004}$ & $< 1 \times 10^{-3}$ & $<1$ & $<4$ & $0.008$ \\ Tornado & $0.307^{+0.020}_{-0.019}$ & $< 10^{-4}$ & $<0.5$ & $45.1^{+1.1}_{-1.3}$ & $0.017$ \\ G43 & $0.47^{+0.10}_{-0.10}$ & $< 10^{-3}$ & $<2$ & $142^{+20}_{-23}$ & $0.203$ \\ G340 & $<0.5$ & $< 0.03$ & $962^{+171}_{-117}$ & $130^{+25}_{-71}$ & $0.928$ \\ G349 & $0.22^{+0.02}_{-0.02}$ & $< 2 \times 10^{-4}$ & $<0.5$ & $101.6^{+2.4}_{-2.8}$ & $0.114$ \\ \hline \end{tabular} \label{tab:mass} \end{table*} \section{Method} \subsection{Observational sample} We consider a sample of seven core-collapse SNRs: \geleven{} (hereafter G11), \gtwentyseven{} (G27), \gtwentynine{} (G29), G$43.3$-$0.2$ (G43), G$340.6$+$0.3$ (G340), G$349.7$+$0.2$ (G349), and G$357.7$-$0.1$ (the Tornado). {We have previously investigated the Tornado in \citet{chawner2020b}, and G11, G27 and G29 in \citet{priestley2021}. In all four SNRs, the IR SEDs were inconsistent with collisionally-heated dust grains, for gas properties derived from X-ray data. We found that an additional, colder dust component is required to reproduce the far-IR SEDs, but were unable to constrain its properties beyond estimating the dust masses involved.} The other three SNRs (G43, G340 and G349) were selected from the \citet{chawner2019,chawner2020} catalogue of Galactic plane SNRs due to having both clear SNR-associated IR emission {in at least one band (most often {\it Spitzer} MIPS $24 \um$)}, and X-ray emission indicating ongoing shock interation with the surrounding ISM. These were excluded from the analysis in \citet{priestley2021} due to their irregular morphologies; {Figure \ref{fig:img} shows far-IR three-colour images of G11 and G340, with X-ray contours overlaid. While G11 has a coincident shell-like structure in both the far-IR and X-ray data, indicating an interaction between the SNR and the surrounding ISM, G340 is much more confused. Nonetheless, the combination of enhanced X-ray emission and dust temperature (represented by the $70 \um$ flux) within the SNR strongly suggests interaction with ambient material.} Table \ref{tab:snrprop} lists relevant physical properties for the SNRs. {G11 and G29 both show evidence of newly-formed ejecta dust interior to the region of interaction with the ISM, found in the pulsar wind nebulae located at the centres of these SNRs\footnote{{In both cases, the radiation field generated by the pulsar wind nebula is insufficient to power the central dust emission \citep{priestley2020}. As the swept-up dust of interest is located at even greater distances, we assume the impact of the central object is negligble.}} \citep{chawner2019}. For these objects, we take IR fluxes from \citet{priestley2021}, extracted from annuli excluding the central regions, and thus presumably dominated by the swept-up ISM. G27 has little central IR flux, but does have a shell-like X-ray structure, so we again use the \citet{priestley2021} annulus fluxes. Fluxes for the Tornado are taken from \citet{chawner2020b}, and those for G43, G340 and G349 from \citet{chawner2020}, using circular apertures. These last four SNRs have previously-derived dust masses ($\gg 1 \msun$) far in excess of what could be produced by a single CCSN, so we assume the IR fluxes are primarily due to ISM dust, and that any contribution from ejecta dust is negligble. The IR data for each SNR are listed in Appendix \ref{sec:snrsed}.} \subsection{Dust SED model} For each SNR, we calculate dust emission models using \dinamo{} \citep{priestley2019}, which determines the temperature distribution for grains heated by the local radiation field and electron/ion collisions. We assume there are two gas components in each SNR; a `hot' component responsible for the X-ray emission, with typical densities of $\sim 1 \pcc$ and temperatures $\gtrsim 10^6 \kel$, and an additional `cold' component with a much higher pre-shock density ($\gtrsim 100 \pcc$) and thus a much lower post-shock temperature ($\ll 10^6 \kel$). {The two components are assumed to be spatially well-mixed (e.g. cold clumps embedded in a hot diffuse medium) , so that the local radiation field is the same for both.} The gas properties of the hot component are taken from analysis of the X-ray data in the literature, listed in Table \ref{tab:snrprop}, and we assume $\nel = \nh$ for simplicity. Properties for G11, G27 and G29 were derived from modelling X-ray data in \citet{priestley2021}, and those for the Tornado obtained similarly by \citet{sawada2011}. The temperature for G43 is taken from \citet{keohane2007}, and we use the lower end of their quoted range of densities ($1-3.5 \pcc$). \citet{park2010} report a temperature for G340 of $1-1.5 \, {\rm keV}$; we again take the lower limit, and assume a density of $1 \pcc$, typical for the rest of our sample. \citet{leahy2020} provide the temperature and the emission measure for G349, again from X-ray modelling, and we obtain the density from the SNR volume, the emission measure, and the assumption $\nh = \nel = {\rm constant}$ throughout the SNR. For the cold component, we assume $\nh = 1000 \pcc$, $\nel = 0.1 \pcc$ and $T = 5000 \kel$ for all SNRs, as determined for G11 from fits to the \citet{andersen2011} H$_2$ line observations in \citet{priestley2021}. These properties may not be appropriate for the rest of our sample, but they are fairly typical values for SNRs interacting with dense ambient material, as derived from molecular line observations \citep[e.g.][]{reach2005,zhu2014}. {We discuss the importance of this assumption in Appendix \ref{sec:coldprop}.} We assume the local radiation field {in both the hot and cold component} is primarily due to emission from the shock-ISM interaction. We use {\sc mappings} \citep{sutherland2017} to calculate plane-parallel radiative shock models, with the {initial gas density, $n_{\rm ISM}$, and shock velocity, $v_{\rm sh}$, chosen to reproduce the post-shock density and temperature reported by X-ray studies of each object} (the initial temperature is fixed to $10^4 \kel$). We then scale the resulting SED by a factor $f$, such that the X-ray luminosity of a spherical shell of emitting material, with the same radius as the SNR, matches the observed values given by \citet{koo2016}. The Tornado and G340 are not included in \citet{koo2016}, so we adopt the values of G349 and G43 respectively, being the best-matched SNRs in terms of {X-ray derived gas properties.} For a spherical shell of emitting material, the flux at any point within the shell is the same as the flux at the centre by symmetry. We approximate the local radiation field {heating the dust} using this central radiation field. {While swept-up grains located at the shock front are much closer to {\it some} of the shocked ISM generating this radiation field, they are also much further away from most of it. The $r^{-2}$ scaling of the received flux with distance to the source should, roughly, cancel out the $\sim r^2$ growth in the amount of material emitting at a given distance. The typical radiation field experienced by a dust grain should therefore be comparable to that at the centre. While several of the SNRs in our sample clearly deviate from spherical symmetry,} this represents a substantial upgrade on our previous work \citep{chawner2020b,priestley2021}, where we assumed a \citet{mathis1983} ISM field scaled by an arbitrary constant. Typical shock-generated radiation fields are very different from those in the wider ISM, {affecting the resulting dust SEDs. Figure \ref{fig:sed} shows the radiation field produced by this method for G11 compared to the \citet{mathis1983} ISM field, and corresponding SEDs for carbon grains of different sizes heated by the two fields. The higher ultraviolet (UV) flux from the ISM field results in $0.1 \um$ carbon grains being heated to higher temperatures than for the G11 field, whereas the much greater flux of X-ray photons from the SNR causes non-equilibrium effects in $5 \nm$ grains to become more important, reflected in the increased mid-IR grain emission. These effects are not necessarily universal, even for different grain types in the same radiation field (large silicate grains have similar temperatures for both the G11 and ISM fields) - it is essential to consider the local radiation field on a case-by-case basis.} Shock parameters and scaling factors are listed in Table \ref{tab:snrprop}. We calculate dust SEDs for $0.1 \um$ and $5 \nm$ grains, representing the largest and smallest sizes typically present in the ISM \citep{mathis1977}, for the hot and cold components of each SNR. {We demonstrate in Appendix \ref{sec:sizedist} that our results are not sensitive to the specific choice of `large' and `small' grain sizes.} We then fit the observed IR SEDs\footnote{Flux measurements shortwards of $24 \um$ are treated as upper limits, due to the potential for significant non-dust contamination at these wavelengths.}, after convolving with the appropriate filter response curves, with the mass of each dust component (small/large, hot/cold) as the four free parameters. {In the following, we depict the hot/large dust component in blue, hot/small in orange, cold/large in green, and cold/small in purple.} Fitting is done using {\it emcee}, a Monte Carlo Markov chain (MCMC) code \citep{foreman2013}, with $500$ walkers, $5000$ steps per walker, and $500$ burn-in steps, which is sufficient for convergence. For dust properties, we use either carbon grains, with optical constants taken from \citet{zubko1996} and a bulk density of $1.6 \gcc$, or silicates, with MgSiO$_3$ optical constants from \citet{dorschner1995} (extended to far-UV/X-ray wavelengths with values from \citealt{laor1993}) and a bulk density of $2.5 \gcc$. While ISM dust includes both species, the data are insufficient to fit both simultaneously, and {our main conclusions hold regardless of the assumed grain composition}. \section{Results} Figure \ref{fig:g11fit} shows the results of our SED fitting for G11. The cold dust mass is at least three orders of magnitude larger than that of the hot component, consistent with the results from \citet{priestley2021}. We also find strict limits on the mass of small grains that can be present in the hot component, due to their high emissivity around $\sim 10 \um$ (Figure \ref{fig:g11emis}) combined with strong upper limits on the observed flux in this wavelength region, again consistent with our previous results. For either grain composition, we find that a significant mass of small grains are required in the cold component, at least comparable to that in large grains, and possibly much larger. This is in contrast to the MRN size distribution we assumed for this component in \citet{priestley2021}, where most of the mass is in the largest grain sizes, and suggests significant dust processing in the shocked material. Figure \ref{fig:allfits} shows the best-fit carbon grain SEDs for the remaining six SNRs in our sample (results for silicate grains are shown in Appendix \ref{sec:silfit}). While the total dust mass in each SNR ranges from $\sim 1-100 \msun$, in all cases this mass is primarily in the cold component, with hot grains {typically} making up a negligible fraction of the total. The hot component consists only of large grains in all SNRs {for which it has a non-negligible mass}, and for all but one (G27), small grains make up a substantial fraction of the dust mass in the cold component. The parameter distributions returned by the MCMC confirm these findings. Median masses for each dust component\footnote{{These can differ significantly from the best-fit masses in Figures \ref{fig:g11fit} and \ref{fig:allfits}, generally indicating that the mass of that component is poorly constrained due to large observational uncertainties (e.g. G29).}} and the 16th and 84th percentiles, listed in Table \ref{tab:mass}, show that for all seven SNRs, the hot/small dust mass is consistent with zero, and typically constrained to be much lower than the other three dust components, regardless of the assumed grain composition. {The total SNR dust masses for G11, G27, G29 and the Tornado are all within a factor of a few of our previous estimates for these objects \citep{chawner2020b,priestley2021}. For G340 and G349, our masses for carbon grains are in good agreement with those obtained from blackbody fits by \citet{chawner2020}, while with silicate optical properties the dust masses are somewhat larger, due to the lower mass opacity. Our carbon mass for G43 is significantly lower than the \citet{chawner2020} estimate because in this case, the far-IR fluxes can be reproduced by grains in the hot component, with high grain temperature and thus high emissivity. With silicate grains, for which this is not the case, our estimated mass is a factor of a few larger, as with G340 and G349. In general, our model results in dust masses basically consistent with those from previous work. The large ($> 100 \msun$) dust masses found for some of the SNRs are required by the observed far-IR fluxes, for the assumed distances in Table \ref{tab:snrprop} and typical dust mass opacities at these wavelengths.} {We estimate the mass of swept-up ISM in the SNRs using the ambient densities derived from the X-ray data, $n_{\rm ISM}$, and volumes corresponding to the regions the IR SEDs were extracted from, $V$, as given in Table \ref{tab:snrprop}. Assuming an ISM DTG ratio of $0.01$ {and a hydrogen mass fraction of $0.7$}, the total swept-up dust masses for each SNR are given in Table \ref{tab:mass}, if the average density of the ambient ISM is that of the (pre-shock) hot component. These values are {all $<0.1 \msun$. The total observed dust mass exceeds our estimate for all seven SNRs, and by factors of $\gtrsim 100$ for all SNRs except G29. In order to contain this much dust, the average ambient ISM density around the SNRs would have to be larger than the $\sim 1 \pcc$ values in Table \ref{tab:snrprop} by a similar factor, i.e. $\left<n_{\rm ISM}\right> \gtrsim 100 \pcc$, comparable to the typical average densities of molecular clouds on these scales \citep{larson1981}.} Figure \ref{fig:massratio} shows the ratio of the {hot component dust mass to the total dust mass} for each SNR, plotted against the density of the hot component. With a few exceptions, this value is well-constrained to be $< 0.1$, and in most cases $<0.01$ - the dust located in the high-temperature shocked material makes up a percent-level fraction of the total swept-up dust mass. {Due to the higher grain temperatures, this dust component typically contributes most of the mid-IR flux, but it is unrepresentative of the bulk of the swept-up dust mass.} {There is a suggestion of a negative correlation between the hot component dust mass fraction and the gas density. This could indicate either an intrinsically lower mass fraction of this component in denser regions, or more efficient sputtering in denser gas destroying a larger proportion of the initial dust mass.} Figure \ref{fig:sizeratio} shows the mass ratio of small to large grains in both the hot and the cold components, again plotted against density. The mass ratio of grains with radii $\le 10 \nm$ to those $\ge 0.1 \um$ for an MRN size distribution ($\sim 16 \%$) is indicated for comparison. In general, it appears that the hot component is substantially depleted in small grains compared to the undisturbed ISM, whereas the cold component either has a more typical size distribution, or is enhanced in small grains, {to the extent that the mass of large grains is negligible in some SNRs} (note that because grain mass is proportional to $a^3$, the enhancement in grain number is even larger than indicated by Figure \ref{fig:sizeratio}). \section{Discussion} \subsection{Caveats} {The flux measurements we model (Appendix \ref{sec:snrsed}) are not the raw observational fluxes, but the estimated contribution from the SNRs at each wavelength, after the subtraction of background flux from unrelated material. There is no clear consensus on the best way to estimate these background contributions, and even if they are explicitly included as an additional model component, some assumptions have to be made about its properties (such as SED shape; see \citealt{delooze2017,delooze2019}). Given this freedom of choice, it is impossible to rigorously account for the possible impact on our results. We have thus refrained from drawing any conclusions based on single objects in our sample. The consistent qualitative results seen in Figures \ref{fig:massratio} and \ref{fig:sizeratio} across the sample of SNRs suggest that these are real physical phenomena, even if the numerical values for any individual SNR should be viewed with some caution.} {Our sample was selected from the \citet{chawner2019,chawner2020} SNR catalogue based on the presence of co-spatial X-ray and warm dust ($24/70 \um$) emission, taken as a sign of interaction. We are therefore biased towards SNRs which are detectable in both these shock tracers, which likely corresponds to high ambient densities. In fact, all seven of our SNRs show signs of interaction with molecular material (G11, G27, G29, G43 - \citealt{kilpatrick2016}; G340 - \citealt{green1997}; G349 - \citealt{lazendic2010}; Tornado - \citealt{hewitt2008}). These interactions are not rare; $\sim 20 \%$ ($88/383$) of the SNRs in the \citet{ferrand2012} catalogue are listed as interacting with molecular clouds, and this seems likely to be an underestimate given the requirement for targeted molecular line observations. While our sample may well be biased, we do not appear to be selecting for a particularly uncommon class of object.} The best-fit models for several SNRs (Figures \ref{fig:g11fit}, \ref{fig:allfits} and \ref{fig:silfits}) have either large $\chsq$s, indicating a poor correspondance between model and data, or $\chsq < 1$, which suggests that the model has overfit the data. In the latter case, it is clear that the SEDs of some objects can be fit with only two, or even one, dust component, rather than the four we use. However, the MCMC fitting approach accounts for degeneracies between model parameters. Figures \ref{fig:massratio} and \ref{fig:sizeratio} include all the information from the chain, so we are confident that the conclusions drawn from them are robust against overfitting issues. For those SNRs with high $\chsq$ values, we note that these are typically driven by relatively small uncertainties on the far-IR fluxes. This is the wavelength range where background subtraction is most uncertain, as it contains the emission peak of cold ($\sim 20 \kel$) ISM dust, so there are likely to be significant additional systematic uncertainties not accounted for by our model. This wavelength range is also where the least well-constrained model parameters, such as the cold gas properties (Appendix \ref{sec:coldprop}), have the most impact on the model dust SEDs. {Modest variations in the assumed cold gas properties would almost certainly be able to obtain statistically-good fits to the data, without substantially altering our main conclusions.} {We have assumed that the radiation field in both components is solely due to shock interactions, but Figure \ref{fig:sed} suggests that heating by the ISM radiation field could also be relevant, particularly for those SNRs located at small Galactocentric distances where the typical radiation field strength is higher \citep{mathis1983}. If we include an additional \citet{mathis1983} radiation field in the dust heating model, multiplied by a factor of five to approximate the stronger $5 \kpc$ field, the best-fit models (shown in Figure \ref{fig:g11ref}) have much lower small/large mass ratios in the cold component than those in Figure \ref{fig:g11fit} without the additional ISM field. However, the upper limit on this quantity (as given by the 84th percentile of the MCMC chain) is $0.36$ for carbon grains and $2.2$ for silicates, still consistent with values above the $0.16$ of the MRN size distribution (and the silicate median of $0.25$ is also higher than the ISM value). Depending on the details of the dust heating model, significant masses of small grains in the cold component may not be necessary to fit the observed SEDs, but they do not seem to be ruled out either, in contrast to the situation in the hot component.} {Another implicit assumption in our models is that all IR flux comes from dust which has already been swept-up by the SNR. An alternative possibility is that the far-IR emission comes from dust grains ahead of the shock, which would explain the large dust masses relative to the inferred swept-up gas masses, and the grain size distributions being consistent with that in the ISM. We disfavour this explanation, as the shock radiation field should only penetrate a short distance into the ambient medium \citep{docenko2010}, and so the volume of dust preheated above ambient temperatures is quite small. The average density in this layer would then have to be orders of magnitude larger than our estimate above, to a somewhat implausible degree (see discussion in \citealt{priestley2021}). In any case, this possibility does not alter our conclusions about the grain sizes in the hot component, or that the mass of this component - in both gas and dust - is a negligible fraction of the total in the immediate surroundings of the SNR.} \subsection{Implications} The overall picture we find for our sample of SNRs is that the high-temperature, diffuse material contains a relatively small mass of large ($\gtrsim 0.1 \um$) grains, while dust in the denser, cooler shocked gas makes up virtually all the total swept-up dust mass, and {may} include a substantial {mass} of small ($\lesssim 10 \nm$) grains. This is {similar} to the situation in the Cassiopeia A reverse shock \citep{priestley2022}, and suggests that the physical processes affecting dust grains in shocks do not differ between the ISM and metal-enriched CCSN ejecta. {The observed distributions of grain sizes in the two gas components are suggestive of the two main processes responsible for processing dust grains in shocks:} small grains in the hot component are rapidly destroyed by sputtering; large grains in the cold component are efficiently converted into smaller grains via shattering in grain-grain collisions \citep{kirchschlager2019}. {Although there is} {some} evidence for shattering ({large small-grain mass fractions}) in the dense, cool ejecta {(subject to the caveats discussed above)}, we find strict upper limits on the mass of small grains in the hot component for all SNRs. The hot component small/large mass ratios in Figure \ref{fig:sizeratio} suggest that either shattering is inefficient in this phase, or is at least subdominant to sputtering (i.e. newly-produced small grains are destroyed by sputtering on shorter timescales than they are produced by shattering of large grains). {Most theoretical studies of dust destruction \citep{jones1996,slavin2015,kirchschlager2022} assume ambient ISM densities of $\sim 0.1-1 \pcc$, similar to the hot component densities in Table \ref{tab:snrprop}, but find much more efficient shattering than appears to be the case in our SNRs (see e.g. the post-shock grain size distributions from \citealt{slavin2015}). These models may be overestimating the importance of grain-grain collisions in the processing of shocked ISM dust.} {The severe mismatch between the estimated total swept-up dust masses, if the ambient ISM densities are those derived from X-ray measurements, and the observed present-day dust masses in Table \ref{tab:mass}, effectively requires that the total (gas plus dust) mass of the hot component is a small fraction of that swept up by the SNRs. Making the conservative assumption that no dust has been destroyed, the implied swept-up {(or soon to be swept-up)} gas masses for a DTG ratio of $0.01$ are (with the exception of G29) hundreds of times larger than those of the SNR hot components (estimated using the gas densities and volumes in Table \ref{tab:snrprop}). Any post-shock reduction in dust mass, or a hot component filling factor lower than unity, will make this discrepancy larger. When considering the overall effect of the SNR on the surrounding ISM, what occurs in (or to) the hot component - the only phase typically modelled by theoretical work - is effectively negligible. As the gas temperatures in {the cold component} are unlikely to be high enough for thermal sputtering\footnote{{Kinetic sputtering may be effective regardless of gas temperature for a sufficiently strong shock, although in higher-density gas the shock strength is also necessarily reduced, while the coupling between gas and grain motions is increased. We would thus expect the kinetic sputtering rate to also be lower in the cold component.}} to be effective {($> 10^5 \kel$; \citealt{biscaro2016})}, and grain collisions themselves mostly reprocess rather than destroy dust \citep{kirchschlager2019}, it is conceivable that {a significant fraction} of the dust in this component has survived being shocked. {It is generally thought that the majority of the mass in the ISM is contained in much denser, colder phases than those typically investigated by models of dust destruction \citep{mckee1977,jones2011}. There is observational evidence, in some cases, for a large fraction of the total SN momentum and energy going into these phases \citep[e.g.][]{cosentino2022}, rather than the more diffuse ISM. If the destruction efficiency in the dense ISM is {in fact} lower, it seems likely that models assuming a uniform, low-density ISM are overestimating the rate of dust destruction, {particularly if grain shattering in the low-density ISM is also being overestimated (which may have a huge impact on the destruction efficiency; \citealt{kirchschlager2022})}.} \section{Conclusions} We have modelled the IR dust emission from the shocked ISM for a sample of seven SNRs, taking into account the multiphase nature of the shocked material, and the uncertain (and likely non-standard) grain properties. We find consistent results {across the} sample: grains located in the {cooler} ($\sim 1000 \kel$) shocked gas make up $> 90\%$ of the total surviving swept-up dust mass; only large ($\gtrsim 0.1 \um$) grains have survived in the hot ($> 10^6 \kel$) phase of the shocked ISM; {grain size distributions in the colder phase are consistent with those in the ISM, or possibly even biased towards small ($\lesssim 10 \nm$) grains}. We {suggest} that this indicates efficient sputtering in the hot phase, and {either} efficient shattering {or generally inefficient dust processing} in the cold phase. The lack of evidence for grain shattering in the hot, {diffuse swept-up ISM} is contrary to models of dust destruction in shocks. {Most theoretical {predictions of the} dust destruction efficiency {in SNRs} assume a uniform, low-density ambient medium. The multi-phase nature of the observed SNRs, with a very small fraction of the total dust mass in the low-density material, suggests that these predicted values may be significantly overestimating the dust destruction efficiency of SNe.} \section*{Acknowledgements} {We are grateful to Florian Kirchschlager for comments on a draft version of this paper.} FDP is supported by a consolidated grant (ST/K00926/1) from the Science and Technology Facilities Council. HC and HLG acknowledge support from the European Research Council (ERC) grant COSMICDUST ERC-2014-CoG-647939. IDL acknowledges support from ERC starting grant 851622 DustOrigin. MJB acknowledges support from the ERC grant SNDUST ERC-2015-AdG-694520. \section*{Data Availability} The data underlying this article will be made available upon request. \bibliographystyle{mnras} \bibliography{hotdust} \appendix \section{SNR dust fluxes} \label{sec:snrsed} Table \ref{tab:flux} lists the IR fluxes used in the dust modelling for each SNR in our sample. \begin{table*} \centering \caption{IR SEDs for our SNR sample, with fluxes given in Jy and filter effective wavelength in $\um$. Data for G11, G27 and G29 are taken from \citet{priestley2021}; G43, G340 and G349 from \citet{chawner2020}; and the Tornado from \citet{chawner2020b}.} \begin{tabular}{ccccccccc} \hline SNR & IRAC $8$ & WISE $12$ & MIPS $24$ & PACS $70$ & PACS $160$ & SPIRE $250$ & SPIRE $350$ & SPIRE $500$ \\ \hline G11 & - & $<6.2$ & $26.0 \pm 3.1$ & $124.8 \pm 22.4$ & $176.8 \pm 85.6$ & $65.0 \pm 52.1$ & $33.4 \pm 21.9$ & $15.3 \pm 7.7$ \\ G27 & - & $<1.2$ & $13.0 \pm 0.2$ & $25.6 \pm 21.6$ & $124.7 \pm 70.0$ & $94.4 \pm 29.5$ & $47.1 \pm 13.2$ & $16.6 \pm 4.5$ \\ G29 & - & $<1.9$ & $10.0 \pm 2.1$ & $101.5 \pm 39.1$ & $16.1 \pm 112.4$ & $11.4 \pm 47.4$ & $1.3 \pm 17.3$ & $2.2 \pm 8.2$ \\ Tornado & $<1.66$ & - & $4.3 \pm 0.2$ & $164.5 \pm 11.5$ & $151.2 \pm 10.6$ & $63.2 \pm 3.5$ & $25.9 \pm 1.4$ & $8.2 \pm 0.5$ \\ G43 & $<61.0$ & - & $80.1 \pm 6.0$ & $744.5 \pm 61.2$ & $283.6 \pm 135.2$ & $74.4 \pm 113.7$ & $20.9 \pm 53.8$ & $5.3 \pm 23.1$ \\ G340 & $<24.5$ & - & $23.2 \pm 7.8$ & $220.6 \pm 29.4$ & $755.0 \pm 117.8$ & $506.5 \pm 85.1$ & $207.5 \pm 36.7$ & $80.2 \pm 14.1$ \\ G349 & $<12.6$ & - & $47.8 \pm 3.5$ & $588.4 \pm 41.5$ & $456.5 \pm 32.8$ & $158.8 \pm 9.1$ & $60.1 \pm 3.5$ & $19.6 \pm 1.2$ \\ \hline \end{tabular} \label{tab:flux} \end{table*} \section{Cold gas properties} \label{sec:coldprop} {For collisional heating in the cold component, we have assumed gas properties derived from H$_2$ line observations by \citet{priestley2021} for G11. There is no guarantee that these properties are appropriate for the other SNRs in our sample. Figure \ref{fig:coltest} shows the influence of the assumed cold component properties on the resulting grain fluxes. We consider three cases: the G11 properties ($\nh = 1000 \pcc$, $\nel = 0.1 \pcc$, $T = 5000 \kel$); radiative heating only, with collisional heating turned off; and $\nh = \nel = 10 \pcc$ with $T = 10^4 \kel$, representing more diffuse, fully-ionised gas. The emissivity of $0.1 \um$ grains is almost completely unaffected by the assumed cold component gas properties, as the grain heating is dominated by the radiation field in all cases. For $5 \nm$ grains, there is a significant difference between the case with no collisional heating and the two cases where it is included, but the grain SEDs are very similar for the two sets of gas properties we consider.} {While investigating the full three-dimensional collisional heating parameter space ($\nh, \nel, T$) is beyond the scope of this paper, it appears that for a range of post-shock dense gas properties, dust SEDs are similar enough to be identical for our purposes. Even if we neglect collisional heating entirely, our main conclusions are qualitatively unchanged. Figure \ref{fig:g11radfit} shows best-fit dust masses for G11, with the cold component only heated by the radiation field. While the dust masses differ from those in Figure \ref{fig:g11fit}, regardless of grain composition we still find that the majority of the dust mass is in the cold component, that the mass of small grains is comparable to that of large grains in this component, and that grains in the hot component (if present) must be large.} \section{Grain size distributions} \label{sec:sizedist} {Our choice of $0.1 \um$ to represent large grains, and $5 \nm$ to represent small ones, is somewhat arbitrary, although informed by the sizes typically thought to be present in the ISM. Figure \ref{fig:g11dist} shows the impact of using a size {\it distribution} for each component, rather than a single representative grain size, using carbon grains in G11 as an example. We replace small and large grains with power-law distributions, with an MRN exponent of $-3.5$, spanning the ranges $5-10 \nm$ and $0.1-0.3 \um$ respectively. Both the SEDs of the individual dust components, and the results of the MCMC fit to the G11 data, are affected by at most a factor of a few. This is a minor source of uncertainty compared to others in our modelling procedure, and is not sufficient to qualitatively change our conclusions to any significant extent.} \section{Silicate grain SED fits} \label{sec:silfit} Figure \ref{fig:silfits} shows the best-fit silicate grain SEDs for our sample of SNRs, excluding G11, which is presented in Figure \ref{fig:g11fit}. {While the different grain composition results in non-trivial changes to both the total dust mass and its distribution between components, we find identical qualitative results across the sample of SNRs as for carbon grains: the hot component is typically a small fraction of the total dust mass, and rarely (if ever) contains any substantial mass of small grains, whereas the cold component requires a non-negligible mass of small grains.} \bsp % \label{lastpage}

Title: Possible counterpart signal of the Fermi bubbles at the cosmic-ray positrons

Abstract: The inner galaxy has hosted cosmic-ray burst events including those responsible for the gamma-ray Fermi bubbles and the eROSITA bubbles in X-rays. In this work, we study the AMS-02 positron fraction and find three features around 12, 21 and 48 GeV of which the lowest energy has a 1.4 to 4.9-$\sigma$ significance, depending on astrophysical background assumptions. Using background simulations that explain the cosmic-ray positron fraction, positron flux and electron plus positron flux, by primary, secondary cosmic rays and cosmic rays from local pulsars, we test these spectral features as originating from electron/positron burst events from the inner galaxy. We find the 12 GeV feature, to be explained by an event of age $\tau \simeq 3 - 10$ Myr; in agreement with the proposed age of the Fermi bubbles. Furthermore, the energy in cosmic-ray electrons and positrons propagating along the galactic disk and not within the Fermi bubbles volume, is estimated to be $10^{51.5}-10^{57.5}$ ergs, or $O(10^{-4}) -O(1)$ the cosmic-ray energy causing the Fermi bubbles. We advocate that these positron fraction features, are the counterpart signals of the Fermi bubbles, or of substructures in them, or of the eROSITA bubbles.

https://export.arxiv.org/pdf/2208.07880

\title{Possible counterpart signal of the Fermi bubbles at the cosmic-ray positrons} \author{Ilias Cholis} \email{cholis@oakland.edu, ORCID: orcid.org/0000-0002-3805-6478} \affiliation{Department of Physics, Oakland University, Rochester, Michigan, 48309, USA} \author{Iason Krommydas} \email{ik23@rice.edu, ORCID: orcid.org/0000-0001-7849-8863} \affiliation{Physics Division, National Technical University of Athens, Zografou, Athens, 15780, Greece} \affiliation{Department of Physics and Astronomy, Rice University, Houston, Texas, 77005, USA} \date{\today} The \textit{Fermi} bubbles \cite{Su:2010qj, Fermi-LAT:2014sfa}, discovered in \textit{Fermi} Large Area Telescope (\textit{Fermi}-LAT) \cite{Gehrels:1999ri, fermiURL} gamma-ray observations, have finite size and well defined limits. This suggests their origin is cosmic-ray activity from the inner galaxy, possibly associated to its supermassive black hole. This activity may leptonic \cite{Crocker:2010qn, Cheng:2011xd, Mertsch:2011es, Guo:2011eg, Guo:2011ip, Yang:2012fy, Carretti:2013sc, Lacki:2013zsa} or hadronic \cite{Crocker:2010qn, Crocker:2010dg, Lacki:2013zsa, Crocker:2014fla}. Moreover, these bubbles may be connected to the galactic center excess in gamma rays \cite{Goodenough:2009gk, Hooper:2010mq, Abazajian:2010zy, Hooper:2011ti, Hooper:2013rwa, Gordon:2013vta, Daylan:2014rsa, Calore:2014xka, Zhou:2014lva, Fermi-LAT:2015sau, Huang:2015rlu, Linden:2016rcf, DiMauro:2021raz, Cholis:2021rpp}, by both excesses resulting from successive cosmic-ray bursts \cite{Petrovic:2014uda, Carlson:2014cwa, Cholis:2015dea}. Additionally, the observation of ``cocoons'' in the southern \cite{Su:2010qj, Fermi-LAT:2014sfa} and the northern \cite{Balaji:2018rwz} galactic hemispheres within the \textit{Fermi} bubbles, and of bubbles in keV X-rays by the \textit{eROSITA} telescope \cite{Predehl:2020kyq}, suggests that the inner galaxy undergoes episodic events of enhanced cosmic-ray injection. Furthermore, \textit{ROSAT} and \textit{Suzaku} X-ray observations found related signals of the \textit{Fermi} bubbles \cite{1997ApJ...485..125S,2013ApJ...779...57K, Tahara:2015mia}, with connections to microwave observations as well \cite{2003ApJ...582..246B}. Finally, a similar excess by morphology was known as the \textit{WMAP} haze \cite{Finkbeiner:2003im, Dobler:2007wv}. This is likely the microwave counterpart signal of the \textit{Fermi} bubbles \cite{Dobler:2009xz}; and was confirmed by \textit{Planck} \cite{Dobler:2012ef, Planck:2012opn, Planck:2015ica}. All these discoveries, point to the \textit{Fermi} bubbles coming from the inverse Compton scattering (ICS) of cosmic-ray electrons ($e^{-}$) and positrons ($e^{+}$) \cite{Crocker:2010qn, Cheng:2011xd, Mertsch:2011es, Guo:2011eg, Guo:2011ip, Yang:2012fy, Carretti:2013sc, Yang:2013kca, Lacki:2013zsa, Yang:2017tjr}. If positrons are contributing to the bubbles emission, a fraction of the originally injected positrons could escape the bubbles regions that expand perpendicularly to the galactic disk. These cosmic-ray positrons would reach us, contributing to the local positron flux. In this \textit{letter} we claim a hint of that event leading to a feature in the positron fraction around energies of 12 GeV (see also \cite{Cholis:2021kqk}). We also find a second feature at 21 GeV, that could come from a more recent burst event from the inner galaxy. We propose that these two features in positrons are associated in energy and age to either the \textit{eROSITA} and \textit{Fermi} bubbles, or the \textit{Fermi} bubbles and the cocoons within them. A clear rise of the positron fraction ($e^{+}/(e^{+}+e^{-})$) above 5 GeV has been discovered the Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (\textit{PAMELA}) and the Alpha Magnetic Spectrometer (\textit{AMS-02}) \cite{PAMELA:2013vxg, AMS:2014bun, AMS:2019iwo, AMS:2021nhj}. That rise is in tension with expectations from positrons produced in inelastic collisions of cosmic-ray protons and nuclei with the interstellar medium (ISM) gas \cite{Moskalenko:2001ya, Kachelriess:2015wpa, GALPROPSite, Strong:2015zva, Evoli:2008dv, DRAGONweb, Evoli:2011id, Pato:2010ih}, and indicates local sources of high-energy positrons. Those could be near-by pulsars \cite{1987ICRC....2...92H, 1995PhRvD..52.3265A, 1995A&A...294L..41A, Hooper:2008kg, Yuksel:2008rf, Profumo:2008ms, Malyshev:2009tw, Kawanaka:2009dk, Grasso:2009ma, 2010MNRAS.406L..25H, Linden:2013mqa, Cholis:2013psa, Yuan:2013eja, Yin:2013vaa, Cholis:2018izy, Evoli:2020szd, Manconi:2021xom, Orusa:2021tts, Cholis:2021kqk}, local and recent supernova remnants (SNRs) \cite{Blasi:2009hv, Mertsch:2009ph, Ahlers:2009ae, Blasi:2009bd, Kawanaka:2010uj, Fujita:2009wk, Cholis:2013lwa, Mertsch:2014poa, DiMauro:2014iia, Kohri:2015mga, Mertsch:2018bqd} (see however \cite{Cholis:2013lwa, Mertsch:2014poa, Cholis:2017qlb, Tomassetti:2017izg}) or particle dark matter \cite{Bergstrom:2008gr, Cirelli:2008jk, Cholis:2008hb, Cirelli:2008pk, Nelson:2008hj, ArkaniHamed:2008qn, Cholis:2008qq, Cholis:2008wq, Harnik:2008uu, Fox:2008kb, Pospelov:2008jd, MarchRussell:2008tu, Chang:2011xn, Cholis:2013psa, Dienes:2013xff, Finkbeiner:2007kk, Kopp:2013eka, Dev:2013hka, Klasen:2015uma, Yuan:2018rys, Sun:2020dla}. In this work, we assume that local pulsars are responsible for the observed cosmic-ray positron flux and fraction measurements following \cite{Cholis:2021kqk}. Pulsars are energetic sources with their properties probed from radio wavelengths to TeV gamma-rays \cite{1968Natur.217..709H, 1998MNRAS.301..235G, 1999ApJS..123..627S, Weisberg_1999, Everett:2000yj, McLaughlin:2003zz, Weisberg:2003ud, 2005AJ....129.1993M, 2009A&A...504..525S, 2012A&A...544A.100M, 2012A&A...540A..28D, 2013MNRAS.433.3325S, 2019A&A...629A.140S, 2009A&A...504..525S, 2012A&A...544A.100M, 2012A&A...540A..28D, 2013MNRAS.433.3325S, 2019A&A...629A.140S, 2019ApJ...871..246M, 2021NatAs...5..552A, 1971ApJ...167L..67G, Becker:2002wf, Gentile:2013yka, 2013MNRAS.433.3325S, Guillot:2019vqp, Zhao:2021ilq, 1994ApJS...90..789U, 2010ApJS..187..460A, Fermi-LAT:2013svs, Buhler:2013zrp, Cholis:2014noa, Fermi-LAT:2019yla, 2009ApJ...700L.127A, MAGIC:2015ggt, HESS:2017lee, Abeysekara:2017hyn, HAWC:2019tcx}. Moreover, they may produce multiple spectral features at high energies \cite{Malyshev:2009tw, Cholis:2017ccs} that dark matter or SNRs may not. Hypothesizing local pulsars being responsible for the overall rise of the positron fraction spectrum, is the conservative assumption when claiming that spectral features in the data are associated to activity from the inner galaxy, instead of coming from an old and energetic local pulsar. In Fig.~\ref{fig:PF_Features}, we show the positron fraction, fitted by two models that each explains the features at 12 and 21 GeV respectively. We use the \textit{AMS-02} cosmic-ray positron fraction \cite{AMS:2019iwo}, -where systematics as instrumental exposure ones may cancel out-, to test the significance of spectral features and constrain the properties of cosmic-ray bursts. We have tested the positron flux \cite{AMS:2019rhg}, and find the features to be present there as well; with however, the cosmic-ray bursts properties be less well constrained. Thus, we focus on positron fraction. Defining the 12 and 21 GeV features as signal, the backgrounds from Ref.~\cite{Cholis:2021kqk}, were fit to the \textit{AMS-02} positron fraction \cite{AMS:2019iwo}, positron flux \cite{AMS:2019rhg} and total $e^{-} + e^{+}$ flux \cite{AMS:2021nhj}; and the \textit{DAMPE} and \textit{CALET} total $e^{-} + e^{+}$ fluxes \cite{DAMPE:2017fbg, Adriani:2018ktz}. Those background simulations include primary electrons accelerated by conventional cosmic-ray sources as SNRs, secondary electrons and positrons produced in inelastic collisions between cosmic-ray protons and nuclei with the ISM gas and local galactic pulsars~\cite{Cholis:2021kqk}. The cosmic-ray propagation, spatial distribution of sources and averaged injection spectra of cosmic-ray nuclei assumptions are in agreement with the \textit{AMS-02} observations \cite{Cholis:2021rpp}, and compatible with~\cite{Trotta:2010mx, Pato:2010ih}. \textit{Our simulations} of~\cite{Cholis:2021kqk}, include $(5-18)\times 10^{3}$ unique pulsars within 4 kiloparsec (kpc) from the Sun. These backgrounds, model uncertainties on a) the stochastic nature of the neutron stars' birth in space and time, with each pulsar having a unique combination. We assumed a total birth rate of 0.6-2 pulsars per century for the Milky Way~\cite{1999MNRAS.302..693D, Vranesevic:2003tp, FaucherGiguere:2005ny, Keane:2008jj}, and relied on observations and modeling of the pulsars' spatial distribution from \cite{Manchester:2001fp, FaucherGiguere:2005ny, Lorimer:2003qc,Lorimer:2006qs}. Our simulations also account for b) each pulsar having a unique initial spin-down power, following a distribution that while uncertain is constrained by radio observations \cite{FaucherGiguere:2005ny, Manchester:2004bp, ATNFSite}. Furthermore, we account for c) the pulsars' time-evolution uncertainties, by testing distinct values for all pulsars' braking index $\kappa$, and characteristic spin-down timescale $\tau_{0}$. The combination of $\kappa$ and $\tau_{0}$ values simulated where in agreement with expected surface magnetic field and period distributions from~\cite{FaucherGiguere:2005ny}. Another modeled uncertainty is d) the fraction $\eta$ of the pulsars' total rotational energy to cosmic-ray $e^{\pm}$ injected to the ISM, and the relevant injection spectral index $n$. Each pulsar has a unique set of $\eta$ and $n$, following alternative choices for a log-normal distribution and a uniform distribution respectively. The spectra from all pulsars have an exponential cut-off at 10 TeV (see Ref.~\cite{Cholis:2021kqk} for details). Finally, these background simulations accounted for e) cosmic-ray propagation uncertainties through the ISM and the Heliosphere. For the ISM propagation, the most significant assumptions are those associated to their diffusion and their energy losses. For isotropic and homogeneous conditions, diffusion is described by a rigidity ($R$)-dependent coefficient, $D(R) = D_{0} (R / (1 \, GV))^{\delta}$, where $D_{0}$ is the normalization at 1 GV and $\delta$ is the diffusion index \cite{1941DoSSR..30..301K, 1967PhFl...10.1417K}. We propagate cosmic rays within a cylinder of radius 20 kpc of half-height $z_{L}$, centered at the galactic center. For cosmic-ray $e^{\pm}$ of $O(10)-O(100)$ GeV, ICS and synchrotron radiation dominate the energy losses. At these energies for ICS the Thomson cross-section \cite{1929ZPhy...52..853K} approximates the Klein-Nishina well \cite{1970RvMP...42..237B}. Thus, the energy-loss rate scales as, $dE/dt = -b (E/(1 \, GeV ))^{2}$. The parameter $b$, scales with the energy density in the CMB and interstellar radiation field photons and the local galactic magnetic field. Diffusive reacceleration \cite{1994ApJ...431..705S} and convective winds affect mostly the cosmic-ray nuclei relevant for the calculation of cosmic-ray secondaries and are included. We use models ``A'', ``C'' and ``E'' from Refs.~\cite{Cholis:2021kqk} and \cite{Cholis:2021rpp} that fit the observed \textit{AMS-02} cosmic-ray hydrogen, helium, carbon, oxygen fluxes and the beryllium-to-carbon, boron-to-carbon and oxygen-to-carbon ratios. For the secondary fluxes \path{GALPROP} v54 \cite{galprop, GALPROPSite}, has been used. We use three choices for the averaged ISM energy losses burst $e^{\pm}$ experience in traveling from the galactic center the Sun. Those are denoted as ``2'', ``4'' and ``5'' \footnote{We retain and extend the notation of~\cite{Cholis:2021kqk} for easier reference.}. These ISM models are given in Table~\ref{tab:ISMBack}. For the local pulsars contributing to the background, only option ``2'' with $b=8.02 \times 10^{-6}$ GeV$^{-1}$ kyrs$^{-1}$ is used, representing the upper limit of local energy-loss uncertainties \cite{galprop, GALPROPSite, Porter:2017vaa}. As we move closer to the galactic center, both the radiation field and the magnetic field amplitude increase (on average). Thus, assumptions of higher energy losses than the local ones are required to study the burst events. Options ``4'' and ``5'' model higher energy losses (than ``2'') and are explicitly used to model cosmic rays originating from the inner galaxy bursts. All cosmic-ray spectra are affected by solar modulation. We implement the time-, charge- and rigidity-dependent formula for the solar modulation potential from \cite{Cholis:2015gna} following the allowed ranges on the $\phi_0$ and $\phi_1$ modulation parameters that describe it from \cite{Cholis:2020tpi, Cholis:2022rwf}. \begin{table}[t] \begin{tabular}{ccccc} \hline Model & $z_{L}$ (kpc) & $b$ ($\times 10^{-6}$GeV$^{-1}$kyrs$^{-1}$) & $D_{0}$ (pc$^2$/kyr) & $\delta$\\ \hline \hline A2 & 5.7 & 8.02 & 140.2 & 0.33 \\ A4 & 5.7 & 16.04 & 140.2 & 0.33 \\ A5 & 5.7 & 24.12 & 140.2 & 0.33 \\ C2 & 5.5 & 8.02 & 92.1 & 0.40 \\ C4 & 5.5 & 16.04 & 92.1 & 0.40 \\ C5 & 5.5 & 24.12 & 92.1 & 0.40 \\ E2 & 6.0 & 8.02 & 51.3 & 0.50 \\ E4 & 6.0 & 16.04 & 51.3 & 0.50 \\ E5 & 6.0 & 24.12 & 51.3 & 0.50 \\ \hline \hline \end{tabular} \vspace{-0.3cm} \caption{The cosmic-ray propagation parameters for the ISM models that we use.} \vspace{-0.5cm} \label{tab:ISMBack} \end{table} \textit{In fitting} the background models, there are seven fitted parameters. There are three normalization factors, for primary $e^{-}$, secondary $e^{\pm}$, and pulsar $e^{\pm}$ fluxes. Two more parameters allow for the spectral hardening or softening of the primary and the secondary cosmic rays, to model uncertainties in the ISM gas and the production efficiency and injection spectra of primary cosmic rays. Finally, the solar modulation parameters $\phi_0$ and $\phi_1$ are marginalized over. Once including the cosmic-ray burst component, an additional normalization is used to set the overall energy deposited in cosmic-ray $e^{\pm}$, that escaped the inner galaxy along the galactic disk; assuming energy equipartition between the two species. We use a combination of \path{SciPy}'s \cite{2020SciPy-NMeth} \path{least_squares} routine from the \path{optimize} module and \path{iminuit} \cite{iminuit,James:1975dr} to fit our models (see Ref.~\cite{Cholis:2021kqk} for further details). \textit{We test four basic properties for the cosmic-ray bursts}. These are the age $\tau$ of the event, defined from the start of the ejection event, and the duration of that event $\Delta \tau$, during which time $90\%$ of the energy was released. The third parameter is the total energy released into cosmic-ray $e^{\pm}$ \textit{escaping the Fermi bubbles, or cocoons, or eROSITA bubbles volume} and reaching us. The forth parameter is the spectral index $n_{\textrm{burst}}$ of the released cosmic rays into the ISM, where, \begin{equation} \frac{dN}{dE}_{\textrm{burst}} \propto E^{-n_{\textrm{burst}}} \textrm{Exp} \left\{ -\frac{E}{E_{\textrm{cut}}}\right\}, \label{eq:InjSpect} \end{equation} with $E_{\textrm{cut}} = 10$ TeV. As cosmic rays cool, the $\textit{AMS-02}$ data cannot probe the exact value of that exponential cut-off. We test values for $n_{\textrm{burst}}$ within $[1.4, 1.9]$. Thus, the total energy is set by the high end of the injected energy. The energy output of the burst can be probed by the cosmic-ray data, through the amplitude of that component as shown in Fig.~\ref{fig:PF_Features}. While a burst has an intrinsic duration $\Delta \tau$, that gives a width to its positron fraction spectral feature, other effects can increase that width. The ICS is a stochastic process (see \cite{John:2022asa}). Also, cosmic rays reaching us diffuse through unique paths. Thus, their travel time varies and they experience unique energy losses as the energy-loss rate is position-dependent within the galaxy. To account for these additional propagation-induced smearing of the burst's flux, we follow Ref.~\cite{Malyshev:2009tw}, where the propagation-induced width $\Delta E$ of an otherwise $e^{\pm}$ with propagated energy $E$ is $\Delta E/E \simeq 0.05 (E/ (1 \textrm{TeV}))^{-1/3}$. For a positron flux feature, this causes a slight shift of its peak to higher energies, which for a positron fraction feature over a falling $e^{\pm}$ primary and secondary only background fluxes as those shown in Fig.~\ref{fig:PF_Features}, (by dot-dashed lines) is further enhanced. We consider the un-smoothed features to be a simplistic approximation, while the smeared ones a more realistic description of the burst's signal on the positron fraction. However, as we show the effect on the final information we derive from the positron fraction on the properties of the burst(s) is minimal. The exact prescription on the propagation smearing is of minor importance. In Fig.~\ref{fig:PF_ISM_assump}, we show the impact that different ISM assumptions have on the exact location of the peak and spectral shape of the burst. The lines have arbitrary normalizations. The normalization of the burst signal changes as faster energy losses or diffusion will suppress the observed amplitude at the Sun. For ISM models ``2'', a burst event contributing at 12 GeV has $\tau \simeq 10$ Myr. Higher energy-loss assumptions from the inner galaxy to the Sun, move the feature's peak from a burst of a given age $\tau$ to lower energies. Thus, higher energy losses to those of the ISM models ``2'', as expected for the inner galaxy will require bursts of younger age. For the feature at 12 GeV, that makes the burst events to be $\tau \simeq 5$ Myr for the ``4'' and $\tau \simeq 3.5$ Myr for the ``5'' ISM models. \textit{After fitting the galactic burst models}, we find a hint for at least one such event present in the positron fraction spectrum. The 12 GeV feature is the most prominent with a statistical significance of $\Delta \chi^{2} \simeq 10$, with a $\Delta \chi^{2}$ range between 2 to 24 (i.e. 1.4 to 4.9 $\sigma$ significance) depending on the background model used. The 21 GeV feature has a significance of $\Delta \chi^{2}$ up to 6.1 (2.4 $\sigma$); while a feature at 48 GeV has a significance of $\Delta \chi^{2}$ up to 8.1 or 2.8 $\sigma$ \footnote{ The quoted $\Delta \chi^2$ values are accurate within $\pm 2$ depending on the minimization starting point.}. Our burst simulations are tested with 20 different background models from~\cite{Cholis:2021kqk}. Each background model gives a good fit to the positron fraction, the positron flux and the $e^{+}+e^{-}$ flux measurements. The bursts do not help explain energy ranges beyond the features themselves. Given these results, we focus on the properties of the burst associated with the most significant and robust 12 GeV feature. A burst event younger by a factor of 0.6 and 1/4th would be responsible for the 21 and 48 GeV features respectively. In terms of its total energy we find the 21 GeV feature to be roughly similar to the 12 GeV feature, which properties we describe next. In Fig.~\ref{fig:12GeVFeature}, we show for the 12 GeV feature the burst's age $\tau$ and $e^{\pm}$ injected energy parameter ranges. We show results where we ignore the propagation smearing (left panel) and where we include it (right panel). This smearing does not significantly change the burst's inferred properties, with the ranges for $\tau$ and the injected energy being marginally wider for the models with the smearing included. The other two burst tested parameters, i.e. $\Delta \tau$ and $n_{\textrm{burst}}$ are not constrained by the positron fraction. We tested values of $\Delta \tau$ from 10 kyr (effectively instantaneous injection) and up to $\Delta \tau /\tau \simeq 1/3$. The presence of smearing reduces slightly the statistical significance of the feature. The $1 \sigma$ ranges are constructed based on the combination of all 20 background models, while the best-fit points show the best-fit model. In this \textit{letter}, we used the cosmic-ray positron fraction observations from \textit{AMS-02} and find a statistically significant 12 GeV feature that can be explained by an inner galaxy burst event of age $\tau \simeq 3-10$ Myr and energy output to cosmic-ray $e^{\pm}$ escaping along the galactic disk (and reaching us) of $10^{51.5}$ to $10^{57.5}$ erg. The larger ages of $\tau \simeq 10$ Myr come from fitting our low energy losses models ``2'', while more realistic assumptions for the inner galaxy (``4'' and ``5'') give $\tau \simeq 3-5$ Myr. This result is in exceptional agreement with estimates for the age of \textit{Fermi} bubbles, as from simulations in \cite{Guo:2011eg, Guo:2011ip}. Furthermore, the \textit{Fermi} bubbles, if of leptonic origin, are expected to have originated by a burst of cosmic rays moving away from the disk with an energy of $10^{56} - 10^{57}$ erg. The 12 GeV feature could be the result of a fraction of that cosmic-ray $e^{\pm}$ energy, escaping from the bubbles region and propagating along the galactic disk, thus reaching our location. The $10^{51.5}$ to $10^{57.5}$ erg is in agreement with that picture, with the ratio of cosmic-ray energy along the disk to the energy within the bubbles being in the very approximate range of $O(10^{-4}) -O(1)$. There are two statistically less significant features at 21 and 48 GeV in the positron fraction. If astrophysical and from the inner galaxy, these features would be associated to more recent burst events. If the 12 GeV feature is the \textit{Fermi} bubbles' counterpart signal in the local positrons, then the smaller in size and later in time gamma-ray cocoons could be associated to one of those higher energy features. Alternatively, the \textit{eROSITA} bubbles with a similar energy output to the \textit{Fermi} bubbles but likely different underlying cosmic-ray spectrum could be the earlier event explaining the 12 GeV feature with the 21 GeV feature being connected to the \textit{Fermi} bubbles. A better understanding of the nature of these inner galaxy bursts detected in gamma rays and X rays will help constrain their properties. Also, with further cosmic-ray measurements, we will attain a better understanding on the local cosmic-ray $e^{\pm}$ fluxes and scrutinize these spectral features. The background simulations from Ref.~\cite{Cholis:2021kqk}, account for primary $e^{-}$, secondary $e^{\pm}$ and $e^{\pm}$ from pulsars; and give good fits to the $e^{\pm}$ measurements from \textit{AMS-02}, \textit{DAMPE} and \textit{CALET}. They account for the stochastic nature of the neutron stars' birth in space and time, uncertainties on their birth rate, initial spin-down power, on the evolution of the pulsars' spin-down, on their injected $e^{\pm}$ fluxes and on cosmic-ray propagation uncertainties. Pulsars can also give spectral features at high energies. For an invisible in electromagnetic observations local pulsar within $\sim 0.5$ kpc, the 12 GeV feature would require it to be of age $\simeq 10$ Myr and with a high initial spin down power at least 3 times that of the Crab pulsar. We consider that an alternative explanation to be probed by multi-wavelength observations. In a separate paper \cite{Krommydas:2022}, we explore the possible impact these lower-energy features have on dark matter limits. We make publicly available our burst simulations in \cite{ZENODOFermiBubbles}, while our background simulations are available through \cite{Cholis:2021kqk, ZENODOBackgrounds}. \textit{Acknowledgements:} We thank Bhaskaran Balaji, Patrick J. Fox, Dan Hooper and Samuel D. McDermott for useful discussions. We acknowledge the use of \texttt{GALPROP} \cite{GALPROPSite, galprop} and the \path{Python} \cite{10.5555/1593511} modules, \path{numpy} \cite{harris2020array}, \path{SciPy} \cite{2020SciPy-NMeth}, \path{pandas} \cite{reback2020pandas,mckinney-proc-scipy-2010}, \path{Jupyter} \cite{Kluyver2016jupyter}, and \path{iminuit} \cite{iminuit,James:1975dr}. IC acknowledges support from the Michigan Space Grant Consortium, NASA Grant No. 80NSSC20M0124. IC acknowledges that this material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0022352. \bibliography{FermiBubbles_Positrons}

Title: BeyondPlanck X. Planck LFI frequency maps with sample-based error propagation

Abstract: We present Planck LFI frequency sky maps derived within the BeyondPlanck framework. This framework draws samples from a global posterior distribution that includes instrumental, astrophysical and cosmological parameters, and the main product is an entire ensemble of frequency sky map samples. This ensemble allows for computationally convenient end-to-end propagation of low-level instrumental uncertainties into higher-level science products. We show that the two dominant sources of LFI instrumental systematic uncertainties are correlated noise and gain fluctuations, and the products presented here support - for the first time - full Bayesian error propagation for these effects at full angular resolution. We compare our posterior mean maps with traditional frequency maps delivered by the Planck collaboration, and find generally good agreement. The most important quality improvement is due to significantly lower calibration uncertainties in the new processing, as we find a fractional absolute calibration uncertainty at 70 GHz of $\delta g_{0}/g_{0} =5 \cdot 10^{-5}$, which is nominally 40 times smaller than that reported by Planck 2018. However, the original Planck 2018 estimate has a non-trivial statistical interpretation, and this further illustrates the advantage of the new framework in terms of producing self-consistent and well-defined error estimates of all involved quantities without the need of ad hoc uncertainty contributions. We describe how low-resolution data products, including dense pixel-pixel covariance matrices, may be produced directly from the posterior samples without the need for computationally expensive analytic calculations or simulations. We conclude that posterior-based frequency map sampling provides unique capabilities in terms of low-level systematics modelling and error propagation, and may play an important role for future CMB B-mode experiments. (Abridged.)

https://export.arxiv.org/pdf/2208.14293

\def\setsymbol#1#2{\expandafter\def\csname #1\endcsname{#2}} \def\getsymbol#1{\csname #1\endcsname} \def\Planck{\textit{Planck}} \def\HeJT{$^4$He-JT} \def\allearlypapers{\nocite{planck2011-1.1, planck2011-1.3, planck2011-1.4, planck2011-1.5, planck2011-1.6, planck2011-1.7, planck2011-1.10, planck2011-1.10sup, planck2011-5.1a, planck2011-5.1b, planck2011-5.2a, planck2011-5.2b, planck2011-5.2c, planck2011-6.1, planck2011-6.2, planck2011-6.3a, planck2011-6.4a, planck2011-6.4b, planck2011-6.6, planck2011-7.0, planck2011-7.2, planck2011-7.3, planck2011-7.7a, planck2011-7.7b, planck2011-7.12, planck2011-7.13}} \def\alltwentythirteenresultspapers{\nocite{planck2013-p01, planck2013-p02, planck2013-p02a, planck2013-p02d, planck2013-p02b, planck2013-p03, planck2013-p03c, planck2013-p03f, planck2013-p03d, planck2013-p03e, planck2013-p01a, planck2013-p06, planck2013-p03a, planck2013-pip88, planck2013-p08, planck2013-p11, planck2013-p12, planck2013-p13, planck2013-p14, planck2013-p15, planck2013-p05b, planck2013-p17, planck2013-p09, planck2013-p09a, planck2013-p20, planck2013-p19, planck2013-pipaberration, planck2013-p05, planck2013-p05a, planck2013-pip56, planck2013-p06b, planck2013-p01a}} \def\alltwentyfifteenresultspapers{\nocite{planck2014-a01, planck2014-a03, planck2014-a04, planck2014-a05, planck2014-a06, planck2014-a07, planck2014-a08, planck2014-a09, planck2014-a11, planck2014-a12, planck2014-a13, planck2014-a14, planck2014-a15, planck2014-a16, planck2014-a17, planck2014-a18, planck2014-a19, planck2014-a20, planck2014-a22, planck2014-a24, planck2014-a26, planck2014-a28, planck2014-a29, planck2014-a30, planck2014-a31, planck2014-a35, planck2014-a36, planck2014-a37, planck2014-ES}} \newbox\tablebox \newdimen\tablewidth \def\leaderfil{\leaders\hbox to 5pt{\hss.\hss}\hfil} \def\endPlancktable{\tablewidth=\columnwidth $$\hss\copy\tablebox\hss$$ \vskip-\lastskip\vskip -2pt} \def\endPlancktablewide{\tablewidth=\textwidth $$\hss\copy\tablebox\hss$$ \vskip-\lastskip\vskip -2pt} \def\tablenote#1 #2\par{\begingroup \parindent=0.8em \abovedisplayshortskip=0pt\belowdisplayshortskip=0pt \noindent $$\hss\vbox{\hsize\tablewidth \hangindent=\parindent \hangafter=1 \noindent \hbox to \parindent{$^#1$\hss}\strut#2\strut\par}\hss$$ \endgroup} \def\doubleline{\vskip 3pt\hrule \vskip 1.5pt \hrule \vskip 5pt} \def\L2{\ifmmode L_2\else $L_2$\fi} \def\dtt{\Delta T/T} \def\DeltaT{\ifmmode \Delta T\else $\Delta T$\fi} \def\deltat{\ifmmode \Delta t\else $\Delta t$\fi} \def\fknee{\ifmmode f_{\rm knee}\else $f_{\rm knee}$\fi} \def\Fmax{\ifmmode F_{\rm max}\else $F_{\rm max}$\fi} \def\solar{\ifmmode{\rm M}_{\mathord\odot}\else${\rm M}_{\mathord\odot}$\fi} \def\Msolar{\ifmmode{\rm M}_{\mathord\odot}\else${\rm M}_{\mathord\odot}$\fi} \def\Lsolar{\ifmmode{\rm L}_{\mathord\odot}\else${\rm L}_{\mathord\odot}$\fi} \def\inv{\ifmmode^{-1}\else$^{-1}$\fi} \def\mo{\ifmmode^{-1}\else$^{-1}$\fi} \def\sup#1{\ifmmode ^{\rm #1}\else $^{\rm #1}$\fi} \def\expo#1{\ifmmode \times 10^{#1}\else $\times 10^{#1}$\fi} \def\,{\thinspace} \def\lsim{\mathrel{\raise .4ex\hbox{\rlap{$<$}\lower 1.2ex\hbox{$\sim$}}}} \def\gsim{\mathrel{\raise .4ex\hbox{\rlap{$>$}\lower 1.2ex\hbox{$\sim$}}}} \let\lea=\lsim \let\gea=\gsim \def\simprop{\mathrel{\raise .4ex\hbox{\rlap{$\propto$}\lower 1.2ex\hbox{$\sim$}}}} \def\deg{\ifmmode^\circ\else$^\circ$\fi} \def\pdeg{\ifmmode $\setbox0=\hbox{$^{\circ}$}\rlap{\hskip.11\wd0 .}$^{\circ} \else \setbox0=\hbox{$^{\circ}$}\rlap{\hskip.11\wd0 .}$^{\circ}$\fi} \def\arcs{\ifmmode {^{\scriptstyle\prime\prime}} \else $^{\scriptstyle\prime\prime}$\fi} \def\arcm{\ifmmode {^{\scriptstyle\prime}} \else $^{\scriptstyle\prime}$\fi} \newdimen\sa \newdimen\sb \def\parcs{\sa=.07em \sb=.03em \ifmmode \hbox{\rlap{.}}^{\scriptstyle\prime\kern -\sb\prime}\hbox{\kern -\sa} \else \rlap{.}$^{\scriptstyle\prime\kern -\sb\prime}$\kern -\sa\fi} \def\parcm{\sa=.08em \sb=.03em \ifmmode \hbox{\rlap{.}\kern\sa}^{\scriptstyle\prime}\hbox{\kern-\sb} \else \rlap{.}\kern\sa$^{\scriptstyle\prime}$\kern-\sb\fi} \def\ra[#1 #2 #3.#4]{#1\sup{h}#2\sup{m}#3\sup{s}\llap.#4} \def\dec[#1 #2 #3.#4]{#1\deg#2\arcm#3\arcs\llap.#4} \def\deco[#1 #2 #3]{#1\deg#2\arcm#3\arcs} \def\rra[#1 #2]{#1\sup{h}#2\sup{m}} \def\page{\vfill\eject} \def\dots{\relax\ifmmode \ldots\else $\ldots$\fi} \def\WHzsr{\ifmmode $W\,Hz\mo\,sr\mo$\else W\,Hz\mo\,sr\mo\fi} \def\mHz{\ifmmode $\,mHz$\else \,mHz\fi} \def\GHz{\ifmmode $\,GHz$\else \,GHz\fi} \def\mKs{\ifmmode $\,mK\,s$^{1/2}\else \,mK\,s$^{1/2}$\fi} \def\muKs{\ifmmode \,\mu$K\,s$^{1/2}\else \,$\mu$K\,s$^{1/2}$\fi} \def\muKRJs{\ifmmode \,\mu$K$_{\rm RJ}$\,s$^{1/2}\else \,$\mu$K$_{\rm RJ}$\,s$^{1/2}$\fi} \def\muKHz{\ifmmode \,\mu$K\,Hz$^{-1/2}\else \,$\mu$K\,Hz$^{-1/2}$\fi} \def\MJysr{\ifmmode \,$MJy\,sr\mo$\else \,MJy\,sr\mo\fi} \def\MJysrmK{\ifmmode \,$MJy\,sr\mo$\,mK$_{\rm CMB}\mo\else \,MJy\,sr\mo\,mK$_{\rm CMB}\mo$\fi} \def\microns{\ifmmode \,\mu$m$\else \,$\mu$m\fi} \def\micron{\microns} \def\muK{\ifmmode \,\mu$K$\else \,$\mu$\hbox{K}\fi} \def\microK{\ifmmode \,\mu$K$\else \,$\mu$\hbox{K}\fi} \def\muW{\ifmmode \,\mu$W$\else \,$\mu$\hbox{W}\fi} \def\kms{\ifmmode $\,km\,s$^{-1}\else \,km\,s$^{-1}$\fi} \def\kmsMpc{\ifmmode $\,\kms\,Mpc\mo$\else \,\kms\,Mpc\mo\fi} \providecommand{\sorthelp}[1]{}