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Title:
Contribution of spicules to solar coronal emission |
Abstract: Recent high-resolution imaging and spectroscopic observations have generated
renewed interest in spicules' role in explaining the hot corona. Some studies
suggest that some spicules, often classified as type II, may provide
significant mass and energy to the corona. Here we use numerical simulations to
investigate whether such spicules can produce the observed coronal emission
without any additional coronal heating agent. Model spicules consisting of a
cold body and hot tip are injected into the base of a warm ($0.5$ MK)
equilibrium loop with different tip temperatures and injection velocities. Both
piston- and pressure-driven shocks are produced. We find that the hot tip cools
rapidly and disappears from coronal emission lines such as Fe XII $195$ and Fe
XIV $274$. Prolonged hot emission is produced by pre-existing loop material
heated by the shock and by thermal conduction from the shock. However, the
shapes and Doppler shifts of synthetic line profiles show significant
discrepancies with observations. Furthermore, spatially and temporally averaged
intensities are extremely low, suggesting that if the observed intensities from
the quiet Sun and active regions were solely due to type II spicules, one to
several orders of magnitude more spicules would be required than have been
reported in the literature. This conclusion applies strictly to the ejected
spicular material. We make no claims about emissions connected with waves or
coronal currents that may be generated during the ejection process and heat the
surrounding area.
| https://export.arxiv.org/pdf/2208.05240 |
\title{Contribution of spicules to solar coronal emission}
\correspondingauthor{Shanwlee Sow Mondal}
\email{shanwlee@prl.res.in\\
shanwlee.sowmondal@gmail.com}
\author[0000-0003-4225-8520]{Shanwlee Sow Mondal}
\affil{Astronomy and Astrophysics Division, Physical Research Laboratory, Ahmedabad 380009, India}
\affil{Indian Institute of Technology, Gandhinagar, Gujarat 382355, India}
\author[0000-0003-2255-0305]{James A. Klimchuk}
\affil{Heliophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Rd., Greenbelt, MD 20771, USA}
\author[0000-0002-4781-5798]{Aveek Sarkar}
\affil{Astronomy and Astrophysics Division, Physical Research Laboratory, Ahmedabad 380009, India}
\keywords{methods: numerical, Sun: corona, Sun: chromosphere, Sun: atmosphere, Sun: magnetic fields, Sun: UV radiation}
\section{Introduction}\label{sec:introduction}
Defying decades of continued efforts, many aspects of coronal heating remain unanswered~\citep{Klimchuk_2006, Klimchuk_2015, Viall_2021}. Even the basic mechanism is a matter of debate. Despite the fact that all the coronal mass is sourced at the chromosphere, agreement on how the chromospheric mass is heated and transported up to the corona has not been reached. An early observation of the solar chromosphere reported the existence of several small jet-like features~\citep{Secchi1877}. They were later named~\textit{spicules}~\citep{Roberts45}. With improved observations, these spicules were seen to propagate upwards~\citep{Pneuman_1977, Pneuman_1978} with speed $20$ - $50$ km s$^{-1}$. They were also seen to survive for about $5$ to $10$ minutes and carry almost $100$ times the mass needed to balance the mass loss in the solar corona due to the solar wind. Further studies of the spicules~\citep{Athay_1982} suggested a pivotal role in transferring energy from the inner layers of the solar atmosphere to the lower solar corona. However, the proposal was not pursued further because these traditional spicules lack emission in the Transition Region (TR) and coronal lines~\citep{Withbroe_1983}.
About a decade ago, using high-resolution imaging and spectroscopic observations from the Hinode and Solar Dynamic Observatory missions, \cite{Pontieu_2007, Pontieu_2011} further discovered jet like features traveling from the chromosphere to the corona. These features appear all over the Sun with a lifetime between $10-150$ s and a velocity of $50-150$ km s$^{-1}$. \cite{Pontieu_2007} termed them type II spicules and suggested that they are capable of connecting the relatively cooler solar chromosphere with the hot corona.
Since their discovery, multiple observations have identified type II spicules and reported on their characteristics. However, nothing conclusive has yet been established about their origin. Only recently,~\cite{samanta_19} have identified the near-simultaneous origin of spicules and the emergence of photospheric magnetic bipoles. The tips of the originated spicules eventually appear in the coronal passband, suggesting that the plasma is heated to coronal temperatures. A 2.5D radiative MHD simulation of type II spicules~\citep{Martinez_2017} has reproduced many of their observed features. This simulation also suggests that ambipolar diffusion in the partially ionized chromosphere may play a crucial role in the origin of type II spicules. On the other hand, a recent work~\citep{Dey_22} based on radiative MHD simulation and laboratory experiment suggests that quasi-periodic photospheric driving in the presence of vertical magnetic fields can readily generate spicules in the solar atmosphere. Their work, devoid of any chromospheric physics, can still account for the abundance of wide varieties of spicules, as seen in the observations.
The evolution of spicules during their propagation is understood through multi-wavelength studies~\citep[e.g.,][]{Pontieu_2011, Skogsrud_2015}. Observations of ~\cite{Pontieu_2011} suggest that spicule plasma emanating from the chromosphere gets heated to transition region (TR) temperatures and even up to coronal temperatures. Such heating may happen for two reasons:
\begin{enumerate}
\item[(a)] Spicule propagation can produce shocks, compressing the material lying ahead of it. In such a scenario, it is not the ejected spicule material but the pre-existing coronal material in front of it that gets compressed by the shock to contribute to the hot emission ~\citep{Klimchuk_2012, Petralia_2014};
\item[(b)] The tip of the spicule may get heated during the ejection process, on-site, through impulsive heating and produce emissions in the coronal lines. In the latter scenario, the emission indeed comes from the ejected spicule material ~\citep{Pontieu_2007}.
\end{enumerate}
The radiative MHD simulations of ~\cite{Martinez_2018} suggest that spicules and surrounding areas get heated by ohmic dissipation of newly created currents and by waves. Note, however, that the currents in the simulations are relatively large-scale volume currents and would not be dissipated efficiently at the many orders of magnitude smaller resistivity of the real corona. Heating in the real corona involves magnetic reconnection at thin current sheets, of which there are at least $100,000$ in a single active region \citep{Klimchuk_2015}. It is not known whether the ohmic heating in the simulations is a good proxy for the actual reconnection-based heating.
\cite{Klimchuk_2012} considered a simple analytical model for the evolution of spicules with a hot tip. He argued that if a majority of observed coronal emission were from such hot tips, it would be inconsistent with several observational features (see also \cite{Tripathi_2013, Patsourakos_2014}). The result was further supported by hydrodynamic simulations~\citep{Klimchuk_2014}. Using these simulations, they studied the response of a static loop to impulsive heating in the upper chromosphere, which produces localized hot material that rapidly expands upward and might represent the hot tip of a spicule. Noticing the inability of a single hot spicule tip to explain the observations, \cite{Bradshaw_2015} further explored the role of frequently recurring chromospheric nanoflares. The study was motivated by the suggestion that rapidly repeating type II spicules might accumulate enough hot plasma to explain the coronal observations \citep{Pontieu_2011}. However, the simulations were still inconsistent with observations.
In both the analytical model and the simulations, the dynamics of the hot material is due entirely to an explosive expansion from the locally enhanced pressure. There is no additional imposed force to bodily eject the material. The consequences of such a force were investigated by
\cite{Petralia_2014}. Their study involves injecting cold and dense chromospheric material into the corona with an initial velocity. The result indicates that the production of a shock can give rise to coronal emission. However, the emission is from the preexisting coronal material rather than the spicule itself. The injected material has no hot component.
The studies mentioned above have investigated the dynamics of either the hot tip of a spicule without any initial velocity or a spicule with a cold tip and finite injection velocity. Our work combines these two effects. The spicule is now injected in a stratified flux tube with high velocity and consists of both a hot tip and a cold body ($T = 2 \times 10^{4}$~K). We further investigate the possibility that most of the observed hot emission from the corona can be explained by such spicules. Through forward modelling, we quantitatively compare the simulations with observations to answer this question.
The rest of this paper is organized as follows. The numerical setup is described in Section~\ref{sec:setup}. We report on the simulation results in Section~\ref{sec:result}. Finally we summarize and discuss our results in Section~\ref{sec:summary}.
\section{Numerical Setup}\label{sec:setup}
Spicules are seen to follow magnetic field lines. To simulate their dynamics, we consider a straight 2D magnetic flux tube consisting of uniform $10$~G magnetic field. We impose a gravity corresponding to a semi-circular loop such that the vertical component of the gravitational force is maximum at both ends and smoothly becomes zero in the middle of the tube. Two ends of the tube are embedded in the chromosphere. The loop is symmetric about the center, which corresponds to the apex. We use Cartesian coordinates, and therefore the loop actually corresponds to an infinite slab. This is a reasonable approximation because we are interested in how the plasma evolves within an effectively rigid magnetic field appropriate to the low $\beta$ corona. The slab dimension corresponding to the loop length is $100$~Mm. The other dimension is $0.42$~Mm, but this is not relevant. Rigid wall boundary conditions are imposed at the sides, and the evolution is essentially equivalent to 1D hydrodynamics, as discussed below. The first 2 Mm of both ends of the loop are resolved with a fine uniform grid with 10 km cells, while the coronal part is resolved with a stretched grid containing 1500 cells on each side. The fine grid close to both the footpoints allows us to resolve the steep transition region more accurately.
The spicule simulation begins with an initial static equilibrium atmosphere obtained with the double relaxation method described in Appendix~\ref{append:steady_state}. We choose a relatively low temperature and low density loop because we wish to test the hypothesis that the observed coronal emission comes primarily from spicules. The apex temperature of the loop is $0.5$ MK. Figure~\ref{initial_density_temp} shows the background loop profile that is used in most of our simulations. The chromosphere is $470$ km deep - approximately half a gravitational scale height. It merely acts as a mass reservoir. Detailed chromospheric physics like partial ionization and optically thick radiation are not implemented in the code as we are solely interested in coronal emission. We use a modified radiation loss function to maintain a chromospheric temperature near $2 \times 10^{4}$~K, as described in Appendix~\ref{append:steady_state}.
The propagation of a spicule in the loop is emulated through an injection of dense material from the left footpoint. The injected material follows specified density, velocity and temperature profiles in time which are described below. At this injection boundary, all plasma parameters, except the density and pressure, are set to their initial values once the injection is over. The density is set to have the prescribed value at the end of the injection phase, and pressure is determined from the ideal gas equation of state. On the other hand, at the right footpoint, all the plasma parameters maintain the initially prescribed values throughout the entire simulation.
We solve the compressible MHD equations inside our simulation domain using the PLUTO code \citep{2007ApJS..170..228M} with ideal gas environment. Plasma inside the domain is cooled by radiation and field aligned thermal conduction. The CHIANTI \citep{chianti} radiative loss function for coronal abundance is used to model the radiative cooling. For anisotropic conduction, the thermal conductivity, $\kappa_{\parallel}$ = $5.6 \times 10^{-7} T^{5/2}$ erg s$^{-1}$ K$^{-1}$ cm$^{-1}$ is considered along the magnetic field lines, whereas $\kappa_{\perp}$ is taken to be zero. Also, for the saturated conductive flux used in PLUTO, $F_{sat}$ = 5$\phi \rho C_{s}^{3}$, where we have considered the value of the free parameter $\phi$ to be 0.9, which represents effective thermal conduction in the system, and $C_{s}$ is the isothermal sound speed. The MHD equations are solved in Cartesian coordinates.
The photospheric magnetic flux is found to be localized and clumpy, whereas, in the corona, it fills out space uniformly. Such nature of the magnetic flux at different layers of the solar atmosphere dictates that the flux tubes expand laterally at the junction of the chromosphere and corona, where the plasma $\beta$ changes from being greater than one to less than one. This type of expansion of flux tubes is realized in 2D MHD simulations of coronal loops~\citep[e.g.,][]{Guarrasi_14}. Through an area expansion factor, this has also been incorporated in 1D or 0D models \citep{Mikic_13, Cargill_22}. We do not include expansion in our model because we are interested in the spicule dynamics in the corona, and the simplification should not affect our results significantly. We note that the plasma $\beta$ is less than unity throughout the evolution, so no expansion from the spicule injection would be expected. Additionally, the initial atmosphere and injection profile are uniform along the horizontal (cross-field) axis. Hence the plasma remains nearly uniform in the lateral direction, effectively making our simulations similar to 1D hydrodynamic simulations. Nevertheless, we ran all our computations using the 2D MHD set up because of our familiarity with the powerful PLUTO code. The limited number of grid points in the cross-field direction keeps the computational demands relatively low.
Two main components of our simulations are: (a) a background loop in hydrostatic and energy equilibrium representing a tenuous coronal atmosphere, and (b) the propagation of injected material resembling spicule propagation along the loop. Our experimental spicule consists of a hot dense tip followed by cold dense material injected from the base of the model. Here we investigate how changing the temperature of the hot tip and injection speed can alter the intensities and profiles of the Fe XII (195 \AA) and Fe XIV(274 \AA) coronal spectral lines.
We have performed six sets of simulations where the spicule tip temperatures are considered to be at $2$, $1$, and $0.02$ MK, followed by cold material with a temperature of $0.02$ MK. All the runs are performed with two injection velocities: $50$ and $150$ km s$^{-1}$ (see Table~\ref{tab:table1}). Since we assume that spicules might have been generated deep inside the chromosphere, we inject a high-density material in the loop to emulate the spicule. The density scale height of the spicule is chosen to be six times the gravitational scale height at the base of the equilibrium loop. To impose such conditions on the ejected spicule, its density follows a time profile given by,
\begin{equation}
\label{rho_profile}
\rho(t)=
\begin{cases}
\rho_{0}\exp\Big[\frac{v(t) t}{6H}\Big], & \ 0 < t \le t_{5} \\
\rho(t_{5}), & \ t_{5} < t \\
\end{cases}~,
\end{equation}
where $\rho(t)$ and $\rho_{0}$ are the injected density at time $t$ and the base density of the equilibrium loop, respectively. The time profile of the injection velocity is given by,
\begin{equation}
\label{vel_profile}
v(t)=
\begin{cases}
v_{inj} \times \Big(\frac{t}{t_{1}}\Big), & \ 0 < t \le t_{1} \\
v_{inj}, & \ t_{1} < t \le t_{4}\\
v_{inj} \times \Big(\frac{t_{5}-t}{t_{5}-t_{4}}\Big), & \ t_{4} < t \le t_{5} \\
0, & \ t_{5} < t \\
\end{cases}~,
\end{equation}
where $v_{inj}$ corresponds to $50$ or $150$ km s$^{-1}$ (depending on the simulation). $H$ represents the gravitational scale height given by
\begin{equation}
H = \frac{k_{B}T_{base}}{\mu m_{H} g_{\odot}}~,
\end{equation}
where $T_{base}=0.02$~MK is the base temperature of the loop, $k_{B}$ is the Boltzmann constant, while $m_{H}$ and $g_{\odot}$ represent mass of the hydrogen atom and solar surface gravity, respectively, and $\mu = 0.67$ denotes the mean molecular weight of the plasma.
The temperature of the ejected spicule also follows a time profile given by
\begin{equation}
\label{tmp_profile}
T(t) =
\begin{cases}
T_{base} + (T_{tip}-T_{base})\times\Big(\frac{t}{t_{1}}\Big), & \ 0 < t \le t_{1} \\
T_{tip}, & \ t_{1} < t \le t_{2}\\
T_{tip} + (T_{base}-T_{tip})\times\Big(\frac{t-t_{2}}{t_{3}-t_{2}}\Big), & \ t_{2} < t \le t_{3} \\
T_{base}, & \ t_{3} < t \\
\end{cases}~,
\end{equation}
where $T_{base}$ is the temperature of the cold material (bottom part) of the spicule ($=0.02$ MK) and $T_{tip}$ is the spicule tip temperature which can take values $2$, $1$, or $0.02$~MK depending on the run being performed. In all the above equations, times $t_{1}$, $t_{2}$, $t_{3}$, $t_{4}$ and $t_{5}$ are chosen to be $2$, $10$, $12$, $90$ and $100$ s, respectively. Times are chosen so that the top $10\%$ of the spicule's body emits in coronal lines as is generally observed~\citep{Pontieu_2011}. The total injection duration is also motivated by the observed lifetime of type II spicules \citep{Pontieu_2011}.
The ramping up of velocity, density, and temperature ensures a smooth entry of the spicules into the simulation domain. Similarly, the ramping down at the end of the injection avoids any spurious effects. Figure~\ref{pulse} shows one such example of velocity, density, and temperature profiles when the spicule is ejected with velocity $150$ km s$^{-1}$, and its hot tip is at $2$~MK. Likewise, different injection time profiles have been used for other injection velocities and temperatures. The initial equilibrium loop remains the same in all cases, unless specified. \\
\section{Results}\label{sec:result}
The large velocity of the spicule and high pressure compared to the ambient medium give rise to a shock, which propagates along the loop and heats the material ahead of it. Depending on the sound speed of the ambient medium (i.e., the preexisting loop plasma) and the temperature of the injected spicule material, the generated shock turns out to be of different kinds: (a) Piston driven shock -- in which case the shock speed is nearly equal to the injection speed (e.g., simulation with $T_{tip} = 0.02$~MK), and (b) Pressure driven~ shock -- in which case the shock speed exceeds the injection speed (e.g., when $T_{tip} = 2$ and $1$~MK). Emission from the shock heated plasma differs depending on the nature of the shock.
We compare different simulations to understand the coronal response to spicules with different injection parameters. Our discussion starts with the results from Run1 where the hot tip of the injected spicule has a temperature $T_{tip} = 2$~MK and injection velocity $v = 150$ km s$^{-1}$. The injection profiles are those already shown in Figure~\ref{pulse}.
\subsection{Dynamics and heating}
Insertion of dense, high temperature plasma ($T_{tip} = 2$ MK) with high velocity ($v = 150$ km s$^{-1}$, Run1) into the warm loop produces a shock. Figure~\ref{shock} shows the temperature, density and plasma velocity along the loop at $t = 70$~s. The dashed lines mark the location of the shock front. It is evident from the figure that the high compression ratio exceeds the ratio of an adiabatic shock. The compression ratio of an adiabatic shock should always be $\leq 4$. To understand the nature of the shock, we perform a shock test with Rankine-Hugoniot (RH) conditions, which read
\begin{equation}\label{RH_rho_vel}
\frac{\rho_{2}}{\rho_{1}} = \frac{\gamma + 1}{\frac{2}{M^{2}} + (\gamma - 1)} = \frac{v_{1}}{v_{2}}~.
\end{equation}
Here $\rho_{1}$ and $\rho_{2}$ are the pre- and post-shock plasma mass densities respectively, and $v_{1}$ and $v_{2}$ are likewise the pre- and post-shock plasma velocities in the shock rest frame. Furthermore, $\gamma$ is the ratio of the specific heats, $c_{s} = \sqrt{\frac{\gamma P_{1}}{\rho_{1}}}$ is the upstream sound speed, where $P_{1}$ is the upstream pressure, and finally $M=v_{1}/c_{s}$ is the upstream Mach number in the shock reference frame. Injection of high temperature plasma accelerates the shock with a speed much larger than the injection speed of the spicule material giving rise to a pressure driven shock front.
Figure~\ref{shock} demonstrates an abrupt change in plasma variables at the shock. The shock speed at this instant is $562$ km s$^{-1}$. It also shows that at the discontinuity location ($s = 36.9$~Mm, $s$ being the coordinate along the loop), the density and velocity ratios are $10.7$ and $0.094$, respectively, in the shock rest frame. The inverse relationship of these ratios indicates a constant mass flux across the shock front, in accordance with equation~\eqref{RH_rho_vel}. The Mach number in the shock frame at the same location is $3.37$. With this Mach number, the RH condition gives density and velocity ratios in accordance with those in the simulation ($10.5$ and $0.095$) when $\gamma = 1.015$. In other words, consistency is achieved with this value of $\gamma$. Being close to unity, it implies a nearly isothermal shock. Efficient thermal conduction carries a large heat flux from the shock front to its surroundings, giving rise to the locally smooth, near-isothermal temperature profile in Figure~\ref{shock}. It is worth mentioning here that RH-jump conditions do not consider any heat loss/gain, such as thermal conduction or radiative loss. However, our system includes these sink terms in the energy equation. It is because of these loss functions the shock-jump is larger. Limited thermal conduction would bring the jump condition closer to the adiabatic approximation but would also affect the thermal profile ahead and behind the shock. Our result is consistent with that of~\cite{Petralia_2014}, where the signature of the shocks in front of the spicule has been reported. As we show later, the initially hot material in the spicule tip cools dramatically. Only ambient material heated by the shock is hot enough to produce significant coronal emission.
Interestingly, the high compression ratio at the shock front depends more on the temperature difference and corresponding pressure difference between the injected and ambient plasma material than on the velocity with which it is injected. Table~\ref{tab:table1} shows a study of how the compression ratio (or the shock strength) varies when the tip of the spicules are at different temperatures and are injected with different velocities. As mentioned earlier, the injection conditions give rise to two different types of shocks. When the injected plasma temperature is high (e.g., spicule tips with temperatures $2$ and $1$~MK), the excess pressure gives rise to a pressure-driven shock. On the other hand, injection of a cold material (tip temperature equal to that at the loop footpoint, i.e., $0.02$~MK) produces a piston-driven shock. Our test runs identify both kinds of shocks. For example, when we inject spicules with a fixed injection velocity of $150$~km s$^{-1}$, but with different tip temperatures (viz.\,$2$, $1$ and $0.02$~MK), the average shock speed is $520$, $400$ and $210$ km s$^{-1}$, respectively (see Figure~\ref{shock_speed}). The first two shocks are pressure-driven as the average shock speeds exceed the injection speed by a wide margin. The third shock maintains a speed close to the injection speed and can be categorized as a piston-driven shock. The shock speed depends not only on the injected tip temperature, but also on the properties of the ambient material in which it is propagating, which vary along the loop. This is discussed further in Appendix~\ref{append:shock_speed}.
\begin{deluxetable}{cccc}
\tablenum{1}
\tablecaption{Dependence of compression ratio on the injected hot tip temperature and speed.\label{tab:table1}}
\tablehead{
\colhead{Run} & \colhead{\hspace{1cm}$T_{tip}$\hspace{1cm}} & \colhead{\hspace{1cm}$v$\hspace{1cm}} & \colhead{\hspace{1cm}Compression\hspace{1cm}}\\
\colhead{} & \colhead{\hspace{1cm}(MK)\hspace{1cm}} & \colhead{\hspace{1cm}(km s$^{-1}$)\hspace{1cm}} & \colhead{\hspace{1cm}ratio\hspace{1cm}}
}
\startdata
1 & \hspace{1cm} 2 & \hspace{1cm}150 & \hspace{1cm}11.2\\
2 & \hspace{1cm}2 & \hspace{1cm}50 & \hspace{1cm}8.9\\
3 & \hspace{1cm}1 & \hspace{1cm}150 & \hspace{1cm}8.7\\
4 & \hspace{1cm}1 & \hspace{1cm}50 & \hspace{1cm}6.2\\
5 & \hspace{1cm}0.02 & \hspace{1cm}150 & \hspace{1cm}3.6\\
6 & \hspace{1cm}0.02 & \hspace{1cm}50 & \hspace{1cm}1.7\\
\enddata
\end{deluxetable}
\subsection{Loop emission}\label{sec:forward}
Thermally conducted energy from the shock front heats the material lying ahead of it. Therefore, a magnetic flux tube subjected to spicule activity could produce hot emission from newly ejected material at the spicule's hot tip and from pre-existing coronal material in both the pre and post-shock regions. We now examine the contributions from these three different sources. We identify the leading edge of the hot spicule tip by finding the location in the loop where the column mass integrated from the right footpoint equals the initial column mass of the loop. Recall that the spicule is injected from the left footpoint. The spicule compresses the material in the loop, but does not change its column mass. We identify the trailing edge of the hot material in a similar manner, but using the column mass at time $t=10$~s, when the injection of hot material ceases and the injection of cold material begins.
Figure~\ref{fe12_10_70} shows emission along the loop in the Fe XII and Fe XIV lines at $t = 10$ and $70$~s, evaluated from Run1. The orange region is the hot spicule tip, while the red region is the shock-heated material ahead of it. The shock front is the dot-dashed black vertical line. The dark orange curve is temperature in units of $10^5$ K, with the scale on the left. The blue curve is the logarithm of density, with the scale on the right. The yellow and green curves are the logarithms of Fe XII and Fe XIV intensity, respectively, with the scale on the left. The variation of intensity is enormous; a difference of 10 corresponds to 10 orders of magnitude. The intensity is what would be observed by the Extreme ultraviolet Imaging Spectrometer (EIS; \cite{Culhane_2007}) onboard Hinode \citep{Kosugi_2007} if the emitting plasma had a line-of-sight depth equal to the EIS pixel dimension, i.e., if observing an EIS pixel cube. This can be interpreted as normalized emissivity.
At $t=10$~s, the emission in both lines comes primarily from the injected hot plasma (orange region). On the other hand, at $t=70$~s it comes primarily from the shock heated plasma (red region). The transition happens very early on. Shortly after the injection of the hot material stops ($t = 10$ s), emission from the shock heated material starts dominating the total emission from the loop. This is evident in the time evolution plot of the loop-integrated emission in Figure~\ref{emission_evolution_fe12_fe14}. Shown are the intensities that would be observed by EIS, assuming that the loop has a cross section equal to the pixel area and that all of the loop plasma is contained within a single pixel. This corresponds to a loop that has been straightened along the line of sight and crudely represents a line of sight passing through an arcade of similar, out of phase loops. The black curve shows the evolution of the total emission contributed by the spicule and pre-existing plasma. Subtracting the spicule component (red curve) from the total gives the evolution of the emission coming solely from the pre-existing (non-spicule) loop material (green curve). Soon after the hot tip of the spicule completes its entry into the loop (at $t=10$~s), the emission from the spicule falls off rapidly. This is because the hot material at the spicule tip cools rapidly as it expands in the absence of any external heating. It is far too faint to make a significant contribution to the observed coronal emission, as emphasized earlier by \cite{Klimchuk_2012} and \cite{Klimchuk_2014}.
For better comparison with observations, we construct synthetic spectral line profiles. The methodology is explained in Appendix~\ref{append:Forward_Modelling}. To construct these profiles, we imagine that the loop lies in a vertical plane and is observed from above. We account for the semi-circular shape when converting velocities to Doppler shifts. We then integrate the emission over the entire loop and distribute it uniformly along the projection of the loop onto the solar surface. We assume a cross section corresponding to an EIS pixel, and thereby obtain a spatially averaged EIS line profile for loop. Finally, a temporal average is taken over the time required for the shock to travel to the other end of the loop ($\approx 190$~s in this case). Such spatially and temporally averaged line profiles from a single loop (e.g., Figure~\ref{spectral_line_fe12_fe14}) is equivalent to an observation of many unresolved loops of similar nature but at different stages of their evolution \citep{Patsourakos_2006,Klimchuk_2014}.
Asymmetric coronal line profiles with blue wing enhancement are manifestations of mass transport in the solar corona. Type II spicules are often suggested to be associated with such a mass transport mechanism ~\citep{Pontieu_2009, Pontieu_2011, Martinez_2017}. However, the extreme non-Gaussian shapes of the simulated Fe XII and Fe XIV line profiles (Figure~\ref{spectral_line_fe12_fe14}) are significantly different from observed shapes~\citep{Pontieu_2009, Tian_2011, Tripathi_2013}. Also, the very large blue shifts are inconsistent with observations. Observed Doppler shifts of coronal lines tend to be slower than $5$ km s$^{-1}$ in both active regions ~\citep{Doschek_2012,Tripathi_2012} and quiet Sun ~\citep{Chae_1998,Peter_1999}. In contrast, a shift of $150$ km s$^{-1}$ is evident in the simulated spectral lines (Figure~\ref{spectral_line_fe12_fe14}).
Our simulation is not reliable after the shock reaches the right boundary of the model. Because of rigid wall boundary conditions, it reflects in an unphysical manner. One might question whether the emission after this time could dramatically alter the predicted line profiles. We estimate the brightness of this neglected emission using the loop temperature profile shortly before the shock reaches the chromosphere at $t = 190$~s. The temperature peaks at the shock, and there is strong cooling from thermal conduction both to the left (up the loop leg) and, especially, to the right (down the loop leg). We estimate the cooling timescale according to:
\begin{equation}
\tau_{cond}= \frac{21}{2}\frac{k_{B}n_{e}l^{2}}{\kappa_{0\parallel}T^{5/2}}~,
\end{equation}
where, $k_{B}$ is the Boltzmann constant, $\kappa_{0\parallel}$ is the coefficient of thermal conductivity along the field lines, $T$ is the temperature at the shock, $n_{e}$ is the electron number density behind the shock, and $l$ is the temperature scale length. We do this separately using the scale lengths on both sides of the shock, obtaining $\tau_{cond}=1290$~s and {7}~s for the left and right sides, respectively. Radiative cooling is much weaker and can be safely ignored. We estimate the integrated emission after $t = 190$~s by multiplying the count rate at that time by the longer of the two timescales, thereby obtaining an upper limit on the neglected emission in our synthetic line profiles. The result is $10565$ DN pix$^{-1}$ for Fe XII and $2206$ DN pix$^{-1}$ for Fe XIV. These are about $0.97$ and $2.76$ times the temporally integrated emission before this time, for Fe XII and Fe XIV, respectively.
The factors are much smaller using the shorter cooling timescale. Even the large factors do not qualitatively alter our conclusions. The profile shapes and Doppler shifts would still be much different from observed. The conclusions we draw below are also not affected by neglecting the emission after the shock reaches the right footpoint.
\subsection{Comparison with observations}
We now estimate the spicule occurrence rate that would be required to explain the observed coronal intensities from active regions and quiet Sun. We have already seen that, in the absence of any external (coronal) heating, the hot material at the tip of the spicule cools down rapidly. However, we are concerned here with the total emission, including that from pre-existing material that is heated as the spicule propagates along the loop. Consider a region of area $\mathcal{A}_{reg}$ on the solar surface, large enough to include many spicules. If the spatially averaged occurrence rate of spicules in this region is $\mathcal{R}$ (cm$^{-2}$ s$^{-1}$), then one may expect $\mathcal{N}_{reg} = \mathcal{R}\tau\mathcal{A}_{reg}$ spicules to be present at any moment, where $\tau$ is the typical spicule lifetime. Since we are averaging over large areas, the orientations of the spicule loops does not matter, and we can treat the loops as straightened along the line of sight, as done for Figure 5. If $\mathcal{I}_{sp}$ (DN s$^{-1}$ pix$^{-1}$) is the temporally averaged intensity of such a loop (the full loop intensity divided by 190 s in Fig. 5), then the expected intensity from a corona that only contains spicule loops is $\mathcal{I}_{obs} = \mathcal{N}_{reg}\mathcal{I}_{sp}\mathcal{A}_{sp}/\mathcal{A}_{reg} = \mathcal{I}_{sp}\mathcal{R}\tau\mathcal{A}_{sp}$,
where $\mathcal{A}_{sp}$ is the cross-sectional area of the loop.
The typical intensities ($\mathcal{I}_{obs}$) observed by EIS in active regions and quiet Sun are, respectively, $162$ and $34$ DN s$^{-1}$ pix$^{-1}$ in Fe XII (195 \AA) and $35$ and $4$ DN s$^{-1}$ pix$^{-1}$ in Fe XIV (274 \AA) ~\citep{Brown_2008}. On the other hand, the temporally averaged intensities from our simulation ($\mathcal{I}_{sp}$) are $56.36$ and $4.22$ DN s$^{-1}$ pix$^{-1}$ for Fe XII and Fe XIV, respectively. Considering $\tau$ to be $190$~s, the time it takes for the shock to travel across the loop, we derive an occurrence rate ($\mathcal{R}$) of spicules as a function of their cross-sectional area ($\mathcal{A}_{sp}$). Results are shown in Figure~\ref{count_fe12_fe14} for the two lines.
Following our earlier logic, we may also argue that at any given time there are $\mathcal{N}_{\odot}=\mathcal{R}\tau \mathcal{A}_{\odot}$ spicules on the solar disk, where $\mathcal{A}_{\odot}$ is the area of the solar disk. Using the estimated value of the occurrence rate of the spicules ($\mathcal{R}$), and taking $\tau$ to be $190$~s as before, the number of spicules on the solar disk is related to the other quantities as per $\mathcal{N}_{\odot} = (\mathcal{I}_{obs}/\mathcal{I}_{sp})(\mathcal{A}_{\odot}/\mathcal{A}_{sp})$. This formula represents $\mathcal{N}_{\odot}$ as a function of the spicule cross-sectional area $\mathcal{A}_{sp}$ (Figure~\ref{count_QS_AR}). Considering the fact that the typical observed widths of spicules lie between $200-400$~km~\citep{Pereira_2011}, we find that the full disk equivalent number of spicules required to explain the observed intensities exceeds $10^7$ in the quiet Sun and $10^8$ in active regions, as indicated by the green shaded region in Figure~\ref{count_QS_AR}. However, observational estimations for the number of spicules on the disk vary between $10^5$~\citep{Sterling_2016} and $2 \times 10^7$~\citep{Judge_2010}. There is a large discrepancy. Far more spicules than observed would be required to produce all the observed coronal emission. For the quiet Sun, $100$ times more spicules would be needed, while for active regions, $10-10^3$ times more would be needed. These are lower limits based on Run1. Our other simulations imply even greater numbers of spicules (see Table~\ref{tab:table2}).
We should mention here that the larger the height the spicule rises, the longer the time it compresses the ambient material, and thus the brighter the time averaged emission. The spicules in our simulations with $150$ km s$^{-1}$ injection speed reach a height of about $23$~Mm, which is much larger than the typically observed spicule height ($\sim 10$~Mm). Therefore, we are likely to overestimate the emission coming from spicule loops, and so the discrepancy between the required and observed number of spicules is even greater. It should also be noted that the values estimated by ~\cite{Sterling_2016, Judge_2010} consider both type I \& II spicules. The discrepancy thus increases further if one considers type II spicules alone.
Analysis of our simulated observations thus suggests that spicules contribute a relatively minor amount to the emission and thermal energy of the corona. Through the generation of shocks, they may heat the local plasma, but that too cools down rapidly due to expansion and thermal conduction. Therefore, synthetic spectra derived from our simulation show a high discrepancy with observed spectra. However, this does not rule out the possibility of spicules contributing significantly to the coronal mass. The ejected spicule material may still get heated in the corona through some other heating mechanism -- a source that exceeds the initial thermal and kinetic energy of the spicule. However, observational evidence of such a process is still lacking. Analyzing the excess blue wing emission of multiple spectral lines hotter than $0.6$~MK, \cite{Tripathi_2013} have concluded that the upward mass flux is too small to explain the mass of the active region corona. Their observations indicate that spicules hotter than $0.6$~MK are not capable of providing sufficient mass to the corona.
So far, we have allowed our spicules to propagate within a warm ($T =0.5$ MK), relatively low density loop in order to determine whether they, by themselves, can explain the observed hot emission. Our simulations indicate that this is not viable. Therefore, there must be some other heating mechanisms at play that produce the hot, dense plasma. Setting aside the issue of heating mechanisms, in the following section we simply test the response of a spicule in a hot and dense flux tube.
\subsection{Spicule propagation in a hot loop}
We have considered a static equilibrium loop with apex and footpoint temperatures of approximately $2$ and $0.02$ MK, respectively. A spicule with a tip temperature of $2$ MK followed by a cold, dense material with temperature $0.02$ MK is injected with a velocity of $150$ km s$^{-1}$ from the bottom boundary, similar to our previous spicules. The velocity profile of the injected spicule is the same as shown in Figure~\ref{pulse}. The injected spicule generates a shock that takes about $180$~s to traverse the loop.
The spatio-temporal averaged spectral line profiles are obtained following the method described in Appendix~\ref{append:Forward_Modelling}. However, in this case, because of the high background temperature, the loop itself emits significantly in the Fe XII and Fe XIV coronal lines. We consider the situation where the line of sight passes through many loops. Some contain spicules and some are maintained in the hot equilibrium state. We adjust the relative proportions to determine what combination is able to reproduce the observed red-blue (RB) profile asymmetries, which are generally $< 0.05$ \citep{Hara_2008, Pontieu_2009, Tian_2011}. For an asymmetry of $\approx 0.04$, we find that the ratios of spicule to non-spicule strands are $1:150$ for the Fe XII line and $1:72$ for the Fe XIV line. Again the conclusion is that spicules are a relatively minor contributor to the corona overall, though they are important for the loops in which they occur.
\begin{deluxetable*}{ccccccccc}
\tablenum{2}
\tablecaption{Summarizing the number of spicules (width $\sim 300$ km) required to explain the quiet Sun and active region intensities as predicted from the test runs.}\label{tab:table2}
\tablecolumns{9}
\tablewidth{0pt}
\tablehead{
\colhead{Run} & \colhead{T$_{tip}$} & \colhead{v} & \multicolumn2c{Loop integrated counts} & \multicolumn2c{Quiet Sun} & \multicolumn2c{Active Region} \\
\colhead{} &\colhead{(MK)} & \colhead{(km s$^{-1}$)} & \multicolumn2c{(DN s$^{-1}$ pix$^{-1}$)} &
\multicolumn2c{(Required number of spicules)} & \multicolumn2c{(Required number of spicules)} \\
\colhead{} & \colhead{} & \colhead{} & \colhead{Fe XII} & \colhead{Fe XIV} & \colhead{Fe XII} & \colhead{Fe XIV} & \colhead{Fe XII} & \colhead{Fe XIV}
}
\startdata
1&2 & 150 & 0.6405 & $4.8\times10^{-2}$ & $4.02\times10^{7}$ & $6.31\times10^{7}$ & $1.92\times10^{8}$ & $5.52\times10^{8}$\\
2&2 & 50 & 0.2336 & $1.47\times10^{-2}$ & $1.1\times10^{8}$ & $2.06\times10^{8}$ & $5.25\times10^{8}$ & $1.8\times10^{9}$ \\
3&1 & 150 & $6.7\times10^{-2}$ & $1.3\times10^{-3}$ & $3.86\times10^{8}$ & $2.26\times10^{9}$ & $1.84\times10^{9}$ & $1.98\times10^{10}$\\
4&1 & 50 & $5.9\times10^{-3}$ & $7.81\times10^{-6}$ & $4.3\times10^{9}$ & $3.8\times10^{11}$ & $2.06\times10^{10}$ & $3.4\times10^{12}$ \\
5&0.02 & 150 & $4.4\times10^{-4}$ & $1.5\times10^{-7}$ & $5.89\times10^{10}$ & $2.02\times10^{13}$ & $2.8\times10^{11}$ & $1.77\times10^{14}$ \\
6&0.02 & 50 & $1.02\times10^{-7}$ & $1.1\times10^{-13}$ & $2.5\times10^{14}$ & $2.7\times10^{19}$ & $1.2\times10^{15}$ & $2.4\times10^{20}$ \\
\enddata
\end{deluxetable*}
\section{Summary and discussion}\label{sec:summary}
The solar atmosphere displays a wide variety of spicules with different temperatures and velocities. It has been suggested that type II spicules are a major source of coronal mass and energy~\citep{Pontieu_2007,Pontieu_2009,Pontieu_2011}. In this work, we numerically investigate the role of spicules in producing observed coronal emissions. In particular, we examine whether, in the absence of any external heating, the hot tips of the spicules and the shock-heated ambient plasma can explain the observed coronal emission. For this, we inject spicules with different temperatures and velocities into a coronal loop in static equilibrium. We choose a relatively cool equilibrium so that the loop does not itself produce appreciable emission in the absence of a spicule. Each of our injected spicules consists of a hot tip followed by a cold body. We consider three different temperatures for the hot tips, viz., $2$, $1$ and $0.02$~MK, while the cold, dense chromospheric plasma that follows the tip has a temperature of $0.02$~MK. Six different simulations are run by injecting each of these spicules with an initial velocity of either $50$ km s$^{-1}$ or $150$ km s$^{-1}$ (see Table~\ref{tab:table1}). We also have constructed spectral line profiles and estimated the spicule occurrence rate required to explain the observed intensities from the quiet Sun and active regions. Our main results are summarized as follows.
\paragraph{Shock formation during spicule propagation} All six runs described above suggest the formation of shocks due to the injection of spicule material into the coronal flux tubes. The shocks are stronger when the temperature differences and therefore pressure differences with the ambient plasma are higher. Table~\ref{tab:table1} shows the variation of the compression ratio (measure of shock strength) with changing temperature of the spicule tip. The nature of the shock depends on the tip temperature. Spicules with a hotter tip produce a pressure-driven shock that propagates with a speed larger than the injection speed. Spicules with a cold tip (i.e., $T_{tip} = 0.02$ MK) produce a piston-driven shock which propagates with a speed close to the injection speed. The intensities and shapes of spectral line profiles depend on the nature of the shock. The formation of shocks during spicule injection agrees well with previous studies~\citep{Petralia_2014, Martinez_2018}.
\paragraph{Rapid cooling of the hot spicule tip} Our simulations show that, in the absence of any external heating, the hot tip of the spicule cools rapidly before reaching a substantial coronal height. Consequently, the tip emission from coronal lines like Fe XII (195 \AA) and Fe XIV (274 \AA) is short lived (Figure~\ref{emission_evolution_fe12_fe14}) and confined to low altitudes. The result is consistent with earlier studies by~\cite{Klimchuk_2012} and \cite{Klimchuk_2014}.
\paragraph{Relative emission contributions of hot tip and shock heated plasma} Our simulations show that the pre-existing material in the loop gets heated through shock compression and thermal conduction. However, the time-integrated emission from this heated pre-existing material is less than that from the hot tip, as shown in Figure~\ref{emission_evolution_fe12_fe14}. The tip plasma is hot for a much shorter time, but it is inherently much brighter because of the greater densities (it is injected in a dense state).
\paragraph{Line profile discrepancies} The shapes of our synthetic spectral line profiles show significant discrepancies with observations. The simulated profiles are highly non-Gaussian and far more asymmetric than observed. A strong blue shift ($\sim 150$ km s$^{-1}$) of the synthetic lines is also inconsistent with the mild Doppler shifts ($< 5$ km s$^{-1}$) observed in the quiet Sun and active regions.
\paragraph{Excessive number of spicules required to explain observed intensities} The spatially and temporally averaged intensities from our simulations (Figures~\ref{spectral_line_fe12_fe14}) imply that far more spicules are required to reproduce the observed emission from the solar disk than are observed (Figure~\ref{count_QS_AR}). The discrepancies are up to a factor of $100$ for the quiet Sun and factors of $10-10^3$ for active regions. These factors apply specifically to Run1, where a spicule with a $2$ MK tip is ejected at a velocity of $150$ km s$^{-1}$. As listed in Table \ref{tab:table2}, the loops in our other simulations with different combinations of tip temperature and ejection velocity are fainter, and therefore more of them would be required to reproduce the observed disk emission, exacerbating the discrepancy.
\paragraph{Ratio of loops with and without spicules} Under the assumption that the corona is comprised of hot loops with and without spicule ejections, red-blue spectral line asymmetries similar to those observed ($0.04$) require far more loops without spicules than with them. The spicule to non-spicule loop number ratio is $1:150$ for the FeXII line and $1:72$ for the Fe XIV line.
Our simulations indicate that spicules contribute a relatively minor amount to the mass and energy of the corona. Such a claim had already been made by \cite{Klimchuk_2012}, where it was shown that hot tip material rapidly expanding into the corona is unable to explain the observed coronal emission. However, a bodily ejection of the spicule was not considered, and the emission from ambient material effected by the expansion was not rigorously investigated (though see Appendix B in that paper). Later, \cite{Petralia_2014} argued that the shock-heated material in front of an ejected cold spicule might be erroneously interpreted as ejected hot material. They did not compare the brightness of the shock-heated material with coronal observations. Our numerical simulations improve on both of these studies. We show that neither the expanding hot tip nor the shock-heated ambient material of a bodily ejected spicule can reproduce coronal observations. A number of discrepancies exist. The existence of some coronal heating mechanism - operating in the corona itself - is required to explain the hot corona. It is not sufficient to eject hot (or cold) material into the corona from below.
We emphasize that our conclusion does not rule out the possibility that waves may be launched into the corona as part of the spicule ejection process, or that new coronal currents may be created outside the flux tube in which the ejected material resides, as suggested by ~\cite{Martinez_2018}. Such waves and currents would lead to coronal heating and could explain at least some non-spicule loops. It seems doubtful, however, that this could explain the many non-spicule loops implied by observed line profile asymmetries. It seems that some type of heating unrelated to spicules must play the primary role in explaining hot coronal plasma.
\begin{acknowledgments}
We thank the anonymous referee for her/his comments to improve the clarity of the paper. SSM \& AS thank Dr. Jishnu Bhattacharyya for many useful discussions. Computations were carried out on the Physical Research Laboratory's VIKRAM cluster. JAK was supported by the Internal Scientist Funding Model (competed work package program) at Goddard Space Flight Center. \\
\end{acknowledgments}
\appendix
\section{Static equilibrium configuration from double relaxation method}\label{append:steady_state}
We inject spicules in a magnetic structure that is in static equilibrium. Such an equilibrium is achieved recursively, and the final equilibrium profile is obtained through two stages of relaxation. First, we obtain the density and temperature profiles by solving the hydrostatic and energy balance equations~\citep{Aschwanden_2002} assuming a steady and uniform background heating $Q_{bg}$. The CHIANTI radiative loss function $\Lambda(T)$ is used to describe the loop's radiation in the energy balance equation. The desired looptop temperature is achieved by adjusting the value of $Q_{bg}$. However, due to lack of exact energy balance, the temperature and density profiles derived in this way do not achieve a perfect equilibrium state. Rather these derived profiles are then used to calculate the final equilibrium loop profile , such that the resulting temperature profile never drops below the chromospheric temperature $T_{ch}$ ($2 \times 10^4$~K), and the system does not generate any spurious velocity either. In the following, we explain these two stages in detail.
\subsection{Heating and cooling in Relaxation-I:}
Starting with the initial profiles described above, the loop is allowed to relax under gravity with the constant background heating $Q_{bg}$. To avoid numerical artifacts, from this stage onward, we smoothly reduce the radiative cooling of the chromosphere to zero over a narrow temperature range between $T_{ch}$ and $T_{min}$, where $T_{min} = 1.95 \times 10^4$~K is a conveniently chosen temperature slightly less than $T_{ch}$. This is achieved by the radiative loss function $\lambda(T)$, defined as
\begin{equation}\label{rad_relax1}
\lambda(T) =
\begin{cases}
\Lambda(T), & \text{if}\, T \ge T_{ch} \\
\left(\frac{T - T_{min}}{T_{ch} - T_{min}}\right) \Lambda(T_{ch}), & \text{if}\, T_{min} < T < T_{ch} \\
0, & \text{if}\, T \le T_{min} \\
\end{cases}~.
\end{equation}
Here $\Lambda(T)$ denotes the optically thin radiative loss function from CHIANTI. The modified function $\lambda(T)$ is plotted in Figure~\ref{chrom_heat_cool}. As the loop relaxes, material drains from the corona and accumulates at the footpoints. The resulting high density of the loop footpoints gives rise to excessive cooling and brings down the footpoint temperatures below $T_{min}$, along with generating short lived velocities. However, the loop eventually achieves a steady-state, and we use the enhanced footpoint density at that time ($n_{base}$) to estimate the additional heating required to keep the chromospheric temperature above $T_{min}$. This is carried out in the next relaxation stage.
\subsection{Heating and cooling in Relaxation-II:}
Once again, we start with the initial density and temperature profiles from the beginning of the first stage. However, this time we apply additional heating in the chromosphere above the constant background heating $Q_{bg}$. This prevents the plasma from cooling below $T_{min}$ and instead lets it hover between $T_{ch}$ and $T_{min}$. The total heating function $Q$ is given by
\begin{equation}\label{heat_relax2}
Q =
\begin{cases}
Q_{bg}, & \text{if}\, T \ge T_{ch} \\
\left(\frac{n}{n_{base}}\right)^{2} Q_{ch} \left(\frac{T_{ch}-T}{T_{ch}-T_{min}}\right) + Q_{bg}, & \text{if}\, T_{min} < T < T_{ch} \\
\left(\frac{n}{n_{base}}\right)^{2} Q_{ch} + Q_{bg}, & \text{if}\, T \le T_{min}
\end{cases}~,
\end{equation}
where $Q_{ch} = n_{ch}^{2} \Lambda (T_{ch})$ is the heat required to balance the radiative losses from the footpoint plasma of the initial loop profile at temperature $T_{ch}$ and density $n_{ch}$. Figure~\ref{chrom_heat_cool} graphically depicts the radiative loss and heating functions that are maintained throughout the simulation.
\section{Variation of shock speed with height}\label{append:shock_speed}
For a pressure driven shock, the shock's speed primarily depends on the pressure difference between the spicule's tip and the ambient medium in which it is propagating. Lower pressure close to the loop apex provides lesser resistance to the shock propagation, and hence the shock speed increases. On the other hand, high pressure close to the footpoints provides greater resistance and thus the shock speed reduces. For a better understanding, we track the shock front along the loop and derive its speed during its propagation. The shock front at any instant can be identified from the density jump moving ahead of the injected spicule material. To track it, we identify the jump in density at each time, which is also associated with the maximum temperature of the loop. Once the locations of the shock front along the loop are identified, a derivative of the same gives the instantaneous shock speed as a function of loop coordinates. Figure~\ref{shock_speed} shows the variation of shock speed as a function of loop coordinates for three different shocks, all ejected with velocity $150$ km s$^{-1}$ but with three different tip temperatures, viz. $2$, $1$ and $0.02$ MK. Though the shock speeds increase at the loop apex for all three shocks, velocity amplitudes depend on the injection temperatures and thus pressures. The larger the tip temperature, the higher the spicule tip pressure and hence larger is the shock speed.
\section{Forward Modelling}\label{append:Forward_Modelling}
Spectral profiles provide a wealth of information about the plasma dynamics along the line of sight (LOS). Adapting the method outlined in~\cite{Patsourakos_2006}, synthetic spectral line profiles are constructed at each numerical grid cell using the cell's density, velocity and temperature.
At any given time, $t$, and location along the loop, $s$, the line profile is
\begin{equation}
I(s,t) = \frac{I_{0}}{\sqrt{\pi} v_{\text{width}}}\exp\left[\frac{-(v - v_\text{shift})^{2}}{v_\text{width}^{2}}\right]~,
\end{equation}
where $I_{0}$ is the amplitude, $v_\text{shift}$ is the Doppler shift, and $v_\text{width}$ is the thermal line width. The amplitude is given by
\begin{equation}
I_{0}(s,t) = n_{e}^{2} G(T)ds~,
\end{equation}
where $n_{e}$, $T$ and $ds$ denote the electron number density, temperature, and length of the cell. The contribution function $G(T)$ for the line is taken from the CHIANTI atomic data base \citep{chianti}. The Doppler shift equals the line of sight velocity of the cell,
\begin{equation}
v_\text{shift} = v_\text{los}~
\end{equation}
in wavelength units, and the thermal width is given by
\begin{equation}
v_\text{width} = \sqrt{\frac{2k_{B}T}{m_{ion}}}~,
\end{equation}
where $m_\text{ion}$ is the mass of the ion.
Once the line profile at each grid point is constructed, spatial averaging is performed by summing the profiles along the loop and dividing by its projected length assuming that it lies in a vertical plane and is viewed from above:
\begin{equation}
\langle I(t) \rangle_{\text{spatial}} = \frac{\pi}{2L}\sum_{s} I(s,t) \times d
\end{equation}
where $L$ is the loop length and $d$ is the pixel dimension. The loop is assumed to have a cross section of $d^2$. Finally the spatially averaged line profiles are temporally averaged over a time $\tau$, which is taken to be the travel time of the shock along the loop; this yields
\begin{equation}
\langle I \rangle_{\text{spatial, temporal}} = \frac{1}{\tau}\sum_{t}\langle I(t) \rangle_{\text{spatial}}
\end{equation}
\bibliography{spicule}
|
Title:
Pushchino multibeams pulsar search. First results |
Abstract: Since the discovery of pulsars, dozens of surveys have already been conducted
with their searches. In the course of surveys in the sky, areas from thousands
to tens of thousands of square degrees are explored. Despite repeated
observations of the same areas, new pulsars are constantly being discovered. We
present Pushchino Multibeam Pulsar Search (PUMPS), having a sensitivity that is
an order of magnitude higher than the sensitivity of all previously made
surveys on pulsar search. In PUMPS daily round-the-clock observations are
carried out of the area located on declinations $-9^o < \delta < +42^o$. The
survey is carried out on 96 beams of a Large Phased Array (LPA) at a frequency
of 111 MHz. During the observation period of August 2014 - August 2022, the
survey was repeated approximately 3,000 times. The expected sensitivity in the
survey reaches up to 0.1 mJy. The paper considers some tasks that can be solved
when processing the received data.
| https://export.arxiv.org/pdf/2208.04578 |
\section{Introduction}
The surveys on the pulsar search are conducted since their discovery in 1967 \cite{Hewish1968} and they have already led to the discovery of more than 3,300 pulsars (https://www.atnf.csiro.au/ research/pulsar/psrcat/, \cite{Manchester2005}). The estimate of the expected number of observed radio pulsars is 30,000 pulsars with a luminosity higher than 0.1 mJy per kpc$^2$ (\citeauthor{Lorimer2006}, \citeyear{Lorimer2006}). The estimate of the number of pulsars available for observations on the radio telescope Square Kilometer Array (SKA) under construction is 20,000 pulsars (\citeauthor{Cordes2004}, \citeyear{Cordes2004}). That is, about 10-15\% of the radio pulsars available for observation have been discovered so far.
In the paper \citeauthor{Wilkinson2007} (\citeyear{Wilkinson2007}), it was shown that the number of new major discoveries in pulsar searches increases as a natural logarithm of the number of discovered pulsars. For each subsequent discovery, it is necessary to discover many times more pulsars than they were known at the time of the previous discovery. A natural question arises about the point of conducting new surveys. Having more and more time and financial expenditures for conducting the observations themselves and processing them, the experimental scientist has less and less chance for a major discovery.
Surveys conducted with high sensitivity make it possible not only to discover new types of pulsars (\citeauthor{McLaughlin2006} (\citeyear{McLaughlin2006}), \citeauthor{Caleb2022} (\citeyear{Caleb2022})), but also to study pulsar populations in detail, to explore the interstellar medium both in the Galactic plane and in the halo. Therefore, if expenditures of time and other resources are acceptable, the surveys should be carried out, because they improve our knowledge about the evolution of pulsars and their properties.
Surveys are conducted, as a rule, on an area of the sky accessible to the telescope. That is, on about half of the celestial sphere. Taking into account the small, as a rule, dimension of the antenna pattern and the number of simultaneously available beams, the survey is an extremely long and, taking into account the cost of an hour of observations, also an extremely expensive task for almost all large telescopes. However, for the Large Phased Array (LPA) radio telescope located in Pushchino Radio Astronomy Observatory (PRAO), a survey of the entire sky is a daily routine task. In this paper, we consider PUshchino Multibeams Pulsar Search (PUMPS) and some of the problems that can be solved along the way in this survey.
\section{Survey and tasks}
The survey on the LPA transit radio telescope (observation frequency 111 MHz) is carried out daily, around the clock since August 2014 on 96 beams, and since January 2022 on 128 beams aligned in the meridian plane and covering declinations $-9^o < \delta < +55^o$. The data are recorded in the 2.5 MHz band, in 32 frequency channels of 78 kHz width with a point reading time of 12.5 ms. The amount of data recorded per year is almost 45 terabytes. Work on the pulsar search started in 2015, and to date 42 pulsars and 46 rotating radio transients (RRAT) have been discovered up to date (see site https://bsa-analytics.prao.ru/en/ and references in it). Since we have not had a good computation server, processing of all the data was impossible. We expect that the new server being purchased, which has a terabyte of RAM and 128 full-fledged cores, will start operation this year and will allow processing in a reasonable time the accumulated data with a volume of about 300 terabytes.
Based on PUMPS data, the search for classical second-duration pulsars and transients is carried out. As was shown in the paper \citeauthor{Tyulbashev2022} (\citeyear{Tyulbashev2022}), the sensitivity of the LPA in one 3.5 minutes duration observation session, when the source passes through the meridian at half the power of the radiation pattern, is inferior to the surveys conducted on the aperture synthesis Low Frequency Array (LOFAR) system and on the Five-hundred-meter Aperture Spherical Telescope (FAST).
Accumulation of the signal by summing up power spectra and periodograms allows to improve the sensitivity by tens of times on the dispersion measures $DM<100$ pc/cm$^3$. For the search for transients, instantaneous sensitivity is primarily important, and this sensitivity is provided by a large effective area of the LPA equal to 45,000 sq.m.
In the paper \citeauthor{Tyulbashev2022} (\citeyear{Tyulbashev2022}), we considered sensitivity when searching for seconds duration pulsars and limited ourselves to PUMPS sensitivity estimates $DM \le 200$ pc/cm$^3$. However, the discovery of the pulsar with a period of 77 seconds and a pulse half-width of 300 ms (\citeauthor{Caleb2022} (\citeyear{Caleb2022})) allows us to seriously consider the search for pulsars at significantly larger $DM$. For such $DM$, the main factor reducing the sensitivity of the search is interstellar scattering ($\tau_s$). In experimental dependencies $\tau_s$(DM) it can be seen that for the same DM the scattering can differ by three orders of magnitude (\citeauthor{Cordes2002} (\citeyear{Cordes2002}), \citeauthor{Bhat2004} (\citeyear{Bhat2004}), \citeauthor{Kuzmin2007} (\citeyear{Kuzmin2007}), \citeauthor{Pynzar2008} (\citeyear{Pynzar2008})). So, in observations on the frequency 111 MHz the scattering can be from ten seconds to ten minutes. We have recalculated the sensitivity curves up to $DM=1,000$ pc/cm$^3$. The following formula was used for estimating the scattering (\citeauthor{Cordes2002}, \citeyear{Cordes2002}):
\begin{equation}
\log(\tau_s) = 3.59 + 0.129\log(DM) + 1.02\log(DM)^2 - 4.4log(f),
\label{eq:1}
\end{equation}
where $\tau_s$ is obtained in microseconds, $DM$ is expressed in pc/cm$^3$, $f$ is the central frequency of observations in GHz. Since the estimates of $\tau_s$ may be significantly higher than those obtained from formula~\ref{eq:1}, this may lead to a deterioration in the sensitivity assessment both in single observation sessions ($S_{min}$) and when summing up power spectra and periodograms ($S_{min-sum}$).
Fig.~\ref{fig:fig1} shows sensitivity estimates for pulsars with different periods after evaluation of $\tau_s$ using formula~\ref{eq:1}. The sensitivities in a pulsar search with different periods and dispersion measures shown in the figure differ slightly from the sensitivities shown in fig.4 in the paper \citeauthor{Tyulbashev2022} (\citeyear{Tyulbashev2022}). These differences are related to the fact that in the paper \citeauthor{Tyulbashev2022} (\citeyear{Tyulbashev2022})2 the scattering was estimated according to the empirical formula from the paper \citeauthor{Kuzmin2007} (\citeyear{Kuzmin2007}).
The sensitivity in PUMPS, equal to 0.2-0.3 mJy, is about 16 times better than sensitivity equal to 3-4 mJy in the survey LOTAAS made on LOFAR (Sanidas2019), when recalculated into the frequency 111 MHz (\citeauthor{Tyulbashev2022} (\citeyear{Tyulbashev2022})). This means that, all other things being equal, 4 times weaker pulsars can be detected on LPA than on LOFAR. In addition, there is a possibility in principle to find pulsars with long (>10-20 s) periods at high DM. Since the luminosity decreases in proportion to the square of the distance, and the volume increases in proportion to the cube of the distance, the number of pulsars available in the survey can grow up to 64 times. There are 73 pulsars discovered in the LOTAAS survey, and the possible number of new pulsars in the PUMPS survey may reach $64 \times 73 \approx 4,500$. This fantastic assessment is most likely very far from reality. The sensitivities indicated in Fig.~\ref{fig:fig1} are achieved only for pulsars that do not have pulse gaps and with very stable (not variable) radiation. Let us note, however, that when we conduct the search, the sensitivity is several times lower than expected with the accumulation of 8 years of observations (see site https://bsa-analytics.prao.ru/en/, references in it and the paper \citeauthor{Tyulbashev2022} (\citeyear{Tyulbashev2022})), for the same areas in the sky, we detect \textbf{all} pulsars discovered on LOFAR and detect almost the same number of new pulsars in these areas that were not detected in the LOTAAS survey (see Fig.6 in the paper \citeauthor{Tyulbashev2022} (\citeyear{Tyulbashev2022})). Our conservative scenario is the discovery of 200-300 new pulsars in PUMPS, the optimistic scenario is the discovery of 1,000-1,500 new pulsars.
The sensitivity in the search for pulsed radiation of transients for the LPA radio telescope is fixed, because we cannot change neither the area of the antenna, nor the temperature of the system, nor the band. However, for a year of observations, taking into account the 3.5-minute passage through the meridian at half power, approximately 20 hours of recording are accumulated for each point in the sky. The survey started in August 2014 and is planned at least until December 2024. This means the accumulation of approximately 8.5 days of data for each point entering the observation area. In the papers \citeauthor{Logvinenko2020} (\citeyear{Logvinenko2020}), \citeauthor{Tyulbashev2022a} (\citeyear{Tyulbashev2022a}), the existence of RRAT is shown, between the pulses of which 10 or more hours can pass. To detect and study such transients, it is necessary to conduct very long-term observations, which appear during monitoring.
The field of view of the LPA on 128 antenna beams is approximately 50 sq.deg.. Field of view estimates of other large telescopes used for RRAT detection are: for 64-meter mirror Parks (Australia) it it is 0.7 sq.deg. on 13 beams at the frequency 1.4 GHz; for 100-meter mirror Green-Bank (USA) it is 0.35 sq.deg. on one beam at the frequency 350 MHz; for 300-meter mirror Arecibo (USA) it is 1 sq.deg. on 7 beams at the frequency 327 MHz; for 500-meter mirror FAST (China) it is 0.16 sq.deg. on 19 beams at the frequency 1.2 GHz. Instant sensitivity for FAST (\citeauthor{Han2021}, \citeyear{Han2021}) after recalculation of the frequency 1.2 GHz into the frequency 111 MHz with an assumed spectral index 1.7 ($S\sim\nu^{-\alpha}$) exceeds the sensitivity of the LPA by about an order of magnitude. However, if we talk about RRAT, for which hours can pass between the appearance of successive pulses, the second main factor for searching, after instantaneous sensitivity, becomes the time of observations at one point in the sky. If the average time between RRAT pulses is one hour, then the FAST radio telescope will need to view half of the sky once [20,000 sq.deg. (half of the celestial sphere)/0.7 sq.deg. (FAST field of view )] $\times$ 1 hour = 28,570 hours or 3.3 years of round-the-clock observations. Due to the FAST availability, even for a single examination of the sky, the task looks hardly realizable.
Thus, both for the search for second-duration pulsars and for the search for RRAT, the LPA radio telescope turned out to be surprisingly suitable, despite all its obvious disadvantages: observations in one linear polarization (whole classes of tasks fall out plus a loss of sensitivity 2$^{1/2}$ times); narrow full band (leads to low accuracy of DM estimation and deterioration of sensitivity obtained on modern broadband recorders); the lack of direct ascension diagram control (leads to low accuracy in determining the pulsar period at an interval of 3.5 minutes, there are problems in obtaining timing, it is impossible or very difficult to investigate weak sources); the dimension of one LPA beam is too large $0.5 \times 1$ deg. (it leads to low coordinate accuracy of detected objects).
Despite the mentioned disadvantages of the tool, the data obtained at the LPA can be used for research for many scientific tasks. We list some of the planned PUMPS tasks without going into details of their solutions. Search tasks: search for pulsars with periods from 25 ms up to minutes, search for RRATs and Fast Radio Bursts (FRBs), search for pulsars in nearby galaxies, search for pulsars with small DM down to 0 pc/cm$^3$, search for pulsars with sporadic radiation.
Research of the interplanetary, interstellar, intergalactic environments: pulsar variability induced by scintillations (diffraction and refraction of radio emission in different medium), pulse scattering of pulsars and FRB, Faraday rotation. Research of pulsed and periodic radiation sources: a nature of RRAT and FRB, statistics of pulsars with inter-pulses, inter-pulse radiation of pulsars, pulsars with fading radiation, intrinsic variability, targeted search for gamma, X-ray and other radio-quiet pulsars, spatial distribution of pulsars in the Galaxy as a whole and for different samples, pulse energy distribution at a frequency of 111 MHz, timing and others.
Let's look at three examples of using monitoring data:
- there are opposite hypotheses about the evolution of the pulsar's "magnetic axis" relative to its axis of rotation. There are hypotheses according to which, over time, the directions of the pulsar's "magnetic axis" and its axis of rotation become perpendicular to each other (orthogonal rotator), other hypotheses suggest that the direction of the axes coincides with time (coaxial rotator) (see \citeauthor{Arzamasskiy2017} (\citeyear{Arzamasskiy2017}) and references there). For the orthogonal rotator, the pulse takes the smallest possible fraction of the period, for coaxial rotators, on the contrary, the pulse takes the maximum possible fraction of the period. Pulsars with long periods are old pulsars (see, for example, the hand book \citeauthor{Lorimer2004} (\citeyear{Lorimer2004})). Their evolution took longer time compared to ordinary second-duration pulsars. Therefore, a simple comparison of the relative pulse duration (duty cycle), that is, the fraction of the period occupied by the pulse for ordinary second-duration pulsars and for pulsars with extra-long periods (>10-20 s), which will be discovered in the survey, should give an answer to the question whether radio pulsars become coaxial or orthogonal rotators by the end of their life in the active phase;
- since pulsars are formed during supernova explosions, and the exploded stars are in the plane of the Galaxy, then pulsars at birth should be located there as well. As a result of the explosion, pulsars can acquire a velocity component perpendicular to the plane of the Galaxy and go into the halo. Pulsar lifetime in the active phase (as a radio pulsar) can be from millions to tens of millions of years, and the pulsar can move away from the plane of the Galaxy by some distance. However, the farther away from the Galactic plane, the fewer pulsars should be detected. Fig.~\ref{fig:fig2} presents histograms showing the number of pulsars and RRATs at different galactic latitudes (at different elevations above the plane of the Galaxy). It is obvious that the distributions for second-duration pulsars with small DM located in the same area where all Pushchino RRATs were detected (see https://bsa-analytics.prao.ru/en/transients/rrat/), and the distribution for Pushchino RRATs are different in appearance. We do not discuss this difference, which can be explained within the framework of hypotheses from insufficient RRAT statistics and selection effects to the discovery of relic pulsars inherited from the previous Universe (\citeauthor{Gorkavyi2021}, \citeyear{Gorkavyi2021}). We are only presenting here one of the problems associated with the strange dependence of the RRAT distribution over Galactic latitudes;
- daily observations allow us to obtain an averaged profile for hundreds of pulsars. The peak or integral flux density estimated from the average profile allows us to construct the dependence of the estimate of the observed flux density on time. The observed variability may be related to scintillations on interstellar plasma. If the variability cannot be explained by the interstellar medium, it is related to internal factors. Fig.~\ref{fig:fig3} shows the "light curve" of the pulsar J0323+3944 (B0320+39). The figure shows changes in the flux density over time. We do not investigate in this paper the causes of apparent variability of J0323+3944, but we only show that the task of studying the variability can be solved on the PUMPS data.
In the presented paper there are no solutions of the problems discussed above in the examples, this is a matter of the future. We only show the fundamental possibility of performing various tasks based on the data received in PUMPS.
\section{Conclusion}
Up to date, 88 pulsars have been discovered in the PUMPS survey (see cite https://bsa-analytics.prao.ru/en/). Having a sensitivity an order of magnitude higher than in the surveys conducted up to date, we can expect the detection of more than 1,000 new pulsars. The main thing, in our opinion, is that with a radical increase in sensitivity, we begin to exploit the area that is called "unknown-unknown" in the paper \citeauthor{Wilkinson2015} (\citeyear{Wilkinson2015}). When working in this area, there is no guarantee that major discoveries will be made, however, as we believe, this kind of work is the real academic science.
\acknowledgments
The study was carried out at the expense of a grant Russian Science Foundation 22-12-00236, https://rscf.ru/project/22-12-00236/.
|
Title:
Structure and evolution of ultra-massive white dwarfs in general relativity |
Abstract: We present the first set of constant rest-mass ultra-massive oxygen/neon
white dwarf cooling tracks with masses larger than 1.29 Msun which fully take
into account the effects of general relativity on their structural and
evolutionary properties. We have computed the full evolution sequences of 1.29,
1.31, 1.33, 1.35, and 1.369 Msun white dwarfs with the La Plata stellar
evolution code, LPCODE. For this work, the standard equations of stellar
structure and evolution have been modified to include the full effects of
general relativity. For comparison purposes, the same sequences have been
computed but for the Newtonian case. According to our calculations, the
evolutionary properties of the most massive white dwarfs are strongly modified
by general relativity effects. In particular, the resulting stellar radius is
markedly smaller in the general relativistic case, being up to 25% smaller than
predicted by the Newtonian treatment for the more massive ones. We find that
oxygen/neon white dwarfs more massive than 1.369 Msun become gravitationally
unstable with respect to general relativity effects. When core chemical
distribution due to phase separation on crystallization is considered, such
instability occurs at somewhat lower stellar masses, greater than 1.36 Msun. In
addition, cooling times for the most massive white dwarf sequences result in
about a factor of two smaller than in the Newtonian case at advanced stages of
evolution. Finally, a sample of white dwarfs has been identified as ideal
candidates to test these general relativistic effects. We conclude that the
general relativity effects should be taken into account for an accurate
assessment of the structural and evolutionary properties of the most massive
white dwarfs.
| https://export.arxiv.org/pdf/2208.14144 |
\title{Structure and evolution of ultra-massive white dwarfs in general relativity
\thanks{The cooling sequences are publicly available at
\url{http://evolgroup.fcaglp.unlp.edu.ar/TRACKS/tracks.html}}}
\author{Leandro G. Althaus\inst{1,2}, Mar\'ia E. Camisassa\inst{3}, Santiago Torres\inst{4,5}, Tiara Battich\inst{6}, Alejandro H. C\'orsico\inst{1,2}, Alberto Rebassa-Mansergas\inst{4,5}, Roberto Raddi\inst{4,5} }
\institute{Grupo de Evoluci\'on Estelar y Pulsaciones.
Facultad de Ciencias Astron\'omicas y Geof\'{\i}sicas,
Universidad Nacional de La Plata,
Paseo del Bosque s/n, 1900
La Plata,
Argentina
\and
IALP-CCT - CONICET
\and
Applied Mathematics Department, University of Colorado, Boulder, CO 80309-0526, USA \and
Departament de F\'\i sica,
Universitat Polit\`ecnica de Catalunya,
c/Esteve Terrades 5,
08860 Castelldefels,
Spain
\and
Institute for Space Studies of Catalonia,
c/Gran Capita 2--4,
Edif. Nexus 104,
08034 Barcelona,
Spain
\and
Max-Planck-Institut f\"{u}r Astrophysik, Karl-Schwarzschild-Str. 1, 85748, Garching, Germany
}
\date{Received ; accepted }
\abstract{Ultra-massive white dwarfs ($M_{\star} \gtrsim 1.05 M_{\sun}$)
are of utmost importance in view of the role they play in type Ia
supernovae explosions, merger events, the existence of high magnetic
field white dwarfs, and the physical processes in the Super
Asymptotic Giant Branch phase.} {We present the first set of
constant rest-mass ultra-massive oxygen/neon white dwarf cooling
tracks with masses $M_{\star} > 1.29 M_{\sun}$ which fully take into
account the effects of general relativity on their structural and
evolutionary properties.}
{We have computed the full evolution sequences of 1.29, 1.31, 1.33,
1.35, and 1.369 $M_{\sun}$ white dwarfs with the La Plata stellar
evolution code, {\tt LPCODE}. For this work, the standard equations
of stellar structure and evolution have been modified to include the
effects of general relativity. Specifically, the fully general
relativistic partial differential equations governing the evolution
of a spherically symmetric star are solved in a way they resemble
the standard Newtonian equations of stellar structure. For
comparison purposes, the same sequences have been computed but for
the Newtonian case.}
{According to our calculations, the evolutionary properties of the most
massive white dwarfs are strongly modified by general relativity
effects. In particular, the resulting stellar radius is markedly
smaller in the general relativistic case, being up to 25$\%$ smaller
than predicted by the Newtonian treatment for the more massive
ones. We find that oxygen/neon white dwarfs more massive than 1.369
$M_{\sun}$ become gravitationally unstable with respect to general
relativity effects. When core chemical distribution due to phase
separation on crystallization is considered, such instability occurs
at somewhat lower stellar masses, $\gtrsim 1.36 M_{\sun}$. In
addition, cooling times for the most massive white dwarf sequences
result in about a factor of two smaller than in the Newtonian case
at advanced stages of evolution. Finally, a sample of white dwarfs
has been identified as ideal candidates to test these general
relativistic effects.}
{%
We conclude
that the general relativity effects should be taken into account for an accurate assessment of the
structural and evolutionary properties of the most massive white dwarfs. These new ultra-massive white dwarf models constitute a considerable improvement over those computed in the framework of the standard Newtonian theory of stellar interiors. }
\keywords{stars: evolution --- stars: interiors --- stars: white
dwarfs --- stars: oscillations (including pulsations) --- Physical data and processes: Relativistic processes}
\titlerunning{Relativistic ultra-massive white dwarfs}
\authorrunning{Althaus et al.}
\section{Introduction}
\label{introduction}
White dwarf stars are the most common end point of stellar
evolution. Therefore, these old stellar remnants contain valuable
information on the stellar evolution theory, the kinematics and the
star formation history of our Galaxy, and the ultimate fate of
planetary systems \citep[see][for
reviews]{2008ARA&A..46..157W,2010A&ARv..18..471A,2016NewAR..72....1G,
2019A&ARv..27....7C}. Furthermore, given the large densities that
characterize the white dwarf interiors, these compact objects are
considered reliable cosmic laboratories to study the properties of
baryonic matter under extreme physical conditions
\citep{2022FrASS...9....6I}. Among all the white dwarfs, of special
interest are the so-called ultra-massive white dwarfs, defined as
those with masses larger than $\sim 1.05 M_\odot$. Ultra-massive white
dwarfs play a key role in constraining the threshold above which stars
explode as supernova to create neutron stars and they are involved in
extreme astrophysical phenomena, such as type Ia supernovae,
micronovae explosions, radio transients via an
accretion-induced collapse \citep{2019MNRAS.490.1166M} as well as
stellar mergers. Ultra-massive white dwarfs constitute
also powerful tools to study the theory of high density
plasmas and general relativity.
The theoretical evolution of ultra-massive white dwarfs with masses up
to $1.29\, M_\odot$ has been studied in detail in
\cite{2019A&A...625A..87C,2022MNRAS.511.5198C}. These studies provide
white dwarf evolutionary sequences with oxygen-neon (O/Ne) and carbon-oxygen
(C/O) core-chemical composition, considering realistic initial chemical
profiles that are the result of the full progenitor evolution
calculated in \cite{2010A&A...512A..10S} and \cite{ALTUMCO2021},
respectively.
This set of ultra-massive white dwarf
evolutionary models provides an appropriate tool to study the
ultra-massive white dwarf population in our Galaxy, subject to the
condition that white dwarf masses do not exceed $1.29\, M_\odot$.
During the last years, observations of ultra-massive white dwarfs have
been reported in several studies
\citep{2004ApJ...607..982M,2016IAUFM..29B.493N,2011ApJ...743..138G,2013ApJS..204....5K,2015MNRAS.450.3966B,2016MNRAS.455.3413K,2017MNRAS.468..239C,2021MNRAS.503.5397K,Hollands2020,2021Natur.595...39C,2022MNRAS.511.5462T}. In
particular, \cite{2018ApJ...861L..13G} derived a mass of
$1.28\pm0.08\,M_{\odot}$ for the long known white dwarf GD 50. The
number of ultra-massive white dwarfs with mass determinations beyond
$1.29\, M_\odot$ is steadily increasing with recent observations.
\cite{2020MNRAS.499L..21P} discovered a rapidly-rotating ultra-massive
white dwarf, WDJ183202.83+085636.24, with $M=1.33\pm0.01\,M_{\odot}$ meanwhile
\cite{2021Natur.595...39C} reported the existence of a
highly-magnetized, rapidly-rotating ultra-massive white dwarf, ZTF
J190132.9+145808.7, with a mass of $\sim 1.327 - 1.365 \,
M_\odot$. \cite{2021MNRAS.503.5397K} studied the most massive white
dwarfs in the solar neighborhood and concluded that other 22 white
dwarfs could also have masses larger than $1.29\, M_\odot$, if they
had pure H envelopes and C/O cores. Furthermore,
\cite{2022RNAAS...6...36S} has confirmed the existence of a branch of
faint blue white dwarfs in the {\it Gaia} color magnitude diagram,
some of them also reported in \cite{Kilic2020},
which is mainly composed by
ultra-massive white dwarfs more massive than $1.29\, M_\odot$.
In addition to all these observations, gravity($g$)-mode pulsations
have been detected at least in four ultra-massive white dwarfs
\citep{1992ApJ...390L..89K,2013ApJ...771L...2H,2017MNRAS.468..239C,2019MNRAS.486.4574R}. Although
these stars have masses slightly below $1.29~M_\odot$, we expect that
more massive pulsating white dwarfs will be identified in the coming
years with the advent of huge volumes of high-quality photometric data
collected by space missions such as the ongoing {\sl TESS} mission
\citep{2015JATIS...1a4003R} and {\sl Cheops}
\citep{2018A&A...620A.203M} mission, and the future {\sl Plato} space
telescope \citep{2018EPSC...12..969P}. This big amount of photometric
data is expected to make asteroseismology a promising tool to study
the structure and chemical composition of ultra-massive white dwarfs
\citep{2019A&A...621A.100D, 2019A&A...632A.119C}. In fact, several
successful asteroseismological analyzes of white dwarfs have been
carried out employing data from space thanks to the {\sl Kepler/K2}
mission \citep{2010Sci...327..977B,
2014PASP..126..398H,2020FrASS...7...47C} and {\sl TESS}
\citep{2022arXiv220303769C}.
The increasing number of detected ultra-massive white dwarfs with masses beyond
$1.29\,M_{\odot}$ as well as the immediate prospect of detecting
pulsating white dwarfs with such masses, demand new appropriate
theoretical evolutionary models to analyze them. Recently,
\cite{2021ApJ...916..119S} has studied the evolution of white dwarfs
more massive than $1.29\, M_\odot$ with the focus on neutrino cooling
via the Urca process, showing that this process is important for age
determination of O/Ne-core white dwarf stars. These models were
calculated employing the set of standard equations to solve the
stellar structure and evolution under the assumption of Newtonian
gravity. However, the importance of general relativity for the
structure of the most massive white dwarfs cannot be completely
disregarded. This was recently assessed by \cite{2018GReGr..50...38C},
who solved the general relativistic hydrostatic equilibrium equation
for a completely degenerate ideal Fermi electron gas. They demonstrate
that for fixed values of total mass, large deviations (up to 50$\%$)
in the Newtonian white dwarf radius are expected, as compared with the
general relativistic white dwarf radius. The impact of a non-ideal
treatment of the electron gas on the equilibrium structure of
relativistic white dwarfs was studied by \cite{2011PhRvD..84h4007R}
and \cite{2017RAA....17...61M}, who derived the mass-radius relations
and critical masses in the general relativity framework for white
dwarfs of different core chemical compositions. These studies conclude
that general relativistic effects are relevant for the determination
of the radius of massive white dwarfs. \cite{2014PhRvC..89a5801D} and,
more recently, \cite{2021ApJ...921..138N} have investigated the
general relativity effects in static white dwarf structures of
non-ideal matter in the case of finite temperature. While
\cite{2014PhRvC..89a5801D} focused their work on the effects of finite
temperature on extremely low-mass white dwarfs,
\cite{2021ApJ...921..138N} studied the stability of massive hot white
dwarfs against radial oscillations, inverse $\beta-$decay and
pycnonulcear reactions. They find that the effect of the temperature
is still important for determining the radius of very massive white
dwarfs.
Despite several works have been devoted to the study of the effects of
general relativity on the structure of white dwarfs, none of these
works has calculated the evolution of such structures. Moreover, in
all of the works mentioned above, the white dwarf models are assumed
to be composed by solely one chemical element. The exact chemical
composition determines both the mass limit of white dwarfs and the
nature of the instability (due to general-relativity effects or to
$\beta$-decays, e.g. \citealt{2011PhRvD..84h4007R}). In this paper we
compute the first set of constant rest-mass ultra-massive O/Ne white
dwarf evolutionary models which fully take into account the effects of
general relativity on their structural and evolutionary
properties. Furthermore, we consider realistic initial chemical
profiles as predicted by the progenitor evolutionary history. We
employ the La Plata stellar evolution code, {\tt LPCODE}, to compute
the full evolutionary sequence of 1.29, 1.31, 1.33, 1.35, and 1.369
$M_{\sun}$ white dwarfs. The standard equations of stellar structure
and evolution solved in this code have been modified to include the
effects of general relativity. For comparison purposes, the same
sequences have been computed but for the Newtonian gravity case.
We assess the resulting cooling
times and provide precise time dependent mass-radius relations for
relativistic ultra-massive white dwarfs. We also provide magnitudes
in Gaia, Sloan Digital Sky Survey and Pan-STARRS passbands, using the
model atmospheres of \cite{2010MmSAI..81..921K,2019A&A...628A.102K}.
This set of cooling sequences, together with the models calculated in
\cite{2019A&A...625A..87C} and \cite{2022MNRAS.511.5198C}, provide a
solid theoretical framework to study the most massive white dwarfs in
our Galaxy.
This paper is organized as follows. In Sect. \ref{equations}
we describe the modifications to our code to incorporate the effects of general relativity.
In Sect. \ref{models} we detail the main constitutive physics
of our white dwarf sequences. Sect. \ref{results} is devoted to describe the impact of general relativity effects on the relevant evolutionary properties of our massive white dwarfs. In this section we also compare and discuss the predictions of our new white dwarf sequences with observational data of ultra-massive white dwarfs, in particular with the recently reported faint blue branch of ultra-cool and ultra-massive objects revealed by {\it Gaia} space mission.
Finally, in Sect. \ref{conclusions} we summarize the main finding of the paper.
\section{The equations of stellar structure and evolution in general relativity}
\label{equations}
Our set of ultra-massive O/Ne white dwarf evolutionary sequences has been computed with
the stellar evolution code {\tt LPCODE} developed by La Plata group, which has been widely used
and tested in numerous stellar evolution contexts of low-mass stars and particularly in white dwarf stars \citep[see][for details]
{2003A&A...404..593A,2005A&A...435..631A, 2013A&A...555A..96S, 2015A&A...576A...9A,
2016A&A...588A..25M,2020A&A...635A.164S,
2020A&A...635A.165C}. For this work, the stellar structure and evolution equations have
been modified to include the effects of general relativity, following the formalism given in \cite{1977ApJ...212..825T}. Within this formalism, the fully general relativistic partial differential equations governing the evolution of a spherically symmetric star are presented in a way they resemble the standard Newtonian equations of stellar structure \citep{2012sse..book.....K}. Specifically, the structure and evolution of the star is specified by the Tolman-Oppenheimer-Volkoff (TOV) equation of hydrostatic equilibrium, the equation of mass distribution, the luminosity equation, and the energy transport equation:
\begin{equation}
\frac{\partial P}{\partial m}= -\frac{G m}{4 \pi r^4}\ \mathscr{H G V} \ ,
\label{TOV}
\end{equation}
\begin{equation}
\frac{\partial r}{\partial m}= (4 \pi r^2 \varrho\ \mathscr{V})^{-1} \ ,
\label{MD}
\end{equation}
\begin{equation}
\frac{1}{\mathscr{R}^2}\frac{\partial (L \mathscr{R}^2)}{\partial m}= -\varepsilon_\nu - \frac{1}{\mathscr{R}}
{\frac{\partial u} {\partial t}} + \frac{1}{\mathscr{R}} {\frac{P} {\varrho^2}\frac{\partial \varrho} {\partial t}}\ ,
\label{lumistandard}
\end{equation}
\begin{equation}
\frac{\partial (T \mathscr{R})}{\partial m}= -\frac{3}{64 \pi^2 ac}\frac{\kappa L}{r^4 T^3}\mathscr{R} \qquad {\rm if} \quad \nabla_{\rm rad} \leq \nabla_{\rm ad} \ ,
\label{T_rad}
\end{equation}
\begin{equation}
\frac{\partial \rm{ln} T}{\partial m}= \nabla\ \frac{\partial \rm {ln} P}{\partial m} \qquad {\rm if} \quad \nabla_{\rm rad} > \nabla_{\rm ad} \ ,
\label{T_conv}
\end{equation}
where $t$ is the Schwarzschild time coordinate, $m$ is the rest mass inside a radius $r$ or baryonic mass, i.e., the mass of one hydrogen atom in its ground state
multiplied by the
total number of baryons inside $r$, and $\varrho$ is the density of rest mass. During the entire cooling process, the
total baryonic mass remains constant. $c$ is the speed of light, $u$ is the internal energy per unit mass, and $\varepsilon_\nu$ is the energy lost by neutrino emission per unit mass. $\mathscr{H, G, V,}$ and $\mathscr{R}$ are the dimensionless general relativistic correction factors, which turn to unity in the Newtonian limit. These factors correspond, respectively, to
the enthalpy, gravitational acceleration, volume, and redshift correction factors, and are given by
\begin{align}
\mathscr{H} &= \frac{\varrho^t}{\varrho} + \frac{P}{\varrho c^2},\\
\mathscr{G} &= \frac{ m^t + 4 \pi r^3 P/c^2} {m}, \\
\mathscr{V} &= \left(1 - \frac{ 2 G m^t}{ r c^2}\right)^{-1/2}, \\
\mathscr{R} &= e^{\Phi/c^2}, \\
\label{R-fact}
\end{align}
\noindent where $m^t$ is the mass-energy inside a radius $r$ and includes contributions from the
rest-mass energy, the internal energy, and the gravitational potential energy, which is negative. $\varrho^t$
is the density of total nongravitational mass-energy, and includes the density of rest mass plus contributions from
kinetic and potential energy density due to particle interactions (it does not include the gravitational potential energy density), that is $\varrho^t= \varrho + (u \varrho)/ c^2 $.
Since the internal and
gravitational potential energy change during the course of evolution, the stellar
mass-energy is not a conserved quantity. $\Phi$ is the general
relativistic gravitational potential related to the temporal metric coefficient. At variance with
the Newtonian case, the gravitational potential
appears explicitly in the evolution equations. We note that
the TOV hydrostatic equilibrium equation differs
markedly from its Newtonian counterpart, providing a steeper
pressure gradient. Also we note that the presence of $\mathscr{V}$ in that equation prevents $m^t$ from being larger than $rc^2/2G$.
The radiative gradient $\nabla_{\rm rad}$ is given by
\begin{equation}
\nabla_{\rm rad} = \frac{3}{16 \pi ac G}\frac{\kappa L P}{m T^4}\frac{1}{\mathscr{H G V}} + \left(1 - \frac{\varrho^t/\varrho}
{\mathscr{H}} \right) \ .
\end{equation}
In Eq. (\ref{T_conv}), $\nabla$ is the convective temperature gradient, which, in the present work, is given by the solution of the mixing length
theory. We mention that in ultra-massive white dwarfs the occurrence of convection is restricted exclusively to a very narrow
outer layer\footnote{This may not be true if neutrino cooling via the
Urca process is considered, in which case an inner convection zone is expected, see \cite{2021ApJ...916..119S}.}, being mostly adiabatic. We follow \cite{1977ApJ...212..825T} to generalize the mixing length
theory to general relativity. In Eq. (\ref{lumistandard}) we have omitted the energy generation by nuclear reactions since these are not happening in our models. However, they should be added when taking into account Urca processes.%
To solve Eqs. (\ref{TOV})-(\ref{T_conv}) we need two additional equations that relate $m^t$ and $\Phi$ with $m$. These two equations, which
are not required in the Newtonian case, have to be solved simultaneously with Eqs. (\ref{TOV})-(\ref{T_conv}). These extra equations are given
by \citep[see][]{1977ApJ...212..825T}
\begin{equation}
\frac{\partial m^t }{\partial m}= \frac{\varrho^t}{\varrho} \frac{1}{\mathscr{V}}\ ,
\label{grav_mass}
\end{equation}
\begin{equation}
\frac{\partial \Phi}{\partial m}= \frac{G m}{4 \pi r^4 \varrho}\ \mathscr{G V} \ .
\label{phi}
\end{equation}
\subsection{Boundary conditions}
The rest mass, total mass-energy, and radius of the star correspond, respectively, to the values of $m$, $m^t$, and $r$ at the surface of the star. We denote them by
\begin{equation}
M_{\rm WD}=m \ , \qquad M_{\rm G}=m^t \ , \qquad R=r \qquad {\rm at \ the \ surface} \ .
\end{equation}
$M_G$ is the total gravitational mass, i.e., the stellar mass that would be measured by a distant observer, which turns out to be
less than the total baryonic mass of the white dwarf.
Outer boundary conditions for our evolving models are provided
by the integration of
\begin{equation}
\frac{d P}{d \tau}= \frac{g^t}{\kappa} \ ,
\label{atm}
\end{equation}
\noindent and assuming a gray model atmosphere. $\tau$ is the optical depth and $g^t$ is the "proper" surface gravity of the star (as measured on the
surface) corrected by general relativistic effects and given by
\begin{equation}
g^t= \frac{G M_G}{R^2} \mathscr{V} \ .
\label{grav}
\end{equation}
In addition, the general relativistic metric for spacetime in the star interior must match to the metric outside created by the star (Schwarzschild
metric). The match requires that $\Phi$ satisfies the surface boundary condition
\begin{equation}
\Phi= \frac{1}{2} c^2 \ln \left(1-\frac{2 G M_G }{R c^2} \right)\qquad {\rm at} \quad m=M_{\rm WD} \ .
\label{phi_sup}
\end{equation}
At the stellar center, $m=0$, we have $m^t=0$, $r=0$, and $L=0$.
\section{Initial models and input physics}
\label{models}
We have computed the full evolution of 1.29, 1.31, 1.33, 1.35, and
1.369 $M_{\sun}$ white dwarfs assuming the same O/Ne core abundance
distribution for all of them. The adopted core composition
corresponds to that of the 1.29 $M_{\sun}$ hydrogen-rich white dwarf
sequence considered in \cite{2019A&A...625A..87C}, which has been
derived from the evolutionary history of a 10.5 $M_{\sun}$ progenitor
star \citep{2010A&A...512A..10S}. In this work, we
restrict ourselves to O/Ne-core massive white dwarfs, thus extending
the range of O/Ne white dwarf sequences already computed in
\cite{2019A&A...625A..87C} in the frame of Newtonian theory of stellar
interior. O/Ne core white dwarfs are expected as a result of
semi-degenerate carbon burning during the single evolution of
progenitor stars that evolve to the Super Asymptotic Giant Branch
\citep{1997ApJ...485..765G,2005A&A...433.1037G,2006A&A...448..717S,2010MNRAS.401.1453D,2011MNRAS.410.2760V}.
Recent calculations of the remnant of a double white dwarf merger also
predict O/Ne core composition as a result of off-center carbon burning
in the merged remnant, when the remnant mass is larger than 1.05
M$_\sun$ \citep[see][]{2021ApJ...906...53S}. In particular, it is
thought that a considerable fraction of the massive white dwarf
population is formed as a result of stellar mergers
\citep{2020A&A...636A..31T,2020ApJ...891..160C,2022MNRAS.511.5462T}.
We note however that the existence of ultra-massive white dwarfs with
C/O cores resulting from single evolution cannot be
discarded \cite[see][]{2021A&A...646A..30A, 2022arXiv220202040W}.
The adopted input physics for our relativistic white dwarf models is the
same as that in \cite{2019A&A...625A..87C}. In brief, the equation of
state for the low-density regime is that of
\cite{1979A&A....72..134M}, and that of \cite{1994ApJ...434..641S} for
the high-density regime, which takes into account all the important
contributions for both the solid and liquid phases. We include
neutrino emission for pair, photo, and Bremsstrahlung processes using
the rates of \cite{1996ApJS..102..411I}, and of \cite{1994ApJ...425..222H}
for plasma processes. The energetics
resulting from crystallization processes in the core has been
included as in \cite{2019A&A...625A..87C}, and it is based on the
two-component phase diagram of dense O/Ne mixtures appropriate for
massive white dwarf interiors, \cite{2010PhRvE..81c6107M}. As shown
by \cite{2021ApJ...919...87B}, $^{23}$Na and $^{24}$Mg impurities have
only a negligible impact on the O/Ne phase diagram and the
two-component O/Ne phase diagram can be safely used to assess the
energetics resulting from crystallization. We have not
considered the energy released by $^{22}$Ne sedimentation process,
since it is negligible in O/Ne white dwarfs \citep{2021A&A...649L...7C}.
\section{General relativity effects on the evolution of massive white dwarfs}
\label{results}
\begin{table*}[t]
\centering
\begin{tabular}{lccccccc}
\hline
\hline \\[-4pt]
$M_{\rm WD}$ & $M_{\rm G}$ & $R^{\rm Newt}$ & $R^{\rm GR}$ & log g$^{\rm Newt}$ & log g$^{\rm GR}$ & $\varrho_c^{\rm Newt}$ & $\varrho_c^{\rm GR}$\\
$M_\odot$ & $M_\odot$ & km & km & cm s$^{-2}$ & cm s$^{-2}$ & g cm$^{-3}$ & g cm$^{-3}$ \\
\hline \\[-4pt]
1.29 & 1.28977 & 2685.40 & 2608.86 & 9.375 & 9.401 & $ 6.71\times 10^{8}$ & $7.51 \times 10^{8}$ \\
1.31 & 1.30976 & 2426.04 & 2326.17 & 9.470 & 9.507 & $ 9.98 \times 10^{8}$ & $1.17 \times 10^{9}$ \\
1.33 & 1.32974 & 2156.90 & 2004.60 & 9.579 & 9.643 & $ 1.57 \times 10^{9}$ & $2.06 \times 10^{9}$ \\
1.35 & 1.34972 & 1829.29 & 1542.51 & 9.728 & 9.878 & $ 2.90 \times 10^{9}$ & $5.36 \times 10^{9}$ \\
1.369 & 1.36871 & 1408.77 & 1051.16 & 9.961 & 10.217 & $7.42 \times 10^{9}$ & $2.11 \times 10^{10}$ \\
\hline
\\
\end{tabular}
\caption{Relevant characteristics of our sequences a $T_{\rm eff}$=10,000K. $M_{\rm WD}$: total baryonic
mass. $M_{\rm G}$: total gravitational mass. $R^{\rm Newt}$: stellar radius in the Newtonian case. $R^{\rm GR}$:
stellar radius in the general relativity case. g$^{\rm Newt}$: surface gravity in the Newtonian case. g$^{\rm GR}$: surface
gravity in the general relativity case. $\varrho_c^{\rm Newt}$: central density of rest mass in the Newtonian case. $\varrho_c^{\rm GR}$:
central density of rest mass in the general relativity case. }
\label{table1}
\end{table*}
Here, we describe the impact of general relativity effects on the
relevant properties of our constant rest-mass evolutionary tracks. We
begin by examining Fig. \ref{factors}, which displays the general
relativistic correction factors $\mathscr{H, G, V}$, and $\mathscr{R}$
(black, blue, red, and pink lines, respectively) in terms of the
fractional radius for the 1.29, 1.33, 1.35, and 1.369$M_{\sun}$ white
dwarf models at log $L/L_{\sun}=-3$. Dashed lines in the
bottom right panel illustrate
the run of the same factors for a 1.369$M_{\sun}$ white dwarf model at
log $L/L_{\sun}=-0.4$ (log $T_{\rm eff}=5$).
We recall that these
factors are unity in the Newtonian limit. As expected, the importance
of general relativistic effects increases as the stellar mass is
increased. We note that $\mathscr{V}$ is unity at the center and
attains a maximum value at some inner point in the star. The
relativistic factor $\mathscr{R}$ decreases towards the center,
departing even more from unity, meanwhile the other factors,
$\mathscr{G}$ and $\mathscr{H}$ increase towards the center of the
star. The behavior of the relativistic correction factors can be
traced back to curvature effects, as well as the fact that the
pressure and the internal
energy appear as a source for
gravity in general relativity. For maintaining hydrostatic
equilibrium, then, both density and pressure gradients are steeper
than in Newtonian gravity. This makes the factors $\mathscr{G}$ and
$\mathscr{H}$, which depend directly on density and pressure, to
increase towards the center of the star. The relativistic factor
$\mathscr{V}$, which can be interpreted as a correction to the volume,
would be unity at the center of the star where the volume is zero, and
increase because of the increasing of density in general relativity
respect to the density in Newtonian gravity. However, as the
departures from the Newtonian case decrease towards the surface of the
star, $\mathscr{V}$ decreases towards the outside, achieving a
maximum value in between. We note that the relativistic factors
depend slightly on the effective temperature.
The impact of relativistic effects on the mass-radius relation
at two different effective temperatures
can be appreciated in Fig.\,\ref{mr}. We note that for
the most massive white dwarfs, at a given
gravitational mass, the radius is markedly smaller in the case that
the general relativity effects are taken into account.
At a stellar mass of 1.369 $M_{\sun}$ the
stellar radius becomes only 1050 km, 25\% smaller than predicted by the
Newtonian treatment (see Table \ref{table1}). As in the Newtonian
case, the effect of finite temperature on the stellar radius is still
relevant in very massive white dwarfs. We mention that general relativistic
corrections become negligible for stellar masses smaller than
$\approx$ 1.29 $M_{\sun}$. In particular, for stellar masses below that value,
the stellar radius results below 2
\% smaller
when general relativity effects are taken into account .
In our calculations, O/Ne white
dwarfs more massive than 1.369 $M_{\sun}$ become gravitationally
unstable (which occurs at a given finite central density) with respect
to general relativity effects, in agreement with the findings for
zero-temperature models reported in \cite{2011PhRvD..84h4007R} for a
pure-oxygen white dwarf (1.38024 $M_{\sun}$) and
\cite{2017RAA....17...61M} for white dwarfs composed of oxygen (1.3849
$M_{\sun}$) or of neon (1.3788 $M_{\sun}$), although their values are
slightly higher\footnote{Preliminary computations we performed for
oxygen-rich core white dwarfs show that they become
unstable at 1.382 $M_{\sun}$.}. We mention that for the 1.369
$M_{\sun}$ white dwarf model, the central density in the
general relativity case reaches $2.11 \times 10^{10}$ g cm$^{-3}$ (see
Table \ref{table1}). Such density is near the density threshold for
inverse $\beta-$decays. We have not considered that matter inside our
white dwarf models may experience instability against the inverse
$\beta-$decay. O or Ne white dwarfs are expected to become unstable
against the inverse $\beta-$decay process at a stellar mass near the
critical mass resulting from general relativity effects, of the order
of 1.37 $M_{\sun}$ \citep[see][]{2011PhRvD..84h4007R,
2017RAA....17...61M}.
The inner profile of rest mass and density of rest mass for the
1.369$M_{\sun}$ white dwarf model in the general relativity and
Newtonian cases are shown in the upper and bottom panel of
Fig. \ref{mass-density}, respectively. For such massive white dwarf
model, general relativity effects strongly alter the stellar
structure, causing matter to be much more concentrated toward the
center of the star and the central density to be larger than in the
Newtonian case. The impact remains noticeable towards lower stellar
masses, although to a lesser extent, as can be noted for the case of
1.35$M_{\sun}$ white dwarf model shown in the bottom panel of
Fig. \ref{mass-density} (dotted lines). In view of this, the run of
the gravitational field versus radial coordinate for the general relativity
case differs markedly from that resulting from the Newtonian
case. This is shown in Fig. \ref{gravity} for 1.369$M_{\sun}$,
1.35$M_{\sun}$, and 1.29$M_{\sun}$ white dwarf models. In particular,
the gravitational field in the general relativistic case as measured
far from the star is given by
\begin{equation}
g^{\rm GR}= \frac{G m}{r^2} \mathscr{G} \mathscr{V}^2 \ .
\label{grav_sup}
\end{equation}
Clearly, the gravitational field in the most massive of our models is
strongly affected by general relativity. In the stellar interior,
large differences arise in the gravitational field due to the
inclusion of general relativity effects. We note that such differences
do not arise from the relativistic correction factors $\mathscr{G}
\mathscr{V}^2 $ (see Fig. \ref{factors}) to the Newtonian
gravitational field $g^{\rm New}= G m /r^2$ that appear explicitly in
Eq. (\ref{grav_sup}), but from the solution of the relativistic
equilibrium instead, which gives a different run for $m(r)$ compared
to the Newtonian case.
Additionally, the surface gravity and stellar radius are
affected by the effects of general relativity. These quantities are
shown in Fig. \ref{grav-surface} in terms of the effective temperature
for all of our sequences for the general relativity and Newtonian
cases, using solid and dashed lines, respectively. In the most massive
sequences, general relativity effects markedly alter the surface
gravity and stellar radius. In this sense, we infer that
general relativity
effects lead to a stellar mass value about 0.015$M_{\sun}$ smaller
for cool white dwarfs with measured surface gravities of log g $\approx$ 10.
The photometric measurements of
\cite{2021MNRAS.503.5397K} for the radius of the ultra-massive white
dwarfs in the solar neighborhood are also plotted in this figure. For
the more massive of such white dwarfs, the stellar radius results
2.8-4$\%$ smaller when general relativity effects are taken into
account.
We note that most of our sequences display a sudden increase in their
surface gravity at high effective temperatures. As noted in
\cite{2019A&A...625A..87C}, this is related to the onset of core
crystallization (marked with blue filled circles in each sequence
depicted in Fig. \ref{grav-surface}), which modifies the distribution
of $^{16}$O and $^{20}$Ne. Specifically, the abundance of $^{20}$Ne
increases in the core of the white dwarf as crystallization proceeds,
leading to larger Coulomb interactions and hence to denser cores, and,
therefore, to higher surface gravities. This behavior can also be
regarded as a sudden radius decrease (bottom panel of
Fig. \ref{grav-surface}). In this context, we note that the density
increase due to the increase in the core abundance of $^{20}$Ne during
crystallization eventually causes O/Ne white dwarf models with stellar
masses larger than $\gtrsim 1.36 M_\sun $ to become gravitationally
unstable against general relativity effects. In order to explore the
mass range of stable white dwarfs in the absence of this processes,
the 1.369$M_{\sun}$ relativistic sequence was computed disregarding
the effect of phase separation (but not latent heat) during
crystallization.
\subsection{General relativity effects on the white dwarf cooling times}
The cooling properties of the ultra-massive white dwarfs are also
markedly altered by general relativity effects, in particular the
the most massive ones. This is illustrated in Fig. \ref{age},
which compares the cooling times of our models for the general
relativity and Newtonian cases, solid and dashed lines
respectively. The cooling times are set to zero at the beginning of
cooling tracks at very high effective temperatures. Gravothermal
energy is the main energy source of the white dwarfs, except at very
high effective temperatures where energy released during the
crystallization process contributes %
to the budget of the star. As noticed in
\cite{2021A&A...649L...7C}, ultra-massive O/Ne-core white dwarfs
evolve significantly fast into faint
magnitudes. General relativity effects cause
ultra-massive white dwarfs to evolve faster than in the Newtonian
case at advanced stages of evolution. In particular, the $1.369
M_\sun$ relativistic sequence reaches $\log(L/L_\sun)$=-4.5 in only
$\sim 0.5$ Gyrs, in contrast with the $\sim 0.9$ Gyrs needed
in the Newtonian case. The larger internal
densities inflicted by general relativity make the Debye cooling phase
more relevant than in the Newtonian case at a given stellar mass, thus
resulting in a faster cooling for the sequences that include general
relativity effects.
The fast cooling of these
objects, together with their low luminosity and rare formation rates,
would make them hard to observe. The trend in
the cooling behavior is reversed at
earlier stages of evolution, where white dwarfs computed in the
general relativity case evolve slower than their Newtonian
counterparts. This is because white dwarfs computed in the general
relativity case crystallize at higher luminosities (because of their
larger central densities), with the consequent increase in the cooling
times at those stages. In the 1.369$M_{\sun}$ relativistic sequence,
the whole
impact of crystallization on the cooling times results smaller, due to
the fact that we neglect the process of phase separation during
crystallization in that sequence.
We mention that we neglect the neutrino emission resulting from Urca
process, which is relevant in O/Ne white dwarfs at densities in
excess of $10^{9}$ g cm$^{-3}$ \citep{2021ApJ...916..119S} . In our modeling, such densities are
attained at models with stellar masses $\gtrsim 1.33 M_\sun $, see
Table \ref{table1}. Hence, the depicted cooling times for the
sequences with stellar masses above this value may be overestimated at
high and intermediate luminosities. A first attempt to include Urca
cooling process from $^{23}$Na-$^{23}$Ne urca pair in our stellar code
leads to the formation of a mixing region below the Urca shell, as
reported by \cite{2021ApJ...916..119S}. Because of the
temperature inversion caused by Urca process, our most massive
white dwarf models develop off-centered crystallization.
We find numerical
difficulties to model the interaction of crystallization and
the Urca process-induced mixing that prevent us from a consistent computation of white
dwarf cooling during these stages. As recently shown by
\cite{2021ApJ...916..119S}, the cooling of such massive white dwarfs
is dominated by neutrino cooling via the Urca process during the first
100 Myr after formation. Our focus in
this work is on the effects of general relativity on
ultra-massive white dwarfs, so we leave the problematic treatment of
Urca-process impacts on the structure of relativistic white dwarfs for an
upcoming work.
\subsection{Observational constrains on ultra-massive white dwarf models}
The ESA {\it Gaia} mission has provided an unprecedented wealth of
information about stars \citep[see][and references therein]{GaiaEDR32021}. In
particular, nearly $\approx$359,000 white dwarf candidates have been
detected \citep{Fusillo2021}, being estimated that the sample up to
100 pc from the Sun can be practically considered as complete
\citep{Jimenez2018}. The extreme precision of astrometric and
photometric measures allow us to derive accurate color-magnitude
diagrams where to test our models. Some unexpected peculiar features
have been already observed in the {\it Gaia} white dwarf
color-magnitude diagram \citep{GaiaDR22018}.
In particular, the Q branch, due to crystallization and sedimentation
delays, has been extensively analyzed
\citep{Cheng2019,2019Natur.565..202T,2021A&A...649L...7C}. However, a
new branch, called faint blue branch has been reported by
\cite{2022RNAAS...6...36S}. This faint blue branch is formed by nearly
$\sim$60 ultracool and ultra-massive objects, which have been
astrometric and photometric verified and cross validated with the {\it
Gaia} catalogue of nearby stars \citep{GaiaNSC2021} and the white
dwarf catalogue of \cite{Fusillo2021}. It is important also to
mention that some of these objects that form this peculiar feature in
the color-magnitude diagram have already been reported \cite[][and
references therein]{Kilic2020}. Most of these white dwarfs exhibit a
near-infrared flux deficit that has been attributed to the effects of
molecular collision-induced absorption in mixed hydrogen-helium
atmospheres, \cite{Bergeron2022}. Some issues still remain to be
clarified under this assumption and not all the objects in
\cite{2022RNAAS...6...36S} are present in the analysis of
\cite{Bergeron2022}. Consequently, for our purpose here, which is not
in contradiction with the analysis done in \cite{Bergeron2022}, we
adopted hydrogen-pure atmosphere models for the analysis of the whole
\cite{2022RNAAS...6...36S} sample, where particular objects are
treated individually.
In the left panel of Fig. \ref{gaia} we show a color-magnitude diagram
for the 100 pc white dwarf {\it Gaia} EDR3 population (gray dots)
together with the faint blue branch objects from
\cite{2022RNAAS...6...36S} (solid red circles). The color-magnitude
diagram selected is absolute magnitude $G$ versus $G_{\rm BP}-$G,
instead of $G_{\rm BP}-$G$_{\rm RP}$, minimizing in this way the
larger errors induced by the $G_{\rm RP}$ filter for faint objects.
We also provide the magnitudes for our relativistic and Newtonian
models (black and cyan lines, respectively) in {\it Gaia} EDR3 passbands (DR2,
Sloan Digital Sky Survey, Pan-STARRS and other passbands are also
available under request) by using the non-gray model
atmospheres of
\cite{2010MmSAI..81..921K,2019A&A...628A.102K}. Isochrones of 0.25,
0.5, 1 and 2, Gyr for our relativistic model are also shown (dashed
black line) in Fig. \ref{gaia}. An initial inspection of the {\it
Gaia} color-magnitude diagram reveals that our new white dwarf
sequences are consistent with most of the ultra-massive white dwarfs
within 100 pc from the Sun.
In addition, the relativistic white dwarf sequences are fainter than
Newtonian sequences with the same mass. Therefore, general relativity
effects must be carefully taken into account when determining the mass
and stellar properties of the most massive white dwarfs through {\it
Gaia} photometry. Not considering such effects would lead to an
overestimation of their mass and an incorrect estimation of their
cooling times. Finally, we check that faint-blue branch objects do not
follow any particular isochrone, thus ruling out a common temporal
origin of these stars.
A closer look to the faint blue branch is depicted in the right panel of Fig. \ref{gaia}. The vast majority of faint blue branch white dwarfs appear to have masses larger than $\sim 1.29\, M_\odot$. Thus, this sample is ideal for testing our models, in particular, those objects which present the largest masses or, equivalently, the smallest radii. Hence, for the analysis presented here and for reasons of completeness we estimated the error bars for those objects which lie on the left of the Newtonian 1.369 $M_{\sun}$ track. Errors are propagated from the astrometric and photometric errors provided by {\it Gaia} EDR3. Although correlations in {\it Gaia} photometry are very low we have assumed that some correlation may exist between parameters. This way errors are added linearly and not in quadrature, thus obtaining an upper limit estimate of the error bars. The parameters corresponding to the 20 selected ultra-massive white dwarf candidates of the faint blue branch are presented in Table \ref{t:FBcandidates}. In the first column we list the {\it Gaia} EDR3 source ID with a label for an easy identification in Fig. \ref{gaia}. Columns second to fifth present the parallax, apparent and absolute $G$ magnitudes, and color $G_{\rm BP}-G$ with their corresponding error, respectively. Columns sixth and seventh represent the observational distance within the color-magnitude diagram measured in $\sigma$ deviations to the limiting 1.369$M_{\sun}$ cooling track when the general relativity model or the Newtonian model, respectively, is used. Finally, the last column is a 5 digits number flag. The first digit indicates if the relative flux error in the G$_\mathrm{BP}$ band is larger or equal to 10\% (1) or smaller (0). %
The second digit indicates if the relative flux error in the G$_\mathrm{RP}$ band is larger or equal to 10\% (1) or smaller (0). %
The third digit indicates if the $\beta$ parameter as defined by \citet{Riello2021} is $\geq 0.1$ (1) or $<0.1$ (0); if 1 then the object is affected by blending. The fourth digit is set to (0) if the renormalized unit vector ruwe \citep{Lindegren2018} is $<1.4$ (indicative that the solution corresponds to a single object) or set to (1) if it is $\geq 1.4$ (bad solution or binary system). The fifth digit indicates if the object passes (1) or not (0) a 5$\sigma$ cut on the corrected G$_\mathrm{BP}$ and G$_\mathrm{RP}$ flux excess ($C^{*}$; \citealt{Riello2021}). %
An ideal case will show a 00000 flag.
The detailed analysis of the color-magnitude distance to the limiting 1.369$M_{\sun}$ relativistic and Newtonian tracks shown in the sixth and seventh columns, respectively, indicates that, on average, the selected faint blue branch objects are more compatible with the general relativistic model than with the Newtonian model. Six of them $\{a,b,c,f,g,m\}$ lie below the limiting 1.369$M_{\sun}$ relativistic track while they are $1\sigma$ compatible with the Newtonian model. Moreover, up to four objects $\{h,j,n,s\}$ are compatible with the relativistic model at the $1\sigma$ level, but only marginally at a $2\sigma$ level with the Newtonian model. In particular, objects $\{j,s\}$ are ideal candidates to confirm relativistic models given that they present a 00000 flag, which is indicative of a reliable photometry and astrometry. The rest of objects $\{d,i,k,o,p,r,t\}$ lie at a distance $2\sigma$ or $3\sigma$ (the last two) for the relativistic model, but at larger distances for the Newtonian model (up to $4\sigma$). According to our study, these objects with such a small radius or larger masses should be unstable against gravitational collapse. However, any conclusion on this should be taken with caution. On one hand,
although some of these objects belong to the sample analyzed by \cite{Bergeron2022} ($d$, J1612$+$5128; $j$, J1251+4403, also named WD1248+443 \citep{Harris2008}; $o$, J1136$-$1057; and $s$, J0416$-$1826) and some near-infrared flux deficit has been reported for them, a more detailed spectroscopic analysis for all of our candidates is deserved for a precise mass and radius estimation.
On the other hand, the presence of strong internal magnetic fields or a rapid rotation, not considered in this paper, could allow these objects to support the enormous gravity. It has been shown, in the general relativity framework, that including strong magnetic fields and/or a rapid rotation could lead to a smaller radius and/or a larger limiting-mass for the most massive white dwarfs \citep[e.g.][]{2013ApJ...762..117B,2016MNRAS.456.3375B,2015MNRAS.454..752S}. Indeed, the existence of super-Chandrasekhar white dwarfs, with masses $2.1-2.8\,M_\odot$ has been proposed as a possible scenario to explain the over-luminous Type Ia supernovae SN 2003fg, SN 2006gz, SN 2007if, SN 2009dc (e.g. Howell et al. 2006; Hicken
et al. 2007; Yamanaka et al. 2009; Scalzo et al. 2010; Silverman et al.
2011; Taubenberger et al. 2011). A detailed follow up of these objects is, in any case, deserved and, at the same time, general relativistic models as the ones presented in this work but for white dwarfs with carbon-oxygen cores are expected to play a capital role in the understanding of the true nature of these objects.
\begin{table*}[t]
\centering
\begin{tabular}{cccccccl}
\hline
\hline \\[-4pt]
{\it Gaia} EDR3 & $\varpi\pm\sigma_{\varpi}$ & $G\pm\sigma_{\rm G}$ & $M_{\rm G}\pm\sigma_{M_{\rm G}}$ & ($G_{\rm BP}-G)\pm\sigma{_{\rm (G_{BP}-G)}}$ & Rel. & New. & flags \\
source ID & (mas) & (mag) & (mag) & (mag) & model & model & \\
\hline \\[-4pt]
$6565940122868224640^a$ & $ 11.717 \pm 0.592 $ & $ 20.275 \pm 0.005 $ & $ 15.619 \pm 0.115 $ & $ -0.008 \pm 0.100 $ & $<1$ & 1 & 00100 \\
$1983698716601024512^b$ & $ 10.761 \pm 0.934 $ & $ 20.549 \pm 0.009 $ & $ 15.708 \pm 0.198 $ & $ -0.001 \pm 0.084 $ & $<1$ & 1 & 01000 \\
$6211904903507006336^c$ & $ 15.411 \pm 0.501 $ & $ 19.893 \pm 0.006 $ & $ 15.832 \pm 0.076 $ & $ -0.014 \pm 0.079 $ & $<1$ & 1 & $00000^1$ \\
$1424656526287583744^d$ & $ 11.523 \pm 0.685 $ & $ 20.668 \pm 0.009 $ & $ 15.976 \pm 0.138 $ & $ -0.236 \pm 0.150 $ & 2 & 2 & $11000^1$ \\
$3585053427252374272^e$ & $ 16.874 \pm 0.464 $ & $ 20.054 \pm 0.005 $ & $ 16.190 \pm 0.065 $ & $ -0.022 \pm 0.070 $ & 1 & 1 & $01000^1$ \\
$4377579209528621184^f$ & $ 14.828 \pm 0.860 $ & $ 20.379 \pm 0.007 $ & $ 16.235 \pm 0.133 $ & $ 0.043 \pm 0.078 $ & $<1$ & 1 & 01000 \\
$1505825635741455872^g$ & $ 29.084 \pm 0.190 $ & $ 19.022 \pm 0.004 $ & $ 16.340 \pm 0.018 $ & $ 0.041 \pm 0.029 $ & $<1$ & 1 & $00100^{1,2}$ \\
$3480787358063803520^h$ & $ 13.189 \pm 1.365 $ & $ 20.769 \pm 0.010 $ & $ 16.370 \pm 0.235 $ & $ -0.116 \pm 0.132 $ & 1 & 2 & 11000 \\
$4461423190259561728^i$ & $ 12.908 \pm 2.082 $ & $ 20.829 \pm 0.011 $ & $ 16.383 \pm 0.361 $ & $ -0.135 \pm 0.096 $ & 2 & 2 & 01000 \\
$ \bf{5064259336725948672^j}$ & $\bf 30.638 \pm 0.219 $ & $ \bf 19.005 \pm 0.004 $ & $ \bf 16.436 \pm 0.019 $ & $ \bf 0.027 \pm 0.026 $ & \bf{1} & \bf{2} & $\bf{00000}^1$ \\
$534407181320476288^k$ & $ 15.218 \pm 0.640 $ & $ 20.533 \pm 0.008 $ & $ 16.445 \pm 0.099 $ & $ -0.127 \pm 0.080 $ & 2 & 3 & $01000$ \\
$5763109404082525696^l$ & $ 16.279 \pm 0.949 $ & $ 20.424 \pm 0.007 $ & $ 16.482 \pm 0.134 $ & $ -0.021 \pm 0.136 $ & 1 & 1 & $11000^1$ \\
$2858553485723741312^m$ & $ 16.357 \pm 0.715 $ & $ 20.452 \pm 0.009 $ & $ 16.521 \pm 0.104 $ & $ 0.026 \pm 0.109 $ & 1 & 1 & $01000^1$ \\
$6178573689547383168^n$ & $ 17.098 \pm 0.946 $ & $ 20.362 \pm 0.009 $ & $ 16.527 \pm 0.129 $ & $ -0.057 \pm 0.116 $ & 1 & 2 & $01000^1$ \\
$3586879608689430400^o$ & $ 17.572 \pm 1.299 $ & $ 20.369 \pm 0.007 $ & $ 16.593 \pm 0.168 $ & $ -0.193 \pm 0.114 $ & 2 & 3 & $01000^1$ \\
$1738863551836243840^p$ & $ 19.444 \pm 0.933 $ & $ 20.296 \pm 0.007 $ & $ 16.740 \pm 0.112 $ & $ -0.117 \pm 0.093 $ & 2 & 3 & 01000 \\
$6385055135655898496^q$ & $ 16.607 \pm 0.924 $ & $ 20.670 \pm 0.009 $ & $ 16.771 \pm 0.129 $ & $ 0.013 \pm 0.130 $ & 1 & 1 & 11000 \\
$283928743068277376^r$ & $ 27.731 \pm 0.332 $ & $ 19.636 \pm 0.004 $ & $ 16.850 \pm 0.030 $ & $ -0.133 \pm 0.052 $ & 3 & 4 & 00100 \\
$\bf{1528861748669458432^s}$ & $ \bf 20.585 \pm 0.614 $ & $\bf 20.325 \pm 0.006 $ & $ \bf 16.892 \pm 0.070 $ & $ \bf -0.043 \pm 0.082 $ & \bf 1 & \bf 2 & $\bf{00000}^{1,3}$ \\
$1674805012263764352^t$ & $ 19.661 \pm 1.347 $ & $ 20.792 \pm 0.015 $ & $ 17.260 \pm 0.164 $ & $ -0.131 \pm 0.076 $ & 3 & 4 & 01000 \\
\hline
\end{tabular}
\caption{Ultra-massive white dwarf candidates selected from the sample of faint blue white dwarfs of \cite{2022RNAAS...6...36S}. Sixth and seventh columns indicate the distance within the color-diagram of Fig. \ref{gaia} measured in $1\sigma$ deviations form the selected objects to the limiting 1.369 $M_\odot$ cooling tracks for relativistic and Newtonian models, respectively. Objects $j$ and $s$, marked in bold, are ideal candidates with no flags to confirm relativistic models. See text for rest of columns and details. }
\label{t:FBcandidates}
\begin{minipage}{\textwidth}
$^{1}$\cite{Bergeron2022}, $^2$\cite{Gates2004} $^3$\cite{Harris2008}
\end{minipage}
\end{table*}
\section{Summary and conclusions}
\label{conclusions}
In this paper, we present the first set of constant rest-mass ultra-massive O/Ne white dwarf cooling tracks with masses
$M_{\star} > 1.29 M_\sun$, which fully take into account the effects of general relativity on their structural and evolutionary properties. Ultra-massive white dwarfs are relevant in different astrophysical contexts, such as type Ia supernovae explosions, stellar merger events, and
the existence of high magnetic field white dwarfs. In addition, they provide insights into the physical processes in the Super Asymptotic Giant Branch phase preceding their formation. In
the last few years, the existence of such ultra-massive white dwarfs in the solar neighborhood has been reported in several studies, including the recent discover of a branch of faint blue white dwarfs in the color-magnitude diagram \citep{Kilic2020,2022RNAAS...6...36S}. Although some of these objects present an infrared flux deficit, it is also thought to be composed by ultra-massive white dwarfs with masses larger than $1.29\, M_\odot$.
It should be noted that shortly, it is very likely that $g$-mode pulsating ultra-massive white dwarfs with masses $M_{\star} \gtrsim 1.29 M_\sun$ will be discovered thanks to space missions such as {\sl TESS} and {\sl Plato} space telescopes, and it will then be possible to study them through asteroseismology.
We have computed the complete evolution of 1.29, 1.31, 1.33, 1.35, and 1.369 $M_{\sun}$ hydrogen-rich white dwarfs models, assuming an O/Ne composition for
the core. Calculations
have been performed using the La Plata stellar evolution code, {\tt LPCODE}, for which the
standard equations of stellar structure and evolution have been modified to include the effects of general relativity. To this end, we have followed the formalism given in \cite{1977ApJ...212..825T}. Specifically, the fully general relativistic partial differential equations governing the evolution of a spherically symmetric star are solved in a way they resemble the standard Newtonian equations of stellar structure. For comparison purposes, the same sequences have been computed but for the Newtonian case. Our new white dwarf models include the energy released during the crystallization process, both due to latent heat and the induced chemical redistribution. We provide cooling times and time dependent mass-radius relations for relativistic ultra-massive white dwarfs. We also provide magnitudes in Gaia, Sloan Digital Sky Survey and Pan-STARRS passbands, using the model atmospheres of \cite{2010MmSAI..81..921K,2019A&A...628A.102K}.
This set of cooling sequences, together with those calculated in \cite{2019A&A...625A..87C} and \cite{2022MNRAS.511.5198C} for lower stellar masses than computed here, provide an appropriate theoretical framework to study the most massive white dwarfs in our Galaxy, superseding all existing calculations of such objects.
As expected, we find that the importance of general relativistic effects increases as the
stellar mass is increased. According to our calculations, O/Ne white dwarfs more
massive than 1.369 $M_{\sun}$ become gravitationally unstable with respect to general relativity effects. When core chemical distribution due to phase separation on crystallization is considered, such instability occurs at somewhat
lower stellar masses, $\gtrsim 1.360 M_\sun $.
For our most massive sequence, the stellar radius becomes 25\% smaller than predicted
by the Newtonian treatment. The evolutionary properties of our ultra-massive white dwarfs are also modified by general relativity effects. In particular, at advanced stages of evolution, the cooling times for our most massive white dwarf sequence result in about a factor of two shorter than in the Newtonian case. In addition, not considering general relativity effects when estimating the properties of such objects through photometric and spectroscopic techniques would lead to an overestimation of their mass of 0.015$M_\sun$ near the critical mass.
We have compared in the color-magnitude diagram our theoretical sequences with the white dwarfs composing the faint blue white dwarf branch \citep{2022RNAAS...6...36S}. We conclude that, regardless the infrared deficit flux that some particular objects may exhibit, several white dwarfs of this branch can present masses larger than $\sim 1.29 M_\sun $ and that it does not coincide with any isochrone nor with any evolutionary track. We found that seven of the white dwarfs in this branch should have a smaller radius than our most massive cooling sequence and should be gravitationally unstable against collapse. However, apart from the need of a more detailed spectroscopic study to accurately characterize the possible effects of the infrared flux deficit in some of these objects, the presence of strong magnetic fields and a rapid rotation, not considered in this study, could favor the stability of such objects, thus supporting the existence of super-Chandrasekhar white dwarfs, that, in the case of CO-core white dwarfs, should likely be the progenitors of the over-luminous Type Ia supernovae SN 2003fg, SN 2006gz, SN 2007if, SN 2009dc. Consequently, a detailed follow-up of these seven objects is required within the framework of the general relativity models exposed here.
\\
As discussed throughout this work, our new ultra-massive white dwarf
models for O/Ne core-chemical composition constitute an improvement
over those computed in the framework of the standard Newtonian theory
of stellar interiors. Therefore, in support of previous studies, the
effect of general
relativity must be taken into account to ascertain the true nature of
the most massive white dwarfs, in particular, at assessing their
structural and evolutionary properties.
\begin{acknowledgements}
We thank Detlev Koester for extending his atmosphere models to the high surface gravities that characterize our relativistic ultra-massive white dwarf models.
We also thank the comments of an anonymous referee that improved the original
version of this paper.
Part of this work was supported by PICT-2017-0884 from ANPCyT, PIP
112-200801-00940 grant from CONICET, grant G149 from University of La Plata, NASA grants 80NSSC17K0008 and 80NSSC20K0193. ST and ARM acknowledge support from MINECO under the PID2020-117252GB-I00 grant. ARM acknowledges support from Grant RYC-2016-20254 funded by MCIN/AEI/10.13039/501100011033 and by ESF Investing in your future. This research has made use of NASA Astrophysics Data System. 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.
\end{acknowledgements}
\bibliographystyle{aa}
\bibliography{ultramassiveCO}
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"Title:\r\nComparing Reflection and Absorption Models for the Soft X-ray Variability in the NLS1 AG(...TRUNCATED) | "\r\nAbstract: We present a spectral analysis of two XMM-Newton observations of the\r\nnarrow-line S(...TRUNCATED) | https://export.arxiv.org/pdf/2208.12177 | "\n\\label{firstpage}\n\\pagerange{\\pageref{firstpage}--\\pageref{lastpage}}\n\n\n\n\n\n\n\n\\begin(...TRUNCATED) |
Title:
Neutrinos from near and far: Results from the IceCube Neutrino Observatory | "\r\nAbstract: Instrumenting a gigaton of ice at the geographic South Pole, the IceCube\r\nNeutrino (...TRUNCATED) | https://export.arxiv.org/pdf/2208.01226 | "\n\n\n\n\\begin{center}{\\Large \\textbf{\nNeutrinos from near and far: Results from the IceCube Ne(...TRUNCATED) |
Title:
Addition of tabulated equation of state and neutrino leakage support to IllinoisGRMHD | "\r\nAbstract: We have added support for realistic, microphysical, finite-temperature\r\nequations o(...TRUNCATED) | https://export.arxiv.org/pdf/2208.14487 | "\n\n\n \\title{Addition of tabulated equation of state and neutrino leakage support to \\igm}\n\(...TRUNCATED) |
Title:
Star formation inefficiency and Kennicutt-Schmidt laws in early-type galaxies | "\r\nAbstract: Star formation in disk galaxies is observed to follow the empirical\r\nKennicutt-Schm(...TRUNCATED) | https://export.arxiv.org/pdf/2208.03735 | " command.\n\n\\newcommand{\\vdag}{(v)^\\dagger}\n\\newcommand\\aastex{AAS\\TeX}\n\\newcommand\\late(...TRUNCATED) |
"Title:\r\nConstraining Accreted Neutron Star Crust Shallow Heating with the Inferred Depth of Carb(...TRUNCATED) | "\r\nAbstract: Evidence has accumulated for an as-yet unaccounted for source of heat located\r\nat s(...TRUNCATED) | https://export.arxiv.org/pdf/2208.03347 | "\n\\label{firstpage}\n\\pagerange{\\pageref{firstpage}--\\pageref{lastpage}}\n\n\n\n\n\n\n\n\\begin(...TRUNCATED) |
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| Title | Annotation | Latex | |
|---|---|---|---|
| Axion bremsstrahlung from collisions of global strings | We calculate axion radiation emitted in the collision of two straight globalstrings. The strings are supposed to be in the unexcited ground state, to beinclined with respect to each other, and to move in parallel planes. Radiationarises when the point of minimal separation between the strings moves fasterthan light. This effect exhibits a typical Cerenkov nature. Surprisingly, itallows an alternative interpretation as bremsstrahlung under a collision ofpoint charges in 2+1 electrodynamics. This can be demonstrated by suitableworld-sheet reparameterizations and dimensional reduction. Cosmologicalestimates show that our mechanism generates axion production comparable withthat from the oscillating string loops and may lead to further restrictions onthe axion window.... | https://export.arxiv.org/pdf/astro-ph/0310718 | ... |
This dataset consists of many csv format files, the name of each of which contains the category of scientific articles presented in this file. Each file consists of 1024 articles.
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